control handbook

Handbook of

Pi and P I D Controller Tuning Rules

2nd Edition

Aidan O*Dwyer

Imperial College Press

Handbook of

PI and PID Controller Tuning Rules

2nd Edition

This page is intentionally left blank

Handbook of

P I and P I D Controller Tuning Rules

2nd Edition

Aidan O'Dwyer Dublin Institute of Technology, Ireland

ICP Imperial College Press

Published by

Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE

Distributed by

World Scientific Publishing Co. Pte. Ltd.

5 Toh Tuck Link, Singapore 596224

USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601

UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

HANDBOOK OF PI AND PID CONTROLLER TUNING RULES (2nd Edition)

Copyright © 2006 by Imperial College Press

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 1-86094-622-4

Printed in Singapore by B & JO Enterprise

Dedication

Once again, this book is dedicated with love to Angela, Catherine and Fiona and to my parents, Sean and Lillian.

Preface

Proportional Integral (PI) and Proportional Integral Derivative (PID) controllers have been at the heart of control engineering practice for seven decades. However, in spite of this, the PID controller has not received much attention from the academic research community until the past fifteen years, when work by K.J. Astrom, T. Hagglund and F.G. Shinskey, among others, has sparked a revival of interest in the use of this "workhorse" of controller implementation.

There is strong evidence that PI and PID controllers remain poorly understood and, in particular, poorly tuned in many applications. It is clear that the many controller tuning rules proposed in the literature are not having an impact on industrial practice. One reason is that the tuning rules are not very accessible, being scattered throughout the control literature; in addition, the notation used is not unified. The purpose of this book is to bring together and summarise, using a unified notation, tuning rules for PI and PID controllers. The author restricts the work to tuning rules that may be applied to the control of processes with time delays (dead times); in practice, this is not a significant restriction, as most process models have a time delay term.

It is the author's belief that this book will be useful to control and instrument engineering practitioners and will be a useful reference for students and educators in universities and technical colleges.

I would like to thank the School of Control Systems and Electrical Engineering, Dublin Institute of Technology, for providing the facilities needed to complete the book. Finally, I am deeply grateful to my mother, Lillian and my father, Sean, for their inspiration and support over many

VII

viii Handbook of PI and PID Controller Tuning Rules

years and to my family Angela, Catherine and Fiona, for their love and understanding.

Aidan O'Dwyer

Contents

Preface vii

1. Introduction 1 1.1 Preliminary Remarks 1 1.2 Structure of the Book 2

2. Controller Architecture 5 2.1 Introduction 5 2.2 PI Controller Structures 6 2.3 PID Controller Structures 7

2.3.1 Ideal PID controller structure and its variations 7 2.3.2 Classical PID controller structure and its variations 11 2.3.3 Non-interacting PID controller structure and its

variations 12 2.3.4 Other PID controller structures 17 2.3.5 Comments on the PID controller structures 19

2.4 Process Modelling 20 2.5 Organisation of the Tuning Rules 24

3. Tuning Rules for PI Controllers 26 3.1 FOLPD Model 26

3.1.1 Ideal controller - Table 3 26 3.1.2 Ideal controller in series with a first order lag

-Table 4 61 3.1.3 Ideal controller in series with a second order filter

- Table 5 62 3.1.4 Controller with set-point weighting - Table 6 63 3.1.5 Controller with proportional term acting on the output 1

- Table 7 65 3.1.6 Controller with proportional term acting on the output 2

- Table 8 66

ix

x Handbook of PI and P1D Controller Tuning Rules

3.2 FOLPD Model with a Positive Zero 67 3.2.1 Ideal controller - Table 9 67 3.2.2 Ideal controller in series with a first order lag

- Table 10 68 3.3 FOLPD Model with a Negative Zero 69

3.3.1 Ideal controller in series with a first order lag -Table 11 69

3.4 Non-Model Specific 70 3.4.1 Ideal controller - Table 12 70 3.4.2 Controller with set-point weighting - Table 13 75

3.5 IPD Model 76 3.5.1 Ideal controller - Table 14 76 3.5.2 Ideal controller in series with a first order lag

-Table 15 83 3.5.3 Controller with set-point weighting - Table 16 84 3.5.4 Controller with proportional term acting on the output 1


-Table 18 87 3.5.6 Controller with a double integral term - Table 19 88

3.6 FOLIPD Model 89 3.6.1 Ideal controller - Table 20 89 3.6.2 Controller with set-point weighting - Table 21 92 3.6.3 Controller with proportional term acting on the output 1


- Table 23 101 3.7 SOSPD Model 102

3.7.1 Ideal controller - Table 24 102 3.7.2 Controller with set-point weighting - Table 25 120

3.8 SOSIPD Model - Repeated Pole 122 3.8.1 Controller with set-point weighting - Table 26 122

3.9 SOSPD Model with a Positive Zero 123 3.9.1 Ideal controller - Table 27 123

3.10 SOSPD Model (repeated pole) with a Negative Zero 125 3.10.1 Ideal controller in series with a first order lag

- Table 28 125 3.11 Third Order System plus Time Delay Model 126

3.11.1 Ideal controller - Table 29 126 3.11.2 Controller with set-point weighting - Table 30 128

3.12 Unstable FOLPD Model 130 3.12.1 Ideal controller-Table 31 130

Contents XI

3.12.2 Controller with set-point weighting - Table 32 135 3.12.3 Controller with proportional term acting on the output 1


- Table 34 140 3.13 Unstable FOLPD Model with a Positive Zero 141


-Table 36 142 3.13.3 Controller with set-point weighting - Table 37 143

3.14 Unstable SOSPD Model (one unstable pole) 145 3.14.1 Ideal controller-Table 38 145

3.15 Unstable SOSPD Model with a Positive Zero 147 3.15.1 Ideal controller - Table 39 147 3.15.2 Controller with set-point weighting - Table 40 148

3.16 Delay Model 149 3.16.1 Ideal controller - Table 41 149

3.17 General Model with a Repeated Pole 152 3.17.1 Ideal controller - Table 42 152

3.18 General Model with Integrator 153 3.18.1 Ideal controller - Table 43 153

4. Tuning Rules for PID Controllers 154 4.1 FOLPD Model 154


-Table 45 181 4.1.3 Ideal controller in series with a second order filter

-Table 46 183 4.1.4 Ideal controller with weighted proportional term

-Table 47 185 4.1.5 Ideal controller with first order filter and set-point

weighting 1 - Table 48 186 4.1.6 Controller with filtered derivative - Table 49 187 4.1.7 Blending controller - Table 50 193 4.1.8 Classical controller 1 - Table 51 194 4.1.9 Classical controller 2 - Table 52 208 4.1.10 Series controller (classical controller 3) - Table 53 209 4.1.11 Classical controller 4 - Table 54 211 4.1.12 Non-interacting controller 1 - Table 55 212 4.1.13 Non-interacting controller 2a - Table 56 213 4.1.14 Non-interacting controller 2b - Table 57 215

Handbook of PI and PID Controller Tuning Rules

4.1.15 Non-interacting controller based on the two degree of freedom structure 1 - Table 58 217



4.1.18 Non-interacting controller 4 - Table 61 224 4.1.19 Non-interacting controller 5 - Table 62 226 4.1.20 Non-interacting controller 6 (I-PD controller)

- Table 63 227 4.1.21 Non-interacting controller 7 -Table 64 230 4.1.22 Non-interacting controller 11 - Table 65 231 4.1.23 Non-interacting controller 12 - Table 66 232 4.1.24 Industrial controller - Table 67 233 Non-Model Specific 235 4.2.1 Ideal controller - Table 68 235 4.2.2 Ideal controller in series with a first order lag

- Table 69 241 4.2.3 Ideal controller in series with a second order filter

-Table 70 243 4.2.4 Ideal controller with weighted proportional term

- Table 71 245 4.2.5 Controller with filtered derivative - Table 72 246 4.2.6 Ideal controller with set-point weighting 1 - Table 73 . . . . 252 4.2.7 Classical controller 1 - Table 74 253 4.2.8 Series controller (classical controller 3) - Table 75 254 4.2.9 Classical controller 4 - Table 76 255 4.2.10 Non-interacting controller based on the two

degree of freedom structure 1 - Table 77 256 4.2.11 Non-interacting controller 4 - Table 78 257 4.2.12 Non-interacting controller 9 - Table 79 258 BPD Model 259 4.3.1 Ideal controller - Table 80 259 4.3.2 Ideal controller in series with a first order lag

-Table 81 262 4.3.3 Ideal controller with first order filter and set-point

weighting 2 - Table 82 263 4.3.4 Controller with filtered derivative - Table 83 264 4.3.5 Controller with filtered derivative and dynamics on

the controlled variable - Table 84 265 4.3.6 Classical controller 1 - Table 85 266 4.3.7 Classical controller 2 - Table 86 268

Contents xiii

4.3.8 Classical controller 4 - Table 87 269 4.3.9 Non-interacting controller based on the two

degree of freedom structure 1 - Table 88 270 4.3.10 Non-interacting controller based on the two

degree of freedom structure 3 - Table 89 272 4.3.11 Non-interacting controller 4 - Table 90 273 4.3.12 Non-interacting controller 6 (I-PD controller)

- Table 91 274 4.3.13 Non-interacting controller 8 -Table 92 276 4.3.14 Non-interacting controller 10 -Table 93 277 4.3.15 Non-interacting controller 12 -Table 94 278

4.4 FOLIPD Model 279 4.4.1 Ideal controller - Table 95 279 4.4.2 Ideal controller in series with a first order lag

- Table 96 282 4.4.3 Ideal controller with weighted proportional term

-Table 97 284 4.4.4 Controller with filtered derivative - Table 98 285 4.4.5 Controller with filtered derivative with set-point

weighting 1 - Table 99 286 4.4.6 Controller with filtered derivative with set-point

weighting 3 - Table 100 287 4.4.7 Ideal controller with set-point weighting 1 - Table 101 . . . 289 4.4.8 Classical controller 1 - Table 102 290 4.4.9 Classical controller 2 - Table 103 292 4.4.10 Series controller (classical controller 3) - Table 104 293 4.4.11 Classical controller 4 - Table 105 294 4.4.12 Non-interacting controller based on the two

degree of freedom structure 1 - Table 106 295 4.4.13 Non-interacting controller 4 - Table 107 297 4.4.14 Non-interacting controller 6 (I-PD controller)

- Table 108 298 4.4.15 Non-interacting controller 8 - Table 109 303 4.4.16 Non-interacting controller 11 - Table 110 304 4.4.17 Non-interacting controller 12 - Table 111 305 4.4.18 Industrial controller - Table 112 306 4.4.19 Alternative controller 1 - Table 113 307 4.4.20 Alternative controller 2 - Table 114 308

4.5 SOSPD Model 309 4.5.1 Ideal controller - Table 115 309 4.5.2 Ideal controller in series with a first order lag

-Table 116 332


4.5.3 Ideal controller in series with a first order filter -Table 117 336

4.5.4 Ideal controller in series with a second order filter -Table 118 337

4.5.5 Controller with filtered derivative - Table 119 338 4.5.6 Controller with filtered derivative in series with

a second order filter - Table 120 339 4.5.7 Ideal controller with set-point weighting 1 - Table 121 . . . 340 4.5.8 Ideal controller with set-point weighting 2 - Table 122 . . . 341 4.5.9 Classical controller 1 - Table 123 342 4.5.10 Classical controller 2 - Table 124 351 4.5.11 Series controller (classical controller 3) - Table 125 352 4.5.12 Non-interacting controller 1 - Table 126 354 4.5.13 Non-interacting controller based on the two

degree of freedom structure 1 - Table 127 363 4.5.14 Non-interacting controller 4 - Table 128 368 4.5.15 Non-interacting controller 5 - Table 129 370 4.5.16 Non-interacting controller 6 - Table 130 371 4.5.17 Alternative controller 4 - Table 131 372 I2PD Model 374 4.6.1 Controller with filtered derivative with set-point

weighting 2 - Table 132 374 4.6.2 Controller with filtered derivative with set-point

weighting 4 - Table 133 376 4.6.3 Series controller (classical controller 3) - Table 134 378 4.6.4 Non-interacting controller based on the two

degree of freedom structure 1 - Table 135 379 4.6.5 Industrial controller - Table 136 380 SOSIPD Model (repeated pole) 381 4.7.1 Non-interacting controller based on the two

degree of freedom structure 1 - Table 137 381 SOSPD Model with a Positive Zero 383 4.8.1 Ideal controller - Table 138 383 4.8.2 Ideal controller in series with a first order lag

-Table 139 385 4.8.3 Controller with filtered derivative - Table 140 386 4.8.4 Classical controller 1 - Table 141 387 4.8.5 Series controller (classical controller 3) - Table 142 388 4.8.6 Classical controller 4 - Table 143 389 4.8.7 Non-interacting controller 1 -Table 144 390 4.8.8 Non-interacting controller based on the two

degree of freedom structure 1 - Table 145 391

Contents xv

4.9 SOSPD Model with a Negative Zero 392 4.9.1 Ideal controller - Table 146 392 4.9.2 Controller with filtered derivative - Table 147 393 4.9.3 Classical controller 1 - Table 148 394 4.9.4 Classical controller 4 - Table 149 395 4.9.5 Non-interacting controller 1 - Table 150 396 4.9.6 Non-interacting controller based on the two

degree of freedom structure 1 - Table 151 397 4.10 Third Order System plus Time Delay Model 398


- Table 153 399 4.10.3 Controller with filtered derivative - Table 154 400 4.10.4 Non-interacting controller based on the two

degree of freedom structure 1 - Table 155 401 4.11 Unstable FOLPD Model 403


- Table 157 408 4.11.3 Ideal controller with set-point weighting 1 - Table 158 . . . 410 4.11.4 Classical controller 1 -Table 159 414 4.11.5 Series controller (classical controller 3) - Table 160 415 4.11.6 Non-interacting controller based on the two

degree of freedom structure 1 - Table 161 416 4.11.7 Non-interacting controller 3 - Table 162 420 4.11.8 Non-interacting controller 8 - Table 163 421 4.11.9 Non-interacting controller 10 - Table 164 423 4.11.10 Non-interacting controller 12 - Table 165 425

4.12 Unstable SOSPD Model (one unstable pole) 427 4.12.1 Ideal controller - Table 166 427 4.12.2 Ideal controller in series with a first order lag

- Table 167 430 4.12.3 Ideal controller with set-point weighting 1 - Table 168 . . . 431 4.12.4 Classical controller 1 - Table 169 432 4.12.5 Series controller (classical controller 3) -Table 170 433 4.12.6 Non-interacting controller 3 - Table 171 434 4.12.7 Non-interacting controller 8 - Table 172 439

4.13 Unstable SOSPD Model (two unstable poles) 442 4.13.1 Ideal controller - Table 173 442 4.13.2 Ideal controller with set-point weighting 1 - Table 174 . . . 444

4.14 Unstable SOSPD Model with a Positive Zero 445

xvi Handbook of PI and PID Controller Tuning Rules

4.14.1 Ideal controller in series with a first order lag -Table 175 445


4.15 Delay Model 449 4.15.1 Ideal controller - Table 177 449 4.15.2 Ideal controller in series with a first order lag

-Table 178 450 4.15.3 Classical controller 1 -Table 179 451

4.16 General Model with a Repeated Pole 452 4.16.1 Ideal controller - Table 180 452 4.16.2 Ideal controller in series with a first order lag

-Table 181 453 4.17 General Stable Non-Oscillating Model with a Time Delay 454

4.17.1 Ideal controller - Table 182 454 4.18 Fifth Order System plus Delay Model 455

4.18.1 Ideal controller - Table 183 455 4.18.2 Controller with filtered derivative - Table 184 457 4.18.3 Non-interacting controller 7 - Table 185 460

5. Performance and Robustness Issues in the Compensation of FOLPD Processes with PI and PID Controllers 462 5.1 Introduction 462 5.2 The Analytical Determination of Gain and Phase Margin 463

5.2.1 PI tuning formulae 463 5.2.2 PID tuning formulae 466

5.3 The Analytical Determination of Maximum Sensitivity 469 5.4 Simulation Results 470 5.5 Design of Tuning Rules to Achieve Constant Gain and

Phase Margins, for all Values of Delay 475 5.5.1 PI controller design 475

5.5.1.1 Processes modelled in FOLPD form 475 5.5.1.2 Processes modelled in IPD form 477

5.5.2 PID controller design 480 5.5.2.1 Processes modelled in FOLPD form

- classical controller 1 480 5.5.2.2 Processes modelled in SOSPD form

- series controller 482 5.5.2.3 Processes modelled in SOSPD form with a

negative zero - classical controller 1 482 5.5.3 PD controller design 483

5.6 Conclusions 484

Contents xvii

Appendix 1 Glossary of Symbols Used in the Book 485

Appendix 2 Some Further Details on Process Modelling 493

Bibliography 507

Index 535

Chapter 1

Introduction

1.1 Preliminary Remarks

The ability of proportional integral (PI) and proportional integral derivative (PID) controllers to compensate most practical industrial processes has led to their wide acceptance in industrial applications. Koivo and Tanttu (1991), for example, suggest that there are perhaps 5-10% of control loops that cannot be controlled by single input, single output (SISO) PI or PID controllers; in particular, these controllers perform well for processes with benign dynamics and modest performance requirements (Hwang, 1993; Astrom and Hagglund, 1995). It has been stated that 98% of control loops in the pulp and paper industries are controlled by SISO PI controllers (Bialkowski, 1996) and that, in process control applications, more than 95% of the controllers are of PID type (Astrom and Hagglund, 1995). The PI or PID controller implementation has been recommended for the control of processes of low to medium order, with small time delays, when parameter setting must be done using tuning rules and when controller synthesis is performed either once or more often (Isermann, 1989).

However, despite decades of development work, surveys indicating the state of the art of control industrial practice report sobering results. For example, Ender (1993) states that, in his testing of thousands of control loops in hundreds of plants, it has been found that more than 30% of installed controllers are operating in manual mode and 65% of loops operating in automatic mode produce less variance in manual than in automatic (i.e. the automatic controllers are poorly tuned). The situation

l

2 Handbook of PI and PID Controller Tuning Rules

does not appear to have improved in recent years as Van Overschee and De Moor (2000) report that 80% of PID controllers are badly tuned; 30% of PID controllers operate in manual with another 30% of the controlled loops increasing the short term variability of the process to be controlled (typically due to too strong integral action). The authors state that 25% of all PID controller loops use default factory settings, implying that they have not been tuned at all.

These and other surveys (well summarised by Yu, 1999, pages 1-2) show that the determination of PI and PID controller tuning parameters is a vexing problem in many applications. The most direct way to set up controller parameters is the use of tuning rules; obviously, the wealth of information on this topic available in the literature has been poorly communicated to the industrial community. One reason is that this information is scattered in a variety of media, including journal papers, conference papers, websites and books over a period of seventy years. The author has recorded 408 separate sources of tuning rules since the first such rule was published by Callender et al. (1935/6). In a striking statistic, 293 sources of tuning rules have been recorded since 1992, reflecting the upsurge of interest in the use of the PID controller recently.

The purpose of this book is to bring together, in summary form, the tuning rules for PI and PID controllers that have been developed to compensate SISO processes with time delay. Tuning rules for the variations that have been proposed in the 'ideal' PI and PID controller structure are included. Considerable variations in the ideal PID controller structure, in particular, are encountered; these variations are explored in more detail in Chapter 2.

1.2 Structure of the Book

Tuning rules are set out in the book in tabular form. This form allows the rules to be represented compactly. The tables have four or five columns, according to whether the controller considered is of PI or PID form, respectively. The first column in all cases details the author of the rule and other pertinent information. The final column in all cases is labelled "Comment"; this facilitates the inclusion of information about the tuning

Chapter 1: Introduction 3

rule that may be useful in its application. The remaining columns detail the formulae for the controller parameters.

Chapter 2 explores the range of PI and PID controller structures proposed in the literature. It is often forgotton that different manufacturers implement different versions of the PID controller algorithm (in particular); therefore, controller tuning rules that work well in one PID architecture may work poorly on another. This chapter also details the process models used to define the controller tuning rules.

Chapters 3 and 4 of the book detail, in tabular form, tuning rules for setting up, respectively, PI controllers and PID controllers (and their variations), for a wide variety of process models. P controller and I controller tuning rules are also defined (as subsets of PI controller tuning rules), and PD controller tuning rules are defined (as a subset of PID controller tuning rules), for processes whose model includes an integrator. One hundred and eighty-three such tables are provided altogether. To allow the reader to access data readily, the author has arranged that each table start on its own page of the book; each table is preceded by the controller used, together with a block diagram showing the unity feedback closed loop arrangement of the controller and process model.

In Chapter 5 of the book, analytical calculations of the gain and phase margins of a large sample of PI and PID controller tuning rules are determined, when the process is modelled in first order lag plus time delay (FOLPD) form, at a range of ratios of time delay to time constant of the process model. Results are given in graphical form.

An important feature of the book is the unified notation that is used for the tuning rules; a glossary of the symbols used is provided in Appendix 1. Appendix 2 outlines the range of methods that are used to determine process model parameters; this information is presented in summary form, as this topic could provide data for a book in itself. However, sufficient information, together with references, is provided for the interested reader.

Finally, a comprehensive reference list is provided. In particular, the author would like to recommend the contributions by McMillan (1994), Astrom and Hagglund (1995), Shinskey (1994), (1996), Tan et al. (1999a), Yu (1999), Lelic and Gajic (2000) and Ang et al. (2005) to the


interested reader, which treat comprehensively the wider perspective of PID controller design and application.

Chapter 2

Controller Architecture

2.1 Introduction

The ideal continuous time domain PID controller for a SISO process is expressed in the Laplace domain as follows:

U(s) = Gc(s)E(s) (2.1)

with

Gc(s) = Kc(l + -^- + Tds) (2.2)

and with Kc = proportional gain, Tj = integral time constant and Td = derivative time constant. If T; = oo and Td - 0 (i.e. P control), then it is clear that the closed loop measured value, y, will always be less than the desired value, r (for processes without an integrator term, as a positive error is necessary to keep the measured value constant, and less than the desired value). The introduction of integral action facilitates the achievement of equality between the measured value and the desired value, as a constant error produces an increasing controller output. The introduction of derivative action means that changes in the desired value may be anticipated, and thus an appropriate correction may be added prior to the actual change. Thus, in simplified terms, the PID controller allows contributions from present controller inputs, past controller inputs and future controller inputs.

Many variations of the PI and, in particular, the PID controller structure have been proposed. As Tan et al. (1999a) suggest, one

5


important reason for the non-standard structures is due to the transition of the controllers from pneumatic implementation through electronic implementation to the present microprocessor implementation. The variations in the controller structures are detailed below.

2.2 PI Controller Structures

Tuning rules have been detailed for seven PI controller structures:

1. Ideal controller Gc(s) = K( i+-L V T i s 7

(2.3)

2. Ideal controller in series with a first order lag:

Gc(s) = Kc 1 + T:S

1

1 + Tfs (2.4)

3. Ideal controller in series with a second order filter (also labelled the 'generalised PID' controller (Lee and Shi, 2002)):

Gc(s) = Kc 1 + 1

v T>sy

2 ^ 1 + bfls + bf2s

l + afls + af2s j (2.5)

4. Controller with set-point weighting:

U(s) = Kc 'i-L^ V T i s /

E(s)-aK cR(s) (2.6)

5. Controller with proportional term acting on the output 1:

U(s) = ^ E ( s ) - K c Y ( s ) Ts

(2.7)

6. Controller with proportional term acting on the output 2: r

U(s) = K 1 \

V 1+

Tfsy

E(s)-K,Y(s) (2.8)

7. Controller with a double integral term:

Chapter 2: Controller Architecture 7

Gc(s) = Kc ' 1 1 ^ 1+ +

V ^ilS Ti2S j

(2.9)

2.3 PID Controller Structures

Forty-six such structures have been considered. The labelling of these structures has not been consistent in the literature; for example, the first PID controller structure considered (labelled the ideal controller below) has also been labelled the 'non-interacting' controller (McMillan, 1994), the 'ISA' algorithm (Gerry and Hansen, 1987) or the 'parallel non-interacting' controller (Visioli, 2001). The controller structures have been divided into four types, as described below.

2.3.1 Ideal PID controller structure and its variations

1. Ideal controller: Gc(s) = Kc

f 1 A

1 + — + Tds V T . s j

(2.10)

A variation of the controller is labelled the 'parallel' controller structure (McMillan, 1994). This variation has also been labelled the 'ideal parallel', 'noninteracting', 'independent' or 'gain independent' algorithm:

Gc(s) = K c + - i - + Tds (2.11) TiS

The controller structure is used in the following products: (a) Allen Bradley PLC5 product (McMillan, 1994) (b) Bailey FC19 PID algorithm (EZYtune, 2003) (c) Fanuc Series 90-30 and 90-70 independent form PID algorithm

(EZYtune, 2003) (d) Intellution FIX products (McMillan, 1994) (e) Honeywell TDC3000 Process Manager Type A, non-interactive

mode product (ISMC, 1999) (f) Leeds and Northrup Electromax 5 product (Astrom and

Hagglund, 1988)

(g) Yokogawa Field Control Station (FCS) PID algorithm (EZYtune, 2003).

2. Ideal controller in series with a first order lag: f \ 1

Gc(s) = Kc l + + Tds v T i s ; Tfs + l

(2.12)

3. Ideal controller in series with a first order filter: /

Gc(s) = Kt 1

1 + — + Tds V T i s J

bfls + l

afls + l (2.13)

4. Ideal controller in series with a second order filter: /

Gc(s) = Kt 1 \

1 + — + Tds v T i s J

l + bfls

l + a n s + af2s (2.14)

5. Ideal controller with weighted proportional term: ( 1 A

Gc(s) = Kt b + — + THs V T i S

J (2.15)

6. Ideal controller with first order filter and setpoint weighting 1: r

U(s) = Kc 1 >

i + — + T d s V T i s J

1 Tfs + 1

R ( S ) 1±^_ Y ( S ) 1 + sT

(2.16)

7. Ideal controller with first order filter and setpoint weighting 2: /

U(s) = Kc 1 \

V T i s

1

Tfs + 1 R<.)l±2^i-Y(.,

l + sTr

K0Y(s)

(2.17) 8. Ideal controller with first order filter and dynamics on the controlled

variable: /

U(s) = Kt 1 A

1 + — + Tds _J^TV t ( l ± K) _JC_ Tfs + 1 • Ts + 1 Ts + 1

9. Controller with filtered derivative:

Gc(s) = Kc 1 + — + - d

T*s l + s ^ N

(2.19)

This structure is used in the following products: (a) Bailey Net 90 PID error input product with N = 10 (McMillan,

1994) and FC156 Independent Form PID algorithm (EZYtune, 2003)

(b) Concept PIDP1 and PID1 PID algorithms (EZYtune, 2003) (c) Fischer and Porter DCU 3200 CON PID algorithm with N = 8

(EZYtune, 2003) (d) Foxboro EXACT I/A series PIDA product (in which it is an

option labelled ideal PID) (Foxboro, 1994). (e) Hartmann and Braun Freelance 2000 PID algorithm (EZYtune,

2003) (f) Modicon 984 product with 2 < N < 30 (McMillan, 1994;

EZYtune, 2003) (g) Siemens Teleperm/PSC7 ContC/PCS7 CTRL PID products with

N = 10 (ISMC, 1999) and the S7 FB41 CONTC PID product (EZYtune, 2003).

10. Controller with filtered derivative in series with a second order filter: ( \

Gc(s) = Kt T>s 1 +

T< V N

l + bfls + bf2s

l + afls + af2s2 (2.20)

11. Controller with filtered derivative with setpoint weighting 1: f \

Gc(s) = K( , 1 TdS 1 + — + %r

T:S - L l + ' d

V s

N j

l + bfls

l + afls R(s)-Y(s) (2.21)

12. Controller with filtered derivative with setpoint weighting 2:


f

Gc(s) = Kc T . s l + ^ s

N

l + b f ls + b f2s"

l + a f ls + a f 2s2 -R( s ) -Y(s ) (2.22)

13. Controller with filtered derivative with setpoint weighting 3: r \

Gc(s) = K( 1 + — + — h -Xs . X

v 1 + ^ s

N ;

1 + b » S + b " S2

2 + b » S3

3 R ( s ) - Y ( s ) A

(2.23)

14. Controller with filtered derivative with setpoint weighting 4:

Gc(s) = Kc 1 + — + • d

Xs , . Td 1 + ^ s N j

' 1 + b " S + b ^ + b " S 3 + b " S4

4 R ( s ) - Y(s) ' 1 + Sf-jS H-S-f-^S ~H S ^ T S + 3 . ^ 4 8

(2.24)

15. Controller with filtered derivative and dynamics on the controlled variable:

Gc(s) = Kt 1 1 T d s

1+- --^ d Xs , . . Xd 1 + s-

N

E ( s ) - K 0 Y ( s ) (2.25)

16. Ideal controller with setpoint weighting 1:

U(s) = Kc (FpR(s) - Y ( s ) ) + ^ ( F i R ( s ) - Y(s))+ KcXds(FdR(s) - Y(s))

(2.26) 17. Ideal controller with setpoint weighting 2:

( U(s) = K,

1 \ 1 + — + Tds

V T i s j

1

T i V + T j S + 1 R ( s ) - Y ( s ) (2.27)

18. Blending controller:

Gc(s) = Kc

f 1 >

Ts - (2.28) s

This structure is used in the Yokogawa EFCS/EFCD Field Control Station product (Chen et al., 2001).

2.3.2 Classical PID controller structure and its variations

19. Classical controller 1: This controller is also labelled the 'cascade' controller (Witt and Waggoner, 1990), the 'interacting' or 'series' controller (Poulin and Pomerleau, 1996), the 'interactive' controller (Tsang and Rad, 1995), the 'rate-before-reset' controller (Smith and Corripio, 1997), the 'analog' controller (St. Clair, 2000) or the 'commercial' controller (Luyben, 2001).

Gc(s) = Kt ' 1 ' l + STd (2.29) l + J-Ts T

N The structure is used in the following products: (a) Honeywell TDC Basic/Extended/Multifunction Types A and B

products with N = 8 (McMillan, 1994) (b) Toshiba TOSDIC 200 product with 3.33<N<10 (McMillan,

1994) (c) Foxboro EXACT Model 761 product with N = 10 (McMillan,

1994) (d) Honeywell UDC6000 product with N = 8 (Astrom and

Hagglund, 1995) (e) Honeywell TDC3000 Process Manager product - Type A,

interactive mode with N = 10 (ISMC, 1999) (f) Honeywell TDC3000 Universal, Multifunction and Advanced

Multifunction products with N = 8 (ISMC, 1999) (g) Foxboro EXACT I/A Series PIDA product (in which it is an

option labelled series PID) (Foxboro, 1994).


20. Classical controller 2: Gc(s) = Kc 1 + -T,Sy

l + NTds

1 + Tds j (2.30)

21. Series controller (Classical controller 3): This controller is also labelled the 'interacting' controller, the 'analog algorithm' (McMillan, 1994) or the 'dependent' controller (EZYtune, 2003).

Gc(s) = Kt 1 + T.Sy

(l + sTd) (2.31)

The structure is used in the following products: (a) Turnbull TCS6000 series product (McMillan, 1994) (b) Alfa-Laval Automation ECA400 product (Astrom and

Hagglund, 1995) (c) Foxboro EXACT 760/761 product (Astrom and Hagglund,

1995).

22. Classical controller 4: This controller is also labelled the 'interacting' controller (Fertik, 1975).

Ge(s) = Kt 1

l + - TdS

1 + s ^ N J

(2.32)

The structure is used in the following products: (a) Bailey FC156 Classical Form PID product (EZYtune, 2003) (b) Fischer and Porter DCI 4000 PID algorithm (EZYtune, 2003)

2.3.3 Non-interacting PID controller structure and its variations

23. Non-interacting controller 1: This controller is also labelled the 'reset-feedback' controller (Huang et al., 1996).

U(s) = Kt 1 + ]_

T i S ,

E(s) % - Y ( s )

1 + -N

This structure is used in the following products:

(2.33)

(a) Bailey Fisher and Porter 53SL6000 and 53MC5000 products with N = 0 (ISMC, 1999).

(b) Moore Model 352 Single-Loop Controller product (Wade, 1994).

24. Non-interacting controller based on the parallel controller structure: 24a. Non-interacting controller 2a:

f 1 A U(s) = Kt 1 + E(s)-

Tds

N

Y(s)

24b. Non-interacting controller 2b: /

U(s) = 1 A

v

K c + TjSy

E(s)-Tds

N

Y(s)

(2.34)

(2.35)

25. Non-interacting controller based on the two degree of freedom structure 1: This controller is also labelled the 'm-PID' controller (Huang et al, 2000) the 'ISA-PID' controller (Leva and Colombo, 2001) and the 'P-I-PD (only P is DOF) incomplete 2DOF algorithm' by Mizutani and Hiroi (1991).

U(s) = Kt , 1 TdS

v N

E(s)-K c ct + PTds

1 + ^ N

R(s) (2.36)

This structure is used in the following products: (a) Bailey Net 90 PID PV and SP product (McMillan, 1994) (b) Yokogawa SLPC products with <x = - l , (B = - 1 , N=10

(McMillan, 1994) (c) Omron E5CK digital controller with p = 1 and N=3 (ISMC,

1999).

26. Non-interacting controller based on the two degree of freedom structure 2:


U(s) = K( 1 + - U - ^ T « 8 l + ^ s

I N ,

E(s)-K (

f \ PTds X

a +

V l + ^ s 1 + T i s

N

R(s)

(2.37)

27. Non-interacting controller based on the two degree of freedom structure 3:

U(s) = K j ( b - l ) + (c-l)Tds]R(s)

( 1 + K, 1 + — + Tds

v T . s R(s)

(l + sTf): -Y(s) (2.38)

28. PD-I-PD (only PD is DOF) incomplete 2DOF algorithm (Mizutani andHiroi, 1991):

f V

U(s) = Kc a-_ 1 _ %Tds 1

1+-5 -S ^ N

Td_ 1 + Xs R(s)

• K , , 1 TdS 1 + — + — -

T ' s 1 + ^ s N

Y(s) (2.39)

However, no tuning rules are available for this controller structure.

29. PI-PID (only PI is DOF) incomplete 2DOF algorithm (Mizutani and Hiroi, 1991):

U(s) = K( a + 1 P

T;s TiS(l + TiS). R(s)-K£ 1+ + =7-

T;s 1 + ^ - s N

Y(s)

(2.40) However, no tuning rules are available for this controller structure.

Chapter 2; Controller Architecture 15

30. Super (or complete) 2DOF PID algorithm (Mizutani and Hiroi, 1991):

U(s) = Kc a- Ts 1 — J _ + _£?>_

1 + Ts I+3L N

R(s)

- K ,

N v

Y(s) (2.41)

Mizutani and Hiroi (1991) state that 0 < a < 1, p > 1 and % ^ 1 , and suggest starting values of a = 0.4, P = 1.35 and % = 1.25. However, no tuning rules are available for this controller structure.

31. Non-interacting controller 3:

U(s) = Kt 1 + — . TiSy

E(s)-KcTds

1 + ^ L N

Y(s) (2.42)

This structure is used in the following products: (a) Allen Bradley SLC5/02, SLC5/03, SLC5/04, PLC5 and

Logix5550 products (EZYtune, 2003). (b) Modcomp product with N = 10 (McMillan, 1994).

32. Non-interacting controller 4: This controller is also labelled the 'PI+D' controller (Chen, 1996).

U(s) = Kt 1 + E(s)-KcTdsY(s) (2.43)

This structure is used in the following products: (a) ABB 53SL6000 Product (ABB, 2001). (b) Genesis product (McMillan, 1994) (c) Honeywell TDC3000 Process Manager Type B, non-interactive

mode product (ISMC, 1999) (d) Square D PIDR PID product (EZYtune, 2003)



U(s) = Kc b + -T,s

E(s)-(c + Tds)Y(s) (2.44)

34. Non-interacting controller 6 (I-PD controller):

U(s) = - ^ E ( s ) - K c ( l + Tds)Y(s) T:S

(2.45)

This structure is used in the Toshiba AdTune TOSDIC 211D8 product (Shigemasa et al, 1987) and the Honeywell TDC3000 Process Manager Type C non-interactive mode product (ISMC, 1999).


K „ U(s) = - ^ E ( s ) - K ,

T:S n T's

1 - T " N

Y(s) (2.46)


U(s) = K c | l + - j - + Tds |E(s)-K i(l + Tdis)Y(s) (2.47) V T,s

37. Non-interacting controller 9: /

U(s) = Kt 1 \

1 + — + Tds v T.s ;

E(s)--^sY(s) N

(2.48)


U(s) = Kc

\ T i sy E(s)-K f -^-s + 1

VTm J Y(s) (2.49)


U(s) = Kt T=s

(l+Tds)E(s)-K i(l + Tdis)Y(s) (2.50)

40. Non-interacting controller 12: f 1 A

U(s) = Kc V T i s j 1 + Tds

E(s)-K,Y(s) (2.51)


U(s) = Kc

v T;sy E(s)- Tds

1 + I l s + I^s2 Y(s) (2.52)

N 2N This structure is used in the Foxboro IA PID product, with 10 < N <50(EZYtune, 2003); however, no tuning rules are available for this controller structure.

2.3.4 Other PID controller structures

42. Industrial controller (Kaya and Scheib, 1988): ( ^

U(s) = Kc 1+ — . T,Sy

1 +T s R(s)-i±MY(s)

V N

(2.53)

This structure is used in the following products: (a) Fisher-Rosemount Provox product with N = 8 (ISMC, 1999;

McMillan, 1994) (b) Foxboro Model 761 product with N=10 (McMillan, 1994) (c) Fischer-Porter Micro DCI product with N = 0 (McMillan, 1994) (d) Moore Products Type 352 controller with 1 < N < 30 (McMillan,

1994) (e) SATT Instruments EAC400 product with N=8.33 (McMillan,

1994) (f) Taylor Mod 30 ESPO product with N=16.7 (McMillan, 1994)


(g) Honeywell TDC3000 Process Manager Type B, interactive mode product with N = 10 (ISMC, 1999)

43. Alternative controller 1: Gc(s) = Kc

f V

1 + Ts

1 + ^ v N

1 + Tds

y\ N j

(2.54)

44. Alternative controller 2:

Gc(s) = Kc 1 + TjS

T l + - ^ s

N j \

2 „ 2 l + 0.5xms + 0.0833xm s

N

(2.55)


K „

/

U(s) = - ^ R ( s ) - K ( T:S

1+ . T,sy

l + sTd

N ;

Y(s) (2.56)

This structure is used in the following products: (a) Honeywell TDC3000 Process Manager Type C, interactive mode

product with N = 10 (ISMC, 1999) (b) Honeywell TDC3000 Universal, Multifunction and Advanced

Multifunction products with N = 8 (ISMC, 1999) However, no tuning rules are available for this controller structure.


U(s) = Kt

N

E(s) + R(s) (2.57)


2.3.5 Comments on the PID controller structures

In some cases, one controller structure may be transformed into another; clearly, the ideal and parallel controller structures (equations (2.10) and (2.11)) are very closely related. It is shown by McMillan (1994), among others, that the parameters of the ideal PID controller may be worked out from the parameters of the series PID controller, and vice versa. The ideal PID controller is given in Equation (2.58) and the series PID controller is given in Equation (2.59).

f 1 ^ 1 + -Gcp(s) = Kcp

Gcs(s) = Kc 1 + T~I

+ Tdps

(l + sTds)

(2.58)

(2.59)

Then, it may be shown that (

K c p = l + _d!L

X. ^•cs ' *ip ~~ V i s + * d s / ' *dp _ X.

X.S+X*

Similarly, it may be shown that, provided X- > 4T( dp •

K„5 = 0.5K„ 1 +Jl-4-l d P

X. X. = 0.5X, 1+11-4 -

X dp

Xds=0.5Xdp 1 - 1 - 4 -LdP

Astrom and Hagglund (1996) point out that the ideal controller admits complex zeroes and is thus a more flexible controller structure than the series controller, which has real zeroes; however, in the frequency domain, the series controller has the interesting interpretation that the zeroes of the closed loop transfer function are the inverse values of Tis and Tds. O'Dwyer (2001b) developed a comprehensive set of tuning rules for the series PID controller, based on the ideal PID controller; as these are not strictly original tuning rules, only representative examples are included in the relevant tables.

In a similar manner, the parameters of an ideal controller in series with a first order filter may be determined from the parameters of the classical controller 1 (Witt and Waggoner, 1990) and the parameters of


the non-interacting controller 10 structure may be determined from the parameters of the ideal PID controller (Kaya et al, 2003).

2.4 Process Modelling

Processes with time delay may be modelled in a variety of ways. The modelling strategy used will influence the value of the model parameters, which will in turn affect the controller values determined from the tuning rules. The modelling strategy used in association with each tuning rule, as described in the original papers, is indicated in the tables (see Chapters 3 and 4). These modelling strategies are outlined in Appendix 2. The following models are defined:

K e~STm

1. First order lag plus time delay (FOLPD) model ( G m (s) = — )

2. FOLPD model with a positive zero ( G m ( s ) = - mV m3J

l + sTml

K m ( l + sTm 3)e-3. FOLPD model with a negative zero ( G m (s) = — ^ ^ )

1 + sTml

4. Non-model specific

K e"SXm

5. Integral plus time delay (IPD) model ( G m (s) = —2 ) s

6. First order lag plus integral plus time delay (FOLIPD) model

(Gm(S)= v r v s(l + s T j

7. Second order system plus time delay (SOSPD) model K e~SXm K e~SXm

(0- (' )=T.1v;^T . , .+ ,°r0-w-(1+TjxuT. I .)> 8. Integral squared plus time delay ( I 2 P D ) model ( G m ( s ) = —m- )

s 9. Second order system (repeated pole) plus integral plus time delay

(SOSIPD) model ( G m (s) = fm& "" ) s(l + sT m ) 2


10. SOSPD model with a positive zero

( G (s)- Kj l -sT^y"- Km(l-sTm3)e-( m ( ) (l + sTml)(l + s T m 2 ) ° r G m ( S ) T m l V + 2 ^ T i n l s + l )

11. SOSPD model with a negative zero

( G m ( s ) =K - ( l + S > > - " ; o r G m ( s ) , Km(l + STm3)e

- S T „

(l + sTml)(l + sTm2) m Tml

2s2+2^mTmls + l

12. Third order system plus time delay model

,^ , -. " m \ - • J,S + b , S 2 + b . S , .

(Gm(s)= mA ' 2 , 3 A or Km(l + b1s + b2s2+b3s3)e"ST"

(l + a,s + a2s2 + a3s3)

K e""m

G m ( S ) = ( l + sTml)(l+msTm2)(l + S T m 3 ) )

13. Fifth order system plus delay model

Km(l + bIs + b 2 s 2 + b 3 s 3 + b 4 s 4+ b s s 5 ) e - ^

(Gm(s) = r j 5 4 n ' \1 + ats + a2s + a3s + a4s + a5s J

K e~STm

14. Unstable FOLPD model (Gm(s) = —* ) Tms-1

15. Unstable FOLPD model with a positive zero Km(l-sTm 3)e-

(Gm(s)= m \ mi ) Tms-1

16. Unstable SOSPD model (one unstable pole)

K e"STm

( G m ( s ) = 7 - ™ ) (TmlS-lX1 + sTm2)

17. Unstable SOSPD model with a positive zero

' ~ T m l V - 2 S m T m l s + l (Gm(s)= J Y ^ T m 3 i e . )

18. Unstable SOSPD model (two unstable poles)

K e~STm

(G (s)= -^- ) ( m ( ) (T m l s- lXT m 2 s- l ) )

19. Delay model (Gm(s) = Kme_s'm )

K e"s'tm

20. General model with a repeated pole (Gm(s) = — ) (l + s T J n


21. General model with integrator

K n(Tm , i s + i)n(T

m / s 2 + 2^„iTm 2 i

s + i)

j i

22. General stable non-oscillating model with a time delay

Tables 1 and 2 show the number of tuning rules defined for each PI or PID controller structure and each process model. The following data is key to the process model type in Tables 1 and 2: Model 1: Stable FOLPD model Model 2: Non-model specific Model 3: IPD model Model 4: FOLIPD model Model 5: SOSPD model Model 6: Other models

Table 1: PI controller structure and tuning rules - a summary

Process model Controller Equation

(2.3) (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) Total

1

172 3 2 7 1 2 0

187

2

42 0 0 2 0 0 0

44

3

46 2 0 4 9 1 1

63

4

10 0 0 3 14 1 0 28

5

32 0 0 3 0 0 0

35

6

62 2 0 13 7 2 0 86

Total

364 7 2

32 31 6 1

443


Table 2: PID controller structure and tuning rules - a summary


(2.10,2.11) (2.12) (2.13) (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) (2.20) (2.21) (2.22) (2.23) (2.24) (2.25) (2.26) (2.27) (2.28)

Subtotal

(2.29) (2.30) (2.31) (2.32)

Subtotal

(2.33) (2.34) (2.35) (2.36) (2.37) (2.38) (2.42) (2.43) (2.44) (2.45) (2.46) (2.47) (2.48)

1 2 3 4 5 6 Total

Ideal structure + variations 100 7 0 2 2 1 0 1 16 0 0 0 0 0 0 0 0 1

130

51 2 0 2 2 0 0 0 11 0 0 0 0 0 0 1 0 0 69

22 1 0 0 0 0 1 0 2 0 0 0 0 0 1 0 0 0

27

12 5 0 0 1 0 0 0 2 0 1 0 2 0 0 1 0 0

24

55 13 1 1 0 0 0 0 3 1 0 0 0 0 0 1 2 0 77

36 9 0 0 0 0 0 0 5 0 0 1 0 2 0 9 0 0 62

276 37 1 5 5 1 1 1

39 1 1 1 2 2 1 12 2 1

389 Classical structure + variations

54 1 6 2 63

5 0 5 1

11

8 2 0 1

11

5 1 1 1 8

20 2 7 0 29

9 0 6 2 17

101 6

25 7

139 Non-interacting structure + variations

1 5 6 12 4 2 0 10 2 10 1 1 0

0 0 0 1 0 0 0 3 0 0 0 0 1

0 0 0 5 0 1 0 4 0 8 0 1 0

0 0 0 6 0 0 0 2 0 11 0 2 0

5 0 0 7 0 0 0 3 1 1 0 0 0

2 0 0 13 0 0 4 0 0 0 1 4 0

8 5 6

44 4 3 4 22 3

30 2 8 1



(2.49) (2.50) (2.51)

Subtotal

(2.53) (2.54) (2.55) (2.57)

Subtotal

1 2 3 4 5 6 Total

Non-interacting structure + variations (continued) 0 0 1

55

0 0 0 J

1 0 1

21

0 1 1

23

0 0 0

17

2 0 1

27

3 1 4

148 Other controller structures

8 0 0 0 8

0 0 0 0 0

0 0 0 0 0

1 2 1 0 4

0 0 0 2 2

1 0 0 0 1

10 2 1 2 75

Total | 256 | 85 | 59 | 59 | 125 | 107 | 691

Table 1 shows that 82% of tuning rules have been defined for the ideal PI controller structure, with 42% of tuning rules based on a FOLPD process model.

The range of PID controller variations has lead to a less homogenous situation than for the PI controller; Table 2 shows that 40% of tuning rules have been defined for the ideal PID controller structure, with 37% of tuning rules based on a FOLPD process model. Many of the controller structures are used in a variety of manufacturers' products (as outlined in Section 2.3); clearly a range of tuning rules are available for use with these products.

2.5 Organisation of the Tuning Rules

The tuning rules are organised in tabular form in Chapters 3 and 4. Within each table, the tuning rules are classified further; the main subdivisions made are as follows: (i) Tuning rules based on a measured step response (also called process

reaction curve methods), (ii) Tuning rules based on minimising an appropriate performance

criterion, either for optimum regulator or optimum servo action, (iii) Tuning rules that give a specified closed loop response (direct

synthesis tuning rules). Such rules may be defined by specifying the


desired poles of the closed loop response, for instance, though more generally, the desired closed loop transfer function may be specified. The definition may be expanded to cover techniques that allow the achievement of a specified gain margin and/or phase margin,

(iv) Robust tuning rules, with an explicit robust stability and robust performance criterion built in to the design process,

(v) Tuning rules based on recording appropriate parameters at the ultimate frequency (also called ultimate cycling methods),

(vi) Other tuning rules, such as tuning rules that depend on the proportional gain required to achieve a quarter decay ratio or to achieve magnitude and frequency information at a particular phase lag.

Some tuning rules could be considered to belong to more than one subdivision, so the subdivisions cannot be considered to be mutually exclusive; nevertheless, they provide a convenient way to classify the rules. In the tables, all symbols used are defined in Appendix 1.

Chapter 3

Tuning Rules for PI Controllers

3.1 FOLPD Model Gm(s) = K„e"

l + sT„

3.1.1 Ideal controller Gc(s) = Kt

v T> sv

Table 3; PI controller tuning rules - FOLPD model Gra(s) = l + sT„

Rule

Callender et al. (1935/6).

Model: Method 1

Kc Ti Comment

Process reaction '0.568/Kmxm

20.690/Kmxm

3.64xm

2-45xm

^=- = 0.3 T

m

Decay ratio = 0.015; Period of decaying oscillation = 10.5xm

' Decay ratio = 0.043; Period of decaying oscillation = 6.28xn

26

Chapter 3: Tuning Rules for PI Controllers 27

Rule

Ziegler and Nichols (1942).

Model: Method 2

Hazebroek and Van derWaerden(1950).

Model: Method 2

Oppelt(1951). Model: Method 2

Moros (1999). Model: Method 1

Chienefa/. (1952)-regulator.

Model: Method 2;

0.1 <^2-< 1.0 T

Chiene/a/. (1952)-servo.

Model: Method 2;

0.1 <^2-< 1.0 T

Kc

0.9Tm

K m t m

x l T m

Km tm

T;

3.33xm

x 2 x m

Comment

Quarter decay ratio.

T

Coefficient values

T

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x l

0.68 0.70 0.72 0.74 0.76 0.79 0.81 0.84 0.87

x2

7.14 4.76 3.70 3.03 2.50 2.17 1.92 1.75 1.61

x, = 0 . 5 ^ + 0.1

3KC(1)

0.8Tra/Kmxm

0.91Tm/Kraxm

0-6Tm

K m T m

0.7Tm

Kmxm

0.35Tm

0-6Tm

K r a x m

x m

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

x 2 = -

x i

0.90 0.93 0.96 0.99 1.02 1.06 1.09 1.13 1.17

x2

1.49 1.41 1.32 1.25 1.19 1.14 1.10 1.06 1.03

*m

•6tm-1.2Tm

1.66xm

3.32Tra

3tm

3.3xm

4xm

2-33xm

1.17Tm

Tm

T

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4

x i

1.20 1.28 1.36 1.45 1.53 1.62 1.71 1.81

x2

1.00 0.95 0.91 0.88 0.85 0.83 0.81 0.80

^2->3.5

xm » Tra

^ m « T m

attributed to Oppelt

attributed to Rosenberg

0% overshoot

20% overshoot

0% overshoot

20% overshoot

3 K,(1> K „

. 5 7 0 8 ^ + 1 I recommended K (0 K „

0.77-

Rule

Reswick(1956). Model: Method 2;

K c , Tj deduced

from graphs.

i s - = 6.25 T

m

Cohen and Coon (1953).

Model: Method 2

Two constraints method - Wolfe

(1951). Model: Method 4

Two constraints criterion - Murrill (1967) -page 356. Model: Method 5

Heck (1969). Model: Method 2

Kc

0.20/Km

0.30/Km

0.15/Km

0.20/Km

J _ 0 . 9 ^ + 0.083

" XlTm

KmTm

Ti

0.20xm

0.22Tm

0.16xm

0.14xm

4 T ( 2 ) i

X 2 T m

Coefficient values

T m

0.2

0.5

0.925

* 1

4.4 1.8

x2

3.23 2.27

/ N 0.946

l X m J

0.8Tm

K m T m

0

T

1.0 5.0

*1

0.78 0.30

x 2

1.28 0.53

x ( A0 '583

1.078 [Tm J

3 i m

x 2 Km T m

Comment

0% overshoot -regulator

20% overshoot -regulator

0% overshoot - servo

20% overshoot -servo

Quarter decay ratio;

0 < - ^ _ < 1.0 T

Decay ratio is as small as possible;

minimum error integral (regulator

mode).

Quarter decay ratio; minimum error

integral (servo mode).

0.1<-^2L<1.0 T

m

Integral controller

Representative coefficient values - from graph - "Stoning"

1B. T

0.11

0.13 0.14

0.17

X 2

2.94 2.86

2.78 2.63

T m

0.20

0.25 0.29 0.33

x 2

2.50

2.38 2.35 2.30

IHL T

1 m

0.40 0.50 0.56

0.63

x 2

2.17

2.13 2.08 2.00

T

0.71

x 2

1.96

• ( 2 )

3.33- -0.31

1 + 2.22-

2\


Rule

Heck (1969)-continued.

Fertik and Sharpe (1979).

Sakai et al. (1989). Model: Method 2

Borresen and Grindal (1990).

Model: Method 2

Klein et al. (1992). Model: Method 1

McMillan ( 1 9 9 4 ) -page 25.

Model: Method 5 St. Clair ( 1 9 9 7 ) -

page 22. Model: Method 5

Hay (1998) -pages 197-198.

Model: Method 1;

Coefficients of KC ,T;

deduced from graphs

Shinskey (2000), (2001).

Model: Method 2

Kc Tf Comment

Representative coefficient values - from graph - "Fuhrung" T m

T

0.11

0.13 0.14

0.17

x2

3.92

3.85 3.64

3.51

T m

0.20

0.25 0.29

0.33

0.56

K m

1.2408Km

T T

1.0Tm

K m t m

0.28Tm

K m ( t m + 0 . 1 T r a )

3

0.333Tm

K m T m

x , / K m

x2

3.33

3.08 2.94

2.82

T

0.40

0.50 0.56

0.63

x2

2.63 2.47 2.41 2.35

0.65Tm

0.5xm

3^ r a

0.53Tra

m

T

X 2 t m

T

0.71

x2

2.27

Model: Method 3

Also described by ABB (2001)

Time delay dominant processes

l a . > 0.33 T

Coefficient values - servo tuning

Tm/Tm

0.1 0.125

0.167 0.25 0.5

x i

6.0 4.5 3.0 2.0 0.8

x2

9.5 7.5 5.4 3.4 1.7

"Cm Am

0.1 0.125

0.167 0.25

0.5

x l

4.0 3.2 2.2 1.3 0.5

x2

10.0 8.0 6.0 4.0 2.0

Coefficient values - regulator tuning

^ m / T m

0.1 0.125 0.167 0.25

0.5

X I

9.5 7.0 5.0 3.4 1.8

0.667Tm

KmTm

x2

3.2 3.2 3.0 2.8 2.0

tm/Xi 0.1

0.125 0.167 0.25 0.5

3.78xra

x i

6.8 5.6 4.3 2.8 1.2

x2

2.8 2.8 2.6 2.3 1.6

T m

Rule

Liptak(2001). Model: Method 2

Minimum IAE -Murrill(1967)-pages 358-363.

Minimum IAE -Frank and Lenz


Minimum IAE -Pemberton (1972a), Smith and Corripio (1997) -page 345.

Minimum IAE - Yu (1988).

Model: Method 1. Load disturbance

K e~STu

model = —- . l + sTL

Kc

0.95Tm

Kmxra

T i

4^m

Comment

Minimum performance index: regulator tuning f N 0.986

0-984 (lm\

— | x , + x 2 - ^

T

0.608

( \

T

0.707

x 2 I T m J

Model: Method 5;

0 . 1 < ^ - < 1 . 0 T

Representative coefficient values - deduced from a graph

* m / T m

0.2 0.4 0.6 0.8

x l

0.37 0.42 0.44 0.45

T

KmTm

5KC(3)

6 K c ( 4 )

x 2

0.77 0.85 0.91 0.98

Tm/Tm

1.0 1.2 1.4

T

T ( 3 )

T ( 4 )

x i

0.46 0.46 0.46

x 2

1.05 1.11 1.17

Model: Method 1;

0.1 <-^2-< 0.5 T

m

^ - < 0.35 Tm

5 K <3> = 0.685

K „ T V m

T „

X ( 3 ) = -0.214 T,

\ 1.977-!=—0.55 . T I

K W _ 0.874

K „

N -0.099+0.159—

T V 1 m J

m J

N - 1 . 0 4 1

T •^-1 , ^ < 2.641^2- + 0.16.

T T

T V m 7

-4.515—+0.067,

- ( 4 )

0.415 T T , 2.641-SL + 0 . 1 6 < - ^ - < l .

T T

Rule

Minimum IAE - Yu (1988)-continued.

Minimum IAE -Shinskey(1988)-



page 38. Model: Method 1 Minimum IAE -Mar l in (1995) -pages 301-307.

Model: Method 1.

K c , Tj deduced from

graphs: robust to

±25% process model

parameter changes.

Minimum IAE -Edgar e/a/ . ( 1 9 9 7 ) -

pages 8-14,8-15. Model: Method 1 Minimum IAE -

Edgar e/a/ . ( 1 9 9 7 ) -page 8-15.

Kc

7 K C( 5 >

8 j / (6)

1.00Tm /Kmxm

1.04Tm /K r axm

l - 0 8 T m / K m t m

1.39Tm /Kmxm

0.95T m /K m T m

0.95T m /K m T m

l-4/Km

l-8/Km

l-4/Km

l-0/Km

0.8/Km

0.55/Km

0.45/Km

0.35/Km

0-4/Km

0-94Tm /KmTm

0 .67T m /K m x m

T,

T ( 5 ) i

T ( 6 ) i

3.0tm

2.25im

l-45t ra

"Cm

3.4xm

2-9xm

0.24xm

0.52tm

0.75tm

0.68xm

0-71im

0.60t ra

0.54tm

0.49xm

0.5xm

4xB

3-5im

Comment

^ r a / T m > 0 . 3 5

T m / T m = 0 . 2

T m / T r a = 0 . 5

^ m / T m = l

^ m / T m = 2

T r a / T m = 0 . 1

T m / T m = 0 . 2

* m / T m = 0 . 1 I

t m / T r a = 0 . 2 5

t m / T m = 0 . 4 3

t m / T m = 0 . 6 7

x m / T m = 1 . 0

x m / T r a = 1 . 5 0

x m / T m = 2 . 3 3

T m / T m = 4 . 0

Time delay dominant

Time constant dominant

Model: Method 2

K (5) 0.871 f ^ -0 .015+0 .384^ f _ ^-1.055

Ti

K „ T T V m J

<(5)

0.444 T,

\ -0.217-!!— 0.213/ % 0.867

I T" I X ) T ,

T T

K (6) _ 0.513 K „ T V T m

» i 0.670 T

V m J

-0.003—-0.084,/

T V m y

Rule

Minimum IAE -Huang et al. (1996).

Model: Method 1




page 167.


page 121. Minimum ISE -


Model: Method 2

Kc

9KC(7>

0.58KU

0.54KU

0.48KU

0.46KU

Ku

T 3 .05-0 .35-^

0.55KU

. . K c ( 9 ,

Ti

T ( 7 ) i

0.81TU

0.66TU

0.47TU

0.37TU

10 j (8)

0.78TU

1.43Tm

Comment

0 . 1 < ^ - < 1 Tm

i m / T m = 0 . 2

xm /Tm=0.5

v / L =i fm/T m =2

Model: Method 1

Model: Method 1; x m /T m =0.2

^ m /T m <0.2

' K ™ K „

6.4884 + 4 .6198^- + 0.8196 ^ T

V m

K „

-5.2132

-7.2712 -0.7241e' 'm J

•y (') T-1

0.0064 + 3 . 9 5 7 4 ^ - 6 . 4 7 8 9 ^ 2 - 1 +9.4348

10 y (8) _ ~ 0 .87-0 .855-^ + 0.172

12\

T -10.7619

f \4

T V ra J

+ T „ 7 . 5 1 4 6 ^ •l.lttb f \ 6

T

» K ' 9 ) = ^ M o . 7 4 + 0.3-T"

Rule

Hazebroek and Van derWaerden(1950)-

continued.

Minimum ISE -Haalman(1965). Model: Method I Minimum ISE -Murrill(1967)-pages 358-363.

Model: Method 5

Minimum ISE -Frank and Lenz


Minimum ISE -Zhuang and Atherton

(1993). Model: Method I

Minimum ISE - Yu (1988).

Model: Method 1.

Kc

x l T m

K m T m

T;

^ m

Comment

Coefficient values

Li. T

0.2 0.3 0.5

X l

0.80 0.83 0.89

x2

7.14 5.00 3.23

0.67Tm

Kmxm

f N 0.959

1.305 f T O

Km IT J

i f T ^ — x 1 + x 2 - s -

Xm

T

0.7 1.0 1.5

x i

0.96 1.07 1.26

x2

2.44 1.85 1.41

Tm

T (

0.492 [

T m

X 2 v

T m "

T

.0.739

x , + x 2 ^ l

T

2.0 3.0 5.0

x l

1.46 1.89 2.75

x2

1.18 0.95 0.81

Mmax =1.9; Am =

2.36; *„, = 50°

0.1 < ^ s - < 1.0 T


TmAm

0.2 0.4 0.6 0.8

1.279

1.346

x l

0.53 0.58 0.61 0.62

/ \ 0.945

—l / s 0.675

—] 12 K (10)

x 2

0.82 0.88 0.97 1.05

Tm/Tm

1.0 1.2 1.4

0-535 \jmj

T ( m

0.552 [

0.586

\0.438

T ( 1 0 )

x l

0.63 0.63 0.62

x 2

1.12 1.17 1.20

0 .1<^_<1 .0 T

\.\<^<2.0 T

^S-<0.35 T

K e~ 12 Load disturbance model = —-— 1 + sT,

K (10) 0.921 ( TL

0.181-0.205—,- N -1.214

T

0.430

TL

N 0.954—-0.49

V m J

, - ^ < 2.3 1 0 ^ L +0.077. T T

Rule

Minimum ISE - Yu (1988)-continued.

Minimum ITAE -Murrill(1967)-pages 358-363.

Model: Method 5

Minimum ITAE -Frank and Lenz


Kc

13 K (11)

14 K (12)

15 K (13)

/ N 0.977

0-859 (lm\

— (x, + x 2 i -

Ti

T ( i i )

T ( 1 2 ) 1 i

T ( 1 3 )

, .0.680

Tm O 0.674 | j m J

^ f x 1 + x 2 ^ x2 V ' » ;

Comment

-^2- < 0.35 T

-^2- > 0.35 T

0 .1<^L<1.0 T

Representative coefficient values - deduced from a graph T m / T m

0.2 0.4 0.6 0.8

x i

0.32 0.35 0.37 0.38

x 2

0.77 0.80 0.86 0.93

Tm/Tra

1.0 1.2 1.4

x l

0.38 0.39 0.39

x 2

1.00 1.05 1.10

1.157 K „ T

T ( U ) = .

0.359 T,

-2.532-=!-0.292 .

T V 1 m J

, 2 . 310^ + 0.077 i

(T \

0.596

0.372-55-- 0.44 ^ .0.46

T V m J , T V m y

Rule

Minimum ITAE - Yu (1988).

Model: Method 1. Load disturbance

K e"STL

model = — . l + sTL

Minimum ITSE -Frank and Lenz


Kc

16 ^ (14)

17 K (15)

18 K (16)

19 K (17)

\ ( T ^ — X , + X 2 ^

Tj

j ( 1 4 )

T ( 1 5 )

T ( 1 6 )

T ( 1 7 )

x2

( T ^ X, +X2^2-

T

Comment

^ - < 0 . 3 5 T

m

•^E->0.35 T

m


Tm/Tm

0.2 0.4 0.6 0.8

x i

0.46 0.50 0.53 0.55

x 2

0.84 0.89 0.97 1.04

^m/Tm

1.0 1.2 1.4

x l

0.55 0.54 0.54

x 2

1.09 1.15 1.20

0.598 K „

/ N 0.272 -0.254-

T T V m J

.(14) T „ /• s 0.304-5^-0.112^ x 0.196

0.805 > T ^ < 2 .385-^ + 0.112.

17 j r (15) 0.735 , x-0.011-1.945-

K, T V m J

T

-1.055

'(15) T m T,

18 K (16) _ 0.787 T T L

Km {Tm

0.425

0.084+0.154

T V m J

-5.809^+0.241 , N 0.901

2 .385^- + 0 . 1 1 2 < ^ < l . T T

T

, N - 0 . 1 4 8 - ^ - 0 . 3 6 5 / N 0.901

-(16)

0.431 , T T V l m J

, l < i . < 3 . T

19 K (17) 0.878 (T L

Km [Tm T V 1 m

. (17) T m

/ \C228-15—0.257

0.794 T T

Rule

Minimum ISTSE -Zhuang and Atherton



(1993).


(1993).

Minimum ISTES -Zhuang and Atherton


Nearly minimum IAE, ISE, ITAE -

Hwang (1995).

Kc

1.015

K m

1.065

/ \ 0.957

V T nJ r T \0-673

VTnJ

20 K (18)

21 K (19)

1.021

1.076

K m

, N 0.953

Tjn.1 V T m J

(T ^ 0.648

22 K (20)

T i

T

0.667

f x 0.552

V m j

T /" ^ - 4 2 7

0-687 {Tm j

T ( 1 8 ) i

T ( 1 9 ) 1 i

T x m

0.629

T

0.650

( \ T m

f \

T

0.546

0.442

K c( 2 0 ) / u 2 K u co u

Comment

0 .1<^2L<1 .0 T 1 m

1.1 < ^ s - < 2.0 T Am

Model: Method 1;

0.1 < ^ s - < 2.0 T

m Model: Method 31;

0 . 1 < ^ S - < 2 . 0 T Am

O . l ^ ^ - S l . O T

m

L I S — < 2 . 0 T

Model: Method 35;

0 . 1 < ^ - < 2 . 0 T Am

20 K ( ,8 )_f l -892K m K u +0.244 , y T ( 1 8 )

3.249KmKu +2.097 Ku, V

K, (19)

A

0.706KmKu-0.227 \

0.7229KmKu+1.2736 J

4.126KmKu-2.610

5.848KmKu-1.06 K u , ^

(19) _ 5.352KmKu-2.926

x5.539KmKu + 5.036

22 ^ = ( 1 - ^ , , * = 1 .14( l -0 .482 m .T m ^0 .068a ,^ ) K u K m / l + KuKm

0.0694(-l + 2.1muTm-0.367co2uT^)

T u .

H2 K u K m / 1 + K u K n

Decay ratio = 0.15.

Rule

Nearly minimum IAE and ITAE - Hwang

and Fang (1995). Model: Method 25

Gerry (2003) -minimum error: step

load change. Model: Method I

Minimum IAE or ISE - ECOSSE Team

(1996a). Model: Method 1.

Coefficients of Kc, Tj deduced from

graphs

Minimum IE -Devanathan (1991). Model: Method 1.

Coefficients of Kc, Tj deduced from

graphs

Kc

23 K (21)

0.3

Kra

x,/Km

Ti

T ( 2 1 ) i

0.42xm

X 2 ( t m + T m )

Comment

0.1 < -^2- < 2.0; T

m

decay ratio = 0.12

x ±ZL>5 T

Coefficient values

fm/Tm

0.11 0.25 0.43 0.67 1.0 1.50 2.33 4.0

X l

1.1 1.8 1.1 1.0 0.8 0.59 0.42 0.32

24 K (22)

x2

0.23 0.23 0.72 0.72 0.70 0.67 0.60 0.53

f m / T m

0.11 0.25 0.43 0.67 1.0

x 2 T u

x i

5.8 3.1 2.1 1.7

0.91

x2

0.5 0.6 0.7 0.8 0.9

Coefficient values

t m / T m

0.2 0.4 0.6 0.8 1.0

x2

0.35 0.29 0.26 0.23 0.21

^m/ T m

1.2 1.4 1.6 1.8

x2

0.19 0.18 0.18 0.17

0 .515-0 .0521^- + 0.0254 T

2 ^

K u ,

, ( 2 1 ) (K c( 2 1 ) /K u « u )

0.0877 + 0.0918-2L-0.0141 T T

m _

2 ^

2; cos 24 K (22)

tan 1.57DD

Kmcos tan

0.16

6.28T

Rule

Minimum IAE -Gallier and Otto


Minimum IAE -Rovira etal. (1969).

Model: Method 5

Minimum IAE - Bain and Martin (1983). Model: Method 1 Minimum IAE -Marlin(1995)-pages 301-307.

Model: Method 1


Model: Method I Minimum IAE -

Smith and Corripio {1991)-page 345. Model: Method 1 Minimum IAE -Huang and Jeng


Kc

Minimum x, /Km

Ti Comment

performance index: servo tuning x 2 ( t m + T m )

Representative coefficient values - deduced from graphs

fmAm

0.053 0.11 0.25

x l

12 6.4 2.9

, % 0.861

0.758 fT m ^ K m l T m J

1

Km

0.3 + 0.3^2-

1.4

Kra

x2

0.95 0.91 0.84

•Cm/Tm

0.67 1.50 4.0

T m

1.020-0.323^-Tra

Tm+0.4xm

0.72tm

Xl

1.15 0.65 0.45

x2

0.75 0.66 0.55

0 .1<Is-<1.0 T

m

-^ ->0 .2 T

^ - = 0.11 T

m Kc , Tj deduced from graphs; robust to ±25% process model

parameter changes

25 K (23)

0-6Tm

KmTm

0.59Tm

K r atm

T ( 2 3 ) i

T m

T m

0.1<^-<1 T

0.1<^-<1.5 T

m

Minimum IAE = 2.1038xm;

Tm /Tm<0.2

25 K (23) _ _ j _ •13.0454 - 9.0916— + 0.3053 -T T

1 +

K „

N - 1 . 0 1 6 9

•1.1075 f N3.5959

-2.2927

T ,

f •, 3.6843

Tm 4.8259eTm

T V m

0.9771 - 0.2492 - ^ + 3.4651 T T

-7.4538 ^ M + 8.2567 ^

-4.7536 U H L ] + 1 .1496^-

Rule

Minimum IAE -Tavakoli and Fleming


Small IAE - Hwang (1995).

Model: Method 35

Minimum ISE -Zhuang and Atherton


Minimum ISE - Khan and Lehman (1996).

Model: Method 1

Kc

26 K (24)

Ti

T ( 2 4 )

Minimum Am = 6dB ; minimum

27 K (25)

f x 0.892

0-980 (lm\

1.072

Km

(T ^ m

l T m J

0.560

28 ^ (26)

Kc(25)/n2Ku<»u

Tm

0.690-0 .155^-T

T m

0.648-0.114-^-T

m

T ( 2 6 ) i

Comment

0.1<^2_<10 Tm

* m =60°

Decay ratio = 0.1,

0 . 1 < ^ L < 2 . 0 T

m

0.1< — < 1 . 0 T

1.1<^2.<2.0 T

0.01 < ^2-< 0.2 T

m

26 j , (24)

K „ 0.4849-2- + 0.3047 T (24) _ T

' i n 0.4262^2. + 0.9581

T

r = 0.5;

27 Kc(25) = (i _ ^ ) K u , n, = 1-07(l-0.466muTm+0.0667co2ux^) ^

K u K m / 1 + K u K m

_ 0.0328(- 1 + 2.21couTm - 0.33&a>lz2m)

^ ~ K u K m / l + K„Km

l.ll(l-0.467<ouTm + 0.0657co2,T2n)

K u K m / 1 + K u K m

^ = 0.0477(-1 + 2.07c)uTm - O . ^ ^ x j ) ^_ K u K m / 1 + K u K m

1.14(l-0.466ffluTm+0.0647ffl2,T2J K u K m / 1 + K u K r a

= 0.0609(-l + 1.97c i)uTm-0.323(n;T2n)>r_1|

K u K m/ 1 + KuKm

r = parameter related to the position of the dominant real pole.

0.7388Tm+0.3185xm

= 0.75;

28 K (26) f 0.7388 ^ 0.3185 "I T m T ( 2 6 ) = T

tm T IK ' ' -0.0003082Tm +0.5291x„

Rule

Khan and Lehman (1996) - continued.

Minimum ITAE -Rovirae/a/. (1969).

Model: Method 5

Approximately minimum ITAE -

Smith (2002).




(1993).

Minimum ISTES -Zhuang and Atherton


Kc

29 K (27)

0.586 (Tm^ K m V T m ;

0.916

0.586 Tm

0.712 fTm "> K m V T m ;

0.921

, N 0.559

0-786 (lm\

0.361KU

31 K (29)

, s 0.951

0-569 (lm\

Km I T J

0.628

Km

, N 0.583

KXm J

Ti

T ( 2 7 ) i

T m

1.030-0.165 — T •* m

T

T

0 .968-0 .247^-T

m

T

0.883-0.158^2-T

m

30-r, (28) M

T ( 2 9 )

T

1.023-0.179^-T

T 1 m

1.007-0.1 6 7 ^ L T

Comment

0.2 < i s - < 20 T

0 .1<i2 .<1.0 T

m

Model: Method 1

0.1<is -<1 .0 T 1 m

1.1< — < 2 . 0 T

Model: Method 1;

Q.\< — <2.0 T

m Model: Method 31;

0.1 < i = - < 2.0 T

0.1 < i 2 - < 1.0 T

m

1.1< — < 2 . 0 T

m

^0.808 0.511 0.255 29 j , (27)

VT~T7

- ( 2 7 ) . 0.808Tm+0.51lTm-0.255VT~^

0.095Tm+0.846xm-0.38lVf^

30 T (28) T/28 ' =0.083(l.935KmKu+l)Tu

31 K (29)

f A A

1.506KmKu-0.177

3.341Km K„ + 0.606

K u , T(29) = 0.0551 3.616Km Ku+1 |TU

Rule

Nearly minimum IAE and ITAE - Hwang

and Fang (1995). Model: Method 25

Simultaneous servo/regulator tuning

- Hwang and Fang (1995).

Model: Method 25

34 Minimum performance index -

Wilton (1999). Model: Method I.

Coefficients of Kc

estimated from graphs

Kc

32 K (30)

T(

T ( 3 0 )

Comment

0.1 < -^2- < 2.0; T

decay ratio = 0.03

Minimum performance index: other tuning

33 K (31)

x,/Km

T ( 3 1 ) * i

T

0 . 1 < - i < 2 . 0 ; T

m

decay ratio = 0.05

Coefficient values

^m/Tm

0.1 0.1 0.1 0.1

0.4 0.4 0.4 0.4 0.4

x i

750.0 17.487 4.8990 2.1213

1.6837 1.5297 1.2247 0.9192 0.7668

b

0.0001 0.04 0.2 1

0.008 0.04 0.2 1 5

fm/Tm

0.2 0.2 0.2 0.2 0.2

0.5 0.5 0.5 0.5

x i

3.3675 2.2946 1.7146 1.1313 0.8764

150.01 7.1386 2.6944 1.1314

b

0.008 0.04 0.2 1 5

0.0001 0.04 0.2 1

32 K (30) 0.438-0.110-2-+ 0.0376 T

•}2\

Ku ,

- ( 30 ) (Kc(30)/Kucou)

0.46-0.0835-21-+ 0.0305 T

2 ^

K „

p (31) = _

0.0388+ 0.108-2!--0.0154 T

(KC(31,/KU(DU)

2 ^

0.0644 + 0.0759— - 0.0111 — T T

2V

' Performance index = r[e2( t ) + b2Km2(y(t)-yocl)

2]dt

Rule

Wilton (1999)-continued

Minimum ITAE -ABB (2001).

Model: Method 7

Bryant et al. (1973). Model: Method 1

CKmetal, (1973). Model: Method 1

Mollenkamp et al. (1973).

Model: Method I

Kc T i Comment

Coefficient values - continued

t m / T m

0.6 0.6 0.6 0.6 0.6

1.0 1.0

0.8591 r

*1

1.1225 1.1218 0.9798 0.7778 0.6572

72.004 1.6657

N 0.977

b

0.008 0.04 0.2 1 5

0.0001 0.2

1.4837Tm

Tm/Tm

0.8 0.8 0.8 0.8 0.8

1.0 1.0

T

0.68

x.

0.8980 0.9178 0.7348 0.7071 0.6025

3.6713 0.8485

b

0.008 0.04 0.2 1 5

0.04 1

^ - < 0 . 5 T

Direct synthesis: time domain criteria 0.37Tm

K m t m

0.40Tra

K m T m

0

T

T

x2KmT r a

Coefficient values - deduced fr

^m/X 0.33 0.40 0.50 0.67 0.33 0.40 0.50 0.67

x2

12.5 11.1 11.1 9.1 7.1 6.3 5.6 4.3

^Tm

Kjl + XxJ

^m/Tm

1.0 2.0 10.0

1.0 2.0 10.0

x 2

6.7 4.2 3.0

3.2 2.1 1.6

T

X = 1.0 , OS<10%, Model: Method 10 (Po 2004).

T m

K m(T C L+0 T m

Critically damped dominant pole

Damping ratio (dominant pole) = 0.6

om graph



X variable; suggested values: 0.2,

0.4,0.6, 1.0. ulin and Pomerleau,

Appropriate for "small xm"

Rule

Keviczky and Csaki (1973);

OS values in first row; Tm/Tm values

in first column; K m = l .

5% overshoot - servo - Smith e< a/. (1975) - deduced from graph

Model: Method 1 1% overshoot - servo - S m i t h s a / . (1975) - deduced from graph

Bain and Martin (1983).

Model: Method 1

Suyama (1992). Model: Method 1

5% overshoot - servo - Smith and Corripio (1997) page 346. Model: Method 1

Viteckova (1999), Viteckova et al.

(2000a) (Model: Method 9 -

Viteckova et al. (2000b)).

Kc

0 Ti

* 2 * m

Comment

Model: Method 1

x2 coefficient values (estimated from graph)

^m/Tm OS

0.1 0.2 0.5 1.0 2.0 5.0 10.0

0%

30 18 8 5 4 3

2.5 0.52Tm

KmTm

5%

22 11 6 4 3

2.4 2.1

10%

16 9 5

3.5 2.5 2

1.8

T

15%

12 7.5 4 3

2.2 1.8 1.7

20%

10 6

3.5 2.5 2

1.6 1.5

0.04 0.2 T

"Very conservative tuning for small

delay"

OS = 10%

<j>m = 60° ; 0.25 < i s - < 1 (Voda and Landau (1995)) m

* i T m

Kmxm T

Closed loop response overshoot (OS)

shown Coefficient values

x i

0.368 0.514 0.581 0.641

OS (%)

0 5 10 15

Xl

0.696 0.748 0.801 0.853

OS (%)

20 25 30 35

x i

0.906 0.957 1.008

OS (%)

40 45 50

Rule

Mann etal. (2001). Model: Method 31 or

Method 33; Closed loop response overshoot (step input)

< 5 %

Viteckova and Vitecek (2002)

Model: Method 1

Pinnella e? a/. (1986). Model: Method 20

Kc

0.5 lTm

K r

35 K (32)

T,

Tm

T ( 3 2 )

x, = 0.4, x2 =3.68

x,=0.8, x2 = 3.28

x,=1.2, x2 = 2.88

0

37 K (34)

36 j <33> •M

T ( 3 4 )

Comment

0<-^2-<2

K t m / T m < 2

2 < T m / T m < 4

4 < ^ m / T m

For 0.05 < — < 0.8 , T; = Tm ; overshoot is under 2% (Viteckova

and Vitecek (2003))

0 38 K (35) Critically damped

servo response

35 K (32)

K „ 0.56 + 1.02-

, ( 3 2 )

1.0404 ^ - - x , x„.

+ x ,

0.56 + 1.02- I m _ h . 0 4 0 4 ^ 2 - - x , + x 2

0 . 0 4 - 1 . 0 2 ^ + 1 ^ 2 - 1 . 0 4 0 4 ^ - x , | + x, M

• ( 3 3 ) K „

1 0.25 1 0.5 - + X " T ' T

m m m m

T T -J2- +0.25 - - = - + 0.5

37 K (34) ^ - [ T m T m x 12 + ( 2 T m + x m ) x 1 + l ] e ^ >

2 0.5 2 0.25 „, X, = + T + r , T :

(34)

T T V T 2

T 2

\ 2

^ m T m x 12 +(2T m +T m )x l +l

X ^ m 1 ™ * ! +Tm + 1 J

38 K (35)

4Kmxm ' T T m J

T J 2 -T T

2

+ 1 0.5

e

Rule

Schneider (1988), Bekker etal. (1991), Khan and Lehman


Regulator - Gorecki et al. (1989).

Model: Method 1

Hang ef a/. (1991). Model: Method 1

McAnany (1993). Model: Method 24

Sklaroff(1992), Astrom and Hagglund (1995) -pages 250-

251.

Kc

0.368 T m

Km tm

0.4 T

13 m

Kraxm

T 1.571 m

39 K (36)

40 K (37)

T,

Tm

T

T

T (36) 1 i

T ( 3 7 )

Comment

4 = 1

4 = 0.6

4=0.0

Pole is real and has maximum attainable

multiplicity

0.16 < ^2-< 0.96 T

Servo response: 10% overshoot, 3% undershoot 41 K (38)

3

Km fl + ^ 1

T ( 3 8 ) i

T m

TCL = 1.67Tm

Model: Method 28; Honeywell UDC 6000 controller

39 K (36> = 2 T

2 + 2T

V m

- 1 2Tm 2Tm

. ( 3 6 ) 2T

3 + v 2T , 2T

2 + 2T

V m

2 + 2T

40 K (37) = :

12 + 2

1 1 - ^ + 13 T

3 7 ^ - - 4 T

15 + 14

1 1 ^ - + 13 T

3 7 ^ - - 4 T

K U ,T , (37) _ 0.2

11^2- + 13 T

3 7 ^ - 4 m

+ i

y

41 K (38) _ (l-44Tm +0.72TmTro -0.43xm -2.14) (3g)

K ^

Km(5

72xm+0.36Tmz+2 4T +1 .28T-2 .4

Rule

Fruehauf et al. (1993).

Model: Method 2

Kamimura et al. (1994).

Model: Method 1

Zhang (1994). Model: Method 1

Davydov et al. (1995).

Model: Method 6

Kuhn(1995). Model: Method 14

Kc

0.5556Tm

TmKm

0.5Tm

T m K m

42 K (39)

43 K (40)

44 K (41)

45 K (42)

0.5

1

T i

5tm

Tm

T ( 3 9 )

T ( 4 0 ) 1 i

T ( 4 1 )

T

T ( 4 2 )

0.5(Tra+Tra)

0.7(Tm+xm)

Comment

i a -<0 .33 T

-^2-> 0.33 T

m Servo response: 10%

overshoot Servo response: 0%

overshoot "Quick" servo

response: "negligible" overshoot

"Good" servo response; xm is

"small" 4 = 0.9;

0 . 2 < x m / T m < l .

"Normale Einstellung"

"Schnelle Einstellung"

42 K (39) 0 - 5 ( T m + Q

Km [0 .5xm (2Tm+xJ + 0.10]

0.375(Tm+Tm)

T (39) _ T

' li 1 i r 0.5xm(2Tm+xm)+0.10

Tm + t m

43 jr (40) _

44 K (41)

Km [o .5xm (2Tm+xJ + 0.0781]

0-425(Tm+Tm)

Km[0.5xm(2Tm+xm)+0.083] :

T ( 4 0 ) _ T + T 0.5xm(2Tm+xm)+0.0781

T-(41)=T +x 0.5x r a(2Tm +xm) +0.083

45 K (42) ,(42)

K „ 1.905--- + 0.826 T

0.153-^ + 0.362 T

T .

Rule

Khan and Lehman (1996).

Model: Method 1

Abbas (1997). Model: Method 1

Valentine and Chidambaram

(1997a). Biefa/.(1999).

Model: Method 13

Bi et al. (2000). Model: Method 13

Ettaleb and Roche (2000).

Model: Method 29

Kc

46 K (43)

47 K (44)

48 j r (45)

1 0.43-0.97^2-

T

0.5064Tm

K m T m

0.5064Tm

Kraxra

1

Ti

T (43) i

T (44 ) i

Tm+0.5xm

0.413 + 0.8^2-T

m

Tm

TmKm

0.5064

T m + T m

Comment

0.01 < ^ L < 0.2 T

0.2 < -^2. < 20 T

0 < V < 0.2 ;

0 . 1 < ^ - < 5 . 0 T

Model: Method 1. \ = 0.707 49

TC L = Tm + ^m

46 T^ (43) _ _J_ni

K „ MM1,0,723 ^

-0.404tm2+0.723Tmxm-0.3852Tm

2

-0.525Tm2+0.4104Tmtm-0.00024Tm

47 K (44)

K „

0.404 0.256 0.1275 T_ VTmT.r

0.404Tm+0.256tm-0.1275VTV^ 1 0.0808Tm +0.719xm - 0 . 3 2 4 7 ^

48 K (45)

0.148 + 0.186 - ^

Km (0.497- 0.464V0590)

s -1.045

49 ±1% Ts = 2.5Tra , 0.1 < -^s. < 0.4 ; ±1% Ts = 5Tm , 0.5 < - ^ - < 0.9 .

Rule

Chen and Seborg (2002).

Skogestad (2003). Model: Method 45

Gorez (2003). Model: Method 1

Kc

50 K (46)

0.5Tm

51 K (47)

52 p, (48)

Ti

T ( 4 6 ) 1 i

min(Tm,8Tm)

T

( l - V > m + T m

Comment

Model: Method 1

o < ^ = — < i m ^ m + T m

I < T m < 1 ^m + Tm

50 K (46) 1 TmTm + 2 T mT C L ~ T c L T (46) _ J m ^ m + . 2 T m T C L - T C L

Km (TC L +xJ2 T m + T m

0<T r < T m + ^ T + T x T/^se -v

d desired K c ( T C L S + l ) 2

XK m T„ with

1 1 . 5 M / - 2 x = — - + M s ( M s - l ) 0 . 3 2 M S

2 ( M S - 1 x + T

V 5x„

t m +

0.5

MS(MS

TmX/M7j

2.5x„

v (xm+TmyMsy

52 K (48)

XKmTn withv = 1-0.5 T m + T m

-0.4JM

1-0.4, M

0.1M„

M - 1 7.5-

-0.4, M

1-0.4, M

Rule

Sree et al. (2004), Chidambaram et al.


Haen (2005). Model: Method 1

Hougen(1979)-page 335.

Model: Method 1; Coefficients of Kc

estimated from graphs.

Regulator - Gorecki etal. (1989).

Model: Method 1

Kc

, x-0.8915

0 . 9 1 7 9 ( 0

54 K (50)

T i

5 3 T ( 4 9 )

0.96Tm+0.46xm

Comment

0 . 1 < ^ - < 1 T

m

0 < I H L < 4 T

Direct synthesis: frequency domain criteria x,/Km T

Coefficient values

OTm

0.1 0.2 0.3 0.4 0.5

X l

7.0 3.5 2.2 1.8 1.4

55 K (51)

0 T m 0.6 0.7 0.8 0.9 1.0

x l

1.1 1.0 0.8 0.75 0.7

T ( 5 1 ) i

Maximise crossover frequency

Low frequency part of magnitude Bode

diagram is flat

53 j (49) _ j 10.59 -J2- - 2 . 3 5 8 8 - ^ + 0.8985 T T

V m J m

0.1 < - 2 - < 0.4; T

0.7719 -3.6608 + 6.5791 O -5.1652-^1- + 2.8059 T J T

0.4<-J2-<1.5 T

54 v (50) _ 1

K . K „ 0.407+ -

6.282

1 + 14.956 T

V in

55 K (51) 1 l + 3(TmAm)+6(Tm /xm)2 + 6 ( T m / Q :

K „ l + 3(Tm/Tm)+3(Tm/Tm)2

T ( 5 , ) =x 1 + 3 T, „A m )+6(T m /T m ) 2 +6(T m /x m ) 3

1 + 2 (Tm/xm)+2(Tm/xm)2]

Rule

Somaraeffl/. (1992). Model: Method 1;

Suggested A m :

1.75 < A m <3 .0

Hang et al. (1993a), (1993b) -pages 61-

65.

Kc

* , / K m A m5 6

T,

x 2 * m

Comment

Coefficient values X m / T m

0.0375 0.1912

0.3693 0.5764 0.8183 1.105 1.449

x i

36.599 7.754

4.361 3.055 2.369 1.950 1.672

57 K (52)

o p T m

K m A m

x2

3.57 3.33 3.13 2.94 2.78 2.63 2.50

T m / T m

1.869 2.392

3.067 3.968 5.263 7.246 10.870

T (52)

58j(53)

x i

1.476 1.333 1.226 1.146 1.086 1.042 1.011

x2

2.38 2.27 2.17 2.08 2.00 1.92 1.85

o.\<^-<\ T

m Model: Method 31

or Method 33

• F o r i a L < 1 , K e . o ^ i Tm K m A m \(

• + 1 or

1.886 + 4—si- -1 .373 T

K, 0.9806

1 + 1 .886^1 ; f o r ^ > l , K * - ^ l + 8. VTr. K r a A m

669

57 K (52)

K m A m

0.80 + 1.33- T (52) __ 5T^

1.04 0.80 + 1.33-

1

2co p- + — P n T

; - ^ < 0.3 , Yang and Clarke (1996).

Rule

O'Dwyer (2001a). Model: Method 1;

i s -> 0.3, T

m

Yang and Clarke (1996).

O'Dwyer (2001a). Model: Method 1

Chen et al. (1999b). Model: Method 1

Gain margin and maximum closed loop magnitude - Chen et

al. (1999a). Symmetrical

optimum principle -Voda and Landau


Kc

K m T m

T,

Tm

Comment

Am = 7t/2x, ;

4m = V 2 - x , 5 9

Representative results x i

1.048

0.7854

Am

1.5

2

x l T m K u T u Tm VV+^X 2

x j m

^mKm

<L 30°

45°

Xi

0.524

0.393

Tra

Tm

Am

3

4

<L 60°

67.5°

Am = V 2 x i ;

<t>m = V 2 - X l

Coefficient values

Xl

0.50

0.61

0.67

0.70

Am

3.14

2.58

2.34

2.24

•I'm

61.4°

55.0°

51.6°

50.0° r l » r T m

Km

1

3.5C }P ( j ° w )

1

2.828 G p ( J ° W )

Ms

1.00

1.10

1.20

1.26

x i

0.72

0.76

0.80

Am

2.18

2.07

1.96

1

4 2 1

4.6

4

ffli 35°

<L 48.7°

46.5°

44.1°

Ms

1.30

1.40

1.50

Model: Method 1 60

^ - < 0.1 T

0 - « / n u T

'Given Am , ISE is minimised when <|>ra = 68.8884-34.3534Am + 9.1606-

for servo tuning (Ho et al. (1999)); 2 < Am < 5 ; 0.1 < Tm/Tm < 1 :

Given Am , ISE is minimised when <|>m = 45.9848Ama2677(Tm/Tm)a2755 for

regulator tuning (Ho et al. (1999)); 2 < Am < 5 ; 0.1 < Tm/Tm < 1

7t 2r ,An

4 x „ , A m ) 2 - l

Rule

Voda and Landau (1995)-continued.

Friman and Waller (1997).

Model: Method I

Schaedel (1997), Henry and Schaedel

(2005) -Optimal servo

response.

Henry and Schaedel (2005).

Model: Method 22; Optimal regulator

response; —— < 0.2

Kc

61 K (54)

0.1751

Gp(j»135<.)

62 0.5T,<55»

K m ( T , - T < 5 5 > )

63 0.375T/56 '

K ^ T . - T , ^ )

0.375Ti(57)

K m ( T l - T < " > j

64 K (58)

T,

T ( 5 4 ) i

1

03135°

T ( 5 5 ) i

T ( 5 6 ) i

63 T (57) i

T ( 5 8 ) * i

Comment

0 . 1 5 < ^ - < 1

^ > 2 . A m > 3 m

"Normal" design -Butterworth filter. Model: Method 21

"Sharp" design -Tschebyscheff filter

(0.5dB). Model: Method 21

Minimum ITAE (Henry and Schaedel,

2005). Model: Method 22

"Normal" design -Butterworth filter

61 ^ (54) ___

4.6

1 , ( 5 4 )

Gp(ja>11,„)-0.6Kn

1.15JGp(jm135o)| + 0.75Kn,

(Bl35„[2.3|GI)(j(D135„)|-0.3Kra]'

6 2 T i( 5 5 ) = A / T , 2 - 2 T 2

2 ,

1 + 3.45

— + Tm,T22=1.25Tmia

m + I n L _ T m + 0 . 5 T m2

; 0 < ^ < 0.104;

-3.45-T

0.7Tm

Tm-0.7xam

2 0.7T, - T m > T 2 = T m T m +

Tm-0.7x^ -0.5tm

2 , i s . > 0.104.

= T i - ^ f - > Tj = -0 .64T 1 + 1 .64T , J l -1 .2 | ^ -

64 j , (58) 1 0.5 ,(58)

2 -\

- 1 T Z / ' T 2 ^

T, I T,2 j

Rule

Henry and Schaedel (2005) - continued.

Leva (2001). Model: Method 1

Minimum §m = 50°

Potocnik et al. (2001).

Model: Method 39

Kc

65 £ (59)

66 £ (60)

67 £ (61)

68 j ^ (62)

69 K (63)

70 £ (64)

T ;

T< 5 9 )

rp(60)

T

T ( 6 3 ) i

T ( 6 4 )

Comment

"Sharp" design -Tschebyscheff filter

(0.1 dB) Minimum ITAE

20 M c T m + 5 T m

20 c x +5T

i s -< 0.1 T

0.1 < i s . < 0.15 T

65 ^ (59)

66 £ (60)

0.7 •(59) 2 "\

, T: (59) T = 3 . 1 1 - ^

T, 1-1.43 V

K „

68 „ (62)

0.69

V

"0.69£

K m

"rp (59) "

. T 2

2 "J

- 1

J

20

j (60) _

T m

Tm K m ( T m + 5 T m ) _

' 0.698 lTm

K m T m I<

50T„, ]

' (r *• in \ m

+ 5Tm)J

3 . 8 6 ^ 1 - 1 . 4 6 ^

J

• i

69 T ^ (63) 0.5

|Gp(jW135»] 1.14 + 13.24

• (63)

G p ( jw , 3 5 o)

J135°

1 + 2.48 *M#). in AnpMA ^ + 17.40

70 K (64) K -2|Gp(ico135„]

2 K m [ K m - 2 | G p ( J a W ) | ] '

-(64) . [ K m + V2|G„ (jco135 .)[ ] [2|GP( j iQ135 . )| ~ K m ]

Q)135»|Gp(j£0135»)|[|GP ( jCD135»>|-Kn1 l

Rule

Potocnik e< a/. (2001) - continued.

Cluett and Wang (1997), Wang and

Cluett (2000) -page 177.

Model: Method 1;

0.01<-^2-<10 T

m

Kc

71 K (65)

72 jr (66)

73 v (67)

74 jr (68)

75 jr (69)

76 K (70)

T,

T(65) i

T(66)

T(67) i

T(68) i

T(69)

T(70)

Comment

0 . 1 < ^ - < 1 Tm

TCL = 4xm .

Am e [7.50,8.28],

(t)me[78.50,79.70]

TCL = 2xm.

A m e [4.35,5.02],

<S)m e\l0.l\74.0°]

TCL = 1.33xm.

Am e [3.29,3.89],

(|)me[64.40,70.60]

TCL = Tm •

A m e [2.74,3.30],

(t)mg[57.80,68.50]

TCL = 0.8tm .

A m e [2.39,2.93],

* m e [49.6°,67.1°]

71 K (65) 2K„ 0.75 + 1.15

|GpOroi35°) K „

T: (65) 0.75 + 1.15^ = G°w)

K „

72 K (66, _0.019952Tm+0.20042Tm ^ m

73 K (67) _0.055548Tm+0.33639Tm T_(67)

74 K (68) _0.092654Tm+0.43620Tm j m

Kmxm ' '

75 K (69) _0.12786Tm+0.51235Tm T m =

Kmxm

76 Kc(70) _0.1605lTm+0.57109Tm > T

K m T m

0.099508Tm+0.99956Tm

0.99747xm -8.7425.10"5Tm

0.16440Tm +0.99558Tm

0.98607Tm-1.5032.10"4Tn,

0.20926Tm+0.98518Tm

0.96515Tm+4.2550.10"3Tm

0.24145Tm+0.96751Tm

0.93566xm +2.2988.10"2Tm

m T ( T O ) _ 0.26502Tm+0.9429 lTm

0.89868xm+6.9355.10"'Tm

Rule

Cluett and Wang (1997), Wang and

Cluett (2000) -continued.

Modulus optimum principle - Cox et al.


Smith (1998). Model: Method 1

Wang et al. (2000a). Model: Method 31 or

32

Kc

77 K (71)

78 K (72)

0.5Tm

0.35

Km

79 K (73)

T|

T ( 7 1 ) 1 i

T ( 7 2 ) i

Tm

0.42xm

Tm

Comment

TC L=0.67Tm .

AmG [2.11,2.68],

4>me[38.1°,66.3°]

t —2L<1 T

m I » L > 1

T

6 dB gain margin -dominant delay

process

1.3 <MS <2

77 K (71) 0.19067tm +0.61593Tm . ( 71 ) _ 0.28242Tm + 0.9123 lTm

0.85491-r +0.15937T„

0.5 T mJ + V T m + 0 . 5 T m T ^ + 0 . 1 6 7 T ,

Tm2Tm+TmTm

2+0.667Tm3

3 ^

• (72) _ / T m

3+ T m

2 T m + 0 .5T m x m2

+ 0 .167T m3 ^

Tm2+TmTm+0.5xm

2

79 K (73) 1.451-1.508

M.

Rule

Wang and Shao (2000a).

Model: Method I Chen and Yang


Hagglund and Astrom (2002).

Model: Method 5; M.=1.4


Model: Method I

Kc

80 K (74)

0 7T

v . / i m

KmTm

0.25Tm

^ m T m

0.1Tm 0.15 — +

Kmt r a Km 0.15

Km

81 K (75)

Ti

T . ( 74 )

Tm

0.8Tm

0.3xm+0.5Tm

0-3Tm

T ( 7 5 )

Comment

A.e [1.5,2.5]

Ms =1.26;

A m = 2 . 2 4 ;

* m = 5 0 °

0 . 2 < ^ L < 1 T

Tm

T m

Ms =1.4;-is->0.5

80 K (74) 1

*-fl (WQn« ) f 2 ( W Q n o ) - -

fl (ffl90» ) :

K „

fr + W } ' -[(Tm + { 1 + (O90l>2Tm2}'tm)Sm(-CO90»'Cm ~ t a n ' " V 7 ™ ) ]

(l + co90„2T, 2 ,"1 ,V.5 ko°Tm2 cosC-^so"1", " tan"1 co90„Tm)J,

f2(« 9 0 «) :

1 + ^ X 2 [(Tra +{l + co9o0

2Tm2}Tm)cot(-co90„xm -tan"1 co90„Tm)J

«9o»lTn1+(l + ©Qn»2Tm

2)xm]cos0 + (l + 2co9Oo2Tm2)sin(

m 9 0 o T n i i , 2 T 2 1 + CV T

m

, (74) _ w90° L m 90u m / m J

-co „3Tm2cos9 + co9no

2[Tm +(l + co9no2Tm

2)Tm]sin(

81 j r (75) ,

K „

90 m y m J

0.14 + 0 . 2 8 ^ - 1 , T;(75) = 0.33xm + 6 - 8 T ™ T "

- C 0 9 0 o T n , - t a n " l c 0 9 0 » T m -

10xm+Tm

Rule

Leva et al. (2003). Model: Method 1

Matausek and Kvascev (2003). Model: Method 1

Kristiansson (2003). Model: Method 11

Leva et al. (1994). Model: Method 43

Kc

82 K (76)

83 K (77)

x l T m

Kraxra

85 ^ (78)

86 K (79)

Ti

Tm

T

T 1 m

1.25 T

m T ( 7 9 )

Comment

minimum <j>m ;

maximum coc

minimum ()>m = 50

x ^ O . S r t - ^ 8 4

1.6 <MS <1.9

82 K ( 7 6 ) = minimum ~Tm(0.5n-$m) 5T„

83 K (77) _ mm lmum

0.698T„,

KmTCL

SaT,,,

Kmxra ' K m ( T m + 5 T j ,T C L = T m + 5 T m

; a e [ 4 , 1 0 ] .

84 'Acceptable' values of x, : 0 . 5 < x , < 3 . 'Recommended' values of x, :

0.32 <x , < 0.54 (59° < <|>m <72°). Choose x, = 0.32 when large variations in

process parameters are expected; choose x, =0.54 to give 6%<OS<13% and

1.4 <M < 2 .

85 K (78) _ ^150°

K „ 0.2 + -

0.075 K , , „ o

K „ + 0.05

86 j ^ (79) _ M c n x i l + «c„2Tm2

Km ii+ajnr]]2

tan , (79)

- ^ + T m „

Rule

Tan et al. (1996). Model: Method 30 Xu et al. (2004).

Model: Method 31

Huang et al. (2005). Model: Method 42

Rivera etal. (1986). Model: Method 1

Chien (1988). Model: Method 1

Brambilla et al. (1990).

Model: Method 1

Kc

87 jr (80)

T

^Kn^m

0.50Tm+0.22xra

KmTm

Ti

T ( 8 0 ) 1 i

Tm

0.9Tm+0.4xra

Comment

X e [1.5,2.5]

Robust

^Km T

m

A>1.7xm,

A>0.1Tm .

A, > 1.7xm , X > 0.2Tm (Luyben (2001)); X = 2xm (minimum ISE -

deOliveirae/a/., (1995)) 2Tm+xm

2XKm

Tm

Km(xm+A)

Tm+0.5xra

Km(A + xra)

Tm+0.5xm

Tm

Tm+0.5xm

A>1.7xm,

A>0.1Tm .

A = Tm (Chien

(1988))88

No model uncertainty: A, = xm , 0.1 < xm/Tm < 1 ;

A = [ l -0 .51og 1 0 (x m /T m ) ]x r a ) l<x m /T m <10.

K (80) .

Am )/l + (pTi(80)coJ ' ' K tanl-tan-1 pTmco<, -pxmco<, -<t>j'

T

co,,, < cou. (3 = 0.8,is- < 0.5 ; p = 0.5,^2- > 0.5 . ra ^m

X = 2xm (Bialkowski (1996)). X > Tm + xm (Thomasson (1997)). X e [0.45xm,0.8xm]

(Huang et al. (1998)). A = 2xm (aggressive, less robust tuning) (Gerry, (1999));

X = 2(Tm + xm) (more robust tuning) (Gerry, (1999)); A = a.max(Tm,xm): a > 3 ...

slow design, 2 < a < 3 ... normal design, 1 < a < 2 .. .fast design, a < 1 ... faster design (Andersson (2000) -page 11). Model: Method 10.

A 6 [0.1Tm,0.5T ra],^ < 0.25 ; A = 1.5(xm+Tm),0.25 < ^n_ < 0.75 ;

* = 3(xm + T m ) , ^ - > 0.75 (Leva (2001)). A e [Tm ,xJ (Smith (2002)).

Rule

Pulkkinen et al. (1993).

Ogawa(1995). Model: Method 1;

Coefficients of Kc ,Tj

deduced from graphs

Thomasson (1997). Model: Method 1

Chen etal. (1997). Model: Method 26

Lee etal. (1998). Model: Method 1

Isaksson and Graebe (1999).

Model: Method 1

Chun et al. (1999). Model: Method 34

Zhong and Li (2002). Model: Method 1


T

0.5

1.0

0.5 1.0 0.5 1.0

0.5 1.0

2K

K ra

Tm

K

T

Tm(x

K r

K

Kc

0.7Tra

K m T m

x,/Km

M

Tf

^ m + T m )

X 2 Tm

Coefficient values

x i

0.9

0.6

0.7 0.47 0.47 0.36

0.4 0.33

x 2

1.3 1.6

1.3 1.7

1.3 1.8 1.3 1.8

*m

m ( t m + ^ )

0-5Tm

max(im ,0.5)

2

1 T m

2(A. + 0

+ 0.25xm

m + 2 X ) - ^ 2

„ (x m +X) 2

Tm

ra(a + 0

1 T

T

2.0

10.0

2.0 10.0 2.0 10.0

2.0 10.0

minim

T m

T

Tm(x

*1

0.45 0.4

0.4 0.35 0.32 0.3 0.3

0.29

x 2

2.0

7.0 2.2

7.5 2.4

8.5 2.4

9.0

0-5xm

um(T m ,6T m )

3

2

1 T m

2(* + 0 + 0.25im

m+2X)-X2

x m + T m

Tm

Tm

Comment

Model: Method 1

20% uncertainty in process parameters




*m » Tm i

^ = TCL

x r a > 0 . 5

T m < 0 . 5

Desired closed loop

specified

Representative A.:

X = 0.4Tm

a e [At m ,1 .4At m ] ;

Axm = time delay

uncertainty.

Rule

McMillan (1984). Model: Method 2 or

Method 44

Boe and Chang (1988).

Model: Method I

Geng and Geary (1993).

Model: Method 1

Feric etal. (1997). Model: Method 40

Matsubaefa/. (1998). Model: Method 1

Huang et al. (2005). Model: Method 42

Kc T, Comment

Ultimate cycle

89 K (81)

0.54KU

0.7 lTm

K m ^ m

Ku tan2 4>m

Vl + tan2c()m

0-99 JX^ 0.9

90 K (82)

T ( 8 1 ) 1 i

1.935Tm [ Xm

T

0.654

3.33xm

0.1592 Tu

tan<t>m

2 . 6 3 T J ^ \ lm J

0.097

T (82)

Tuning rules developed from

K u ' T u

Quarter decay ratio

i a - < 0.167 T

m

0.1 < ^ 2 - < 1.0 T

m

89 ^ (81) 1-881 Tm

K m Tm

L T ;( 8 1 , = 1 . 6 7 T J I + | T "

90 K (82) _ 0-55

Km

0.9, K„ - 1

TC - tan K „ - 1

+ 0.4

X ( 8 2 ) =0.159T„ 0.9, K„ - 1 + 0.4-U - tarT K„ - 1


3.1.2 Ideal controller in series with a first order lag

Gc(s) = K, 1 + -T j S ,

1 Tfs + 1

R(s) > E(s)

y i

K, 1

1 + Tfs

U(s)

• Kme-"»

l + sTm

V

^

Table 4: PI controller tuning rules - FOLPD model Gm (s) = K„,e~

l + sT„

Rule

Umetal. (1985). Model: Method I

Rivera and Jun (2000).

Model: Method 1

Kc T; Comment

Direct synthesis: time domain criteria

Tm

Km(24TCL2+xm) T T 2

j XCL2 1 2 ^ T C L 2 + - C

m

Robust

Tm

Km(2Tm+*.)

T T f =

TmX ; 2Tm+X

X not specified

Desired closed loop transfer function = TC L 2V+2^TC L 2s + l

3.1.3 Ideal controller in series with a second order filter

Gc(s) = Kc

E(s)

R(s) ®+K + X

1 +

U(s)

J_ T i S y

2 A

l + b f l s + b f 2 s

1 + -T i S y

l + b f l s + b f 2 s 2 ^

v 1 + 3. j-jS H~ 3. j -2^ /

K e"

l + sTm

Y(s)

Table 5: PI controller tuning rules - FOLPD model Gm (s) = Kme~ l + sT„

Rule

Tsange/a/. (1993). Model: Method 15

Lee and Shi (2002). Model: Method I

Kc Ti Comment

Direct synthesis: time domain criteria x l T m

K r a x m T 2

Coefficient values x i

1.851 1.552 1.329 1.160

\ 0.0 0.1 0.2 0.3

x l

1.028 0.925 0.841 0.768

\ 0.4 0.5 0.6 0.7

x l

0.695 0.622 0.553

\ 0.8 0.9 1.0

Robust 3 K (83) T

Y

2T l < y < ^ 2 _

2 b f l = 0.5xm , b f 2 = 0.0833xm2, afI = 0.2xm, af2 = 0.01xm

2

3 K (83) «>bw T m

Km(cobwyTm+lJl + ^ ^ t a n

tm+2cobwTmTm+2Tm

;b f l =T m +0.5T m ;b f 2 =0.5T m T r a ;

2 KwY^ m +l ) ;a f2

Tmxro

2(cobwYTm +1) , suggested co =

0.6

3.1.4 Controller with set-point weighting

U(s) = K

<xK„

+ R(s) t ~ E(s) K,

v T i s ;

v T>sy E(s)-aKcR(s)

+® + U(s)

Kme~ 1 + sT

Y(s)

Table 6: PI controller tuning rules - FOLPD model G ra (s) = Kme-

l + sT„

Rule

Servo/regulator optimisation

- Taguchi and Araki (2000).

Model: Method 1

Chidambaram (2000a).

Model: Method 1

Kc T i Comment

IVlinimum performance index: other tuning

4 K (84) T(84)

Tm /Tm<1.0.

Overshoot (servo step) <20%

a = 0.6830 - 0.4242-21- + 0.06568 -^-

2

Direct synthesis: time domain criteria T-i 'T' (85)

T 2 +T ( 8 5 ) x 1 C L + M xm

5 j (85) Choose TCL,^

K (84)

K „ 0.1098 + -

0.7382

2™ T

• 0.002434

. ( 8 4 ) = T „ 0.06216 + 3.171—s-- 3.058

T T _ m _

2

+ 1.205 T

_ m _

3 \

J

5 j (85) = TmTm + 2 T C L ^ - TC L Tj(85) - £Jt

, ( 8 5 ) ^ o r a = "C"J!1 K . K J T ^ ' + T J - T / 8 5 '

-(85)

Rule

Srinivas and Chidambaram (2001).

Astrom and Hagglund (1995) - dominant pole design-page

204-208. Model: Method 5 or

14

Astrom and Hagglund (1995)-modified Ziegler-

Nichols - page 208.


Model: Method 5; Ms =1.4;

0 < a < 0 . 5


Model: Method 1

Kc

6 0.9Tra

K m T m

Ti

3.33xm

Comment

Model: Method 1

Direct synthesis: frequency domain criteria

0.29e-2'7t+3jT!Tm

K m x m

g _ 9 T i n e -6 .6 , + 3.0T» Q r

0.79Tme-L4l+2-4t2

0.14<^=-<5.5; Tra

M.=1.4

a = l-0.81e°'73,+L9T2

0.78e-4lT+5'7t2Tm

K m t m

8.9xme^+3-0T2 or

0.79Tme-L4x+2-4T2

0 . 1 4 < ^ L < 5 . 5 ; Tm

M s = 2 . 0

a = l-0.44ea78t-°-45t2

0-4Tm

K m T m 0.7Tm

0.1 < — <2 T

m a =0.5; Model: Method 5 or 14

0.35Tm 0.6

0.35Tm 0.6 K m T m K m

0.25Tm

7 R (86)

7xm

0.8Tm

0.8Tm

T ( 8 6 )

-^=-<0.11 T

m

0.11<^=-< 0.17 T

0 . 1 7 < ^ - < 0 . 2 T

^2-<0.5 T

m Ms =1 .4 ;0<a<0 .5

Sample K c , Tj taken by the authors (corresponding to the process reaction curve

tuning rules of Zeigler and Nichols (1942)); oc = - — - — = 0.30-1.11-^2 K c K m T m

for

these tuning rules.

7 K (86) — 0.14 + 0 .28^ -1 , T-Km 1 T

(86) : 0 .33 t „ 10xm+Tm


3.1.5 Controller with proportional term acting on the output 1

K „ U(s) = - ^ E ( s ) - K c Y ( s )

T:S

R(s)

+ ?) • F- E(s)

Kc

T;S

U(s) +

P V s J - *

Kc

t

Kme-™

l + sTm

Y(s)

^-

Table 7: PI controller tuning rules - FOLPD model G ra (s) = K e-

l + sT„

Rule

Minimum ITAE -Setiawan et al.


Chien etal. (1999). Model: Method 1

l a T

0.5 0.75 1.0 1.5 2.0

Kc Ti Comment

Minimum performance index: other tuning x l T m

Km tm

x l

0.554 0.570 0.594 0.645 0.720

x2

1.805 1.373 1.117 0.822 0.679

Direct si

8 K (87)

^ m

Coefficient values

Ira. T

3 4 5 6

X l

0.900 1.090 1.290 1.520

x2

0.547 0.480 0.445 0.428

l a T

m 7 8 9 10

x i

1.740 1.930 2.120 2.410

x2

0.413 0.398 0.385 0.389

,'nthesis: time domain criteria

T (87) Underdamped system response - ^ = 0.707 .

*•» > 0.2Tm

K (87) -TCL/+1.414TCL2Tm + tmTm r(87)

K m T f , , /+1 .414T r , ,T m +T„

-1.414TCL2Tm+Tmxn

T „ + T „



U(s) = Kc

V T , s 7 E(s)-K,Y(s)

R(s) E(s)

>6* » + 1 i

( i "1 K c 1 +

U(s) +

T

K m e-""

l + sTm

Y(s)

' P"

Table 8: PI controller tuning rules - FOLPD model Gm (s) = Kme"

1 + sT,.

Rule

Lee and Edgar (2002).

Model: Method I

Kc T, Comment

Robust

9 K (88) T (88) Kj =0.25KU

K, K m (^ + Tm)

Tm-0.25KuKmTm ^ T

2(X + t m )

. ( 8 8 ) m -0 .25K u K m x m

l + 0.25KuKm 2{X + zJ'

3.2 FOLPD Model with a Positive Zero Gm(s) K m ( l - s T m 3 ) e -

1 + sT, ml

3.2.1 Ideal controller Gc(s) = Kc

v T i s y

R(s) E(s)

< 2 > — Kc v T i sy

U(s) . Km ( l -sTm 3>-"-

l + sT„

Y(s)

Table 9: PI controller tuning rules - FOLPD model with a positive zero

K m ( l - s T m 3 > - ^ Gm(s) =

l + sT„

Rule

Sree and Chidambaram

(2003a). Model: Method 1

Kc Ti Comment

Direct synthesis: time domain criteria i YTml

K m T m 3 T (89)

K „ T m 1 y = value of the inverse jump of the closed loop system; y<— m m3

T m i

• (89) _ T m 3

^ [ T m 3 ( l - x l ) - 0 . 5 x m ( l + x,)]

yTm

Lm3

- ( l - X , ) - X ,

x, e 0 . 9 5 — — ^.98—ITJHI— Y T

m l + T m 3 Y T m l + T m

YTml 0 3_ YTm, 0 . 1 -

yTmi+Tm3 yTmi+Tm

0.1 < - = ^ < 0.3;

- > 1



Gc(s) = Kc 1 + — 1

R(s) E(s)

•®->K

U(s)

v Ti sv 1 + Tfs ^ K ro(l-sTm3)e-""

TjsJl + TfS

Y(s)

l + sT„

Table 10: PI controller tuning rules - FOLPD model with a positive zero

K m ( l - s T m 3 ) e - ^ Gm(s) =

1+sT^

Rule


(2003a). Model: Method I

Kc T; Comment


Tmi Km(2Tm3+X + Tm) Tmi

Tf = T m 3 ^ - 0

2Tm 3+^ + tm


3.3 FOLPD Model with a Negative Zero Gm(s) _ K m ( l + sTm3)e-

1 + sT, ml


Gc(s) = Kc

f 1 A

1 + — v T iSJl + Tfs

1

^ 6 ? S K v T i s ;

l

1 + T fs

U(s) K m ( l + sTra3)e-

l + sT„

Y(s)

Table 11: PI controller tuning rules - FOLPD model with a negative zero

Km(l + s T m 3 > - ^ Gm(s) =

l + sTm

Rule

Change/al. (1997). Model: Method 2

Kc

Tm, Km(TCL + Tra)

Ti

Robust Tml

Comment

Tf = Tm3


3.4 Non-Model Specific

3.4.1 Ideal controller Gc (s) = Kc

v T i sy

R(s) l> m> y *

L

Kc f i+-Ll I TiSJ

U(s) Non-model

specific

Y(s)

Table 12: PI controller tuning rules - non-model specific

Rule


Hwang and Chang (1987).

Parr (1989) -page 191.

Parr (1989)-page 192.

Hanged a/. (1993b)-page 59.

Pessen(1994).

McMillan (1994)-page 90.

Astrom and Hagglund (1995) -pages 141-

142.

Kc Ti Comment

Ultimate cycle 0.45KU

0.45KU

0.83TU

1 ("5.22 5.22'

P. 1 Tu T i , T

Quarter decay ratio

p, ,T, = decay rate, period measured under proportional control

when Kc = 0.5KU

0.5KU

0.33KU

0.25KU

0.25KU

0.3571KU

0.4698K„

0.1988KU

0.2015KU

0.43TU

2T

0.25TU

0.167TU

T

0.4373TU

0.0882TU

0.1537TU

dominant time delay process

dominant time delay process

A m = 2 , ( j ) m = 2 0 0

Am = 2.44,

Am = 3.45,

4>m = 46°

Rule

Calcev and Gorez (1995).

Edgar e/a/. (1997)-page 8-15.

ABB (2001).

Robbins (2002).

Wang et al. (1995b).

VranCicefa/. (1996), Vrancic(1996)-

page 121.

Vrancic(1996)-page 140.


Kc

0.3536KU

0.59KU

0.5KU

0.3KU

0.3KU

Ti

0.1592TU

0.81TU

0.8TU

1 T (90)

1 T(91)

Direct synthesis 2 ^ ( 9 2 )

0.5A3

(A ,A 2 -K r a A 3 )

0.45KU

0.4830

Gp(j f f l150»)

T(92)

A3

A2

A3

A2

3.7321

Comment

4>m= 45°, smallxra

(j)ra= 15°, largexm

Minimum IAE -regulator

Minimum IAE -servo

Good response -regulator

Am>2,<t>m>60°

(Vrancic et al. (1996))

Modified Ziegler-Nichols method

A m > 2 ,

K > 15°

1 T/90 ' = (0.24 + 0.11 lKuKm )TU ; T;(91) = (0.27 + 0.054KuKm )ru . 2 For a stable process, Km is assumed known; for an integrating process, Km and i

are assumed known.

Gp(jfflCL){l-GCL(jcoCL)}J'

GP(JCOCL){1-GCL(JCOCL)}_

J W C L G C L G«>CL) I Gp(jffi>cL){l-GcL(jfflCL)}_

K c ( 9 2 ) = ^ _ I m

. (92)

Im

Rl

Rule

Kristiansson and Lennartson (2000).

Kristiansson and Lennartson (2002).

Kristiansson (2003).

Aikman(1950).

Rutherford (1950).

MacLellan (1950).

Young (1955).

Kc

1.18KuKm-1.72

K u K m

0.50KuKm-0.36

K U Km

2 0 K 1 3 5 „ K m - 1 6 0

K,35oKm

5.4K 1 3 5 „K m -13 .6

K135 o Km

0.33KuKm -0.15

Km(0.18KuKm+l)

5KC<97»

6 K (98)

Ti

3 T (93) i

3 T (94)

4T(95) i

4 T (96)

KuKm

cou(0.18KuKm+l)

T(97)

T(98)

Comment

0.1<K uK r a<0.5

K u K m >0.5

K u K m < 0 . 1 ;

K135oKm<0.1

K u K m < 0 . 1 ;

K135oKm>0.1

K u K m <10

K u K m >10

1.6 <MS <1.9

Other tuning rules

0-81K37%

[0.63K33„/o,0.97K33„/o]

0.90K37%

1.19K37%

1.02K37%

^ 3 7 %

1.29T37% J

[1.1T33%>1.3T33%

T37%

0.92T37%

0.92T37%

<1.1TU

37% decay ratio

33% decay ratio

37% decay ratio

37% decay ratio

37% decay ratio

37% decay ratio

3 j (93) _ 1.18KuKm -1.72 (94)

• (0.33KuKm-0.17)cou ' ' 0.50KuKm-0.36

(0.33KuKm-0.17)o)u

4 T ( 9 5 )

5 v (97)

2 0 K 1 3 5 o K m - 1 6 0 - ( 9 6 ) 5 .4K 1 3 5 „K m -13 .6

K

(0.315K1 3 5„K r a-0.175) f f l u (1 .32K 1 3 5 .K m -3 .2 )m u

0 . 0 9 K , „ „ K m + 0 . 2 5 135° 0 .09K,„ 0 K m +1.2

, ( 9 7 ) K135oKm

,5ol0.09K1 3 5„Km +1.2)

6 K (98)

K „ 0.2+ -

0.075

150- +0.05 K „

>T i( 9 8 )=<D 150°

0.06 + 1.6-K,

Kn

-0.06 ^ 1 5 0 °

V K m J

Rule

Chesmond(1982) -page 400.

Parr (1989) -page 191.

McMillan ( 1 9 9 4 ) -pages 42-43.

Wade (1994) -page 114.

ECOSSE team (1996b).

Hay (1998) -page 189.

Bateson (2002) -page 616.

Astrom (1982).

Leva (1993).

Astrom and Hagglund (1995) -page 248.


Cox et al. (1994).

Jones ef a/. (1997). Model: Method 2

Kc

K x %

0.5 + 0.3611nx

0.999K25%

0.667K25%

0.42K25%

0.33K25%

0.7515K25%

0.65K50%

K 2 5 %

K2 5%

sin(|)m

G p ( j a v )

7 K C< 9 9 )

0.5

Gp(jra135o)

0.25

Gp(jco135„)

0 . 2 5 K U

0.2hT u s i n ( t> m

A P U

0.5787 A P

Ti

Tx%

1.2Vl+0.0251n2x

0.814T25„ /o

T25%

^25%

^25%

0.7470T25„ /o

0 .7917T 5 0 %

^25%

T

tan(|)m

co90»

T (99) 1 i

4

1.6

co 1 3 5 „

0.2546 Tu

0.16Tu tan*m

aTu

Comment

x = 100(decay ratio)

Example: x = 25 i.e. quarter decay ratio

Quarter decay ratio

'Fast ' tuning

'Slow' tuning

'Modified' ultimate cycle method

tan((j)m - (j)ra - 0.5TI)

> 0 Alfa-Laval

Automation ECA400 controller

Alfa-Laval Automation ECA400

controller - large delay

Model: Method 2

a e [0.7958,1.5915]

tan(<L-<!>. • , -0-5rc) T ( •») = tan((|)m - <|)m - 0.5TC)

| G p ( j a ) ) ^ t a n 2 ( ^ - < ! . , „ - 0 . 5 T I ) ' ' m

74 Handbook ofPIandPID Controller Tuning Rules

Rule

NI Labview (2001). Model: Method 2

Tang et al. (2002). Model: Method 4

Kc

0.4KU

0.18KU

0.13KU

2.5465hsin<f>m

Apx, co90o Am

^

0.8TU

0.8TU

0.8TU

tan<|)m

C 0 9 0 °

Comment

Quarter decay ratio

'Some' overshoot

'Little' overshoot

0.81 <x, < 1 ;

x, = 1, 'small' xm;

x, =0.91, 'large'Tm

Chapter 3: Tuning Rules for PI Controllers


75

U(s) = Kc

aK„

v Tisv E(s)-aK cR(s)

+ <?>

R(s) - E(s)

*K, v T i s / + U(s)

Non-model specific

Y(s)

Table 13: PI controller tuning rules - non-model specific

Rule

Astrom and Hagglund (1995) -page 215.

Vrancic (1996)-page 136-137.

Kc Ti Comment

Direct synthesis

0.053Kue2-9K-2-6K2 0.90T„e-4'4K+2'7K2 Ms =1.4

a = l - l . l e - 0 0 0 6 l K + L 8 K 2 ; 0<K m K u <oo .

0.13KueL9K-L3Kl O^OT^-4 '4"2-7^ Ms = 2.0

a = 1 - o.48e0'40^017K2; 0 < KmKu < oo .

8 K (100) rp (100)

a = [0.2,0.5] - good servo & regulator action (Vrancic

(1996), page 139)

8 K (100)

c " ( l - b 2 K 2 A 3 +

A , A 2 - K m A 3 - x

A, - 2 K m A , A 2 J

. K m A 3 - A , A 2 < 0 ;

K (100) A , A 2 - K r a A 3 + x

( l - b 2 J ( K m2 A 1 + A , 3 - 2 K m A 1 A , ^m " 3 T ^ l y

K m A 3 - A , A 2 > 0 ; b = l - a ;

X = V ( K m A 3 - A 1 A 2 ) 2 - ( l - b 2 ) A 3 ( K m2 A 3 + A 1

3 - 2 K m A 1 A 2 ) ;

A, . (100)

K m + -K (I00)K 2

2K (100) (,-„>)

76

3.5 IPD Model Gm(s)


3.5.1 Ideal controller Gc(s) = Kc T,Sy

R(s)

+

E(s) K 1 + -

Xs

U(s) Kme" Y(s)

Table 14: PI controller tuning rules - IPD model Gm (s) = Kme"

Rule


Model: Method 2



Coon (1956), (1964). Model: Method I

Astrom and Hagglund (1995) -page 13. Model: Method 1

Hay (1998)-page 188.

Hay (1998)-page 199.

Model: Method 1 Kc and T>

deduced from graphs

Kc T,- Comment

Process reaction 0.9

Kmxm

0.6

K m t m

0.87

K m t m

3.33xm

2.78xm

4.35xm

Quarter decay ratio

Decay ratio = 0.4

Decay ratio is as small as possible

Minimum error integral (regulator mode). 1.0

0.63 K m T m

0.42

K m X m

7.5

3.5

2.5

2.2

0

3-2xm

5.8xm

4.5Kmxm

Quarter decay ratio; Kc,Tj deduced from

graphs Ultimate cycle Ziegler-Nichols

equivalent

Model: Method 3

K m t m = 0 . 1

K m x m =0.2

KmTm=0.3

K m x m =0.4


Rule

Hay ( 1 9 9 8 ) -continued.

Bunzemeier (1998). Model: Method 4

c Kp

" s(l + sT p ) n

Minimum IAE -Shinskey ( 1 9 8 8 ) -

page 123.


page 74.


page 148. Model: Method 1 Minimum ISE -


Minimum ISE -Haalman(1965). Model: Method 1

Minimum ITAE -Poulin and Pomerleau



1

1

* i

0.18

0.24 0.22

0.21

0.20 0.20

(

1

(

1

C

(

(

(

Kc

2.0

1.7

* i

^ m

x 3

( J n ,

x 2

1.350

3.050 1.910

1.690 1.584 1.557

Minimu

).9524

).9259

.61KU

1.5

).6667

X5264

).5327

K m T r a

Minir

0.28

K m T m

Ti

X 2 T m

Comment

K m t m = 0 . 5

K m x m = 0 . 6

0% overshoot (servo)


Coefficient values

x 3

0.23 0.40 0.30

0.28 0.26

0.26

n

0 1 2

3 4

5

x i

0.20

0.19 0.19

0.18 0.18

x 2

1.477

1.463 1.453

1.385 1.378

X 3

0.25 0.24 0.24

0.24 0.24

n

6

7 8

9 10

m performance index: regulator tuning

num

4T

4T

Tu

5.56xm

0

4.5804xm

3.8853xm

Model: Method 1

Model: Method 1

Also specified by Shinskey (1996),

page 121.

Model: Method 2

M , = 1 . 9 ,

A m = 2 . 3 6 ,

* m = 5 0 °

Process output step load disturbance

Process input step load disturbance

performance index: other tuning

7 t m Mm a x =1.4


Rule



Tyreus and Luyben (1992). Model:

Method 1 or 9


Rotach(1995). Model: Method 5

Wang and Cluett (1997).

Model: Method 1

Cluett and Wang (1997).

Model: Method 1

Poulin and Pomerleau (1999).

Model: Method 1

Kc

0.404

K m T m

0.49

T,

7xm

3.77xm

Comment

Mm a x=1.7

Mmax = 2.0

Direct synthesis: time domain criteria 0.487

Kraxm

0.31KU

0.5

0.75

Kmxm

i K (ioi)

2 K (102)

0.9588/Kmxm

0.6232/Kmxm

0.4668/Kmxm

0.3752/Kmxm

0.3144/Kmxm

0.2709/Kmtm

2.13 0.34KU or

KmTu

8.75xm

2.2TU

5xm

2-4lTra

T ( i o i ) i

•j- (102)

3.0425xm

5.2586xm

7.2291xm

9.1925xm

11.1637xm

13.1416xm

1.04TU

Maximum closed loop log modulus

= 2dB; TCL =

Model: Method 2

Damping factor for oscillations to a

disturbance input = 0.75.

4 = 0.707;

T c L ^ m ^ x J

4 = 1; TCL 6k.>16xJ

TcL = Tm

TCL = 2xm

TCL = 3xm

TCL = 4tm

TCL = 5xra

TCL = 6xm

Mmax = 5 dB

i K (ioi)

Kmim(0.7138TCL+0.3904)

1

Kmxm(0.5080TCL+0.6208)

, T,m) = (l.4020TCL +1.2076)xm .

Tj002 ' = (l.9885TCL +1.2235)xm .

Rule

Viteckova(1999), Viteckova et al.


Chidambaram and Sree (2003).

Model: Method I

Huba and Zakova (2003).

Model: Method I

Skogestad (2003), (2004b).

Model: Method 1


Model: Method 1 Chidambaram (1994),

Srividya and Chidambaram (1997).

Gain and phase margin - Kookos et

al. (1999). Model: Method 1

Kc

X l / K m T m

x l

0.368 0.514 0.581 0.641

OS

0% 5% 10% 15%

1.1111

Km tm

0.23 K m T m

0.281

K m T m

1

K m ( T C L + 0

0.5

m m

T; 0


0.696 0.748 0.801 0.853

OS

20% 25% 30% 35%

4-5xm

2.914xm

3.555xm

4?2(TCL+0

8tm

Comment

x i

0.906 0.957 1.008

OS

40% 45% 50%

Suggested % = 0.7 or 1

'good' robustness -

T C L = x m , 5 = l

Direct synthesis: frequency domain criteria 3 K (103)

0.67075 K r a T m

m p

AmKm

0-942/Km xm

0.698/Kra xm

0.491/Km rm

0.384/Kmxm

0

3.6547tm

1

C0p (0.571-COpTm)

Representative results 4.510xm

4.098tm

6.942xm

18.710xm


Model: Method 6; A m = 2

Am =1.5;

<L = 22.5°

A m = 2 ; 4 , m = 3 0 °

Am = 3; *„,= 45°

Am = 4 ; ^ m = 6 0 0

3 Values recorded deduced from a

^ m ^ m

K (103)

0.1

7.0

0.2

3.5

0.3

2.2

graph: 0.4

1.8

0.5

1.4

0.6

1.1

0.7

1.0

0.8

0.8

0.9

0.75

1.0

0.7

Rule

Chen et al. (1999a). Model: Method 10

Cheng and Yu (2000).

Model: Method 1


Kc

•L+fta-i) K m T r a (A m

2 - l )

5 K (105)

0.5236/Km xm

1 * 1

< m T m

T;

4rp(104)

j ( 1 0 5 )

8^m

* 2 * m

Comment

f 1.0607~] Am

A m =2 .83 ;

* m = 46.1°

[6]

Representative coefficient values

* 1

0.558

0.484

0.458

x2

1.4

1.55

3.35

Am

1.5

2.0

3.0

4>m

46.2°

45.5°

59.9°

x i

0.357

0.305

x2

4.3

12.15

Am

4.0

5.0

4>m

60.0°

75.0°

4 T (104) __n_ A mz - 1

• 0 . 5 i r ( A m - l ) 0.57t-

+ 0 . 5 7 t ( A m - l )

5 r , (105) T l 7 1 ! 2 ^ , , , - l ) - ( 1 0 5 ) _ _

4 K m x m ( r 12 A m

2 - l ) ' ; C r 12 A m ( r 1 A m - 2 X 2 r 1 A m - l )

A m - 1

^ ( r . ' A ^ - l ) 2

A „

x 2

4 x ,

Tt i" . ,

IT--\

7t

x2

2

1 + -71 7t 7t

"2 +^T_^7

<t>m = t a n ' 1 ^0 .5x ! 2 x 22 + 0 . 5 x 1 x 2 ^ / x , 2 x 2

2 + 4 0.5x,2 + 0 . 5 ^ J - A / x 12 x 2

2 + 4 M

Rule


Chien (1988). Model: Method I

Ogawa(1995).

Model: Method 1;

Coefficients of Kc

and Tj deduced from

graphs

Alvarez-Ramirez et al. (1998).

Model: Method 1 Zhong and Li (2002).

Model: Method 2

K c

aK„

T,

bTu

Comment

[']

Robust

1 ( 2X + zm ^

X l / K m Tra

x l

0.45

0.39

0.34

0.30

0.27

x 2

11

12

13

14

15

K m lT C L TmJ

2 a + i m

K m ( a + t m ) 2

2X + rm

X 2 t m

8



40%) uncertainty in process parameters



T m + ^ m

2 a + xm

TCL > t m

a not specified

7 Am =

2b Tea

71

4b

1 + -71

—+ 2

7t

4b

<h tan 27:2a2b2+27tabV47t2a2b2+4 - . . 0 .125TCV +—V47i 2 a 2 b 2 +4 11 16b

X = [l/Km ,xm] (Chien and Fruehauf( 1990)); X = 3xm (Bialkowski (1996));

I > tm +Tm (Thomasson (1997)); I = [l.5xra,4.5Tm] (Zhang era/. (1999)). From a

graph, X = 1.5tm ....Overshoot = 58%, Settling time = 6xm

,....Overshoot = 35%, Settling time = 1 lxm

, ....Overshoot = 26%, Settling time = 16xm

.Overshoot = 22%, Settling time = 20xm .

X = emax /100Km ; emax = maximum output error after a step load disturbance

(Andersson (2000) -page 12);

X = 2.5Tm

X = 3.5Trn

X = 4.5xm

„„V/100K„

Xse

Kc(^s + l) • (Chen and Seborg (2002)); X = xra (Smith (2002)).


Rule


Skogestad (2004a). Model; Method 1

Penner(1988). Model: Method 1

Kc

1

Km^m

9 K (106)

Ti

%

4

Kc(1°6)Kra

Comment

Other methods 0.58

KmTm

0.8

KmTm

10tm

5.9xra

Maximum closed loop gain = 1.26

Maximum closed loop gain = 2.0

9 K (106) > l A d 0 l

I A Y max I

|Ad0| = maximum magnitude of a sinusoidal disturbance over all frequencies,

I Ay maX | = maximum controlled variable change, corresponding to the sinusoidal

disturbance.



Gc(s) = K(

U(s)

83

1+-T i S y

1

R(s) E(s) K, 1 + —

v T i s /

1

1 + Tfs

Kme"

1 + Tfs

Y(s)

Table 15: PI controller tuning rules - IPD model Gm (s) = Kme"

Rule

H„ optimal -

Tan et al. (1998b). Model: Method 1


Model: Method 1

Kc

0.463X. +0.277

Kmxm

1 K (107)

Ti

Robust

t m

0.238X +0.123

2(x1B+X)

Comment

A. = 0.5;

' 5.750A. + 0.590

X not specified

1 £ (107) 2{Tm+X) , T f = -

x ^

Km(2xm2+4zmX + X2)' f 2xm

2+4TmX + X2

U(s) = Kc

KaK V T > s /

E(s)-aKcR(s)

R(s)

K8> - E(s)

K, v T . sy

xg> U(s)

Kme" Y(s)

Table 16: PI controller tuning rules - IPD model Gm (s) = -Kme"

Rule

Taguchi and Araki (2000).

Model: Method I

Minimum ITAE -Pecharroman (2000). Model: Method 10

Chidambaram (2000b).

Model: Method 1


Model: Method 2

Kc T i Comment

Minimum performance index: servo/regulator tuning

0.7662

KmTm

x,Ku

4.091xm

X 2 TU

a = 0.6810, Tm /Tm<1.0.

Overshoot (servo step) < 20%

K m = l

Coefficient values

* i

0.049

0.066

0.099

0.129

0.159

0.189

x2

2.826

2.402

1.962

1.716

1.506

1.392

a

0.506

0.512

0.522

0.532

0.544

0.555

0-64/Kra xm

0.487/Km xm

In general, a ~ 0.

0.35 K T

4>c

-164°

-160°

-155°

-150°

-145°

-140°

*1

0.218

0.250

0.286

0.330

0.351

x2

1.279

1.216

1.127

1.114

1.093

a

0.564

0.573

0.578

0.579

0.577

4>c

-135°

-130°

-125°

-120°

-118°

Direct synthesis 3.333Tta

8.75xm

a = 0.65

a = 0.557

6 (on average) or a = 0.'

7?m

T i

Ms = 1.4;

0 .3<a<0.5


3.5.4 Controller with proportional term acting on the output 1 K „

U(s) = -£-E(s)-K cY(s)

R(s) • 6 ^ *\ F- E(s)

Kc

T,s ~ L 6 ? V — * -+ U(s)

Kc

4

Kme-"» s

Y(s)

•

Table 17: PI controller tuning rules - IPD model Gm (s): Kme"

Rule

Minimum ISE -Arvanitis et al.

(2003a).

Arvanitis et al. (2003a).

Model: Method 1

Minimum ISE -Arvanitis et al.

(2003a).


Model: Method 1

Kc T; Comment

Minimum performance index: regulator tuning 0.6330

Kra^m

2 0.6114

KmTm

3 0.6146

Kmxm

Minimum | 0.6270 K m x m

2 0.6064

K m t m

3 0.6130 K m T r a

2.3800im

2.7442xm

2.6816xra

Model: Method I

jerformance index: servo tuning

2.4688xm

2.8489tm

2.7126tm

Model: Method I

1 Minimum f[e2(t) + K m V(t ) ]d t .

Minimum 1 rdu^2

e 2 ( t ) + K - 2 U t dt.

Rule

Chiene/a/. (1999). Model: Method 2


Model: Method 1


Model: Method 1

Kc

4 j r (108)

4

(8-P)Kmxm

5 K (109)

Ti

Direct synthesis 1.414TCL2+tm

4t m

P

Robust

0.(712 - a j

Comment

Underdamped system response - £, = 0.707 .

P%Al

4 v (108)

^r'=—r 1.414TCU+Tm

J . , , 2 +1 .414T r , I ,T m +x„ 2 ) ' CL2

5 K (109) 4K1

[(8-a>i2 +a2]Kr

CL2 '•m T l m

with 0 < a < %2 :

recommended a = — 2

1 - 1 ' T I 2 ( 4 5 2 + I )

, £> 0.3941



R(s) E(s)

+ * K, 1 + -

T;s

U(s) = Kc 1 +

U(s)

Ti^y E(s)-K,Y(s)

"KgH-K„e- S T ™

Y(s)

K,

I Table 18: PI controller tuning rules - IPD model Gm (s) =

K „ , e "

Rule


Model: Method 1

Kc Ti Comment

Robust

6 K (HO) T(110) i

K, =0.25KU

6 K (HO) 0.25K,,

fc + O K u K m 2(X + T m ) T(110) _ _ i _ _ T , Tm

KuKm m 2(\ + T J "

3.5.6 Controller with a double integral term

Gc(s) = K

E(s)

' l l ^ 1 + — + — -

T;,s X9s2

R(s) < g ^ K, \ 1 1 ^ 1 + +

V Tiis T i 2s2y

U(s) K„e" Y(s)

Table 19: PI controller tuning rules - IPD model Gm (s) = K „ e -

Rule

Belanger and Luyben (1997).

Model: Method 2

Kc Ti Comment

Ultimate cycle

0.33KU

T„ = 2.26TU ;

Ti2 = 20.5TU2

Maximum closed loop log modulus =

+2dB.


3.6 FOLIPD Model Gm(s) - K e "

>(l + sTm)

3.6.1 Ideal controller Gc (s) = K(

v T i sy

R(s) ?) E ( s ) , y *

k

Kc

I TisJ

U(s) K m e " ^

s(l + sTm)

Y(s)

Table 20: PI controller tuning rules - FOLIPD model Gm (s) = K„e"

s(l + sTm)

Rule

Coon (1956), (1964). Model: Method 1; Coefficients of Kc

deduced from graphs


page 75. Minimum IAE -

Shinskey(1994)-page 158

Kc

x i

K m (^ m +T m )

^m/Tm

0.020 0.053 0.11

x i

5.0 4.0 3.0

Minimum pe 0.556

K m (x m +T m )

0.952

K m ( T m + T j

^ Comment

Process reaction

0 Quarter decay ratio

criterion

Coefficient values

t m / T m

0.25 0.43 1.0

*1

2.2 1.7 1.3

tm/Tm

4.0

x i

1.1

rformance index: regulator tuning

3.7(xra+Tm)

4(Tm + 0

Model: Method I

Model: I Method 1


Rule






Model: Method 1 Poulin and Pomerleau


Kc

i K (I")

Ti

x , (x m +T m )

Comment

Kc and Ti

coefficients deduced from graph

Coefficient values Tm/Tm

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

*1

5.0728 4.9688 4.8983 4.8218 4.7839 3.9465 3.9981 4.0397 4.0397 4.0397

x2

0.5231 0.5237 0.5241 0.5245 0.5249 0.5320 0.5315 0.5311 0.5311 0.5311

tm Am 1.2 1.4 1.6 1.8 2.0 1.2 1.4 1.6 1.8 2.0

* 1

4.7565 4.7293 4.7107 4.6837 4.6669 4.0337 4.0278 4.0278 4.0218 4.0099

x2

0.5250 0.5252 0.5254 0.5256 0.5257 0.5312 0.5312 0.5312 0.5313 0.5314

Direct synthesis

2 1.26X! K m T m

2 13 0.34K,, or

KmTu

0

1.04TU


Mmax = 5dB

1 K (111) X 2 T m

° _ K m ( x m + T m ) ^ X l ( T m + T m ) 2

2 x, values recorded deduced from a graph:

* m / T m

x,

1

0.31

2

0.19

3

0.14

4

0.11

5

0.09

6

0.078

7

0.068

8

0.06

9

0.053

10

0.05

Rule

Huba and Zakova (2003).

Model: Method 1

McMillan (1984). Model: Method 1

Peric etal. (1997). Model: Method 8

Kc

3 K (112)

^ 0

Comment

Representative tuning rules; tuning rules for other xm/Tm values

may be obtained from a plot.

0.193/Kraxm

0.207/Kmtm

0.219/Kmxm

3.717xm

3.386xm

3.127xm

t m / T m = 0 . 5

Tm Am =1

^ m / T m = 2

Ultimate cycle 4 K (113)

Ku tan2 <L

yj\ + tan2 <L

-p (113)

0.1592 Tu

tan<L


KU,TU

3 v (112) - 2 T m + ^ / x m2 + 4 T m

2

JS.. — 1— 2Tm+xm->/Tni

2+4Tm

4 K . ( H 3 ) _ l - 4 7 7 T m

K m x m2

1

1 + | —

• , Ti1"-" = 3.33xm 1 + ri Y'65

m I

U(s) = Kc 1 + -TiS,

E(s)-aKcR(s)

* aK c

R(s) + T E(s) K, 1 + -

Xs + U(s)

K„e"

s(l + sTm)

Y(s)

Table 21: PI controller tuning rules - FOLIPD model Gm(s) = Kme"

s(l + sTm)

Rule


Model: Method I

Minimum ITAE -Pecharroman (2000).

Model: Method 4

Km = l ; Tm = l

0.05 <Tm <0.8

Kc

Minimum

5

Xl

0.049

0.066

0.099

0.129

0.159

0.189

K c 0 M ,

X 1 K U

x2

2.826

2.402

1.962

1.716

1.506

1.392

Tj Comment

performance index: servo/regulator tuning

T(114)

x2Tu

^n/T r a <1.0 . Overshoot (servo

step) < 20%

Coefficient values a

0.506

0.512

0.522

0.532

0.544

0.555

4>c

-164°

-160°

-155°

-150°

-145°

-140°

x i

0.218

0.250

0.286

0.330

0.351

x2

1.279

1.216

1.127

1.114

1.093

a

0.564

0.573

0.578

0.579

0.577

i -135°

-130°

-125°

-120°

-118°

K (114)

K „ 0.1787 + -

0.2839

-0.001723

. ( 1 1 4 ) V " " =4.296 + 3.794-^- + 0.2591 - ^ T T

a = 0.6551+ 0.01877-

Rule

Astrom and Hagglund (1995) -pages 210-

212. Model: Method 3

0.14 <^2-< 5.5 T

Kc

Q 4 1 e -0 .23T+0.019t 2

K m ( T m + i J

a g l e - l . l t + 0 . 7 6 T '

K m (T m + T m )

Ti

Direct synthesis

5.7xrae,J-°-69T2

a = l -0 .33 e2 5 ' - '^ 2

3 4 T e ° - 2 8 i - 0 0 0 8 9 ^

a=l-0.78e-L 9 T + 1 '2 x !

Comment

Ms =1.4

M s = 2 . 0

3.6.3 Controller with proportional term acting on the output 1 K„

U ( s ) = - * - E ( s ) - K c Y ( s ) T:S

R(s)

+ T d • y ' F - E(s)

K«

T,s ^ 6 ? i k-. f U(s)

Kc

T

K m e ^

s(l + sTm)

Y(s)

•

Table 22: PI controller tuning rules - FOLIPD model Gm (s): Kme"

s(l + sTra)

Rule

Arvanitis et al. (2003b).

Model: Method 1

Kc Ti Comment

Minimum performance index: regulator tuning 4

(8-p)Kmxm

4xm

P

Minimum ISE 6

3P = 2.6302-2.1248 i

, T + 2.5776

T •1.8511

T V m J

+ 0.82557 T

-0.23641 • I ' m /

-1.7164.10" , T V m J

-0.044128 -si-VTm J

+ 3.1685.10"7

-0.0053335^2- + 0.00040201^1-V m J V m 7

10

±2L , 0 4.1.

Rule


Model: Method 1 (continued)

Kc Ti Comment

Minimum performance index 7 ,8

Minimum performance index 9 , l 0

7 Performance index = He2 (t) + Km V (t)Jdt .

!B = 2.1054-0.76462^=- +0.79445 f^=- -0.51826

K IT, ' T + 0.21771

-0.06 T

-4.132.10"

+ 0.010927 ' T *

T -0.0013006

T -9.717.10"

T

T

('_ A + 7.6235.10"

T , 0 < - S - < 4 . 6 ;

T * m

B = 1.69667833406 - 0.00148768 T

V m

-4.6 •>4.6.

Performance index: e2(t) + K m2 f dt

6 = 1.6505623 - 0.0034983 - ^ - - 8 > 2 ;

8 = 2.1806-0.6027: V m J

+ 0.3875 X

T -0.14955

T + 0.034535

T V m

-0.0042243 T

V, m J

+ 8.2395.10" > T V m

+ 5.0912.10"5 |^- -7.283.10" T

V m J

+ 4.2603.10"7 (, \9

T V m J

-9.5824.10 I — , -21- range not defined.

Rule


Model: Method 1

Kc

II K (H5)

T i

T ( H 5 )

Comment

Minimum ISE 12

Minimum performance index '3, 14

n K ( i i 5 ) _ . 12 .5664(T m +Tj-4

Km[2(xm +Tm)(l2.5664(xm +Tm j - 4 ) - (l.5Tra2 + 3xmTm + 2Tra

2^J

with a = [0.5708 +

12 % = 0.99315 + 1.9536

3.4674Tm+3.1416Tm- l k ; T | ( 1 1 5 ) _ 1 2 . 5 6 6 4 ( x m + T m ) - 4

, T -2.5157

T 1.9163

( \3

T 0.8893

, T , V m /

+ 0.26167 - E -

+ 2.0083.10"

-0.049809-^2-V m J

+ 0.0061098-2-T

- 0 .00046593^-V m J

T - 3.7368.10-7

, T 0<^_<6 .0 ;

T

T Xm 6 1, -^2- > 6.0 . X = 1.96705757 + 0.01308189

' Performance index = l[e2 (t) + KmV(t)]di

X = 0.73114 + 2.346ip2- -2 .8509PS Un, J ( j r a

+ 2.0965 A-1

(~ \ -0.95252

m / T

V m J

+ 0.27639 ^ L _ 0.052089 — l T m J {Tm j

+ 0.0063413 f ^7

- ^ L | -0.00048064 T, >J

V m J

+ 2.0611.10~5 f. V

T , -3.8178.10~7

fx^

T V m J

X = 1.916423116 + 0.018175439^- -4 .5 V ' i n

, 0<-!2-<4.5 ; T

• > 4.5 .

http://-3.8178.10~7

Rule




Model: Method 1

Kc T, Comment

Minimum performance index 15, 16

Minimum i 4

(8-p)KmTm

performance index: servo tuning 4T

P

Minimum ISE "

' Performance index = f e2(t) + Km2 — I dt

J raUJ| o L J

X = 0.60896+ 3.0593

+ 0.38696

, T -3.9339 ' T *

, T , V m /

+ 2.933 V m

-1.3353 T

V m

T V m J

-0.07277: r \6

x„

T + 0.0088395

T -0.0006686 it

-2.8623.10" T

V m

-5.2941.10' > T

0<-m_<4.5; T

X = 1.9283751 + 0.016846129 T

V m

-4.5 - ^ - > 4 . 5 . T

17 p = 1.997- 0.82781| - ^ - |+1.0002 ^ *

T V m J

0.72106 M M +0.32269 [ ^ T , T

-0 .092452^- +0.017215 v T T

-0.0020701 \TmJ

-0.00015505 T

-6.5691.10" T

V m J

+ 1.2025.10" T

V m J

, 0 < ^ - < 5 . 5 ; T

(3 = 1.6281745-0.001171 T

- - 5 . 5 ->5.5.

Rule



Kc Ti Comment

Minimum performance index l8,19


00

1 Performance index = f[e2 (t) + K m V (t)]dt

'(3 = 1.5693 + 0.1066^2-1-0.10101 f \ 2

T V 1 m J

+ 0.05217 -0.016147 e \ 4

+ 0.0030412 — I -0.00032187 V m

-^2-1 +1.2181.10"5

VTm , T

T V m J

+ 9.9698.10"

•1.2274.10"7

T

r \ + 3.7365.10"

T

10

, 0 < - ^ - < 4 . 6 . T

T

Performance index = *"**.'i£ dt

P = 0.780651417 + 4.7464131 -^--0.1 T

0<^_<0.3; T

P = 1.729934 -0.051535 -^--0.3 T

V 1 m J

, 0.3<-SL<2. T

Rule


Model: Method 1

Kc

22 K (116)

Ti

i

Comment

Minimum ISE 23

Minimum performance index 24,25

22 v (116) 12.5664(xm+Tm)-4

Km[2(xm +TmXl2.5664(Tm + T m ) -4)- ( l .5x m2 +3xmTm + 2Tm

2>]

wkha = [0.5708 + i p Z l ^ H l l K E i ] x , T(..« = 12.5664(xm+Tm)-4

23 x = 0.79001 + 2.114|^2-Tmy

-2 .5601P 2 - ! +1.8971

IT,

r A3

T V 1m J

0.86871 f \ 4

x „

T V m y

+ 0.25371 -0.048068

.9231.10" T

V m

T

-3.5718.10"

+ 0.0058771 • ^ M -0.00044707 ^ M Y m 7 V m 7

T V 1m J

0< JL<4 .5 ; T

X = 1.903177576 + 0.0181584p2--4.5 , ^ - > 4 . 5 . V m / m

00

1 Performance index = f[e2(t) + K r aV(t)jdt .

25 x = 0.48124 + 2 .5549^-1-2 .896

\5

T + 2.0495

vTm

+ 0 . 2 6 0 9 4 ^ -Y m )

- 0.048775 ' T ^

i T , V m J

+ 0.0059064 T

-0.911151 —E-

-0.00044618

+ 1.9095.10" T V m j

-3.5335.10" T

V 1m J

0<-^ -<4 .0 ; T

X = 1.87054524 + 0.021802| — - 4 ->4 .0 .

Rule




Model: Method 1

Kc T i Comment


Direct synthesis 4

(8-P)Kmxm

28K ( U 7 )

4t m

p

(l + 4 ? 2 ) k + T j

p = ^

00 f

' Performance index = I e2(t) + K m2 —

X = 0.31373 + 3.7796 ' x ^

T V m J

- 5.0294 T

V m

dt.

+ 3.8589 f ^ I™. T

1.7956 ( \4

T

+ 0.52906 r A5

T 0.10079 -S2-

\

+ 4.0678.10-5|-E- -7.5694.10

+ 0.012369 ^ 2 - | -0.00094357 f \

T

T V 1 m J

T 0 < - E - < 3 . 5 ;

T

X = 1.86144723+ 0 . 0 2 1 9 8 9 ^ - 3 . 5 , — > 3.5 . V * m j *m

(l + 4 4 2 k + T m )

K r a k 2 ( 0 . 5 + 842)+xraTm(l + 16^ )+8^ 2 T m2 ] '

28 j r (117)



( 1 1 U(s) = Kc 1 + — E(s)-K,Y(s) V T.sy

R(s) b(s)

k I TisJ

+ vy

f 1

t

»;

•"W

K m e - "

s(l + sTm)

Y(s)

w

Table 23: PI controller tuning rules - FOLIPD model Gm(s) = K e~

s(l + sTm)

Rule


Model: Method 1

Kc Ti Comment

Robust

29 K (118) T ( 1 1 8 ) K, =0.25KU

29 ^ (118) 0.25K,,

(A- + 0 KuKm 2(X + T„ >T;

(118)

2( + 0 '


3.7 SOSPD model Kme" K e"

or-T m l V + 2 ^ T m l s + l (l + TmlsXl + Tm2s)

3.7.1 Ideal controller G (s) = K 1 + — V T i s 7

R W ^ E ( S ) . + 1 i

Kc f.+-L) U(s) SOSPD model

Y(s)

Table 24: PI controller tuning rules - SOSPD model

K - e - " - o f K m e - ' -

T m l V+2i ; m T m l s + l (l + TmlsXl + Tm2s)

Rule


page 158.

Minimum IAE -Shinskey (1996)


-^2-= 0.2 Tml

Kc Ti Comment

Minimum performance index: regulator tuning

1 K (119)

0.77Tm,

0.70Tml

K m X m

0.80Tml

K m T m

0.80Tral

Kmxm

T ( 1 1 9 ) i

2-83(Tm+Tm2)

2.65(Tm+Tm2)

2.29(tm+Tm2)

l-67(Tm+Tm2)

Model: Method 1

T - H i = 0.1 Tml

^Jsl = 0.2 Tml

^ 2 1 = 0.5 Tml

T i==- = 1.0 Tml

1 K (119) 100T„.

K m (x m +T m 2 ) 50 + 55 ! _ e ^ + T -

-019) _ . 0.5 + 3.5 1-e

3Tm,

Rule


Model: Method 1

Kc

2 K (HO)

T,

T (120)

Comment

0 < I n i 2 . < i ; T 1 m l

0.1<-±2-<l Tra,

K

Note: equations continued into the footnote on page 104

(120) _ 1

Kn

1

Km

J_ Km

1

Km

J_

1

K „

6.4884 + 4 .6198 -^2 - - 3.49i_i!i2__ 25.3143 TmT™2

T m l T m l T m l

( _ A -0.9077

0.8196 V T m i y

•5.2132

-0.063

-7.2712

f \ 0.5961 X,

Lml J V T m i y

-18.0448

•sO.7204 , x

^L\ + 5 . 3 2 6 3 ^ -vTmi

0 . 4 9 3 7 ^ - + 1 9 . 1 7 8 3 ^

l T r a l

^ n , |

1.0049 , \ 1.005

+ 13.9108—212-

+ 12.2494-VTml V

8.4355-1 m 2

' m l

-17.6781-T V ^ m l

-0.7241eT"» -2.2525eT m l +5.4959eT""2

, (120) = T t m , „ im2 <c /1-709 J E " 0.0064 + 3.9574—2- + 4.4087-2^- - 6.4789 T T

+ T„

+ Tml

+ T„

+ Tml

-12.8702 T m T m 2 -1.5083 T 1 ml

' ^ 2 i l | +9.4348 V ^ m l V^ml

17.0736-VTml 7

+ 15.9816-VTml J

- 3 . 9 0 9 T "l 1 m 2

VTml y

•10.7619

-6.6602

V •'ml

x„ X

•10.864-Tm, Tm,

-22 .3194 V ^ m l V ^ m l

^ml V t"1 J

+ 6.8122|- i2l T,

^ + 7 .5146-21

ml J ml /

Rule

Minimum IAE -Huang etal. (1996).

Model: Method 1

Kc

3 K (121)

T,

T ( 1 2 1 )

Comment

0.4 < 5m < 1;

0.05 < — <1

+ T„

+ Tm

+ T„

+ T„.

2.8724-

.2174,

T V ml

11.4666 V *-m\

2 , N 3

V ^ m l

+ 11.1207 \2 ( V

^m t m T

V ^ m l

l ml J

-4.3675 ml J

5

(T x m 2

V * m l

- 0 . 1 1 2 ^ - ^ 2 - + 1 . 0 3 0 8 - ^

-2.2236

•1.9136 V ^ m l V ' ' m l

-3.4994U^3- I 1-=^-

lT, , , T ml y \ xml y

-1.5777 T

T-m2 I + 1.1408^4 i 2 -^m] V ml .

' Note: equations continued into the footnote on page 105.

K (121)

K „ 10.4183 - 20.9497^2- - 5.5175^m - 26.5149^m - ^

l m l ITT

K „ 42.7745

s N 1.4439 T

V lml

+ 10.5069 f \0.1456

Tm ! +15.4103[ T" V * m l

+ — [34.32364m3'7057 -17 .8860C 5 3 5 9 -54.0584^m

L9593] K m

-0.0541 f -\ 4.7426

+ — I 22.4263^mf ^=_) +2.7497^, ' T' K „

K „

T

50.2197^m'-8i88 - 2 - -17.1968—=s-4m2-7227 + 1.0293£,m -JBL

Kn

5 m ^

, ( 1 2 1 )

•16.7667e'ml +14.5737e?m -7.3025e

11.447 + 4.5128-S- - 75.2486^m - 110.807 V ^ m l

12.282^n

+ T„. 345.3228^m2 +191.9539 ^ | +359.3345^n


Rule



Minimum ISE -McAvoy and Johnson

(1967). Model: Method 1;

Coefficients of Kc and Tj deduced

from graphs

^

1 1 1

Kc

0.48KU

* i / K m

Ti

0.83TU

X2^m

Coefficient values T 1 m l

0.5 4.0 10.0

x i

0.8 5.7 13.6

X 2

1.82 12.5 25.0

^

4 4

Tm,

^m

0.5 4.0

x i

4.3 27.1

x2

3.45 6.67

^m

7 7

Comment

^ - = 0.2, Tm, T - ^ - = 0.2 Tm i

Iml

•Cm

0.5 4.0

x i

7.8 51.2

x2

3.85 5.88

+ T„.

+ T„

+ T„

+ T„

-158.7611—4m2 -770.2897^m

3 -153.633 Tm i

-412.5409£,n l m l J

-414.7786^m2 U * - + 4 8 5 . 0 9 7 6 ^ m

3

864.51954m +55.4366 - s - | + 222.2685i;n

275.116^ 21 Tm ' ' - 2 0 5 . 2 4 9 3 ^ - \J -479.5627 Um, ' ' m l V^ml

-473.1346^m -6.547 5 c e n _Tn

T - 43 .28224 m | - ^

' m l

A + 99.8717£,n

+ Tm

+ Tm

- 7 3 . 5 6 6 6 1 ^ 1 4m2 "56.4418 ^ - j ^J -37.4971 ^ | ^

1 6 0 . 7 7 1 4 ^ m5

' m l

Rule

Minimum ITAE -Lopezes / . (1969). Model: Method 7;

Coefficients of Kc and T; deduced

from graphs

Minimum ITAE -Chao etal. (1989). Model: Method 1

Minimum ITAE -Hassan (1993).

Model: Method 7

Kc

x,/Km

^

x2Tm

Comment

Coefficient values

lm

0.5 0.5 0.5

Tml

0.1 1.0 10.0

*1

3.0 0.2 0.3

x2

2.86 0.83 4.0

4 K (122)

Sm

1 1 1

Tml

0.1 1.0 10.0

x l

7.0 0.95 0.35

x2

2.00 2.22 5.00

T(122) i

4-

4 4 4

*m

Tm, 0.1 1.0

10.0

Xl

40.0 6.0

0.75

x2

0.83 3.33 10.0

0.7 < £ < 3 ; 0.05 < Tm < 0.5 2^mTml

5 K (123) T (123)

0.5<£. m <2,

0 . 1 < ^ S - < 4 Tra,

4 R ( m ) _ 0.09578+ 0.6206£,m-0.1069i;n

K „

-1.108+0.2558l;m-0.0579l;m

-0.6441+0.76265m-0.11935„'

V^Sm ml J

T,<122) = 2i;mTml(l.31-0.0914£.m + 0.07464J;m2 J - I s _

V^Sra 'ml ,

5 Formulae correspond to graphs of Lopez et al. (1969). Kc(123) is obtained as follows:

log[KmKc<123)]= -0.0099458 + 1.9743700E,m - 3.8984310^2- - 1.1225270E,m

2

,(123) ,

+ 2.3493280 — VTml

+ 1.9126490§m -is_ + 0.0754568£,m2

V ml .

-0.5594610^, f \2

VTml J - 0.7930846£,iT1

2 - i * - + 0.2412770£,m3

Tmi

-0.3567954 V 'ml J

-0.0024730£,n

-0.1354322E.m3-^--1.1756470£.m

3

' m l vTmi

+ 0.04785934n

- 0.0096974^,

/

V ' m l .

3 X„

ml /

T; is obtained as follows (equation continued into the footnote on page 107):

• (123) '

log = 0.9321658-0.7200651E,ra -3.4227010^2- + 0.0432084£,m2

Tmi

Rule

Minimum ITSE -Chao etal. (1989). Model: Method 1


Hwang (1995). Model: Method 10

Kc

6 K (124)

Tf

T(124)

Comment

0.7 < £m < 3 ; 0.05 < — ^ — < 0.5

7 K (125) T (125) i

Decay ratio = 0.2

T V m

-1.6011510£,n

+ 1.4965440-^2- +5.3636520^m-^--0.0848431£m2

f ^

V^ml - 2.1777800^m

2 -2- + 0.0404586^m3

-0.09120084m -21--0.1769621-^2-l T n , l ,

+ 0.3033341^ra3 -^2- + 0.1753407^m

3

Tml

+ 0.15014744n 2 t „

K (124) 0.3366 + 0.6355E,m -0.1066^n

K „

r^-) -0.0622614^m3f^2

V Iml 7 \lmlJ

-1.0167+0.1743^1T,-0.03946^m

2^mTm

T/124' =2^mTml(l.9255-0.3607^m +0.1299^n 2 ^ T r a

-0.5848+0.72945m -0.1136l;m

7 0.2 < —— < 2.0 , 0.6 < £ra < 4.2 . Equations continued into the footnote on page 108.

K (125) _ 0.622[l-0.435coHTm +0.052(coHTm)2 [

KHKm / ( l + K H K m ) K t

. (125) Kc(125)(l + K H K m )

K (125)

0.0697fflHKra(l

0.724[l-0.469coHTm +0.0609(coHTm)2|

KHKm / ( l + K H K m )

l + 0.752coHTm-0.145coH2xm

2 •, s < 2.4 ;

K,

,(125) Kc(125)(l + K H K m )

0.0405(DHKm(l + 1.93(oHTm-0.363coH2Tm

2j 2 . 4 < e < 3 ;

K (125) 1.26(0.506)mHl'"[l-1.07/s + 0.616/s2]'

KHKm / ( l + K H K m ) K^

Rule

Minimum IAE -Galher and Otto

(1968). Model: Method 19.


Model: Method 1

Kc ^ Comment

Minimum performance index: servo tuning x, /Km

x 2( Tm i + T m 2 + x m ) T - T

' m l x m2 Representative coefficient values - deduced from graphs

T m

2Tml

0.053 0.11 0.25

x l

3.6 2.3 1.35

8 j r (126)

x2

1.35 1.17 0.93

^m

2Tml

0.67 1.50 4.0

T (126)

Xl

0.76 0.53 0.43

x2

0.72 0.60 0.51

0 < I E 2 . < I ;

Tm,

0 . 1 < ^ - < 1 T

.(125) KC(I25)(1 + K H K J

K (125)

-. = =rf- r—-5, 3 < s < 20 : 0.066 koHKm(l + 0.824 ln[coHtm])(l + 1.71/s-1.17/E2 j

x 1.09[l-0.497a)Hxm+0.0724((oHTm)2lN

K „ K j ( l + K H K j K H ,

X (125) Kc<125)(l + KHKm)

0.054fflHKm(-1 + 2.54a,HTm -0.457coH2xm

2)


, 8 > 20 .

K (126)

K,

1

1

K m

1 Km

1

1

,T, -13.0454 - 9.0916^2- + 2 . 6 6 4 7 - ^ + 9.162 m2 T T

0.3053 + 1.1075

' m l

3.5959

•2.2927

-31.0306 U ^ -l T m l

13.0155|^sl X ml /

l m l J

+ 9.6899 i ^ -l T m l

- 0.6418^2-+ 18.9643^2 I xm T T ' m l V ' m l

-39.7340-' m l V ml .

28.155^-1 -V^- -2.0067-T T

V ' m l J

T„, T „ , ; 1 m T m .

4.8259eT"" + 2.1137eT"" +8.45lie T""2

Rule


Model: Method 1

Kc

9 R (127)

Ti

T ( I 2 7 ) 1 i

Comment

0 - 4 < ^ m < l ;

0.05 < - ^ 2 - <1 Tml

.(126) 0.9771 - 0 . 2 4 9 2 - ^ - + 0 . 8 7 5 3 - ^ + 3.4651 - ^ ' m l ' m l V rr

3

+ 1 1 . 6 7 6 8 - ^ Tmi / Tr

- 3.8516 T m T T 2

7.5106 T

-7.4538 V ^ m l

+ T„

-10.9909-

-18.1011-

^ - 1 -16.1461 - ^ + 8 . 2 5 6 7 ^ V ^ m l ) V^n-v"Tmi

V ^ m l

+ 6.2208 V ^ m l

-21.9893-V T m l .

15.8538 V ^ m l

-4.7536 + 14.5405-

+ T„

+ Tml

+ Tm

-2.2691 V T m l J

m

l ml J

-8.387

16.651 T T

V ml ) \ 1 m l

ml J

4

2 f ^ 3

t m 1 T„

' m l / i ' m

5

Is!_ If I s L I -7.1990 I n n ] + u496p2_

-4.728-T

V ml

+ 1.1395 V ' m l

-0.6385 - ^ s -l T m l V ^ m l

.08851 ^ _ I Y^A + 3.16151 ^ Tm, J I Tral J { T „

2 / r 1 m 2

V ' m l ml / V ml

' Note: equations continued into the footnote on page 110

+ 4.5398-

K (127) — - 10.95 - 1 . 8 8 4 5 ^ 2 - - 3.41234m + 4.59544m - ^ L . - 1.7002

T V ml

\ Xml J

K „ - 2 . 1 3 2 4 1 J J - * -

^ml y

-14.4149^m2 | ^ 1-0.7683^, 3


K„

/• N0.421

7 . 5 1 4 2 ^ 2 -VTml j

+ 3.7291 V^ml

lis-1 +5.3444| J V ml j

+ J _ [ _ 0 . 0 8 1 9 ^ m1 9 ' 5 4 1 9 - 3 . 6 0 3 ^ m

1 0 7 4 9 + 7 .1163^ m

1 1 0 0 6 ]

K„ 3.206^n -7.8480£n - ^ 5 - | + 11.3222!; L9948

V *ml V ml .

K„

T (127) _ T 2i _ 1ml

2.4239eT»' + 3.4137e^ +1.0251e T°" - 0 . 5 5 9 3 ^ ^ 2 1

2.4866 - 2 3 . 3 2 3 4 - ^ - + 5.3662£m + 65.6053 Tmi

r \2

Tml

+ T„

+ Tm

+ T„

+ T„

+ T„

+ T„

+ T m

+ Tm

29 .0062^ m ^- -24 .1648) ; m2 - 8 3 . 6 7 9 6 ^

Tml I Tm

-135.9699£n + 43.1477-^a-^m2 + 5 1 . 9 7 4 9 ^

Lml J

86.0228 Tral

+ 70.4553£„ r A3

VTml^ + 153 .4877^ 2 * n

Trol

- 1 2 5 . 0 1 1 2 ^ ^ m3 - 6 8 . 5 8 9 3 ^ m

4 -62 .7517 Tmi T

V ml

2 7 . 6 1 7 8 ^ m - J 2 -f \ 3

-152.7422^ n l m l J VTml J

- 2 0 . 8 7 0 5 ^ = - ] £„

54.0012 U ^ i - C + 5 8 . 7 3 7 6 C + 1 3 . 1 1 9 3 | ^ 'ml .

2 0 . 2 6 4 5 ^ | ^ | - 2 3 . 2 0 6 4 4 m6 - 6 1 . 6 7 4 2

V Xml

f \ 3

136.2439 V^mlV

f - \ Z.J -95 .4092

Tm, C+20 .4168^C

lml

Rule

Minimum ISE -Keviczky and Csaki


Representative K,.,^ coefficients

estimated from graph; Kra = 1

Minimum ITAE -Chao etal. (1989). Model: Method 1

Minimum ITSE -Chao etal. (1989). Model: Method 1

^m/X 0.1 0.2 0.5 1.0 2.0 5.0 10.0

Tm, 0.2 0.5 0.5 1.0 1.0

10

l i

Kc

x i

Ti

x 2 * m

Coefficient values

%

I

0.1

x i

0.26 0.21 0.15 0.12 0.1 0.1 0.15

x2

1.5 1.3 1.2 1.1 1.0 1.2 1.6

0.2

X l

0.55 0.4

0.25 0.2 0.16 0.16 0.2

x2

2 1.5 1.3 1.2 1.2 1.4 2.0

0.5

x i

2.2 1.5 0.8 0.65 0.4 0.4 0.45

x 2

3.5 2.8 2

1.8 1.9 3.0 5.5

Comment

1.0

x i

6 3 2 1.2 0.9 0.65 0.6

Other coefficient values

x i

30 13 30 7 14

x2

15 14 30 14 30

K (128)

\

5.0 5.0 10.0 5.0 10.0

Tml

2.0 2.0 5.0 5.0 10.0 10.0

x i

4 8

1.9 3.4 1.1 2.0

T (128)

0.7<^m < 3 ; 0 . 0 5 < — ^ — <0. 2i;mTml

K c ( . 2 9 , x (129)

0.7<£m < 3 ; 0 . 0 5 < — ^ — <0. 2^mTml

x2

6 4.5 3.5 3.3 3.5 5.0 7.5

2.0

x i

14 9 5 3

1.8 1.0 0.7

x2

9 8 7

6.5 7 8 10

x2

15 30 17 32 19 34

4 5.0 10.0 5.0 10.0 5.0 10.0

5

5

10 K (128) 0.2763 + 0.3532^m - 0.069554n

It T

0.3772-0.24141;m +0.033421;m

ml J

Tj"28 ' = 2^mTml (o.9953 + 0.07857^m + 0.005317£,m2| — -

-0.3276+0.3226^,-0.05929^„

11 v (.29) 0.3970+ 0.43764m-0.08168^ K „ ^ m T m

- ( 1 2 9 ) ^ r a T r a l ( l 1.3414-0.1487^m +0.07685E,n

2^mTral

0.5629-0.11874m+0.01328^„,2

-0.42B7+O.I981^m-0.01978E.m

^mr„

Rule


Hwang (1995). Model: Method 10

Kc

12 ^ (130)

T, T ( 1 3 0 )

i

Comment

For^m <0.776 + 0.0568-^- + 0.18 ' m l V


0.822 X 8

-0.441 X15

-0.986 x 2 2

-0.466

x2

-0.549 x9

0.0569 X16

0.558 x 2 3

0.0647

X 3

0.122 X 10

0.0172 X 17

0.0476 x 2 4

0.0609

x4

0.0142 x n

4.62 x I 8

0.996 x 2 5

1.97

X 5

6.96

X 1 2

-0.823 X | 9

2.13 X 26

-0.323

X 6

-1.77 X 13

1.28 X 2 0

-1.13

X 7

0.786 X14

0.542 X 21

1.14

12 Decay ratio = 0.1; 0.2 < -^2- < 2.0 ; 0.6 < £,m < 4.2 . Tmi

( K (130) Xl[l + x2a)HTm +x3(coHTm)2J

K H K m / ( l + K H K j K t

• ( 1 3 0 ) Kc(l30)(l + K H K m )

X4aHKm(l + x5coH

r, E < 2 . 4 ;

t m +X 6 f f l H T„

K (130) X7[l + X8(0HTm+X9(cOHTm)2J

KHKm / ( l + K H K m ) K H .

. (130) Kc(l30)(l + K H K m )

K (130)

f ^ ^ 7—r\, 2.4 < E < 3 ; X l 0 « H K m l l + X„C0HT r a + X | 2 C 0 H Tm j

x13x14m"T"'[l + x l 5 /s + x 1 6 / e 2 f

KHKm /( l + K H K m ) K H ,

•(130) Kc(130)(l + K H K m )

X 1 7 ( 0 H K m ( 1 + XI8 l n[C OH-Cm]X1 + X 1 9 / S + X 2 o / s 2 ) 3 < s < 20 ;

K (130) X21[l + X22(0HTm +X23(cOHTm)2J

KHKm / ( l + K H K m )

,(130)

Kv

KC(130)(1 + K H K J

^ H K r a ( - 1 + X25COHTm + X26COH2Tm

2 j e>20.

Rule

Nearly minimum IAE, ISE, ITAE -Hwang (1995)-

continued

Kc T i

For lm > 0.889+ 0 . 4 9 6 ^ - + 0.26 Tmi

Comment

2


0.794 X 8

-0.415 X 15

-0.959 x 2 2

-0.466

x2

-0.541 X 9

0.0575 X 16

0.773 X 2 3

0.0667

X 3

0.126 X 10

0.0124 X 17

0.0335 x 2 4

0.0328

x4

0.0078 x u

4.05 X 18

0.947 X 25

2.21

x 5

8.38 x 1 2

-0.63 x19

1.9 x 2 6

-0.338

X 6

-1.97 X 13

1.15 x 2 0

-1.07

x 7

0.738 X 1 4

0.564

x2)

1.07

For £m >0.776 + 0.0568^2- + 0.18 — M and - ' i l l i-p ryi

ml V 1ml J

Em < 0.889 + 0 . 4 9 6 ^ - + 0 . 2 6 ^ - :


0.789 x 8

-0.426 X15

-0.978 X 2 2

-0.467

X 2

-0.527 x9

0.0551 X16

0.659 X 23

0.0657

x 3

0.11 x 1 0

0.0153 X17

0.0421 x 2 4

0.0477

x4

0.009 x u 4.37

X 18

0.969 X 25

2.07

X 5

9.7 X 1 2

-0.743 x19

2.02 X 2 6

-0.333

X 6

-2.4 X 13

1.22 X 2 0

-1.11

X 7

0.76 X 1 4

0.55 X21

1.11


Rule


K c T| Comment

Direct synthesis 0

x i

K m X m

x 2 K m T m l

Tm ,

T m i > T m 2

Tmi > T m 2

Coefficient values - deduced from graph

tm/Tml

0.33

0.40 0.50 0.67 0.33

0.40 0.50

0.67

x i

0.08

0.09 0.09 0.11 0.14 0.16

0.18 0.23

tm/Tml

1.0

2.0 10.0

1.0 2.0 10.0

x i

0.15

0.24 0.33

0.31 0.47 0.64



13 Representative x2 values, summarized as follows, are deduced from graphs:

x 2 value

^

0.4 0.6 0.8 0.9 1.0 1.1 1.2

1.5 1.9

•tmAml

0.5

----

7.94 8.93

9.80 12.2 15.4

1.0

-

5.13 5.99 5.81 9.01 9.90

10.8 13.3 16.7

2.0

5.81 6.94 7.30 7.30 11.5 12.3 13.2

15.4 18.9

3.0

7.46 8.40 9.17 10.0 13.9 14.5

15.4 18.2 21.3

4.0

9.17 10.0 11.9 14.1 16.4 16.9

17.9 20.0 23.8

5.0

10.9 11.8 15.4 19.6 18.9 19.6

20.4 22.7 25.6

6.0

12.7 14.1

20.0 28.6

---

-

-


Rule

Bryant e/a/. (1973) -continued

Hougen(1979)-pages 345-346.

Model: Method 1

Kc

1 4 X l / K m

15 K (131)

T| x2KmTml

T ( 1 3 1 )

Comment

Tmi > Tm2


1 Representative x, values, summarised as follows, are deduced from graphs:

x, value

km 0.4 0.6 0.8 0.9

T m / * m l

1.0

--

0.055 0.139

2.0

0.006 0.039 0.110 0.147

3.0

0.076 0.093 0.127 0.122

4.0

0.103 0.114 0.114 0.093

5.0

0.115 0.122 0.093 0.070

6.0

0.122 0.120 0.075 0.052

Representative x 2 values, summarised as follows, are deduced from graphs:

x2 value

5m 0.4 0.6 0.8 0.9

* m / T m l

1.0

-5.13 5.99 5.81

2.0

5.81 6.94 7.30 7.30

3.0

7.46 8.40 9.17 10.0

4.0

9.17 10.0 11.9 14.1

5.0

10.9 11.8 15.4 19.6

6.0

12.7 14.1 20.0 28.6

15 K (131)

K „ 1+-

1ml *m2

0.7Tm2(Tml+T ra2)+Tm, 2~\

Tm, +0.7T„

Ti ( 1 3 1 )=(Tm l+0.7Tm 2 | 1 + 0.1

x, values are obtained as follows:

x, value

Tm2/Tml 0.1 0.3 0.5 1.0

' m2 /Tml 0.1 0.3 0.5 1.0

0.1

0.31 0.6 0.65 0.7

0.6

0.085 0.2 0.29 0.34

0.2

0.19 0.42 0.5 0.6

0.7

0.075 0.18 0.26 0.32

c r a / ^ m l

0.3

0.15 0.35 0.45 0.5

C m / ' m l

0.8

0.07 0.16 0.24 0.31

0.4

0.11 0.28 0.46 0.45

0.9

0.065 0.155 0.23 0.29

0.5

0.09 0.23 0.32 0.38

1.0

0.06 0.15 0.22 0.28

Rule

Hougen(1988). Model: Method I

Somanie?a/. (1992). 17

Kc

16 K (132)

X | / K m A m

T i

T (132) i

x 2 T m

Comment

Five criteria are fulfilled

Coefficient values - see page 117

Equations for Kc deduced from graph;

K (>32,=_1_ ] 0! 731og,0 -=H +0.65

:0.1

K (132) 0.611ogl0 -1H1 +0.29

K „ = 0.2;

K c < 1 3 2 » = ^ 1 0 0.48 l o g j - ^ +0.05

= 0.5;

K (132)

For

K „

K , 25TmlT.

Ti

0.9806

K m A j

-10 0.41 l o g j - m i -0.06

^ . = l.T i(132)==Tml+0.7Tm2+0.12Tn

' m l

« 1 .

<X ^ 2.944 - tan"

1 m l Am

1 + 2.944 - tan" J_m_L + J_m2_

K * ^ - U i + ' T . "2

2.944 -T T

m ''m

1 + T T

2,944 - ml - m2

For 25T ,T

X • » 1, K

0.9806 c K A „

1 + (T

0.5 V ml T m 2 j

+ 0.945+ ,0.25 V^ml Tm2

- 0 . 9 4 5 ^ + — l T m l T m 2 J

0.5 + m , wm

T T V ml 1 m 2 J

+ 0.945-,|0.25| ^is_ +JEJS_ Tmi Tm2 j

-0.9451 -^=- + -^2-Trai Tm2

Rule

Sorrnmi etal. (1992) - continued.

Model: Method 1;

Suggested A m :

1.75 < A m <3 .0

Tan et al. (1996). Model: Method 2

?ou\in etal. (1996). Model: Method 1

Kc Ti Comment

Coefficient values

^ m

Tm2

0.1 0.1 0.1 0.1 0.1 0.1

0.125 0.125

0.125

0.125 0.125 0.125 0.25

0.25 0.25

0.25 0.25

0.25

0.5 0.5 0.5 0.5 0.5

0.147 0.440 0.954

1.953 4.587

13.89 0.121 0.404

0.897 1.842

4.219 11.628 0.064

0.256 0.667

1.406 3.021

6.452

0.062 0.376

0.909 1.880

3.401

x i

11.707 10.091 10.315

11.070 12.207

13.337 11.318 8.613

8.510 8.997 9.840

10.707 14.494

6.504 5.136 4.961 5.174

5.502

14.918 4.353

3.274 3.025

3.038 18 jr (133)

Tm, K m ( T m l + T m )

x2

12.5 8.33 6.25

5.00

4.17 3.70 12.5 8.33

6.25 5.00 4.17

3.70 11.11

8.33 6.25

5.00 4.17

3.70

8.33 6.25

5.00 4.17

3.70

Tra2

1.00 1.00 1.00

1.00

1.67

1.67 1.67 1.67

1.67 2.5 2.5 2.5 2.5 5.0 5.0 5.0 5.0 5.0 10.0 10.0

10.0 10.0

10.0

T m

0.082

0.437 1.016

2.075

0.169 0.581 1.242 2.445

4.651 0.156

0.338 0.559 1.182

0.075 0.238

0.433 0.667

0.951 0.073 0.235 0.424 0.649

0.917 -p (133)

T ml

x i

12.389 3.462

2.370

2.087

6.871 2.773 1.930 1.624

1.957 7.607

4.015 2.797

1.848 16.187 5.623 3.448 2.52

2.014

17.650 5.996 4.524 2.641

2.093

x2

6.25 5.00 4.17

3.57

5.00

4.17 3.57 3.13

3.13 4.55

4.17 3.85 3.33 4.17 3.85

3.57 3.33

3.13 3.85 3.57

3.33 3.13 2.94

T - T iml 'm2

Tmi > 5Tm2 ;

Min lmum §„ = 58°

18 K (133) P ^ S V 1 ^ 7 - " * ) ' t p _ a s > t m

Am)/l + (3T<'33>coJ ' " T - >

Rule

Schaedel (1997). Model: Method 1

Pomerleau and Poulin (2004).

Model: Method 5

Leva et ah (2003). Model: Method 1

Brambilla et ah (1990).

Model: Method 1

X values obtained from graph

Kc

19 K (134)

20 K (135)

Tml

K m (T m l +t n J Tml

2K m (T m 2 +T,J

Tm l+Tm 2+0.5Tm

Kmxm(2A + l)

Tm

'ml + T m 2

0.1

0.2

0.5

1.0

Tj

T (134) i

T (135) i

l-STml

4(Tm 2+xm)

Robust

Tm l+Tm 2+0.5Tm

Coefficient values

X

3.0

1.8

1.0

0.6

*m

Tm ,+T,

2.0

5.0

10.0

Comment

"Normal" design

"Sharp" design

Tm l=Tm 2 ;OS<10%

' m l >> Tm2 + Tm '

Symmetrical optimum principle

0.1 < Tm <10 Tmi + Tm2

n2 X

0.4

0.2

0.2

19 K (134) (134) 0.5T;

Ti°34) = V & m T m l + x , J -2(T ra l2 +2^mTm l tm +0.5T,,,2).

20 K (.35) _ 0.375 4Z,m'Tml* +2^mTmlTm +0.5T,/ - T m l2

Km Tm l2+24mTm lTm+0.5Tm

2

.(135) . 4 s m ' m l + ^ S m ' m l T m + ^ - 5 Tm ' m l

2smTmi + Tm


Rule

Krist iansson (2003) . Model: Method 1

Panda et al. (2004) . Model: Method 1

Kc

21 K (136)

22 K (137)

23 K (138)

25 K (139)

T i

2 ^ m T m l

j(138)

T(139)

C o m m e n t

Suggested M m a x = 1 . 7 , p = 10

24

21 K (136)

22 v (137)

2^mT„ a - la2 - 1 + -

M „ , a = l+ - 1 - 1

M „

K 2^raTral

Kmxra

1 + 0 . 1 9 1 3 ^ - - J o . 3 4 6 + 0.38261s!2-+ 0.0366-is2-

23 v (138) _ 8 ^ m T r o l + 2 ^ m T m + T r o l T ( n 8 ) _ 4 ^ r o T m l + 4 m T m + 0 - 5 T m l K „

X, > max imum

2MC

0 . 8 2 8 + 0 .812-2! 2 - + 0 .172T m l e

2 ^ m

-(69T„,2/Tml) , 7

0.828 + 0 . 8 1 2 ^ ^ + 0 .172T m l e -(6.9Tm;/Tml) , 0 -558T m 2 T m

25 v (139) _ K

A.K„ T m l + 1 . 2 0 8 T m 2 | 0.Sxn

T i(139)=0.828 + 0 . 8 1 2 ^ + 0.172Tm,e-(6-9T^ /T-'+ ° - 5 5 8 T ^ T " " + 0 . 5 t n

Tml Tm,+1.208Tm2

U(s) = Kc

v T i s y E ( s ) - a K c R ( s )

* a K „

+ T R(s)

E(s) * K ,

v T.sy + U(s)

K e"

T ^ V + l ^ T ^ s + l

Y(s)

Table 25: PI controller tuning rules - SOSPD model Kme"

Tml2s2+2i;mTmls + l

Rule


Model: Method I

Kc T i Comment


1 K (140) T (140) 5m = i ;

T m / X < 1 . 0 ;


K (140)

K „ 0.3717 + -

0.5316

+ 0.0003414

•(140) 2.069-0.3692-S1- + 1.081 T

-0.5524

f ~ \ a = 0.6438 - 0.5056^2- + 0.3087

T T -0.1201

U ^ T

V X m J

Rule

Taguchi and Araki (2000) - continued

Minimum ITAE -Pecharroman (2000),

Pecharroman and Pagola (2000).

Model: Method 11

K m = 1 i T m i = 1 •>

Kc

2 ^ (141)

x,Ku

x i

0.147

0.170

0.1713

0.195

0.210

0.234

0.249

0.262

0.274

x2

1.150

1.013

1.0059

0.880

0.720

0.672

0.610

0.568

0.545

1

T ( 1 4 1 ) 1 i

x2 T


0.411

0.401

0.4002

0.386

0.342

0.345

0.323

0.308

0.291

i -146°

-140°

-139.7°

-133°

-125°

-115°

-105°

-94°

-84°

* i

0.280

0.291

0.297

0.303

0.307

0.317

0.324

0.320

x2

0.512

0.503

0.483

0.462

0.431

0.386

0.302

0.223

Comment

^ m =0.5

a

0.281

0.270

0.260

0.246

0.229

0.171

0.004

-0.204

4>c

-73°

-63°

-52°

-41°

-30°

-19°

-10°

- 6 °

2K <141> - -K „

0.1000 + -0.05627

- +0.06041]2

4.340-16.39^2-+ 30.04 T

-25.85

a = 0.6178 - 0.4439-^ - 7.575 T

f \ l

3 r - i4^

+ 8.567 - ^ T

3

T + 9.317

vTm

•3.182 T

V i m .

3.8 SOSIPD Model - Repeated Pole s(l + Tmls)2


U(s) = Kc ' .+-L' v T i s y

E(s)-aK cR(s)

aK

R(s) &

E(s) *K,

V T i S V x8>

U(s)

K „ e "

s(l + Tmls)2

Y(s)

•

Table 26: PI controller tuning rules - SOSIPD model Gm (s) = K e"

Rule


Model: Method 1

Minimum ITAE -Pecharroman (2000).

Model: Method 2

Km = 1; Tml = 1

0.1 <xm <10

Kc

Minimum

4

Xl

0.049

0.066

0.099

0.129

0.159

0.189

K c ( . 4 2 ,

x,Ku

x2

2.826

2.402

1.962

1.716

1.506

1.392

T;

s(l + TraIs)

Comment

performance index: servo/regulator tuning

8.549 + 4 . 0 2 9 ^ -Tm,

X 2 T u

^ / T m l < 1 . 0 ;

Overshoot (servo step) <20%


0.506

0.512

0.522

0.532

0.544

0.555

* c

-164°

-160°

-155°

-150°

-145°

-140°

x i

0.218

0.250

0.286

0.330

0.351

x2

1.279

1.216

1.127

1.114

1.093

a

0.564

0.573

0.578

0.579

0.577

4>c

-135°

-130°

-125°

-120°

-118°

K (142)

K „ 0.07368 + -

0.3840

- + 0.7640 , a = 0.6691+ 0.006606-

3.9 SOSPD Model with a Positive Zero Gm(s) = K m ( l -T m 3 s)e-^

(l + sTmlXl + sTm2)

3.9.1 Ideal controller Gc (s) = Kc ' i+-L^ v T i s y

R(s)

+ < ^

E(s) c i N\

K, 1+-T j S y

U(s) Km(l-Tm3s)e-

(l + TralsXl + sTm2)

Y(s)

— •

Table 27: PI controller tuning rules - SOSPD model with a positive zero

K m ( l - T m 3 s ) r ^ Gm(s) =

(l + sTmlXl + sTm2)

Rule


Model: Method 3

Kc Tj Comment

Direct synthesis: time domain criteria 1 Tml

KmTm l+T r a 3+Tm l-5Tml OS<10%;Tm l=Tm 2

Rule

Khodabakhshian and Golbon (2004).

Model: Method 1

Luyben (2000). Model: Method 1

T m 3 / T m i v a l u e s

x, values

Kc ^ Comment

Direct synthesis: frequency domain criteria 5 K (143) T (143) Recommended

M r a a x =0dB;

Tmi > Tm2 Ultimate cycle

Ku

x i Km

0.2

3.9

3.8

4.1

4.3

4.8

5.2

5.5

6.0

0.4

3.3

3.4

3.6

3.8

4.3

4.6

5.0

5.4

6 Tu

T1

0.8

2.8

3.0

3.2

3.4

3.7

3.9

4.4

4.7

1.6

3.0

2.9

3.1

3.2

3.3

3.4

3.7

3.8

Tmi - Tm2

T ra/T ra3=0.2

^m /T r a 3=0.4

xm /T r a 3=0.6

xm /T r a 3=0.8

xm/Tm3=1.0

Tm/Tm3=1.2

x r a/Tm3=1.4 X J T m3 =1-6

5 K (143) 4 ' f f l 2 T,"43) J(T r o lTm 2)V+(Tm l

2+Tm 22)to

Km \ ( T m J m 3 ) 2

r a4 + ( T m , 2 + T m 3

2 > 2 + l with o> given by solving

•10 -0.1M„„ A

= 0.5TI + tan"1 (Ti<143)co)- tan"1 (Tm3co)- tan"1 (Tm2co)

-tan-'(Tmlco)-Tmco:

.(143) __J_ i T

1 + 0.175—S2- + 0.3| A m l

+ 0.2- -<2:

,(143) (T \

0.65 + 0.35—— + 0.3 VTml J

+ 0.2- ->2.

0.5 + 1 . 5 6 ^ - + 3.44-1.56-Tm3 ' Tm

3.10 SOSPD Model (repeated pole) with a Negative Zero Km(l + Tm3s)e-

Gm(s) : (l + sTml)

2


Gc(s) = Kc 1 + -T|S y 1 + Tfs

R(s) ^ E ( s ) - •

Table 28: PI controller tuning rules - SOSPD model with a negative zero

Km(l + T r a 3 s>-^ Gm00 =

(l + sTml)2

Rule


Model: Method 3

Kc T i Comment

Direct synthesis: time domain criteria 1 Tml

K m T m l + T m l-5Tml Tf =Tm3;OS<10%

3.11 Third Order System plus Time Delay Model

Gm(s) = Kr l + b,s + b2s2 +b3s3

l + a,s + a2s3 +a3s

Gm(s) =

'or

K „ e "

(l + sTmlXl + sTm2Xl + sTm3)

3.11.1 Ideal controller Gc(s) = Kc 1 + —

E(s)

R(s)

+ %-* ?_

Kc \ 0 1 + — I TisJ

U(s) Third order system

plus time delay model

Y(s)

Table 29: PI controller tuning rules - Third order system plus time delay model

Kme-"" „ , , „ l + b,s + b,s 2 +b-,s3 _,, _ , N

Gm(s) = Km ! 2~ ^ e st" or Gm(s) = l + a,s + a,s +a,s

(l + sTmlXl + sTm2Xl + sT m 3 ) -

Rule Kc T, Comment

Direct synthesis

Hougen (1979) -pages 350-351.

Model: Method 1

0.7

Km \Xm )

0.333

I K (H4)

T ( H 4 )

^ - > 0.04 ;

T m l - '•ml - T m 3

•^2- < 0.04 ; Tml

T m l > T m 2 > T m 3

1 K (144)

2Kn

0.7|i=i- + 0.8 Tmi + Tm2 + Tm3

vTmiTm2Tm3 r


Rule

Vrancic e/a/. (1996), Vrancic (1996)-


Vrancic et al. (2004a).

Model: Method 1

Vrancic et al. (2004b).

Kc

0.5A3 2

(A ,A2-K m A 3 )

0.5A3

( A , A 2 - K m A 3 )

0

4 R (146)

T;

A3

A2

3 x (145) i

A,A3

A[A2 - KmA3

T (146)

Comment

A m > 2 , 4 L > 6 0 0

(Vrancic et al. (1996))

Model: Method 1

2 A, = K r a ( a 1 - b 1 + x m ) , A 2 = K m ( b 2 - a 2 + A , a , - b , t m + 0 . 5 x m2 ) ,

A3 = Km(a3 - b 3 + A2a, - A,a2 +b2xm -0.5b,x ra2 +0.167xm

3).

A,A3(A,A2 - KmA3)) 3 T (145)

4 K (146)

(A,A2 - K m A 3 ) 2 +KmA 3 (A IA 2 - K m A 3 ) + 0.25Km2A3

2

A,A2 - A 3 K m -[sign(A,A2 - A3Km)]A,VA22 - A,A3

A , K m2 - 2 A , A , K m + A , 3

-(146) _ 2A, A, A2 - A3Km - [sign(A1 A2 - A3Km )]A, JA2

2 - A, A3

(A3Km2 - 2A,A2Km + A,3)(l + KC

(146)KJ2

12 8 Handbook of PI and PID Controller Tuning Rules


( 1 "1 U(s) = Kc 1 + — V T i s J

aK„

R(s) E(s)

K, v T i sy

U(s)

E(s)-aKcR(s)

Y(s)

Third order system plus time delay model

Table 30: PI controller tuning rules - Third order system plus time delay model

O . W K • * b - + - - : + b - : . - - . , o . w . l + a,s + a2s3 +a3s3 l + sTmlXl + sTm2Xl + sTm3) '

Rule


Model: Method 1

Kc Ti Comment

Minimum performance index: servo/regulator tuning 5 K (147) T (147)

i ^2-<1.0 T

Overshoot (servo step) < 20% ; Tra] = Tm2 = Tm3

5 K (147)

K „ 0.2713 + -

0.7399

- + 0.5009 , a = 0.4908-0.2648-^ + 0.05159

T

f \2

T V m J

T (147) = T 2.759-0.003899-^2-+ 0.1354 T

Rule

Vrancic(1996)-page 136-137.

Model: Method 1

Kc T; Comment

Direct synthesis

6 K (148) T (148) i

b = [0.5,0.8]-good servo and regulator

response (page 139); b = l-oc

6 j r (148)

V^¥ A , A 2 - K r o A 3 - X K m A , - A , A , < 0 ;

n2 A 3 + A 1

3 - 2 K m A 1 A 2 j •~mni n \ n l

K (148)

¥^¥

X = V(KmA3 - A,A2)2 - ( l -b2]A3(K r a2A3 + A,3 - 2 K m A , A 2 ) ;

A , A 2 - K m A 3 + x K m A 3 - A , A , > 0 . 2 A , + A , 3 - 2 K „ A , A , ) ' ~ ™ - 3 — ' " 2

.(148)

K m + -

m rt3-r^i.| z . x v m ^ , r v 2

A,

1 K <148>K J with

2K (148) &-„')

• K . ( b ' ) . A, = K m ( a i - b , + T m j , A2 =K m ^b 2 - a 2 +A 1 a 1 -b 1 T r a +0 .5T m

A3 =Km(a3 - b 3 + A2a, - A,a2 +b2xm - 0 . 5 b , x j +0.167xm3).

Kme-sx™ 3.12 Unstable FOLPD Model Gm(s)

T „ s - 1

3.12.1 Ideal controller Gc(s) = Kc

R(s)

1 + -TiSy

?) E ( S ) r i

Kc M I Ti sJ

U(s) K m e - -

T r a s -1

Y(s)

Table 31: PI controller tuning rules - unstable FOLPD model Gm (s) = -Kme-

Tms-1

Rule

Minimum ITSE -Majhi and Atherton

(2000). Model: Method 1 Minimum ITSE -

Majhi and Atherton (2000).

Model: Method 3

Kc

Minimum i

1 K (149)

2 K (150)

T, Comment

Derformaiice index: servo tuning

T (149) i

x ( I S O ) i

0<I™-< 0.693 T

0 < I s . < 0.693 ; Tm

K = - A p / K m h

1 v (149) _ _J_ K „

- ' A 0.889 +

- 0.064

v eT-/T™ - 0.990

2 K (150) J_ 0.889 + 1

K(I + K) T (150)

T(i49) _2.6316Tm(eT-/T- - 0 96e)

' (e- ' -^"-0.377)

TmK(l + K)

0.5(l-0.6382K)tanh_1 K '

Rule

De Paor and O'Malley (1989). Model: Method 1

Venkatashankar and Chidambaram (1994).

Model: Method 1 Chidambaram

(1995b). Model: Method 1

Chidambaram (1997).

Valentine and Chidambaram (1998).

Model: Method 1

Kc T; Comment

Direct synthesis

3 K (151)

4 K (152)

i f T ^ — 1 + 0.26-^-K „ A ^J

1.678, In

Km

2 .695 -1.277Tra/Tm

Km

T ( 1 5 1 ) i

25(Tm-xJ

25Tm-27xm

0.4015Tme5'8t'"/T"'

0.4015Tme58T"'/T"'

Ara = 2 ; ^ < l m

^S-<0.67 T

m

- ^ - < 0 . 6 T

m

Model: Method 1

5 ^ = 0.333;

^2-<0.7 T

K (151)

K „ cos

/ T m jT m \ t „

1--=- | sin T, T T

-(151) , <|> = t a n T V Tmy

4 ^ (152) = _

K m |

/ 0.98J1 +

0.04T^

( T m - x m ) 2

25 |p(Tm-xm)

1 + P2T2

l + P 2 ^ | ( T m - x J 2

B = 1.373, — < 0.25 ; B = 0.953, 0.25 < — < 0.67 .

5 +1% Ts = 7.5Tm , xm/Tm = 0.1; ±1% Ts = 8.75Tm, xm/Tm = 0.2 ;

±1% Ts = H.25Tm, xm/Tm =0 .3 ; ±1% Ts = 15.0Tm, xm/Tm =0.4;

±1% Ts= 17.5Tm , xm/Tm =0.5 ; ±1% Ts = 20.0Tm , xm/Tm =0.6.

Rule

Ho and Xu (1998). Model: Method 1

Luyben(1998). Model: Method I

Gaikward and Chidambaram (2000).

Model: Method I Chidambaram

(2000c). Model: Method 1

Kc

» p T m

AmK.m

Ti 6 T ( 1 5 3 )

i

Comment

^ < 0.62 T

m

Representative results 0.5775Tm

K m T m

0.6918Tm

K r a T m

x ] T m / K m xm

T T

0.3212Tm-xm

T T

0 3359T - T

-T

Good servo response. Am = 2.3 ,

* m =25°

Good regulator response. Am = 1.9 ,

<L=22.5°

Other representative results: coefficient values x i

1.0472

0.7854

Am

1.5

2

0.31KU

K 30°

45°

X l

0.5236

0.3927

2.2TU

Am

3

4

*m

60°

67.5°

TCL = 3.16xm

Maximum closed loop log modulus = 2 dB

7 K (154)

8 ^ (155)

x (154)

T (155)

0.1<^2-<0.6 T

'small' ^ -Tm

6 j (153)

1.57cop-cop xm - —

7 v (154) _ _J_ Km

K „ 0.2619 + 0.7266-^- ,(154) = T 0.193 + 4.19—2-4-8.214 T

r \1

T

8 K (155) -(155)

0.04TS242+Ti(155)tr

- T-(155) _ 0 .04T S

2 ^ 2+ T m t m + 0 .4T S ^

Rule

'Chandrashekar et al. (2002).

Model: Method 1

Coefficients of

Kc and Tj recorded

in tables; TCL2 as in

footnote

Sree et al. (2004). Model: Method 1

Rotstein and Lewin (1991).

Model: Method 1

Kc

49 .535 e-21.644xm/Tm

Km

/• \ -0.8288

0.8668 f x j

Km tTmJ

Ti

0.1523Tme7'9425T">^

Comment

i s - < 0.1 T

tn

0.1 < i s - < 0.7 T

Alternative tuning rules 6.0000/Km

3.2222/Km

2.2963/Km

1.8333/Km

1.5556/Km

1.3265/Kra

1.1875/Kn

0-8624 f x j

Km | j m J

-0.9744

0.6000Tm

1.4500Tm

2.6571Tm

4.4000Tm

7.0000Tm

14.6250Tm

31.0330Tm

10 T (156)

xm /Tm=0.1

Tm /Tm=0.2

xm /Tm=0.3

x m /T m =0.4

* m /T m =0.5

t m / T m = 0 . 6

Xm /Tm=0.7

0.01 -"- , (TCL2S + 0 2 '

0.025 + 1.75- T m , i 2 . < 0 . 1 ; T C L 2 = 2 x m , 0 . 1 < i 2 . < 0 . 5 ;

T C L 2=2x m +5T m is-_o.5 , 0.5<is-<0.7. T T

10 J (156) _ rj, 143.34 -73.912 + 19.039 - 2 - -0.2276 T

Rule

Rotstein and Lewin (1991) - continued

Tan et al. (1999c). Model: Method 1

Kc

Km uncertainty =

30%

11 K (157)

Ti

*m /Tm =0.2

*m /Tm =0.4

^m /Tm =0.6

T ( 1 5 7 )

Comment

^G[0.5Tm,4.5Tm]

^e[1.5T r a ,4.5TJ

^6[3.9Tm,4.1Tra]

^*-<0.5 T

K (157) , ( 1 5 7 )

K „ 0.1486-^ + 0.6564 T

12.7708—2- + 3.8019 T

0.1486-2!- +0.6564 T

135

E(s)-aKcR(s)

Table 32: PI controller tuning rules - unstable FOLPD model Gm (s) = Kme-

T m s -1

Rule

Chidambaram (2000c).

Model: Method 1

Jhunjhunwala and Chidambaram (2001).

Model: Method 1

Kc T, Comment

Direct synthesis 12 K (158)

13 2 . 6 9 5 -1.277tm/Tm

Kra

14 K (159)

f \ -0.898

0 . 8 7 6 ^

T (158) i

0.4015Tme5-8x™/T'"

T (159) _ i

6.461tm

0.324Tme T"

'small' xm/Tm

'Dominant pole placement' method;

E, = 0.333

^2_<0.2 T

m

0.2 < -is- < 0.6 T

m

12 j r (158) -(158)

O.O^s^HTi"58 '!,, • . T i

(.58) 0 .04T s ^ 2 +T m T m + 0.4T s 4 2

<x = l-0.2T s4

2(Tm-Tm)

0.04TS2^2+Tmxm+0.4TS^

a = 0.733+ 0.6^2--0.36 T

r \2

T V ra J

0.1 < - 2 - < 0.6. T

14 v (159) _ 1 K„ 10.7572-35.

K c " " ' K m ( 2 - T j + ( K c ' l " ' K m - l ) r i " ^

2K<159)K

Rule

15 Chandrashekar et al. (2002).

Model: Method I

Coefficients of Kc and Tf recorded

in tables; TCL2 as in

footnote


Model: Method 1; M, =1.4;

0 .3<a<0.5

Kc

4 9 . 5 3 5 -21.644Tm/Tm

e ra' m

Kra

0.8668 f O

Km [ T J

Kc

6.0000/Km

3.2222/Km

2.2963/Km

1.8333/Kra

1.5556/Km

1.3265/Km

1.1875/Km

0.28/Kml

0.2/KmT

0.28/KJ

0.13/KJ

0.28/KJ

0.28/KJ

-0.8288

Ti

0.1523Tme7'9425t"/T™

Comment

i s - < 0 . 1 T

m

0.1 < -^s- < 0.7 T

Alternative tuning rules

Ti

0.6000Tm

1.4500Tra

2.6571Tm

4.4000Tm

7.0000Tra

14.6250Tra

31.0330Tm

m

m

m

m

m

m

a

0.6667

0.7246

0.7742

0.8182

0.8571

0.8974

0.9323

8T m

7.8xm

l l ^ m

10tm

7tm 19tm

135tm

^ m /T m =0.1

^ m /T m =0.2

xm /Tm=0.3

i m / T m = 0 . 4

t m /T m =0 .5

^ m /T m =0.6

t m / T m = 0 . 7

•^2- - 0.02 T

m

i ^ - 0 . 0 5 T

i s - - o . l O Tm

^m /Tm=0.15

15 Desired closed loop transfer function = — — (TCLZS + I ) 2

TCL2 = 0.025 + 1.75- T m , ^ < 0 . 1 ; T C L 2 = 2 x m , 0 . 1 < ^ < 0 . 5 ;

T C L 2 = 2 t m + 5 x m | ^ - 0 . 5 , 0 . 5 < - ^ < 0 . 7 . T

a = 0.505 + 3 . 4 2 6 ^ -10.57 ^ T T

• < 0 . 2 :

a = 0.713+ 0.232-^-, 0 .2<i3-<0.7 T T

U(s) = ^ E ( s ) - K c Y ( s ) T:S

+ d r - E(S)

Kc

T i s

^6?V——fc.

- X u(s)

t

Kme-St™

T m s - 1

Y(s)

•

Table 33: PI controller tuning rules - unstable FOLPD model Gm (s) = Kme"

Tms-1

Rule

Minimum ISE -Paraskevopoulos et


Kc T; Comment

Minimum performance index

1 K (160) i . 9 i 6 , m +4 - ; ^ 2

•'rn

0.88--^-T

2.725xra + L ? ^ 2

0 . 8 8 - ^ T

m

0.05 < ^ - < 0.85; T

regulator tuning

0.05 < ^ s - < 0.85 ; T

servo tuning

£ (160) _ / ] £ (min)j^

with K (min)

K „

m m i n T i ) l + m m i n 2 T m 2 R (max) _ fi>maxTi ^ " m a x ^ m 2 ^

l + m m i „ T i K „ 1 + w m a x 2 T i 2

T T +T . ™n

K~

2xm

T rt

T i ^m

T i + ^ m T m j

T i ^

Rule

2 Paraskevopoulos et al. (2004).

Model: Method 1

f Minimise

iraskevopoulos al. (2004).

dt

et

Paraskevopoulos et al. (2004).

Model: Method I

Kc

Minimum |

1 K (160)

(page 137)

1 j , (160)

(page 137)

1 K (160)

(page 137)

X l

0.7269 1.7016 3.0696

x2

2.5362 3.0585 3.7509

Ts Comment

performance index: other tuning 3 T ( 1 6 0 )

i

3 r-p (161) i

2.58,m + 3 - 6 V

T

0 . 8 8 - ^ -m

Direct synthesis - j . (min) 4

i

. l-35xm

V m J

X l ^ m + X 2

2

VTmTm

Coefficient values

4 0.5 0.75

1

x i

6.975 12.452

0.01 < — < 0.3 T

m

0.3 < i s - < 0.85 T

0.03 < ^ - < 0.85; T

m

Model: Method 1

0.02 <^S-< 0.7 T

m

0 . 1 < ^ - < 0 . 9 T

i s - < 0.3 T

x2

5.7053 8.4358

S 1.5 2

: Minimise performance index = l[e2(t) + (u ( t ) -u„ ) 2 jit

3 j (160) _ ~ 0.747 + 3 . 6 6 - ^ + 18.64

T

f \ 3

T V m J

. T ( 1 6 1 ) _ _

1 (% \ 4.52-!2- +0.476 - ^

T T

0.88-

Ti(min) =Tm [1.3146-^- - 2 . 3 8 0 5 ^

( - \ + 57.622

T V m J

( . -158.99

T

+ 173.41 ' ^

T

Rule

Paraskevopoulos et al. (2004) -continued.

Paraskevopoulos et al. (2004).

Model: Method 1

Zhang and Xu (2002). Model: Method 1

Kc 1 K (160)

(page 137)

Ti

Xi t m +x 2 T m + 6 T -5

X 3 T m 1 m

Comment

0.2 < ^2- < 0.7 T

Coefficient values X l

4.59 6.16 8.21

x2

-0.19 0.028 0.37

X 3

8.28 9.90 16.88

1 K (160)

(page 137)

I 0.5

0.75 1

x i

12.41 19.91

x2

1.89 3.44

5 T (162)

X 3

51.95 95

4 1.5 2

^2L<0.2 T


0.409 0.717 1.142

x2

0.2817 0.6542 1.177

1 K (160)

(page 137)

TcL2 0.5 0.75

1

T

x i

2.353 4.07 8.88

x, +x 3

r \

T

3~

x2

2.67 4.76 10.73

TcL2 1.5 2 3

0.2 < ^ 2 . < 0.7 T

m


1.518 3.34 5.81

x2

10.54 16.89 29.17

TcL2 0.5 0.75

1

x i

13 23.08 51.94

x 2

49.66 82.87 176.8

*CL2

1.5 2 3

Robust

6 K (163) T (163) ^ < Tmi i for

^2 -<0 .5 , T

X. = 2 -5x m

• (162) = T „

T

6 v (163) ^ + 2XTm + T m T m (163) i V ; T-, > ^i

^+2XTm+TmTm

K ^ + T J

E ( s ) - K , Y ( s )

Table 34: PI controller tuning rules - unstable FOLPD model Gm(s) = Kme-Tms-1

Rule

Chidambaram (1997). Model: Method 1


Model: Method 1

Kc T, Comment

Robust T m - t m

8 K (165)

7 j (164) i

T (165)

X > 1 . 7 ; i s - < 1 T

m

7 j (164) _

8 K (165)

-p (165)

(T ' T - - 5 5 -

1 111

0.5

K / l 6 5 , K m - l

Km(X + Tm)

T m - K , ( I 6 5 ) K

n ^T v 1 .

1 v . - , . m , x M

L Kr>Km-i 2

T T

K , ( 1 6 5 ) K m - l

K (165) 1

K m

2(X

C ° S \

+ ' J '

T T V m J * m

/ „ \0 .5

x m2 1

2(^ + TJ_

1

'

T m , H i

3.13 Unstable FOLPD Model with a Positive Zero

Gm(s) =

141

Km(l-sTm3)e-

T m l S - 1

3.13.1 Ideal controller Gc (s) = K

E(s)

1 + -Ti^y

R(s) <g)— + T

f i >\

K, 1

V T i S V

1 + -U(s)

K r a ( l - sT m 3 ) e -

T m l s - 1

Y(s)

Table 35: PI controller tuning rules - unstable FOLPD model with a positive zero •

K m ( l - s T m 3 > - " -Gm (s) = ~ :

Rule



Kc Ti Comment


1 Y T ml

K r a T r a 3 T (166)

y = the negative of the value of the inverse jump of the closed loop system; y > —I! -Tmi

YTm -(166) _ i m 3

-iTm3(l-x,)-0.5Tm(l + x,)]

^ - ( l - x j + x , , X , >

YTrol

YTml-Tm3

G c ( s ) = Kc 1 + — V T>SV 1 + T fs

R(s) E(s)

+ T

U(s)

K, 1 + v T i s y

l

1 + T fs K m ( l - s T m 3 ) e -

T m , s - 1

Y(s)


Rule



Kc T; Comment


2 K (167) T (167) ISL. - JEHL. < o.62 Tmi Tm3

( l - s T )(l + T.(167)s)e~ 2 Desired closed loop transfer function = -—*—^-^——-1 EL, (I + T&SJ(I + T&S) '

K (167) .(167)

K r a lT( 'L>2 +T<2L)2 +Tm

, (167) )•

. (167) _ Tml jTCL2TCL2 ~ Tm3Tm + [TCL2 + TCL2 + Tm3 + Xm Fml j Tml2+T r a3Xm-(-C r a+Tm 3)Tm l

T _ T(167) 1m3Tmxi

T (T(l) + T ( 2 ) + T + T _ T ( 1 6 7 ) V iml^CLJ + XCL2+ 1 m 3 + T m li )

Tf =

U(s) = Kc

v T i s y

143

E(s)-aK cR(s)

aK„

R(s) + T - E(s) Kr

v T i sy

U(s)

v8» K m ( l - s T m 3 ) e -

T m l s - 1

Y(s)


K m ( l - sT m 3 ) r "» Gm(s) = -

Tm ls-1

Rule



Kc Ti Comment


3 K (168) ry (168) -^s—-^2-< 0.62 Tmi Tm3

(1_ST )(l + T.(168)sVst 3 Desired closed loop transfer function = -—*—m3?\ y '—,„; \

(l + T&sJ l + T&s)

K (168) .(168)

K ^ l (i) , T < 2 ) 'TCL2 + T C L 2 + T m 3 + T m ~ *i

. (168)

. (168)

T, =

T ( T ( D T ( 2 ) _ T T , | T ( l ) + T ( 2 ) + T + t ( r ) 1 m U 1 C L 2 1 C L 2 1 m 3 1 : m + I ^CI^ + XCL2 + i m 3 + xm\lm\]

f = :a

Tmi + T m 3 t m - ( x m + T m 3 J T m |

T T ( 1 6 8 )

a = 1-0.5 IT") + T ( 2 ) + T + T _ T ( 1 6 8 )

i 1 C L 2 + ^CL2 + im7,+Xm l\ )

T 4- T^1) 4- T ( 2 ) i m 3 + XCL2 + XCL2 ,(168)

V

Rule



Kc

4 yTm ,

KraTra3

T|

T (169)

Comment

4 y = the negative of the value of the inverse jump of the closed loop system;

^ - [ T m 3 ( l - x 1 ) - 0 . 5 T m ( l + x1)] > T m 3 ^, (169) _ T n l 3

_ T m 1 ' ' " ^ - ( l - x j + x , 1 m 3

yTm

yT m l -T m

ct = l -0 .5 1 m 3 "*" XCL2 "*" 1 CL2

T <169)

Desired closed loop transfer function = (l-sTn3)(l4.T,»»),)b-".

(Sree and Chidambaram (2003b)).


3.14 Unstable SOSPD Model (one unstable pole)

Gm(s) =

145

K e"

(Tmls-l)(l + sTm2)

3.14.1 Ideal controller Gc(s) = K(

R(s) . o . E(s)

1 + -^J

K e ~ (Tmls-lXl + Tm2s)

Y(s)

Table 38: PI controller tuning rules - unstable SOSPD model (one unstable pole)

Kme"s t" Gm(s) =

(Tmls-lXl + sTm2)

Rule





Kc Ti Comment

Minimum performance index: regulator tuning 1 K (170)

Tm/Tm

0.05 0.10 0.15 0.20 0.25

x i

0.9479 1.0799 1.2013 1.3485 1.4905

4Tm l(Tm+Tm 2) X l T m l - 4 ( T m + T m 2 )

x2

2.3546 2.4111 2.4646 2.5318 2.5992

tm/Tm 0.30 0.35 0.40 0.45 0.50

Coefficients of KC,T;

deduced from graph

x i

1.6163 1.7650 1.9139 2.0658 2.2080

x2

2.6612 2.7368 2.8161 2.9004 2.9826

0.05 0.10 0.15 0.20 0.25

1.1075 1.2013 1.3132 1.4384 1.5698

2.4230 2.4646 2.5154 2.5742 2.6381

0.25 0.35 0.40 0.45 0.50

1.6943 1.8161 1.9658 2.1022 2.2379

2.7007 2.7637 2.8445 2.9210 3.0003

X 1 ! „ .

K m ( x l T r r

4(Tm+Tm2

-4 [xm+T^

Rule

Pouline/a/. (1996). Model: Method 1

McMillan (1984). Model: Method 1

Kc

<°Tml

3.53

K r a

3 K (172)

T;

Direct synthesis 2 T ( 1 7 1 )

i

Tml

Ult imate cycle T(172)

Comment

minimum i?m = 25

T m 2 + t m <0.16TmI;

<t>me[25°,90°]


KU>TU

2 x( '71) 4Tm l (Tm+T r o 2 )

1 .3T m l -4 (T m + T m 2 )

where phase of Gm(jco)Gc(j(n) is maximum

. 0.16Tm, < Tm2 + xm < 0.3Tml ; co = frequency

3 £ (172) 72) 1.477 T^T K m Xm

2

1 + 0.65

l ^ ro l + ^m2/^ml ^m2

(Tml _ Tm2 XTral _ Tm i1!! _

Umi + T m 2 j T m l T m 2 T:(172) = 3.33xj l + lTml T m 2 j ( T m l - T m JT r

3.15 Unstable SOSPD Model with a Positive Zero

Gm(s) _ Km( l-Tm 3s>-

2 „ 2 T m l V - 2 2 ; m T m l s + l

3.15.1 Ideal controller G c ( s ) = K

R(s)

1+ —

+ 3 E ( s ) > f- Kc

U(s) Km( l -Tm 3s)e-^

Tml2s2-2^mTmls + l

Y(s)

— •

Table 39: PI controller tuning rules - unstable SOSPD model with a positive zero

K m ( l -T m 3 s ) e -^ G n ( s ) =

Tm l2s 2 -2^mTm ls + l

Rule

Sree and Chidambaram (2004).

Model: Method 1

Kc Ti Comment


4 R (173) T (173)

( l - sT yi + T('73)s)e"STm

4 Desired closed loop transfer function = -f —^ 7-7-17 r . (l + s T ^ l + s T ' ^ l + sx,)

(173)

J (173) _ Tml \TCL + TCL + X l + Tm3 + Tm j ~ TCLTCL X l ^ ^

I i r ^ m ~~ ^ml

x T m , (T^+Ta ) +T m 3 +T m ) ( -2^ m T m 3T m +T m l (T 3 + T m ) ) - x 2

X ' (- 2^raTml + Tra3 + xm Jr^Tg, ' - Tml2 + (Tra3xm - Tm l

2)(Tg + T<2> + 2^mTm ]) '

and with x2 = (Tm3xm - T ^ ) ( T # T & > -TmTm 3 ) ; 2^m

V Tmi < 0.62 .

1m3 /

U(s) = Kt

a K v T ; s y

E(s)-aK cR(s)

+ R(s)

®— E(s)

K, v T i s y

• < g ^ U(s)

K m ( l - s T m 3 ) e - " -

T m l2 s 2 - 2 ^ m T m l s + l

Y(s)

Table 40: PI controller tuning rules — unstable SOSPD model with a positive zero

K m ( l - T m 3 s ) r ^ Gm(s) =

T m ,V-25 r a T m I s + l

Rule

Sree and Chidambaram (2004).

Model: Method 1

Kc T; Comment


5 K (174) T (174) i

1 Desired closed loop transfer function = (l + s T ^ l + s T ^ l + sx , ) '

K (174) _

-(174)

K ^ | T S + T g . ) + x 1 + T i n 3 + T i n - T i (174)

'ml T m 3 y

< 0.62 ;

T 2 ( T ( 1 ) +T < 2 ) + X 4-T + T 1 - T ( I ) T ( 2 ) I •p (174) _ ' m l VXCL + JCL + X 1 + ^ 3 +Xm) XCL XCL X l ^

*m3 X m — ' m l

., T m , ( T ^ + T g + T m 3 + T m ) ( - 2 ^ T m 3 T m + T m l ( T + T J ) - X 2

(- 2^mTml + Tm3 + xm )r^>Tg.> - Tml2 + (Tm3xm - Tml

2 )(T<'L> + T<2L> + 2^mTm l) '

a n d w i t h x 2 4 m 3 T m - T m , 2 f e ^ ^ ^


3.16 Delay Model Gm(s) = Kme

3.16.1 Ideal controller Gc(s) = K

R(s)

'i+-L^ v T i s y

Table 41: PI controller tuning rules - G m (s) = K me

Rule


Model: Method 1



Haalman(1965)-minimum ISE.

Model: Method 1 Gerry (1998)-

minimum error: step load change.

Gerry and Hansen (1987) - minimum

IAE.

Kc Ti Comment

Process reaction '0.568/KmTm

20.65/Kraxm

30.79/KmTm

40.95/Kmtm

0.2 K m T m

3.64xm

2.6xm

3.95im

3.3xm

0.3im

Representative results deduced from graphs

Decay ratio is as small as possible

Minimum error integral (regulator mode)

Minimum performance index: regulator tuning 0 l-5xm

M , = 1 . 9 ; A m = 2 . 3 6 ; i)m = 50°

0.3

Km

0.345

Km

0.42tm

0.455im

Model: Method I

Model: Method 1

Decay ratio = 0.015; Period of decaying oscillation = 10.5xn

2 Decay ratio = 0.1; Period of decaying oscillation = 8xm

3 Decay ratio = 0.12; Period of decaying oscillation = 6.28xm

4 Decay ratio = 0.33; Period of decaying oscillation = 5.56im

Rule

Shinskey (1988)-minimum IAE - page

123. Shinskey (1994)-

minimum IAE. Model: Method 1 Shinskey (1988)-

minimum IAE -page 148.

Shinskey (1996)-minimum IAE -page

167.

Minimum IAE -Huang and Jeng

(2002). Model: Method 1 Minimum ISE -

Keviczky and Csaki (1973)

Model: Method 1

Fertik(1975). Model: Method I

Astrom and Hagglund (2000).

Model: Method 1

Skogestad (2001). Model: Method 1 Skogestad (2003). Model: Method 1 Skogestad (2001). Model: Method 1


Model: Method 1

Kc

0.43

Km

0-4/Km

0

0.4255

Km

0.4KU

^

0.5xra

0.5xm

1.6Kmxm

0.25TU

0.25TU

Comment

Model: Method 1

page 67

page 63

Model: Method 1

Model: Method 1

Minimum performance index: servo tuning

0.36

Km

0

0.5

Minimum i 0.35/Km

0.25

Km

0.16/Km

0.200/Km

0.26/Km

x,/Km

0.47xm

1.33xm

0.635xm

Minimum IAE = 1.377xm. A r a = 2 ;

<L=60° .

Minimum ISE = 1.53xm; K r a = l

K m = l

jeri'ormance index: other tuning 0-4xm

0.35xm

0.340xra

0.318xm

0.306xm

x2 tm

Kc,Tj values

deduced from graph Maximise

Kc/T i subject to

M m a x =2

Mm a x=1.4

Mm a x=1.6

Mm a x=2.0

Coefficient values

* i

0.057 0.103

0.139

0.168

0.191

x2

0.400 0.389

0.376

0.363

0.352

Mmax

1.1 1.2

1.3

1.4

1.5

*1

0.211 0.227

0.241

0.254

0.264

x2

0.342 0.334

0.326

0.320 0.314

M max

1.6 1.7

1.8

1.9 2.0


Rule

Van der Grinten (1963).

Model: Method 1

Buckley (1964). Model: Method 1


Keviczky and Csaki (1973).

Model: Method 1; Representative Kc,Ti values

estimated from graphs; Km = 1

Kamimura et al. (1994).

Model: Method 1

Hansen (2000). Model: Method 1

Bequette (2003) -page 300.


Skogestad (2003).

Kc Ti Comment

Direct synthesis 0-5/Km

5 K (175)

0

0.075/Km

0.53/Km

0.55/Km

0

0

0

0.5xm

T (175) i

l-45Kmxm

0-Hm

T m

1.59im

2-70Kmxm

2-50Kmim

X2^m

Step disturbance

Stochastic disturbance Mm a x=2dB

Representative results;

Mm a x=2dB


Damping factor (dominant pole) = 0.6

Coefficient values x2

0.64

OS

105%

x2

1.00 x i

OS

50%

x2

1.5

OS

17%

2xm

x2

2

OS

4%

Coefficient values x>

0.68

OS

5%

x i

0.72 0

0

0

0-2/Km

OS

10%

x! 0.75

OS

15% 2Km tm

2.6667Kmtm

2.3529Kmxm

0.3tm

x i

0.78

OS

20% OS = 10%

OS = 0%

"Quick" servo response; "negligible"

overshoot

Robust

V Km(2^ + xm)

0.294/Km

0.5xm

0.5xm

Model: Method 1

IMC based rule

Other tuning 0.25/Km 0.333xm Model: Method 1

(•75) = e - m ^ A p s e - ^ J T (175) = _ L s _ (t _ 0.5e-»^»).

3.17 General Model with a Repeated Pole

/ 3.17.1 Ideal controller Gc (s) = Kc

1

R(s) <S>

E(s)

+ T K,

f 1 ^ 1 + -

T;s

1 +

U(s) Kme"

d + sTm)"

Y(s)

Table 42: PI controller tuning rules - general model with a repeated pole

K m e - n -Gm(s) =

d + sTm)n

Rule


Kc T i Comment

Direct synthesis: frequency domain criteria 0.5

Km[r

0.375

Km

. 0 5 - l J "n + l"

_ n - l .

x +T m ar

(n + l)(Tm+Tar)

2n

'Normal' design

'Sharp' design

3.18 General Model with Integrator

n ( Tm , i s + i ) n ( T

m , 2 s 2 + 2 ^ i Tm 2 i S + i )

Gm(s) = K,

s n(Tm3i s+ i)n(T

m/ s2+2^diTm4is+i)e

3.18.1 Ideal controller Gc (s) = K

E(s)

1 + -T i s y

Table 43: PI controller tuning rules - general model with integrator

n(T^ s + in(T- / s 2 + 2^ i T^ s + i) Gm(s) =

K „

s n (T->'S+^n t v + 2 ^ « . . i T m , i s + ! ) Rule

Loron(1997). Model: Method 1

Kc Ti Comment


K c ( 1 7 7 ,

a J Q

Symmetrical optimum method Non-symmetrical optimum method

6 K (176)

KmVo7TQ

l + sin<t

cos<|>n

T Q = ^ T m j i + 2 ^ ^ m d i T m 4 i - ^ T m | i + 2 ^ ^ m n i T , + T „

K (177) X =

KmV a2TQ

M „

M / - 1 , Mc = desired closed loop resonant peak,

1 + SlnA^l .A^cos^A).

Chapter 4

Tuning Rules for PID Controllers

4.1 FOLPD Model Gm(s) = K e~

l + sT„

/ 4.1.1 Ideal controller Gc(s) = Ki

1 \

V T i s J

R(s) ,o E ( s )

Table 44: PID controller tuning rules - FOLPD model G m (s) = Kme~

l + sT„.

Rule


Model: Method 1 Ziegler and

Nichols (1942). Model: Method 2

Kc Ti Td Comment

Process reaction

i 1.066 K m T m

aTm

KmXm

a s [1.2,2]

1.418xm

2 T m

0.353xra or

0.47xra

0.5xm

^ - = 0.3 T

Quarter decay ratio

Decay ratio = 0.043; Period of decaying oscillation = 6.28xn

154

Chapter 4: Tuning Rules for PID Controllers 155

Rule

Chien et al. (1952)-regulator.

Model: Method 2

Chien et al. (1952)-servo.

Model: Method 2

Cohen and Coon (1953).

Model: Method 2

Three constraints method - Murrill

(1967) -page 356.

Model: Method 5

Astrom and Hagglund(1988) -pages 120-126.

Parr (1989)-page 194.

Borresen and Grindal (1990).

Kc

0.95Tm

K m t m

l-2Tm

Kmxm

0.6Tm

K m x m

0.95Tm

1 K (178)

1.370

Km —

l T nJ

Ti

2.38xm

2xra

Tm

1.36Tm

T ( 1 7 8 ) i

T

1.351

r x 0.738

x m J

Td

0.42xm

0.42xm

0.5xra

0.47xm

0.37tm

1 + 0 . 1 9 ^ -T

m

2 T (179)

Comment

0% overshoot;

0.1<^2-<1 T

m

20% overshoot;

0 . 1 < ^ - < 1 T

m

0% overshoot;

0 . 1 < ^ - < 1 T

20% overshoot;

0 .1<-^L<1 T

Quarter decay ratio.

0 < ^ - < l Tm

KcKraTd = Q

T

0 .1<^-<1 T

m Quarter decay ratio; minimum integral error (servo mode). 3

Km

^90% 0.5Tm Model: Method 23

T90% = 90% closed loop step response time (Leeds and Northrup

Electromax V controller). 1.25Tm

K m t m

Tm

2.5xn

3^m

0.4tm

0.5tm

Model: Method 2

Model: Method 2

K (178)

K „ 1.35—!2- + 0.25

,(178)

2 . 5 - ^ + 0.46 T

rx ^

T \ m /

1 + 0.61-

2 ^ ( 1 7 9 ) = 0 3 6 5 T I I


Rule

Sain and Ozgen (1992).

Conne l l (1996) -page 214.

Model: Method 2

Hay ( 1 9 9 8 ) -pages 197.198.

Model: Method 1

Coefficients of

K c ,T j ,T d

deduced from graphs

Moros (1999). Model: Method 1

Liptak(2001).

ControlSoft Inc.

(2005).

Kc

3 K (180)

l-6Tm

K m T n i

x , / K m

T

0.125

0.167

0.25

0.5

x i

6.5

5.0

3.5

1.8

IlIL T

0.125

0.167

0.25

0.5

x i

9.8

7.2

4.5

2.2

l-2Tm

K m x m

1.2Tm

Kmx r a

0.85Tra

K m x m

2

K m

Ti

T (180) i

1.6667xm

x 2 T m

Td

T (180) *d

0-4Tm

X 3 t m

Coefficient values - servo tuning

x 2

9.0

6.5

4.2

2.2

x 3

0.47

0.45

0.40

0.35

Coefficient values

x2

2.0

1.8

1.6

1.4

x 3

0.52

0.50

0.45

0.35

2 t m

2^m

l-6xm

^ m + T m

T

0.125

0.167

0.25

0.5

x l

5.0

3.7

2.5

1.1

- regulator tuning

T

0.1

0.125

0.167

0.25

0.5

x i

10.0

8.0

6.0

4.0

2.0

0.42Tm

0.44xm

0.6xm

4 T (181) : d

Comment

Model: Method 24

Approximate quarter decay

ratio

x 2

9.8

7.0

4.8

2.5

x 3

0.34

0.32

0.30

0.27

x 2

2.2

2.2

2

1.8

1.4

x 3

0.35

0.35

0.34

0.32

0.28 attributed to

Oppelt

attributed to Rosenberg

Model: Method 2

Model: Method 10

K „ 0.6939-=!-+ 0.1814 | , T,

(180) 0.8647Tm +0 .226T„

^ + 0.8647

. (180) 0.0565T^

4 -T, (181)

Ty ' =max

Tm Tm

0.8647^-+ 0.226

(slow loop); Td =min tm Tm (fast loop).

Rule

Minimum IAE -Murrill(1967)-pages 358-363.

Minimum IAE -Marlin(1995)-pages 301-307.

Model: Method 8

Minimum IAE -Peng and Wu

(2000). Model:

Method 17

Minimum IAE -Pessen (1994).

Model: Method 1 Modified

minimum IAE -Cheng and Hung

(1985).

Kc Ti Td Comment


1.435

Km —]

l-3/Km

l-0/Km

0.85/Km

0.65/Km

0.45/Km

0-4/Km

T i ^ 0 • 7 4 9

0.878 [xm J

0.7xm

0.68xm

0.65xra

0.65xm

0.55xm

0.50tm

5 T (182) x d

0.01xm

0.05xm

0.08xm

0.10xm

0.14xm

0.21xm

Model: Method 5;

0 . 1 < ^ - < 1 Tm

xm /Xm=0.43

xm/Xm=0.67

^ / T m = 1 . 0

T m / T m = 1 . 5

xm/X r a=2.33

*m/Tm=4.0

Coefficients of K c , T ; , Xd estimated from graphs; robust to ±25%

process model parameter changes 6.558Kra

2-31 lKm

1.479Km

1.117Km

0.947Km

0.825Km

0.731Km

0.695Kra

0.630Km

0.651Km

0.7KU

3 K m

'T ^ m

X

0.921

0.925Xm

0.677Xm

0.633Xm

0.647Xm

0.705Xm

0.752Xm

0.792Xm

0.871Xm

0.895Xm

1.030Xm

0.4XU

T ( A0'749

0-878 \Tm )

0.061Xm

0.189Xm

0.233Xm

0.250Xm

0.268Xm

0.289Xm

0.312Xra

0.316Xm

0.326Xm

0.306Xm

0.149XU

6 T (183)

xm/X r a=0.1

i m /X m =0.2

xm /Xm=0.3

xm /Xm=0.4

t m /X m =0.5

xm /Tm=0.6

xm /Xm=0.7

x r a/Xm=0.8

V / T r a = 0 . 9

*m/Tm=1.0

0.1<^2-<1.0 X

m

Model: Method 12

5 T (182) f N1137

T

6 T d d S 3 ) = 0 4 g 2 T n

X V m

Rule

Minimum ISE -Murr i l l (1967)-pages 358-363.

Minimum ISE -Zhuang and

Atherton(1993). Model: Method 1

Minimum ISE -Ho et al. (1998). Model: Method 1

Minimum ITAE -Murr i l l (1967) -pages 358-363.

Minimum ISTSE - Zhuang and

Atherton(1993). Model: Method I

Kc

1.495 f T j

Kra l x m J

1.473

K m

1.524

K m

0.945

, N 0.970

VTnJ

(T ^ m

0.735

8 K (186)

1.357 ( T J Km {zmj

1.468 f T j K m \xm)

1 .515 fT j

Km W)

0.947

0.970

0.730

T ,

T x m

1.101

T m

1.115

T Am

1.130

r \

T

0.771

( V' 7 5 3

XJ , N 0.641

T ( l 8 6 ) i j

T Am

0.842

/• \ 0.738

V 1 m V

Tm O 0-942 ( j m J

T ( A0598

Tm O 0.957 Tm J

Td

0.56Tm

, x 1.006

r 7 T (184)

i d

7 T (185) i d

T (186) i d

9 T (187) Xd

10 T (188) i d

10 rp (189) i d

Comment

0 . 1 < ^ s - < l ; T

Model: Method 5

0 . 1 < ^ - < 1 . 0 T

1.1 < ^ 2 - < 2.0 Tm

A ra 6 [2,5],

<L40 0 , 60° ] ,

0.1 < ^ 2 - < 1.0 T Am

Model: Method 5

0 . 1 < ^ s - < l T

m

0.1 < - ^ < 1.0 T

m

l . l < ^ 2 - < 2 . 0 T

7 ^ ( 1 8 4 ) = 0 _ 5 5 0 T n

/ \ 0.948

T V 1m J

, T d( 1 8 5 ) =0.552T n

T

8 K (186) _ 1-0722 <|>n

Km A . T T-

.86) l-2497Tm^n

0.2099 T

(186) _ 0.4763Tmf -0.328

0.0961 T A m V *m J 0.2035 / / T , \0.3693

Given A r a , ISE is minimised when <t>m = 46.5489Am°'2035(Tm/Tn ,)°

(Hoe / al. (1999)); 2 < A m < 5 ; 0.1 < t m / T m < 1.0 .

9 T (187) = 0.381Tm | ^ s _

10 T (188) = Q . 4 4 3 T

T V i m

, Td(189) = 0.444T;

T

Rule


Atherton(1993).

Minimum ISTES - Zhuang and


Minimum error -step load change -Gerry (1998).

Nearly minimum IAE, ISE, ITAE -Hwang (1995).

Kc

11 K (190)

12 K (191)

1.531

1.592

Km

VXm J

(T ^

0.960

0.705

0.3

16 ^ (194)

Ti

•p (190)

T ( 1 9 1 ) i

Tm

0.971

T

0.957

, x 0.746

, N 0.597

0.5xm

T (194)

Td

0.144TU

0.15TU

15 T (192)

15 T (193) x d

T

0.471KU

K m © u

Comment

0.1 <^2-< 2.0; T

m

Model: Method 1

0.1 < -^s- < 2.0; T

m

Model: Method 31

0.1<i3-<1.0 T

m

1.1 < ^ s - < 2.0 T

m

T i s - > 5 ; T

Model: Method 1 6, < 2.4

Model: Method 35; Decay ratio = 0.15; 0.1 < i s - < 2.0 T

„ K (190) 4 .434KmKu- 0.966 (I90)

5.12KmKu+1.734

12 (191) 6 .068KmK u -4 .273* „ ( , 9 „ 1.1622Km K „ - 0.748 « J v c — K u , l j — ~ l u .

5.758KmKu-1.058

15T<192)=0.413T, r A 0.933

2.516KmKu-0.505

N 0.850

T , Td

(193)=0.414Tn T

V 1 m J

16 K (194) ' 0.674JL - 0.447coHltm + 0.0607fflHI2Tm

2 f Km/(l + KH 1Km)

,(194) . K.r>(l + K H I K m ) coH1Km0.0607(l + 1.05(DHItm-0.233coH,2Tm

2)'

Rule

Nearly minimum IAE, ISE, ITAE -Hwang (1995)

- continued.

Nearly minimum IAE, ITAE -

Hwang and Fang (1995). Model:

Method 25

Kc

17 K (195)

18 K (196)

19 K (197)

20 j ^ (198)

T i

T (195) i

T ( 1 9 6 ) i

T (197)

T ( 1 9 8 ) 1 i

Td

T (198) ' d

Comment

2.4 <e, <3

3<e , <20

e, >20

0 . 1 < - ^ - < 2 . 0 ; T

decay ratio = 0.12

, 7 K < 1 9 5 ) = k , 0.778[l-0.467(oH,Tm + 0.0609coH/x 2T 2 h HI ^m

Km/(l + KH1Km)

18 ^ (196)

j(195) Kc (l + K H 1 K m )

»HIKm0.0309(l + 2.84WH1xm - 0.532o>H12Tm

2)"

1.3l(0.519)m"|T" [1-1.Q3/S) +0.514/e l2f

Km /(l + KH ,Km) J'

KC"96)(1 + KH1K )

)H1Km0.0603(l + 0.9291n[coH1xm])(l + 2.01/s1-1.2/8 l2) '

(196)

19 K (197) K t

izi 1.14|l-0.482coH,xm +0.068(DH,2Tm2]N

Km/(l + KH1Km)

• (197) Kc(""(l + K H 1 K j

)H,Km0.0694(-l + 2.1coH,Tm-0.367WHI2Tm

2)'

20 K (198) 0.802 -0.154 — + 0.0460 T

' x ^

i T , V m J

K „

,(198) K (198)

K „ 0 ) u 0.190 + 0.0532-52- - 0.00509 T

f ^2

T V 1m

• (198) . 0.421 + 0.00915-21--0.00152P!! T I T

Rule

O'Connor and Denn(1972).

Model: Method 1; K m = T m ;

^ 2 - < 0 . 2 T

m

Syrcos and Kookos (2005).

Model: Method 1

Minimum IE -Devanathan

(1991). Model:

Method 1; Coefficients of

K c , T; provided

are deduced from graphs

Kc

2i axm2 +

(2 + x , )

Ti QO

Minimum 0.5

0

2x,

22 K (199)

Minimum

Td Comment

; ' " K ^ axm

2 + 2 X l

2Tma

T (199) 1 i

x l x m

xm2a + 2x,

T (199) Xd

performance index - constraints are input and output signals

23 K (200)

Tm/Tm

0.2 0.4

0.6

0.8

1.0

(

x 2

0.31 0.29

0.29

0.29

0.28

132TU

* 2 T u

0.08TU

x3T„

Coefficient values

x 3

0.08 0.07 0.07

0.07

0.07

T m / T m

1.2 1.4

1.6

1.8

Recommended

a e 0.6 1

2 ' 2

2 < I i 2 L < 5 T

placed on the process

x 2

0.26 0.26 0.24

0.24

x 3

0.07

0.07 0.06

0.06

2 + -

22 K (199) = _

K „ 0.31 + 0.6- x (199) _ T

^COS

23 K (200)

tan 1.57D„ 0.16

D R J

K cos tan 6.28Tm

0.777 + 0.45-2!-T

(199) (~ 0.44 - 0.56

Rule


(1968). Model:

Method 1

Minimum IAE -Rovira et al.

(1969).

Minimum IAE -Mar l in (1995) -pages 301-307.

Model: Method 1

Minimum IAE -Sadeghi and Tych (2003).

Wang et al. (1995a).

Model: Method 1

0.05 < ^ - < 6 "T

m

Kc Ti Td Comment


x , / K m x2(Tm + O x3(Tm + 0


tm/Tm

0.053

0.11 0.25

* 1

12.3

6.8 3.4

/ \ 0.869

1 0 8 6 | T 1

Km m

l-3/Km

l-0/Km

0.8/Km

0.7/Km

0.55/Km

0.48/Km

0-4/Km

Coefficients of

25KC<202)

26 K (203)

26 K (204)

x2

0.95

0.93 0.88

x3

0.0215 0.041 0.083

T

0 .740-0.13—

0.93xm

0.80xm

0.7hm

0.73xra

0.64xm

0.56xm

0.52xm

Cc ,Tj ,Td estima

process model pa

T (202) i

Tm + 0.5xm

•tm/Tn,

0.67 1.50 4.0

* i

1.45 0.85 0.57

24 T (201)

0.02xm

0.05xm

0.06xm

0.09xm

0.12xm

0.15xm

0.21xm

x2

0.75 0.63 0.53

X 3

0.155 0.21

0.19

Model: Method 5

0.1< — < 1

xm /Tm=0.25

xm /Tm=0.43

xm /Tm=0.67

xm /Tm=1.0

t m /T m =1 .5

xm /Tm=2.33

t m / T m = 4 . 0

ted from graphs; robust to ±25%

rameter changes.

1.45933xm

^ - + 3.27873 Tm

0.5Tmxm

Tm + 3.5xm

Model: Method 1

0 . 1 < ^ - < 1 . 0 T

Minimum IAE

Minimum ISE

24 T (201)

25 v (202)

= 0.348T„

f -,0.914

T

K

26 K (203)

K „ 0.26266 + 0.82714- T (202) T

0.7645 + 0.6032

* m / T m

0.28743-S L +1.35955 .

(T m +0.5x m )

Km(Tm + xm) >KC

(204)

0.9155 +

T

0.7524 (Tm+0.5xra)

K ra(Tm + xm)

Rule


Atherton (1993). Model: Method 1

Minimum ISE -Hoe/al. (1998). Model: Method 1

Minimum ISE -Sadeghi and Tych (2003).

Minimum ITAE - Rovira et al.

(1969).

Kc

1.048

1.154

fry } m

X

0.897

f \ 0.567

-

29 j r (207)

3oK c( 2 0 8»

0.965 i- \ 0.85

, X m J

T,

27 T (205)

28 j (206)

T (207) i

T (208)

31 T (209)

Td

T (205)

T (206) *d

T (207)

1.93576xm

^ + 3.83528 T

T (209) x d

Comment

0 .1<^_<1 .0 T

1.1 < ^s_ < 2.0

Ame[2J5],

0 . 1 < ^ - < 1 . 0 . T

0 . 1 < ^ - < 1 . 0 ; T

Model: Method 1

0 . 1 n

Km Am° ,T, (207) 0.4899Tm<t>m

0.1457 ( \ 1.0264

A „ T

T(207) 0.021 lTm[l + 0.3289Am +6.4572^)m +25.1914(xm/Tm)],

' l + 0.0625Am-0.8079(),m+0.347(xm/Tm)

Given Am , with 2 < Am < 5 and 0.1 < xm/Tm < 1.0 , ISE is minimised when

^ ^ 2 9 . 7 9 8 5 + ^ l ^ +4 0 - 3 1 8 2 T - - ^ g ^ - ( H O e / a / . ( 1 9 9 9 ) ) .

K „ 0.40455 + 0.96441- ry (208) _ ry 0.43770-2-+ 1.39588

T

31 j (209)

0.796-0.1465-

Td<209) = 0.308Tn

r N 0.929

X„

T V my

Rule

Minimum ITAE - Wang et al.

(1995a). Model: Method 1 Minimum ITAE - Sadeghi and Tych (2003).

Modified minimum ITAE

- Cheng and Hung (1985).

Modified minimum ITAE - Smith (2003).


Atherton (1993). Model: Method 1

Kc

32 K (210)

3 3 K C < 2 > "

11

0.965

1.042

T 1

(T N

m T

3.855

0.855

/ x 0.897

—1 i.i42 rTm

N 0.579

T,

Tm+0.5xm

T ( 2 1 1 ) i

3 4 T ( 2 1 2 ) i

1.26Tm

35 j (213)

36 q-. (214)

Td

0-5TmTm

Tm+0.5xm

1.55237Tm

^ - + 4.00000 Tm

T (212)

0.308xm

T (213) 1 d

T (214) 1 d

Comment

0.05 < ^ - < 6 T

m

0 . 1 < - ^ - < l ; T

m

Model: Method 1 £, = 0.707;

Model: Method 12

Model: Method 1

0.1 < ^2-< 1.0 T

m

1.1 < i=-< 2.0 T

32 K (210)

0.7303-0.5307

t m /T m

(Tm+0.5xm)

33 K (2") = _ 1 K „

34 ~ (212)

K m (T m +x m )

0.20360 + 0.80451- T,(211) =T I 0 . 2 9 3 4 9 ^ + 1.29110

0.796-0.147-T <212) = 0.308Tn

, \ 0.929

T

35 j (213)

0.987-0.238-TH

(213) = 0.385T„ / \ 0.906

, T V. m

36 j ( 2 1 4 ) , (214)

0.919-0.172-, Td

l i" ' = 0.384Tn

/ -,0.839

T ,

Rule


Atherton(1993).


Atherton(1993). Model: Method I


Model: Method 35

Kc

0.509KU

0.604 Ku

/• \ 0.904

0.968 Tm)

1.061

Km

/ „ \ 0.583

—l 40 ^ (219)

41 K (220)

T,

37 T (215)

37 T ( 2 1 6 )

38 T (217)

39 T (218)

T (219) i

T (220) i

Td

0.125TU

0.130TU

T (217)

T (218) 1 d

0.471KU

K m » u

Comment

0 .1<^S-<2 .0 ; T

Model: Method 1

0 . 1 < ^ - < 2 . 0 ; T

Model: Method 31

0.1 <-^s_< 1.0 T

l . l < i s - < 2 . 0 T

m

8, <2.4

2 .4<6 ,<3

37 Tj(2i5) = 0 .05l(3.302KmKu+l)Tu , T^2'6' =0.04| 4.972Kra Ku + 1 |T„.

T f \ 38 T ( 2 1 7 ) _

0.977-0.253-, T d

u l " =0.316Tn

0.892

39 T (218) _ • T

(218) :0.315T„

T

, N 0.832

0.892-0.165-T

V m J

40 K (219) 0.822[l-0.549coH1Tm+0.112a)H12Tm

2]N

Km/(l + KH 1Km)

,(219) K,<"»>(1 + KH 1KB)

2JT+

41 K (220) 2 l\\

coH1Km0.0142^ + 6.96o)HITra-1.77coH1 T„

( 0.786[l-0.441fflH1Tm +Q.Q569coH1

KH1

Kc'22°>(l + KH1Km)

K J ( I + K „ , K J

T (220) _

J

coH1Km0.0172(l + 4.62coH1xm-0.823a)H,2T„21 u H l l n T

Rule

Nearly minimum IAE, ISE, ITAE -Hwang (1995)

- continued

Nearly minimum IAE, ITAE -

Hwang and Fang (1995).

Kc

42 K (221)

43 j , (222)

Ti

T (221)

T (222) i

Td Comment

3 < e, < 20

E, >20

Decay ratio = 0.1; 0.1 < — < 2.0 m

44 K (223) T (223) T (223) Model: Method 25

0.1 < ^ - < 2.0 ; decay ratio = 0.03

42 K (221) 1 .28(0 .542) ' ° H ' X ' " [ I -0 .986 /E 1 +0.558 /S 12 ]

H' Km/(l + KH 1Km)

43 K (222)

44 K (223)

T (221) K c ( 1 + KH|Km)

raH1Km0.0476(l + 0.9961n[coHITm])(l + 2.13/s1 -1 .13/e , 2 ) '

' K 1.14[l-0.466mH1Tm+0.0647coH12Tm

2lN

Km/(l + KH 1Km)

.(222) Kc (l + KH 1Km)

' coH1Kra0.0609(-l + 1.97coH1Tm-0.323coH12xm

2)'

K „ , 0.537 - 0.0165^2- + 0.00173 -^ T T

T, (223) K (223)

K u » u 0.0503 + 0.163—m-- 0.0389 T

f \ 2

T V 1 m J

• (221) _ 0.350 - 0.0344-^- + 0.00644 - ^ T T

Rule

Simultaneous Servo/regulator tuning - nearly minimum IAE, ITAE - Hwang

and Fang (1995).

Servo or regulator tuning - minimum IAE

- Huang and Jeng (2003).


Model: Method 1

Kc Tj

Minimum performance

45 K (224)

0.36 + 0 . 7 6 ^

46 £ (225)

x l ^ r o + X 2 T m

X l

0.057

0.103

0.139

0.168

0.191

x2

0.139

0.261

0.367

0.460

0.543

_

T (224)

0.47xm+Tm

T (225)

x 3 T m + x4*m


0.400

0.389

0.376

0.363

0.352

x4

0.923

0.930

0.900

0.871

0.844

X 5

0.012

0.040

0.074

0.111

0.146

Td Comment

index: other tuning

T (224)

0.47Tmtm

0-47Tm+Tm

T (225)

XeTm^m

t m + X 7T m

X 6

1.59

1.62

1.66

1.70

1.74

X 7

4.59

4.44

4.39

4.37

4.35

0 . 1 < ^ - < 2 . 0 ; T

m

decay ratio = 0.05;

Model: Method 25 Model: Method 37;

^2- < 0.33 T

1SL>O.33

M m a x = l . l

Mm a x=1.2

Mm a x=1.3

Mm a x=1.4

Mm a x=1.5

45 K (224) 0.713-0.176-2!-+ 0.0513 T

- (224)

f ^ Is. T

K u ,

K (224)

Kumu 0.149 + 0.0556-21- - 0.00566 T

( \2 x.

T V m J

0.371 - 0.0274 -2- + 0.00557 T I T co„K

(224)

46 K (225) 0.8194Tn+0.2773Tn. v 0.9738T 0.0262

, T^"> =1.0297Tm + 0 . 3 4 8 4 T „

T (225) _ 0.4575Tm+0.0302Tn.

1.0297Tm+0.3484t my

Rule

Astrom and Hagglund (2004)

- continued.


Model: Method 1

Regulator -Gorecki et al.


Kc Ti Coefficient values

X l

0.211

0.227

0.241

0.254

0.264

x2

0.616

0.681

0.740

0.793

0.841

x 3

0.342

0.334

0.326

0.320

0.314

x4

0.820

0.799

0.781

0.764

0.751

X 5

0.179

0.209

0.238

0.264

0.288

Td

X 6

1.78

1.81

1.84

1.87

1.89

X 7

4.34

4.33

4.32

4.31

4.30

Comment

Mm a x=1.6

M r aax=1.7

M r aax=1.8

Mm a x=1.9

Mm a x=2.0


_ L ( o . 5 + ^ l

47 K (226)

48 K (227)

Tm+0.5tm

T (226)

T (227)

Tra^m ^ m +2T m

T (226)

T (227)

Step disturbance

Stochastic disturbance

Pole is real and has maximum

attainable multiplicity; "tm/Tn, <2

47 K (226) _ e_

K „

- (226)

In

1 - 0.5e"°,',x"' +-la-e-<»d^ x „

(226) (l - 0.5e""jT- )ln

T

'K (227) _ 2 \

2T„ - x , - 9 -

v 2 Tm y 2T

2Tm

T ( 2 2 7 ) = T 1 i L n

21 + 3 ^ - + T

1

2T

— 1 N 2 T J _

X

x, - 9 - - ^ - -2T

, - 36 -4 .5 - ^ -T

'O l 2 T m ;

(2T

2

^

r A3

—

l2 TJ T < 2 2 7 ) = 0 . 5 x „

A A 2

3 +

6 + -2T

x , - 9 -2T 2T

2T

Rule

Saito et al. (1990). Model:

Method 18

Suyama(1992). Model: Method 1


Model: Method 2

Juang and Wang (1995).

Model: Method I

Abbas (1997) Model: Method 1

Camacho et al. (1997).

Kc

0 . 2 1 5 t m + T m

1.37Kmxm

50 K (229)

0.56Tm

TmKm

0-5Tm

TmKm

51 K (230)

52 £ (231)

1 T m + x m

T|

0 .315T m +T m

T m + 0 . 3 0 9 t m

5 i m

Tm

T (230) • M

T m + 0 . 5 t r a

4Tmxm

T r a + x m

Td

0.315Tmxm

0.315x r a+Tm

49 x (228)

T (229)

<0.5Tm

<0.5xm

T (230)

T m * m

2 Tm + tm

T m+*m

Comment

2 H L < 1

T m

1 < ^ - < 1 0 T

m OS=10%

^ 2 - < 0.33 T

m

— >0.33 T

TCL = « T m ,

0 < a < l

0 < V < 0 . 2 ;

0.1 < -^2- < 5.0 T

m

Model: Method 1

( 2 2 8 ) _ 0 . 3 1 5 x m T m + 0 . 0 0 3 T „

0 .315T r a +T m

50 v (229) _ 1

K . K „ 0 . 7 2 3 6 - 2 - + 0.2236 . T,

(229) 2-236TraTm

51 K (230) T T

K m a

7 .236T m +2.236x n

,2

. (230)

a + IsL + o.5 U s T I T

a + -

(230) _ V i m J V 0.5 a + ^ - O . S a ^

T T

a + i s_ a + Ii!L + o.5 Tm Tm

2 ^

52 K (231)

0.177 + 0.348 i "

Km(0.531-0.359Va7 '3)

Rule

Cluett and Wang (1997), Wang

and Cluett (2000). Model:

Method 1;

0.01 < -^2- < 10 T

Kc

53 K (232)

Ti

T (232)

Td

T (232) ld

Comment

TCL = 4xm , Am E [7.97 ,8.56], *m e [79.67° , 79.70°]

54 ^ (233) T (233) T (233) ld

T C L = 2 x m , A r a e [4.84,5.31], *„ e

55 K (234) T (234) i

73.9°,74.14°

T (234) ld

TCL=1.33Tm,AmE[3.81,4.16],<t.ra6

56 j , (235) T (235) i

69.84°,70.70°

T (235)

TCL= Tm, Am G [3.30,3.56], <))m e [65.86°,68.50°

57 K (236) T (236) T (236)

T C L =0 .8 t m , Am 6 [3.00,3.19], * m e 60.60°,67.21°

53 „ (232) _0.019952xm+0.20042Tm (232)

54 v (233)_0.055548Tm+0.33639Tm (233)

0.099508xm + 0.99956Tm

0.99747x-8.7425.10_5T„

(232) -0.0069905xm+0.029480T„

0.029773xm +0.29907Tm

0.16440xm +Q.99558Tm

0.98607x-1.5032.10~4T„, m

(233) -0.01665 IT m + 0.09364 lTm

0.093905xm +0.56867T„

55 v <234) _0.092654xm+0.43620Tm T (234)

K m T m 'K ,

0.20926xm + 0.98518Tm

0.96515xm+4.2550.10-3T„

(234) . -0.024442xm +0.17669Tm

0.17150xm +0.80740T„

56 K (235) _0.12786xm+0.51235Tm ^ T<235)

K m T m

57 „ (236) _0.1605lTm+Q,57109Tm T(236) K c . l i

0.24145xm+Q.9675 lTm

0.93566xm + 2.2988.10~2T„,

(235) -0.030407xm +0.27480Tm

0.25285xra+1.0132Tm

0.26502xm +0.94291Tm

0.89868xm+6.9355.10"2Tm m '

T (236) = -0.035204Tm+0.38823Tm d 0.33303xm+1.1849Tm

Rule

Cluett and Wang (1997), Wang

and Cluett (2000) - continued.


(1997a). Morilla et al.

(2000). Model:

Method 28

Kc

58 K (237)

T|

T (237) *i

Td

T (237) *d

Comment

TCL= 0.67xm, Am e [2.80,2.95], ^m e [52.35°,66.45° j

59 K (238)

60 K (239)

T (238) i

T (239)

T (238)

al™

\ = 0.707 ; Model: Method I

a = 0.1; 60 = [0.2,0.5]

58 v (237) 0.19067tm+0.61593Tm T<237) _ 0.28242Tm +0.91231Tm ^ Kmx„ ' ' 0.85491T+0.15937T n K „

(237) -0.039589Tm +0.5194 IT,,

0.40950T„ + 1.3228Tm

59 K (238) _ 1 0.746-1.242 In T

T

rr\ {£Jd) rr\ 0.3106 +1.372-^1- _ 0.545 T

r ^2

T V m J

-T, (238) T

16.015Tm -11.702x„

±1% Ts = 2.5Tm, 0 . 1 < - ^ - < 0 . 5 ; ±1% Ts = 3.0Tm, - ^ - = 0.6; +1% Ts

3.75Tm , ^=- = 0.7; ±1% Ts = 4.3Tm ,^- = 0.8; ±1% Ts = 5.0Tm , ^ = 0.9 .

60 K (239) T^'V , (239)

K r a [28 0 + r o n 0 (x r a -T j( 2 3 9 ) ) l ' ' 1 + V T ^ 4 ^ '

• 5„Tm + V» 02 T m » + T m (T B , -T i '

r o >)-ar r ,<° ! "] '

( ^ - T ^ j - a P ^ ] 2 Tm (?m " T i

with 80

1 + 2TT

logc[b/a]

2 a = desired closed loop response decay ratio.

Rule

Mann et al. (2001). Model:

Method 31 or Method 33


Kc

61 j r (240)

62 ^ (241)

«K C < 2 4 2 >

Ti

T (240) i

T (241) i

T (242) i

Td

T (240)

x (241)

T (242)

Comment

OS < 5%;

0 < * m / T m < l

OS < 5%; l ^ m / T m < 2

Model: Method 1

61 K (240)

0.770+ 0.245^2-l T n W

0.854"

Tm

T ( 2 4 0 ) ' i 1.262 + 0 . 1 4 7 ^

V ir

0.262 + 0.147 (240) _

, N 0.854

V m J

0.770 + 0 . 2 4 5 ^

62 j , (241)

0.603 + 0.275 T

V m ,(241)

K m ' C n i

1.1618 + 0.165 r N-2.4

T V m

T m '

(241) _

0.1618 + 0.165|-^

0.603 + 0.275

63 K (242) 1 (2TmTm + 0.5xm2)(3TCL + 0 . 5 x J - 2 T C L

3 -3TCL2xm

Km 2TC L3+3TC L

2xm+0.5xm2(3TC L+0.5xJ

T(242, _ (2Tmtm +0.5Tm2)(3TCL + 0 .5xJ -2T C L

3 -3TCL2xn

t m ( 2 T m + T m j

Td(242) . 3TcL2xmTm + 0-5TmTm

2 (3TCL + 0.5xm)- 2(Tm + xm )TC

(2Tmxm +0.5xm2)(3TCL + 0 .5xJ -2T C L

3 -3TCL2xm

T<242>s(l + 0 .5x m s ) r -

K ^ ' ^ s + l)3

Rule

Viteckova and Vitecek (2002).

Model: Method 1


Sree et al. (2004).

Model: Method 1

Kc

64 K (243)

T,

T (243) i

Td

T (243) l6

Comment

For 0.05 < S- < 1.6 , T; = 1.2Tm ; overshoot is under 2% (Viteckova

and Vitecek (2003)).

( l -v )x m +T r a

XKmTm

1 [

1.377

T 1 -^2-+ 0.5 T m J

(T ^

T V m J

0.8422

( l - v ) r m + T m

Tm+0.5xm

67 T (245)

( l -v)xmTm

0-v)xm+Tm

66Td< 2 4 4 )

T (245)

65

0 . 1 < ^ _ < 1 T

m

K w = ^ l [ T m 2 T m X | 3 + ( 3 T m T m + T m2)X i2 + T m X ] _ 1 ] ; 64 v (243) _ e

K „

3 0.5 | 3 0.25 x, = +

"m ^m x m T m A | T

2 T 2

m m

• (243)

(243)

^ m ' ^ X , 3 + ( 3 T m T m + T m2 ) x 1

2 +T X, - 1

("cm2Tmx, + 2 t m T m +xj)

T/"J '=-0.5 ^ m

2 T m x , 2 +(4x m T m + x m2 ) x 1 + 2 x m + 2 T m

T m 2 T m X l 3 + ( 3 t m T m + T m2 ) x , 2 +TmX, - 1 '•m 1 m T Lm r-\ T v m ^ l

' F o r O ^ T " <r i V = > T m + T m2 + 0 . 5 x m

2

X =

F o r T m < ^ ^ < 1 , v = ^ Tm

+ T ™ 2 + 0 - 5 ^ 2

( ^ m + T r a ) 2

1 | ( 1 .3M S2 -1 ) 1.56Tm

2(0.5xm+3T,

2 M , ( M , - l ) 2 M . ( M . - l ) ( i m + T m ) 3

0.65M,

( ^ m + T m ) 2 M . - l

66 x (244) 0 . 5 i m ( T m + 0 . 1 6 6 7 t m )

T m + 0 . 5 x m

67 T < 2 4 5 > = i . 0 8 5 T „ T

V m J

,T, (245) _ . 0 . 3 8 9 9 - ^ + 0.0195

T

Rule

Regulator -Gorecki et al.

(1989).

Somani et al. (1992). Model:

Method 1; Suggested A m :

1.75 <Am <3.0

Kc ^ Td Comment

Direct synthesis: frequency domain criteria 68 j r (246) T (246) T (246

Low frequency part of magnitude Bode 69 K (247)

^m/Tm

0.0375 0.1912 0.3693 0.5764 0.8183 1.105 1.449

X 2 T m * 3 * m


3.57 3.33 3.13 2.94 2.78 2.63 2.50

X 3

0.71 0.67 0.63 0.59 0.56 0.53 0.50

* m / T m

1.869 2.392 3.067 3.968 5.263 7.246 10.870

Model: Method 1

diagram is

x2

2.38 2.27 2.17 2.08 2.00 1.92 1.85

flat.

x 3

0.48 0.45 0.43 0.42 0.40 0.38 0.37

68 K (246)

K m 2 _ x„

- (246) - + 1

V "m /

7 + 42-^ - + 135 T ( 2 4 6 ) - T

T m

rT ^ + 240

(T + 180

(y \4

15

T (216) _

T 2 - ^ - + l l + 3 ^ _ + 6

/ T 'N2

1 + 7 - ^ + 27 xm

+ 60 ' T „ °

+ 60

7+ 42-2!-+ 135 xm

(T \ + 240 + 180

fy A4

For -EL < 1 , K T

(247) 0.7809

KmAm

+ 1 or

|0.803 + 4^s-+0.896

K (247) 0.7809

KmAm1 1 + 0.803 , T v K (247) 0.7809

KmAm1

1 + 0.455 • > 1 .

Rule

Chidambaram (1995a). Model:

Method 1


Model: Method 1

Kc

70 ^ (248)

*i/KmAm

Iss. T

0.062 0.083 0.150 0.196 0.290 0.339

x l

29.770 22.100 12.470 9.670 6.686 5.784

1.20Tm

K m t m

1.03Tm

K m T m

Ts

T (248)

X 2 t m

Td

T (248)

x 3 ^ m

Coefficient values

x2

2.34 2.33 2.29 2.27 2.23 2.21

X 3

0.37 0.37 0.37 0.36 0.36 0.35

2-4xm

2-4xm

ISL m

0.389 0.543 1.03 2.92 3.69

x i

5.104 3.781 2.262 1.209 1.105

0.38xm

0.38tm

Comment

Alternative

x2

2.19 2.14 2.00 1.72 1.67

Ara

Am =

X 3

0.35 0.34 0.32 0.28 0.27

= 1.5

= 1.75

70 j r (248)

•(248)

KmAm

3.42 • + 0.85

4.45 + 4 -S - -2 .11

5Tm

0 - 8 5 , A AC

or ,1+4.45

5T„

/ T \2

1.17 3.42

+ 0.85

T <248> ~ yd

4.45 + 4 -5 i - -2 . i l Tm

0.8Tm

1.17 3.42

+ 0.85

4.45 + 4-21--2.11

(T \ 4.45 -0.15

0.8T

4.45 / T \2

0.15

http://-5i--2.il

Rule

Chidambaram (1995a)-continued.

Branica et al. (2002). Model:

Method 1; M.=1.8



Kc

0.90Tm

KmTm

71 K (249)

72 ^ (250)

73 K (251)

7 4 mK u cosfaJ

Ti

2.4xm

T (249) i

T ( 2 5 0 )

T ( 2 5 1 ) i

T ( 2 5 2 ) 1 i

Td

0.38xm

T (249) Xd

T (250)

T (251) x d

T (252) i

a

Comment

A m =2.0

0.1< — <0.5 T

0.5 <-^S- < 1.1 T

1.1 <^2-< 2.0 T

* m = 60°

71 v (249) = 1 K, 0.021 + 0.788- , T i

( 2 4 9 ) =1 .534T n, T "

K „ 72 v (250) _ _J_[ 0.292 + 0.671-^2- | , Ti(250) =1 .239T„

(249)

f \ -0.554

/- \ -0.055

v T V m J

(250) : 0 .230T„

T V 1 m J

r \ -0.069

T

73 v (251) _ 1 K „ 0.338 + 0.623^2- T<25» = 1.228xm - ^

x m T „

\-0.480

Td(251)=0.23kJ^H

T

i = 0.614(1 - 0.233e-°'347K"'K»), <|>m = 33.8°(l - 0.97e"°'45K"'K"), 0.1 < ^ - < 2.0 ,

a = 0.413(3.302KmKu +1), T; tan(c|>J+ - + tan 2(<|>J

+1), T/252' = a 1 « 2co„

Rule


Atherton(1993) - continued.

Uet al. (1994).

Model: Method 1

Tan et al. (1996). Model:

Method 30

Kc

7SmKucos(4)m)

76 j r (253)

<L 15°

20°

25°

Ara

2

1.67

1.67

n 2.8

3.2

3.5

4h

7tApAm

-7—cos<t>m

Ara

Ti T (252)

(page 176)

4 T d ( 2 5 3 ,

4>m

30°

35°

40°

Am

1.67

1.43

1.43

il

3.8

4.0

4.2

0.3183T

aTd(254>

78 ry (255)

Td

T (252) i

a

T (253) *d

4>m

45°

50°

55°

Am

1.25

1.25

1.25

n 4.6

4.9

5.2

0.0796T

77 x (254) ld

T (255) *d

Comment

A m = 2 ,

* m = 6 0 °

4>m

60°

65°

Am

1.11

1.11

TI

5.4

5.5

simplified algorithm

a chosen arbitrarily

Arbitrary A m ,

<L at <»,,,

'Model: Method31; m = 0 .613( l -0 .262e" u ^^" ) , <(>m = 33.2U l-1.38e -0.68KmK„

a = 1.687KmKu, 0.1 < - ^ < 2.0 .

76 K (253) 4hcos0 (253)

7 l A „ fiT^ + T*™ 471 T1 + J T 1 2 + -

tan(|)m -_

}*[*,

(b/VA.'-b1

^ . • - " • • J .

JJ with ±b = deadband of relay, T = limit cycle period.

77 T (254) _ T

i n — I n

K 2 tancj>m + J — + tan i

47C I recommended Am = 2 , d)m = 45

Rule


Model: Method 1;

A m > 2

Rivera et al. (1986).

Model: Method 1

Brambilla et al. (1990).

Model: Method 1

Lee et al. (1998) Model: Method 1

Gerry (1999). Model-

Method 10

Kc

0.25KU

0.4830

GpO r a .5o°]

T;

1.1879TU

2.6064

Td

0.2970TU

0.6516

Comment

t m / T m < 0 . 2 5 ,

$m > 60°

0.25 < -i=- < 2.0 Tm

* m > 4 5 °

Robust

T m + 0 . 5 t m

K m ( ^ + 0.5xm) T m + 0 . 5 T m

Tn^m 2 T r a + x m

X > 0 . 1 T m ,

^ > 0 . 8 x m .

0.5xm < X < 3xm (Zou et al. (1997))

79 Tm + 0-5xm

Km(A. + i J

T m +0 .5x r a

2 T m + x m

0.1 < -^s -< 10 T

m No model uncertainty - X « 0.35xm

T (256)

K ra(X + x m )

81 K (257)

Tm

K m ( ^ + 0.5xm)

80 y (256)

T (257)

T

T (256)

T (257) 1 d

0-5xm

T C L - ^

X = 2xm (aggressive, less robust tuning);

X = 2(Tm + x m ) (more robust tuning)

X£[0.1Tm ,0 .5Tm] ,^<0.25;

^ = 1 .5(x m +T m ) ,0 .25<^< 0.75 ; X = 3 ( x r a + T m ) , ^ > 0.75 (Leva (2001)).

80 y (256)

81 K (257)

= Tm + 2(^ + t m )

, (257)

(256)

2{X + Tm) 1- -

. (256) 3T-V J 1 i J

K r a ( 2 T C L + x r a - a ) ' l i

0 1fi7r 3 -T m a -

T / ^ T + a ^ + a ^ - 0 . 5 x n

2TCL + xm - a

0.167xm3-0.5axm

2

(257) 2Trl + xm - a T r l 2 + axm - 0.5x 2

, (257) 2Tr, +xm - a

a = T, m - T j l - ^ M e

2 _ISL

Rule

McMillan (1984).

Model: Method 2 or Method 44

Geng and Geary (1993).

Model: Method 1

Wojsznis and Blevins(1995).

Peric et al. (1997). Model:

Method 40

Matsuba et al. (1998).

Model: Method 1

Wojsznis et al.


Kc T, Td Comment

Ultimate cycle

82 K (258)

0.94Tm

Kmx r a

T m + 0 . 5 x m

a K r a ^ m

K u « > S ( | > m

84 K (260)

Kucos(|)m

A m

T (258) 1 i

2 t m

T m + 0 . 5 t r a

aT d( 2 5 9 )

T (260) i

85 j (261)

T (258)

0.5xm

T m t m

2 T m + x m

83 T (259)

x (260) Xd

a T . ( 2 6 1 )


Ku>Tu

•^2- < 0.167 T

1.3<a < 1.6;

Model:

Method 36

Recommended

ot = 4

0.1 < ^ s - < 1 . 0 T

82 K (258) _ 1.415 Tm

1 +

. T.(258) = x 1 + T + x

(258) = 0 . 2 5 T J 1 +

T +x V m m y

/ 83 x (259)

84 K (260)

tan<L + J— + tan2 <|>n a

1.31 f T, ,0.9

Km Vt m y

2.19 IT,

K r a ^ m

T

47t

0.9

T i( 2 6 0 )=1.59x„

T <260) = 0.40x„

= (tan<j)m + y 4 a + tan2<|> I—-J 4na

Rule

Wojsznis et al. ( 1 9 9 9 ) -

continued.

Wojsznis et al. (1999).

Model: Method 36

Kc

0.5Kucos<t>m

0.38KU

0.27KU

87 K (263)

T| 86ry (262)

i

1.2T U

0.87TU

ry (263) i

Td

0.15Ti(262)

0.18TU

0.13TU

0.125Tj<263)

Comment

Default values

^m /Tm<0.2

A m = 3 ,

4>m=33°;

xm /Tm*0.25

a, e [0.3,0.4];

a2 E [0.25,0.4];

0.2<^-<0.7 T

m

T i( 2 M ) = 0 . 5 3 T „ ( t m ( | > m + 7 a 6 + • tan

87 ^ (263) _ K 1.0-a, 0.6

1 + e

r-p (263) T-. 0.6

- - ^ - 7 . 0

l + e ^ T m -j

4.1.2 Ideal controller in series with a first order lag f

G c ( s ) = Kc 1 \

V T i s j

1

l

1 + Tfs

U(s)

• K m e ' " "

l + sT„

T fs + 1

Y(s)

Table 45: PID controller tuning rules - FOLPD model G m (s) = K m e

l + sT„

Rule

Schaedel (1997). Model:

Method 21

Horn et al. (1996).

Model: Method 1

Kc Ti Td Comment

Direct synthesis: frequency domain criteria 1 K (264) T,2 - T 2

2

T T (264)

T (264) Tf =Td(264)/N

5 < N < 20

Robust Tm+0.5xm

Tm+0.5xm

T r a ^ m

2Tm+Tm

T ^ m

' 2^ + x J ' ^e[xm,Tra]

X = max(0.25Tm,0.2Tm)(Luyben(2001)); X>0.25tm (Bequette,

(2003)).

1 ^ (264) 0.375T (264)

K „ T, +-(264)

N , (264)

_ j (264) _ T j T 3

^ ' d - T, T22

1 + 3.45-

- + T m , T 2 = 1 . 2 5 T m T > - T „ , + 0 . 5 T ' T , = 0 ,

1 + 3.45-

0.7T -+^m -T2 = T mC +

0.7T - + 0.5t„

T„-0.7xam "" ' T m - 0 . 7 <

0.952Tm2(C-0. l) , 0-35Tm

2tm2

0<-^-< 0.104: T

0.167xJ, — > 0.104. T „ , - 0 . 7 T * T m - 0 . 7 <

Rule

H^ optimal -

Tan et al. (1998b).

Model: Method I Lee and Edgar


Huang et al. (2002). Model:

Method 1

Zhang et al. (2002).

Kc

2 K (265)

Tj

Tm+0.5xra

Td

x T

^m+2Tm

Comment

X = 2 : 'fast' response; X = 1: 'robust' tuning; X = 1.5 : recommended

3 K (266)

Tm+0.5xm

Kj^ + xJ

T (266) M

Tm+0.5xm

T (266) Xd

Tm

2Tm+xm

i-p r-p (266) l f - l d

' 2(X + xJ X = 0.65xm - minimum ISE; X. = 0.5xm - overshoot to a step input

< 5% , Am = 3 . In general, Ara = 2{X/xm +1).

4 K (267) Tm+0.5xm 2Tm+x ra Model: Method 1

2K(265)_0-265X + 0.307rTm

K„ lx m

1

. T f 5.314X. +0.951

3 K (266)

Km(^ + xm) Tm+- T 3 T m - x m xn

"X„ I—7- r + -2( \ + T J m\6(*. + x J 4( + x r a ) 2

. (266) _ry X „ 3 T m - ' c m + . t „

2(X + xm) m ^ + x J 4(^ + x r a)2 '

T (266) _

[6(X + x J 4(X + xJ 2

K (267) _ _ 1 Tm+0.5x„

Km 2X. + 0.5X. 2X. + 0.5x„ orTf_AE^m

x m + 2 T

4.1.3 Ideal controller in series with a second order filter r 1 A

Gc(s) = K£

V T i s J

l + bfls

l + afls + af2s

R(s) E(s)

K, l + — + Tds V T i S

u(S; l + b f ls

l + af,s + af2s

K „ e "

l + sT„

Y(s)

Table 46: PID controller tuning rules - FOLPD model Gm (s) = K„e"

l + sT„

Rule

Kristiansson (2003). Model:

Method 11

Kc Ti Td Comment


5 K (268) rp (268) T (268) 0.05<^S-<2; T

m

1.6 <MS <1.9

' Equations continued into the footnote on page 184;

K (268) _

l . l ^ - O . l 0.06 + 0 . 6 8 — - 0.12 T

f \2

T,

T

K „ f ~ \

0.68+ 0.84-=!--0.25 T T

2T„

. (268)

0.06 + 0.68 -5=- -0.12 T T

0.68 + 0.84 -^- -0.25 T T

Rule

Kristiansson (2003) -

continued.

Horn et al. (1996).

Model: Method 1

Kc

6 K (269)

7 K (270)

Ti

T (269) i

Td

T (269)

Robust

Tro+0.5Tm

T m T m

^ m +2T m

Comment

0.7 < — < 2 ; T

m

1.6<MS <1.9

*• > ^ - *• < T m •

(268) _

0.68 + 0 . 8 4 ^ - 0 . 2 5 ^ T T

T r = -

0.06 + 0 . 6 8 ^ - -0.12 T

f \ 2 x

ra )

-|2

b l f = 0 , a l f =0.8T f , a 2 f =T f with

T 1 . 1 ^ - 0 . 1

6 K (269)

. (269)

min-l 5+ 2 - ^ , 2 5 x „

0.6667(Tm+Tm 1.1 - - 0 . 1 X m

0.68+ 0.84-2!--0.25 T

' O 2

0.6667(Tm+xm

0.68+ 0 .84^ - -0 .25 T

m

) ( ^2

\LmJ

T V m /

n . T , (269) 0.1667(Tm+xm)

0.68 + 0.84 ^~ T

(. \ 2 ]

-°ii) \ (Tm +xJ2

b l f = 0 , a,,- = 0.8Tf, a2f = Tf with Tf = -

1 . 1 ^ - 0 . 1 xm

9T„ 5 + 2-V m y

7 K (270) 2T m +x m _A.2xm+2Tmxm(xm-A.) , 2*,(2Tm->.) - , D f l - -, ; +

2(2A. + x m - b „ ) K n Tm(xm+2X) (xm+2X) '

2XTm+2X2+bf,Ta, « f l — TT~. : \ — i df

^ x „ fl 2(2X + x r a - b f l ) ' f2 2(2X + T m - b f i :

Chapter 4: Tuning Rules for PID Controllers

4.1.4 Ideal controller with weighted proportional term

185

Table 47: PID controller tuning rules - FOLPD model Gm (s): Kme"

l + sT„

Rule

Astrom and Hagglund(1995) -pages 208-210.

Model: Method 5 or Method 14


Kc

3 . 8 e - " , + 7 J t I T m

K m T m

8.4e-* 6 t + M ' 2Tm

K m T m

0 - v > m + T m

xKraTm

Ti Td

Direct synthesis

5 . 2 x m e - — '

or

0.46Tme2 '8T-2 'u2

0.89Tme-037t-4w2

or

0.077Tme50t-4 '8l !

b = 0.40e0'18t+2'8x2

3.2Tme-'-5T-a93T2

or

0.28Tme18T"''6t2

0.86Tme-'9t-04412

or

0.076Trae3 '4t-MT!

b = 0.22ea 6 5 , + 0 0 5 l T 2

0-v) t m +T m ( l - v > m T m

( l - v ) x m + T m

b _ ( X - I ) t m - T m : 1

( l - v ) t m + T m

Comment

M , = 1 . 4 ;

0.14 < ^ 2 - < 5.5 Tm

M s = 2 . 0 ;

0.14 < ^ 2 - < 5.5 T

8

' For 0 < -^m+Tm

• < T V =

Fori < V < 1 ; V _^T m + T m2

+ 05T m2 _

TmTm+Tm2+0.5Tm

2

(^m+Tm)2

1 { (1.3MS2-1) 1.56Tm

2(0.5Tm+3Tm),

2 M , ( M . - l ) 2 M . ( M , - l ) (Tm+Tm)3

0.65M„

^m+Tm frm+Tj M - 1


4.1.5 Ideal controller with first order filter and set-point weighting 1

f

K(s)

w

l + 0.4Trs

1 + Trs

U(s) = K

E(s)

1 \ 1+ — + Tds

V T > S J

1

Tfs + 1

D , , l + 0.4Trs R(s) — ^ J - - Y(s)

^ - ^

( K„

1 \ 1 + — + Tds

V T i s J

1

1 + Tfs

1 + sT,

U(s)

K e~

l + sT„

Y(s)

Table 48: PID controller tuning rules - FOLPD model G m (s) = K e -

l + sTm

Rule

Normey-Rico et al. (2000).

Model: Method I

Kc Ti Td Comment

Direct synthesis

0.375(xm +2Tj K m T r a

Tm+0.5xm

T m ^ r a

2Tm+xm

Tf=0.13xra

T r=0.5xm

4.1.6 Controller with filtered derivative Gc(s) = K£ , l T d s

T;s . X,

Table 49: PID controller tuning rules - FOLPD model G m (s) l + sT„

Rule

Tavakoli and Tavakoli (2003).

Model: Method 1;

0 . 1 < i s - < 2 ; T

N = 1 0

Kc T( Td Comment

Minimum performance index: servo tuning 1 K (271)

2 K (272)

1

Km

( \

0.8

i s - + 0.1 T

T (271)

T ( 2 7 2 )

T m

( T ^ 0.3 +-si-

l Xm J

0.01 HTm

x m

( \

0.06

i s - + 0.04 T

Minimum IAE

Minimum ISE

Minimum ITAE

1 K (271)

2 K (272)

1

" K r o

1

~ K m

1

i s - + 0.2 T

V m )

0.3i°- + 0. T

i=- + 0.0; T

T < 2 7 ' ) ->

75

)

T ( 2 7 2 )

0 . 3 - ^ + 1.2 T

I Tm

> x i ^ - -"-m

+ 0.08 J

f

1

T

"\

4

m

0.06

i s - + 0.04 T

Rule

Davydov et al. (1995).

Model: Method 6

Kuhn(1995). Model:

Method 14; N = 5

Kc

3 K (273)

4 K (274)

VKm

2/Kra

T, Td Comment

Direct synthesis T(273)

T (274) i

0.66(Tm+Tm)

0 .8 (T m +Tj

T (273)

T (274) 1 d

0.167(xm+Tm)

0.194(xm+Tm)

^=0.9;

0 . 2 < t m / T m < l

"Normale Einstellung"

"Schnelle Einstellung"

3 K (273)

K J 1.552-^ + 0.078

, (273) 0.186-E- + 0.532 T

Td(273) = 0.25 0 .186-^ + 0.532 |T„

•(274)

K „ 1.209^2-+ 0.103 T

0.382-^s- + 0.338 Tm T

(274) 0.4 0.382^2-+ 0.338 T

Rule

Schaedel (1997)

Goodwin et al. (2001) -page

426.

Kc

5 K (275)

6 j , (276)

Ti

T (275)

T (276) i

Td

T (275)

T (276)

Comment

Model: Method 21

Model: Method 1

0.375T (275)

T

T, + -(275)

N

,(275)

•(275)

1 + 3.45-

- + T m , T / = 1 . 2 5 T n X +

T Z ryi Z FT* £, r-p J

1 ~ X2 T (275) l2 h T T (275) ' *d T T 2

- T m + 0 . 5 T m M 3 = 0 ,

1 + 3.45-

0 < ^ - < 0.104: T

0.7T, 2

? r a , V = T m x am

0.7T„

Tm-0.7x...

3_0.952Tm2(xa

m-0.l)

T -0.7xa •+o.V,

TV Tm-0.7x a

+ TmTmTan + °-35Tro Tm +0.167Tm

3, ^ - > 0 . 1 0 4 m m m _ A _ a m ' r p

T m - 0 - 7 T m lm r (275)

N: .(275) A

iii^l+o. . (275)— (275)

3 7 5 - ^ Kv

0.05 < a < 0.2 , Kv = prescribed overshoot in the manipulated variable for a step

response in the command variable.

6 v (276) . _ (4f ;TC L 2+Tm ) (xm+2TJ-4TC L 22

Km(l6^2TCL22+8^TCL2x ra+Tm

2) '

T (276) _ (4^TCL2 +Tm)(xm +2T m ) -4T C L 2 , +i_ ^

2(l6i;2TCL22+8i;TCL2Tm+Tm

2) ^

Td( 2 7 6 )=(l6^2TC L 2

2+84TC L 2xm+T r a2)

l(4^TCL2 + xm )2imTm - 2TCL22(4^TCL2 + xm Xxm + 2Tm)+ 8TC

(4^TCL2 + xm )3 [(4^TCL2 + xm Xxm + 2T m ) - 4TCL22 ]

N = 4 4 T C L 2 ^ m T ( 2 7 6 ) w , t h G c L ( s ) = . '

2Tr TCL2V+2!;TCL2s + l

Rule


Morari and Zafiriou(1989).

Model: Method 1 Gong et al.

(1996). Model: Method 1 Maffezzoni and Rocco (1997).

Kasahara et al. (1999).

Kc Ti Td Comment

Robust Tra+0.5xm

Km(^ + 0.5xm)

7 j r (277)

8 K (278)

9 K (279)

10 K (280)

Tra+0.5xm

Tm+0.5xm

Tm+0.3866xm

T (279)

T (280)

Tmtm

2Tm+xm

Tm^m

2Tm+xm

0.3866Tmxm

Tm+0.3866xm

T (279)

T (280)

(Chien and Fruehauf

(1990)); N=10 ^.>0.25xm,

\>0.2Tm

N = [3,10]

Model: Method 12

Robust to 20% change in plant

parameters

Model: Method 1; N = 10; 0 < ^ - < 1 T

K (277) _ 1 T m +0.5x„

8 j , (278)

K (279)

... . ^ + Tm

Tm+0.3866xm

Km(x + 1.0009xm)

= 2 T m ( T m + X ! ) + T m

2 K m ( t m + x , ) 2

N = 2Tm(^ + xm)

^ (2T m +x m ) '

. (0.1388+ 0.1247N)Tm+0.0482NTn , *• = — r - r . . . _ . - T „

- T (279)

0.3866(N - l)Tm +0.1495Nxn

2

= T + 2 ( T n + x , ) '

T(279)_ xm2[2T ( x m + x , ) - x m x 1 ] M _ 2T m (x m +x 1 ) -x m x 1

2( t m +x , ) | 2T m (x r a +x 1 )+x m2 J ' x,[2Tm(xm+x1)+xm

2]

x Xj = 0.25tm for uncertainty on xm only; xx >0.1Tm>—— <0.4 for uncertainty on the

minimum phase dynamics within the control band.

10 K (280> = 0.6K, 0.718-0.057- , T^8U ' = 0.5TU 0.296 + 0.838-

Td(280) = 0.125Tu 0.635-1.091-^

T

Rule

Kasahara et al. (1999)-

continued.

Leva and Colombo (2000).

Kc

11 K (281)

12 K (282)

13 K (283)

14 K (284)

Ti

x (281)

T (282)

T (283)

2

T I T m

m 2(X + T J

Td

T (281)

T (282)

T (283)

T (284) x d

Comment

Robust to 30% change in plant

parameters Robust to 40% change in plant

parameters Robust to 50% change in plant

parameters A. not specified;

Model: Method 1

11 K<281)=0.6K, 0.690-0.076- , Ti(281) = 0.5TU 0.274 + 0.788-

Td(281) = 0.125TU 0.526-0.057-!2-

T

12 v (282) K/"" ; =0 .6K, 0.331 + 0.149- • (282) _ 0.5T„ 0.041 + 0.857-E-T

Td(282) =0.125TU 0.495-0.064-

13 K <283) = 0.6K, 0.195 + 0 .266-^ T

X(283) = 0.5T„ -0.013 + 0.731-

(283) :0.125T„ 0.489-0.065-52-

T

14 v (284) _ 1 2 ( X + T m ) T m + T m2

(284) _ Tm{l + tm)

Km 2(X + TmY [2(^m)Tm+Tra2]'

N = 2Tm(X + Tm)2

x[2(A. + Tm)Tm+Tm2J'

Rule

Leva (2001). Model: Method I

Leva et al. (2003).


Kc

15 j r (285)

16 j , (286)

1 7 j , (287)

Ti 2

T T i m m 2(* + TJ

2 T

T i m

m 2(x + xJ

T (287)

Td

T (285)

T (286) 1 d

T (287) x d

Comment

Model: Method 1

Model: Method 1

5 v (285) _ 1 2(X + Xm)Tm +T n K

K m 2(X + Xm)2 N:

2Tm(X + t J 2

«.[2(X + T , ) T m + x „

V2S5)=rJ ^ Y } 2 ^ ' ^ 2 |A e [0.1Tm ,0.5Tm] ,^0.25; 2(X + Tm)[2Tm(^ + Tm)+Tm

2J Tm

>. = 1 . 5 ( t m + T m ) , 0 . 2 5 < ^ - < 0 . 7 5 ; A. = 3 ( T m + T m ) , i s - > 0 . 7 5 .

16 v (286) _ 1 2(^ + Tm)Tm + T m T (286) __ TmTm(X + Tm) K „

Km 2(X + T m ) 2 [2(^ + x l )T m + T r a

2 ] ;

N = r 2T™(X + X™> ^ . Xe[0.1Tm,0.5Tm], i 2 - < 0 . 3 3 ; 42(^ + x m )T m + t m

2 Tra

X = 1 . 5 ( x m + T m ) ) 0 . 3 3 < ^ < 3 U = 3 ( x m + T j , ^ > 3 .

0.5xm+T -~A-17 K (287) m m

(287) d

N Km(2* + 0.5xJ

, (287) :0.5xm+Tm

(287)

N

(287) _ l m i m x m T „ T, (287) T (287)

2T(287) N ' N 2?,+0.5x„ orN=10.


f 4.1.7 Blending controller Gc (s) = Kc

E(s)

1 \

V T i S J

R(s) ®— + T

f K,

1 \ 1 + — + Tds

U(s) K e"

1 + sT

Y(s)

Table 50: PID controller tuning rules - FOLPD model Gm (s) = l + sTm

Rule

Chen et al. (2001).

Model: Method 1

Kc Ti Td Comment

Direct synthesis: frequency domain criteria 18 K (288) T (288) T (288)

Representative X values

^ = 9.56tm) Mmax =1.20; ^ = 4.68xra, Mmax =1.26;

^ = 2.19xm, Mmax =1.50;^ = 1.40tm, Mmax =2.00;

^ = l . l lT r a ,M r a a x =2.50;X = 0.95Tm, M r a a x=3.00.

K (288) 1 T + T + 2 A . ,(288) _

K „ (*• + *.? T m +T r a +2X ,T (288)

T„+Tm+2X.


4.1.8 Classical controller 1 Gc(s) = K

Table 51: PID controller tuning rules - FOLPD model G m (s) = K m e "

l + sTm

Rule

Witt and Waggoner


Witt and Waggoner


Kc Ti Td Comment

Process reaction

K m T m

X! 6 [0.6,1.0]

1 K (289)

*m

T ( 2 8 9 )

X m

T (289)

Equivalent to Ziegler and

Nichols (1942); N e [10,20]

Preferred tuning; N e [10,20]

1 Tuning equivalent to Cohen and Coon (1953).

K (289)

1 . 3 5 0 ^ _ + 0.25+ ^ . 0 . 7 4 2 5 + 0 .0150^3-+ 0.0625 V2

T

2K„

*(289)

1.350^2- + 0 . 2 5 - i s - 0.7425 + 0.0150-^=- + 0.0625 ^ t m Tm V Tm T

(289)

1 . 3 5 0 - ^ + 0.25 + - E - 0 . 7 4 2 5 + 0.0150^ ! ! -+ 0.0625 t m xm V Tm

r \2 xn

T V 1 m J

Rule

Witt and Waggoner (1990)

(continued).

St. Clair (1997) -page 21.

Model: Method 2

Harrold(1999). Model:

Method 10; N not specified

Shinskey (2000). Model: Method 2

Minimum IAE -Huang and Chao

(1982).

Kc

2 K (290)

Tm

K m x m

0-5Tm

K m x m

1/Km

0-5/Km

0.25/Km

0.889Tm

K m x m

T,

T ( 2 9 0 )

5xra

5xm

Tm

l-75xra

Minimum performance

0.817

Km \Xm J

0.982

3 T ( 2 9 1 )

Td

x (290)

0.5xm

0.5xm

<0.25Tm

0.70xm

Comment

Alternate tuning; N e [10,20]

'aggressive' tuning;

N > 1

'conservative' tuning;

N > 1

^m/T r a^0.25

x m /T m «0.5

x m /T r a >l

^2-= 0.167; T

N not specified

index: regulator tuning

x (291) Xd

N = 8; Model: Method 1

2 K (290)

1 . 3 5 0 ^ - + 0.25 - ^ i 0.7425 + 0 . 0 1 5 0 ^ - + 0.0625 x m x m V T m

f \2 x„

VTmV

2K„

X (290)

1 . 3 5 0 ^ + 0.25 + — . 0 . 7 4 2 5 + 0 .0150-^ -+ 0.0625 xm xm V Tm

f \ 2 x„

T

(290)

1.350^2- + 0.25 - — .0 .7425 + 0 . 0 1 5 0 ^ + 0.0625 x„ xm V Tm T

s 0.954

3 T(291) = Q.903T T

, Td( z y" = 0.602T,,

, T ,

Rule

Minimum IAE -Kaya and Scheib

(1988). Model: Method 5 Minimum IAE -

Witt and Waggoner


Minimum IAE -Shinskey(1988)

-page 143. Model: Method 1

Kc

4 ^ (292)

5 K (293)

6 j r (294)

T i

T (292) 1 i

T (293)

T (294)

Td

T (292) 1 d

T (293)

x (294)

Comment

0 < i l L < l ;

T

N=10 Preferred tuning

Alternate tuning

0.1<-^2- < 0.258; N e [ l 0 , 2 0 ] m

0 .95T m /K m i r a

0.95T m /K m x m

1.14Tm /KmTm

1.39T ra/K raTm

1.43Tm

l-17Tm

1.03xm

0.77xm

0.52xm

0.48tm

0.40xm

0.35xm

^ / T r a = 0 . 2

xm /Tm=0.5

tm/Tm =1

T m /T m =2

4 K (292) _ 0.98089 (^ ^ 0.76167

K „

, (292) T , N 1.05211

0.91032 T V m J

Trf<292) = 0.59974T I s

5 £ (293) 0.718 | T„ K m VTmy

f ,. A 1 + , 1 - 1.693

0.964T„

T V 1 m J

r A 1 1 3 7

, (293) VTm V

1 - , 1-1.693 IlIL , T

6 ^ (294) 0.718

K „ T

-0.921

1 - , 1-1.693

0.964T„ r(294)

T V m J

f \1.137

Im. T

0.964T„ (293)

l + J l + 1.693

0.964Tn

T \ m J

( \ 1 1 3 7

• (294) T

l - J l + 1.693

/• -,1.886

T V m J

l + J l - 1 . 6 9 3 T

Rule


-page 119. Minimum IAE -

Edgar et al. (1997) - pages 8-

14. 8-15. Minimum IAE -

Edgar et al. (1997) -page 8-

15. Minimum IAE -

Smith and Corripio(1997)-

page 345.


-page 148. Model-

Method 1


— page 167. Minimum ISE -Huang and Chao

(1982).

Minimum ISE -Kaya and Scheib


Kc

0.96Tm/Kmim

0.94Tm/Kmxm

0.88Tm/Kmxm

Tm

K m T m

0.56KU

0.49KU

0.51KU

0.46KU

Kuxm

3xm-0.32Tu

/ - m \ ° - 8 8 1

I.IOI r -o

9 K (297)

T i

l-43tm

1.5Tm

l-8xm

Tm

0.39TU

0.34TU

0.33TU

0.28TU

7 T (295) i

8 T (296) i

j (297)

Td

0.52xm

0.55xra

0.70xm

0-5xm

0.14TU

0.14TU

0.13TU

0.13TU

0.14TU

T (296)

T (297) i

Comment

Model: Method 1 x m /T m =0.2

Model: Method 1 Time constant

dominant; N=10 Model:

Method 2; N=10

Model: Method 1;

0.1 < ^2-< 1.5

x m /T m =0.2

xm /T r a=0.5

T m /T r a =l

^ m / T m = 2

Model: Method 1;

N not specified Model:

Method 1; N = 8

0 < - ^ - < l ; T

m

N=10

0.15—^-0.05

T ( 2 9 6 ) = U 3 4 T

> T

0.883

(296)

, K (297) 1.11907 TTm

K m {xm

• (297)

= 0.563T„

T

v T

0.7987 T

• (297) = 0.54766T, T

V ' • J

Rule

Minimum ITAE - Huang and Chao (1982).

Minimum ITAE - Kaya and

Scheib (1988).

Minimum ITAE - Witt and Waggoner

(1990).

Kc

0-745 Tm^

11 K (299)

T i

10 T (298)

T (299)

Td

T (298) ' d

T (299)

Comment

Model: Method I;

N = 8

0 0 T ( 2 9 8 ) = a 7 7 I t „ , Td(298)=0.597Trt

„ K (299) 0.77902 ( X

TT\? >T; (299) T

f \ 1.006

I™. , T

• A 0.70949 t .

1.14311 T

TH(299) = 0 . 5 7 1 3 7 T j i

12 K (300) _ 0-679

K „ T 1 + , 1-1.283

0.762T,

, 0.1 < -**- < 0.379; T

f N 0 . 9 9 5

.(300) T 0.762T„

/• -v 0.995

(300) T V 1m J

1-J l -1 .283 T

V m y

l + J l + 1.283 r V-7 3 3

13 K (301) _ 0.679

K „ T V 1 m J

1-J l -1 .283

0.762T„

-1.733

,(301) T 0.762T„

T V m J

, N 0 . 9 9 5

• (301) > T

1 -J l+ 1.283 1 + , 1-1.283 ( A1'733

T V ' m y

Rule

Minimum ITAE -Ou and Chen

(1995).

Minimum ITSE -Huang and Chao

(1982).

Minimum IAE -Huang and Chao

(1982).



Kc

Kc

1.357

Km

0.994

Km

(302) _

f N 0.947

Is]

(T v 0.907

Ti T (302) _

Tm O 0-842 [Tm J

15 j (303)

Td

14 T (302)

T (303) ' d

Comment

Model: Method 46

Model: Method 1;

N = 8

Minimum performance index: servo tuning f N-1.00

0.673 U \ Km I j J

K m ^ x m j

.04432

16 T (304)

17 T (305)

T (304)

X (3"5) lA

Model: Method 1;

N = 8

0 < ^ - < l ; T

N=10

14 x <302>

N = ± (302)x (302) 0.75K r aKcl^'Td

T i « 3 0 2 > + K m K c ^ ( T d ( 3 ° 2 ) + T i ^ - T m - T j N > 0 .

15 T ( 3 0 3 ) = 1 - 0 3 2 T Is. | , Td<303> = 0.570T„

V m T

16 T <304)

17 T . ( 3 0 5 >

0.148 'm

• > T , (304) = 0.502T,

+ 0.979 T

V m J

0.9895 + 0.09539-- .T,

(305) = 0.50814T v T

Rule

Minimum IAE -Witt and

Waggoner (1990).

Model: Method 2

Minimum IAE -Smith and

Corripio(1997)-page 345.

Model: Method 1

Kc

18 K (306)

19 K (307)

0.83Tm

K m x m

Ti

j (306)

T (307) i

T m

Td

T (306)

T (307)

0.5xm

Comment

Preferred tuning;

0 . 1 < ^ _ < 1 ; T

Ne[l0,20] Alternate tuning

0.1 < ^ - S 1.5; T

N not specified

18 ^ (306) 1.086

K „ T V ' m /

,(306)

1 + J l -1 .392 0.74-0.13-

0.696T , N 0 . 9 1 4

T V m J

1-J l -1 .392 / \u.»oy r ' T„ 1 I

T ^ 0 . 7 4 - 0 . 1 3 -

0.696T„ (306)

1 + J l -1 .392 , x 0.869

T

T V m J

0.74-0.13-

19 j , (307) 1 - J l - 1.392 , N 0.914

T

T V 1 m J

0.74-0.13-

0.696T . (307)

1 + J l -1 .392 0.74-0.13-

0.696T„ / \ 0.914

(307) T

l - J l - 1 . 3 9 2 f \ 0.869

T 0.74-0.13-

Rule

Minimum ISE -Huang and Chao

(1982).

Minimum ISE -Kaya and Scheib

(1988). Model: Method 5 Minimum ITAE

- Huang and Chao (1982).

Minimum ITAE -Kaya and Scheib


Kc

0.787 fTm^

21 K (309)

0.684

Km

/ \ 0.995

VXnJ

23 K (311)

Ti

20 ry (308)

T (309)

22-p (310)

T (311)

Td

T (308) x d

T (309)

T (310) 1 d

T (311)

Comment

Model: Method 1;

N = 8

0 < I H . < 1 ; T

N=10

Model: Method 1;

N = 8

0 < ^ 2 - < l ; T

m

N=10

20 T (308) 1.678T, TH(308) =0.594T„

f \ 0.971

T V ' m J

2, K (309) _ 0.71959 K „

T (309) _ _

1.12666-0.18145-

,0.86411

T <309) =0.54568Tm U *

22 T (310)

23 v (311) K

0.114| -!=-| + 0.'

1.12762 fTm

(310) = 0.491T„ v T \ m

0.80368

,(311)

0.99783 +0.02860-^ T

.(311)

Rule

Minimum ITAE -Witt and

Waggoner (1990).

Model: Method 2

Kc

24 K (312)

25 K (313)

T;

T ( 3 1 2 )

T (313)

Td

T (312)

T (313)

Comment

Preferred tuning;

0 . 1 < - ^ - < l ; T

N g [10,20]

Alternate tuning

24 K (312) = 0-965 fxm

K m I T„ 1 + , 1 -1 .232

T i 0.796 -0.1465-JH-

0.616T, •(312)

1 - , 1-1.232

r ^0.85 X

T V 1m J

0 .796-0 .1465-

0.616T„ (312)

1 + J l - 1 . 2 3 2

, N 0 . 8 5

T V 1m J

0 . 7 9 6 - 0 . 1 4 6 5 - ^ -T

2 5 j c ,3,3) _ 0.965

K „ T 1 - J l -1 .232

f N 0.929 r

^ 2 - \ 0.796 - 0 . 1 4 6 5 ^

v T m J 1 Tra

0.616T„ .(313) T

V m

1 + , (1-1 .232

f N 0 . 8 5

T 0 .796-0 .1465-

0.616T„ (313)

( . \ T V m 7

1 - , 1-1.232 0 .796-0 .1465-


Rule

Minimum ITAE - Ou and Chen

(1995).

Minimum USE -Huang and Chao

(1982).

Tsang et al. (1993). Model:

Method 15

Tsang and Rad (1995). Model:

Method 15 Smith and


Model: Method 1

Kc

K c ( 3 1 4 , =

0.965 ( l ^

0.718 f

K m „

T 1 X m J

0.85

1.00

K r a tm

* i

1.6818 1.3829 1.1610

\ 0.0 0.1 0.2

0.809Tm

Km tm

0-5Tra

Kro^m

T i

26 T (314)

27 T (315)

Td

T (314) x d

T (315)

Direct synthesis

T 0.25xra

Coefficient values

* i

0.9916 0.8594 0.7542

\ 0.3 0.4 0.5

T

T

x i

0.6693 0.6000 0.5429

\ 0.6 0.7 0.8

0-5tm

0-5xra

Comment

Model: Method 46

Model: Method 1;

N = 8

N = 2.5

x i

0.4957 0.4569

£ 0.9 1.0

Overshoot = 16%;N=5

Servo - 5% overshoot; N not

specified

26 T ( 3 1 4 )

0.796-0.1465--. T,

(314)

T

N = ±-

•• 0 . 3 0 8 T „

0.75KmK ( 3 1 4 ) T (314)

(314) + KmKclJ""lV"+T i

27 T (315) T

(314) | T (314) + T ( 3 1 4 )

028

N > 0 , '^m-TrnJ

, ( 3 1 5 ) : 0.596T„.

0.063 ^ + 1.061 , T

Rule

Schaedel(1997) Model:

Method 21

O'Dwyer (2001a).

Model: Method 1

Wang et al. (2002).

Model: Method 1

Kc 28 K (316)

X l T m

K m X . r

Ti

T 2

T , - - i 2 -T,

N

Td

-y/T,2-2T22

T

Representative result

0.079Tm

K m T m

1

K4 i+2r] 0-65 f - Q

1.04432

30 K (318)

0.1Tm

Tm

4 + ^ 29 j (317)

x (318)

Tm

0.5Tm

T (317)

T (318)

Comment

N not defined

A m = w / 2x ,N ;

K = 0.57c-x,N

A m = 2 . 0 ;

K = 45°

Labeled IMC; N = 1 0

Labeled IAE setpoint; N = 1 0

Labeled IAE load; N = 10

0.375 ( T 2A

28 K (316)

T,

T,

Kn

T

N T, v ' 2T,

J

1 + 3.45-

- + Tm,T,2=1.25Tmxam+ I i L _ T m + o . 5 T r a \ 0 < ^ < 0.104;

1 + 3 . 4 5 ^ -

L m > l2

T.= ° ' 7 T m + T m , T 22 = T m T a

m + _ ° - 7 T m _ + 0 . 5 x m2 , j ^ > 0.104.

Tm-0.7C T m - 0 . 7 C

2 9 ^ ( 3 1 7 ) (317) , Td

,J " =0.50814

, s 1.08433

0.9895 + 0.09539^2-T

T

30 K (3.8) _ 0,98089

K „ — I , Ti(318) =1.09851Trt

/• >, 1.05211

, T

, (318) : 0.59974Tm

Rule

Wang et al. (2002) -

continued.


Method 42

Kc

31 K (319)

32 K (320)

33 K (321)

34 K (322)

0.65Tm

Kmxm

T, -p (319)

ry (320) i

T (321)

T (322) M

T m

Td

T (319)

T (320) 1 d

T (321)

T (322) 1 d

0.4xm

Comment

Labeled ISE setpoint; N = 1 0

Labeled ISE load; N = 10

Labeled ITAE setpoint; N = 1 0

Labeled ITAE load; N = 10

N=20 A m = 2 . 7 ,

* m =65°

The above is a representative, default, tuning rule; other such rules may be deduced from a graph provided by the authors for other Am ,4)m

values.

31 K (319) : 0.71959 ( T,

^ •m v ' m ;

,(319) _

1.12666-0.18145-

32 „ (320) _ 1.H907 K r - -

K „

(320) = ] 252QT ' X n

T (319) ' d

N 0 . 9 5 4 8 0

: 0.54568 f \ 0.86411

Td<320) = 0.54766T,,

T V m J

f \ 0.87798

T V m J

33 K (321) m 1-12762

K „ , T= (321)

0.9978 + 0.02860-

34 ^ (322) ;

, x 1.06401 f

0.77902 T^ (322) = 0 8 7 4 8 1 T N

T (321) x d

0.70949

= 0.42844

Td(322) =0.57137Tm

Rule

Harris and Tyreus(1987).

Model: Method 1


Zhang et al. (1996). Model:

Method 38 Zhang et al.

(2002). Shi and Lee


Lee and Shi (2002).

Model: Method 1

Kc Ti Td Comment

Robust

Tm

K m ( x m + ^ )

Tm

K n (X + 0.5Tn)

0.5xra

K m ( ^ + 0.5xm)

35 K (323)

36 K (324)

37 K (325)

38 K (326)

39 g (327)

T

Tm

0.5xm

0.5xra

T m

T m

Tm m

T

40 j (328)

0-5xm

0.5xm

Tm

Tm

0.5xm

0.5xra

1 tan

» g I 2 J

0. " m

N = ( x m + ^ ) A ; X > max imum [0.45xm)0.1Tm]

(Chien and Fruehauf

(1990)); N=10

^e[0.2xm ,

Model: Method 1

Suggested 0.6

Suggested 0.6

35 K (323) _ _ 0 - 5 ^ N _ T m ( 2 X + 0.5xm) Km(o.5xm+2\)' tf

36 K (324) =

K m ( 0 . 5 x m + 2 ^ )

Tm

K m ( 2 ^ + 0.5xra)

, N = 0.5xm(2^ + 0.5xm)

N = 1 0 o r N = K

38 ^ (326)

39 v (327)

"bvyTm

K„ l + ^ t a n f e ^ zr , N = l +

0.5xm(2X + 0.5xm)

2co

< » „

-tan g O v 2 j

K J°bwTm

K „ 1 + ^ t a n V 2 ,

, N = 1 + ^ t a n [ ^

4 0 T ( 3 2 8 ) = J J ! L ^ 1 + T n L + _ l _

Tm m b w t r a

Rule


Method 42

Kc Ti Td Comment

Ultimate cycle

41 K (329) T (329) T (329)

N=20 A m = 2 . 7 ,

The above is a representative, default, tuning rule; other such rules may be deduced from a graph provided by the authors for other Am ,<|>m

values.

41 ^ (329) _ 0-65 K „ - 1

TI - tan" K,

,T il " ! "=0 .159T u J Ku - 1

Td(329) = 0.064 T„ 7t-tan ' J K„

4.1.9 Classical controller 2 Gc (s) = Kc

E(s) R(s)

<g>-K 1 + -1 Yl + NTHs

XsA 1 + Tds

1

U(s)

T i s y

1 + NTds

V 1 + Tds j

K „ e '

l + sT„

Y(s)

Table 52: PID controller tuning rules - FOLPD model Gra (s) = -K„,e~

l + sT„

Rule

Hougen(1979)~ page 335.

Model: Method 1

Kc T; Td Comment


1 j r (330) T 0.45^E-N

Maximise crossover frequency; N e [10,30]

1 Values deduced from graph. v (330) n A/T^ _ I T Kc

(330) =9.0/K ra,xm/Tm =0.1;KC(330) =4.2/Km,xm/Tm =

Kc(330)=2.8/Km,xm/Tm=0.3;Kc

(330»

K/330)=1.7/Km)x r a/Tm =

0.2;

K (330)

£ (330)

= 2.1/Km,xm/Tm=(

0.5;KC(330)

: m /T m =0 .8 ;

0.9;Kc(330 )=0.85/Km ,Tm/Tm=1.0.

.,l^m,.ml,m-^,^c =1.45/Km,Tm/T ra=0.6

= 1.2/K r a ,xm/Tm=0.7 ;Kc( 3 3 0 )=l.l/Km ,x

= 0.95/Km,Tm/Tm=0.9;Kc(330) =

4.1.10 Series controller (classical controller 3)

209

(l + sTd)

Table 53: PID controller tuning rules - FOLPD model G m (s) = Kme"

l + sT„.

Rule

Kraus(1986). Model:

Method 19 Tan et al.

(\999a) - page 25.

O'Dwyer (2001b).

Chien et al. (1952) equivalent

- regulator

O'Dwyer (2001b).

Model: Method 2

Kc

0.833Tm

Kraxm

0.6Tra

KmXm

x j m

Kmxm

* 1

0.7236

0.84

x2

1.8353

1.4

Representatii 2 K (331)

T; Td

Process reaction

l-5xm

*m

x 2 T m

0.25xm

^m

X 3 t m

Representative results X 3

0.5447

0.6

0% overshoot; 0.1 <

Comment

Foxboro EXACT controller pre-

tuning

Model: Method 2

Model: Method 2

^m/T m <l

20% overshoot; 0.1 < i m /T m < 1

/e results - Chien et al. (1952) equh T(331) T (331)

alent - servo 0% overshoot; Tm /Tm<0.5

K (331) 0.3T„ 1-J1

2x„ .(331) 1 + J l -

2x„

Rule

O'Dwyer (2001b)-continued.

Tsang et al. (1993). Model:

Method 15

Pessen (1994). Model: Method 1

Kc

3 K (332)

x j r a

x l

1.8194 1.5039 1.2690

S 0.0 0.1 0.2

0.35KU

Ti

T (332)

Td

T (332) 1d

Direct synthesis

T m

x i

1.0894 0.9492 0.8378

\ 0.3 0.4 0.5

0.25xm

x i

0.7482 0.6756 0.6170

% 0.6 0.7 0.8

Ultimate cycle

0.25TU 0.25TU

Comment

20% overshoot; * m / T m < 0.7234

x i

0.5709 0.5413

I 0.9 1.0

0.1<^s-<l T

m

3 j , (332) _ 0 .475T,

Km tm

1+ 1-1.3824- T;(332) =0.68T„ 1 + . 1-1.3824-

Td(332) = 0.68Tn 1 - 1 - 1 . 3 8 2 4 -


4.1.11 Classical controller 4 Gc(s) = Kt 1 + - ^ L

1 + ^ N ;

Table 54: PID controller tuning rules - FOLPD model K „ e ~

l + sTm

Rule

Hang et al. (1993b) -page

76. Model: Method 2


Kc

0.83Tm

K m T m

Tm

Km(X + 0.5Tm)

0.5xm

Km(X + 0.5xm)

Tf Td

Process reaction

l - 5 T m 0.25xm

Robust

Tm

0.5xm

0.5tm

Tra

Comment

Foxboro EXACT controller

'pretune'; N not specified

^ 6 [ T m. T m] (Chien and Fruehauf

(1990)); N=10


4.1.12 Non-interacting controller 1

U(s) = K, v T i s y

E ( s ) — ^ - Y ( s )

N

R(s) E(s) K, '.+-0

V T i s /

U(s) — •

Ke~

1 + sT

Tds T

1 + ^ - s N

Y(s) — •

Table 55: PID controller tuning rules - FOLPD model Gm (s) = Kme^

l + sT„

Rule


Method I

Kc T i Td Comment

Direct synthesis

0.8(Tm+0.4xm)

Kmxm

0.6(Tm+0.4Tm)

""m m

Tm+0.4xm 0-4Tmxm

Tm+0.4Tm

N = min[20,20Td];

regulator tuning

N = min[20,20Td];

servo tuning

4.1.13 Non-interacting controller 2a

U(s) = K

213

v T i s y E(s) % - Y ( s )

1 + -N

R(s) / C > E(s) K,

U(s)

^ + < g > - 1 + sT

Tds T

N

Y(s)

Table 56: PID controller tuning rules - FOLPD model Gm (s) = K „ e "

l + sT„.

Rule




Atherton(1993).

Kc T( Td Comment


1.260

Km

/ \ 0.887

—] 1.295 f T j

0.619

/ x 0.930

1.053 fT r a j

Km [ x j

4 T ( 3 3 3 ) i

5 T (334)

6 ^ ( 3 3 5 )

T (333)

T (334)

T (335) Xd

0.1<^a-<1.0 T

1.1 < ^2 -< 2.0 T

m

0.1 <^S-< 1.0 ; T

m

Model: Method 1

4 T (333) _ _

0.701-0.147-, Td

(333) = 0.375T„ (. ^

T V 1 m j

,N=10.

5 T (334) _ _

0.661-0.110-

-,Td<334)=0.378Tm ' T" , N=10.

6 j ( 3 3 5 )

0.736-0.126-, Td

<335) = 0.349T,, / N0 .907

X,

T ,N=10.

Rule


Atherton (1993) (continued)


Atherton (1993). Model: Method I


Atherton (1993).

Kc

1.120

0.942

1.001

(T ^

\ T m J

0.625

0.933

r N 0.624

10 £ (339)

11 K (340)

Ti

7 T (336)

8 j ( 3 3 7 ) i

9 T ( 3 3 8 ) i

•p(339)

0.271KuTuKm

Td

T (336)

T (337)

T (338) ld

0.112TU

0.116TU

Comment

1.1 < ^2-< 2.0; T

N=10

0.1 <-^=-< 1.0 T

1.1 < ^2-< 2.0 T

0 .1<^2-<2.0 ; T

m

Model: Method 1

0.1 <-^2-< 2.0; T

m

Model: Method 31

7 j (336)

• (337)

9 j (338)

0.720

0.770

-0.114^2-T

T

-0.130^2-T

T m

, Td<336) = 0.350T„ ' x . ^

(337) : 0.308T„

T V m J

, N 0.897 X„

T V ni J

N=10.

0.813

, Tdl"°' = 0.308Tm \—\ , N=10. (338)

0.754 -0.116^2-T

m

l l K c ( 3 4 o ) = 2 . 3 5 4 K m K u - 0 . 6 9 6 £ u ) N = 1 0

3.363KmKu + 0.517

4.1.14 Non-interacting controller 2b

215

R(s) E(s)

Xs

U(s) = Kc + E ( s ) - ^ ! _ Y ( s )

N

+ o U ( s )

*® > K„e"

1 + sT

Y(s)

T„s

T

N

Table 57: PID controller tuning rules - FOLPD model l + sTm

Rule


(1988). Minimum ISE -

Kaya and Scheib (1988).

Kc T( Td Comment


12 K (341)

13 K (342)

T (341)

T (342) i

T (341) i d

X <342>

Model: Method 5

12 K (341) _ 1-31509 K „

13 K (342) _ 1.3466 [ Tra

x(3"l) _ 'm

'' 1.2587

(341)

T

T / " ' =0.5655Tm| —2-

^ 0.4576

0 < -SL < 1 ; N=10. T

Km VTmJ

,0.9308

. T i (342)

1.6585 T

. (342) T d ^ z ' = 0.79715T„

f x0.41941

T V 1m J

0<-!S-<l;N=10. T

Rule

Minimum ITAE -Kaya and Scheib

(1988).


(1988). Minimum ISE -Kaya and Scheib

(1988). Minimum ITAE -Kaya and Scheib

(1988).

Kc

14 K (343)

T,

T (343) 1 i

Td

T (343)

Comment

Model: Method 5


15 K (344)

16 K (345)

17 K (346)

T ( 3 4 4 )

T (345)

T (346)

T (344)

T (345) ld

T (346) x d

Model: Method 5

14 K (343) 1.3176

K „

- (343)

1.12499 [Tm

15 K (344) 1.13031 I Tm

Km l T m /

Td(343) = 0.49547T„

• (344) _ _

, T 0 < - ^ - < l ; N = 1 0 .

T

5.7527-5.7241-

1.26239 (l„ . T j

(344)

(345)

/• \ 0.17707

T V m /

, 0 < - S - < l ; N = 1 0 . T

6.0356-6.0191-

Td( 3 4 5 )=0.47617Tm ;^ 0 < - ^ - < l ; N = 1 0 .

T

,7 „ (346) 0.98384 (Tm~)

K „ l T m , T-

(346) _ _

2.71348-2.29778-

• (346) 0.21443T,

T

N 0.16768

0 < - a - < l ; N = 1 0 . T

4.1.15 Non-interacting controller based on the two degree of freedom structure 1

f \ f \

U(s) = Kc 1 T„s

1+ — + -

v N

E(s)-Kc

J

a + PTds

1 + ^ s N

R(s)

( \ PTds

a + T 1 + ^ s

N ;

K

-V& +

R(s) E(s)

K, 1 T„s

1 + — + T's l + i s

N ;

U(s)

* g > — •

Y(s)

K^e" 1 + sT

Table 58: PID controller tuning rules - FOLPD model G m (s) = K m e "

l + sTm

Rule

Hiroi and Terauchi (1986)

- regulator. Model:

Method 2;

a = 0.6; (3 = 1

Kc T, Td Comment

Process reaction

0.95Tm

Km T m

1.2Tm

K m T m

2.38xm

2 t m

0.42tm

0.42tm

0% overshoot;

0 . 1 < ^ - < 1 T

m 20% overshoot;

0.1<-^=-<l T

Rule


Shen (2002). Model:

Method 5; N=0, (3 = 0,

M s < 2

Kc Ti Td Comment

Minimum performance index: Servo/regulator tuning 1 £ (347) T (347)

i T (347) 1 d


Minimum performance index: other tuning oo oo

Minimum f|r(t) - y(t)|dt + f|d(t) - y(t)|dt

0 0

2 K (348) T (348) T (348) 0 < I J 2 L < O O

T

1 v (347) _ _J_ K, 0.1415 + -

1.224

-0.001582

• (347) 0.01353+ 2.200-2--1.452 T

+ 0.4824

(347) 0.0002783 + 0.4119—E-- 0.04943 T

2~\

a = 0.6656 - 0.2786-2- + 0.03966 T

, N fixed (but not specified);

P = 0.6816 -0.2054-J2- + 0.03936 T

2 K (348)

.(348)

K m T m exp 2.94-11.63

T + T

•< 1.0. Model: Method 1.

+ 11.15 VTm+T

ra

(348)

: xm exp

: xm exp

1.88-3.63 V ^ m "•" ^m J

f + 0.86

VTm+T r a y

•0.25-0.06 VTm+Tm y

-1.99 ( , 1 m + ' n i ;

a = 1 - exp -0.22-0.90 ^ + T m ;

+ 1.45 * m + T m /

Rule


Model: Method 1

Huang et al. (2000).

Model: Method 1

Srinivas and Chidambaram

(2001).

Kim et al. (2004). Model:

Method 1

Kc

3 K (349)

^Tm+0.4T t t

Kmxm

a

4 K (350)

x,/Km

T m / T m

0.1 0.2 0.3

T T

Ti Td

Direct synthesis •p. (349)

+ 0.4xm

T (349)

Comment

0-4Tmtra

Tm+0.4xm

Closed loop transfer function is 4th order; N = 0

For X = 0.80 , A r a=2.13

= ^ 1, P = 0 , N = min[20,20Tdl.

X]

12.5 6.1 4.1

T (350)

x2Tm

T (350)

x 3 T m


0.22 0.41 0.57

x3

0.04 0.08 0.11

Model: Method 1;

N = oo

a

0.68 0.63 0.62

P 0.75 0.70 0.70

3 K (349)

K.

0.2xm3 + 1.2x 2T + 5 . 4 T T 2 + 9.6T 3

Tm(xm+3Tm)2

T i( 3 4 9 ) =1 .667x m i ! i + 3 T

-1.389T„

(349)

frm+3Tm)3

x m + 2 T m - " • - (x m + 2T m ) 4

T m2 T m (T m +3Tj

1 .2 (x m +2T r a ) 3 -x m (x m + 3T m ) 2 '

ct = l-0.723x„

P = l-0.261x r

^m+3Tm

V ^ + 2 T m

T „ +2T„

1.667x„ *m+3Tm) 1 „„ n T m

2 (x m +3T m ) 3

1%,+2T„ -1.389-

(xm+2Tm)4

1

X ( 3 4 9 ) T (349) •

'Sample Kc<350), j . * 3 5 0 ) ^ * 3 5 0 ' taken by the authors correspond to the process reaction

tuning rules of Zeigler and Nichols (1942); a =

T „ T „

— - 4 = 0.5-0.83^

and P = • (350) K„<350,Km 2T: (350)

T ( 3 5 0 )

= -0.16 for this tuning rule

Rule

Kim et al. (2004) - continued

Leva and Colombo (2004).

Hang and Astrom (1988a).

Kc

T m / T m

0.4 0.5 0.6 0.7 0.8 0.9 1.0

OS < 20% (

5 K <35

6o.6K,

70.6K

* 1

3.1 2.5 2.11 1.82 1.61 1.44 1.33

(servo step ^hien et al.

)

i

T, Td


0.71 0.83 0.94 1.05 1.13 1.22 1.26

x 3

0.15 0.18 0.21 0.24 0.28 0.31 0.34

Comment

a

0.59 0.58 0.56 0.54 0.51 0.48 0.49

P 0.69 0.69 0.69 0.69 0.68 0.69 0.66

i; settling time < that of corresponding rule of 1952); a cost function is minimized.

Robust T (351) T (351)

Ultimate cycle

0.5TU

ry (352)

0.1251

0.125T

Model: Method 41

u

u

Model: Method 2

5 K (351) Tm + 2(Tm+X) Km(xm+X)

T-(351) = Tm + 2 ( x m + ^ ) '

(351) = V . l ' , ^ ; " » N _ 2Tm(tm + f }

2Tm(T ra+X)+Tm2 2 (T m + X) ' x[2Tm{xm+X)+xm

2}

3AT a not specified; (3 = 1. A, > — , Aim = maximum variation in the time delay, with

only variation in the time delay considered (Leva and Colombo (2004)); X chosen so

that nominal cut-off frequency = <x>u (Leva, 2005).

a = l - 2 ( x - 0 . l ) - 1 , 6 6 T m , i s - < 0 . 3 ; a = l - 2 x - ^ - , 0 . 3 < ^ - < 0 . 6 ;

x = % overshoot. N=10, P = 1.

7 T (352 ) 0.5 1.5-0.83- T„. a = - ^ - - 0 . 6 , 0.6 < ^2-< 0.8; T T

a = 0.2, 0 .8<-S-<1 .0 . T

Rule

Hang and Astrom (1988a)-continued

Hang et al. (1991) - servo.

Hang and Cao (1996). Model:

Method 31

Kc

0.6KU

8 0.6K„ 9 0.6K U

0.6KU

T i

0.335TU

0.5TH

0.222KTU

10 y (353)

Td

0.125TU

0.125TU

0.125TU

T (353) 1d

Comment

-^2->1.0; T

a = 0.2 Model:

Method 2; (5 = 1

0 . 1 < ^ L < 0 . 5 ; T

N=10; (5 = 1

0 . 1 6 < ^ - < 0 . 5 7 ; N = 10. a = l r , 10% overshoot; Tm 15 + K

a = 1 5 i _ , 20% overshoot with K = 2\ 1 1 ^ - / ^ - " J + 1 3

27 + 5K l37 [T m /T m ] -4 /

9 0.57 < - ^ s - < 0.96 ; N =10. ce = l - — f - K ' + l 1, 20% overshoot, 10% undershoot. 1 7 1 9

10 T.(353) =1 0.53-0.22 T T U > X (

(353) l T

0 . 5 3 - 0 . 2 2 — — . a takes on values of L 4

1.0, 0.2 and 0.40(xm/Tm)2 -0.05(xm/Tm)+0.58 at different points on the open loop

process step response.

f \ f \

U(s) = Kc 1+ — + - a

X s , . Td

V 1 + ^ s

N ;

E ( s ) - K £

( \

pTds x__

l + ^ s 1 + T*S

v N

a + - K „

+

R(s) E(s)

K „ , l T d s

1+ — + - d T.8 l + i8

N .

a + PTds X

Td„ l + TiS 1 + ^ s N

U(s)

+

K e"

l + sTm

R(s)

Y(s)

Table 59: PID tuning rules - FOLPD model Gm (s) = Kme-l + sT„

Rule

Hiroi and Terauchi (1986) -

regulator. Model: Method 2

Kc

0.95Tm

K m T m

l-2Tm

Kmxm

a

Ti Td

Process reaction

2.38tm

2tm

0.42xm

0.42xm

= 0.6; p = l or (3 = -1.48 ; x =

Comment

0% overshoot;

0.1<^H-<1 T

m 20% overshoot;

0 . 1 < ^ - < 1 T

= 0.15

U(s) = K c [ ( b - l ) + (c- l )T ds]R(s) + K ( Xs

R(s)-1

(l + sT f ): -Y(s)

K c[(b-l) + (c-l)Tds]

+ ir U(s) Y(s)

K e"

1 + sT

(l + sT f)2

Table 60: PID controller tuning rules - FOLPD model Gm (s): Kme"

l + sT„

Rule



Kc Ti Td Comment

Minimum performance index: other tuning 11 K (354) T (354)

i 0-5xmTm

°-3*m +Tm

Model: Method 1

No precisely defined performance specification 12 j r (355) T (355) T (355) Model:

Method 45 No precisely defined performance specification

11 K (354) = _

K „ 0.2 + 0.45-

Xm+0.1Tm m T

b = l , - 2 L > l ; c = 0; Tf = 0.1xn

12 K (355) = _

K „ 0.2 + 0.45 - T - + 0 - 5 r '

Tm+0.5Tf

, b = l , c = 0. Tf =-, (355) d

N ,Ne[4,10];

355) = 0 .4xm +0.8Tm +0.6T f

xm+0.1Tm+0.55Tf

T<355) _ 0.5(Tm+0.5Tf)(Tm+0.5Tf) . xm ? }

0.3xm+Tm+0.65Tf ' Tra



U(s) = Kc 1

v T>sv

R(s) E(s)

+ X K,

v T . sy

1 +

U(s)

E(s)-KcTdsY(s)

—+®~2—-K „ e ~

1 + sT

Y(s)

K J d s

Table 61: PID controller tuning rules - FOLPD model Gm (s) = Kme"

l + sT„

Rule

VanDoren (1998).

ABB (2001). Model: Method 2


- page 117.



Minimum IAE -Edgar et al.

(1997) -pages 8-14, 8-15.



Kc T, Td Comment

Process reaction l-5Tm

Kmxm

1.25Tm

Kraxm

2.5xra

2^m

0-4xm

0-5Tm

Model: Method 2


1.32Tm/Kraxra

l-32Tra/Kratra

1.35Tm/KmTm

1.49Tm/Kmin

l-82Tm/K ratm

1.30Tm/KmTm

l-33Tm/Kmtm

1.80xm

l-77tm

l-43xm

1.17xm

0.92xm

1.8Tm

2.1xm

0.44xm

0.41xm

0.41xm

0.37xm

0.32xm

0.45xra

0.63xm

T m /T m =0 .1 ; Model: Method 1

x m /T m =0.2

i m /T m =0 .5

^ m / T m = l

^ r a / T m = 2

Model: Method 1

Model: Method 2

Minimum performance index: other tuning 0.77KU

0.70KU

0.66KU

0.60KU

0.48T„

0.42TU

0.38TU

0.34TU

0.11T„

0.12TU

0.12TU

0.12TU

^ m /T m =0.2

xm /Tm=0.5

x m / T m = l

x m / T m = 2

Rule

Shinskey (1994) -page 167.

ABB (2001).

Haeri (2002)

Kc

1 K (356)

2 j . (357)

3 K (358)

T;

T 2

0.125-^-

T (357)

Direct s T (358)

i

Td

0.12TU

T (357)

vnthesis T (357)

Comment

Minimum IAE; Model: Method I

Model: Method 7

Model: Method 1

1 K (356) K „

3 . 7 3 - 0 . 6 9 - i T

_ > i < 2 . 7 ; Kc<356)

T „ T „

K „

2 v (357) = 1.3570 (Tm K r T/3 5 7 ' =1.1760T„

2 . 6 2 - 0 . 3 5 - ^ *m

^ T

0.738

Xm

±->2.1.

(357)

N 0.995

: 0 . 3 8 1 0 T J - ^ - , ^ - > 0 . 5 .

3 K (358) = _J_ 6.84 - + 0.64

1 + 2.33 T

•7.82j

T (358) _ 0.95+ 2 .58 -S- +3.57 T

( \2

T V m J

0 < - ^ - < 0 . 2 9 , T

,(358) _ ,

(358)

T (358) _ T 1 d _ T n

2.15 - 0 . 7 6 ^ 2 - + 0 . 3 3 - ^ Tm T„.

0.29 ^m """ T m

+ 3.94 V T m + ^

0.29 < — < 4 ; T

- 4.65 T

m + t m

, 0 < - 2 L < 1 . 7 9 ; T

0.87| -^2- - 0 . 4 9 ^ - 1 +0.09 > T

1 . 7 9 < - 2 - < 4 . T

U(s) = K

R(s)

*-(b- l )K c

E(s)

+ T K,

v T i s v

b + — v T i s y

E(s)-(c + Tds)Y(s)

+

U(s)

>® K „ e "

l + sT„

Y(s)

(c + Tds)

I Table 62: PID controller tuning rules - FOLPD model Gm (s) = -

K„,e~

l + sT„

Rule


Model: Method I

Kc

4 K (359)

5 l-2xm

KmT ra

T| Td

Direct synthesis T(359)

0-5Tm

T (359)

0.15xm2

KraTm

Comment

c=Kc(359>

1.2T c - ra

K m T m

4 K (359) =

K „ 4.114-6.575-J2- + 3.342

T 'O*

T

,(359) 0.311 + 1.372^2--0.545 T

r \ 2

T V m J

(359) K (359)T(359)

16.016-11.7

b = 0.793-3.149^L + 8 . 4 0 5 ^ - -8.431 T T A m V m J

b = 0.0584 + 0.44^2. o.5 < - < o.9 . T T

, T •<0.5,

5 b = 0.3238 + 0.4332^- Im. < 0.8 ; b = 0.8507 , ^ - = 0.9 , T T T

m m Jm


4.1.20 Non-interacting controller 6 (I-PD controller)

R(s)

-J® E(s) K .

Xs

U(s) = E ( s ) - K c ( l + Tds)Y(s) T:S

U(s) Kme"

1 + sT

K c ( l + Tds)

Z3Z

Y(s)

Table 63: PID controller tuning rules - FOLPD model Gm (s) = Kme"

l + sT„

Rule

Suyama(1993).

Shigemasa et al. (1987),

Suyama (1993).

Kc Ti Td Comment

Direct synthesis: time domain criteria 6 R (360) rj, (360)

i T (360) *d

Closed loop transfer function is 4lh order 7 j r (361) T (361) T (361)

*d

Closed loop transfer function is 4th order Butterworth

Model: Method 1

Model: Method 1

6 K (360)

K „

0.2xm3 + 1.2xm

2Tra + 5.4xmTm2 +9.6Tm

3

^m(^m+3T ra)2

, (360) : 1 . 6 6 7 t mT m + 3 T m - 1 . 3 8 9 x m

2 j T m + 3 T m j ' m x m + 2 T m ( x m + 2T m ) 4

(360) T m2 t m ( t m +3T m )

1.2(xm+2Tm)3-xm(xm+3T r a)2

7 K (361) 1.3319(xm+2Tm)3 1

Kmxm(xm+3Tm) 2 Km '

T ( 3 6 l ) = 1 2 5 3 2 x m + 3T m _ L 6 9 ^ 2 ( x m + 3 T m )

(361)

T. .+2T, ™ ( T m + 2 T m r

1.5095(xm +2Tm)2 -(x r a +TmXtm +3Tm) : t m ( x r a + 3 T m ) 1.3319(x r a+2Tm)3-xm(xm +3T„


Rule

Shigemasa et al. (1987),

Suyama (1993) (continued).

Model: Method 1

Kc

8 K (362)

Ti

T (362)

Td

T (362) *d

Closed loop transfer function is 4th order ITAE 9 K (363) T (363)

i T (363)

Closed loop transfer function is 4th order Bessel 10 K (364) T ( 3 6 4 ) T (364)

Closed loop transfer function is 4th order Binomial

Comment

Toshiba AdTune TOSDIC211D8

product

K (362) 2.6242(xm+2Tm)3 1

K m t r a (T m + 3T r a ) 2 Kr

T i( 3 6 2 ) = 1 . 8 8 9 8 x m i ^ ^

' m T J-9T

T (362) Xd

K (363)

.8898xm T m + 3 T m -0.720x 2 ym+yTm]t ,

^ r a + 2 T m (x m + 2T m ) 4

_ T (z + T r ^.3130(xm + 2 T m ) 2 - ( x m + T m X x m + 3 T m )

2 .6242(x m + 2T r a ) 3 -x m (T m + 3T m ) 2

0.9450(xm+2Tm)3 1

K m x r o ( x m + 3 T j 2 Km

T(363) = 3.3333xm T m + 3 T m

1 T + 2 T

333xm 1X12M._3.527xm2 .(Tm+fm|] ,

T(363) / , _ U.3444(x m +2T m ) 2 - (x m +1

1.9450(xm+2Tm)3-xmO

2 T m ) 2 - ( x m + T m X x m + 3 T m )

1.9450(xm+2Tm)3-xm(xm+3Tm)2

K (364)

m

_ 0 . 5 6 2 2 ( x m + 2 T j 3 1

K m x r a ( x m + 3 T j 2 Kr

T/364' = 5.3337xm T m + 3 T m - 9.488xm

2 ( T " + 3 T " )3

1 111 ."IT-. Ill / _

^ m +2T m (xm+2T,

T (364)

m x m + 2 T m ( x m + 2 T m ) 4 '

T (T + 3T ^ •1244(x m + 2T m ) 2 - (x m + T m XT m +3T r

0.5622(x r a+2Tm)3-xm(xm +3Tm)2

Rule

Chien et al. (1999).


Model: Method 1

Ozawa et al. (2003).

Minimum ISE -Argelaguet et al.

(2000).

Kc

11 K (365)

12 j r (366)

13 K (367)

14 K (368)

T,

T ( 3 6 5 )

ry (366) i

T (367) i

T ( 3 6 8 ) i

Td

T (365)

T (366)

T (367) 1 d

T (368)

Comment

Model: Method 1

OS = 0%

OS = 10%

Model: Method I

OS e[0%,10%]

Minimum performance index: servo tuning 2 T m + *m

2K ratm Tm+0.5xm

T X

2T m +t m Model: Method 1

Underdamped system response - Z, = 0.707 . xm > 0.2T,,,

K (365) 1.414TC L 2Tm+TmTm +0.25Tm2-Tc

2

" ^ X.,,2+0.707T r , ,Tm+0.25Tm2

CL2 "

(365)

'CL2 l m

^ C L 2 T m "T l - m 1 m 1.414Tri,T +xmTm+0.25xmz

(365)

Tm+0.5xm

_ 0.707TroTCL2Tro +0.25Tmxm2 -0.5xmTCL2

2

Tmxm +0.25xm2 +1.414TCL2Tm -T C L 2

2

1 2 K ( 3 6 6 ) _ 0.28l(xm+2Tm)3 1

K m x m ( x m + 3 T m ) 2 Kn

, (366) :5.33x„ t m + 3 T „

-18.98xn (xm+3Tm)3

(366)

t m + 2 T m - ' ( x m + 2 T m ) 4 '

,0.562(xm + 2 T m ) 2 - ( x m + T m X x m + 3 T m ) : tm(^m+3Tm)-0.28l(xm+2T r a)3-xm(x r a+3Tn l

13 K (367) 1.066Tm -0.467xn •(367) _ 4.695xm(1.066Tm-0.467xm)

xm+2Tm

T (367) _ i d ~~ ln

0.331Tm-0.334xn

1.066T„ -0.467x„

K „ 14 v (368) .

1.6193(xm+2Tm)3 1

Kmxm(xm+3Tm) 2 Km ' xm + 3Tm

, (368) = 2.7051x„ •1.6706x„ (xm+3Tm

x m +2T m - •" ( x m + 2 T m ) 4 '

T <368> _ T IT J - T T )

1.7793(xm +2T m ) 2 - (x m + TmXx ro+3Tm) i d " T m l T m m j 1.6193(xm+2Tj3-xm(xm+3Tj2 '

U(s) = - ^ E ( s ) - K t

T,s 1 +

T -

R(s)

T

N

Y(s)

E(s) K

T iS

+ ^ U ( S ) Kme"

l + sTm

Y(s)

Table 64: PID controller tuning rules - FOLPD model l + sT„

Rule

Ogawa and Katayama

(2001).

Kc Ti Td Comment

Direct synthesis: time domain criteria 15 K (369) T (369)

i T (369) N=10

Minimum ISE - servo. Model: Method 1

15 e

Desired closed loop response = (l + sTCL)2 K (369) 1

~ K „

T „ T

XCL . T ^ C L

T T V ra x m J

j (369) _ T

<v T v XCL f 2 C L

T T V *m

l c k _ 2 ^ k + 4 m A

l CL + 4

V

( f

(369) _ . T T

m V m J

• T T IcL__2lck + 4 T T

^ e [0.01,1.00];

- ^ = -0.1902 T T

- 0.6974 - SL + 0.007393 , - ^ e fo.05,1.00l. 1 Tm T ' J



231

U(s) = Kc

>

v T i s E(s)-K1(l + Tdls)Y(s)

j

R(s) E(s) 1 + x-> U ( s )

Kc(l+ — + ! » _±*(g) „ K e "

l + sTm

KiG + T^s)

Y(s)

Table 65: PID controller tuning rules - FOLPD model G m (s) = Kme"

l + sT„

Rule


Model: Method I

Kc T, Td Comment

Robust

16 jr (370) T (370) T (370) Ki =0.25KU,

T d i =0

16K (370) l + 0.25KuKm

Kra(^ + t m )

Tm-0.25KuKmxm | xn

l + 0.25KuKra 2{X + xn

T(370) T m -0 .25K u K m x m | x 1 ; = — + l + 0.25KuKm 2(X + xm)'

T^10) _ l + 0.25KuKm 0.25KuKmxm2 3 4

Km(X + Tm) l2(l + 0.25KuKm) 6 (^ + T J 4{X + xm)2

, xm2 (Tm -0 .25K uKm im )

2(l + 0.25KuKmX^ + O '


4.1.23 Non-interacting controller 12 f

U(s) = K{ T,s 1

Tds + 1 E(s)-K,Y(s)

R(s) E(s)

+ i i

Table 66: PID tuning rules - FOLPD model Gm (s) = Kme"

l + sT„

Rule


Model: Method 1

Kc T, Td Comment

Robust

17 K (371) T (371) T (371) x d

K, =0.25KU

n £ (371) _ l + O^KyKn, fTm -0.25K l lKraTm

Km(^ + Tm) l + 0.25KuKm 2(^ + xm)

0.25KuKm im | xm2 | (Tm-0.25KuKmTm)Tm

2

2(l + 0.25KuKra 6(^ + Tm) 4(^ + Tm)2 2(l + 0.25KuKm)(X + xm)

2 T (371) _ Tm ~ 0 - 2 5 K u K m T m

l + 0.25KuKm 2(X + Tm)

0.25KuKm T m T „ (Tm-0.25KuKmT r a)in

2(l + 0.25KuKm 6(X + xJ 4{X + xm)2 2(1+ 0.25KuKm)(^ + xm)

(371

0.25KuKm (T ra-0.25K l lKmTm)xn

_2(l + 0.25K„Km 6(X + xm) 4(X + xm)2 2(1+ 0.25KuKra)(^ + Tm)

4.1.24 Industrial controller U(s) = K 1+ T,s

1 + T s R ( s ) - - ^ ^ Y ( s )

1 + ^ N

R(s) K, 1 + -

XS

U(s) K e~

l + sT„

Y(s)

1 + Tds

1 + ^ s N

Table 67: PID controller tuning rules - FOLPD model Gm (s) = Kme^

l + sT„

Rule


Model: Method 5

Minimum IAE Minimum ISE

Minimum ITAE


Model: Method 5

Minimum IAE Minimum ISE

Minimum ITAE

Kc Ti Td Comment


x i M1

,0 T

X 3

f \

T

x 4

xsTm T

x6

0 < ^ - < l ; T

m

N=10 Coefficient values

x i

0.91 1.1147 0.7058

x2

0.7938 0.8992 0.8872

X 3

1.01495 0.9324 1.03326

x4

1.00403 0.8753

0.99138

X 5

0.5414 0.56508 0.60006

X 6

0.7848 0.91107

0.971 Minimum performance index: servo tuning

x, TT

K m l T m

V2

J

Tm

* 3 - X 4 ^

(O x5Tm - ^

" 6

0 < - ^ - < l ; T

N=10


0.81699 1.1427 0.8326

x2

1.004 0.9365 0.7607

X 3

1.09112 0.99223 1.00268

x4

0.22387 0.35269 0.00854

x 5

0.44278 0.35308 0.44243

x 6

0.97186 0.78088 1.11499


Rule

Harris and Tyreus (1987).

Model: Method 1

Hoetal. (2001). Model: Method 1

Kc T, Td Comment

Robust Tm

K m (0 .5 t m + ^)

18 K (372)

1.551Tm

K -m A m T m

T

T

T

0.5tm

0.5xm

0.5xra

X > max imum [0.8im,Tm];N not specified

N not specified

cog = frequency at which the Nyquist curve has a magnitude of unity.


4.2 Non-Model Specific

235

4.2.1 Ideal controller Gc (s) = K£

R(s)

+ 0 E(s) K.

A

1 + — + THs

V T,s

)

U(s) Non-model specific

Y(s)

Table 68: PID controller tuning rules - non-model specific

Rule


Farrington (1950).

McAvoy and Johnson (1967).

Atkinson and Davey(1968).

Carr (1986); Pettit and Carr

(1987).

Parr (1989)-pages

190,191,193.

Tinham (1989).

Blickley(1990).

Kc T, Ta Comment

Ultimate cycle [ 0 . 6 K U , K J

[0.33KU,0.5KJ

0.54KU

0.25KU

0.5TU

Tu

T

0.75T„

0.125TU

[0.1TU,0.25TJ

0.2TU

0.25TU

Quarter decay ratio

20% overshoot - servo response

Ku

0.6667 Ku

0.5KU

0.5K„

0.5KU

0.5KU

0.4444KU

0.5KU

0.5T,

Tu

1.5TU

T

T 4 U

0.34TU

0.6TU

Tu

0.125TU

0.167TU

0.167TU

0.2TU

0.25TU

0.08TU

0.19TU

[0.125TU,

0.167TJ

Underdamped

Critically damped

Overdamped

£ =0.45

Less than quarter decay ratio response

Quarter decay ratio

Rule


Astrom (1982).


Hang and Astrom (1988b).


- page 60.

Astrom and Hagglund (1995) -page 141-142.

DePaor(1993).

McMillan (1994) - page 90. Calcev and

Gorez(1995).

Kc

0.75KU

Ku cos<|>m

0.87KU

0.71KU

0.50KU

Ara

Kucos<j>m

Kusin<|)m

0.35KU

0.4698KU

0.1988KU

0.2015K„

0.906KU

0.866KU

0.5KU

0.3536KU

4>m

T:

0.63TU

1 T (373)

Td

0.1T„

aT,(373)

Representative results 0.55TU

0.77TU

1.19TU

arbitrary

aTd<374)

Tu(l-cos<|)m)

0.77TU

0.4546TU

1.2308TU

0.7878TU

0.5TU

0.5TU

0.5TU

0.1592TU

= 45° , small xm

0.14TU

0.19TU

0.30TU

T

47l2T;

2 T (374)

Tu(l-cos<|)ra)

47tsin(j>m

0.19TU

0.1136TU

0.3077TU

0.1970TU

0.125TU

0.125TU

0.125TU

0.0398T;

<L =15°, large

Comment

Quarter decay ratio

* m = 3 0 °

l m = 4 5 °

K = 60°

Specify Am

Specify <j>m

A m > 2 ,

* m ^ 4 5 0

Am = 2,

<L=20°

Am = 2.44,

* m = 6 1 °

Am = 3.45,

4>m = 25°

4>m = 30°

tm

, (373) V" = | tan<|)m + .J4a + t a n 2 c t > m ] — • a = 0.25 (Astrom, 1982); N J 4cac

2 T (374) . tan <L+V^ -tan

a = 0.5 (Seki et al., 2000); 0.1 < a < 0.3 (Seki et al., 2000). \T


Rule

ABB Commander

300/310(1996).

Luo etal. (1996).

Karaboga and Kalinli (1996).

Luyben and Luyben ( 1 9 9 7 ) -

page 97.

Tan et al. (1999a ) -page

26.

Tan et al. (1999a) -page

27.

Wojsznis et al. (1999).

Yu (1999) -page 11.

Tan et al. (2001).

Chau (2002) -page 115.

Robbins (2002).

Smith (2003).

Rutherford (1950).

Kc

0.625KU

0.48KU

|p .32Ku ,0 .6Ku]

0.46KU

0.4KU

0.5KU

T,

0.5TU

0.5TU

[0.213TU,

1.406TJ

2.20TU

0.5TU

Tu

Td

0.083TU

0.125TU

[0.133TU,

0.469TJ

0.16TU

0.125TU

0.125TU

Comment

'P+D' tuning rule

Based on work by Tyreus and Luyben (1992)

Tighter damping than quarter decay ratio tuning

0.4KU

0.33KU

0.2K„

K c (375 ,

0.33KU

0.2KU

0.45KU

0.45KU

0.75K„

0.333TU

0.5TU

0.5TU

3 T (375)

0.5TU

0.5TU

T ( 3 7 6 ) 4 l j

4 T (377) i

0.625TU

0.083TU

0.125TU

0.125TU

Q 2 5 T _(375)

0.333TU

0.333TU

0.25TU

0.25TU

0.1T„

Some overshoot

No overshoot

"Just a bit o f overshoot

No overshoot

Minimum IAE -step change in

setpoint "Good" response - step change in

load

Other rules

1.25K37%

! - 1 2 K 3 7 %

T37%

T37%

° - 1 2 5 T 3 7 %

0.25T37% 37% decay ratio

c „ „ „ „ , (375), 2 2 , ' i (375)

0.25[Ti ( 3 7 5 ) l2Q)„2+l = 0.3183T„tan

0.571 + Z G ; Q ( D U )

- (376)

Gc(jff lu) = desired frequency response of controller Gc(jffl) at cou

= (0.24 + 0.11 l K m K u )TU T/ 3 7 7 ' = (0.27 + 0.054KmK„ )TU

Rule

MacLellan (1950).

Harriott (1964) -pages 179-180. Lip tak (1970) -pages 846-847.

Chesmond ( 1 9 8 2 ) - p a g e

400.

Parr ( 1 9 8 9 ) -page 191.

McMillan (1994) -page 43

Wade ( 1 9 9 4 ) -page 114.

Hay ( 1 9 9 8 ) -page 189.

ECOSSE team (1996b).

Bateson (2002) -page 616.

Rotach(1994).

Kc

1.77K37%

1.67K37%

1.49K37%

K 2 5 %

K 2 5 %

5 K (378)

0.999K25o/o

K 2 5 %

0-83K25%

0.67K25„/o

[1.002K25%>

1.671^] K 2 5 %

0.8497K50%

K 2 5 %

6 K (379)

Ti

1-29T37%

1.29T37„/o

° - 9 8 T 3 7 %

0.167T25./o

0.667T25%

j ( 3 7 8 )

0.488T25„/o

0.67T25%

0.5T25%

0.5T25%

0.45T25%

0.667T25%

0.475T50%

0.6667TU

T ( 3 7 9 )

Td

0.16T37./o

0.16T37„/o

0.24T37%

0.667T25%

0-167T25%

0.25Ti(378)

0.122T25%

0-17T25%

0.1T25o/o

0-lT25%

0.1125T25%

0.167T25„/o

0.1188T50%

0.1667TU

_0.5 ^ = -

Comment

37% decay ratio

Quarter decay ratio

Quarter decay ratio

x = 100(decay ratio)

Example: x = 25 i.e. quarter decay

ratio

Quarter decay ratio

'Fast' tuning

'Slow' tuning

'modified' ultimate cycle

5 K (378)

6 K (379)

0.5 + 0.3611nx

, (378) Tx%

2Vl +0.025 In2

M „

A/M^

• G . ( j < D M _ ) , Tj' (379)

dco.

Rule

Rotach(1994)-continued. Garcia and

Castelo (2000).

Lloyd (1994). Model: Method 3

Jones et al. (1997).

Kc

7 j , (380)

8 K (381)

0.4KU

0.4 Ku

0.7490

Ti

ry (380)

T (381)

0.3333TU

Tu

aT u ,

Td

T (380)

0.25Ti(381)

0.083TU

0.159TU

0.125TU

Comment

CO, < C 0 U

Self-regulating processes Non self-regulating processes

Model: Method 2

a e [0.2526,0.5053]

7 K (380) M „

Mraax - 1 •|Gp(jCDr)| .

, (380)

(380)

d ^ , . i f • -1 1 cor -•n + tan jt-sin + i

dcor Mmav

n 7 C -tan 7C-sin M „

0.5,

cor r dcor

1 + tan' 7i - s in + M,

tan 7t - sin Mn

K (381) cos(l80°+<t.m-ZGp(jco1)) ^ (381) 2[sin(l80° + +„ -ZGp( jm1))+ 1]

Gp(jco,) j(381) .

co1cos(l80°+(|)m-ZGp(jco1))

Rule

Shin et al. (1997).

NI Labview (2001).

Model: Method 2

Tang et al. (2002).

Model: Method 4

Dutton et al. (1997).

Kc

9 j £ (382)

0.6 Ku

0.25 K„

0.15KU

2.5465hsin<j>m

PX1 a,9(f ^ m

T,

T (382) i

0.5TU

0.5TU

0.5TU

5^(383) ^

5 e [l.5,4]

Td

aT/382*

0.12TU

0.12TU

0.12T„

10 T (383) 1 d

Comment

Model: Method 2

Quarter decay ratio

'Some' overshoot

'Little' overshoot

0.81 <x, <1

x, = 1 , 'small' xm ; x, = 0.91, 'large' Tm

u0.5Keu

T e 1 u 0.25Tu

e pages 576-8.

y2 = — s i n | ZGp(jcou) | . Typical a : 0.1; typical i,: [0.3,0.7]. Ku

10 T (383) _ -cot4>m+Vcot2c|)m +4/5

2co9 0 ,

Te = 6.283

1 -55

1110 with r = ratio of the height of the

first peak to the height of the first trough of the closed loop response, when controller

gain Kc is applied; cod = damped natural frequency.

4.2.2 Ideal controller in series with a first order lag r

R(s) E(s)

- < g > K 1 + — + T « S

V T i S

Gc(s) = K£

U(s),

1 \ 1 + — + Tds

v T i s j

1

Tfs + 1 Non-model

specific

Tfs + 1

Y(s)


Rule


Kc Ti Td Comment

Other tuning: minimum performance index 12 K (384) T (384) T (384)

"Almost" optimal tuning rule; K)J0„ > 0.03 ; 1.6 < M s < 1.9

12 K (384)

,(384)

0.60

K„ K

150" K„

-0.07 0.32 + 1.6-K-,5no

K„

K 1 5 0 °

1.5

0.32 + 1.6-

(384)

(384) _ _

K 1 5 0 °

0.6667

-0.8 ^ 1 5 0 "

0.32 + 1.6-. K-nnO

K „ -0.8

K,

V K m J

0.4

K l 5 » ° - - 0 . 0 7 K „

K-,<„0 - J ^ l ; n > 0.32 + 1 . 6 — ^ _ - 0 . 8 |

K„ K „

-or

min-^ 6 -K„

K I 5 0 °

,25

(384)

20- 150° + 0.5 0.32 + 1 . 6 — ^ - - 0 . 8 | K,

K.


Rule

Wang et al (1995b).

Kc T: Td Comment

Direct synthesis 13 jr (385) T(385) T (385)

' d

13 For a stable process, Km is assumed known; for an integrating process, Km and T„

are assumed known. K (385) -Im • CL

JMCLGCLQCOCL)

G P O ^ C L ^ - G C L Q C O C L ) }

. (385)

Im 1

Rl

JCOCLGCL(jfflCL)

GpGttcLXi-GcLGaa.)}

J«CLGCL(J®CL)

_Gp(jfflCLXl-GCL(jQ)CL)}

1 + —

3

JMCLGCLCPCL) Rl •Rl Gp(j(ocJl-GCL(j«CL)jj 1 Gp 02(00. Xl-Go. 020)0.)]

j2coCLGCL(j2co

(385)

Rl f JMcLGcLptfcL) -Rl J2(»CLGCL(J2C0CL) 1

Gp(jcoCLIl-GCLOraCL)]J [Gp02coCL)[l-GCL02coCL).

Im JOJCLGCLO^CL)

GpOfflCLl1-GCLO(BCL)]j

4.2.3 Ideal controller in series with a second order filter f 1 A

G c ( s ) = Kc

V T i s J

l + bfIs

l + afls + af2s

R(s) E(s)

-+®-+ + x

U(s)

K, l + — + Tds V T i s

l + b f l s

l + a f ,s + a f 2s

Non-model specific

Y(s)


Rule

Kristiansson et al. (2000).

Kc Ti Td Comment

Minimum performance index: other tuning l K (386) T (386) rj, (386) M . * l . 7

"Almost" optimal tuning rule

1.6

l ^ (386)

. (386)

Gp(j(Du) Gp(jcou) 1.6-L^ ^-L-2.3J—- L + l.l

K „ K „

Km

1.6

0.37 + Gp(jcou)|

K „

0.37 + G p ( jcoJ

=r,T, (386)

, b l f - 0 , alf - T f , a2f - Tf'

0.625

K „ 0.37 +

G p O u

K „

, ,M1.2 .3W+ 1 , K „ K „

0.37 + Gp(jcou)

K „ 13-20

Gp(ja>„)

Kn

|Gp(jcoj|

Km

< 0.5 or

! ,Ml_,3ra + 1 , K „

3co„ 0.37-Gp(j(Bu)

K~

K „

13-20 G P ( K )

Km

|Gp(jfflj|

Km

> 0 . 5 .

Rule


Kc

2 K (387)

Ti

T (387) i

Td

T (387) x d

Comment

"Almost" optimal tuning

rule; K,5 0„>0.03;

1.6 <MS <1.9

2 K (387) 0.60

K „ K 1 5 0 ° -0.07 0.32 + 1.6-

K,

K „ -0.8

V K m J

,(387)

(387)

1.5

0.32 + 1.6—^--0.8 Kn.

0.6667

K,

V K m J

0.32 + 1.6-K,,„o 150"

K „ -0.8 ^ 1 5 0 °

V Km J

b l f = 0 ,

0.8

2 0 ^ + 0.5 0.32 + 1 . 6 ^ ^ - 0 . ^ K l 5 0 ' ^ K „ K „

2 0 ^ 1 ^ + 0.5 0.32 + 1 . 6 ^ L - O.gf K ' s ° u

K „ K „

Gc(s) = Kc

245

R(s)

K c ( b - 1 )

+ T

Y(s) E(s)

> K r 1 A

1 + — + Tds v T i s J

+ T U(s)

+ Non-model

specific

v T i s

Y(s)


Rule


-page 217.

Streeter et al. (2003).

Kc Ti Td Comment


4 j r (389)

5 K (390)

ry (388)

T ( 3 8 9 ) i

T (390) 1 i

T (388)

T (389)

T (390)

0 < K m K u < c o

Mmax = 1.4

0 < K m K u < c o

Mmax =2.0

0 < K m K u < c o

Mmax =2.0

Kc(388) =(p.33e-°J l M 2)cu , T,(388) = (o.76e-L6K-036K2 )ru, (o^e"031*-*2 )KU

T ^ ^ O . n e ^ - ^ J i ^ b ^ . S S e - '

Kc<389) = (o.72e-L6K+1 2K2 )KU , Ti'389' = (o.59e-'-

3K+0-38K2 )ru ,

Td(389) =(o.l5e

1.3K+0.38

1.4K+0.56I

6

" % » = 0.25eu

Kc(390) =(o.72e-''6K+L2l<2)K:u -0.0012340TU -6.1173.10"

(390) _ (o.59e-'-3K+a38K2 ](o.72e-'6K+'-2lc2 )K„ - 0.0012340T - 6.1173.10"6 \ru

0.72e~

(390) 0.108e" KUTU - 0.00266401

0.040430e"1JK

log(l.6342'

•38K ] K U

w ° 0.72K,1e"L6K+,'2l<2 -0.0012340T,, -6.1173.10~6

K „

4.2.5 Controller with filtered derivative Gc(s) = K t 1 + — + — —

R(s) E(s)

<8M + T K „

, 1 T d s

1 + — + d T*s l + ^ s

N .

U(s)

Non-model specific

N

Y(s)


Rule

Leva (1993). N not specified

Yi and De Moor (1994).

Kc Ti Td Comment

Direct synthesis 6 R (391)

7 K (392)

8 K (393)

T {391)

aTd(392)

T (393) *i

6.283/Pco

T (392)

T (393) Xd

(3 = 10

6 < a < 1 0

N not defined

6 K (391) 0)T: (391)

Gp(j<4 1-coT, (391) 2TC

pj + W

2[T i<391)]2

X (391)

Y + ytan(<l>m-<l>o>-0-57c)

. <h> > < L " * •

7 K (392) COT: (392)

|Gp(ja,)|V(l-a,2T i^Td^)2+fl)

2[T i«M2>]2

(392) • aco + y a2©2 + 4aco2 tan2 (<(>m - <|>a - 0.5TC)

2aa>2 tan((j)m - <j>m - 0.57t)

K l J „ , 0.5cos[225°-ZGp(ja)^J T < 3 9 3 ) = ^ ( 393 ) (393)

Gp(jW^)

(393) tan[2250-ZGp(jffl^)]+A/4 + tan2[225°-ZGp(jffl^)

2®A

Rule

Vranfiic(1996)-page 130.

Vrancic (1996)-pages 120-121.

Vrancic (1996) -page 134.

Vrancic et al. (1999).

Kc T (394)

2(A,-T, (394 ))

10 J , (395)

1 1 K (396)

12 K (397)

T|

9 T (394)

T (395)

T (396) i

T ( 3 9 7 )

Td

T (394)

A 3 A 4 - A 2 A 5

A3 _ A , A 5

^ ( 3 % )

T (397)

Comment

N<10

N>10

N = 10;

1 = [0.2,0.25]

8 < N < 20

9 T (394)

A 2 - T d ^ > A , -(394)-.2

N

(394) -(A32 - A5A,) +J (A 3

2 - A 5 A J - 1 ( A 3 A 2 - A5XA5A2 - A4A3)

N ( A 3 A 2 - A 5 )

10 v (395) _ *r \."*> 0.5A3\A3 A,A5 j C ~ (A 3

2 -A 1 A 5 ) (A 1 A 2 -A 3 K r a ) - (A 3 A 4 -A 2 A 5 )A 12 '

T(395) = A 3 ( A 32 - A | A 5 )

' ( \ 32 - A , A 5 ) A 2 - ( A 3 A 4 - A 2 A 5 ) A 1

A2 - V A 22 - 4 X A , A 3 11 K (396)

12 K (397)

(396) 0.5T;

A . - K , , , ^ (396) T

(396)

2XA,

A 1 A 2 - A 0 A 3 - T d < 3 9 7 , A 12 - I I i -

- (397) .

N

(397)

-A0A, J

A - T ( 3 9 7 ) A [T, (397)-, 2

I J__

N

, T6 is obtained by solving:

K „

V M T (397) M , A ,A 3 L (397)}

W^ J + 1 ^ [ d J ApA5 A2A3

N [Td(397)f

+ (A 32 - A, A5lTd

<397)]+ (A2A5 - A3A4) = 0 .

Rule

Lennartson and Kristiansson

(1997).

Kristiansson and Lennartson

(1998a).


(1998b).

Kc

13 K (398)

14 j , (399)

15 K (401)

16 K (402)

Ti

T (398) i

T (399)

T (400) •M

T (401)

T (402)

Td

0 4 T . ( 3 9 8 )

0 4 T , ( 3 9 9 ,

0 4 T ( 4 0 0 )

04T . (40„

0.333T;(402)

Comment

KuKm>1.67

N = 2.5

13 v (398) v v 12KU Km -35K u K m +30 k c - N ^ n T ; ; 1 .

K„Km +2.5|l2K„2Km2 -35K u K m +30J

• (398) _

u m ' '— ' |_* "" * - u " m

K (398)

-0.053co„3 +0.47(0„2 -0.14co„ +0.11

N: 2.5 12 55_+ 30

KmKu K 2 K 2K 2

m K u

14 v (399) _ v v

, (399)

12K 2 K m2 - 3 5 K u K m + 3 0

K u K m + x , [ l 2 K u2 K m

2 - 3 5 K u K m + 3 o ] '

K (399)

-0.525CO, +0.473cou2 -0.143cou +0.113 KUK

±— < 0.4;

, (400) K (400)

-0.185couJ+1.052cou -0.854cou +0.309 KUK

, - ^ > 0 . 4 ;

N = - i l - 35 30 12 — + -

15 K (401)

Km K u Km

2Ku2

7.71 9.14

x , = 3 , K u K m > 1 0 ; x , = 2 . 5 , K u K m < 1 0 .

, , - + 3.14, KuKra > 1.67 , ii—>0.45; K 2K„2 KmK„ m K„K„ ••m " - U m " i

,(401) K, (401)

• 0.63co„3 + 0.39co„2 + 0.15co„ + 0.0082

16 v (402) 12K„3K 2 - 3 5 K „ 2 K m + 3 0 K „

" Km3Ku

3 +2.5[l2Ku2Km

2 -35K u K m +3o] '

,(402)

j Kc

(4°2)Km3Ku

2

cou[0.95Km2Ku

2-2Kr

2.5 — 1 , N =

Ku+1.4j KmKu

1 2 — 1 * - + - 3 0 K

m K u Km2Ku

2

Rule

Rristiansson and Lennartson

(2000).

Kc

17 K (403)

18 K (404)

T;

j (403)

T (404)

Td

T (403)

T (404) ld

Comment

0.1<KuKm

<0.5

K„K r a>0.5

'K (403) ( l . lK u2 K m

2 -2 .3K u K m + 1 .6 ) 2

K r a2Ku(-20 + 13KmKJ(l + 0.37KmKu)2

"l.6(-20 + 13KmKu)(l + Q.37KmKu)

l . lK m2 K u

2 -2 .3K m K u + 1 .6

-(403) _ K m K u ( l . lK u2 K r o

2 -2 .3K u K m +1.6)

cou(-20 + 13KmKJ(l + 0.37KmKu)2

1.6(-20 + 13KmKJ(l + 0.37KmKu) ^

U K m2 K u

2 - 2 . 3 K m K u + 1 . 6

(403) _ KmKu(l . lKu Km -2.3KuKm+1.6)

o)u(-20 + 13KmKu)(l + 0.37KmKu)2

(-20 + 13KmKu)2(l + 0.37KmKu)2

( l . lK m2 K u

2 -2 .3K m K u +1.6) 2

1.6(-20 + 13KmKu)(l + 0.37KmKJ

UK m2 K u

2 -2 .3K r a K u +1 .6

- -1 N: (403)

(403)

r (403) _ KmKu(l . lKu Km -2.3KuKm+1.6) f cou(-20 + 13KmKJ(l + 0.37KmKu)2

K (404) ( l . lK u

2 K m2 -2 .3K u K m +1.6) 2

3G)uKm2Ku

2(l + 0.37KmKu)2

4.8KmKu(l + 0.37KmKJ

U K 2K 2 •2.3K raKu+1.6

, (404) _ (UK u2 K m

2 -2 .3K u K m +1 .6 ) x2

(404) _

3cou(l + 0.37KmKur

( l . lK u2 K m

2 -2 .3K u K m +1.6)

3au(l + 0.37KmKu)^

4.8KmKu(l + 0.37KmKJ

l . lK m2 K u

2 -2 .3K r a K u +1.6

3Km2Ku

2(l + 0.37KmKJ2

( l . lK m2 K u

2 -2 .3K m K u +1.6) 2

4.8KmKu(l + 0.37KmKu)

l . lK m2 K u

2 -2 .3K m K u +1.6

(404)

N: Tf

(404) . T f (404) l . lK u

2 K m2 -2 .3K u K m +1.6

3cou(l + 0 .37K m KJ 2

Rule


(2000) -continued.


(2002).

Kc

19 K (405)

20 K (406)

T;

j(405)

T (406)

Td

T (405)

T (406) Xd

Comment

K u K m <0.1

K u K m <10

19 K (405)

• (405)

(-6 + 3.7Kn,„Km)2 20.8(1.8+ 0.3KmK..,„)

13Kra(1.8 + 0.3KmK135o)2

(-6 + 3.7K135„Km)K135oKm

13co135o(1.8 + 0.3KmK1350)2

(405) _ ( _ 6 + 3-7 KmK135°)KmK- i35»

" 13o3135.(1.8 + 0.3KmK135„)2

(-6 + 3.7KmK135o)

20.8(1.8+ 0.3KmK135o)

(-6 + 3.7KraK1350)

' 169(1.8+ 0.3KmK135,)2

(-6 + 3.7KmKm„)2

20.8(1.8 + 0 . 3 K m K „ )

(405)

N = -(405) .Tf

(405)

l f

(-6 + 3.7KmK135„)

(-6 + 3.7KmK135o)KmK135„

13co135„(1.8 + 0.3KmK|35„)

- 1

2 '

(406) = Ku [l.5(KmKu +4)(0.4KmKu +0 .75)-K m K u X l ]x ,

Km (0.4KmKu+0.75)2(4 + KmK u )

(406) _ KmKu [l.5(KmKu +4)(0.4KmKu + 0.75)-KmK ux,]

CO

(406) K m K u

(0.4KmKu+0.75)2(4 + KmK u )

(4 + K m K J

u [l.5(4 + KraKu )(0.4KmKu + 0.75) - KmKux, \l + N" 1 ) '

N = -(406)

(406)

j (406) _ K m K u

(0.4KmKu+0.75)2(4 + K m K J

x, = 0.13 + 0.16KmKu -0.007Km2Ku

2 .

Rule


(2002) -continued.

Kc

21 K (407)

Ti

T (407)

Td

T (407)

Comment

K uK r a>10

21 K (407) Kn5o [l.S(0.44KmK,35o +1.4)x2 -KmK,35oX3Jx3

K „ (0.44KmK135„+l.4)'x2

,(407) KmK135o [l.5(0.44KmK,35o +1.4)x2 - K m K u x 3 j

(407)

c o 1 3 5 „

K m K i 3 5 °

<B,

(0.44KmK,35o+1.4)2x2

J135" [l.5x2(0.44KmK135„+1.4)-KroK135„x3ll + N - 1 ) '

N: (407)

d (407) . T f

K 2K 2

(407) _ ^ m ^ 1 3 5 °

(0, (0.44KmK,1,„+1.4) ix2

x2 =min(14 + 0.5KmK135„,25),x3 =1.35 + 0.35KmK,35„ - 0 . 0 0 6 K m2 K | 3 /


4.2.6 Ideal controller with set-point weighting 1

U(s) = K c (F pR(s)-Y(s))+^(F iR(s)-Y(s)) + KcTds(FdR(s)-Y(s)) 1 :b


Rule

Mantz and Tacconi(1989).

Kc T i Td Comment

Ultimate cycle 0.6KU ;

Fp=0.17 0.5TU; 0.125T,;

Fd = 0.654

Quarter decay ratio

4.2.7 Classical controller 1 Gc(s) = K 1 T;S

1 + sT,

N

R(s) E(s)

K, 1 + — Xs T

N ;

U(s)

Non-model specific

Y(s)


Rule


(1997) -page 8-15.

Kinney (1983).


St. Clair (1997)-page 17.

N > 1

Harrold(1999).

Kc T; Td Comment


0.56KU 0.39TU 0.14TU N = 1 0

Ultimate cycle 0.25KU

0.6KU

0.5KU

0.25KU

0.67KU

0.5TU

0.5TU

1.2TU

1.2TU

0.5TU

0.12T„

0.125TU

0.125TU

0.125TU

0.125TU

N not defined

10 < N < 20

'aggressive' tuning

'conservative' tuning

N not defined

R(s) E(s) K,

v T i s y (l + sTd)

U(s) = K

U(s)

— •

' i+JL v T ;sy

(l + s T d )

Non-model specific

Y(s)

Ta

Rule

Pessen(1953).

Pessen (1954).

Tan et al. (1999a) -page

26. O'Dwyer (2001b)-

representative results - Astrom and Hagglund

(1995) equivalent -page 142.

Tan et al. (2001).

ble 75: PID controller tuning rules -

Kc

0.25KU

0.2KU

'optimum' re 0.33KU

0.3KU

0.2349KU

0.0944KU

0.1008KU

1 K (408)

Tj

- non-model specific

Td

Ultimate cycle 0.33TU

0.25TU

gulator response -0.5TU

0.15TU

0.2273TU

0.6154TU

0.3939TU

y (408) i

0.5TU

0.5TU

step changes 0.33TU

0.25TU

0.2273TU

0.6154TU

0.3939TU

0 2 5 T _(408 )

Comment

'optimum' servo response

'Some'overshoot

Ziegler-Nichols ultimate cycle

equivalent A m = 2 ,

*m = 20°

A r a=2.44,

<t»m=61°

Ara = 3.45,

+ m =46°

1 K (408) = _

, 4 0 8 ) ^ >u|Gc(jttu)|

.(408)

V[T i<408)]2cou

2+lVo.0625[T i(408)]2Q)u

2

6.25cou2 +4cou

2 tan2 2ZG;(jcou)+7t"

2

+ 1

-2.5cou

co„ tan 2ZG;(JCO U )+7I

Gc(jcou) = desired controller frequency response at oo = cou


/

4.2.9 Classical controller 4 Gc(s) = Kc

v T i s y 1 + - s T d

l + ^ L V N

R(s) E(s)

4- T K,

V T i s 7 1 + -

sT„

1 + s-N

U(s)

Non-model specific

Y(s)


Rule

Hang et al. (1993b)

- page 58.

Kc Ti Td Comment

Ultimate cycle

0.35KU 1.13TU 0.20TU N not specified

f \ f >

U(s) = Kc 1 + — + - a

V

T . s 1 + ^ - s N j

E ( s ) - K c a + PTds

1 + ^ s

N ;

R(s)

( \

PTds

1 + ^ s N

a + K_

+

R(s) E(s)

K, 1+ — + - d

T . s 1 + ^ - s N .

U(s)

><%>-+

Y(s)

Non-model specific


Rule

Shen (2002).

Kc Ti Td Comment

Minimum performance index: other tuning 2 K (409) T(409) T (409)

Ku > 1/Km

! Minimise Jr(t) - y(t)|dt + f|d(t) - y(t)|dt; N=0, (3 = 0, Ms < 2 :

K c( 4 0 9 '=K u exp 0.17-2.62/ KmK„ +1.79,

T, ( 4 0 9 )=Tuexp

Td( 4 0 9 )=Tuexp

-0 .02-2 .62/ K m K u +1.34,

1.70-0.59/ K K „ -0.25,

a = 1 - exp 0.30-0.48/ K m K u +0.93,

( . 2 ^

Km Ku

v j

f /> 2\

Km Ku

V J

f * 2 \ Km Ku

V J

If -2 Km Ku



257

R(s) E(s)

+ X Kr

XS +

U(s) = K 1 + -V T i S /

E(s)-KcTdsY(s)

Non-model specific

Y(s)

K J d s

I Table 78: PID controller tuning rules - non-model specific

Rule


(1997)-page 8-15.

VanDoren (1998).

Kc Ti Td Comment


0.77KU 0.48TU 0.11TU

Ultimate cycle 0.75KU 0.625TU 0.1TU



U(s) = Kc T:S

E(s)-^-sY(s) N

R(s) E(s>

+ X K „ 1+ — + Tds 3

U(s)

•

Non-model specific

Y(s)

N

3: Table 79: PID controller tuning rules - non-model specific

Rule

Bateson (2002) -pages 629-637.

Model: Method 1

Kc Ti Td Comment

Other tuning 3 K (410) T ( 4 1 0 ) 2

0 5 180° N = 1 0

3 K (410) = 1 0 » {{-K^Do^iGp^^H

dB;T i(410)=min[mg20,0.203,^. ]

' ppO^HO0] a n d | G P U ° W ) | are given in

4.3 IPD Model Gm(s)


K e"ST™

259

f 4.3.1 Ideal controller Gc(s) = Kc

1 A

v T i s j

Table 80: PID controller tuning rules - IPD model Gm(s) = Kme"

Rule

Ford (1953). Model: Method 3

Astrom and Hagglund(1995)

-page 139.

Hay (1998)-page 188.

Hay (1998)-page 199.

Model: Method i; KC!Td

deduced from graphs

Visioli (2001). Model: Method 1

Kc Ti Td Comment

Process reaction 1.48

K m x m

0.94

Km tm

2xm

2xm

0.37xm

0.5xra

Decay ratio 2.7:1

Model: Method 1

Ultimate cycle Ziegler-Nichols equivalent 0.4

Km 1™

10.0

4.0

2.5

2.0

1.8

3.2xra

3-2KmTm2

0-8xm

0.55xm

0.30tm

0.25tm

0.25Tra

0.25xm

Model: Method 3

K m x m =0.1

K r axm=0.2

K r axm=0.3

K r axm=0.4

Kmxm =0.5,0.6

Minimum performance index: regulator tuning X i /K r a x m x2Xm

x3xm

Coefficient values 1.37 1.36 1.34

1.49 1.66 1.83

0.59 0.53 0.49

Minimum ISE Minimum ITSE Minimum ISTSE

Rule

Visioli (2001). Model: Method 1

Astrom and Hagglund (2004). Model: Method 1

Leonard (1994). Model: Method 1

Wang and Cluett (1997).

Cluett and Wang (1997).

Model: Method 1

Kc ^ Td Comment

Minimum performance index: servo tuning x l / K m T m

1.03 0.96 0.90

0 x 3 Tm

Coefficient values 0.49 0.45 0.45

Minimum ISE Minimum ITSE

Minimum ISTSE Minimum performance index: other tuning

X l / K m T m

X l

0.139 0.261 0.367 0.460 0.543

x2

76.9 23.3 12.2 7.85 5.78

0.74 K m T m

0.47KU

1 K (411)

2 K (412)

X i / K r a x m

X l

0.9588 0.6232 0.4668

x2

3.0425 5.2586 7.2291

X 2 t m x 3 T m


0.346 0.365 0.378 0.389 0.400

Mmax

1.1 1.2 1.3 1.4 1.5

x i

0.616 0.681 0.740 0.793 0.841

x2

4.58 3.82 3.28 2.89 2.61

Direct synthesis

12.2tm

3.05TU

T ( 4 1 1 )

T (412)

0.4lTra

0.10TU

T (411)

T (412)

K u, Tu deduced from graph

X 2 t m X 3 t m

Coefficient values X 3

0.3912 0.2632 0.2058

x4

1 2 3

X l

0.3752 0.3144 0.2709

x2

9.1925 11.1637 13.1416

X 3

0.410 0.418 0.426 0.434 0.440

Mmax

1.6 1.7 1.8 1.9 2.0

OS (step input) < 10%; Minimum

IAE (disturbance ramp).

Model: Method 1

To. = X 4 T m

X 3

0.1702 0.1453 0.1269

x4

4 5 6

1 K (411)

KmTm(0.7138TCL+0.3904)

T (411) _ X

T/41" = (l.4020TCL +1.2076)rm ,

TCLe[T ra)16Tm], ^ = 0.707.

2 K (412) = 1 Kmim(0.5080TCL+0.6208)

• (412) .

1.4167TCL+1.6999

Ti(412) =(l.9885TCL+1.2235)rm,

.0043TCL +1.8194 T C L e k , 1 6 T m ] , %=\.

Rule

Rotach(1995). Model: Method 5

Chen et al. (1999a).

Chidambaram and Sree (2003).


(2005b). Chen and Seborg

(2002).

Zou etal. (1997), Zou and Brigham


or Method 8

Kc

1.21 K m T m

Ti

1.60xm

Td

0.48xm

Damping factor for oscillations to a disturbance 1.661

A m K r r J m

1.2346 K m T m

0.896

Kmxm

5 K (415)

2

Km(X + 0.5Tm)

0

4-5xm

2.5xm

T (415) i

2 ^ + Tm

3 T (413)

4 T (414) ' d

0.45xm

0.55xm

T (415)

Robust

X + 0.25xm

— x 2X + xm

Comment

input = 0.75.

Model: Method 1

Model: Method 1

Model: Method I

Model: Method 1

0.5xm<^

^ 3 x m

Td<413) = 1.273-cJl- 0.6002Am (l.571 •

4 T / 4 , 4 ) = T J l . 2 7 3 2 riAm

1.6661

i(A _. T,(415)S(l + 0.5Tms)e-^ (4i5) _ T m (3TCL+Q.5tm)

' •' " """ " ' ' ~Km(TcL+0.5xmr v ^ / desired K c (TCL S + 1)'

. ( 4 1 5 ) = 3TCL+0.5Tm, Td

(415) 1.5T 2xm + 0 .75T r , x2 +0.125xm

3 - T 3

xm(3TCL+0.5xm)



Gc(s) = Kc 1 \

1 + — + Tds V T i s j Tfs + 1

R ( V «•>. +1 k

Kc f 1 ^ I Ti s )

1

1 + Tfs

U(s) K m e " ^

s

Y(s)

Table 81: PID controller tuning rules - IPD model Gm (s) = Kme"

Rule

Rice and Cooper (2002).

Model: Method 1

Kc T i Td Comment

Robust

6 K (416) 2X + 1.5xm

0.5xm2+Xxm

2X + 1.5Tm

Suggested X = 3.162tm

6 j , (416) = _ 2X + 1.5tn

K, 1 ^ + 2 X x m + 0 . 5 t m2 )

. Tf = 0.5X2x„

^ + 2 ^ x m + 0 . 5 x m2


4.3.3 Ideal controller with first order filter and set-point weighting 2

U(s) = Kc

v T i s j

1

Tfs + 1

, l + 0.4Trs R(s) — - Y(s)

1 + sT

R(s)

— • 1 + 0.4Ts

1 + T s

+ E(s) f i A

1+ — + Tds V T i S J

1

Tfs + 1

+ U(s)

•K 0 Y(s)

Y(s)

+ & » Kme"

Kn

Table 82: PID controller tuning rules - IPD model Gm (s) = Kme-

Rule


Model: Method 1

Kc Ti Td Comment

Direct synthesis

0.563

Kmt r a l-5xm 0.667tm

T f=0.13xm

Y ~ l

° 2Kraxm

T r=0.75xm


4.3.4 Controller with filtered derivative Gc(s) = K(

R(s)

1 + — + d

N

?> E(s) » L

Kc 1, l , T « s

T>s l + ^ s V iN J

U(s)

s

Y(s)

—P-

Table 83: PID controller tuning rules - IPD model Gm (s) = K m e"

Rule


(2002).


Kc T, Td Comment

Direct synthesis 7 K (417) T (417) T (417)

Model: Method 1

Robust 8 K (418) 2X + xm Tm(X + 0.25xra)

2X + xm

N=10

7 K (417) 2.5(0,,

K „ .0.08 + ^ + ^

K, "i J

• (417) 2.5 2.3 - 2 K

_ (417) _ 2.5

N = -j (417)

(O .Q59 /K I2 )+ (0.055/K,)- 0.08

2.3-2K, 0.4(10+ 1/K,) —+ 1 N

T, (417) >T f

(417) (0.059/K!2)+ (0.055/K))- 0.08

0.16cou(lO + l/K!) , K, e [0.5,1].

8 K (418) 2>. + T„

Km(0.5^ + Tra)2 •; X e [l/Km,Tm] (Chien and Fruehauf (1990)).

2 (o.osVK^+fo.oss/Kj-o.os' 2.3 -2K, 0.4(10 + 1/K,)

2 (0.059/K)2)+ (0.055/K!)- 0.08"

0.4(10 + 1/K,)


4.3.5 Controller with filtered derivative and dynamics on the controlled variable

Gc(s) = Kc 1+ — + 5-

R(s) E(s)

+ T K,

N

N

+ U(s)

E(s)-K0Y(s)

* ® ^ K „

Y(s)

Table 84: PID controller tuning rules - IPD model Gm (s) = K m e ^

Rule


Model: Method I

Kc Ti Td Comment

Direct synthesis - time domain criteria

9 K (419)

0.1716 K m T m

(1.5-yk,

*m

T (419)

0.5xra

Y - X

" ° 2 K m * r a

N = l;

Y - l

" ° 2 K m x m

Recommended y = 0.5 ; non-oscillatory response

•5-Y 4y + 1 - 4^/y2 +0.5y .T, (419)

1.5-y " Y ^ m . N •

l - r ( l •5-Y) (1.5 -y)y

(


V T i s V

1 + Tds

N J

R(s) &

E(s)

+ T

Y(s)

K „

v T i s y

l + sTd

l + s ^ N

U(s) Kme" Y(s)

Table 85: PID controller tuning rules - IPD model G m (s) = Kme"

Rule

Bunzemeier (1998). Model:

Method 4; G P =

K P

s(l + sT p )"


-page 143.


-page 74. Model:

Method 1

Kc

x i

K m T m m m

x4

K m x m

x, 0.21

0.68 0.47

0.35

0.30 0.27 0.25 0.23 0.22

0.22

^ 0.93

K r a x m

0.93

K m T m

0.93

K m x m

x2

0.730

1.570 1.200

0.981

0.866 0.828 0.801

0.782 0.767

0.756 linimum p<

1

1

Tf Td

Process reaction

X2 t ra

Coefficie

x3

0.280

0.796

0.669

0.587 0.530

0.487 0.452

0.424 0.401

0.381

;rformance

•57xm

•60xra

• 4 8 t m

X 3 t r a

nt values

x4

0.24

0.94 0.62

0.46

0.38 0.34

0.32

0.30 0.29

0.27

index: regi

0.58xn

0.58Tn

0.63xn

Comment



N

5.60

5.99 6.03

5.99

6.02 6.01 6.03

5.97 5.99

5.95

n

0 2 3 4 5 6 7 8 9 10

ilator tuning

i

i

i

Model: Method 2;

N not specified

N=10

N=20


Rule


-page 117. Minimum IAE -Shinskey (1996)

-page 121.


Rice and Cooper (2002).

Model: Method 1

Luyben (1996). Model: Method 9

Belanger and Luyben (1997).

Model: Method 1

Kc

0.93 K m X m

0.56KU

10 K (420)

10 K (421)

11 K (422)

0.46KU

3.11KU

Ti

1.57xm

0.39TU

Td

0.56xm

0.15TU

Robust

0.5xm

2X. + 0.5xm

2X + im

2>. + 0.5xm

0.5xm

0.5xm

Ultimate cycle

2.2TU

2.2TU

0.16TU

3.64TU

Comment

Model: Method 2;

N not specified Model:


^ [ l / K m , x m ] (Chien and Fruehauf

(1990)); N=10 Suggested

k = 3.162xm

Maximum closed loop log modulus of +2dB ; N=10

N=0.1

10 K (420) _ 1

Km

0.5x„ K (421)

^ + 0.5xJ2J

K m ^ 2 +2^x m +0 .5x m2 )

K „

2X + 0.5xm

[ + 0.5xm]2

N = X.2+2A.xm+0.5x,

4.3.7 Classical controller 2 Gc (s) = K

E(s) R(s)

<R)->K f 1 "\ l V l + NTds

v T i s A l + T a s J i + -

v T i s y

U(s)

l + NTds

K„e" Y(s)

Table 86: PID controller tuning rules - IPD model Gra (s) = Kme"

Rule

Hougen(l979)-page 333-334.

Model: Method 1

Hougen(1988). Model: Method 1

Kc ^

Direct synthesis - freq

12 K (423)

13 K (424)

00

oo

Td Comment

uency domain criteria

0.45^2-N

]QD°Sro im+0-65]

Maximise crossover frequency Ne[l0,30]

Five criteria are fulfilled; N = 10.

' Values deduced from graph:

Kmxm

K (423)

K r a T m

K (423)

0.1

9

0.6

1.45

0.2

4.2

0.7

1.2

0.3

2.8

0.8

1.1

0.4

2.1

0.9

0.95

0.5

1.7

1.0

0.85

Equations deduced from graph; Kc ' «—— 10 K „ T'



f P + — v T i s y

l + ^ s

1-3* N


Rule


Kc T( Td Comment

Robust 1 K (125)

2 ^ (426)

2>. + 0.5Tm

0.5xm

0.5xm

2X + 0.5xm

^e[l/Km,tmJ (Chien and Fruehauf

(1990)); N=10

K (425)

K „

2? I + 0 . 5 T „

[X + 0.5T m j J

2 K (426) 1 I 0 .5 l „

Km{[x + 0.5xJ



f \ f ^

U(s) = K, R(s)

Table 88: PID controller tuning rules - IPD model G m (s) =

Rule

Minimum ITAE - Pecharroman

and Pagola (2000). Model:

Method 10

Kc Tf Td Comment

Minimum performance index: servo/regulator tuning x,Ku x2Tu * 3 T u


1.672

1.236

0.994

0.842

0.752

0.679

0.635

0.590

0.551

0.520

0.509

x2

0.366

0.427

0.486

0.538

0.567

0.610

0.637

0.669

0.690

0.776

0.810

X 3

0.136

0.149

0.155

0.154

0.157

0.149

0.142

0.133

0.114

0.087

0.068

a

0.601

0.607

0.610

0.616

0.605

0.610

0.612

0.610

0.616

0.609

0.611

4»c

-164°

-160°

-155°

-150°

-145°

-140°

-135°

-130°

-125°

-120°

-118°

Rule


Model; Method 1



Model: Method 1

Chidambaram and Sree (2003). Model: Method 1

Kc

1

Km

f 1.253^

T,

2.388xm

Td

0.4137xm

Comment

^2-<1.0 T

m a = 0.6642 , p = 0.6797 , Overshoot (servo step) < 20% ;

N fixed but not specified Direct synthesis

0.938Kmtm

0.9425 K m T m

1.2346

Kmxro

2.7xm

2xm

4-5Tm

0.313tm

0.5xm

0.45Tm

N = 0; P = l ;

a = 0.833 N = oo; p = l ;

a = 0.707 or a = 0.65

N = 0; p = 0 ;

a = 0.6

U(s) = Kc [(b -1 ) + (c - l)Tds]R(s) + Kc

/ 1 \ 1 + + Tds

T s R ( s ) -

1

(l + sT f ): •Y(s)

K c[(b-l) + (c-l)Tds]

+ R(s)

-+<&->K ( 1 \

1 + — + Tds V T i s J

+ > +

U(s) Y(s)

K e"

(l + sT f)2

Table 89: PID controller tuning rules - IPD model Gm(s): Kme"

Rule

Astrom and Hagglund (2004). Model: Method 1

Kc T; Td Comment


0.45

Km

8xm 0.5xra

No precisely defined

performance specification

b = 0, - ^ - < l ; b = l , ^ - > l . c = 0. Tf = 0 . k m m m



273

U(s) = K v T i s y

R(s)

+

E(s) K,

f P 1 + -

v i i ° y Xs

+ >® U(s)

E(s)-KcTdsY(s)

Y(s) K „ e "

KcTds

T Table 90: PID controller tuning rules - IPD model

Kme"

Rule


-page 143. Model: Method 2 Minimum IAE -Shinskey (1994)

-page 74. Minimum IAE -Shinskey (1988)

- page 148. Minimum IAE -Shinskey (1996)

-page 121.

Kc T i Td Comment


1.28 K m T m

1.28 K m T m

0.82KU

0.77KU

1.90xm

1.90xm

0.48TU

0.48TU

0.48xm

0.46tm

0.12TU

0.12TU

Also defined by Shinskey (1996),

page 117.

Model: Method 1

Model: Method 1

Model: Method 1



U(s) = - ^ E ( s ) - K c ( l + Tds)Y(s)

R(s) >M + 1 k

E(s) K c

T iS

U(s)

r y y r

4 K c ( l + Tds)

t

K . e - ' -

s

Y(s)

— •


Rule

Minimum ISE-Arvanitis et al.

(2003a).



Minimum ISE-Arvanitis et al.

(2003a).


Kc Ti Td Comment

Minimum performance index: regulator tuning 1.4394 K m T m

1.2986

Kmxm

2.4569xm

3.2616Tm

0.3982xm

0.4234xra

Model: Method I

Model: Method 1

oo

Minimise performance index je2 (t) + Km2u2 (t)Jdt

0

1.1259

Kmxm 6.7092xm 0.4627xm

Minimise performance index

0

Model: Method 1

;'«^'(T]" dt

Minimum performance index: servo tuning 1.5521

Kmxm

1.4057

Km tm

2.1084tm

2.5986xra

0.3814xra

0.4038xm

Model: Method 1

Model: Method 1

oo

Minimise performance index [e2 (t) + Km2u2 (t)Jdt

0


Rule


Chien et al. (1999).

Model: Method 2


Kc

1.3332

Kmxm

Ti

3.0010xm

Td

0.4167xm

Minimise performance index 1

0

e2(t) + Km2 |

Direct synthesis

3 K (427)

16(2^2+l)

(32^2+4JKmtm

1.414TCL2+tm

N2+lk,

T (427)

1 6 ^ + 4 _

16 (2^ 2 +l j " m

Comment

Model: Method 1

'duf~ dt

Underdamped system response -

\ = 0.707

Model: Method 1

3 K (427) 1.414TCL2+xm

^ ( W + O J O T T c ^ + O ^ x , , 2 ' ' ±d r . (427) 0.25Tm^+0.707TCL2Tn

1.414TCL2+xm



U(s) = Kt 1 \

1 + — + Tds V T i s J

E(s)-K1(l + Tdis)Y(s)

R(s) E(s)

Table 92: PID controller tuning rules - IPD model Gm (s)

Rule


Model: Method I

Kc T| Td Comment

Robust

4 K (428) T (428) T ( « 8 )

K; =0.25KU,

T d i =0

4 j , (428) _ 0 .25K U

(X- K ; „ K „

T ^ = 0 2 S ^ { Q 2_ T

2{X + xn

3

,(428)

K„Km ' 2(X + i J

xm2( l -0 .25K uKmTm) ,

fr + O 6(X + t r a ) 4{X + xmf 0.5KuKm(X + x J



U(s) = Kc 1 + f- E(s)-K f

277

1+ I Tisy

-^-s + 1 vTm y

Y(s)

R(s) ^^y. E(s) K

V T , s 7

+ /cx U ( s ) K „ e "

Y(s)

K, i i s + l T

V m J I Table 93: PID tuning rules - IPD model Gm (s) =

K „ e "

Rule

Kaya and Atherton(1999),

Kaya (2003).

Kc Ti Td Comment

Direct synthesis

0.313Ku 0.733TU

5 T (429) Xd Model: Method 9

0.753

5 x (429)

( A A Km Ku

V T m T " )

0.333

^nJm K „

0.567

K „

( A A

Km Ku

V Tm T" J

• 0 . 3 1 3 K u



U(s) = Kc 1 A

V T i s j

1

Tds + 1 E(s)-K,Y(s)

R(s) E(s)

+ r — * ® - * Kc(l + -J- + Td

Table 94: PID controller tuning rules - IPD model Gm (s) = Kme-St™

Rule


Model: Method 1

Kc T; Td Comment

Robust

6 K (430) T (430) T (430) K, =0.25KU

6 K (430) 0.25K,, • i . + -

, (430)

(X + xJ'KuKm "' 2{X + zm)

xm 0.5

4 x,

*m , V , (l-0.25K„Kn,Tni)

6ft+ 0 4(X + t m ) 2 0.5KuKm(X + Tm)

0.5

2{x+imy

0.5 ^ + - T" - +

T (430) _

( l-0.25KuKmxm)

6(^ + xra) 4(X + zm)2 0.5KuKm(^ + xm)

-i0.5

0.5 ^ + _ T" • + ( l-0.25KuKmxm)

6(X + xm) 4(X + xm)2 0.5KuKm(X + t m )

4.4 FOLIPD Model Gm(s) = K e"

<l + s T j


f 1 A

1+ — + Tds T i S J

l ( !L^o E ( s )

+ T KJ1+ — + Tds

T:S

U(s) K e-sx"

s(l + sTm)

Y(s)

Table 95: PID controller tuning rules - FOLIPD model Gm (s) = Kme-

<l + sTm)

Rule

Minimum ISE -Haalman(1965).


Model: Method 1

Tachibana (1984).

Kc

Minim

2

3Kmxra

1 K m T m

e - 2 « ) d ^

K m t m

X

Km(l + Xxm)

Tj Td

um performance index: regulator

0 T

A m = 2 . 3 6 ; * m = 5 0 0 ; M 1 = 1 . 9 . Direct synthesis

0

0

0

Tm+0.5xm

1 x (431)

T

Comment

tuning

Model: Method 1

Step disturbance


Model: Method 5

(431) l -O-Se -^^+^ -e - " ' - t m e

Rule

Chen et al. (1999a).

Viteckova (1999),

Viteckova et al. (2000a).

Model: Method 1

O'Dwyer (2001a).

Model: Method I



Model: Method I

Kc

rapTm

K A T <432)

riwrTm

v T (433)

x l / K m X r a

X,

0.368 0.514 0.581

OS

0% 5% 10%

X l T m

K m T m

x l

0.785

A

3 K (434)

Tm+T ra+2?t

Kra(Tm+X)2

Ti

0

0

0

Td

2 T (432)

2 T (433) ld

T

Coefficient values x l

0.641 0.696 0.748

OS

15% 20% 25%

0

x. 0.801 0.853 0.906

OS

30% 35% 40%

T

Representative coefficient values

m

) *m

45°

3TCL+Tra

Rot

T r a +T m +2^

x l

0.524

A

T (434)

)USt

Tm(xm+2X)

x m + T m + 2 ^

Comment

Model: Method 1

Model: Method 1

x. 0.957 1.008

OS

45% 50%

Ara = 0.5/x, ;

<t>m = 0.571-X,

m

s *m

60° Model: Method 1

X not specified

(432) 1

3(1

. ^ p ^ m 1 2co v- + —

P n T

(433)

4mr2xm [ 1

_ 0 25n(2r,A -1)

_(3T C L +x m >e-"" (434).. (3T C L + T m XT m +T m )

~Kc<->(TCLs + i r ' ~ (TCL+xJ3 •

X (434) 3T C L2 T m + 3 T C L T m T m -T C L

3 +T m T m2

(3TCL+xmXTm+Tm)

Rule

McMillan (1984).

Model: Method 1

Peric et al. (1997).

Model: Method 8

Kc T| Td Comment

Ultimate cycle

4 K (435)

KuCOS(|)m

T («5)

aTd<436)

0.25Ti(435)

5 T (436)


Ku>Tu

Recommended a = 4

K (435) 1-111 Tm • (435) 2 l m 1 +

5 T (436) tand)m +,|— + tan a

T 471

Gc(s) = Kc 1 \

1 + — + Tds V T i s j

1

R(s) E(s)

+

r K,

1 \ 1 + — + Tds

v T i s J

1

Tfs + 1 K „ e "

s(l + sTJ

1 + Tfs

Y(s)

U(s)

Table 96: PID controller tuning rules - FOLIPD model G ra (s) = Kme"

<l + sTm)

Rule

H„ optimal-

Tan et al. (1998b).


Model: Method 1

Tan et al. (1998a). Model:

Method 1; X not specified

Kc Ti Td Comment

Robust 6 K (437) T ( 4 3 7 ) T (437) Model: Method I

A. = 0.5 (performance), A, = 0.1 (robustness), A. = 0.25 (acceptable)

7 j r (438) 3 - + Tm+xm

3^ + xm+Tm

1.5xm<A.

^4.5xm

X = 1.5xm , OS = 58%, Ts = 6xm ; X = 2.5xm, OS = 35%, Ts = 1 l i m ;

/. = 3.5xm,OS = 26%, T s= 16xm;A, = 4.5xm,OS = 22%, Ts = 20xm.

8 K (439)

8 ^ (440)

Tra+8.1633xra

Tm + 5 .3677xm

T m * m

0.1225Tm+xm

T T

0-1863Tm+Tm

Tf=0.5549xm

Tf=0.4482xra

6 v (437) (0.463X + 0.277)([0.238X + 0.123]Tm + T )

Kmx„/ 5.750A. +0.590

. (437)

7 K (438)

- T (437) Tmtm

0.238A. +0.123' d (0.238A,+ 0.123)Tm+xn

K ^ l 3^ + T r a+xm

3k'+3\xm+xm< i . T f = -

3k'+3\xm+xm'

K (439) 0.0337T„

K m x „ 1 + -

0.1225T„, , K

(440) 0.0754T„

Kmx m m V 0.1863T„


Rule

Tan et al. (1998a)-continued.


Model: Method 1

Kc

9 K (441)

10 K (442)

Ti

Tm+3.9635xm

2(Tm+V) + Tm

Td

Tmtm

0.2523Tra+xm

2 T m ( t m + ^ )

2(xm+X) + Tm

Comment

Tf = 0.2863xm

X not specified

, K (44i) 0.1344Tm

K m T„ 1+-

0.2523Tm

10 K (442) _ 2(Tm+^)+Tm j _ Tm^

K r a K 2 + 4 T m X + X2)' f 2Tm2+4Tm^ + ^



Table 97: PID controller tuning rules - FOLIPD model Gm (s) = Kme"

s(l + sTm)

Rule

Astrom and Hagglund(1995) -pages 212-213. Model: Method 1

or Method 3

Kc

5.6e-8'8T+68,!

K r a(Tm+Tm)

8 6 e-7.1x+ 5 .4x ;

K m (T m +t m )

Ti Td

Direct synthesis

l . h r a e — ' ! 1.7tme-6-4T+20T2

b = 0.12e6'9t-6W

1.0xme"T-2^ 11 T (443)

Comment

Mmax =1.4

Mmax = 2.0

11 j (443) _ Q 3 C T 0.056T-0.60t2


4.4.4 Controller with filtered derivative G c ( s ) = Kc 1 + -Tds

R(s)

7 ^ E(s)

/

K, 1 + ^ + - ^

N

U(s)

T . S 1 + s ^ N

Y(s) Kme"CT

s(l + sTB)

Table 98: PID controller tuning rules - FOLIPD model G m (s) = K m e - "

s(l + sTm)

Rule


(2002).


Kc

12 ^ (444)

2*. + ^ra+Tm

T, Td

Direct synthesis

T(444) T (444)

Robust

2^ + Tm+xm

Tm(2^ + xm)

2^ + T m +x m

Comment

Model: Method 1; K, e [0.5,1]

^e[Tm,Tm]

(Chien and Fruehauf (1990)); N = 1 0

12 ^ (444) 2.5co„

K „ •0.08 + ^ + ° ^

K, K i J

2 ( O . 0 5 9 / K I 2 ) + ( 0 . 0 5 5 / K I ) - 0 . 0 8

2.3-2K 0.4(10+ 1/K,)

,(444) 2.5 2.3-2K,

(0.059/K,2)+ (0.055/K,) - 0.08'

0.4(10 + 1/K,)

(444) 2.5 ( .2 ( O . 0 5 9 / K | 2 ) + ( 0 . 0 5 5 / K I ) - Q . 0 8 A

N = -(444)

d (444)

l f

2.3-2K, 0.4(10 +1/K,)

(o.osg/K^j+fo.oss/Kj-o.os 0.16cou(lO + l/K,)

N +

(444)


4.4.5 Controller with filtered derivative with set-point weighting 1

Gc(s) = K£ , 1 T d s

TiS 1 + Td

v N

1 + b^R(s)-Y(s) . l + a f l s

R(s) l + b n s

1 + a,, s

+ E(s) /

K, 1 + -U-^ N ;

U(s) Kme"

sd + sTJ

Y(s)

Table 99: PID controller tuning rules - FOLIPD model G m (s) = -Kme-

'(1 + s T j

Rule

Liu et al. (2003). Model:

Method 1; recommended

A, = x

Kc

13 j r (445)

X

0-3Tm

0.5xm

t r a

l-5tm

^ Td Comment

Robust T(445) T (445)

Representative A values deduced from a graph

OS

= 57%

= 33%

= 7%

= 0%

Rise time

= 3Tm

= 3.5xm

= 5xm

- 8.5rm

A

2Tm

2-5Tm

3Tm

3-5Tra

OS

- 0 %

= 0%

= 0%

= 0%

Rise time

= 8Tm

= 10.5xm

= 12.5xm

= 14tm

n K (445) _{3X + xm+Tmi3X2+3Xim+zm2)-X3 ^

Km(3X2+3Xxm+xm2}

T (445) = (3X + xm+TmJ3X2 +3Xxm+ij)-X3

(3X 2 +3^ r a +T m2 ) '

T (445, Tm{3X + xmJ3X2+3Xxm+xm2) A3

(3A + Tm+Tm)(3A2+3Axm+Tm2)-A3 3A2+3ATm+Tm

2 '

Tm{3X + xm) _

^t(3A + T m + Tj(3A 2 +3AT m + T m2 ) -A 3 ]

N = -



Gc(s) = Kc 1 1 T

d s 1 + — + — = -

Ts - X

V 1 + ^ - s

N ;

ri+b f ls±b i ls2

+b f 3s 3_R ( s )_Y ( s )^ 1 + af ,s + af 2 s 2 + af 3s3

R(s)

l + b f l s + b f 2 s 2 + b f 3 s 3

l + a n s + a f 2 s 2 + a f 3 s 3


*(l + sTm

Rule

Liu et al. (2003). Model: Method 1

Representative

X values

(deduced from a graph)

K c

1 ^ (446)

a f l =

X 0-3Tm

0-5xm

t r a

l-5Tm

7X + xm; a

OS

= 43%

= 15%

= 0%

= 0%

T j Td

Robust r(446) rj- (446)

X d

f 2 = X(\6X + 4xm), af

Rise time

= 3.5xm

= 4.5Tm

= 8Tra

= 12-5Tra

X 2Tm

2.5Tm

3Tm

3.5Tra

Comment

= 4X2(3X + xm)

OS

= 0%

= 0%

= 0%

= 0%

Rise time

= 16.5tm

= 20xm

= 24t m

= 28Tm

• K (446, _ (3*. + t m + Tm J3X2 + 3Xxm +xm2)-X3

K ^ + S ^ + T , , , 2 ) 2

Ti(446) = {3X + xm+TmJ3X2 +3Xxm+xm2)-Xi

l 3X 2 +3^ m +T m2 )

(446) Tm{3X + xmJ3X2+3Xxm+xm2)

b f[ = 3X , b f2 = 3X , b f3 = A.

*3

(3X + T m +T m ) (3X 2 +3^ m +T m2 ) -X 3 3X2+3Xxm+Tm

2'

N = Tm(3X + T m ) t

X3 [(3X + xm + Tm )(3X2 + 3Xxm + xm2)- X3 ]


Rule

Liu et al. (2003) - continued.

Kc Ti Td Comment

a f , = 4X + xm ; a f2 = X(3.25X + xm ) , a f 3 = 0.25A2 (3A. + xm )

Representative A. values (deduced from a graph)

A

0.3xm

0.5tm

V

l-5xm

OS

* 6 1 %

» 40%

- 1 5 %

* 7 %

Rise time

- 3 t m

- 3 i m

- 3 . 5 x m

- 5 x m

X

2xm

2.5xm

3xm

3-5xm

OS

* 4 %

* 4 %

* 1 %

» 0 %

Rise time

- 6 . 5 x m

* 8 x m

"10T m

* 1 2 x „

U(s) = Kc(FpR(s)-Y(s))+|^(F iR(s)-Y(s))+KcTds(FdR(s)-Y(s))


s(l + sTm)

Rule

Oubrahim and Leonard (1998).

Model: Method 1

Kc T, Td Comment

Ultimate cycle 0.6KU,

Fp=0.1

0.5TU,

F i = l

0.125TU,

Fd=0.01 0.05 < — < 0.8

T

20% overshoot

v T i s y

1 + Tds

N ;

R(s)

+

E(s) K,

V T i s 7

l + sTd

T 1 + s-i i

U(s) Kme"

s(l + sTm)

Y(s)

Table 102: PID controller tuning rules - FOLIPD model G m (s) = Kme"

S(l+sTm)

Rule


-page 75.

Minimum IAE -Shinskey (1994) pages 158-159.

Model: Method 1;

N not specified

Kc Ti Td Comment


K m ( t m + T J

2 K (447)

3 K (448)

1.38(xm+Tm)

~ (447)

0-66(xm+TJ

0.56tra+0.75Tm

N=10. Model: Method 1

^-<2 T

IHL>2 T

2 j r (447)

3 v (448)

100

108KraTm 1.22-0.03-

- T ( 4 4 7 )=157x 1 + 1.2 1 - e T™

K 100

108Kmxm 1 + 0.4-

Rule

Minimum ITAE - Poulin and Pomerleau

(1996), (1997). Model:

Method 1

Process output step load

disturbance

Process input step load

disturbance

Poulin et al. (1996).

Model: Method 1


Kc

4 R (449)

Ti

X 2 ^ m + T m )

Td

Tm

Comment

0< , T") , < 2 (Tm/N)

3.33<N<10 Coefficient values (deduced from graphs)

Xm

Tm/N

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

x i

5.0728 4.9688 4.8983 4.8218 4.7839 3.9465 3.9981 4.0397 4.0397 4.0397

x2

0.5231 0.5237 0.5241 0.5245 0.5249 0.5320 0.5315 0.5311 0.5311 0.5311

t m

Tm/N

1.2 1.4 1.6 1.8 2.0 1.2 1.4 1.6 1.8 2.0

Xl

4.7565 4.7293 4.7107 4.6837 4.6669 4.0337 4.0278 4.0278 4.0218 4.0099

x2

0.5250 0.5252 0.5254 0.5256 0.5257 0.5312 0.5312 0.5312 0.5313 0.5314

Direct synthesis

S K (450) 2l(0.2Tm+Tm) T N = 5.

Mmax = 1 dB

Robust

Tm

Kra( + 0 2

2X + xm

Km(X + t J 2

T

' *- + *m

2X + xm

Tm

^e[Tm ,Tm]

(Chien and Fruehauf

(1990Y>-N=10

1 4.4.9 Classical controller 2 Gc(s) = K

E(s) R(s)

+ ®-K f , ~\

1+-V T i S y

1 + NTds

V 1 + T d « j

1+-v T i s y

U(s)

l + NTds

1 + Tds j

K „ e - " »

s(l + sTm)

Y(s)


<l+sTm)

Rule


Model: Method 1

Kc Ti Td Comment


60.106x! K m T m

00 0.135Tma7tm

0-3

Maximise crossover frequency; N e [10,30]

x, values deduced from graph:

^m/Tm

x l

^m/Tra

X l

1

6

7

1.1

2

3

8

1.0

3

2.1

9

0.9

4

1.8

10

0.8

5

1.6

11

0.7

6

1.3

12

0.7



Gc(s) = Kc

U(s)

293

1+ v Tisy

R(s)

~ &

E(s) K

v T i sy (l + sTd)

Kme" s(l + sTm)

(l + sTd)

Y(s)

Table 104: PID tuning rules - FOLIPD model Gm(s) =

Rule

Skogestad (2003).

Model: Method I

Kc T; Td Comment

Direct synthesis 0.5

K m X m 8tm T

4.4.11 Classical controller 4 Gc(s) = Kc '.+-L' V T i s /

1 + •?*

1 + -N

R(s)

• &

E(s) Kr

V T i s / 1 + -

sTri

1 + s-N

U(s) Kme-

s(l + s T J

Y(s)

Table 105: PID controller tuning rules - FOLIPD model G m (s) = -Kme-

<l + sT j Rule


Kc

2X + xm

Km(^ + xm)2

Tra

Kmfr + 0 2

Ti Td

Robust

2X + Tm

Tm

Tm

2X + xm

Comment

(Chien and Fruehauf

(1990)); N=10

U(s) = Kc 1 + - U THS

TiS l + (Td/N> E(s)-Kc a +

PTds

l + (Td/N> R(s)

a + PTds

T

N ;

K „

R(s) + E(s)

K, T ' S l + ^ s

N

U(s)

+

K e"ST"

s(l + sT n )

Y(s)


<l + sTra)

Rule


and Pagola (2000). Model:

Method 4; Km = l ;

T m = i ;

P = 1,N=10;

0.05 <xm <0.8

Kc Ti Td Comment

Minimum performance index: servo/regulator tuning x,Ku x2Tu x3Tu

Coefficient values

* i

1.672

1.236

0.994

0.842

0.752

0.679

0.635

0.590

0.551

0.520

0.509

x2

0.366

0.427

0.486

0.538

0.567

0.610

0.637

0.669

0.690

0.776

0.810

X 3

0.136

0.149

0.155

0.154

0.157

0.149

0.142

0.133

0.114

0.087

0.068

a

0.601

0.607

0.610

0.616

0.605

0.610

0.612

0.610

0.616

0.609

0.611

*.

-164°

-160°

-155°

-150°

-145°

-140°

-135°

-130°

-125°

-120°

-118°

Rule


Model: Method 1

Wang and Cai (2001).

Model: Method 1


Xu and Shao (2003a).

Xu and Shao (2003c).

Kc

1 K (451)

2 K (452)

0.6068

Kraxm

0.6189

5 K (455)

1.5xm+0.5Tra

2Kmxm2

Tj

T (451)

Direct s T (452)

5.9784xm

5.9112xm

2(Tm+1.5xm)

T m + t m

Td

T (451) x d

vnthesis T (452) *d

3 T (453)

4 T (454)

(Tm+0.5xm)2

(Tm+1.5Tm)

l-5Tmxm

1.5xra+0.5Tm

Comment

^s-<1.0 T

N=0; p = 0.

Model: Method 1

Model: Method 1

6

Model: Method I

K „ 0.7608+ -

0.5184

[-2- + 0.01308]z

,(451)

(451) = T

0.03330 + 3.997-

0.03432 + 2.058-

-0.5517

-1.774 -0.6878 3A

a = 0.6647 , P = 0.8653 - 0.1277(xm /Tm)+ 0.03330(xm /Tm f,

N fixed but not specified; OS < 20% .

K (452) 1.3608-1.2064

M „

T ,452) = xm(l.3608Ms-1.2064) 1 0.29M,-0.3016

(452) (Tm+0.1xmXl.450Ms-1.508) , a = -

0.2M, , 1.3<M < 2 .

1.3608MS-1.2064 ' 1.3608MS-1.2064

Td(453) = 0.8363(Tra + 0.1xm), a = 0.3296 , Ms = 1.6 (A m > 2.66 , <|)m > 36.4°)

Td(454) = 0.8460(Tm + 0.1xm), N=0, P = 0, a = 0.3232 , Am = 3 , <|>m = 60°.

K (455) 0.5(Tm+1.5x„ ;> a = : K m x m ( T m + x m ) ' Tm+1.5x

, P = 0.6667 , N= oo , Am > 2 , $m > 60°

N=co,p = 0 , A m > 2 , 4 m > 6 0 0

.5T+0.5X



U(s) = Kc 1 + -T , sy

R(s) E(s)

+ X K, 1 + —

V T i S 7

U(s)

E(s)-KcTdsY(s)

Y(s) K m e-

s(l + sTJ

K cT ds

I Table 107: PID controller tuning rules - FOLIPD model Gm (s) =

Kme"

s(l + sTm)

Rule


-page 75. Minimum IAE -Shinskey(1994)

-page 159.

Kc T i Td Comment


Km(x r a+Tm)

7 ^ (456)

l-38(tm+Tm)

-p (456) i

0.55(Tm+Tm)

0.48xm+0.7Tm

Model: Method 1

Model: Method 1

7 K (456) 1.28

Kmxm(l + 0 . 2 4 - ^ - 0 . 1 4

T ( 4 5 6 , = 1 . 9 x „

Tm "\

l + 0.75[l-e Tm]

U(s) = - ^ E ( s ) - K c ( l + Tds)Y(s) T,s

R(s) >6* + 1 L

E(s) Kc

T iS

U(s)

' W "

Kc(l + Tds)

*

Kme-™

s(l + sTm)

Y(s)

— •

Table 108: PID tuning rules -FOLIPD model Gra (s) = Kme-

s(l + sTJ

Rule


Minimum ISE 8

Kc T( Td Comment

Minimum performance index: regulator tuning 16

(16-3P)KmTm

4

(3Tra ( 8 - P ) T

16 m

Minimum j[e2(t) + KmV(t)]dt 9

0

Model: Method 1

P = 1.5841 + 0.3101| — +0.38841 - 0.4622 'O* T

+ 0.219181 —^

-0.060691 T

+ 0.010771 'O6

T V m j

•0.0012466 T

V m

-^H +9.1201.10"5 X,

T V 1m J

•3.8302.10" , T i*- +7.0316.10"8

f \10

T ,

P = 0.066485+ 2.0757 '^^ T

+ 1.0972 M -V m J

( - \ -2.7957

-0.68418 T

+ 0.14763 f ^

1M. , T

•0.019733

T

f ^

+ 1.9019 ( \4

T V m J

T + 0.0016016

t T

-7.2368.10" ^-| +1.397.10"6

T 0<-S-<4.0;

T„

P = 2.128638-0.005608121-^-4.0 •>4.0.

Rule


Model: Method 1

Kc

10 j , (457)

T; T(457)

Td

T (457)

Comment

Minimum ISE ' '


10 K (457) 12.5664(xm +Tm)-4

Kmxm[25.1328(Tm+Tm)-8-(xm+2Tm)a]

r ^ ™ „ 3.4674xm+3.1416T - 1 , with a = [0.5708 + J ^ ^ ]% ,

T(457) . _ 12.5664(xm+Tm)-4 T <457> _ T T 2

' i - ' x d _ 1 m 1 r a 12.5664(xm+Tm)-4

11 x = 1.178 + 4.5663 't A

T V 'my

-7.0371 T

+ 5.5784 -^-\ -2.61031 V m V m

-0.7679 f \ 5

I™. T

V m

(~ \ -0.1458

T + 0.017833 'o7

> T , V m y

-0.0013561 H 2 -> T

+ 5.83.10" T

-1.0823.10" T

0<-2=-<3.5, T

X = 2.2036 - 0.0033077| ^ - - 6.5 , ^ - > 3.5 .

00

: Performance index = |je2(t) + KmV(t)]dt

13 x =-0.1185 + 4.781^2-]-3.9977 f \ x

+ 0.057708 / \5

X „

T V m y

0.0038421

T

V. 1 r a y

X

-1.733 X

T v m y

-0.42245 / \ 4

> T V m y

T V m y

+ 2.6386.10" T-i2-| +1.078.10"5' ^ s -vTm

-4.2486.10" T

V. m

Rule


Kc T, Td Comment

Minimum performance index: servo tuning 16

(16-3B)Kmxm

4

( 3 T m

(8-P) 16 m

Minimum ISE M


Model: Method 1

14 p = 1.2171 + 0.7339Us- +0.083449 \—\ -0 .14905^2- + 0.030257^

lTJ ITJ \Jm) lTmJ + 0.0045567

/ ^5

T

T -0 .0029943p -0.00056821

T V m J

-5.4957.10"

+ 2.7505.lO"6!^1- -5.6642.10"8M2-

Performance index = |e2(t) + Km2u2(t)Jit

'P = 0.083709+ 2.28751 .18739 T

V m J

•1.0059PB IT.

-0.79718 vTJ

-0.3036^2- +0.067859-^2-

l T m J l T m y

(- \ -0.0093104

T + 0.00077161 ' T ,

A

T

-3.5473.10" 5 Lm + 6.9487.10" I™. , o<^2L<4.5

V T J Tm

6 = 2.444547-0.0146274714 •Is.-4.5 1 ^2_>4.5 . T I T

Rule


Model: Method 1

Kc 1 7 K (458)

T;

T ( 4 5 8 )

Td

T (458) 1 d

Comment

Minimum ISE 18


17 K (458) 12.5664(xm+Tm)-4

KmTm[25.1328(xm + T m ) - 8 - ( x m +2Tm)a]

with a = [0.5708-3.4674xm +3.1416Tm - 1

- (458) 12.5664(Tm+Tm)-4 (458) = T - T m *-m 12.5664(x r a+Tm)-4

X = 1.1571 + 5 . 0 7 7 5 ^ . - 8 . 2 0 2 1 ^ - + 6 . 6 3 5 3 ^ - -3.1355

ITJ ITJ UJ + 0.92683-2!-

V m y -0.17635

vTm

f ^ X „

T

T V m J

+ 0.02158 -0.0016402 Xr

T V m y

]X

+ 7.0439.10" 5 l m -1.3056.10" 0<-in-<4,5

T

X = 2.124307 - 0.004844182| i= - - 4.5 , — > 4.5 .

' Performance index = |je2(t) + Km2u2(t)Jdt

20 x =-0.70861+ 7.4212| ^2-1-8.1214 'O' T

V m y

+ 4.8168 r A 3

x.

T V m y

-1.7217

+ 0.38998 ^ V ' m y

•0.056975 — +0 .0053141^- -0 .00030233^ V.T m J V m y V m y

+ 9.3969.10"6 - a - -1.1861.10" v 111 /

X = 2.1069751341 -0 .0017098-^--4 .5

0 < _ E - < 4 . 5 T

•>4.5 .

Rule

Arvanitis et al. (2003b)

(continued).


Model: Method 1

Kc

23 K (459)

24 K (460)

T| Td


Direct synthesis

k2+lk y (460)

16^2 +4

16(2i;2+l)Tm

T (460)

Comment

Performance index = e2(t) + K 2 — J m l dt

dt

22 % = 0.05486 + 0.816621 X

V m J -1.4989pa

V m J

-1.1955 p V m j

+ 0.11011 -0.013386-^-l T m J

^ -0.00059665 'x . N?

T

-0.48464p-

+ 3.8633.10"5 - ^ L

-5.307.10' T

V m J

+ 1.6437.10"7 0 < - 2 L < 6 . 0 ; T

X = 2.1111 -0.001025 -^S--6 T,

- > 6 .

23 JJ- (459)

24 v (460) K

16(2^2+l)

(3242 +4)KmTm '

^ + T m + 4 T m ^ 2

Kmtm2(l+8527

Tr ) ^ m +T m +4T r a ^ ( Tr 0 , = TmTm (l + 4^)

^ + T m + 4 t m ^

303

/ U(s) = Kt

1 \

V T i s J

EC^-KiCl + T^^YCs)

Table 109: PID controller tuning rules - FOLIPD model G m (s) = K m e "

5(l + sT r a)

Rule

Wang et al. (2001b).

Model: Method 7


Model: Method 1

Kc T( Td Comment

Direct synthesis 0.4189 K m T m

4-t 1.25(Tm+0.lTm)

Km^m

Td i=0; Am=3;<t .m=60 0

Robust

1 K (461) T (461) i

T (461)

Ki = 0.25KU ,

T d i =0

i „ (461) 0.25K,,

ft + O • x m +

2(A. + T „ )

, (461) t m +

2(^ + t r a ) :

(461) _ 0-25Ku 4Tm 2 Tm | Tm

T m2 ( l - 0 . 2 5 K u K m T m ) ,

0.5KuKm(^ + x J J

U(s) = K(

v T i sy (l + Tds)E(s)-K i( l + Tdls)Y(s)

R(s) E(s)

+

f 1 ^ <8^KC1 + -L(l + T d s ) ^ 0 > v Tisy

U(s) K „ e "

s(l + sTm)

Y(s)

Ki(l + Tdis)

I Kme"

Rule

Majhi and Mahanta(2001). Model: Method 6

Kc

0.5236

Kmtm

" ' & " " V U 1 " " "

T|

m v /

Td

Direct synthesis

2 x m e ^ T

s(l + sTm)

Comment

T d i =T m ;

K m T m

A m = 3 ;

4»m=60°



305

f U(s) = Kc

1 \ 1 + — + Tds

V T i s j

1

Tds + 1 E(s)-K,Y(s)

R(s) E(s) 1

Tds + 1

+

w

U(s)

• § -K e"

(l + sTj

Y(s)

K,

I Table 111: PID controller tuning rules - FOLIPD model Gm(s) =

K . e " s(l + sTm)

Rule


Model: Method 1

Kc Ti Td Comment

Robust

2 K (462) T(462) T (462) K, = 0.25KU

l K ^ , = 0 J 5 K J L _ J _ _ X m + _ L

^ + t m KuKm

4 T m - + 0.5xJ.

2(^ + xm)

K u K m

C , Tm4 f ( l-0.25KuKmxm)xm

6(^ + xm) 4(X + xm)2 0.5KuKm(^ + x J

-(462) 4 x - - T m + KuKm

2(X + T m )

4 T - - + 0.5xJ-KuK.m

V , C , ( l - 0 . 2 5 K u K m Q T m

6(^ + xra) 4(^ + x ra)2 0.5KuKm(X + xm)

(462) ^ - + 0.5x 2 - X" (l-0.25KuKmxm)x„

6(^ + xm) 4(^ + xm)2 0.5KuKm(A. + Tm)


4.4.18 Industrial controller U(s) = K 1 + — R ( s ) - ^ Y ( s )

N j

R(s)

+ K, 1+-

T i S y

U(s) K e"

s(l + sTm)

Y(s)

1 + Tds

l + ^ s N

Table 112: PID controller tuning rules - FOLIPD model Gm (s) = K„e"

s(l + sTm)

Rule

Skogestad (2004b). Model:

Method 1

Kc Ti Td Comment

Direct synthesis: time domain criteria 1

Km (TC L +x r a ) ^ 2 ( T C L + T J T 'good'

robustness -

TcL = Tm ' 4 =0.7 or 1

N e [5,10]; N = 2 if measurement noise is a serious problem

4.4.19 Alternative controller 1 Gc (s) = Kc 1 + TjS

N j

1 + Tds

N ;


<l + sTm)

Rule

Tsang et al. (1993).

Model: Method 2

Tsang and Rad (1995).

Model: Method 2

Kc T, Td Comment

Direct synthesis x i

Km tm T 0.25xm N = 2.5

Coefficient values

* i

1.6818 1.3829 1.1610

% 0.0 0.1 0.2

0.809

x l

0.9916 0.8594 0.7542

% 0.3 0.4 0.5

T

Xl

0.6693 0.6000 0.5429

% 0.6 0.7 0.8

0.5xm

Xl

0.4957 0.4569

\ 0.9 1.0

Maximum overshoot =

16%; N = 8.33


4.4.20 Alternative controller 2

Gc(s) = K£

\

1 + TjS

V N .

C 2 „ 2

l + 0.5Tms + 0.0833T„ s

N

K,

f \

1 + TjS

1 + ^ N

/ 2 „ 2

l + 0.5xms + Q.0833xm s

l + ^ s N

U(s) Kme"

s(l + sTm)

Y(s)


>(l + sTm)

Rule

Tsang et al. (1993).

Model: Method 2

Kc T i Td Comment

Direct synthesis

* i K

m T m T 0.25xm N = 2.5

Coefficient values

*1

1.8512 1.5520 1.3293

4 0.0 0.1 0.2

x l

1.1595 1.0280 0.9246

% 0.3 0.4 0.5

x i

0.8411 0.7680 0.6953

\

0.6 0.7 0.8

x i

0.6219 0.5527

\ 0.9 1.0


4.5 SOSPD Model

Gm(s) = K fi

l l + s T m l X l + s T m 2 ) orGm(s) =

Kme~

T m l2 s 2 +2^T m l s + l

4.5.1 Ideal controller Gc(s) = Kj 1 + — + Tds T:S

R ( s ) ^ E ( s )

+ S i Kc

{ 1 1 1 + — + Tds

I T i s J

U(s) SOSPD model

Vis)

w

Table 115: PID tuning rules - SOSPD model G m (s) = K m e "

(l + sTm,Xl + sTm2)

G m (s ) = K m e "

T m l2 s 2 + 2 ^ m T m l S + l

Rule

Auslander et al. (1975).

Kc Ti Td Comment

Process reaction 1 K (463) T (463)

i 0.25Tj(463) Model: Method 3

Note: equations continued into the footnote on page 310.

K (463) . 1.2 K m ( T n i l ~ T m 2 )

, Tm lTm 2 , Tjni T + In

111 rr^ ry^ r-r-> lm\ ~ Xra2 Xin2

( T m / Tm2

Tm l -Tm ; 1m2

T

Tml 1 Tml-Tm2

( T m l - T m

T , 1T„„-Tm! m2 ' 5 n 2 . 1 T - - T -

| ! Tm2 j Tm2

Tmi ~ Tm2 ^ Tm,

Trai I Tm2

Tmi Tm2 ^ T m l j


Rule

Minimum IAE -Wills (1962b) - deduced from

graph. Model: Method 1

Minimum I A E -Shinskey (1988)


Minimum ITAE - Lopez et al.

(1969). Model:

Method 17

Kc T i Td

Minimum performance index: regulator

5.8

Km

5

Km

0.62KU

0.68KU

0.79KU

x , / K m

0.36TU

0.29TU

0.38TU

0.33TU

0.26TU

x 2Tm l

0.21TU

0.29TU

0.15TU

0.19TU

0.21TU

x3Tm l

Representative coefficient values (from graphs)

x i

25 0.7

0.35

25 1.8 9.0

x2

0.5 1.3 5

0.5 1.7 2

* 3

0.25 1.2 1.0 0.2 0.7

0.45

$« 0.5 0.5 0.5 1.0 1.0 4.0

^ m / T m i

0.1 1 10 0.1 1 1

Comment

tuning Representative tuning values;

= 0.1Tnl

T

Tm2 + T m

T m2 - 0 5 T m 2 + *m

T m 2 - 0 7 S

T m 2 + 1m

• (463)

2Tm lTm 2 Tml

Tm2 , T-ral T m 2

+ 2x „

1+-

2

( T m , - T m 2 )

x m2

V T m l )

T„,2

T m , - T m 2 (T ~\

1 m 2

VTml J

Tm, 1 T m , - T m 2

m2

Tml _ T m 2

J m 2

T V 'ml )

Tmi Tm2

Xm2

V T m l )

Rule

Minimum ITAE -Bohl and McAvoy (1976b).

Kc

2 K (464)

T,

T (464)

Td

T (464)

Comment

Model: Method I

T 2 _E!l = 0.12,0.3,0.5,0.7,0.9; - ^ - = 0.1,0.2,0.3,0.4,0.5

K, (464) 10-9507

K „

> -1.2096+0.1760 In 1

V Tml J

o^y

10-

0.1044+0.1806 In 10^=2- -0.2071 In 10-^=-l Tm,J { TmlJ

T i< 4 6 4 )=0.2979Tn l

N 0.7750-0.1026 In 10-

10-2 V Tml )

> 0.1701+0.0092 In

10- 2

T V ml /

J 1 (&L +0.0081 In [ 10 -^ -

Td< 4 6 4 )=0.1075Tn ,

v 0.6025-0.0624 In 10-

10-2

V Tml J

T „ ,

10-

0.4531-0.0479lnl 1 0 ^ - |+0.0128In| 10-T„ , T m i ;

' m l )

Rule

Minimum ITAE -Hassan (1993).

Kc

3 R (465)

T,

T (465) i

Td

T (465)

Comment

Model: Method 7

J 0.5 < ^m < 2 ; 0.1 < — ^ < 4 . Kc,40:" is obtained as follows:

Tmi

log[KmKc(465)]= 1.9763370-0.6436825^m - 5.1887660-^2- + 0.43755584„

+ 2.9005550^2-] +3.14680104m^!_-0.16972214n V^ml

-0.81618084n T

1.2048220^m2 - 5 ! - - 0.08103734m

3

-0.4444091-^-1 +0.03194314n

f \ 3

VTml J + 0.1054399^

VTmiy

-0.16528074m3^i_ + 0.11759914m

3[^2-| -0.03752454n 3 ' T"

ml J

T:(4 5) is obtained as follows:

lot . (465)

-0.7865873 + 0.6796885^m + 2.1891540-^2-- 0.3471095^m2

Tmi

-1.9003610 - ^ M -0.70078014m-^- + 0.3077857^n ^Tmi J Tml VTm1 )

-0.8566974^ ' m l y

- 0.2535062^m2 ^ - + 0.0412943^m

3

Tmi

+ 0.3484161 ( \ x

VTml J -0.1626562^ ^ = - -0.06618994m

2

v 1mi y

+ 0.2247806£m3 - $ - - 0.2470783^n ^ + 0.049301 l^m

3U=

(465) Td is obtained as follows (continued into footnote on page 313):

IOE T, (465)

= -0.6726798 - 0.20724774m + 2.6826330^2- + 0.0807474^m2

-1.7707830 -^s- -1.6685140^m-^- + 0.0845958^m2

^Tml J Tml Tmi

Rule

Minimum ITAE - Sung et al.


Kc

4 j , (466)

T i

rj, (466)

Td

T (466)

Comment

0.05 < i s - < 2

(, \ + 0.7159307^n

VTmiy + 0.56314474m

2 i * - - 0.0225269£m3

Tmi

+ 0.2821874 -0.0616288^ m] /

i s - | -0.0626626^ 2

V^rol V^ml

- 0.0372784^m3 i s - - 0.0948097^m

3 i s - | +0.0272541^ 3

V ml .

Formulae continued into footnote on page 314.

K (466)

K „ -0.67 + 0.297

T V ml J

+ 2.189 r \ -0.766

1 „

VTml , Tm, •<0.9 or

K (466) _ __1_ -0.365 + 0.260 T

V iml

-1.4 +2.189 T

V 1 m l

->0.9;

. (466) 2.212 , N 0.520

VTml / -0.3 •<0.4 or

Ti(466) =Tm l {-0.975 + 0.910 •is--1.845 + a} , - i s - > 0.4 with VTml J Tml

a = 1-e 5.25-0.1

T (466) _

1-e 1.45 + 0.969 -1.9 + 1.576 V ^ m l

Rule


Model: Method 10

Kc

5 R (467)

6 K (468)

7 ^ (469)

Ti

T (467)

T (468)

T ( 4 6 9 )

Td

T (467)

Comment

s2 < 2.4

2.4 < e2 < 3

3 < e2 < 20

5 K (467) f K „ , - ^ ;l-0.435<DH,Tm +0.052a)H,2Tm2P

, (467)

Km/(l + K H 2 K m )

Kc'467)(l + K H 2 K j

Km0.0697(l + 0.752coH2tm-0.14503H22Tn,2j'

T d - = ^ ^ f l - ^ l ( l - 0 , 1 2 c o u x m + 0 , 0 3 c o u S m2 ) .

K m (»u I £ n

6 K (468) ' K 0.724[l-0.469(0H2Tm+0.0609coH22Tm

2lN

Km/(l + K H 2 K m )

.(468) K c « 6 8 > ( l + K H 2 K m )

o)H2Km0.0405^1 + 1.93oiH2Tm-0.363cDH2iTm

/

7 K (469) _ 1.26(0.506)'°"2T'"[l-1.07/e2+0,616/e22]N

Km/(l + K H 2 K m )

.(469) T l « " l K c (1 + K H 2 K m )

)H2Km0.066l(l+ 0.824 ln[coH2xm])(l + l . 7 l / E 2 _ l . l 7 / e 22 ) '

Rule

Hwang (1995)-continued

Minimum IAE -Wills (1962b) - deduced from

graph. Model: Method 1


(1968). Model:

Method 1

Minimum ITAE -Wills (1962a)-deducedfrom

graph.

Kc

8 K (470)

Decay ratio

Ti

T (470) i

Td

~ (467)

(page 314) = 0.2; 0.2<xm /Tm l <2.0 and 0.6

Comment

e2 >20

<?im<4.2


6-5/Km

7/Km

x, /K m

Represent fm

2Tml

0.053 0.11 0.25

x i

6.7 4.15 2.35

18

Kra

1.45TU

0.95TU

9 T ( 4 7 1 )

0.14TU

0.22TU

T (471) x d

Representative tuning values;

Tm2 = Tm

= 0.1Tml

Tmi = Tm2

itive coefficient values - deduced from graphs x2

0.89 0.84 0.77

X 3

0.24 0.24 0.24

« T

V 2Tml

0.67 1.50 4.0

* 1

1.05 0.70 0.52

*0.2TU

x2

0.64 0.55 0.50

x3

0.24 0.26 0.20

Tmi = Tm 2 ;

^m=0.1Tm l ;

Model: Method J

K (470) _ ' 1.09[l-0.497(oH2Tm+0.0724a)H22Tm

2lN

H2 Km/(l + K H 2 K j

T(470) K c (l + K H 2 K m )

' fflH2Km0.054(-1 + 2.54aH2xm - 0.457a>H22xm

2)'

"if7" =x2(Tm l + T m 2 + x j , Td<471) =x3(Tml + T m 2 + xj.

Rule

Minimum ITAE - Sung el al.

(1996). Model: Method 9 Nearly minimum IAE, ISE, ITAE -Hwang (1995).

Model: Method 10

Kc

10 £ (472)

11 j r (473)

Tj

j (472)

j (473)

Td

x (472) 1 d

0.471KU

Kmcou

Comment

0.05<-^2-<2

Decay ratio = 0.1;

0.2 < -^2- < 2.0; Tml

0 .6<^ m <4 .2 1 2

10 K (472) _ 1 - 0.04 + 0.333 + 0.949 / N-0.983

5 m <0.9 or

K (472) _ 1 -0.544 + 0.308—i2-+ 1.408 * . T i l i f ' 5" 5 m >0-9 .

Ti(472) =Tml [2.055 + 0.072(xm/Tml)Jm, Tm/Tml <1 or

T/472' =Tml[l.768 + 0.329(Tm/Tml)£m, xm/Tm, > 1 ,

(472)

T , „ i

J _ e 0.870 0.55 + 1.683

11 K (473) K 0.822t-0.549(oH2Tm+0.112coH22Tm

2]N

Km/(l + K H 2 K m )

- (473) K ( ^ ( 1 + KH2K,J (oH2Km0.0142(l + 6.96(DH2Tm-1.77o)H2

2xm2)'

12 ^< 0.613+ 0.613^21-+ 0.117 T ' m l

r ^2

vT m , y

Rule

Hwang (1995)-continued.


Kc

13 j , (474)

15 K (475)

16 K (476)

17 K (477)

Ti

T (474)

T (475)

T ( 4 7 6 )

T (477)

Td

0.471KU

K m m u

0.471KU

K m f f l u

Comment

Decay ratio = 0.1;

0.2 < ^-2-< 2.0 ; Tm,

0 .6<^ m <4 .2 1 4

Decay ratio = 0.1;

0 . 2 < ^ - < 2 . 0 ; Tral

0 .6<4 m <4.2 1 8

13 v (474) _. „ 0-786]! - 0.441coH2Tm + 0.0569coH22Tm

2 h

' Km/(l + K H 2 K j

T(474) _ Kc (l + KH 2Km)

2Km0.0172(l + 4.62WH2xm -0.823coH22tm

2) '

E<0.613 + 0.613-^- + 0.117 U ^ Tml ^Tm

15 K (475) K t 1.28(0.542)c°H;tm[l-0.986/s2+0.558/e2

2}

, (475)

16 j . (476) K t

.H[I

Kc'475>(l + KH 2Km)

2Km0.0476(l + 0.9961n[coH2Tm])(l + 2.13/s2 -1 .13/e 22 ) '

1.1411-0.466coH2xm + 0.0647coH22Tm

2^

T;

Km/(l + KH 2Km)

(476) Kc (l + KH2Km)

17 K (477) *[l-0.: r-coH,Km0.0609 - 1 +1.97co„,xm - 0.323co

— 2 — 5 1 -H2 T m J

0.794|1 - 0.541coH2Tm + 0.126(oH22Tm

2 "

Km/(l + K H 2 K m )

.(477) K<477'(l + K H 2 K j

a>„2Km0.007811 + 8.38coH2Tm - 1.97coH2zxn

£> 0.649+ 0 . 5 8 ^ - - 0 . 0 0 5 Tm, VTml )

Rule



Kc

19 j r (478)

21 j r (479)

22 K (480)

23 K (481)

T,

T (478)

T ( 4 7 9 )

T (480)

T (481)

Td

0.471KU

Kmwu

0.471KU

Comment

Decay ratio = 0.1;

0.2 < -^5- < 2.0 ; Tn,

0.6 < ^m < 4.220

Decay ratio = 0.1;

0.2 <-^2-< 2.0;

0 .6<^ m <4 .2 2 4

19 K (478) _ 0.738[l - 0.415ooH2xm + 0.0575coH22xm

2 \> j ^ H 2 . ,

Km/(l + KH 2Km)

T (478) K (l + K H 2 K m ) 1 aH2Km0.0124(l + 4.05coH2xm-0.63coH2

2xm2)'

X ( T ^ 20 £,> 0.649 + 0 .58-^- -0.005 —s

'ml V^miy

21 K (479) K t

1.15(0.564)m"iT"[l-0.959/e2+0.773/e22]"

Km/(l + K H 2 K m ) J'

T . (479) K c ( 1 + KH2Km)

coH2Km0.0335(l + 0.9471n[coH2xm])(l + 1.9/82-1.07/e22) '

22 K (480) _ i 1.07|l-0.466coH2xro+0.0667mH22xm

2]N

K H 2 - , J

v Km /(l + K H 2 K j

r(480) Kc'^(l + K H 2 K j

23 K (481) ?[i - o.:

mH2Km0.0328(-1 + 2.21C0H2X,,, - 0.338coH22Tm

2)'

0.789|l-0.527(DH2xm+0.11CoH22Tm

2l^ K H 2 —-• =

Km /(l + K H 2 K m )

.(481) K™(1 + K H 2 K B )

)H2Km0.009(l + 9.7co xm-2.4coH , 2Tm2) ' H2 lm

I < 0.649 + 0.58—— - 0.005 T VTml 7

,^>0.613 + 0.613^a- + 0.117 X „

VTml J

Rule


Wilton (1999). Model: Method 1

Kc

25 K (482)

27 K (483)

28 K (484)

Ti

T (482)

rp (483)

T (484)

Td

0.471KU

Kmwu

Comment

Decay ratio = 0.1;

Q.2<^-<2.0; Tmi

0.6 <lm <4.22 6


Minimum

x i

"[e2(t) + x 22Km

2 (y(t) -yJ 2 ]dt; Tm, >Tm2 •0

Tm l+Tm 2

Tm2

1 + Tm2/Tml

Coefficient values (obtained from graphs) T r a2/Tml=0.25

x i

1250.1 22.946 4.8990 2.1213

x2

0.0001 0.04 0.2 1

Tm2/Tml=0.5

x i

20.396 4.8990 2.1213

x2

0.04 0.2 1

Tm2/Tm, =0.75

x i

22.946 4.8990 2.1213

x2

0.04 0.2 1

xm /Tm l=0.1

25 K (482) _ ' 0.76[l-0.426o)H2Tm+0.0551mH22Tm

2lA

Km /(l + KH 2Km)

26 £,< 0.649 + 0.58-^—0.005 Tmi

• (482)

( \2

T

Kc«82>(l + K H 2 K m )

raH2Km0.01531 + 4.37coH2Tra-0.743»H2iTn

VTml V ,4>0.613 + 0 . 6 1 3 ^ - + 0.117

' x > '

VTmiy

27 K (483) 1.22(o.55)"'H'T" [l-0.978/s2 +0.659/e22 j

Km/(l + KH 2Km)

, (483)

28 K (484)

j t io j ; K c (1 + KH2Km )

coH2Km0.042l(l + 0.9691n[coH2xm])(l + 2.02/e2 - l . l l / s 22 ) '

' l.ll[l-0.467coH2Tm +0.0657coH22Tm

2lN

Km/(l + K H 2 K m )

T; (484) Kc'^(l + K H 2 K j

>H2Kra0.0477(-l + 2.07coH2Tnl-0.333fflH22Tm

2)'


Rule

Wilton (1999) -continued.

Keane et al. (2000).

Model: Method 1

Huang and Jeng (2003). Model:

Method 13

Kc

x i

Ti

Tmi + Tm2

Td

Tm2

1 + Tm2/Tml

Coefficient values (obtained from graphs) Tm 2 /Tm l=0.25

x i

180.01 8.1584 2.9394 1.2728 90.004 4.5891 2.0331 0.8768

x2

0.0001 0.04 0.2 1

0.0001 0.04 0.2 1

29 K (485)

Tm2/Tml=0.5

x i

230.01 9.1782 3.1843 1.3435 100.00 4.6911 2.0821 1.0324

x2

0.0001 0.04 0.2

1 0.0001

0.04 0.2

1 T (485)

1 I

Tra2/TMl=0.75

x i

300.01 10.708 3.4293 1.4142 112.01 5.354

2.1311 1.0748

x2

0.0001 0.04 0.2

1 0.0001

0.04 0.2

1 T (485) ' d

Comment

Tm /Tm l=0.5

x m / T m l = l

Minimum ITAE servo plus ITAE regulator, with minimum Mmax and

with good noise attenuation; Tml = Tm2

30 j - (486) 2^mTml T (486) Sn.Sl-1

( l n K j T u + T m , ( l + lnKu)] 29 K (485)

-(485) _

T (485) _ T T ' u ' m l

T u+Tm l ( l + l n K u ) '

(lnKu)[Tu+Tm l(l + lnKu)]

(lnKu)(l + lnKu) + [ lnKu+ln(l + eT»-K-)][ln(Ku+1.33419Tm)-l]

30 K (486) _ 1.2858Tm^m [ T„

KmTm

(486) _ [(-0.0349^m +1.QQ64X +(0.4196^m -0.1100>mP

2T,^m

Rule

Huang and Jeng (2003) -

continued.

Hang et al. (1993a). Model:

Method 8;

^ 2 - > 0 . 3 Tml

(Yang and Clarke (1996)).

H o e ; al. (1994). Model: Method 1

Hoet al. (1995a). Model: Method 1

K c

31 K (487)

Ti

T (487) i

Td

T (487)

Comment

4m > 1-1

Servo or regulator tuning - Minimum IAE

Direct synthesis: Frequency domain criteria

^ m l

A m K r o T m 2Tml 0-5Tml Tml ~~ Tm2

Sample results

1.5708Tml

KmTm

1.0472Tml

K m x m

32 K (488)

M p T m l

An ,K i n

2Tml

2Tral

T ( 4 8 8 ) i

33 j (489)

0-5Tml

0-5Tml

T (488)

Tm2

A r a = 2 . 0 ,

A m = 3 . 0 ,

* m = 6 0 °

T m / T m < 2 ^ m

Tm, > T m 2 .

31 K (487) 0.5890

K m t m y ^ra V

T 0 . 0 0 5 2 - 2 ^ - + 0.8980Tm2 +0 .4877t m + T m l

T/ 4 8 7 ' = 0 . 0 0 5 2 - 2 ^ - + 0.8980Tm2 +0.4877xm + T m l ,

T (487) _ T

T 0.0052 - = ^ - + 0.8980Tm2 +0 .4877i n

0.0052 - = = - + 0.8980Tm2 +0.4877xm + T m l

32 T , (488) _ ' " " p 1 m l 2conTm,2 f „5

7 t A „ , 7 t - 2 0 ) T m

V«pTm,

T (488) _ 2 2 > M - ' m l

(488) _

J

f , ^ i s ! - + t - 2 « p x m

VTml«p TtT.

• + t T m l -2T m ] co p T m

2co„

1

4©D2Tm 1

Rule

Ho et al. (1995a) - continued.

Ho etal. (1997). Model: Method 1

Leva et al. (1994). Model:

Method 15

Kc Ti Td Comment

Sample results 0.7854Tm,

K m T m

0.5236Tml

K m x m

34 K (490)

36 K (491)

Tml

Tm,

T (490) 1 i

T (491)

Tm2

Tm2

T (490)

O^TA491'

A m = 2 . 0 ,

<L=45°

A m = 3 . 0 ,

<K„=60o

35

* m = 7 0 °

(at least).

34 K (490)

tA m K m **>m+K-2-

l m l J

,Tr 0 , =-T m l fe m + , -2 , m ) , 7t

TtT, ml T (490) _ _

2 ( ^ m T m l + 7 t T m l - 2 T m ) ' T, Tm < 1 , 2£ > ^ 2 - .

8™ m ^ - ( A m

2 - l ) - 2 7 t A m ( A m - l )

4A„

36 v (491) _ W c n j i (491)

K l + »c„ 2Tm , 2

K m ^

) 2 +4^ m2 co c n

2 T m l

, (491)™ (4911 2

1-T^'V X" r+[Ti ] w r , ( 4 9 1 ) 0 ^ 2

4.07 - <|>ra + tan" i [2^x m T m l (0 .57 I - c t m )

(0.57t-<^ ra)2Tml2-xm

2

, (491) -tan 0-5 co c n T m +<|> m -0 .57t- tan"

2^mcocnTml

V ^ c n T m l - ! y

l m ^ o r ^ m > l (with 0 . 5 7 1 - ^ > 3 T „

^m+V^2-1

Rule

Leva et al. (1994)-

continued. Wang et al.

(1999).

Wang and Shao (1999). Model:

Method 12

Kc

37 K (492)

SmT r al

K m T m

1 K m X m

Ti

T (492) 1 i

2^raTm,

2^mTml

Td

T (492)

0-5Tml

4 m

T

Comment

Model: Method 12. 38

Some coefficient values

* i

1.571

1.047

Comment

A m = 2 , < L = 4 5 °

A m =3,< t , m =60°

* i

0.785

0.628

Comment

Am = 4 , ^ = 6 7 . 5 °

A r a=5,<j>m=72°

T 2 | C 0 c n 2 + - V ( 2 ^ V ^ 37 ^ (492) _ _ ^ c n j _ m l _ 1 m l V

( 4 9 2 > l l coj+x2

• 1 - 1

(0.5TI -(|>m) Tml - x m

co„

tan fflcnTml

^m+V^n

. (492) Tralz + U m + ^ m

2 - l (492) ,

Sm + ^ m2 - 0 Tmlz + Um + kJ-l

. 5m>l

with 3x„

£m "V^ir < 0.571 -< |> m < Tml

S m + A ^

^ m > 0.7071

or 0.05 < ° - 7 ° 7 h - <0.15, Q J Q 7 l T "- > U „ * 1 Tm l A /24m

2- l T m l ^ - 1

Tml

or 0.05 < % ^ s - < 0.15 , - ^ 2 - > l,^m < 1.

Rule

Wang and Shao (2000b). Model:

Method 12

Chen et ah (1999b).

Model: Method 1

Leva et ah (2003).

Model: Method 1

Kc

39 ^ (493)

X lSmT m i

* i

1.00

1.22

1.34

1.40

1.44

1.52

1.60

^

2^mTml

2^m

Td

Tml

2^m

2^m

Comment

Suggested Mmax = 1.6

Comment

A m = 3 . 1 4 , <L = 61.4°, Ms =1.00

A m = 2 . 5 8 , ^ = 5 5 . 0 ° , M.=1.10

Am =2.34, <L = 51.6°, M. =1.20

Am =2.24, 4>m = 50.0° ,M, =1.26

A m = 2 . 1 8 , *m = 48.7° , M,=1.30

Am =2.07, <L=46 .5° ,M S =1.40

Am =1.96, 4)m =44.1°, M,=1.50

40 K (494)

41 K (495)

16(Tm 2+tm)

5Tm2+4Tm

4(Tm 2+xm)

T (495) x d

3 9 j£ (493) _ " » » m l 2^mTm .451-1.508 M „

40 K (494) :

8Km(T, 4(Tm , ) •

41 rr (495) _ Tm l ( 5 T m 2 + 4 T m ) T (495)

8Km(Tm 2 +Tm)2

4Tm2(Tm2+xm) 5Tm2+4Tm

T , > ' A m l —

8(Tm + x„

Rule

Wills (1962a). Model: Method 1 Servo response;

4 = 1-

•Cm = ° - 1 Tm l


Model: Method 1

Pemberton (1972b).

Pemberton (1972a), (1972b).

Model: Method 1

K T, Td Comment

Direct synthesis: time domain criteria X i / K r a x2Tu

X 3 T u


* i

2.5 2.75 3.2 20 12 1.4 1.5

x2

2.9 2.9 2.9 2.9 2.9 1.4 1.4

* 3

0.035 0.073 0.143 0.285 0.57

0.035 0.073

42 £ (496)

43 j r (497)

2(Tml+Tm2)

3Kmxra

(Tm l+Tm 2)

Kraxm

* 1

1.6 16 10

0.42 0.4 10 2

x2

1.4 1.4 1.4

0.70 0.70 0.70 0.70

Tm l+Tm 2

+0.5Tm

T (497)

Tm l+Tm 2

T r a l+Tm 2

* 3

0.143 0.28 0.72 0.073 0.143 0.36 0.70

*1

0.2 8

0.09 0.1

0.11

T (496) 1 d

T (497)

TmiTm2

Tml +Tm2

TmiTm2

x2

0.48 0.48 0.28 0.28 0.28

* 3

0.143 0.36 0.073 0.143 0.28

Step disturbance


Model: Method I

0 . 1 < ^ - < 1 . 0 Tm2

0 . 2 < ^ - < 1 . 0 Tm2

42 K (496) = 1 f Q 5 i T m l + T m 2 I

K. (496) (Tm ,+Tm 2)xm+2Tm lT l 1 1 m 2

^ + 2 ( T m l + T m 2 )

43 K (497) _ e

K „ 1 - Q.5e-a'x- +-

T „ , + T „

• (497)

• (497)

Tml + T „

X

l-0.5e" ( T m l + T m 2 ) T

(Tml+Tm2) m /

l-0.5e~ *ml + ' m 2 e~MJTm

Rule

Pemberton (1972b).

Model: Method 1

Chiu et al. (1973).

Model: Method 1

Mollenkamp et al. (1973).

Model: Method 1

Smith et al. (1975).

Model: Method 1

Suyama(1992). Model: Method 1

Wang and Clements (1995).

Gorez and Klan (2000).

Model: Method 1

Viteckova (1999),

Viteckova et al. (2000a). Model:

Method 1

Kc

2 ( T r a l + T m 2 )

3K m x m

^T m l + Tm2

K n ( l + XTm)

2smTm i

K r a ( T C L + x r a )

W m l

K m ( l + ^ t m )

X = pol

2 K m x m

2 ^ m T m l

K r a( l + ^ m )

44 K (498)

x , (T m l + T m 2 )

K m t m

x i

0.368 0.514

0.581 0.641

OS (%)

0 5

10 15

Ti

Tmi +Tm2

Tmi + T m 2

2smTm i

Tml

e of specified FO]

Tmi +Tm2

2smT r a i

2smTm i

Tmi + Tra2

x,

0.696 0.748 0.801

0.853

OS (%)

20 25

30

35

Td

T r a , + T m 2

4

TmlT,n2

T m l + T m 2

Tml

2^ m

Tm2

Comment

0 . 1 < ^ k . < 1 . 0 Tm2

0.2 < ^ 2 - < 1.0 Tm2

A. variable; suggested values are 0.2, 0.4, 0.6

and 1.0.

Appropriate for "small xm "

Tm, > Tm2

J D closed loop response.

TmiTm2

T m l + T m 2

24m

Tml

2^ m

Tm iTm 2

T m , + T m 2

x i

0.906 0.957

1.008

OS (%)

40 45

50

OS=10%

Model: Method 16;

Underdamped response

Non-dominant time delay

Closed loop overshoot

(OS) defined

Overdamped process;

T„i > Tm2

K (498) _ 2^mTm l

K m ( 2 ^ m T m l + ! „ , ) •

Rule

Viteckova (1999),

Viteckova et al. (2000a) -continued.

Skogestad (2004b).

Model: Method 1

Vitefikova et al. (2000b).

Model: Method 4 Arvanitis et al.

(2000).

Bietal. (2000). Model: Method 8


Kc

xiSmTmi

KraXm

x, 0.736 1.028 1.162 1.282

OS (%)

0 5 10 15

2^mTml

K m (T C L + x m )

A

0-74Tml

KmTm

45 jj- (499)

l -0128^mTm l

K m x m

46 Tj (500)

Ti

24 m T m l

x i

1.392 1.496 1.602 1.706

OS (%)

20 25 30 35

2 ^ m T m l

Td

Tm l

2 ^ m

* 1

1.812 1.914 2.016

OS (%)

40 45 50

Tm l

2 ^ m

Comment

Underdamped process;

0-5 < ^m < 1

Recommended

TCL = Tm

ra = 3 . 1 4 , <|>m = 6 1 . 4 ° , M m a x =1 .59

2Tm l

2^dm

esTml

l -9747K m x m

T (500)

0-5Tml

Tm l ^edes

0.5064Tm l2

K m t m

x (500)

Tml - Tm2

Model: Method 1

Model: Method 1

45 K (499) 2 ^ s T m

6 K (500) ^ 1 [(Tm| + Tm 2 )xm + Tm |Tm 2](3TC L + T m ) - T C L3 - 3T C L

2 x m

C " K m ( T C L + ^ ) 3

T(500) = [(Tm[ + T m 2 ) x m +T m l T m 2 ] (3T C L + T m ) - T C L3 - 3 T C L

2 t m T m i T m 2 + ( T m i + T m 2 + T m ) T m

T (5oo) = 3T C L2 T m l T r o 2 + T m l T m 2 x m ( 3 T C L + x m ) - ( T m l + T m 2 + x m ) T C L

3

" [(Tm, + T m 2 )x m + Tm ,Tm 2](3TC L + x m ) - T C L3 - 3 T C L

2 x m

.dVdesi

T ( 5 0 0 ) s e - s x „ _ Ti se -

'desired K c ( T C L S + 1 /

Rule

Chen and Seborg (2002) -

continued.

Brambilla et al. (1990). Model:

Method 1

Lee etal. (1998). Model: Method 1

Kc

47 K (501)

48 y- (502)

Ti

T (501) i

Td

T (501)

Comment

Robust T (502) ry, (502)

»d

Representative coefficient values (deduced from graph)

^m T m l + T m 2

0.1 0.2 0.5 T ( 5 0 3 )

Km(2A. + Tn

X

0.50 0.47 0.39

,)

1m

Tml + Tm2 1.0 2.0 5.0

49 j (503)

X

0.29 0.22 0.16

•tm

Tm l+Tm 2

10.0

T (503) J d

X

0.14

47 K„ ( 5 °

..) 1 (2^T m , x m + T m , 2 ) [3T C L + T m ) -T C L3 -3T C L

2 T

^ _ K m ( T C L + X J 3

j (5oi) _ j2^mTmlTm+Tml J j3T C L +t m ) - TCL -3TC L

Tm l2+(2f T . + r

/K~"-CL ' v m / J C

+ (^^m^ml + T m) T i + l 2 Sm 'ml + T m / T m

T (501) _ 3 T g L2 T m l

2 + T m l Tm JTcL

+ Tmlz<3TCL + -3TCL^

IdJ, 48

. /desired K c (TCLS + 1 /

K (502) = Tml + T m 2 + ° - 5 T m T (502)

KmTm(2^ + l) ' ' = Tml +Tm

49 T(503) _ 7 ? T 2X -Z

' ~1^'^ ihx + i

,2+0.5xm

j (502) _ Tm |Tm2 +Q-ATml + Tm2/tm Q | < \ d Tm ,+Tm2+0.5Tm ' ' ~ T m ^

<10. Tm ,+Tm2+0.5Tm Tm l+Tm 2

2 ^ ~ T m T (503) _ T (503) _ , , T Tml _

2(2X + T r a ) ' d " ' ^ m l " » +T ] ( 5 0 3 )

x 3 e-

x"s

—,.-,. .m r ; desired closed loop response = — 6Ti

(503)(2>. + Tm) (Xs + l)2


Rule

Lee et al. (1998). Model: Method 1

Huang et al. (1998), Rivera and Jun (2000).

Model: Method I

Marchetti and Scali (2000).

Kc

T (504) i

Km(2A. + Tm)

T; 50 y (504)

* i

Td

T (504) ' d

Comment

X = max imum[0.25Tm ,0.2Tml ] (Panda et al., 2004)

T (505) i

Kra(^ + Tm) rj, (506) 1 i

Kro( + 0 24mTral

Km(xro+A.)

51 T (505)

52 T (506) * i

2^mTml

T (505) *d

T (506) ' d

Tml

2^m

Desired closed loop response =

(X.s +1)

A, specified graphically for different OS and TR values (Huang et al.

(1998)) 53 ^ (507) j (507) x (507)

' d Model:

Method I

50 T ( 5 0 4 ) _ T T 2 ^ 2 - x m2

2(2A + x m ) T (504) _ T (504) T T

T m l T m 2

• (504)

-rrrrr r ; desired closed loop response = — 6T i

(504)(2A + Tm) (Xs + l)2

( T

3 ^

51 x (505) T i ^ ' = 2 ^ m T m l + 2(^ + Tm)

T (505) _ T (505) ») . T

> 'd ~ li lCsm 'ml 6(A + T J

.(505)

52 T (506) : 1 m 1 + * m ~> "*" 2(X-

T (506) _ T (506) ( T , T -v 1 d ~ M I ' m l + ' n i 2 '

6(X + -,(506)

53 K (507) _ 2^mTml +0-5Tm T(507) = 2 £ J + 0 5t T ( 5°7 ) - Tml(Tml + ^ m O

Km(X- 2^ m T m l +0.5T n

Rule



Wills (1962a). Model: Method 1

Quarter decay ratio tuning. Tmi = Tm 2 ;

^m=0.1Tml

Bohl and McAvoy (1976b).

Landau and Voda (1992).

Model: Method 1

Kc

Tml

Kj^ + O 54 K (508)

Ti

Tml

2^raTral

Td

Tm2

TmI

2^m

Comment

^ e K , . T m 2 ] ,

T m l >T m 2

Model: Method 1

Ultimate cycle x,/Km x2Tu x3Tu

Coefficient values (deduced from graph) x i

10 8 14 33 18

x2

2.9 1.4 2.9 2.9 1.4

x 3

0.036 0.036 0.068 0.143 0.068

55 K (509)

56 £ (510)

3KU

x i

28 6 2

0.6 30

x2

1.4 0.67 0.48 0.28 0.67

T ( 5 0 9 )

4 + (3 1

0.51T„

* 3

0.143 0.068 0.068 0.068 0.143

x i

18 0.9 18 18 13

T (509)

4 1

4 + M,35»

0.13TU

x2

0.48 0.28 0.67 0.48 0.28

X 3

0.143 0.143 0.24 0.24 0.28

Model: Method 1

1 < (3 < 2

(»uTm<0.16

54 v (508) _ 2 ^ m T m l K Kmxm

55 K (509) _ 2 . 2 6 8 9

K „

T:(509)=0.2693T„

Td(509) = 0.0068TU

M „

2.0302+0.9553 In T I IT.

T

. Recommended values of Mraax : 1.4,1.7,2.0.

(V K \1.3135+0.18091n(K„K„)+0.52191np!!.

r \-0.0517-0.0643 In —

T

-4.2613-1.58551n

(K K ) a 5 9 1 4 - 0 0 8 3 2 1 n ( K « K m ) + 0 0 6 8 7 1 n — 1

(K K V11436-°-2658ln(KuKm)-lll001n| ^2-1

56 K (510) 4 + P P 4 2V2|Gp(ja>135„)|

Rule

Landau and Voda (1992)-

continued.

Kc

1.9KU

T;

0.64TU

Td

0.16TU

Comment

0.16<couTm

<0.2

Gc(s) = Kc 1 + "Tds Tfs + 1

R(s) ms;

^ 1 i

K c | l + - J - + Tds I T,s J - •

1 T fs + 1

U(s) S O S P D

mode l

V(s)

^w

Table 116: PID controller tuning rules - SOSPD model Gm (s) = K m e ^

(l + sTm,Xl + sTm2)

Gm(s) Kme"

Rule

Umetal. (1985).

Arvanitis et al. (2000).

Model: Method 1

Panda et al. (2004).

Model: Method 1


Method 14

Hang et al. (1993a).

Model: Method 8

m v /

K c

T m l2 s 2 + 2 4 r a T

Ti

m l S + 1

Td Comment

Direct synthesis: time domain criteria 1 K (511)

2 K (512)

S m * m l

KmTm

T m l + T m 2

2 ^ % ,

2^mTm l

lmlTm2

Tmi + Tral

Tra, -^tdes

Tm,

2^ m

Direct synthesis: frequency domain cri

KmTm 2^mT r a l

Tm,

24m

A m * 3 , < t , m * 6 0 °

Robust

2Tml + Tm

2{X + xm)Km

Tm, +0.5xm ^ m l T m

2 T m l + x m

Model has a repeated pole ( T m l ) .

Model: Method 1

T ^ %m

T T ml

f " 2N^ m '

N not defined teria

T °-025Tml

^ > 0 . 2 5 x m .

T ^ m

' 2(X + i m )

1 K (5U) =

2 ^ (512)

Tm l+Tm

Km(2^Tc • . T f

T, CL2

24TCL2+tn

KJS

*CL2 T l m

= , ^CL2 ~ ' l r n de%,,+0' f 2^esTr,,+T

2CTm, m lCL2 T '•ra/

= 2 ^ T r i , 2 - i „ 2 1

?m »CL2 T l m


Rule


Model: Method

1; X not

specified Lee and Edgar



Model: Method 1

5% overshoot: X = 0.5xm

K c

24 m T m l

X

2^mTm l

K m ( 2 T m + X )

3 K (513)

Tmi + Tm2

K m k+2^)

4 K (514)

T ( 5 1 5 )

K m ( 2 ^ + Tm)

T,

2^mT r a l

2^mT r a l

X (513) i

Tmi + T m 2

I ml Im2

Tmi + T m 2

5 T ( 5 1 5 )

Td

2^ m

Tml

2^ m

T (513)

TmiTm2

Tml + Tm2

T m l + T m 2

T (515)

Comment

T f = - c m

T f = ^ 2xm+X

T * ' • 2A. + t m

H ^ method

T f = Xx™ ; X + 2xm

H 2 method

Tf = 0 ;

Maclaurin method

3 K (513) = _ (2f;mTml

0.5x 2 - ^ 2

Km(2X + xm)K S m ml 2X + x

6{2X + xJ 2 ^ T m l +

0-5xm2-X2^

2X + xm

0.5xJ-X2

2X + x„

J (513) . i t T , li - z S m ' m l +

2A. + T m

6(2^+ x m ) 2^mTm l +

0.5x 2 - ^

2X + x

0 . 5 x „ / - ^

m ; 2X + x„

(513)

4 v (514) K

6(2^+ x m )

Ami l m 9

2^mTm l + 0-5xm

2-X2

2A, + xm

0.5xm2 -X2

2A. + T„

K m ( T m l + T m 2 X ^ + 2 i m ) '

5 T ( 5 1 5 ) : T m l + T „ . 2X2-xJ 2(2X + xm)'

t m l ^rtl

• (515) 12A. + 6xm 2 ^ - x „ -(515) 2(2X + x m ) '


Rule


Model: Method 1

Panda et al. (2004).

Model: Method 1

Kc

6 K (516)

7 K (517)

8 j £ (518)

9 K (519)

Ti

^ S m ^ m l

T ( 5 1 8 )

-p (519)

Td

2^m

T (518)

T (519)

Comment

K (516) 2^mTm

7 K „

Km i r a

2^raTm

a - a - 1 + -M „

1—1 Mn

T T — ml

• f " T ' 1 + 0 . 0 1 9 — - JO.346 + 0.038-!-211- + 0.00036

Suggested M m a x =1.7 ,p = 10.

8 K (518) _24 m X 1 +0-25T m l +0.5^ m T Km(X + 2^mTml)

T(518) _ 4^m Tml +0.STml +4mTm T (518) _ (Tml +2^mTmJTm |^m

' 4^m2Tm, +^ m t m +0.5T m

T f =

2^„

^(Tm l+2^mxm)

44ma. + 44mxm+2Tm

-, X = max imum 0.25 2^n

-0.4^mT,

9 K (519)

0.828 +0.812-2^-+ 0.172Tmle Jm]

-(6.9T„,/Tml) . 0.558Tm2T„,

Tml+1.208Tra2

- + 0 . 5 T „

K „ A, + 0.828 + 0 . 8 1 2 ^ - + 0.172Tmle-(6-9T™;/T"")

Tmi 0.558T ,T

T/ s ' 9 >=0.828+0.812ii^+0.172T . e ^ " " ' ^ ^ u ' 3 3 8 1 ^ 1 " " + 0 .5x n Tml Tml+1.208Tm2

0.828 + 0 . 8 1 2 ^ + 0.172T ,e-(6 '9T-/T">T LU6T^T^ + T Tml »Tm,+1.208Tra2 (519)

1.656 + 1 . 6 2 4 - ^ + 0.344Tm,e-(6-9T"'/T-')+ L 1 1 6 T ^ T " " + T

Tf = 4l-'16Tm,Tm2+Tro(Tm,+1.208Tro2)]

2(^ + xmXTm, +1.208Tm2)+2.232Tm2Tml '

Tml+1.208Tro2

X = max imum °-279Tm2Tml +Q2^ 0 - 1 6 5 6 + 0 - 1 62 4 l !°2. + 0-0344T e-MTm2/Tml)

Tml+1.208Tm2

Rule


Method 14

Kc

10 K (520)

T, Td

Ultimate cycle T(520) T (520)

Am«3,<t ,m«60°

Comment

T (520)

T f = d

20

10 K <520) = 0.080-K„ T„

/ 6.283xn

T V u J

T.(520)=0.159K„ Tu 6.283t„

T

Td(520) = 0.159-?-

K„

1 + K„ cos

f \ 6.283xr

T V u J

6.283T„

T


4.5.3 Ideal controller in series with a first order filter f - ^

Gc(s) = Kc

V T , s j

b f ls + l

a f ls + l

R( s) E(s)

K, 1+ — + Tds l + bf]s

l + afls

U(s)

H K „ e "

T m l V + 2 ^ T m l s + l

Y(s)

Table 117: PID controller tuning rules - SOSPD model Gm(s) = Kme-

T m l V + 2 ^ m T m l S + l

Rule

Jahanmiri and Fallahi (1997).

Model: Method 6

Kc

2sm'mi

K m ( t r a + ^ )

T i Td

Robust

2smTmi 24m

X = 0.25Tm+0.1^mTml

Comment

b f l =0.5Tm,

^ m

" " 2(Tm+^)


4.5.4 Ideal controller in series with a second order filter (

Gc(s) = Kc 1

1 + — + Tds V T i s )

l + bfls

l + afls + af2s

R(s) E(s),—

T:S

U(s

l + b f l s

l + a f ls + a f 2s

Kme"

Tml2s2 + 2^mTmls + 1

Y(s)

Table 118: PID controller tuning rules - SOSPD model Gm (s) = Kme"

T m l V + 2 4 m T m l s + l

Rule


Model: Method 1

Kc Ti Td Comment

Robust 11 K (521) 2^mTmi Tml

2^m

M m a x =1.7 ,

P = 10.

bIf = 0 , a I f =0.08Tm I ,a2 f =0.01Tml2

II ^ (521) _ 2 ^ m T m l 1 + 0 . 0 1 9 ^ - - .0.346 + 0.038-^- + 0.00036-=^

r

4.5.5 Controller with filtered derivative Gc(s) = K,

E(s) R(s)

+ & - K

c , 1 T d s

T i s l + ^ s N

, ! T d s

N J

U(s) SOSPD model

Y(s)

Table 119: PID controller tuning rules - SOSPD model G m ( s ) = K m e "

(l + sTmlXl + sTm2) or

Gm(s) = Kme"

Tml2s2+24mTmls + l

Rule

Hang et al. (1994).

Model: Method 1

Hang et al. (1994).

Model: Method 1 Zhong and Li

(2002).

Kc

WpTml

A m K m

0.7854Tml

K m x m

Tml

K m ( T C L + x m )

13 K (523)

Ti Td

Direct synthesis 12 T (522) Tm2

Sample result

Tml Tm2

Robust

Tmi

T (523)

Tm2

T (523)

Comment

N = 20. Tml > T m 2

A m = 2 . 0 ,

<J)m = 45°

N = 20.

T m l > ™m2

Model: Method 1

1

2co„ -4co„ t m i

n v (523) 2 ^ m T m l ( 2 a + T m ) - a (523) 2 a + x„

K m ( 2 a + x m ) ^

j (523) _ rj, (523)

2^raTml(2a + T m ) - a 2

2^mTm laz(2a + T m ) - a 4

(2a + xm)2 N = 2 0 ^ ^ 5 2 3 ,


4.5.6 Controller with filtered derivative in series with a second order f \

filter Gc(s) = Kc 1 + — + — ^ — T i S l + ^ s

V N

l + b f ls + b f 2s A

\ 1 ~r SfiS T Sf^S /

E(s) R(s)+

K, , 1 TdS 1+ + £— T;s

1 + -N

l + bf,s + bf2s

l + afls + af2s

U(s)

K e "

(l + sTml)(l + sTm2)

Y(s)


(l + sTmlXl + sTm2)

Rule

Shi and Lee (2004).

Model: Method 1

Kc | ^ Td Comment

Robust 14 K (524) T (524)

i T (524)

14 j , (524)

br, =" •m 1

4co 2co_ ' > a f 2 _

0.5 } m - 6 d B ,

,b f l =-2co_

2co„6dB(l + co_6dBxm)'

T (524) = T + T 0.5 T (524) = TmlTm2(p_6dB2 - 0.5[(Tml + Tm2 )co_6dB - 0.5]

[(Tml+Tm2)ffl-6dB-0-5]

N = 0.5[(Tml+Tra2)«_6dB-0.5]

Tm,Tm2a_6dB2 -0.5[(Tml +Tm2>o_6dB -0.5] '

U(s) = Kc (FpR(s) - Y(s))+^(F i R(s) - Y(s))+ KcTds(FdR(s) - Y(s)) T:S

Table 121: PID tuning rules - SOSPD model Gm(s) = K„e"

(l + sTmlXl + sTm2)

Rule

Oubrahim and Leonard (1998).

Model: Method I;

Tmi = Tm2 ;

0 . 1 < ^ - < 3

Kc Ti Td Comment

Ultimate cycle

' 0.6KU

20.6KU

0.5TU

Fi= l

0.5TU

F i = l

0.125TU

0.125TU

10% overshoot

20% overshoot

' F p = 1 . 3 1 6 - K " K " , F d = 1 . 6 9 f 1 6 - K " * ° p 17 + K K d

F„ = -38

17 + K m K u y

1.717

(l + 0.121KmKj

4.5.8 Ideal controller with set-point weighting 2 f 1 Af

R(s)

T W + T i S + l

U(s) = Kc

+ _E(s)[—

V T i S J T i V + T i S + l

wx f

\ \ l + — + Tds

V T i s j

K„e~

R ( s ) - Y ( s )

,Y(s)

T m l2 s 2 +2^ m T m l s + l

Table 122: PID controller tuning rules - SOSPD model Gm (s) = K „ e "

Tm l2s2+2^mTm ls + l

Rule


Kc T i Td Comment

Direct synthesis 3 ^ (525)

4 K (526)

T ( 5 2 5 )

T (526)

T (525)

x (526) Xd

Model: Method 1

3 K (525) X ^ Kraxm

2^m + T 1 m l , 0 < % < 1, with suggested x = 0.4 ;

,(525) =2m_ a2

1 +

a + 2£' u 2

+ 2 y l 7- ' , • n'l1-! . with

T-^ r + E2 - - 1 >0andwith a = l + j — ^ r: XTm](24mTm+Tral) U J XTml(24mTm+T ra l)

(525) _ 1 T m l T m

4 K (526)

.(526)

X 2 L T » +Tml

1 (2i;TCL2 + Tm p^mxm + Tm, )Tml - TmTCL22

Km*m TCL22+T r a(2^TCL2+Tm)

_ (24TCL2 H^mX^- T m +

' m l m^CL2

^ 2 + ^ m2 T m l ( 2 ^ r a + T m l )

(526) Tml PcL2 + Tm (2^TCL2+xm)]

(2^TCL2 +TraX2£mTm +Tml)Tm, - x m T c

4.5.9 Classical controller 1 Gc(s) = Kc

E(s)

1+ v T i s y

1 + Tds

R(s)

+ ®— K,

v T i sy

1 + Tds T

l + - ^ s N

U(s)

SOSPD model

Y(s)

Table 123: PID controller tuning rules - SOSPD model Gm(s) =

Kme"

Kme"

(l + sTmlXl + sTm2)

Rule


-page 149. Model:


Minimum IAE -Chao et al.


m \ J

Kc

Minim 0.80Tral

KmTm

0.77Tml

Km tm

0.83Tml

KmTm

5 K (527)

Tm l2s2+2^mT

Tj

mlS + 1

Td Comment

um performance index: regulator tuning

l-5(Tm2+Tm)

l-2(Tm2+xm)

0.75(Tra2+xra)

T(527)

0.60(Tm2+xm)

0.70(Tm2+xm)

0.60(Tm2+Tra)

T (527)

T m2 n ""\ T m 2 + t m

T m2 =0.5 Tm2 + xm

T 1m2 - 0 75 T m 2 + t m

N = 8

0.8 < £m < 3; 0.05 < — ^ — < 0.5 2^raTml

5 K (527) -0.01133 + 0.5017^m-0.07813^n

K „ 2smTmi ,

Ti<527) =2^mTm ,(l.235-0.41264m +0.09873^n

Td(527) =2^mTm,(l.214-0.6250^ra +0.1358^

-1.890+0.7902^-0.1554^'

-0.03793+0.3975$„, -0.053545m

2smTm i , 0.5696-0.8484^m +0.05055m

!

^Sm^ml ,

Rule


-page 159. Model: Method 1 N not specified


-page 121. Model: Method 1 N not specified

Minimum IAE Shinskey (1996)


-^s - - 0.2; Tm,

N not specified

Minimum ISE -McAvoy and

Johnson (1967).

N = 20

N = 1 0

Kc

6 R (528)

2-5Tm, K m T m

0.59KU

0.85TmI

K m ^ m

0.87Tm,

Km tm

l-00Tml

Kmxm

l-25Tml

Kmxm

x i

i

Ti

T ( 5 2 8 )

M

m+0.2Tm2

0.36TU

1.98xm

2.30xm

2.50xm

2.75xm

x 2 T m

Representative coefficient v

^m

1 1 1

1 1 1

2k xm

0.5 4.0 10.0

0.5 4.0 10.0

X l

0.7 7.6 34.3

0.9 8.0 33.5

x2

0.97 3.33 5.00

1.10 4.00 6.25

X 3

0.75 2.03 2.7

0.64 1.83 2.43

Td

T (528)

xm+0.2Tm2

0.26TU

0.86xm

1.65xm

2-00xm

2-75xm

x3 t m

alues (deducec

S-

4 4 7 7 4 4 7 7

0.5 4.0 0.5 4.0 0.5 4.0 0.5 4.0

Comment

T JjH2_<3 xra

i^>3 xm

-^2- = 0.2, Tml

2k = o.2 T m l

2k = o.i Tml

T - ^ . = 0.2 Tml

2k = o.5

2k = i.o Tml

Model: Method 1

[from x i

3.0 22.7 5.4

40.0 3.2

23.9 5.9

42.9

p-aph) x2

1.16 1.89 1.19 1.64 1.33 2.17 1.39 1.89

X 3

0.85 1.28 0.85 1.14 0.78 1.17 0.78 1.04

6 K (528)

,(528) _ ,

48 + 57

100

1 - e x™ K m T m 1 +0.342k _o.2

x m

T„, "\

1.5-e 1 + 0.9 1-e x-

2 ^

,Td<528)=0.56t„

1.2Tml \

1 - e T™ 0-6Tm2.

Rule

Minimum ISE -Chao et al.


Minimum ITAE - Chao et al.


Kc

7 K (529)

8 K (530)

T,

T(529)

Td

T (529) ld

0.8 < S < 3; 0.05 < — ^ — < 0.5 2i;mTml

T ( 5 3 0 ) T (530)

Comment

N = 8

0 . 8 < 4 m < 3

7 (529) 0.2538 + 0.3875^m -0.04283^,

K „ V Sm 'ml

T;1529* =2^mTn,1(l.8283-0.8083^m +0.1852§m2lf X

-1.652+0.54165m-0.095445m

0.1363+0.23424m -0.01445^ra2

Td<529) =2^raTml(l.077-0.4952^m + 0 . 1 0 6 2 ^ V T

2smTmi j

0.4772+0.04424m+0.0l694^m

2^raTml

8 K (530) _-0 .1033+ 0.53474m -0.07995);, 2 / N-1.798+0.71914m-0.1416^m

2

K „ ^ S n i ' m l J

Ti(530) =2^mTml(o.9096-0.1299^m + 0.04179i;ra

2 J — I

Td<530)=24mTml(l.2226-0.6456^m+0.1373^ 2 ! X

N -0.1277+0.52204m -0.086294m

2^mTmi ,

0.4780-0.03653^m +0.04523^'

2£ T m ml J

Rule


(1989)-continued.

Minimum ITSE -Chao et al.


Kc

9 K (531)

10 K (532)

Ti

T ( 5 3 1 ) i

Td

T (531) 1 d

N = 8 ; 0 . 0 5 < — ^ — <0.5 2smTmi

T (532) i

T (532) 1 d

Comment

0.3<4m<0.8

0 . 8 < ^ m < 3

9 K (531) 1.3292-2.85274m+2.03654„

K „ 2^mTml

-1.6159-0.066t6^m+0.53515m2

T1(53 , )=2^mTmlexp(x1), x, =(3.3368-7.7919^m+4.3556^m

2)

+ (l .4094-4.458 l^m + 3.4857E,m2 )lnj

2^raT, m ml

+ (o.H60-0.8142^m+0.74024m2

; In

Td(53,) = 2^mTml exp(x2), x2 = (2.3054- 6.4328^m + 3.9743^m

2)

+ (l.8243-5.6458^m + 4.6238^m2)ln

2smTm | ,

V SmTml J

(o.2866-1.4727^m+1.2893£m2

; In 24«,T„

10 K (532) 0.1034 + 0.4857i;m - 0.0709^n

K „ 2 ^ m T m i j

Ti(532, =2^mTm ,(l.5144-0.6060^m +0.1414^

-1.713+0.6238^-0.1178!;m

2^mTn

0.1008+0.26255m -0.022035m

Td(532) =2^mTml(l.0783-0.4886^m + 0.1038(;„

ZtuJu

0.4316+0.08163^m +0.009 l73E,m2

Rule K „ Comment

Minimum ITSE Chao et al. (1989)-

continued.

X (533) (533) 0.3<^m<0.8

N = 8; 0.05 < 2^mTml

<0.5

Minimum performance index: servo tuning Minimum IAE -

Chao et al. (1989).

Model: Method 1

12 K (534) T; (534) (534) N = 8

0.8 <^m <3; 0.05 < 24mTml

<0.5

K (533) _ 1.8342-3.8984$m+2.7313; 2 / \-1.5641+O.16285n,+O.093375m

!

<2smTml ,

T,'5") =2^mTmlexp(x1), x, =(3.2026-7.4812^m+4.3686^m2)

+ (0.9475 - 3.6318^ra + 2 .9969C W r ^ -

+ (- 0.06927 - 0.3875^m + 0.4220£m2 y

Td(533) =24mTm lexp(x2) , x2 =(l.7393-3.7971^m+1.7608^m

2)

+ (l .0983 -1.5794^m +1.0744£m2 )ln| — -

In 2^mT, v ' S m 1ml J

2$mTn

+ (o.01553-0.01414^ra -0.009083£m2 | In

12 K (534, _ -0 .1779+ 0.67634m-0.1264^

K.

v2smTmi j

\-0.9941+0.16695„,-0.04381£m2

f T „ ^

2smTm i ,

T/534' =2^mTral(o.2857 + 0.4671^ - 0 . 0 8 3 5 3 ^ m2 ( - ^ -

x, = 0.1665 - 0.3324£m + 0.1776^m2 - 0.02897£m

3;

Td(534) =2^mTml exp(x2), x2 = (-0.8776 + 0.4353^ -0.1237^m

2)

+ (- 2.0552 + 2.786^m - 0.5687£m2 )ln

,2^mTml

+ (- 0.5008 + 0.6232^m - 0.1430£ra2

1 In v 2SmT m i ,

Rule

Minimum ISE -Chao et al.




Kc 13 K (535)

14 K (536)

T,

T (535)

Td

T (535)

0.8 < £m < 3; 0.05 < — ^ — < 0.5

T ( 5 3 6 ) T (536) Xd

Comment

N = 8

0 . 8 < ^ m < 3

13 K (535) _ 0-1446 + 0.4302^ -0 .07501^ f x_ >

K, ^•%m*-m\ /

-1 .214+0.3123? m - 0 . 0 6 8 8 9 5 „ 2

T, (535)=2^mTmlexp(x1), x, =(-0.1256 + 0.1434^m-0.03277^m2)

+ (- 2.2502 + 3.7015£m -1.7699^m2 + 0.2680£,m

3)lnj 2£nJm,

-(-0.4823+ 0.9479^m -0.4913^m2 +0.07792^m

3^ lnf — \ V ' ' S m 1 m l J

Td(535) =2^mTml(0.9105-0.3874^m +0.08662^m

2 J — ^ _

,4 „ (536) - 0.2641+ 0.7736^m-0.1486^2 f

K „ K.

x, = 0.0779 + 0.28554m - 0.01985^m2.

-0.7225-0.04344<m-0.0012474rai

2^mT m i m l J

Ti(536) =2^mTml(o.22257 + 0.7452^ -0.1451$m2] 21 t „

v ^ S m T m l J

x, = 0.03138 -0.04430^m +0.01120^

Td(536) = 2£mTmI exp(x2), x2 =(o.05515-0.7031^ +0.1433^m

2)

+ (-1.2256 +1.8544^m - 0.3366^m2 )ln| — -

2 . m »

24raT, m l >

+ (-0.2315 + 0.3402^ -0 .06757^ 2 ] In %

2smTm i ,

Rule



Minimum ITSE -Chao et al.


Kc

15 K (537)

16 K (538)

T,

T (537)

N = 8; 0.05 <

T (538)

Td

T (537)

— ^ — <0.5

x (538)

Comment

0.3<^m<0.8

0 . 8 < ^ m < 3

5 K (537)

• (537)

1.1295-2.8584^m +2.1176£n

K „

-0.9904-2.531 lS„,+2.

TiWJ" = 2^mTml 18.6286-20.70^m +13.2034 2£ T

(537)

x, = 0.7962-2.4809^m +1.7611^

: 2^mTml exp(x2), x2 = (3.9285 - 12.874^m + 8.6434^m2)

+ (3.2228-12.0344m +9.2547^m2)ln

•(o.6366-3.0208^m+2.4603^m2 In

^ S m ^ml j

16 K (538) 0.0820+ 0.4471^m -0.07797£,n

K „ v2smTmi

-0.9246+0.07955^ro-0.02436^„

1™ = 2i;raTral (o.5930 + 0.1457^m -0.02062!;n 2 T „

(538)

x, =-0.2201 + 0 .1576^-0 .03202^

^ m T r a l e x p ( x 2 ) , x2 =(-0.6534-0.1770^m-0.0240^m2)

+ (-1.0018 + 1 .5914^-0.2831^ r a2 ) lnf- l !B_

V^Smiml J

+ (- 0.2291 + 0.3064^m - 0.06344,/ In 24mTra

Rule

Minimum ITSE -Chao et al. (1989)-

continued.


Model: Method 1


Model: Method 1

Sklaroff(1992).

Kc

17 £ (539)

T:

T (539) 1 i

Td

T (539)

Comment

0 .3<^ m <0.8

N = 8; 0.05 < — ^ — <0.5

Direct synthesis ^Tml

Kn.K.+l)

x l *ml

K m T m

Tml

Tral

Tm2

Tm2

TCL

N not specified

D - T m ' Trai + Tm2

N=10 Coefficient values (deduced from graph)

4m 6

1.75 1.75 1.5 1.0

P

>0.02 0.27 0.07 0.11 0.25

x i

0.51

0.46 0.36 0.33 0.46

18 K (540)

ra

3

1.75 1.5 1.5 1.0

P

>0.04 0.13 0.33 0.08 0.17

Tm i+Tm 2

x i

0.50

0.42 0.44 0.28 0.48

4m 2

1.75 1.5 1.0 1.0

TmiTm2

Tmi + Tm2

P

>0.06 0.09 0.17 0.50 0.13

x i

0.45

0.39 0.38 0.40 0.49

Also given by Astrom and Hagglund (1995) -page 250-251. Model: Method 7. N = 8 - Honeywell UDC6000 controller.

„ K ,539, ^ 0.9666-2.3853^m + 2 . 4 0 9 6 ^ f xn

K „

-1.533-0.054425ra+10.9165,,

.24mTm l .

T,(539) =2^mTml(6.3891-15.773^ + 1 0 . 9 1 6 ^ m2 | - ^ -

x, = 0.04175-1.38605m+1.40534n

Td(539) = 24mTml exP(x2), x2 = (s.2620- 10.7894m + 7.6420!;m

2)

+ 2.277-8.94024m +7.7176^,/lln )lnf T- 1 V^SmTml

(o.3465-1.92744m+1.80304ro2, In

f \ In,

24mT, V^^m 'ml J

18 j r (540)

K j l + 3x„

Tml + T m 2 J


Rule


Model: Method 1


Panda et al. (2004). Model:

Method 7 Panda et al.


Kc

Tml

K m (T m l +x m )

19 K (541)

20 K (542)

2Km tm

Ti

Tml

T(541)

2^mTml

TB,

Td

Tm2

T (541)

2^m

Tm2

Comment

N = 5; Tml ^5T m 2 ;

Minimum

N not defined

N = 8

N not defined

19 jr (541) _ 0 3 7 5

K „

4^ m2 T m l

2 +2^T m l T r o + 0 .5T m2 -T m l

2

(2?mTml + T J 2 +1 i-lj(2?mTml +im)TmlV2(24ra2-l)-x

* = 44m2Tml

2 +2^mTmlxm + 0.5xm2 - T m l

2 ;

• (541) _ 4^ m2 T m l

2 +2^ m T m l T m +0.5T m2 -T m l

2

2 ^ T m , + T m

Td(54" = V t e n J m l +x m ) 2 -2(Tml

2 + 2 ^ T m l x r a +0.5xm2) .

20 K (542)

Km 1 1 ^ „

E(s)

1+ — v T i s y

l + NTds

1 + T d s y

Kme"

(l + TmlsXl + Tm2s)

Y(s)


(l + sTmlXl + sTm2)

Rule

Hougen(1979)-page 348-349.

Model: Method 1

Hougen (1988). Model: Method 1

Kc Ti Td Comment

Direct synthesis

0.8Trala7Tra2

a3

Km

0.84Tm,°-8Tra20-2

tra

22 K (544)

0-5Tml+Tm2

21 j (543)

0.5Tml+Tra2

0.lV^Tm.Tm2

T (543) J d

T (544)

N=10

N=30

N=10;xm<Tm l

2, Tj(543) = a 5 3 T m i +1.3Tm2,Td(543> =0.08(xmTralTm2)a28 .

K „ -10

X|-x2log, m . Coefficients of Kc<544) deduced from graphs:

T m / T m 2

* 1

x 2

0.1

0.9

0.7

0.3

0.42

0.7

0.5

0.24

0.68

1

-0.10

0.70

2

-0.39

0.69

T d( 5 4 4 ,=(Tm ,+Tm 2>0

0.331ogl0| ^ |-1.40

= 0.1

Td(544) = (Tml+Tra2)l0l m-0.3 l l o g , 0 - J ! ! - -1.63

, 0.2<-2^-<l.

Gc(s) = Kc

U(s) v T i s y

R(s) E(s)

+ K,

v T i sy (l + sTd)

K fi

ll+TmlsXl+Tm2s)

0 + Tds)

Y(s)


(l + sTm,Xl + sTm2)

Rule

Minimum ISE -Haalman(1965).


Kc Ti Td Comment

Minimum performance index: regulator tuning 2Tml

3xmKm T r a l Tm2 Model: Method I

T r a l >T r a 2 .M s =1 .9 , A m =2.36 , ^ = 5 0 ° .

1 K (545) T (545) M

T (545)

Minimum ITAE; Model: Method 1

T (545) x (545)

- ^ - = 0.1,0.2, 0.3,0.4,0.5; - ^ = 0.12,0.3 ,0.5,0.7,0.9.

K (545) 5.5030

K „

f N-1.1445+0.1100

1 0 -T V 1ml J

iiot)

( ^0.1648+0.17091n| 1 0 - ^ |-0.21421n| 1

V T m l J

T < 5 4 5 ) = 1 . 8 6 8 1 T T r 10

0.6212-0.0760 In 10-

f \0.3144-0.00621nl 10-=^ 1+0.0085 Inl 10-

l02k T

V 'ml J

Tm,

Rule

Yang and Clarke (1996).

Model: Method 8

O'Dwyer


Skogestad (2003).

Majhi and Litz (2003).

Model: Method 8


K c

fflpTml

K m A m

x iTm i

K m x m

0.785Tml

K m x m

0-5Tral

K m T m

3 K (547)

4 K (548)

Ti Td

Direct synthesis

2 T ( 5 4 6 )

Tml

Tml

Tm2

Sample result

Tml

min(Tml ,8Tm)

T (547)

Tm2

Tm2

TmI

Ultimate cycle

T (548) T (548)

Comment

Tmi = Tm2

A - % • 2x,

*m =0.57C-X1 .

A m = 2.0;

* m = 45°

Model: Method 18

Tmi = Tm2

T (548) T (548)

Model: Method 1

2 T (546) _

4 f f lP Tm 1 2C0_ +

P 71 T, ml

3 v (547) _ ^ m 2$ + i t ( A m - l ) Tm

2(A m2 - l ) Km

, (547)

1 + K1^)1 T.i . A » - ^ [ 2 ^ t , ( A „ - l ) | l - , A " . 1 ( 2 ^ „ + » ( A J , - 1 ) )

4 K (548) 0.5458

K „

N 1.2264+0.4568 In —

T;<548) = 0.0393T, T v u

t ( A m2 - l )

/ „ „ y.2190-0.09581n(K„K|T,)-0.10071n ~

-1.9314-0.62211n

(KuKm) -0.0703-0.1125ln(KoKm)-0.29441nP!!!

E(s) %-Y(s)

Table 126: PID controller tuning rules - SOSPD model G ra (s) = Kme-

(l + sTm,Xl + sTm2) or

Gm(s) = K „ e - -

T m l2 s 2 +2^T r a l s + l

Rule


Model: Method I

Kc Ti Td Comment

Direct synthesis *KC<*»> T ( 5 4 9 )

i T (549)

1ral ' im\ ' im\ '

Servo tuning

X = 0.6^m<\ A. = 0.7, K ^ r a < 2

^ = 0-8»-z;— £m > 2 > Regulator tuning

1 m l

5Kc(549) = x 2 T m l ^ m + 0 . 4 T m Tj(549) ^ ^ + 0 . 4 T l I

' <549> = Tml +0-8T m l 5 m T l t

0.8T m l ^ m T m

Rule

Minimum IAE -Huang et al.

(1996).

Kc T| Td Comment

Minimum performance index: regulator tuning 6 ^ (550) T (550)

i T (550) Model: Method

7;N=10

6 Note: equations continued into the footnote on page 356; 0 < Tm2 /Tml < 1 ;

0 . 1 < x m / T m l < l .

K (550) 1

1

K„

K„

K„

K„

0.1098 - 8.6290-2- + 76.6760- -3.3397 x m T „

1.1863 IHL. Tm>

-23.1098 , N 0.6642

+ 20.3519 , N 2 . 1 4 8 2

X,

V T m i y

-52.0778

, N 0.8405

T V ml J

9.4709-

•19.4944-

T r a i j

Tm, VTml 7

• 12.1033 M 2 2 -

-13.6581-^2-

-28.2766-

*m

T m i

A T \ 1 1 4 2 7

Xm2

T v 1mi y

C!k -19.1463eT"" + 8.8420eT™' +7.4298e T"" - H . 4 7 5 3 ^ 2 -

,(550) = T - 0.0145 + 2.0555^2- + 0 . 7 4 3 5 ^ _ 4.4805 V^ml

^ 2 _ +1.2069-^-^1 xmTm

+ T„

+ Tm,

+ Tm

+ T„

0.2584 Lm2

r m i j + 7.7916 -6.0330-

ml / Tmi l^Tml J

3.9585-

7.0004-

3.1663

l m l

^-\ -3.0626 / T \3

V ^ml

VTml 7

VTml 7 - 7.0263

-2.7755 V^ml VTml

VTmi y

-1.5769 VTmi y 'mi y

VTmi y + 2.4311 -0.9439

1 mi y

1m2

VTml J


+ Tm

+ T „

- 2 . 4 5 0 6 - - 0 . 2 2 2 7

2 /

T I. Jml

1.9228 V ^ml ml J

-0.5494-^S- p 2 2 -Trai V.Tmi

(550) - 0.0206 + 0.9385-^2- + 0 . 7 7 5 9 ^ _ 2.3820 — Tmi Tml vTm l

+ 2.9230 T m T m 2

+ T „ - 3 . 2 3 3 6 | ^ - +7.27741-^5- -9.9017-Tmi vTm l

2.7095-T ( T 'ml V ml

+ 6.1539 V ^ml

l m 2 ' — 11.10181 -^s lml J

+ T m 10.6303-^ml V ml

+ 5.7105 VTml J

+ T „

+ T „

- 7.9490 - i s - p n ? .

Tml V Tml )

- 4 . 4 8 9 7 -

-6 .6597 VTml )

'm2 T

V 'ml J 4

+ 8.0849

- 2.274 V -ml

f N5

V^ml

2.2155

0.519-

'ml V ml J

(T \

• 7.6469 VTml J V^ml

ml / T

V 'ml J

-5.06941-^2-

-1.1295 'm2

VTml J

Lm2 r 'ml 7

+ 4.1225 f-^-l T m l .

•1.6307 T

V 'ml

^m2

VTml J

+ T „ 2.2875

ml /

+ 0.9524 V ml

V ^ \ 3

VTml J - 0 . 9 3 2 1 - T m I T m 2

Rule


(1996).

Kc

7 K (551)

Ti

T ( 5 5 1 )

Td

T (551)

Comment

Model: Method / ; N = 1 0 .

7 Note: equations continued into the footnote on page 358. 0.4 < £,m < 1,

0.05<xm/Tm, < 1 .

K (551)

K „

K „

K „

K „

K „

K „

K „

- 35.7307 + 14 .19^_ + 1.4023^m + 6.86184m -Tmi \ T, ml J

-0.9773 ^ - +55.5898£,n

, \ 0.086 X,

T - 3.3093^n

V^ml

53.8651^=Hm2 +11.491 \\J +0.8778| •

-29.8822

V ml j

-0.6951 / N -0.4762

+ 53.535 ^ j -16.9807^, u 1 9 7

^T m i

-25.4293^m14622 -0.1671^m

58981 +0.0034^n

, N -2.1208

VTml )

. 25.0355^Km3 '0 8 3 6 - 54 .9617 -^Hj ' 2 1 0 3 -0.1398eT-

-8.2721e^ +6.3542e T™' + 1.0479IE-JSL

.(551) 0.2563 + 11.8737-21- - 1.6547^m -16.1913 / A 2

- 9.7061£ 1 ml J

+ T„ 3.5927^m2 + 19.5201 ^ - \ - 14.5581£,n

xn

V T ra l7 + 2.939^

i m l

/ A t m

V ' m l /

+ T. ml -0.4592^m -34.6273 V T m l J

+ 50.5163^n VTml )

+ T „ i.9259CP=-V Xml J

+ 8.6966^-^m3 -6 .9436^m

4

iml


+ T „ 27.2386 f "\5

-Tml 8.5019

VTmlV

X

20.0697^n VTml J

-42.2833^„ ^T m i

Em3 -12.2957 ±S- \m* + 8 . 0 6 9 4 ^ - 2 . 7 6 9 1 ^

1ral J V 1ml .

+ Tm ft V

•7.7887 - E -V T m l ,

-2.3012£n Tml

+ 4.6594 - ^ K m5

8.8984 V T m i y

£„/+10.2494 / V ^ U m

3 - 5.4906 V l m l J

f \ 2

V ' m l

(551) -0.021 + 3.3385—2- +-0.185^m -0.5164 ^ M - 0 . 9 6 4 3 4 m ^

V 1ml ) 'ml

+ Tm

+ T „

+ T „

+ Tm

+ T „

-0.8815^m2+0.584 - ^ - 1 2 . 5 1 3 4 m | ^ L | + 1.3468^

V ml .

2.3181^m3-5.2368-^2-J + 15.3014£n

V*ml VTml 7 -3.1988£,n

H-9607^

-0.8219£n

0.484 if ^ Un.y

^=- - 2.041 \ ^ J +3.4675 f ^ x,

V T m i y

T V 1ml J

•15.07184n V^ml J

•1.8859 f \ 2

±m_] f 3

V 'ml J

C + 2 . 2 8 2 1 4 m5 - 0 . 9 3 1 5 | ^

ml J

+ 0 . 5 2 9 ^ m | - ^ 1ml.

-0.6772^,-1.4212 - ^ U r a2 +7.1176 ^ L . ^

-2.3636 ^ b m4 +0.5497 ^ J

V ' m i y 'ml

Rule


(1996).

Kc T; Td Comment

Minimum performance index: servo tuning 8 K (552) T (552) T (552)

1 d Model: Method

1. N=10

' Note: equations continued into the footnote on page 360; 0 < Tm2/Tml < 1,

0 . 1 < x m / T m l < l .

K (552) _ _

K „ 7.0636 + 66.6512—25- -137.8937- -122.7832-

. T m T „

Kn

K „

K „

K „

K „

26.1928 V *ml

' m l

0.0865

+ 33.6578 / N 2.6405

X

-10.9347 / T \2M5

VTml J

34.3156^521 Tmi

( . \

T V 1 m l J

V T m l J

+ 141.51l| — ' m l j

-70.1035-2^ T,

+ 3.0098 (T

V ^ m l

+ 29.4068-

152.6392-T m l V ml

-47.9791-

T ml V ml J

V ^ m l

-57.9370eTml + 10.4002eT«" +6.7646e T""2 +7.3453

f \ -0.4062

V 1 m J

• (552) _ , 0.9923 + 0.2819—52- - 0.2679: Ami

1.4510 f A2

T.

V T m l J

-0.6712-, T X

0.6424p5Sl +2.504^555-VTmlJ VTml,

m2 | ^m + 2.5324-^ m l V ml J

+ T„

+ T„

+ Tm

2.3641-T *m

T m l V ml J

+ 2.0500 v ' m l

•1.8759 ' x ^

0.8519-

-2.4216-

T m i j

V ^ m l

-1.3496-V T m i y

V T m l J

- 3.4972 V T m l J Lml /

•3.1142 - l m l

V^ml J + 0.5862

VTmlV


+ Tm

+ T„

0.0797 W ^ M +0.985 i ^ -

1.2892

V^ml

( - ^VT

T V m

2 ^ x „ V

VTral J

V T m i y V '"ml + 1.2108

VTml J

(552) 0.0075 + 0.3449^2- + 0.3924^mL - 0.0793 -T

+ 2.7495 ^ V

+ T„ 0.6485 f-r \2

VTml J + 0.8089^2- -9.7483^21

VTml 1

3 . 4 6 7 9 ^ - f ^ - - 5 . 8 1 9 4 | - ^ - ] - 1 . " " " " ' X

T .0884

+ T„ 12.0049-( \ 3

T V 1ml J

-1.4056 V^ml

T V 'ml J

i a i - J -3.7055^ffl-|in2_ V ml J Tm, ^ Tml j

10.0045 / T \

T V 'ml J

+ 0.3520 f \ 5

-3.2980 VTml 7 V 'ml

T

+ 7.0404

. 6 . 3 6 0 3 ^ ml V ml )

V 'ml V^ml

+ T„

+ T„

+ Tm

1.4294 V 'ml

'm2

T V 'ml J - 6.9064

fry \$

T V 'ml J

-0.0471 VTml J

1.1839-Tm l V ml 7

+ 1.7087 / \ 6

VTml 7

+ 1.7444| JEs

ml J

• 1.28171 2 _ P ^ - l -2.128l| T" + 1.5121^IS-LlmL

Tm l V ml ,

Rule


(1996).

Kc

9 K (553)

Ti

T (553) i

Td

T (553)

Comment

Model: Method 1. N=10 .

Note: equations continued into the footnote on page 362. 0.4 < ^m < 1;

0 . 0 5 < x m / T m l < l .

K (553)

K „

K „

- 8.1727 - 32.9042-2- + 31.9179£m + 38.3405£n l m t V T m l J

+ 0.2079 + 29.3215 , xO.1571

V T m l J

+ 35.9456

^ [ - 2 1 . 4 0 4 5 ^ m0 1 3 U +5.1159^m'-9814 -21.9381^m'-737]

K „

K „

K „

T (553) _ T

+ Tm

+ Tm

-17.7448^m - s u -26.8655£n

ml J

, N 1.2008

V ^ m l

T„ 1

52.9156—s-^m11207 -22.4297-S_^ra°'3<i26 -3.3331eT'

^ m

8.5175e^-1.5312e T™' +0.8906 I T

1.1731 + 6.3082—^- - 0.6937^m +8.5271

[Huang (2002)];

^ - 2 4 . 7 2 9 1 ^ ^ V * ml J lm\

•6.7123^m | -^- +7.9559^m2-32.3937 ^ s - | + 5.5268!;, 6

-27.1372 V^ml - ^ - + 166.9272^ m \ +36.3954^„2

V^ml

f \

V T m l J

-94.8879^n V^ml

-^2_ -22.6065^m3 -1.6084^2_^m

3 + 29.9159^

+ T^ 49.6314' -84.3776^n l m l /

-93.8912^ Lml J ml /

+ T „

+ T „

+ T „

(553) _r

( V 110.1706 -^2- £,m

3 -25.1896| KTral J

1™ | ^ m4 -19 .7569^ 5

V 1 ra l

-12.4348 V T m l

•11.75894n

V T m l 7

+ 68.3097 ro U, ml

-17.8663 -^M ^m3 - 22.5926J

;Tmi J

r^r-) C + 9 . 5 0 6 1 ^ m5

V Xml J lm\

0.0904 + 0.8637—— - 0.1301^m + 4.9601 TmJ

+ 14.3899^n

+ Tml 0.7170^m2 -12.53 l l | * " -42.5012^ ^ " 2

V T miy + 17.0952^

+ Tm

+ T „

+ T „

+ Tm

+ T „

-21.4907!;,

102.9447^m

vTmiy -6.95554m -12.3016

' T ^

V T m l 7 + 7.5855^,

V T m i y

VTml

+ 19.i257^-^m3

( - \ 10.8688

V T m i y •17.2130^n

< ' T „ A

V T m l 7 -110.0342£n

V T m l 7

50.6455U=- ^m3 - 1 6 . 7 0 7 3 U ^ I C -16.2013^m

5 +5.4409^n V ml J V ml ,

-0.0979

21.6061

V^ml

^ „ ^

V T m i y

-10.9260^

5m3-24.1917'

V T m i y + 29.4445

TmJ

v T m i y C + 6 . 2 7 9 8 ^ C

r \ f \

U(s) = Kc , 1 T d s 1 + — + —

T:S - X

V 1 + ^ s

N ;

E(s)-K ( ot + PTds

1 + ^ - s

N ;

R(s)

a + pTds

T

N

K „

R(s)

-Mg> + T

E(s)

K, 1 THs

1 + + -

N ;

* U ( s )

Y(s)

+

K e"

Tml2s2+2^mTmls + l

Table 127: PID controller tuning rules - SOSPD model K ^ e - 1 -


Rule

Minimum IE -Shen (1999).

Kc T i Td Comment


1 K (554) j (554) T (554) 2 d

Model: Method 1;

K m = l

1 X l = e x p [ a + b | ^ | + c

V T ml7 + d

V^ml + e % ^ + fi;m+g^2

Tm,

+ h S m m +i r \3

T

V T m i y Sm +j

V lm\ )

K (554) _ T m l • (554) (554) :x x., a = 1-x, ; coefficients of x, are

given below (and continued into the footnote on page 364). 1.4 < Ms < 2 ; |3 = 0; N = 0.

a b c

K c ( 5 5 4 ,

1.40 -11.60 14.17

T (554)

2.43 -2.50 -4.16

T (554)

1.32 -0.51 0.70

a

-1.44 1.27 1.34

Rule


Kc

2 K (555)

Ti

T (555)

Td

T (555)

Comment

Model: Method 1;

K m = l

(continued)

d e f

g h i

J k

K c ( 5 5 4 ,

-6.44 8.28 -0.07 0.13 -4.66 -0.99 -3.03 2.89

T (554) i

2.85 -0.30 0.1

-1.72 15.51 -7.98 -8.45 4.66

T (554) ' d

-1.06 0.02 -0.09 -7.43 1.77 2.64 4.77 -3.74

a

-0.37 1

-5.97 -0.78

-10.63 18.85 -10.97 6.44

2 x,=exp[a + b f — ^ — ] + c f — ^ — ] + d f — ^ — ) +e^ra+f^ra2

Um+Tm lJ l.Tm+Tm,J Um+Tm lJ

( \ ( V c V + g ^ h — L + h - — ^ — 5m+i—Jbr-Um

ltm+TmlJ ^ m + T m l J {lm +Tm\ J

Kc(555) = _ n i . X | j Xf =TmXi,T d = Tmx,, a = 1-x, ; coefficients of x, are

x m

given below; Ms = 2 ; |3 = 0 ; N = 0; 0 < £m < 1; 0 < -^2- < 1. ml

a b c d e f

g h i

J k 1

K c , 5 5 5 ,

1.73 -17.51 30.73 -24.69 0.15 -0.12 3.30 33.07 -37.64 2.63

-34.57 36.88

T (555)

2.28 0.87

-24.00 15.09 -0.01 -0.02 -8.03 45.21 -22.48 3.89

-22.39 8.32

T (555)

1.43 -1.94 14.82 -21.44 -0.25 0.16 -1.06

-43.39 51.16 -3.25 36.04 -34.95

a

-1.60 4.30

-12.16 26.54 1.33 -1.08

-17.56 38.13 -56.45 14.80

-39.33 51.18

Rule


Model: Method 1


and Pagola (2000).


(2000). Model:

Method 11

Kc

3 K (556)

Ti

T (556) i

Td

T (556) *d

Comment

K m = l

Minimum performance index: servo/reguiator tuning

0.7236K, 0.5247TU 0.1650TU

N=10; Model:

Method 11

a = 0.5840, (3 = 1, 4>c =-139.65° ,Km = 1; Tml = 1; ^m = 1

x,Ku x2Tu "3TU

Coefficient values and other data x i

0.803

0.727

x2

0.509

0.524

x 3

0.167

0.165

a

0.585

0.584

4>c

-146°

-140°

x, = exp[a + b| xm+T,

+ g

ml / V T m + T m l + d

V T m + T r a , ^ m + h

V T m + T m l J

^ m + T m i y

^ m + i |

+ e4m+f^„

+ J Tm + T m i j Sm + k Tm + T m ] J Xm + T m ] J

4 m 1 ;

K ( 5 5 6 ) = J J H L X I J T i » o ^ = T m X Tdk"° '=T r ax, , a = l - x , ; coefficientsof x, , (556) (556)

are given below; Ms = 2 ; p

a b c d e f

g h i

j k 1

K (556)

128.9 -560.3 769.2 -338.9 -324.8 194.3 1421

-1981.6 895.0 -848.1 1186.1 -538.1

= 0 ; N = 0; 0.4 < ^m < 1; 1 < -&- < 15 . Tmi

T (556)

92.8 -391 521.6 -225.5 -224.5 137.5 963.9

-1310.6 574.4 -590.8 808.0 -356.7

T (556) x d

-3.1 46.3

-107.4 63.1 5.2 -2.5 -79.6 176.0 -103.6 36.2 -78.4 45.8

a

-119.9 539.5 -785.1 371.8 233.9 -115.7 -1068 1560.5 -741.5 529.2 -774.6 369.2

Rule


(2000) -continued.

K m = l ;

Tm, = l ; P = 1;N=10;

0.1<xm <10


Model: Method 1

Kc T, Td Comment

Coefficient values and other data (continued) x i

0.672

0.669

0.600

0.578

0.557

0.544

0.537

0.527

0.521

0.515

0.509

0.496

0.480

0.430

x2

0.532

0.486

0.498

0.481

0.467

0.466

0.444

0.450

0.440

0.429

0.399

0.374

0.315

0.242

4 K (557)

x 3

0.161

0.170

0.157

0.154

0.149

0.141

0.144

0.131

0.129

0.126

0.132

0.123

0.112

0.084

T (557)

a

0.577

0.550

0.543

0.528

0.504

0.495

0.484

0.477

0.454

0.445

0.433

0.385

0.286

0.158

T (557)

4>. -134°

-125°

-115°

-105°

-93°

-84°

-73°

-63°

-52°

-41°

-30°

-19°

-10°

-6°

S» = i-0

4 K (557) _ 1

Km 1.389 + -

0.6978

- ^ - + 0.02295]2

T (557) _ T

T (557) _ T LA — ^ r r

0.02453 + 4 . 1 0 4 - a - - 3.434 Tm,

0.03459 + 1.852-a-- 2.741

+ 1.231

+ 2.359 -0.7962

- ] 4 \

a = 0.6726-0.1285-2--0.1371

(3 = 0.8665 - 0.2679-E- + 0.02724 Tml

-0.07345

TmI

Rule

Taguchi and Araki (2000) -

continued.

K, T; Td Comment

T (558) T (558) f = 0.5

Tm/Tml < 1.0; Overshoot (servo step) < 20% ; N not specified

5 R (558)

Direct synthesis

Huang et al. (2000) - servo

tuning. Model: Method 1

6K (559) • (559) (559)

X = 0 . 6 , - 2 K m < l A m = 2 . 8 3 .

X. = 0 . 7 , K - = - $ m £ 2 A m = 2 . 4 3 .

X = 0 .8 , -=^ , . > 2 A m =2.13

2Tml^n - 1 . P = -0.5T

2Tml4m+0.4Tm • • Tml '+0.8Tml4mxm

N = min[20,20Td]

5 K (558) _ _

-(558)

K „

= Tral

0.3363-0.5013

[-J2- + 0.01147]2

T m i

• 0.02337 + 4.858-S- - 5.522 Tm,

+ 2.054 3 "\

(558) _ Tm, 0.03392+ 2.023-^--1.161 + 0.2826

Tm,

a = 0.6678 - 0.05413-^- - 0.5680 Tml

+ 0.1699

P = 0.8646-0.1205^!_-0.1212

6 „ (559) . , 2 T m , ^ m + 0 . 4 T K'3X"=X

m x (559)

K m T m T-1 ' = 2T ,E +0 4x

. (559) _ Tm/+0.8Tml^mxn d 0.8Tml^mTm



U(s) = K, 1 + E(s)-KcTdsY(s)

R(s) E(s) K 1 + —

+ *g> U(s)

SOSPD model

Y(s)

K cT ds

T Table 128: PID controller tuning rules - SOSPD model Gra(s) =

K m e " ^

(l + sTmlXl + sTra2)

Gm(s)=-K „ e "

T m l V+2^ m T m l s + l

Rule


-page 119. Model:

Method 1;

- ^ - = 0.2 ' m l

Kc Ti Td Comment

Minimum performance index: regulator tuning 1.18Tml

Kmxm

l-25Tml

KmTm

1.67Tml

Kmxm

2-5Tml

^-m^m

2.20xm

2.20xm

2.40xm

2.15xm

0.72xm

1.10xm

1.65xm

2.15xm

T -2^-= 0.1 Tml T - ^ - = 0.2

• ^ = 0.5 ' m l

• ^ = 1 .0

T m i

Rule





Kc

1 K (560)

3.33Tml

K m ' C n i

0.85KU

T|

T ( 5 6 0 )

^m+0.2Tm2

0.35TU

Td

T (560) ' d

^m+0.2T ra2

0.17TU

Comment

I E I 2 _ < 3 t m

2k_>3

^ - = 0.2, T ' m l

T - ^ - = 0.2 Tm,

1 R (560) 100

38 + 40 1-e x» /

1 + 0 . 3 4 - ^ - 0 . 2

v

^ 2 ^

T„, "\

0.5 + 1.4[l-e 1 5 t m] 1 + 0.48 1-e T-

J; f 1.2Tm

Td(560)=0.42xn 1-e T- + 0.6Tm



U(S):=KC b + — V T i S V

E(s)-(c + Tds)Y(s)

R(s)

> ( b - l ) K c

E(s)

+ I K,

v T i s y

+

U(s)

.+ *® Kme""™

(l + sTml)(l + sTra2)

Y(s)

(c + Tds)

I Table 129: PID controller tuning rules - SOSPD model Gm (s) =

Kme" (l + sTm,Xl + sTra2)

Rule


Kc Ti Td Comment

Direct synthesis

2 K (561)

1.54T.(5«)

1.27T;*561'

1.75T;(561)

T (561)

b = 0.198; nearly zero overshoot b = 0.289;

nearly minimum IAE

b = 0.143; conservative

tuning

2 K (561) 2 k i T m 2 + ( T m l + T m 2 ) r m + 0 . 5 T m2 f

9Km[TmlTra2im +0.5(Tm, + Tm2)xm2 + 0.167tm

3f

T(56.) 3[rm,Tm2Tn, +0.5(Tm, +T m 2 > m2 +0.167Tm

3l

[Tm ,Tm 2+(Tm l+Tm 2)xm +0.5Tm2]

(561) 2k,Tm2+(Tm,+Tm2)rm+0.5Tm2]2

Tr a l + T m 2 + * m

c = K,

3Km[TmlTm2xm +0.5(Tml +T m 2 > m2 +0.167xn

(561) 1

Km '

n — E ;

4.5.16 Non-interacting controller 6 U(s) = —c- E(s) - Kc (1 + Tds) Y(s) T:S

R(s)

>6* + 1 L

E(s) Kc

T i S

U(s)

—4

Kc(l + Tds)

t

K m e-""

TmlV+2^T r a ls + l

Y(s)

^

Table 130: PID controller tuning rules - SOSPD model Gm (s) = Kme-Tml

2s2+2^mTmls + l

Rule


Model: Method 1

Kc Ti Td Comment

Direct synthesis

3 K (562) T(562) Tm, 0 < s < l ;

suggested E = 0.5

3 y- (562) _ s T m |

24m + — T

ETr

m J

T, (562)

2^„

[ |Tm l(2^mTm +Tm l)(442(l-S)+ S) ' with

= - L 8(l-e)2k(2^ i nT in + T m l ) fe2(l-e) + e)iHL(2^T m +Tml)+Tn

e + 2ct2(l-s)2[5=L] (24mxm+Tml)2 >0 . w h e r e Jk . (24mxm +Tm l )+-

K V K m /


f

4.5.17 Alternative controller 4 U(s) = K sT,

E(s) + R(s )

Table 131: PID controller tuning rules - SOSPD model Gra (s) = Kme"

(l + sTm,Xl + sTm2)

Rule

Bohl and McAvoy (1976a).

Model: Method 1

Kc Td Comment

Direct synthesis: time domain criteria 4 K (563) T (563)

*d N=10

Response reaches a new steady state in minimum time, to a servo input, with no observable overshoot. Upper and lower limits exist on the manipulated variable.

-^2- = 0.1,0.2,0.3,0.4,0.5 ; i s ! = 0.12,0.3,0.5,0.7,0.9 .

K (563) 4.486

K „ 10-

-1.240+0.1409 In 10-

,-0.0252+0.1542 In l o i = i -0.16161n 10-^-. T „ , 1 I T,,, j { Tml ,

(563) ,0.3664-0.0476 In 10-

10-l m l J

0.7797-0.07731n| l O ^ i |+0.02701n| 10 -^ - 1

10Im2 T-T

Rule

Bohl and McAvoy (1976a).

Model: Method 1

Kc Td Comment

Ultimate cycle 5 K (564) x (564)

»d N = 1 0

5 K (564) _ 0.2455

K „

, -.-0.4986-0.7510 In - a

v T u y

,0.3875-0.2667 ln(K„Km)-l .1111 I n p s

Td<564) =0.000726TU

N -6.1523-1.4285 i n - ^

T

(K„Kmy

(K K y1-4494+°-0402 |n(KuKm)-0- ' , 6021n —


4.6 I2PD Model Gm(s) = K „ e ~


G c ( s ) = K t

A , 1 T

d s 1 + — + %r

Xs . T, V

1 + ^ - s N

l + bfls + bf2s2

l + afls + af2s2 • R ( s ) - Y ( s )

R(s) l + b f ls + b f 2s

l + a f ls + a f 2s

E(s)

Kc 1 + l

+ ^

I N )

U(s)

— •

s2

Y(s)

Table 132: PID tuning rules - I2PD model Gm (s) = — 2 ^

Rule

Liu et al. (2003). Model: Method I

Kc

1 K (565)

T i Td

Robust T(565) x (565)

Comment

1 £ (565) {AX + zm jlX3 + 6X\n + 4Xzm2 + t m

3 ) - XA

Km(AX3+6X2zm+4Xzm2+zjj

(565) _ (4k + T J4X3 + 6X2Tm + 4ATm2 + T 3 ) - X4

\4Xi+6X2xm+4Xzm'+zm 2_ , A\~ 2 , - 3T

T (565, _ (6X2 + 4Xzm + zj )AX3 + 6X2xm + 4Xz2 + zj )

(4X + Tm ^ + 6X2zm + 4Xxm2 + T m

3 ) - A4

4X3+6X2Tm+4Xtm2+Tm

3 af, = 4X + zm ; af2 = 6X2 + 4Xxm + z„

6X2+4Xzm+zm ——. rf-— r—~ ~—; T\ '. 1 » b n = 2X , b f2 = X X3[(4X + tm)(4X3 +6X2tm +4Xxm

2 +Tm3)-X4]


Rule


Kc

X

0.8xm

*m

l-5xm

2xm

Sampi OS

a 33%

« 20%

a 3%

* 0 %

T, Td

e data (deduced from a Rise time

- 4 x m

*5xm

«Vxra

*8xm

X

2.5xra

3tm

3.5xm

4xm

Comment

graph) OS

* 0 %

«0%

* 0 %

* 0 %

Rise time

«10tm

-12.5xm

*14xm

*16xm

4.6.2 Controller with filtered derivative with set-point weighting 4 f \

Gc(s) = Kc

R(s)

1 + - U ^ T>s 1 + ^ s

N

1 + b f jS + b f 2s + b f 3 s 3 + b f 4 s >

l + afls + af2s2 +a f3s3 + af4s4 R(s)-Y(s)

J U(s) Y (s )

l + b n s + b f9s2 + b f , s3 + b f 4s4

1 + af ls + a f2s2 + a f3s3 + af4s

Table 133: PID tuning rules - I2PD model Gm(s): Kme"

Rule

Liu et al. (2003). Model: Method 1

Kc

2 j ^ <566>

X

0.8xm

tm

1.5Tm

Sampl OS

= 14%

= 4%

= 0%

T, | Td Comment

Robust - , (566)

i T (566)

e data (deduced from a graph)3

Rise time

= 3rm

= 3.5tm

-4 5x

X

2.5xm

3xm

35rm

OS

= 0%

= 0%

= 0%

Rise time

= 7xm

= 8.5T,„

= 10.5xm

K (566) _ (4X + Tm fr3 + 6X2Tm + 4XTJ + Tm3 ) - X4 ^

Km(4X3+6X2Tm+4Xrm

2+xm3j

_ (4A. + T )(4X3 + 6A2xm + 4Axm2 + x 3)-X4

^ 3 + 6 X 2 x m + 4 X x m2 + x m

3 j

_ (6X2 + 4XT + xm2 &A3 + 6X2xln + 4Ax, 2 + xm

3)

(4X + xm X4X3 + 6A2xm + 4Xxm2 + x m

3 ) - X4

X4

, b f l = 4 A , b f 2 =6A2, b f 3 = 4 A 3 , b t 4 = A 4 ,

.(566)

(566)

4X3+6A2xm+4Xxm2+xm

3

l 2 ' " t m + T m2 6A.2+4Ax +x„

X'[(4X + x m ^ +6X\^+4Xxm2 +xm

3)-X4]

af, = 8A + xm , af2 = 26k2 + SXrm + xm2 , af3 = 4x(lOA2 + 20ktm + 4xm

2),

a f 4=4X2(6X2+4Axm +xm2) .


Rule


Kc

2Tm

A

0.8xm

Tm

l-5xm

2Tm

Ti Td Comment

Sample data (deduced from a graph) - continued «0% *6xm 4xm «0% «13Tm

Sample data (deduced from a graph)4

OS

* 46%

* 34%

* 16%

* 8 %

Rise time

~6xm

«13.5xm

*13xra

«16.5xm

A

2.5xm

3xm

3-5xm

4Tra

OS

«4%

- 2 %

* 1 %

«0%

Rise time

*20.5xm

«24xm

- 2 8 i m

«32xm

4 afl =5A + t m , a f2=10.25A2+5ATm+Tm2, af3 = A(7A2 + 4.25Atm + i m

2 ) ,

a f4=0.25A2(6A2+4Axm+Tm2).



R(s)

~ ^ =

E(s) K,

V T i s 7 (l + sTd)

Gc(s) = Kc

U(s)

l + J-](l + sTd) v T i S

K „ e " Y(s)

Table 134: PID tuning rules - I 2 PD model Gm(s) = K „ e - « -

Rule

Skogestad (2003).

Model: Method 1

Kc T; Td Comment

Direct synthesis 0.0625 K m t m

8xm 8tm



f \ ( ^

U(s) = K E(s)-Kc a + PTds

1 + ^ s V N

R(s)

Y(s)

*g)->* + s

U(s)

Table 135: PID tuning rules - I2PD model Gm(s) = ^ ^ -

Rule


Kc T( Td Comment

Direct synthesis

3.75KmTm2 5.4xm 2-5tm N = 0; p = l ;

a = 0.833


4.6.5 Industrial controller U(s) = K(

v Tisy R ( s ) - ^ Y ( s )

1 + -N

Table 136: PID tuning rules - I2PD model Gm{s) = ^ L .

Rule

Skogestad (2004b).

Model: Method 1

Kc Tj Td Comment


0.25

Km(TC L+Tm)2 4^2(TCL+xm) 4(TcL+xm)

'good' robustness -

TcL = T m >

£=0.7 or 1

N e [5,10]; N = 2 if measurement noise is a serious problem

K e"st"'

381

4.7 SOSIPD Model (repeated pole) Gm(s) sfl + T ^


( \ f \

U(s) = K

Table 137: PID controller tuning rules - SOSIPD model (repeated pole) Kme~

s(l+T f f l ls)'

Rule


and Pagola (2000).

Model: Method 2

K m = i ;

T m = i ;

0.1 < t m <10;

P = l , N = 10.

Kc ^ Td Comment

Minimum performance index: servo/regulator tuning x,Ku x i \ x3Tu

Coefficient values

*! 1.672

1.236

0.994

0.842

0.752

x2

0.366

0.427

0.486

0.538

0.567

x3

0.136

0.149

0.155

0.154

0.157

a

0.601

0.607

0.610

0.616

0.605

*c

-164°

-160°

-155°

-150°

-145°

Rule

Pecharroman and Pagola (2000) -

continued.


Model: Method 1

Kc Ti Td Comment

Coefficient values (continued)

*| 0.679

0.635

0.590

0.551

0.520

0.509

* 2

0.610

0.637

0.669

0.690

0.776

0.810

1 K (567)

X 3

0.149

0.142

0.133

0.114

0.087

0.068

T (567) *i

a

0.610

0.612

0.610

0.616

0.609

0.611

T (567)

•. -140°

-135°

-130°

-125°

-120°

-118°

^ - < 1 . 0 ; ' m l

Overshoot (servo step) < 20% ; N

fixed but not specified

I K (567) = _

K „ 0.1778 + -

0.5667

T (567) _ T

- + 0.002325 J

0.2011 + 11.16-^--14.98 •13.70 -4.835

n 4 >

' m l

(567) V =Tm l 1.262 + 0.3620-^- a = 0.6666, ml J

P = 0.8206 - 0 .09750-^ + 0.03845


4.8 SOSPD Model with a Positive Zero K m ( l - sT m 3 ) e -^

Gm(s) = Tml

2s2+2^mTmls + l or Gm(s)

_K m ( l - sT m 3 ) e - " -


4.8.1 Ideal controller Gc(s) = Kc Xs

R(s)

*R\ + Y

T 1

E ( s ) (

Kc f 1 1 1+ — + Tds

T.s v *•" y

U(s)

— • Gm(s) Y(s)

— w

Table 138: PID controller tuning rules - SOSPD model with a positive zero

Gm (S) = M ^ - s T ^ K ^ ^ Kn (1 - sTm3 y^ (l + sTm,Xl + sTra2) Tml¥+2i;mTmls + l

Rule

Minimum IAE -Wang et al.


Kc Ti Td Comment

Minimum performance index: servo tuning i

i P o q i - P i r Km Po Po

p 0 q 2 -P2 Pi

PoQi-Pi Po

Po : T m l + T m 2 + T r a 3 + T m , p , = T m I T m 2 + 0 . 5 T r a ( T m I + T m 2 ) - 0 . 5 T m T m 3 ,

p 2 = 0.5TmlTm2Tm , qi = Tml + Tm 2 + 0.5tm , q2 = Tm lTm 2 + 0.5xm (Tml + Tm 2 ) ,

r = 8 0 + 8 1 - E - + 5 2 'n,2

VTml J

S7 VTml J

+ 5, VTml J

+ 8

+ 83

'T,

'1113

Tml

'O VTml J

VTml J V T m i y

ml J

f

+ 5B

+ 8C

V^ml

T n , + 8,

fT \

VTml J

S0 = 3.5550 ,8, = -3.6167 ,8 2 = 2.1781,83 = -5.5203 ,5 4 = 1.4704 ,

85 = -0.4918,86 = 2.5356 ,8 7 = -0.4685 ,88 = -0.3318 ,8 9 = 1.4746 .


Rule

Minimum ISE -Wang et al.


Minimum ITAE - Wang et al.


Wang et al. (2000b), (2001a). Model: Method 2

Kc

2

i P o q i - P i r

K m Po2

3

i p0qi - P i r

K m Po2

4 ^ (568)

Ti

Po

Po

Direct s

2smTm i

Td

p 0 q 2 - p 2

PoQi -P i

Poq2 - P 2

PoQi - P i

ynthesis

T 2

"•ml

Po

__P_L Po

Comment

M , = 1 . 5

p0 ,Pi,p2 ,q,,q2 ,r defined as on page 383.

S0 = 3.9395 ,5 , = -3.2164 ,6 2 =1.6185 , 5 3 = -5.8240,84 =1.0933 ,

65 = -0.6679 , 56 = 2.5648 , 57 = -0.2383, 58 = -0.0564, 59 = 1.3508. 3 Po>Pi>P2>qi>q2>r defined as on page 383.

80 = 3.2950 ,5 , = -3.4779 , 5 2 = 2.5336,83 = -5.5929,54 = 1.4407 ,

85 = -0.3268 , 56 = 2.7129 , 87 = -0.5712 , 58 = -0.5790 , 8 9 = 1.5340 .

4 K (568) 1 C0S9 2 ^ r oT m l m

Ms cos[anm - t a n - ' ( - T m 3 c o ) J K m ^

co and 9 are determined by solving the equations:

, V + i

tan cos 9

M, -s in0 tan '(-Tra3co)-coTm-0.57t and

l = tan~ ~ T m3 T mt0 2 ~ l)cOs(cOTm ) - C0Tm sin(cOTm )

rTm3^mco2-l)sin(coTm)+coTmsin(coTm)


4.8.2 Ideal controller in series with a first order lag f

R(s) E ( s )

G c ( s ) = K (

U(s)

1 \

V T i s J

1

/ K r

1 1 + — + Tds

v T i s J T fs + 1 j K m ( l - s T m 3 ) e - ^

(l + sTmlXl + sTm 2)

T fs + 1

Y(s)

— •


K m ( l - s T r a 3 > - ^ Gm(s) =

(l + sTmlXl + sTm2)

Rule

Li et al. (2004). Model: Method 1

Kc Ti Td Comment

Direct synthesis: time domain criteria 5 K (569) Tm, +Tm2 TmiTm2

Tm l+Tm 2

5 K (569) Tm l+Tm 2 ; . T f Km (2TCL2 + Tm3 + xra ) '

f 2TCL2 + Tm3 + xra '

desired closed loop transfer function = ( l - s T m 3 > -


4.8.3 Controller with filtered derivative Gc(s) = Kc

R(s)

, 1 Tds

1 + — + -


Gm(s): K m ( l - s T m 3 ) e - o r G m ( s ) = JCm(l-_sTm3 )e

(l + sTmlXl + sTm2) Tml2s2+24mTmls + l

Rule


Kc T i Td Comment

Robust 6 ^ (570)

7 K (571)

j (570)

x ( 5 7 1 )

x (570) 1 d

T (571)

Tml > Tm2 > Tm3; N=10; X = [TmI,xm J (Chien and Fruehauf (1990)).

T m l + T „ , + -T m l x „

6 K (570) • + T m 3 + T m T (570) _ T T T m 3 T m

X + T m , + T„ Km(^ + T m 3 +x m )

T (570) _ T m 3 T m Xd - ~ - +

T m l T r a 2

^•+Tm3+Tra J + x , Tm3Tn

^ + Tm 3+xm

OE T _j m3 m >m ml ^ x X

7 ^ (571) A + 1 m 3 + X m T (571) _ - j . T 1m3Tm

M - / S m 1 m l + K„ Km(X + T m 3 +x m ) k + T n > 3 + * m '

(571) T m 3 T m

^ + T m 3 + * m oK x , " ^ n . 2^mTml + *- + T m 3 + 1 : m

4.8.4 Classical controller 1 Gc(s) = Kc 1+ — v T i s y

1 + Tds

N

R(s) E(s)

K, v T i sy

1 + Tds

l + ^ s V. N .

U(s), K m ( l - s T m 3 ) e -

(l + sTmlXl + sTm2)

Y(s)


K m ( l - s T m 3 ) e - ^ Gm(s) =

(l + sTm,Xl + sTm2)

Rule


Model: Method I

O'Dwyer (2001a).

Model: Method I


Kc T; Td Comment

Direct synthesis

8 K (572)

x i T m i

K m T m

Tml

Tm,

Tm2

Tm2

Minimum

Am =0.57t/x, ;

K = 0.57t-x,;

N = Tm2/Tm3

Sample result 0.785Tml

K r a T m Tm, Tm2

A m = 2 . 0 ;

<L=45°

Robust

Tm2

Km(^ + Tm)

Tml

*•*(*•+ 0

Tm2

Tn,

Tml

Tm2

N=10;

^e[Tm l ,Tm] (Chien and

Fruehauf(1990))

Tml > Tm2 > Tm3

8 K (572) _ Tral j ^ _ Tm2 (Tm, + 2Tm3 j K m ( T m l + 2 T m 3 + x J :

= 56° when Tm, = Tm, = xm . m3 1 m l Lm



Gc(s) = Kt 1 + — (1 + Tds)

R(s)

^

E(s)

K, 1 + — Ts

(l + sTd)

U(s)

• K m ( l - s T m 3 ) e

(l + Tm]sXl + Tm2s)

Y(s)

Table 142: PID controller tuning rules - SOSPD model with a positive zero Km(l-sTm3)T

SI™ Gm(s) =

(l + sTmlXl + sTm2)

Rule

Zhang (1994). Model: Method 1

Kc T; Td Comment

Direct synthesis

Kmxm Tml Tm2

"Good" servo response; xm is

"small".


4.8.6 Classical controller 4 Gc (s) = Kc 1 + — v T i s y

1 + - TdS

N ;

R(s) E(s)

-0— K, V T i S /

1 + . ^

V N ;

U(s) Km(l-sTm3)e-

(l + sTralXl + sTm2)

Y(s)


Km(l-sTm3)e-ST" Gm(s) =

(l + sTmlXl + sTm2)

Rule


Kc T, Td Comment

Robust

Tm2

Km(^ + xm)

Tml

Km(*- + 0

Tm2

Tmi

9 T (573) 1 d

10 T (574)

N=10; X e [T, Tm ] , T = dominant time constant (Chien and Fruehauf

(1990)).

= T „ , + -T m , x „

^ + T m 3 +x m

10 x (574) _ T Tn^Tp. 1^ — AmT -1

^ + T m 3 + Tra

U(s) = Kc

V T i S / E(s) ^ - Y ( s )

i+3fi N

R ( S ) v, ^ + K

V T . S /

U(s) •

Km(l-sTm3>-

T m l V+2^T m l s + l

Y(s)

Tds

l + ^ s N


Km(l-sTm3)e-ST" Gm(s) =


Rule


Model: Method 1

N not specified

Kc Ti Td Comment

Direct synthesis

..Ke(5"> 2Tml^m+0.4Tm T (575)

' d

^ = 0 . 5 , ^ m < l . X = 0 . 6 , K ^ H m < 2 . 1 m l ' Ami '

T 1 0 7 m £ ~> 9 u - ' > T S n , ^ ^ s e r v o tuning

1 m l

^ = 0 . 6 , ^ m < l.X = 0.7, 1 < - I ^ m < 2 . Tral ' T m I

T T . _ n o l ' m e . 9 *• u - ° > T S m ^ ^ regulator tuning

ml

ii K (575) =-^2T ro l^m +0.4xm T (575) _ Tml + 0,8Tmli;mTn

Km(Tm+2Tm 3) 0-8Tml^mTn

U(s) = Kc

f 1 T "\ 1 T d S

1+- + d V " i TjS 1 + Tds/N

E(s)-Kc a + -PTds

1 + Tds/N R(s)

a + PTds

T

N

K „

R(s)

+ E(s)

1 + - U - ^ T>S l + ^ s

N V

U(s)

+

K r a ( l - s T m 3 ) e -


Y(s)


Kra(l-sTra3)e-SI™ Gm(s) =

(l + sTmlXl + sTm2)

Rule K „ Comment

Direct synthesis Huang et al.


N = min[20,

20Td]

12 K (576) , (576) (576) Servo tuning

X = 0.6,^m<\ A m = 2 . 8 3 . X = 0 . 7 , K ^ m < 2 Tml ' Tml

A m = 2 . 4 3 . X = 0 . 8 , ^ r a > 2 A m = 2 . 1 3 .

a = -2T m , ^

2Tml^m+0.4xn • 1 . P = -

0-5Tml2

Tm /+0.8Tm l^mT n

- - 1

,2 ^ ( 5 7 6 ) = 3 L 2 T m , ^ + 0 . 4 T n , T i ( 5 7 6 ) = 2 T m | ^ + Q 4 i n

K m ( t m +2T m 3 )

(576) Tm l2

+0.8Tm l^ r ax ml^rri "m

0-8Tra]^mTn


4.9 SOSPD Model with a Negative Zero

Gm(s) = Km(l + sTm3)e-"-

T m i y + 2 ^ T m l s + l o r G J s ) :

Km(l + sTm3)e-"-

(l+sTmI)(l + sTm2)

/ 4.9.1 Ideal controller Gc (s) = K(

1 \

V T i S J

R(s)

*6 1 ?) E^ »

f-Kcfl + - L + Tds

I T i s J

iU(s; Km(l + sT m 3 >-"-

T m l V + 2 ^ m T m l s + l

V(s)

—v-

Table 146: PID controller tuning rules - SOSPD model with a negative zero

Km(l + sTm3)e-sx™ Gm(s) =

Tm lV+2);mT r a ls + l

Rule

Wang et al. (2000b), (2001a). Model: Method 2

Kc Ti Td Comment

Direct synthesis

1 K (577) 2?mTml T 2

1 m l

M,=1.5

1 K (577) 1 cos 6 2 ^m T ml f f l

Ms cos[mTm-tan-,(Tra3co)] K m ^ T m 3 V + •; co and G are determined by

solving the equations:

tan cos 9

VMS -sinO = tan '(Tm3co) —coxm -0.5TC and

6 = tan"1 T m 3 ^ m m 2 - 1 ) c O s ( ( 0 T m ) - C 0 T m SJn(c0Tm)

l T m 3 ' C m f f l 2 - 1 ) s i n ( a i X m ) + f f > ' C m s i n ( ( a X m ) _


4.9.2 Controller with filtered derivative Gc (s) = Kc

R(s)

+ a

E(s) K „

, 1 Td s

1 + — + • — T 's 1 + ^ s N .

, 1 TdS 1 + — + - d

U(s)

N ; Y(s)

Gm(s)


Gm(s) = Km(l + sTm 3 )e-^ _^ ^ _ Km(l + sTm3)e-ST-

(l + sTmlXl + sTm2) o rG m ( s ) :

Tml2s2+2^mTmls + l

Rule


Kc Ti Td Comment

Robust 2 K (578)

2smTmi ~ Tm3

T (578) i

2smTmi ~Tm3

T (578) *d

3 T (579)

N=10; *-e[Tml,Tm]

(Chien and Fruehauf(1990)).

2 v (578) _ Tml + Tm2 - Tm3 (578) K

Kra(X + t J . V (578)

: Tml + Tm2 Tm3 .

. TralTm2 (Tmi+Tm2 T^JT^ • > Tm| > Tm2 > Tm3.

3 T (579) Tmi (2i;mTml Tm3jTm3

2smTrai _ T m 3

4.9.3 Classical controller 1 Gc(s) - Kc 1 + ^ v T i s y

1 + Tds

I N ;

R(s) E(s)

+ x K, 1 + —

1 + Tds

1 + ^ s N

^ K m ( l + s T m 3 ^

(l + sTralXl + sTm2)

Y(s)


Km(l + sTm 3>-"-Gm(s) =

(l + sTmlXl + sTra2)

Rule

Zhang (1994). Model: Method I


Model: Method 1


Kc T( Td Comment

Direct synthesis

Trol

K m T m Tmi Tm2 N = Tm2/Tm3

"small" xm ; "good" servo response.

Tm, Km(T r a l+Tm)

Tmi Tm2

N = T r a 2 / T m 3 ;

Minimum

*m=90°;<t)m =

6 0 ° , t m = T m l

Robust

Tm2

Km(^ + t m )

Tml

Km( + 0

Tm2

Tm,

Tml

Tm2

Trai >T ra2;N=10; ^e[Tm l ,xmJ (Chien and Fruehauf( 1990)).


/

4.9.4 Classical controller 4 Gc(s) = K v T iV

l + - T ' s

N

R(s) E(s)

K„

v T i s / 1 + - ^

T

N ;

U(s) Km(l + sTm3)e-

Y(s)

(l + sTmlXl + sTm2)


K m ( l + s T m 3 ) e - ^ G m ( s ) =

(l + sTmlXl + sTm2)

Rule

Chien(1988). Model: Method 1

Kc Ti Td Comment

Robust

Tm2

K m (* + Tm)

Tm, Km( + 0

Tm2

Tm,

Tmi - Tm3

Tm2 ~~ Tra3

Tm, > T m 2 > T m 3 ; N = 1 0 a 6 [ T r a l , T m ]

(Chien and Fruehauf (1990)).

4.9.5 Non-interacting controller 1 r

U(s) = Kc E(s)- TdS Y(s)

N

Km(l + sTm3>-"-

Tm,V+2^mTmls + l

Y(s)


Km(l + sT m 3 )e-^ Gm(s) =

T r a l V + 2 ^ T r a l s + l

Rule


Model: Method 1

N not specified

Kc Ti Td Comment

Direct synthesis

4K C( 5 8 0 ) 2Tral^m+0.4Tm T (580)

servo tuning

l = 0 . 6 , ^ m 2 .

regulator tuning

4 K (580) = ? 2Tml£m + 0,4tm (580) _ Tm, +0.8Tml£mTm

Km(Tm+2Tm 3) 0-8Tml^mTn

f \ f \

U(s) = Kc

N

E(s)-Kc a + (3Tds

1 + ^ s N

R(s)

a + PTds

T 1 + ^ s

N

K „

R(s)

E(s)

K, 1 THs

1 + + -T=s 1 + ^ s

N .

U(s)

*g>> K m ( l + sTm3)e-™

T m i y + 2 ^ m T m l s + l

Y(s)


Km(l + sTm3)e-"" Gm(s) =

T m l V + 2 ^ m T m I s + l

Rule K, Comment

Direct synthesis

Huang et al. (2000) - servo


sK (581) 2Tral^m+0.4Tn (581) N = min[20,

20Td]

2Tml$n

2Tml^m+0.4Tn • l . p = -

0-5Tra,2

T r a/+0.8Tml^mTn

• 1 .

X = 0 . 6 „ ^ K m < 1 Am = 2.83 . X = 0.7,1 < m < 2 Ami ' ' Ami '

Am =2.43. X = 0.'< -lm>2 A m =2.13

5 K (581) _.| ZTml m +0-4lm ^ (581) _ Tml +0.8Tml^mTn

K m ( i m + 2 T m 3 ) . d 0.8Tml^mim


4.10 Third Order System plus Time Delay Model

K e~ST|" K e~STm

G (s) = m or G (s) - m

m W l + a1s + a 2 s 2 + a 3 s 3 m W (l + sTml Xl + sTm2 Xl + sTm3)

/ 4.10.1 Ideal controller Gc(s) = Kc

R(s)

1 A

1+ — + Tds

V T i s

U(s)

K „ e "

(l + STmlXl + sTm2Xl + sTm3)

Y(s)

Table 152: PID controller tuning rules - third order system plus time delay model

K „ e - " -Gm(s) =

(l + sTm,Xl + sTm2Xl + sTm3)

Rule

Standard form optimisation -

binomial -Polonyi (1989). Standard form optimisation -

minimum ITAE -Polonyi (1989).

Kc Ti Td Comment


1 K (582)

2 K (583)

T ( 5 8 2 ) i

T (583)

, X m

t m

Model: Method I

Tm, > 10(Tm2+xm)

1 K (582) _

6tm Tm3

T/5821 =4V6T m 2 T r a + t r a .

4.10.2 Ideal controller in series with a first order lag /

Gc(s) = Kc

—JU(s)

1 \

v T i s j

1

K, V T i S

1

T fs + 1

K „ e "

Tfs + 1

Y(s)

l + a,s + a 2 s 2 + a 3 s 3

Table 153: PID controller tuning rules -Third order system plus time delay model

K m e " ^ Gra(s) = -

l + a,s + a2s + a3s

Rule

Schaedel(1997). Model: Method 1

Kc Ti Td Comment

Direct synthesis: frequency domain criteria 3 K (584) T (584) T (584) T f =T d

( 5 S 4 ) /N ;

5 < N < 20

3 K (584) 0.375T (584) , (584) _ a l ~ a 2 + a l T m +0-5'cm

Km U , + T m + -(584)

N - (584) a, + t m - T , (584)

(584) _ a2+a1xm+0.5Tm a3+a2Tm+0.5a,Tm +0.167xn 1 J —

a2+a1Tm+0.5Tn

4.10.3 Controller with filtered derivative G c ( s ) = Kc

U(s)

i 1 Td s

1 + + =r

R(s) E(s)

K, 1 THs

1 + — • + -T ' S 1 + ^ s

N

N

K „ e " " "

l + a,s + a 2 s 2 + a 3 s 3

Y(s)

Table 154: PID controller tuning rules - Third order system plus time delay model

Gra(s) = -Kme~

i j a T « 2 a a 3 a

Rule

Schaedel(1997).

Kc Ts Td Comment

Direct synthesis: frequency domain criteria 4 K (585) T (585) T (585) Model: Method 1

4 K (585)

K „

0.375T,- (585)

(585)

N

T(585) = a , 2 - a 2 + a , T m +0.5Tn

• (585) *i + t m - T d (585)

(585) _ a 2 + a,xm+0.5xm a 3 +a 2 x m +0.5a ,x m +0.167Tn

a, +x„ a 2 +a ,x m +0 .5x (585)

N:

a + T m - T j (585) Jki^flt

(585) T (585)

0.375-L ry yjOJfy

K „

0.05 < a < 0.2 , Kv = prescribed overshoot in the manipulated variable for a step

response in the command variable.

( \ r \ U(s) = Kc

, 1 TdS 1+ — + d

T i s 1 + ^ s N

E(s)-Kc a + -PTds

1 + ^ s N j

R(s)

a + Fds

T N ;

K,

R(s)

+ n

E(s)

K,

N

U(s)

*g» K „ e - " »

d + sTml)3

Y(s)

Table 155: PID controller tuning rules - third order system plus time delay model -

K m e ^ Gm(s) =

0 + sTml)3

Rule


Model: Method 1

Kc Ti Td Comment


5 K (586) ry (586) T (586)

Overshoot (servo step) < 20% ; N

fixed but not specified

' Equations continued into the footnote on page 402; xm /Tml < 1.0 .

K (586)

K „ 0.4020+ -

1.275

- + 0.003273

T (586) rp i " 1 m l 0.3572 + 7.467-a!--12.86

)

^m

T m l .

2

+ 11.77 _ T m i

3

-4.146 . T m ! _

4"l


(586) = T„ 0.8335 + 0.2910- - 0.04000

2\

a = 0.6661 - 0.2509-12- + 0.04773 Tm,

, p = 0.8131 - 0.2303—S3- + 0.03621 T


K „ e " " -4.11 Unstable FOLPD Model Gm(s)

T j - 1

403

f 4.11.1 Ideal controller G (s) = Kc

1 A

V T i s

_RCs)_^E(s)

+

r K,

1 1 + — + Tds

V T i s J

U(s) K e "

T m s - 1

Y(s)

Table 156: PID controller tuning rules - unstable FOLPD model G m (s) = Kme"

T m s-1

Rule

Minimum ISE -Visioli (2001).

Minimum ITSE -Visioli (2001).

Minimum ISTSE -Visioli (2001).

Minimum ISE -Visioli (2001).

Minimum ITSE -Visioli (2001).

Kc Ti Td Comment


1.37

1.37

Km

1.70

f

T

f

T

1

\

I

1

2.42Tra

4.12Tm

\ 1m J

( \ m

T

4-52T m f^ V m j

0.90

1.13

0.60xm

0.55xm

0.50xm

Model: Method 1

Model: Method 1

Model: Method I

Minimum performance index: servo tuning / NO.92

1 . 3 2 ( 0 Km lTmJ 1.38'

T

.90

4 . 0 0 T „ ^ j

4.12Tm

f \

T

0.90

1 T (587)

2 T (588)

Model: Method 1

Model: Method 1

1 T ^ ( 5 8 7 ) = 3 7 8 T ^

2 Td(588) = 3 6 2 T ^

1 - 0.84

1-0.85

/ N0.02

f \09S

T V m

f A 0 9 3

T V m J

Rule

Minimum ISTSE -Visioli (2001).

Jhunjhunwala and

Chidambaram (2001).

De Paor and 0'Malley(1989). Model: Method 1


Model: Method 1

Kc

, N0 .95

1.35fO K ralTJ

Ti

4 . 5 2 T m f ^ 1.13

Td

3 T (589)

Comment

Model: Method I


4 K (590) T (590) T (590) Model: Method 1

Direct synthesis

5 K (591)

6 K (592)

T (591) i

T 25-27^2-

T (591) 1 d

0.46xm

T m

<1

^2_<0.6 T

m

3 x (589)

4 K (590)

: 3.70T„ 1-0.861

13.0-39.712—2 K m l Tm

• < 0 . 2 :

K (590) 1.397

K „ , 0 . 2 < - ^ - < 1 . 4 ;

T

Ti(590> = 0 . 8 5 6 T e T- , 0 <-^- < 1 ; T: (590) = 0.3444T e T- , 1 < ^ L < 1 . 4 :

T (590) _ T

' d ~ Xrr 0.5643-^2- + 0.0075 , 0 < ^ _ < 1 . 4 . T T

5 K (591)

,(591)

K „ cosJU-^Ll^L+J1 T m / / T m s in j f 1 - - T "

T

l - t „ / T m

Tm/Tm

tan(0.75(f))

f \ (j) = tan

v Tm y

Tm K / \

m \ m V m /

6 v (592) _ _ 1 _ K „ 1.3 + 0.3^2-

Rule

Chidambaram (1997).

Model: Method 1


(1997b)-dominant pole

placement. Model: Method 1

Kc

1.165 Am

, \ 0.245

1.165 l T m J

Ti

7 T (593) 1 i

S T (594) ^i

9 T ( 5 9 5 )

10 j (596)

Td

j ( 5 9 3 )

a

T (594) ^d

Comment

T

0.6 <

0.8 <

<0.6

T m m

m

<0.8

<1

7 (593) =0 .176 + 0.360^2- T m ; a = 0 .179-0 .324^-+ 0.161 \ * m / *• m

f \ 2

2m. I t n vT™

<0.6

a = 0.04, 0 .6<^s .<0 .8 ; a = 0 .12-0 .1 -^ - , 0 . 8 < ^ - < 1 . 0 . T T T

, (594)

9 j (595)

10 j (596)

0.176 + 0.36-m J

0 . 1 7 9 - 0 . 3 2 4 ^ + 0.161 — T T

r - T d (594) 0.176 + 0.36-

o.ne + o^-^psT^ 0.176Tm+0.36T„

0 . 1 2 - 0 . 1 ^ -

Rule

Pramod and Chidambaram


Gaikward and Chidambaram

(2000). Model: Method 1 Sree et al. (2004) Model: Method 1

Kc

11 K (597)

13 K (599)

14 v (600)

Ti

T (597)

12 j (598)

T (599)

T ( 6 0 0 )

Td

T (597)

rj, (599)

0.4917tm

Comment

T

0.6 <-

- <0.6

T <0.8

0.1<^L<0.6 T

0.01 < ^ L < 0.9 T

11 K (597) = _

K „ 2.121-1.906-2-+ 0.945

T „

f \2

t,

T

, ( 5 9 7 )

0.176 + 0.36-

0 .179-0 .324-^ + 0.161 T

/" \ 2 ' d (597) _

T

0.176 + 0.36-

12 j (598) 0.176 + 0.36- 2.5T„

Kn

0.299 + 0.726- • (599) = T „

( V 4.104^2.

V m J

+ 2 .099-^ + 0.096 T

(599) 0.822 r \2

t

T V m J

+ 0.419-E- +0.019 T „

14 xj (600) _ 1.4183 (' \ K „ T

V m y

y (600) _ T 16.327 VTm/

+ 5.5778^2- + 0.8158 T

, 0.01 <-^< 0.6; T

196 - 247.28 -J2- + 87.72 T

, 0.6 <-^2-< 0.9. T

Rule

Sree et al. (2004) - continued.

Tan et al. (1999b).

Model: Method 1 Lee et al. (2000) Model: Method 1

Kc

15 pr (601)

Ti

y (601)

Td

T (601) x d

Comment

0.01<is-<1.2 T

Robust v (602)

17 K (603)

rp (602) •M

y (603) •M

T (602)

x (603) J d

-^-<1.0 T

0 < ^ 2 - < 2 Tm

m

15 v (601) _ K

1.2824

K „ T

•. (601) _ , 5.5734-iS-- 0.0063 T

, 0.01 < - a - < 0.5; T

. (601) _

16 if (602)

0.483Trae ' T™ , 0.5 < ^ - < 1.2 ; Td(601) = Tn

1

0.507-21-+ 0.0028 T

Kn 0.0373

, (602)

(602)

X

10.5045

+ 0.1862 ^ - +[0.6508-0.1498 log?.]

+ 0.8032 3.2312

- 0.3445

0.0373 + 0.1862 ^ - + [0.6508- 0.1498 log*.]

0.9065X,0-"'"-12--l

Tm 0.2075X + 2.0335

0 0 3 7 3 +0.1862 | ^ - + (0.6508-0.14981og?0 , X not specified.

17 £ (603)

. (603) _

, (603)

K m ( 2 T C L + x m - a ) .<* = Tn i ck + l

T e*.P._l

• T m + a -T C L

2 +aT m -0 .5T m2

2 T C L + t m - a

, (603)

T a Tm2(0.167xm-0.5a)

2TCL + T m - a TC L2+atm-0.5Tm

2

,(603) 2 T C L + x m - a

4.11.2 Ideal controller in series with a first order lag f •> \

Gc(s) = Kc

U(s) V T i s

1

Tfs + 1

R(s) / 0 v E ( s )

^ ' v "

+ X

/ K,

1 1 + — + Tds

V T i s j

1

Tfs + 1 Kme" T „ s - 1

Y(s)

Table 157: PID controller tuning rules - unstable FOLPD model Gm (s) = Kme"

• o - i

Rule

Chandrashekar et al. (2002).

Kc Ti Td Comment

Direct synthesis 18 j r (604) ry (604) 0-5Tm Model: Method 1

18 v (604) _ _ J _

K „ . K „ 4282 -1334.6-21-+ 101

T 0<-S-<0.2;

T 1 m

K (604) .1161 f N-0.9427

K m V m J

0.2 < -!=- < 1.0. T

y (604) _ - , 36.842 T

-10.3 X

T V m J

+ 0.8288 , 0 < — < 0 . 8 ; T

T: ( 604 )= 76.241T„ , 0.8 < - 2 - < 1.0 T

Tr = 0 .1233-^ + 0.0033 Tm , 0 <-$-< 1.0.

(TIS + l)e~STm

Desired closed loop transfer function = ^~- r ^ ; TCL2 = 4.5115Tn

/• \ 5.661 x „

(TCL2s + l)2 T

0 . 8 < ^ < 1 . 0 , T C L 2

f \ i \

0.0326+ 0.4958-2L +2.03 T T

V m J

Tm , o < - s - < o.: T


Rule


Model: Method 1

Kc

Kc

101.0000

Kra

10.3662

5.3750

3.2655

2.4516

Kra

2.0217

1.7525

1.5458

1.3723

1.2420

1.1400

T, Td


Ti

0.0404Tm

0.3874Tm

0.8600Tm

1.8018Tm

3.0400Tm

4.6500Tm

6.7286Tra

10.337Tm

17.693Tra

33.610Tm

80.620Tm

Td

0.0

0.0435Tm

0.0884Tm

0.1375Tm

0.1868Tm

0.2366Tm

0.2866Tm

0.3380Tm

0.3910Tm

0.4440Tm

0.4969Tm

Tf

0.0

0.0134Tm

0.0250Tm

0.0435Tm

0.0581Tm

0.0696Tm

0.0784Tm

0.0885Tm

0.1005Tm

0.1124Tm

0.1247Tm

Comment

TcL2

0.02Tm

0.10Tm

0.20Tm

0.40Tm

0.60Tm

0.80Tm

1.00Tm

1.30Tm

1.80Tm

2.60Tm

4.20Tm

T m / T m

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

U(s) = Kc (FpR(s) - Y(s))+^(F iR(s) - Y(s))+ KcTds(FdR(s) - Y(s)) T:S

Table 158: PID controller tuning rules - unstable FOLPD model Gm (s) = T m s-1

Rule

Prashanti and Chidambaram


Kc Ti Td Comment


Fp=0.17

T (605) *i

Fi= l

T (605)

Fd = 0.654

Modified method of De Paor and

O'Malley (1989);

h./Tmj

tan(0.75(|») V 1 V / 'm

(j) = tan Tm xra

V m / m \ m \ m /

Rule


(2000)-Alternative

tuning 1. Model:

Method I;

K (605) T ( 6 0 5 )

Td specified

on page 410.


(2000) -Alternative

tuning 2. Model:

Method T, K (605) T ( 6 0 5 )

Td(605> specified

on page 410.

Kc

K (605)

T;

T (605)

Td

T (605) *d

Comment

0 - 0 5 < x m / T m

<0.90

Representative results 2

Fp = 0.309

Fp =0.281

Fp = 0.244

Fp =0.213

Fp =0.182

Fp =0.150

Fp =0.116

Fp = 0.080

Fp = 0.040

Fp =0.0091

K (605)

F i = l

F , = l

F i = l

F; = 1

F i = l

Fj = 1

F , = l

F i = l

F ; = l

F | = l

y {605)

Fd = 2.834

Fd = 2.605

Fd =2.413

Fd = 2.346

Fd = 2.336

Fd = 2.207

Fd = 2.093

Fd =1.884

Fd =1.508

Fd =1.159

T (605) Xd

x m / T m = 0 . 0 5

T m / T m = 0 . 1 0

t m / T m = 0 . 2 0

x m / T m = 0 . 3 0

t r a / T m = 0 . 4 0

x m / T m = 0 . 5 0

T m / T m = 0 . 6 0

i m / T r a = 0 . 7 0

x r a / T m = 0 . 8 0

t m / T m = 0 . 9 0

-6 .405T„ ,

Fp = 0.9726e Tm , F; = 1, Fd = 0 ; 0.05 < ^ - < 0.90 .

Representative results

FP = 0 . 7 7 7 , xjTm =0 .05

Fp = 0 . 4 5 7 , t m / T m =0 .10

Fp = 0 .227, x m / T m =0 .20

Fp = 0 . 1 2 1 , t m / T r a =0 .30

Fp = 0 . 0 7 8 , x m / T m =0 .40

Fp =0.046 , T m / T m =0 .50

Fp = 0 .026, x m / T m =0 .60

Fp = 0 . 0 1 3 , T m /T m =0 .70

Fp = 0.0057, x m / T m =0 .80

Fp = 0.0022, x m / T m =0.90

!Fn = 0.3127-0.3369-^- + 0.0378 ^ _ m \ *m J

2 f V - 0 . 0 4 5 ^

V m j

• F= = 1 .

FH =2.779-2.44-2!- + 5

/ ^2 x„

T V m J

-4.81 T

Rule




(2000) -alternative


Kc

3 K (606)

Fp = 0.542

Fp = 0.437

Fp = 0.366

Fp = 0.303

Fp = 0.263

Fp =0.214

Fp =0.164

F„ =0.151

Fp =0.139

Fp =0.113

Fp = 0.775

Fp = 0.605 , T.

T i

rj, (606) i

Td

^ ( 6 0 6 )

Representative results 3=1 F i = l

F | = l

F j = l

Fi =1

F |= l

F;=l

F ,= l

F ,=l

F i = l

Fd =2.916

Fd = 2.878

Fd = 2.525

Fd = 2.307

Fd =1.948

Fd =1.729

Fd =1.530

Fd =1.297

Fd =1.093

Fd = 0.962

7-1.289^2-+ 0.6267 T

m

0.05 < ^ < T

m

T V xm J 0.90

2


n l /Tm=0.05 Fp = 0.322 , T

Comment

xm /Tm=0.05

t m /T m =0 .10

^m /Tm=0.20

^m/Tm=0.30

xm /Tm=0.40

x r o /Tm=0.50

t r a /T m =0.60

xm /Tm=0.70

t m /T m =0.80

xm /Tm=0.90

1, Fd = 0 ;

m /T m =0.50

3 K C( 6 0 6 ) =—[2.121-1 .906^2- +0.945

Km T

f \ 2 t m I , - , (606)

T ' " •

T a

0.176 + 0.36-

a = 0.179-0.324-^2. +0.161 \ l^-\ , — < 0 . 6 a = 0.04 ,0.6 < 1^- < 0.8 ; T T T T

a = 0.12 - 0.1-^2-, 0.8 < - < 1.0; T T

Fn =0.5165-0.8482^2L +0.5133f T" -0.071 f ^

1M. T

; F i = l :

F d =3.206-3 .461^L+0.984p2- | +0.089 m V ' ,

f A3

V m J

Rule


(2000) -continued.

Jhunjhunwala and

Chidambaram (2001).


Kc T i Td Comment

Representative results (continued) Fp =0.664, Tm/Tm =0.10

Fp = 0.550, Tra/Tm =0.20

FP =0.456, xra/Tra =0.30

Fp =0.395, i m /T m =0.40

4 K (607) T ( 6 0 7 ) * i

Fp = 0.248, xm/Tm =0.60

Fp =0.228, xm/Tra =0.70

Fp = 0.209, xm/Tm =0.80

Fp = 0.169, xra/Tm =0.90

T (607) Model: Method I

Robust 5 K (608)

F p = F i =

y (608)

1 as + 1

ry, (608) x d

T _ l T

m J

0<-^-<2 T

2

e t m / T m ,

4 K (607) = _J_[ 13.0-39.712-T" K „

-^-<0.2: T

K (607) 1.397

K „

2.044tm

, 0.2<-!2-<1.4; T

T(607) = 0.856Tme T- , 0 < ^ - < 1 ; T,'607' = 0.3444Tme T- , 1 < - ^ - < 1.4 ; | III ' rji ' I 111 ' rji

(607) _ . 0.5643— + 0.0075 T

, 0 < - 2 - < 1 . 4 : T

F i = 0 ; Fd = 0 ; F p =0 .6863 , - ^ - = 0.1 ;Fp =0.4772e T™ , 0.2 < - < 1. 1 m * TTI

T; (608)

K m ( 2 T C L + T m - a ) ". a = T n

(T XCL

T + 1 e ^ - 1

,(608) _ -T„ + a •

- T „ a - -, (608) _

TC L2+aT:m-0.5Tm

2

2TCL+Tm-CX

m2(0.167Tm-0.5a) 2 T a + T m - a TCL

2+aTm-0.5Tm2

T(608)


4.11.4 Classical controller 1 Gc(s) = K

E(s) R(s)

®—K, + T

Table 159: PID controller tuning rules - unstable FOLPD model G m (s) = K „ e '

Tms-1

Rule

Shinskey(1988) - minimum IAE

-page 381. Method: Model I

N not specified

Kc

Minim 0.91Tra

Kmt r a

1.00Tm

K m x m

0.89Tm

Kmxm

0.86Tm

KmTm m m

0.83Tra

Km tm

Ti um performance

1.70Tm

1.90xm

2.00xra

2.25xm

2-40xm

Td

index: regulator 0.60xm

0.60im

0.80xm

0.90xm

1.00xm

Comment

tuning Kn/Tra| = 0.1

k /T m | = 0.2

k /T m | = 0.5

k /T m | = 0.67

k/T r a | = 0.8

Gc(s) = K,

415

\+A"

Table 160: PID controller tuning rules - unstable FOLPD model Gm (s) = Tms-1

Rule

Huang and Chen (1997), (1999).

Kc Ti Td Comment

Direct synthesis: time domain criteria 6 ^ (609) y (609) T (609)

Xd Model: Method 4

0 < —— < 1; 'Good' servo and regulator response.

K (609) 0.5

Km

1 + 1.61519 T

(609),

. ^ (609) = K c KroTm [ 2 x l + 2 m .

V <609>K- 1 I T T ^-,- J^-m ~ 1 \ 1 r a ' m y

with recommended x, =Tm -1.3735.10"3 + 0.22091^2-+ 1.6653PS-T T

T (609) _ T 4 , n O'}c/ in m , ^ 1 £ m r \ _ n

1.5006.10"4 +0.23549—E2- + 0.16970I T V 1 m J

U(s) = Kc ' , 1 Tds ^

1 + — + - -v

TiS l + (Td/N>

f E(s)-K t

J a +

PTds

l + (Td/N>_ R(s)

a + -(3Tds

T

N j

K_

R(s) E(s)

* K , 1 + ^ + - ^ T ' S 1 + ^ s

N

U(s)

+

K e"

T s - 1

Y(s)

Table 161: PID controller tuning rules - unstable FOLPD model Gm (s) = Kme"

T m s - 1

Rule

Chidambaram (2000c).

Model: Method I

4 = 0.333; N = oo;(3 = l

Kc Ti Td Comment

Direct synthesis

1.165 (T ~\ 0.245

K m l T m J

7 j (610)

4.4Tm+9xra

8 T ( 6 U ) i

T (610) T

0.6 <

0.8 <

- <0.6

T

T m

<0.8

<0.9

7 j (610)

0.176 + 0.36-

0.179-0.324^2-+ 0.161 T

f A 2 ' d (610)

, T

0.176 + 0 . 3 6 - ^ T

T .

8 Ti(61,) = 0.176Tm+0.36xm; a = 0 7 4 + 0 4 2 I ^ _ 0 2 3

0.12-0.1^2- T™

/ \2

T V ' n /

Rule

Srinivas and Chidambaram

(2001). Model: Method 1 Chandrashekar et


Kc

9 K (612)

10 K (613)

T,

T ( 6 1 2 ) i

T ( 6 1 3 )

Td

T (612)

0.5Tra

Comment

N = oo

N = oo;|3 = 0

'Sample Kc(612), Tj(612) ,Td

(612) taken by the authors correspond to the dominant pole

placement tuning rules ofValentine and Chidambaram (1997b) (see page 405);

•r 1

• ; P = ,(612) K ( 6 , 2 , K (612) K (612)K 2X ( 6 1 2 )

10 ^ (613)

K „ 4282 - r -1334.6-^- +101

T , 0<-!S-<0.2;

T

K (613) 1.1161

K „ , T , V m J

, 0 .2<-S-<1 .0 . T x m

36.842 f N 2

, T •10.3

t T + 0.8288 0<JU2-<o.8;

T

T:(613) = 76.24IT ' Tn 0 . 8 < - ^ < 1 . 0 . T

a = 0.83, - ^ - < 0 . 1 ; a = 0.8053-0.1935-^-, 0.1<i2_<0.7 T T T

Desired closed loop transfer function =

0.0326 + 0.4958- -2.03

( r j s+_ l )e^ .

(TCL2S + 1)2 '

\2^ Tm , 0 < - ^ < 0.8 ;

T r l , =4.5115T T

, 0.8 < - 2 - < 1.0 T


Rule


Model: Method I


Model: Method I

Kc Ti Td 1 Comment


Kc = — , Tj = x2Tm , Td = x3Tm , Tf = x4Tm , TCL2 = x5Tm K m


101.000 10.3662 5.3750 3.2655 2.4516 2.0217 1.7525 1.5458 1.3723 1.2420 1.1400

x2

0.0404 0.3874 0.8600 1.8018 3.0400 4.6500 6.7286 10.337 17.693 33.610 80.620

11 K (614)

x3

0.0 0.0435 0.0884 0.1375 0.1868 0.2366 0.2866 0.3380 0.3910 0.4440 0.4969

x4

0.0 0.0134 0.0250 0.0435 0.0581 0.0696 0.0784 0.0885 0.1005 0.1124 0.1247

T (614) M

X 5

0.02 0.10 0.20 0.40 0.60 0.80 1.00 1.30 1.80 2.60 4.20

a

0.505 0.742 0.767 0.778 0.803 0.828 0.851 0.874 0.898 0.923 0.948

T (614) 1 d

* m / T m

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

N=0; p = 0

(614) 0.5236Tm 0.4764 Pf^ T (6I4) _ 0.5236Tm - 0 . 4 7 6 4 ^ - I ) A; , -

Rule

Xu and Shao (2003b).

Model: Method 1 Sree and


Model: Method 1

Kc

12 K (615)

13 K (616)

Ti

T (615)

j ( 6 1 6 ) i

Td

T (615) *d

T (616) ' d

Comment

N=0; P = 0

N=oo; p = l

12 K (615) _ 0-5

Km

T T 1 m I ' i f

/

T (615)

T IT

(615) 0.5/L

Tm |T„ A ' a =

2 ^

Tm+VT;

mTm ( A r a > 2 , 4 > m > 6 0 ° ) .

13 Sample Kc(616), Tj*6'6' ,Td

<616) taken by the authors correspond to the tuning rules of

Chandrashekar et al. (2002) and Huang and Chen (1999); the latter tuning rule is defined for an unstable SOSPD process model.

iQ-qfc^s + ilr"-Desired closed loop transfer function =

TCL2s2+2£TCLs + l

The optimum value of a is chosen by minimizing the ISE (servo response). Then,

a = l - -

a = l-

2TCL .(616)

, £,= 1; a = l-2TCL5 T (616) , S < 1 ;

.(616)

^+^~i)(-^-^~i)2 J-z+fiT^ + x,

^+Vi^T -^-V^T +x 4>i

with x, =-(-$-fi*~Tj + ^ + A /^T7j ^ _ ^ T 7 j and

x^^-^T^-^V^J^^+V^J^-V^ If it is desired to minimize the overshoot (servo response), for £, > 1, then a = 1.

U(s) = Kc

V T i s / E ( S ) - ^ £ Y ( S )

N R(s)

E(s) K „

v T i s y

U(s)

- * R > >

KcTds

1 + ^ s N

K e"sx™

T „ s - 1

Y(s)

Table 162: PID controller tuning rules - unstable FOLPD model Kme"

T m s -1

Rule

Huang and Lin (1995) -

minimum IAE. Model: Method 2

Kc T: Td Comment


1 K (617)

2 K (618)

-p (617)

rj, (618)

T (617)

T (618) Xd

0.01 <

r>

T

[=10 <0.8;

Servo response: K (617)

K „ - 0.433 + 0.2056-2-+ 0.3135

T

,(617) = T • 0.0018 + 0.8193-^- + 7.7853 T

6.6423 \

(617) = T

v

T 7.6983-^

0.0312 + 1.6333-SL + 0.0399e T™ T

m J

• Regulator response: Kc(618) = 0.2675+ 0.1226-=i- +0.8781

T

-1.004 A

. (618) 0.0005 + 2.4631 -S- + 9.5795

T i(618)

= 0.001 lTm+0.4759x„



421

f U(s) = Kt

1 \

v T ; s j

E(s)-K i( l + Tdls)Y(s)

R(s) E(s)

-*8h* K + f

+ x-x U ( S )

+ Tds) — • ( X ) • K e"

T „ s - 1

KiO + V )

I

Y(s) - •

Table 163: PID controller tuning rules - unstable FOLPD model Gm (s) = Kme~

T m s-1

Rule

Park et al. (1998).

Model: Method 1

Kc Ti Td Comment

Robust

3 K (619) T(619) T (619) K _ i | T 7

' Km Vtm

T d i =0

3 Apply minimum ITAE - regulator tuning rule or minimum ITAE - servo tuning rule defined by Sung et al. (1996) for SOSPD model, ideal PID controller (pages 313, 316). For these formulae, Km , Tml and £m are replaced by the following parameters from the unstable FOLPD model:

K „ • (Unstable FOLPD) -> Km (SOSPD)

- 1

A 7 1 X 0.25 0.75

0-71Tm xm

V T A05 (Unstable FOLPD) -» Tml (SOSPD)

„ 7 , L 0.5 0.5 L 0.25

rT \0-5 • (Unstable FOLPD) -> %m (SOSPD)

- 1

Rule


Model: Method 1

Kc

4 K (620)

T;

y (620)

Td

x (620) l6

Comment

T d i =0

V <620>l<r 1 4 K (620) _ K i K-m _ 1

Km(^ + x J T K ( 6 2 0 ) K - T

X T v (620) „

(620) _ l m - K - i K - m X „

(620)

K i( 6 2 0 ) K m - l 2 ( X + T J

K m ( ^ O ^ K , < 6 2 0 ) K m - l ) 6(X + x J 4(X + T m ) 2

2(K/-»Km-lJ(X + xJ}'

K, (620)

K „ T V m /

1H.+ In f \

T V 1 m J

1 - ^ T ,

V m y

U(s) = K( v T i s y

E(s)-K f

f T ^

VTm J

423

Y(s)

R(s) E(s)

+ X K,

V T i S /

+ xg> U(s)

K„e-"-

Tms-1

Y(s)

Kf -^-S + l VTm J

T Table 164: PID tuning rules - unstable FOLPD model Gm(s) =

Kme"

T m s -1

Rule

Majhi and Atherton (2000)

- minimum ITSE Majhi and

Atherton (2000) -minimum ITSE.

Model: Method 3

Kc T; Td Comment


5 K (621)

6 K (622)

T (621) i

T (622)

0.5tm

ln(l + K)

4tanh-'K m

Model: Method 1

0 < ^ - < 0.693 T

f 2

0.1227+ 1.4550^2--1.271 l ^ r -1 2T

_ 0.801 l(l - 0.9358VRT^)) J _ O Z , K = Ap/(Kmh), Kraln(l + K) KraVln(l + K)' p/V

(622) 0.1227 + 1.4550 ln(l + K)-l.271 l[ln(l + K)f

' 2tanh- 'K

Rule

Majhi and Atherton (2000)

- continued.

Kc

7 K (623)

T,

T(623)

Td

T (623) x d

Comment

0.693 <-^2-<1 T

7 K (623)

K „

0.801 l(K + 0.9946) 0.7497VK + 0.9946

K +0.0682 VK + 0.0682

, (623) _ 0.0145K3 + 0.5773K2 + 2.6554K + 0.3488 ,

K3 + 5.2158K2 + 4.4816K + 0.2817

, (623) = 0.0237(K +34.5338),

" K + 4.1530 Kf

_1_ 12(K +0.9946)

KV K +0.0682

U(s) = Kc 1 \

1 + — + Tds v T i s j Tds + 1

E(s)-K,Y(s)

R(s) E(s)

(1+ +Tds) T;S Tds + 1

+ U(s) K ^ e -

T m s - 1

Y(s)

K,

I Table 165: PID controller tuning rules - unstable FOLPD model Gm (s) =

Kme~

T „ s - 1

Rule


Model: Method I

Kc T i Td

Comment

Robust

8 K (624) T (624) i

T (624)

8 K (624) _ ^ 1 K , | a \ „ - l T - K , « K „ i „ 1 m 1 m ''tn

Km(^ + x J K, ( 6 2 4 ) K m - l 2(X + xm)

K ( 6 2 4 ) K 2M | ^ , (Tm-K, ( 6 2 4 )KmTm)

2(K, ( 6 2 4 , K m - l ) 6(X + xm) 4(X + T J 2 2(K 1( 6 2 4 ,Km- l ) (^ + Tm)

T V ( 6 2 4 ) f T . (624) _ J m ~ ft-1 K m T m , l m

K,<624»K - 1 2(^ + tn

K,(624,K T m T „ ( T m - K , Kmxm)

2(K, ( 6 2 4 )Km - l ) 6(^ + xm) 4(X + Tm)2 2(K, ( 6 2 4 ,Km - l ) (^ + Tm)

(624)

(624), { ( T m - K , ^ K m x m )

2(K1< 6 2 4 )Km-l) 6(^ + xra) 4(^ + Tm)2 2(K1

(624)Km-l)(A. + Tm)

K (624)

K „ cos J 1 m \ m , m

T

f \ 1 —2HL ^m

T T


Rule


Model: Method 1

Kc

9 K (625)

Ti

T (625)

Td

T (625)

Comment

K, =0.25KU

9 K (625) _ l + 0.25KuKm (Tm -0 .25K uKmxm

Km(X + xJ l+0.25KuK ra 2(X + TJ

0-25KuKm _ xm + xj + (T m -0 .25K u K m xJx m2

.(625) _ i n

2(l + 0.25KuKm 6(X + xm) 4(X + xm)2 2(1+ 0.25KuKm)(k + xm)

2

+ Tm -0 .25K uKmxm + x l + 0.25KuKm 2(X + xm)

0.25KuKm (Tm-0.25KuKmxm)xn

2(l + 0.25KuKm 6(X + xm) 4(X + xm)2 2(1+ 0.25KuKm)(^ + xm)

(625)

0.25KuKm (Tm-0.25KuKmxm)T„ 2(l + 0.25KuKm 6(X + xm) 4(X + xm)2 2(1+ 0.25K uKJ(^ + xm)

Chapter 4: Tuning Rules for PID Controllers All

4.12 Unstable SOSPD Model (one unstable pole)

Gm(s) = * * e

(Tmls-l)(l + sTm2)

4.12.1 Ideal controller G c(s) = Kc

R ( s ) > ( C > E ( s )

+ K,

v T i s

1 + ^ - + Tds

U(s) K e~ (Tmls-lXl + sTm2)

Y(s)

Table 166: PID controller tuning rules - unstable SOSPD model (one unstable pole)

Kme"sx» Gm(s) =

(Tmls-lXl + sTm2)

Rule

Wang and Jin (2004).

Kc Ti Td Comment

Direct synthesis 1 K (626) T (626) „ (626) Model: Method 1

1 v (626) _ K m T i ( T m l ~ T m X T m 2 + T m )

(T C L3+t m ) 3 K „

TCL3 + 3xmTCL3 + 3(TmlTm2 + xmTml - xmTm2 JTCL3 + TmTm2 (Tml - xm j .(626) 1 CL3

. (626) _ (Tm2 +Tm ~TmiJTCL3 + 3TmlTm2TCL3(TCL3 + xmJ + TmlTm2xn

Rule

Rotstein and Lewin(1991).

Model: Method 1

Lee et al. (2000). Model: Method I

Kc Ti Td

Robust 2 K (627) T (627)

i T (627)

X determined graphically - sample values provided Tm/Tm=0.2,^e[0.5Tm ,1.9Tm]

T m /T m =0.4Ae[1 .3T m ,1 .9Tj

Tm /Tm=0.2^e[0.4Tm ,4 .3Tm]

xm/Tm=0.4,A.e[l.lT in>4.3Tm]

i m /T m =0.6 ,X6[2 .2T m , 4 .3TJ

3 ^ (628) T (628) i

T (628) ld

Comment

Km uncertainty

= 50%

Km uncertainty

= 30%

0 < ^ - < 2 T

m

(627) _ . — + 2 | + Tm2>

V ' m l

3 £ (628) . (628)

•Km(2X + T m - a )

(627) VTml J

M— + 2 +Tm

, X = desired closed loop dynamic parameter,

a = T . 1 e, - / T - ' - l

T(628)__T _ X1 + axm -0 .5 i m2

, (628) -Tmia + Tm2a-Tm]Tm2

2X + xm - a

xm2(0.167Tm-0.5a)

n(628)

2X + zm-a X2 + axm-0.5xm2

2X + x„ - a

Rule

McMillan (1984).

Model: Method I

Kc T| Td Comment

Ultimate cycle

4 K (629) T(629) ~ (629) Tuning rules

developed from KU,TU

4 ^ (629) 1.111 Tm,Tm

K m T 1 +

lTm| +T m 2 / r m l T a

T < 6 2 9 , = 2 x j l +

Td( 6 2 9 )=0.5xm l +

_ vTml Tm2 ATmi tm/^mj

(Tmi + Tm2JTmlTm2

\Jm\ Tm2XTmi ^m^m.

(Jm! +Tm2jTmlTm2

\}m\ Tm2j(Tml Tm/C„

R(s) - 1 f

Gc(s) = Kc

U(s)

1 \ 1 + — + Tds

V T i s J Tfs + 1

+ <§>* Kr

1 A 1 + — + Tds

V T i s J

1

(Tfs + l) K„e~

(Tmls-lXl + sTm2)

Y(s)


K m e-"" Gm(s) = (Tmls-lXl + sTm2)

Rule

Zhang and Xu (2002).

Model: Method 1

Kc T, T, Comment

Robust 5 K (630) T (630) T (630)

1d *m < T m l

5 j , (630) . Tm2Tmi(Tmi - x J + X3 +3X2Tm, +3XTJ+T Tm

K m (^+3^ 2 T m , +3XTm ,xm+xm2Tm l)

T (630) _ T M - 1m2 +

ml v ml * m m * ml j

ml ~ r-^"Tml + T m T m l ^3+3X.2T , +3VTm,2+T T, 2

^mllTmi XraJ

T,

(630) Tm2(X, + 3X Tmt + 3XTml +TroTm, )

TmlTm2(Tml - xm)+ X3 + 3X2Tml + 3XTml2 + xmTra

(630) _ ^ TmlVml ~ T m )

Tm l(^+3X2Tm ,+3XTm lTm+Tm2Tm l r

f o r ^ < 0 . 5 , A = 2 - 5 ( t n + T m 2 ) .

U(s) = Kc (FpR(s) - Y(s))+^(F,R(s) - Y(s))+ KcTds(FdR(s) - Y(s)) T:S


„ . , K m e ^

Rule


u m V

Kc

y~(Tmls-lXl + sTm2

Ti

)

Td

Robust 6 j r (631)

Fp=Fi =

-p(631)

1 f d - , , ' a

as + 1

T (631)

Tm, V ' m l J

Comment

0 < ^ - < 2 Tm

- / T - , _ !

6 K (631)

.(631)

, (631)

Km(2^ + x m - a ) , X = desired closed loop dynamic parameter,

• i r " = - T m l + T n 2 + a - ; X +a,Tm - 0 . 5 T

m " • " " m

- T m l a + T m 2 a - T m l T m 2 -

2X, + xm - a

xm2(0.167xm-0.5a)

(631) 2>. + x m - a X1 + a t m -0 .5x n2

vm m - (631) 2X + xm - a

(


R(s) E(s) K,

f P 1 + -Xs

1 + Tds

l + ^ - s V N ,

v T i s y

1 + Tds

l + -*-s N

U(s)

H K „ e -

(Tmls-lXl + sTm2)

Y(s)


Gm(s)= K ^ e

(Tmls-lXl + sTra2)

Rule

Minimum ITAE - Poulin and Pomerleau

(1996), (1997).

Process output step load

disturbance

Process input step load

disturbance

Kc ^

Minimum performance 7 K (632)

X m + T m 2

Tml

0.05 0.10 0.15 0.20 0.25 0.05 0.10 0.15 0.20 0.25

T (632)

Td Comment

index: regulator tuning

Tm2 Model:

Method 1 Coefficient values (deduced from graph)

x i

0.9479 1.0799 1.2013 1.3485 1.4905 1.1075 1.2013 1.3132 1.4384 1.5698

x2

2.3546 2.4111 2.4646 2.5318 2.5992 2.4230 2.4646 2.5154 2.5742 2.6381

Tm+T r a 2

Tml

0.30 0.35 0.40 0.45 0.50 0.30 0.35 0.40 0.45 0.50

x i

1.6163 1.7650 1.9139 2.0658 2.2080 1.6943 1.8161 1.9658 2.1022 2.2379

x2

2.6612 2.7368 2.8161 2.9004 2.9826 2.7007 2.7637 2.8445 2.9210 3.0003

X l '-ml

7 K (632)

K r a(x 1Tm l-4[Tm+T, ^\Xm + ^ml} T (632) 4Tm l(xm+Tm 2) _

x , T m l - 4 ( T m + T m 2 ) !

0< (T„/N)

^2;0 .1T m 2 <-^-<0.33T m 2

433

Gc(s) = K( 1 + — v T i s y

R(s)

—i E(s)

K 1 + v T i sy

(l + sTd)

U(s) K „ e "

(Tmls-lXl + sTm2)

(1 + Tds)

Y(s)


K m e-"-Gra(s) =

(Tmls-lXl + sTm2)

Rule

Chidambaram and Kalyan

(1996). Huang and Chen (1997), (1999).

Model: Method 4

Ho and Xu (1998).

Model: Method 1

Kc

8 ^ (633)

9 K (634)

Ti Td Comment


T (633) i

T (634) i

Tm2

Tm2

Model: Method I

'Good' servo and regulator response

Direct synthesis: frequency domain criteria fflpTml

AmKm

0.7854Tml

K m T m

10 T (635) Tm2

Representative result

"Tml Tm2 A m = 2 ,

4>m=45°

8 Kc(633) and T;

(633) are the proportional gain and integral time defined by De Paor and O'Malley (1989), Rotstein and Lewin (1991) or Chidambaram (1995), for the ideal PI control of an unstable FOLPD model [see Table 31].

9 K (634) 0.5 1 + 1 . 1 1 4 1 6 ^

(634), T (634) _ Kc KmTro I 2xt T^

' ' ~ K c( 6 3 4 > K m - l l T m T m y

Recommended x, =0.160367Tme T-' ; 0 < - 2 - < 0.8 . Tml

•57cop-cop xm-

R(s) E(s)

Kr

U(s ) = K

U(s)

v T i s / E(s)-i^Y(s)

i3 N

" ®— K e"

(T r a ls-lXl + sTm 2)

Y(s)

KcTds T

1 + ^ - s N


K „ e - " » Gm(s) =

(Tmls-lXl + sTm2)

Rule

Huang and Lin (1995)-

minimum IAE.

Kc T, Td Comment


1 j r (636) j (636) i

T (636) x d

Model: Method 2;

N not specified

Note: equations continued onto page 435. 0.05 < Tm/Tml < 0.4 ; Tm2 < T ra l .

K (636) 1 Km

K „

Kn

- 174.167 - 3 1 . 3 6 4 - 2 - + 0.4642 Tml VTmiy

-103.069-^-

ST. \ 2 M

-83.916 V T m l )

- 66.962 IEILSL + 59.496 i k i s L T ml /

3n_ i m T m 2 T m

. 7 0 7 9 im2_ + 2 3 121eTml +126.924eT"" +26.944e T""' . T „

. (636) 0.008+ 2.0718-2-+ 6.431 f \ 2

T v 1miy

+ 0.4556-E2- + 0.7503

2.4484 T m 2 T m -18.686 T 1ral

V ^ m l

-2.9978 i m 2

V ^ m l

-21.135

VTml J

1 m 2 x m

V ' m l J

Rule


continued.

Kc

2 K (637)

Ti

- (637) i

Td

T (637)

Comment

N not specified; Model: Method 2

+ T„

+ T„

12.822 T m ' m 2

T 3

V 1ml J

+ 39.001 ' t n ^

VTml J + 22.848

c T 3^ J m 2 T m

T 4

•4.754 rT 3 A

x m2 T m

T "

-0.527 f T 2 2 ^

' m 2 T m

T " + 1.64

VTml 7

(636) = T / \

1.1766-2!--4.4623 Tml V T m i y

,T, + 0.5284-522--1.4281

T „ VTmi y

+ Tm 6Tm2T +nA7J2js_) + L 0 8 8 6 I s l

-0.0301-0.5039

Tm,

/ T 2

T 3

v 1mi y

l T m l

-9.8564

- 5.0229 fT 2 T

T m I m 2

V Tml x„

V^ml -7.528

/ T 3

T " V 'mi y

+ T„ -2.3542 ' x T 3

+ 9.3804 ( 2 T 2 A

T m ^ 2

T 4

V xmi y

-0.1457 f \ 4

T v 1mi y v 1mi y

: Note: equations continued onto page 436. 0.05 < t r a /Tm l < 0.25, Tml < Tm2 < 10Tml ;

K (637) 1 K „

K „

K „

1750.08 +1637.76-=- + 1533.91 VTmi y

-7.917-

6.187 T

v 1mi y

- 6.451 Tm2T,ro +0.002452 T v L i

L3729-sL-l739.77eTml -0.000296eT"" +2.311e T""

j (637) _ T

+ Tm

51.678 - 57.043-SL- +1337.29 Y T

- ^ H + 0 . 1 7 4 2 - 2 ^ - 0.1524

7.7266-. T m , x „

, - 6 0 1 1 . 5 7 — 2 2 I j

T

+ 0.0213

V ml

-9135.52

Rule


minimum IAE.

Kc T i Td Comment


3 K (638) T ( 6 3 8 ) i

x (638) ' d


-65.283 -0.0645 f T 2 \

T m 1 m 2

V ml J

+ 274.851 V Aml J

T T 1 m 2 '•m

T V ' m l J

+ T„

+ T„

0.003926 • m l "Cr.

I T, 4

ml J

-2.0997 fT 2x 2

' m 2 x m T "

V ml J

-0.001077 Tm2

l T , ml J

-49.007eT"" + 0.000026eT"" + 0.2977e ^

T (637) _ T

' d _ ' m l

+ Tm

- 0.0605 + 4.6998^2- - 29.478 — + 0.0117^ TmI lTm

-0.01291-^=^-] + 0.6874 Tm2\m +140.135 T

/ A3

t,

V T m l J

+ T„

+ Tml

0.002455 fT ^ 3

-22-1 +1.4712 V ml

( 2T A X m ' m 2

T J

V ' m l J

-0.1289 ^x T 2^

l m 1 m 2

T 3

V ' m l J

-238.864-^2- +0.6884 l T m l

i m 2 T m

T 4

V ' m l J

-0.007725 i T i fam m2

T 4

V ' m l J

+ T„ -0.1222 f \ 2T 2

T V ' m l J

-0.000135 -Tm2 V

V ml ,

1 Note: equations continued onto page 437. 0.05 < xm/Tml < 0.4 , Tm2 < Tml: -1.344

K (638) . 1

K „

K „

10.741 - 13 .363— + 0.099 — + 727.914—s^-

-708.4811-^21 T,

0.995 T m , + 9.915 Tm2T,m +84.273

T ml J ' m l

' m l 1.031/ N 0.997

1 ^

V ^ m l

- 90.959-2^+ 9.034eT"" -2.386eTml -16.304e T""2

xm


.(638) -149 .685-141 .418- ,717 ' T I T

17 29 T T

V 'ml

+ Tm

+ T„

-12 .82 T m 2 T ,m +3.611 Tm l

2

/ N0.286

V^ml 7 V 'ml

(638)

20.5181 ^ 3 ^ -

l T m , J T m T „ , ; T m ; T m

0 .000805-2^ +141.702eTml -2.032eT"" +10.006eT""2

• 0.4144 + 1 5 . 8 0 5 ^ - - 142.327( - ^ - I + 0 . 7 2 8 7 ^

+ Tm

+ Tm

+ Tm

+ Tml

+ Tm

0.1123

Tm,

-18.317 T m 2 T ,m +486.95 Tm l

2 1SL) _io.542ps2-T

V 'ml

204.009 ( i 2 T

t m '1112 T 3

V 'ml J

+ 41.26 f T 2

Tm *-ml T 3

V 'ml J

-138.038 fx 3 T >*

xm 'm2 T

V 'mi y

+ 52.155 Tm'm2

+ 396.349

3 ^

VTml J

- 646.848 (^ 2 T 2^ lm 'm2

J m l

-289.476

19.302|^22_] - 4 7 3 1 . 7 2 | ^ - | +425.789

V 'ml J

5

T V 'ml 7

Tm im2

TmTm2

T 5

V 'ml 7

-841.807 ^x 3T 2 ^

'm 'm2 T

V 'ml 7

V ml J

+ 1313.72 f. 2T 3A

T m 'm2

T V 'ml J

-3.7688 1m2 T

V 'ml /

( , \ + 6264.79

VTml ) -161.469

Tm 1m2 ry 6

V Xml J

204.689

+ T„ 648.217

f T 5^ Tm ^m2

V Tml J

Tm ^m2

T V 'ml J

+ 25.706

-5.71:

f . 4_. 2 A Xm 'm2

V ^ i J -791.857

r T 2 T 4 \ Lm 1m2

Rule

Huang and Lin ( 1 9 9 5 ) -

minimum IAE -continued.

Kc

4 K (639)

Ti

T (639)

Td

T (639)

Comment


1 Note: equations continued onto page 439. 0.05 < x m / T m l < 0.25 , Tm l < Tm 2 < 10T m l ;

r

K (639) 1

K„

K„

-130.2 + 8 5 . 9 1 4 - 2 - + 34 .82

s -, -0.3055 X „

Tml

+ 10.442 ^ L

•22.547

(» \°-5174 Xm2

V ^ m l J -14.698 Tm2T

2m + 5 2 . 4 0 8 1 ^ -

'ml VTn

(T \

Tml J

- 5 1 . 4 7 (T

s 2 - +53.378eT"« - 0.00000 leT"" +0.286e T™i

, (639) = T 72.806 - 268.146^-- 4.922l[ ^ 2 . Trai J

+ 2468.191 - a . ml J

+ T„

+ T„

+ Tm,

+ T„

+ T„

0 .6724 {j \ 2

VTml J +151.351 Tm2\m -6914 .46 'O*

V T m l J

- 795.465 fx 2 T ^

l m 1 m 2 •14 .27

V 1ml J

T T i ^m x m2 2\

1417.65 x 3 T ^

+ 0 .4057

V 1ml J

T

V 1ml J

X T , 3 ^

+ 5 5 8 0 . 1 7

- 0 . 0 0 9 2 / T "\3

1m2

VTml J

Tml

+ 55.536

V 1ml J

fx 2 T 2 ^ '•m xm2

T " V 'ml J

-0.001119J ^ina_ J _44.903eT"" + 0.000034e T"" -15 .694e T""'

678778 V T m l , V T m l J

T (639) _ ~ Xd - ' m l

+ T „

175.515-86 .2-^2- +348.727 Tm,

( -,1.1798

- ^ - 1 - 0 . 0 0 8 2 0 7 ^ -V ^ m l

- 5 5 . 6 1 9

f NO.1064

_JH2_ + 0,0418 Tm2\m +78.959 VTmlJ Tm, V T m l J V T m l y

439

/ U(s) = Kt

1 \ 1 + — + Tds

V T i s J

E^-K^ l + VJYCs)

R(s) E(s)

K, 1+-J- + Tds V T i S J

+ U(s)

K^e""-

(Tmls-lXl + sTm2)

Y(s)

Ki( l+T d i s )


Gm(s) = (Tmls-lXl + sTm2)

Rule

Kwak et al. (2000).

Model: Method 3

Kc T i Td

Comment

Direct synthesis 5 j , (640) j , (640) T (640) Optimal gain

margin: regulator response

+ Tm 0.005048 'T . ^

-187.01eT"" + 0.000001eT"" -0.0149e T""2

T T 5 Note: equations continued onto page 440. 0 < —— < 1 — —

K (640)

K „ -0.67 + 0.297

, s-2.001

VTml J + 2.189

, ^-0.766

T -<0.9 or

K (640) _ 1

Kn

- 0.365 + 0.260 - i s - -1 .4 VTml J

+ 2.189 V^ml ^ 5» ' T

•>0.9;

,(640) _ . ( - "\

2.212 T

-0 .3 <0.4 or

Rule

Kwak et al. (2000) -

continued.

Kc

6 K (641)

Ti

T (641)

Td

T (641)

Comment

Optimal gain margin: servo

response

T^u, = T m i {_o.975+ 0.910 V ' m l

-1.845 + a} , -J2- > 0.4 with T 1 m l

1-e 5.25-0.: V ^ m l

--2.8

(640)

^ K, 1.45 + 0.969 A T A 1 1 7 1

-1.9 + 1.576 l m l J

x,+x2-^-+x3

2 ^

Tml with

/ \ 2 x

X, =-0.0030 + 0 . 6 4 8 2 - ^ ^ - 2 . 2 8 4 1 ^ + 2 . 6 2 2 1 ^ Tmi ^ Tml J ^ T,

-0.9611 ml y V T m l y

X, = 0.2446 - 1 . 0 4 1 0 - ^ -13.6723 ^

X, = 0.1685+ 0.8289-^2--9.3630

-16.76221-5s2-1 +5.1471 ^=2.

^22-) +2.9855 V ^ m l

f'r- ^ 3

T V ml j

V ml j

+ 7.3803 ^=2. l T m l J

K: = A/|Gm(jcou)(l + jTdlcou)|Kn

T T 6 Note: equations continued onto page 441. 0 < —— < 1 — . T m i

K (641)

Kn

-0.04 + 0.333 + 0.949 T,

•> -0 .983

ml J

lm < 0.9 or

K (641)

K „ -0.544 + 0.308 —

Tmi

+ 1.408 /

. lm > 0-9 .

T^640 =Tm][2.055 + 0.072(Tm/Tml)fem, xm/Tml <1 or

T^641' =Tm l [1.768 + 0.329(xra/Tml )]^m , xm/Tral > 1 ;

T (641) Tml .

1-e

X, + X 2 :

0.870

x.

0.55 + 1.683

2^

Tml

T f T X, =-0.0030 + 0.6482-2^- -2.2841| ^2

Trai

-2.6221 'T , ^

-0.9611

X, = 0.2446-1.0410-^-13.6723 Tml

( -Y ^2

T 16.7622

V^ml

+ 5.1471 V^ml

X, =0.1685 + 0 . 8 2 8 9 ^ - -9.3630 \ — \ + 2 . 9 8 5 5 - ^ - + 7 . 3 8 0 3 - ^ . l T . ml / \ ml j

4.13 Unstable SOSPD Model (two unstable poles)

Gra(s) = K „ e "

(Tmls-lXTm2s-l)

f 4.13.1 Ideal controller G (s) = Kc

1 \ 1 + — + Tds

-+6 TS a E ( s ) >

< Kc

f 1 1 1 + — + Tds

I T i s J

U(s) ^ K r a e - »

(Tm,s-lXTm2s-l)

Y(s)

— w

Table 173: PID controller tuning rules - unstable SOSPD model (two unstable poles)

Kme-sx™ Gm(s) =

(T r o ls-lXTm 2s-l)

Rule

Wang and Jin (2004).

Kc Ti Td Comment

Direct synthesis 7 K (642) T (642)

i T (642) Model: Method 1

7 jr (642) _ K m T i (Tml ~ Tro XTm2 ~ Tm )

• (642)

. (642)

( T C L 3 + ^ ) 3

= TCL33 + 3 T m T CL3 2 + 3 ( T m , T m 2 + t m T m l - T m T m 2 ) T C U + T m T m 2 (T m l - T m )

\Tm l ~ ^m ATra ~ T m 2 )

= (Tm2 + Tm ~ Tml )^CLZ + 3TmlTm2TCL3 (TCL3 + Tm ) + TmlTm21:m2

T r ' ^ - t j T ^ - x J

Rule


Kc

8 j r (643)

T( Td

Robust T (643) T (643)

Xd

Comment

0 < ^ - < 2 T

K (643)

.(643)

Km(4X + x m - a , ) X = desired closed loop dynamic parameter.

.(643)

(643)

6X2 - a 2 + a!Tm -0.5xm

4X + tm - a .

a 2 - ( T m l + T m 2 ) a , + T m l T , 4X 3 +a 2 x m +0 .167 t m

3 -0 .5a 1 t m2

4X + T m - a , • (643)

6X2 - a 2 +a,Tm -0.5xm

with cC|,a2 values determined by solving 1 (q2s2 + oc1s + l);

(Xs + l)4 _j L = 0 .

U(s) = Kc (FpR(s) - Y ( s ) ) + ^ f a R f s ) - Y(s)) + KcTds(FdR(s) - Y(s))

Y(s)

Table 174: PID controller tuning rules - unstable SOSPD model (two unstable poles)

Kme-S^ Gm(s) = (Tmls-lXTra2s-l)

Rule


Kc

9 j £ (644)

F P = F .

1

Ti Td Comment

Robust T (644) T (644)

0< — < 2 T

m

a2s +a , s + l

( a 2 s 2 +a , s + l ) e - v i , , 0

frs + l)4 ls=^^7

9 K (644) , (644)

Km(4A, + T m - a , ) X = desired closed loop dynamic parameter.

, (644) = -Tmi - T r a 2 + a , 6X2 - a , + a , t m -0 .5t 2

4X + x m - a ,

« 2 - ( T m l + T m 2 ) a l + T m l T , (644)

4 ^ + a 2 x m + 0 . 1 6 7 T mJ - 0 . 5 a , T n

4A, + T m - a 1

, (644)

6X. 2 -a 2 +a |T m -0 .5T m2

4^ + T r a - a ,

4.14 Unstable SOSPD Model with a Positive Zero

Gm(s) = K m ( l - T m 3 s ) e - " "

T m l V - 2 S m T m l s + l

4.14.1 Ideal controller in series with a first order lag f - \

R(s) E(s)

+ < & * K

Gc(s) = K(

U(s) v T i s J

1

1 \ 1 + — + Tds

v T i s j Tfs + 1 > Km(l-Tm3s)e-

2 „ 2 T m ,V-2^ m T m l s + l

Tfs + 1

Y(s)

Table 175: PID controller tuning rules - unstable SOSPD model with a positive zero

K m ( l - T r a 3 s ) r ^ Gra(s) =

TmlV-24mTm ,s + l

Rule


(2004).

Kc

10 K (645)

T i Td Comment


11 T(645) i

T (645) x d

Model: Method 1

1' Desired closed loop transfer function = l-sTm3)(l + T,(645)s)e-^

l + sT^Jl + sTg'Jtl + sx,) '

10 v (645) , (645)

( o ^ =

3.5(T

iT& ) +T& ) +x 2 +T i n 3+0.5T i n -T i (645)

) '

(645) 0.5lT;(645) -0.5T,

• (645) » I f -

nl V

0.5TmQ<2L>x2

f " T„,2(T« + T<2> + X , +Tm , +0.5xm - T / 6 4 5 ' ) ' ml \ACL ^ XCL T A 2 T 1 m 3 m I

11 Equations continued into the footnote on page 446.

j (645) _ x l ~ T m l lTCL + TCL + X 2 + Tm3 + T m ]

0.5Tml2Tm3Tm-Tml

4

x, = T M x 2 +0.5xmTm l2 tr^Tg> + T&>T&> + T ^ T ™ ) + ^ T r a l x m T < ^ x 2

with

x _ Tm,4(0.5Tm3Tm - T j f a ' T f f + 0.5xmfc'L> + TC2L> - T m 3 ] r f f + 2 i ; m T m 3 x 3 )

x4

, T m , 4 ( - 2 ^ T m l +T m 3 + 0.5x Jo.5xmTc'L>T,l2L> - T r o 3

2 x 3 ^ x4

x 3 = TCL + TCL + Tm3 + xm ,

x 4 =-Tm l2(o.5Tm 3xm - T m l

2 X r m l2 [ r S + T<2> +0 .5 t m + 24mTj-0.5xmT<1

L)T<2

L>) - ( - 2 £ T r a l + T m 3 + 0 . 5 x m )

fc^M + 0 . 5 x m ( T S +T<2L»)]+4mTmlxmT<'L»T<2) - T m l

4 ) .



f \ r \

R(s)

X0-

U(s) = Kc , 1 T d s

1+ — + ^— T*s 1 + ^ - s

N

E(s)-K t a + PTds

1 + ^ s N

R(s)

a + PTds

N

K „

+ x E(s)

K, T ' s i + ^ s

N ,

U(s)

+ K m ( l - s T m 3 ) e -

T m l V - 2 ^ T m l s + l

Y(s)

Table 176: PID controller tuning rules — unstable SOSPD model with a positive zero

K r a ( l -T m 3 s )e -^ G m (s ) :

T m l V - 2 ^ ^ , 8 + 1

Rule



Kc Ti Td Comment


12 j r (646) 13 T (646) T (646) P = 0; N = oo

( l - sT Vl + T(646)s)e~STra 13 Desired closed loop transfer function = •) 2 i£ '—-x .

(l + sT&lJll + sT&'Xl + sx,)

. (646)

K ^ T ^ + T ^ + x . + T ^ + O . S t , , , - ^ 6 4 6 1 }' (646) O .SJT^ -O.s t j , Tm3+0.5(T^+Tg)

T ( 6 4 6 ) ' T ( 6 4 6 )

13 Equations continued into the footnote on page 448.

(646) _ x l ~ Tml \TCL + TCL + X 2 + Tm3 + Tm j w j ^

"•5T m i i m 3 t m _ l m l

*. = 0 2 + 0.5TmTm,2(T<'L>T<2L> + T&>T&> + T<3

L>T<'>)+ ^ m T m , x m T< '>T< 2 >x 2 ,

v _ Tm,4(o.5Tm3Tm -lj)T&lg +0.5tm[T^ + Tff - T ^ j r g +2^mTm3x3) x2 -

x4

, Tro,4(-2^mTm, +Tm3 + 0.5xmjo.5xmT^T^ -Tm3

2x3)^

x4

X3=T<1L

)+T<2L»+Tm 3+xm ,

x4 =-Tm,2(o.5Tm3Tm -Tml2)(Tml

2[TS + T<2) + 0.5xm +2^mTml]-0.5tmT^T<2L

)) -(-2!;Tml+Tm3+0.5xm)

( ^ [ i M +0.5xrafe'» +T<2»)]HmTm,xmT^T<2L> -Tj).


4.15 Delay Model Gm(s) = Kme"

f 4.15.1 Ideal controller Gc(s) = Kc

1 \ 1+ — + Tds

V T . s J

R ( s > ^ E(s)

+ T K „ ! + — + Tds

U(s) K e-SI™

Y(s)

Table 177: PID controller tuning rules - Delay model Gm (s) = Kme

Rule


Model: Method 1

Kc Ti Td Comment

Process reaction , 0.749

K m' c m

2 1.24 K r a T m

2.734tm

1.31xra

0.303xm

0.303xm

Representative results deduced from graphs

1 Decay ratio = 0.015; Period of decaying oscillation = 10.5xn

2 Decay ratio = 0.12; Period of decaying oscillation = 6.28tm



j « % ^ / K,

1 \ 1 + — + Tds

V T i s J

Gc(s) = K

U(s)

1 A 1 + — + Tds

V T i s J T fs + 1

1

1 + Tfs K e ~

Y(s)

Table 178: PID controller tuning rules - Delay model Gm(s) = Kme

Rule

Bequette (2003) -page 300.

Model: Method 1

Kc

Km(4X + xm)

T(

0.5xra

Td

Robust

0.167xm

Comment

2^2-0.167xm2

4^ + Tm

4.15.3 Classical controller 1 Gc (s) = K

E(s)

'.+-L^ V T i s /

1 + Tds

^ N

R(s)

+ < g > — K

v T i sy

1 + Tds

l + ^ s N

U(s)

* K „ e - " »

Y(s)

Table 179: PID controller tuning rules - Delay model Gra(s) = Kme~

Rule

Hartree et al. (1937).

Model: Method 1

Kc

3 0.70

K m T m

4 0.78

Kmxra

T; Td

Process reaction

2.66xm

2.97xm

*m

0.5xm

Comment


3 N = 2. Decay ratio = 0.15; Period of decaying oscillation = 4.49tm

4 N = 2. Decay ratio = 0.042; Period of decaying oscillation = 5.03xn

Kme" 4.16 General Model with a Repeated Pole Gm(s) =

(l + sTra)n

/ 4.16.1 Ideal controller Gc(s) = Kc

1 \ 1 + — + Tds

v T i s J

RQO > ( 0 vE ( s )

K, 1 \

1+ + Tds V T i s J

U(s) K„e"

0 + sTJ-

Y(s)

Table 180: PID controller tuning rules - general model with a repeated pole

Gm(s)= K ^ e

d + sTra)n

Rule

Skoczowski and Tarasiejski

(1996).

Kc Ti Td Comment

Direct synthesis

5 R (647) T T (647) Model: Method I

5 £ (647) < w g * m ( i+» g

2 T m2 r co„Tm 1 + co/T,

T (647) n - 1

Km ^ l + W82[Td

(647>f " n + 2

2n + 4 xm 4n + 2

Tm , n > 2 with

o „ = •

b = Tm

n 2 - 2 n - 2 + - + <t»n It T m 71

2Tm 2 . _ 4n + 2 t r a 2 n - 2

n - 4 n + 3 + —+ -

±b

-=—, and with

7C T m 71

n 2_2 n_2 +2E±l^+4 n + 2

c = 4(n + 2) 1

7t Tm 7t

2 „ „ 4n + 2 ( T n - 4 n + 3 + -

71 T,

+ c , with

2 n - 2 + <

= specified phase margin.

4.16.2 Ideal controller in series with a first order lag f 1 \

Gc(s) = Kc 1

1 + — + Tds V T i s j

1 Tfs + 1

R(s)

+ 7) •

L

Kc f 1 ^ I T i s )

1

T fs + 1

U(sj K m e ^

d + sTJ"

Y(s)

— •

Table 181: PID controller tuning rules - general model with a repeated pole

Kme-"» Gm(s) =

(l + sTm)n

Rule

Schaedel(1997). Model: Method 2

Kc T, Td Comment


0.563

Km

"n + f

. n - 2 .

6 j (648) T (648) T f =T d( 6 4 8 ) /N ,

5 < N < 20

6 T (648) _. 3(n + 1) / x T (648) _ n + 1 (_ \

5n - 1 6n

4.11 General Stable Non-Oscillating Model with a Time Delay

4.17.1 Ideal controller Gc (s) = K I 1 + — + Tds T:S

R(s) / 0 v E ( s )

+ <&

f K,

1 1 + — + Tds

V T i s j

U(s) General model

Y(s)

Table 182: PID controller tuning rules - general stable non-oscillating model with a time delay

Rule

Gorez and Klan (2000).

Model: Method I

Kc

7 K (649)

Ti Td

Direct synthesis

T ( 6 4 9 ) 1 i

T (649)

T (650)

0 2 5 T j (649)

Comment

alternative tuning

alternative tuning

7 K (649) . (649)

T . ( 6 4 9 ) + T

l + J l + 2 T (649) _ j

T V ar J

(649) _ 1 a r , (649) fr, (SW)+Ta&-

t ! " 2T„ 1 - K < 6 4 9 > ^ 1 +

2 T .

3T, (649)

T (650) _ y (649) 2m_l j _ 2m_

T

\ , T^ = average residence time of the process (which

equals Tm + Tm for a FOLPD process, for example); T^ = additional apparent time

constant; T = 1-K, (649)

1+-2T (649)

4.18 Fifth Order System plus Delay Model ^ _ Km(l + b!S + b2s2 +b3s3 +b4s4 +bss

5)e" m ' S ' ~~ ~ Ti 2 3 4 TV

l + ajS + ajS +a3s +a4s +a5s J


E(s) U(s)

1+ — + Tds Xs

R(s) Y(s)

+ K, 1 + - - + Tds

Km (l + b,s + b2s2 + b3s

3 + b4s4 + b5s

5 p (l + a,s + a2s

2 + a3s3 + a4s

4 + a5s5 T

Table 183: PID controller tuning rules - fifth order model with delay

Gm(s) = Km(l + b1s + b2sz +b3s3 +b4s + bss )e

(l + a,s + a2s2 +a3s3 +a 4 s 4 +a 5s 5)

Rule


Kc Ti Td Comment

Direct synthesis 8 K (651) T (651) T (651) Model: Method I


K (Ml) = a l 3 T a l 2 b l + a l b 2 - 2 a l a 2 + a 2 b l + a 3 ~ b 3 +Yl w i t h

2 K j ^ a 7 2 b , + a , a 2 + a ; b | 2 - a 3 - b , b 2 + b 3 t y j ]

Yi = tm(a i 2 - a 1 b 1 - a 2 +b 2 )+0 .5 ( a l -b l ) r m2 +0 .167x m

3 and

Y2 =(a, - b , ) 2 T m +(a, - b , ) t m2 +0.333xm

3 - ( a , - b , + xm)2Td<651);

T (65i) = a,3 -ai2bi + zxb2 -2a,a 2 +a2b1 +a3 - b 3 +y, .

' [a,2 - a . b , ~a2 +b2 +(B, -b , ) r m + 0.5xra2 -(a, - b , + xm)rd

(651)J '

Td(651> is found by solving the following equation:

Td(651)x, + 360x2 -360xmx3 + 180xm

2x4 -60xm3x5 - 15xm

4x6

- 3 x m5 x 7 + 7 x m

6 ( b 1 - a 1 ) - x m7 = 0 , w i t h

x, = 360[a,4b2 - a t (a2b] +a3 + b!b2 +b3)

+ a,2(a22 +a 2 (b 1

2 -2b 2 ) + 3a3b1 + 2a4 H-b^- ,+b 22 -b 4 )

-a , (a 2 (2a 3 -3b 3 ) + a3(2b,2 - b 2 ) + 3a4b, +a5 - b , b 4 +2b2b3 - b 5 )


- a 2 b ,b 3 +a32 +a3(b,b2 -2b 3 ) + a4b,2 +a5b, —b,b5 +b 3 ]

-360Tm[a,4b, -a , 3 (a 2 + b,2 +2b2) + 3a,2(a3 +b,b2)

+ a,(3a2b2 -3a 3 b, -3a 4 - b , b 3 - 2 b 2 +2b4)

- a 2 (b ,b 2 +b3) + a3(b, - b 2 ) + 2a4b, +a5 - b , b 4 +2b2b3 - b 5 ]

+ 180xm2[a,4 -4a , 3 b, + 3a,2(b,2 +b2) +a1(3a2b, -3a 3 -5b,b 2 +b3)

- a 2 ( b , 2 +2b2) + a3b, +2a4 +b,b3 +2(b22 - b 4 ) ]

+ 60Tm3[4a,3 -9a , 2 b, -a ,(3a2 -5b , 2 - 4b 2 ) +4a2b, - a 3 -5b,b 2 +b3]

+ 15xm4[ 9a,2 -14a,b, - 4 a 2 +5b,2 +4b2] -42xm

5(b, - a i ) + 7Tm6 ,

x 2 = a , b 3 - a , ( a 2 b 2 +a ,b 3 +a 4 +b ,b 3 )

+ a,2(a22b, +a2(2a3 +b,b2 -3b 3 ) + a3(b,2 +b 2 ) + 2a4b, +a s +b 2b 3 - b 5 )

- a , ( a 23 + a 2

2 ( b , 2 - 2 b 2 ) + a2(2a3b, - 2 b , b 3 + b 22 - b 4 )

+ 2a32 +a3(2b,b2 -3b 3 ) + a4(b,2 +b 2 ) + a5b, - b , b 5 +b 3

2 )

+ a22a3+a2[a3(b, - 2 b 2 ) - a 5 - b , b 4 +b5] + a3 b,

+ a3(a4 — b,b3 +b 2 - b 4 ) + a4(b,b2 - b 3 ) + a5b2 - b 2 b 5 +b 3 b 4 ,

x 3 = a , b 2 - a , ( a 2 b , + a 3 + b , b 2 + b 3 )

+ a,2(a22 +a2(b,2 - 2 b 2 ) + 3a3b, +2a4 +b,b3 +b 2

2 - b 4 )

-a,(a2(2a3-3b3) +a3(2b,2 - b 2 ) + 3a4b, +a5 - b , b 4 +2b2b3 - b 5 )

- a 2 b ,b 3 +a32 + a3(b,b2 - 2b 3 ) + a4b,2 +a5b, - b , b 5 + b 3

2 ,

x 4 = a , b, — a, ( a 2 +b , +2b2)

+ 3a,2(a3 +b,b2) + a1(3a2b2 -3a 3b, -3a 4 — b,b3 - 2 b 22 +2b4)

-a 2(b,b 2 + b3) +a3(b, - b 2 ) + 2a4b1 +a5 - b , b 4 +2b2b3 - b 5 ) ,

x 5 = a , -4a , b, +3a, (b, +b 2) + a , (3a2b , -3a 3 -5b ,b 2 +b 3 )

- a 2 ( b , 2 +2b2) +a3b, +2a4 +b,b3 +2(b22 - b 4 ) ,

x6 =4a,3 -9a , 2 b, -a , (3a 2 -5b , 2 - 4 b 2 ) +4a2b, -a 3 -5b ,b 2 + b 3 ,

x7 = 9a,2 -14a,b, - 4 a 2 + 5b,2 +4b2 .

4.18.2 Controller with filtered derivative

Gc(s) = Kt i+ -U Tds

E(s)

R(s) , 1 TdS

1 + — + - d T.s l + ^ s

N .

U(s)

T,s l + (T d /N>

Y(s)

Km (l + b,s + b2s2 + b3s3 + b4s4 + b5s5 p

(l + a ,s + a2s2 + a3s3 + a4s4 + a5s

5


Gm(s) = Km(l + b1s + b2s2 +b3s3 +b 4s 4 + bss

5)e~ST"'

(l + a,s + a2s2 +a3s3 +a 4 s 4 + -7) Rule

Vrancic(1996)-pages 118-119.

Model: Method 1

Kc

ry (652)

2(A,-T i<652)j

T, Td

Direct synthesis

9 j (652) i

T (652)

Comment

N<10

(652) [T, (652),2

N

(652)

-(A32 - A 5 A , ) + .J(A3

2 - A ^ J - ^ ( A 3 A 2 -A5XASA2 -A4A3

N ( A 3 A 2 - A 5 )

A, = K m ( a , - b , + x m ) , A2 = K m ( b 2 - a 2 + A , a 1 - b I T m + 0 . 5 T m2 ) ,

A3 = K m ( a 3 - b 3 + A 2 a , - A,a 2+b 2xm -0 .5b,xm2+0.167xm

3 ) ,

A4 =Km(b4 - a 4 +A3a, - A 2 a 2 +A,

A5 =Km(a5 - b 5 + A4a, - A3a2 + A2a3 - A,a4 +b4xm -0.5b3xm2)

+ Km(o.l67b2xm3-0.042b1xm

4+0.008xm5)

b3Tra +0.5b2Tm2 + 0.167b,xm

3 +0.042xm4)

Rule


Model: Method 1

Kc

10 K (653)

Ti

T (653) i

Td

T (653)

Comment

8 < N < 20

10 K (653) = _

2K,

a, - a , b, +a,b2 -2a ,a 2 + a2b, +a3 - b 3 +y,

^ - a / b j + a . a j + a ^ , 2 - a 3 - b , b 2 +b3 +(a, - b , ) 2 x m +y2

t, with

Yi = t m (a i 2 -a ,b 1 -a 2 +b 2 )+0 .5(a 1 -b , )T: m2 +0.167T m

3 and

y2 =(a, -b , )x m2 +0.333xra

3 - ( a , - b , +xm)2Td

• (653)

(653) r T <6 5 3>12

SV J-(a,-b I +xB); a, - a , b , + a , b 2 - 2 a , a 2 + a 2 b , + a 3 - b 3 + y .

a,2 - a , b , - a 2 +b2 +(a, -b , ) r m +0.5xm2 -(a, - b , +xm)Td

(653) [T, (653)-,2

Td is found by solving the following equation:

[Td(653)]4N-3Z) + [ T d ( 6 »)] 3 N- 2 z 2 +[Td(653)]2N- ,z3 +

TH<653)x, +360x, -360xmx, + 180x 2x< - 60xm

2x< - 15x 4x m A 4 '•m A 6

- 3 t m5 x 7 + 7 T m

6 ( b , - a 1 ) - x r a7 = 0

with x1 ,x2 ,x3 ,x4 ,x5 ,x6 and x7 as defined in Table 183; z,,z2 and z3 are defined as

follows:

z, = 360[a, - a t2 b , +a,(b2 - 2 a 2 ) + a2b, +a3 - b 3 ]

+ 3 6 0 x m [ a 12 - a 1 b 1 - a 2 + b 2 ] + 180xm

2(a ,-b,) + 6 0 x m \

z2 = 360(a, -b , ) [a , - a , b , + a , ( b 2 - 2 a 2 ) + a 2 b , + a 3 - b 3 ]

+ 360tm[2a,3 -3a , 2 b, -a ,(3a2 - b , 2 - 2 b 2 ) + 2a2b, +a3 - b , b 2 - b 3 ]

+ 180xm2(3a,2 -4a ,b , - 2 a 2 +b, 2 +2b2) + 240xm

3(a, - b , ) + 60xm4,

z3 = z4 + z5xm + z6xm2 + z7xm

3 + 135(a, - b,)xm4 + 27xra

5, with

z4 =-360[a,4b, -a , 3 (a 2 + b , 2 + b 2 ) +a,2(2(a3 +b 1 b 2 ) - a 2 b 1 )

+ a,(a22 +a2(b,2 + b 2 ) - a 3 b , - 2 a 4 — b,b3 - b 2

2 +b 4 )

-a 2 (a 3 +b,b2) + a4b, +a5 +b2b3 - b 5 ] ,

z5 =360[a,4-3a,3b, +a,2(2(b,2 + b 2 ) - a 2 ) +a,(3a2b, - a 3 -3b,b 2)

- a 2 ( b , 2 +b 2 ) + a4 +b,b3 +b 22 - b 4 ] ,

z6=540[a, -2a , b, - a , ( a 2 - b , - b 2 ) + b , ( a 2 - b 2 ) ] and

z7 =180[ 2a , 2 -3a ,b , - a 2 + b , 2 + b 2 ] .

Rule

VranCic et al. (2004a), Vrancic

and Lumbar (2004).

Model: Method 1

Kc

11 K (654)

Ti

T (654)

Td

T (654)

Comment

8 < N < 20

Kc( 4>, Tj and Td

(654) are determined by solving the following equations:

K (654) x (654) .

0.5^(A4A, - A 5 K j - 0 . 5 - ^ ( A , A 2 -A3Km) A l ^ m

A<A,Km + A,A,A, - A,A,2 - A,2K„ M ^ l ^ - m T n l n 2 n 3 • r l 4 r M

(654) K 2 L (654) I2

, - - Km lK° ,. J and J r „ 2(A1+K ra

2Kc(654 ,Td

<654))

K, 1 "" l + 2KmKc(654) +

• (654)

K „ A3Km -A,2KmKc

( 6 5 4 )Td '6 5 4 ) - A 2 A , + A2Km2Kc<

654)Td<654> + x

2A,A9K - A , 3 - A , K m2

with

x2 =[(2A,KmKc<654>Td<«4> + A2)A2 - A , A 3 -A 3 (K m

2 + A,2)Kc'«

4»Td<654»]

[A,+K r a2K c< 6 5 4 ,T/ 5 4>],

and with A i = K m ( a i - b l + x m ) , A 2 = K m ( b 2 - a 2 + A , a 1 - b 1 x m + 0 . 5 x m

2 ) ,

A3 =Km(a3 - b 3 + A2a, - A,a2 +b2xm -0.5b,xm2 +0.167Tm

3),

A4 =Km(b4 - a 4 + A3a, - A2a2 +A,a3 -b 3Tm +0.5b2xm2 + 0.167b,xm

3 + 0.042Tm4)

A5 = Km(a5 - b 5 + A4a, - A3a2 + A2a3 - A,a4 +b4xm -0.5b3xm2)

-Km(0.167b: xm3-0.042b,xm

4+0.008xm5).

R(s) E(s) K „

Xs

U(s) = - ^ E ( s ) - K ( 1+ T ' S

1 + ^ s N ;

Y(s)

U(s) Km(l + b,s + b2s2 + b3s3 + b4s4 + b5s5)s

(l + a,s + a2s2 + a3s3 + a4s4 + a5s5 J

Y(s)


Gm(s) = Km(l + b,s + b2s + b3s +b4s +bss )e

(l + a^ + a ^ 2 +a3s3 +a 4 s 4 +a5s5J

5 \ „ - s t m

Rule

Vrancic et al. (2004a), Vrancic

and Lumbar (2004).

Kc Ti Td Comment

Direct synthesis

12 K (655) T (655) T (655)

Model: Method 1; 8 < N < 20

12 Note: equations continued into the footnote on page 461. Kc ' , T;<655) and Td

are determined by solving the following equations:

0.s42-(A4A1 - A5Kn)-0.5^-(A1A2 -A3Km) K

(655) T (655) A, K „

A,A,Km + A,A,A, - A.A 2 - A 2 K „

K c< 6 5 5 ) _ l + 2 K m K c

(655) +

, (655)

Km2[KH:

2V (655)^ (655)

| r v 2 j - t 3 n 4 * - i ,

|2

and

K (655) A 3 K m ~ A l K m K c (655) T (655)

•A,A, + A , K mz K 2V ( 6 5 5 ) x (655)

+ X

2A,A,K - A , 3 - A , K 2 , with

x '=U2A,KmK c >)A2 (655) T (655) .

l d - t - / \ 2 i n 2 n ] n j n 3 i i v m A , A , - A 1 ( K m

2 + A 12 ) K l

( 6 5 5 ) T (655)

[A,+Km2K(

(655)y (655)1


and with

A, =Km(a, - b , + x m ) , A2 = K m ( b 2 - a 2 + A , a , - b , t r a +0 .5x m2 ) ,

A3 = Km(a3 - b 3 +A2a, - A,a2 +b 2 t r a - O ^ t , , , 2 +0.167-cm3),

A4 =Km(b4 - a 4 + A3a, - A2a2 + A,a3 - b 3 t r a + 0.5b2xm2 +0.167b,Tm

3 +0.042xm4)

A5 = K.m(a5 - b 5 + A4a, - A3a2 + A2a3 - A,a4 +b4xm -0.5b3Tm2)

+ Km(o.l67b2Tm3 -0.042b,xm

4 +0.008Tm5).

Chapter 5

Performance and Robustness Issues in the Compensation of FOLPD Processes with PI

and PID Controllers

5.1 Introduction

This chapter will discuss the compensation of FOLPD processes using PI and PID controllers whose parameters are specified using appropriate tuning rules. The gain margin, phase margin and maximum sensitivity of the compensated system as the ratio of time delay to time constant of the process varies, are used as ways of judging the performance and robustness of the system. This work follows the method of Ho et al. (1995b), (1996), in which good approximations for the gain and phase margins of the PI or PID controlled system are analytically calculated. The method will be extended to determine analytically the maximum sensitivity of the compensated system. Insight will be obtained into the range of time delay to time constant ratios over which it is sensible to apply various tuning rules, to compensate a FOLPD process.

The chapter is organised as follows. Formulae for calculating analytically the gain margin, phase margin and maximum sensitivity are outlined in Sections 5.2 and 5.3. The performance and robustness of FOLPD processes compensated with a variety of PI and PID tuning rules are evaluated in Section 5.4. In Section 5.5, tuning rules are designed to achieve constant gain and phase margins for all values of delay, for a number of process models and controller structures. Conclusions of the

462

Chapter 5: Performance and Robustness Issues... 463

work are drawn in Section 5.6. This work was previously published by O'Dwyer (1998), (2001a).

5.2 The Analytical Determination of Gain and Phase Margin

5.2.1 PI tuning formulae

The controller and process model are respectively given by

Gc(s) = Kc

K T i s ;

Gm(s) = K_e~

1 + sT

Then

r cu*c rw> KmG~>aZm K c(J»T i + l ) (jm(jco)Gc(jco) =

(5.1)

(5.2)

(5.3) l + j©Tm jooT;

From the definition of gain and phase margin, the following sets of equations are obtained:

<t>m =arg|Gc(j©g)Gm(ja)g)j+7i

1 Am =

|Gc(jcop)Gm(jcop)

(5.4)

(5.5)

where co and co are given by

|Gc(jcog)Gm(jcog)| = l

arg[Gc(ja> )Gm(ja> ) J = - T I

(5.6)

(5.7)

From Equation (5.3),

K m K c J l + G>2Ti2

G m ( j c o ) G c ( ] c o ) - ^ ^ ! -2 ^ 2 coTj -y/l + co2Tn

Z - 0.57T + tan-1 coTj - tan-1 ©Tm - coxm (5.8)

Therefore, from Equation (5.4)

(|>m = 7t-0.5n + tan~1cogTi -tan_1cogTm -cogTm (5.9)

with co given by the solution of Equation (5.6) i.e.

K m K c J l + co 2T,2

y s = = i (5.io) cogTiA/l + cog

2Tm2

Also, from Equations (5.5) and (5.8), 1 conT; Jl + co 2T 2

A = . - P ' E EL. (5 11) m |Gm(j(Dp)Gc(jcop)| K ^ ^ l + co/Ti2

with co given by the solution of Equation (5.7) i.e. 0.57t + tan ' c o T . - t a n ' co Tm -co xm =-n (5.12)

p i p m p m

From Equation (5.10), co may be determined analytically to be

T,(Kc2Km

2 - l ) + A / ( K c2 K m

2 - l j V +4K c2 K m

2 T m2

^ 2T iTm2

An analytical solution of Equation (5.12) (to determine cop) is not

possible. An approximate analytical solution may be obtained if the following approximation for the arctan function is made:

-1 ?t I I ., , _i 7T 7X I I - . , r- i ^

tan x « — x , x < l and tan x ~ x > l (5.14) 4 ' ' 2 4x M

This is quite an accurate approximation, as is shown in Figures la and lb (page 465). Considering Equation (5.12), four possibilities present themselves if the approximation in Equation (5.14) is to be used. These possibilities, together with the formula for cop that may be determined analytically for each of these cases, are

n± In2 -4m„ T T

(i) c o ^ > 1, copTm > 1: cop = > - ^ '- (5.15)

n±ln2-~-(0.25n7m+Tm) L _ ! (5.1( (ii) co T j > l , c o p T m < l : co = , , (5.16)

2(0.257iTm+Tm)

X T „ (iii) copT; < 1, copTm > 1: (op = \\4T_m

nT (5-17)

(iv) copT,<l, c o p T m < l : c o p = - j — — - (5.18)

The gain and phase margin of the compensated system, for each of the

tuning rules, as a function of xm/Tm , may be calculated by applying

Equations (5.9), (5.11), (5.13) and the relevant approximation for co

from Equations (5.15) to (5.18).

Figure la: Arctan x and its approximation (Equation 5.14)

0 2 4 6 8 10

Figure lb: % error in taking approximation (Equation 5.14) to arctan(x)

40 T

20-

-20

T

10

5.2.2 PID tuning formulae

The classical PID controller structure is considered, whose transfer function is given as

Gc(s) = K, 1 + 1 Y 1 + sT,

T;s 1 + sceT, (5.19)

and the process model is given by Equation (5.2). Substituting Equations (5.2) and (5.19) into Equations (5.4) and (5.5) gives <j>m = 0.5rc + tan-1 cogTi + tan"1 cogTd - tan-1 cogTm - tan-1 cogocTd - cogxm

(5.20) and

_ CQpTj l(l + wp2Tm

2Xl + w p V T d2 )

m " KcKm y (l + cop2

Ti2)(l + cop

2Td2)

From Equation (5.6), (i)g is given by the solution of

(5.21)

KmKc^/l + (ag2Ti

2>/l + (ag

2Td2 _f

4 ' " co„TJl + a)„2T„2^l + a2w„2T,2

The equation for a)g may be determined to be

a)g6+a1o)g

4+a2cog2+a3 =0

(5.22)

(5.23)

with a j = T i

2(Tm2+a2Td

2)-Kc2Km

2T i2Td

T ^ V T , 2

^ - K . X 2 ^ ^ / ) Kc2Km

2

2 ~ , 2™ 2 ^ 2 T 2 3 2 T , 2 „ 2 T 2 T i \ ' a ' T d T i X ' a ' T d

Following the procedure outlined by Ho et al. (1996), the analytical solution of Equation (5.23) is given as

0), =J^/R+VQ T 7R 2 "+^R-VQ 3 + R 2 -— (5-24)

. . ^ 3a, -a . " , „ 9a,a7 - 27a , -2a , with Q = —2- x— and R = —— - —.

9 54 From Equation (5.7), co is given by the solution of

0.5n + tan -1 copTd + tan"1 copT; - tan"1 copTm - tan"1 copaTd - coptm = 0

(5.25) As with Equation (5.12), an analytical solution of this equation is not possible. An approximate analytical solution may be obtained if the approximation detailed in Equation (5.14) is made. Looking at Equation (5.25), twelve possibilities present themselves if the approximation in Equation (5.14) is to be used; these possibilities are

(1) copT; > 1, ffiT > 1

(a) copTd < 1, ©paTd < 1

(b)copTd > 1 , copaTd <1

(c) o pTd > 1. ^paTd > 1

(3) cOpT; < 1, copTm > 1

(a) ©pTd < 1, copaTd < 1

(b) copTd > 1, ©paTd < 1

(c) ©pTd > 1, copaTd > 1

2 ) c o p T i > l , <DpTm<l

a) ©pTd < 1, copaTd < 1

b) copTd > 1, copaTd < 1

c) copTd > 1, copaTd > 1

^ c O p T ^ l , copTm<l

a) copTd < 1, copaTd < 1

b ) a>pTd > 1. <opaTd < 1

c) copTd > 1, copaTd > 1

The formulae for cop that may be determined for each of these cases are

as follows:

(1) copT, > 1, copTm > 1

(a) ©pTd < 1, copaTd < 1:

7 r ± j 7 t 2 - 7 i ( 4 T m - 7 i T d ( l - a ) )

C O P = ' 4 T m - 7 i T d ( l - a )

(5.26)

(b) copTd > 1, © aTd < 1:

® p = -

( 1 1 1 ^ 2TC ± 4TI2 - 7i(4xm + 7iTda) — + - - -

4xm+TcTda (5.27)

(c) copTd > 1, copaTd > 1 : P "^d

t o p = -

J_ J 1 1_ V ^d Tj a T d Tm j

n ±ln -47ix„

4x „ (5.28)

(2)copT i>l, c o T < l : p m

(a)copTd <1, copaTd <1 : p"- J d

2n± Un2 + ^-(nTd-nTm-naTd - 4 x J

cop=-7tTm+4xm+7caTd-7cTd

(5.29)

(b)copTd>l, copaTd<l p^^d

3TT± J9n-n

con

V T d + T i y

( K T m + 7 i a T d + 4 x m )

n T m + 4 x m + ™ T d (5.30)

(c) copTd > 1, copaTd > 1:

2n±J4n +n 1 1 1

co„ V a T d Td T i y

("Tm+4xm)

7tT +4x (5.31)

(3) copTj < 1, coT > 1:

(a) copTd < 1, copaTd < 1: cop =.

(b)copTd>l, copaTd<l:

'Tm(4xm+7r[aTd-Td-T i]) (5.32)

- 7 t ± j 7 l -71

% = "

(TcT i-7taTd-4xm)

rcTj-4xm-7taTd

(5.33)

7t 7t 71

(c)copTd>l, c o p a T d > l : o D = ^ T d T - a T d

7iT. - 4x '^ l m

(5.34)

(4) a)pTj<l, Q) p T m <l :

(a) copTd < 1, copaTd < 1: cop =

(b) co Td > 1, oo aTd < 1:

2 71

7i(aTd-Td)+^(Tm-T i)+4xn

(5.35)

2n±Un2 - A ( „ T m -nT, +naTd + 4 x J X

C O P = -7tT„ - TTT, + 4x„ 7 i a T ,

(5.36)

(c) co Td > 1, oo aTd > 1

n±Jn -n

< » P = -

v a T d T ay ( 7 i T i - 7 i T m - 4 x m )

(5.37)

The gain and phase margin of the compensated system, for each of the tuning rules, as a function of tm /Tm may now be calculated by applying Equations (5.20), (5.21), (5.24) and the relevant approximation for oop

from Equations (5.26) to (5.37). As Equation (5.19) shows, the procedure applies to tuning rules defined for the classical PID controller structure only; however, Ho et al. (1996) suggest that the method may be applied to tuning rules defined for the ideal PID controller structure, by using equations that approximately convert these tuning rules into their equivalent in the classical controller structure.

5.3 The Analytical Determination of Maximum Sensitivity

The maximum sensitivity is the reciprocal of the shortest distance from the Nyquist curve to the (-1,0) point on the Rl-Im axis. It is defined as follows:

1 M m a x = M a x

all ra l + Gm(jco)Gc(jco)

For a FOLPD process model controlled by a PI controller,

|Gm(jco)Gc(jco)| = Vl + oo'Tj2 KcKr

(5.38)

(5.39)


and

arg[Gm (j©)Gc (jco)] = -0.5TT - tan"1 coTm + tan"1 oaT; - coxm (5.40)

For a FOLPD process model controlled by a PID controller,

V(l + co2Tm 2)(l + a V T d

2 ) »T>

and arg[Gm(jco)Gc(jco)] =

- 0.5TC - tan"1 coTm - tan"1 coaTd + tan-1 coT; + tan"1 coTd - coxm (5.42) The maximum sensitivity may be calculated over an appropriate range of frequencies corresponding to phase lags of 100° to 260°.

5.4 Simulation Results

Space considerations dictate that only representative simulation results may be provided; an extensive set of simulation results covering 103 PI controller tuning rules and 125 PID controller tuning rules (for the ideal and classical controller structures) are available (O'Dwyer, 2000). The MATLAB package has been used in the simulations. Figures 2 to 7 show how gain margin, phase margin and maximum sensitivity vary as the ratio of time delay to time constant varies, if some PI tuning rules are used (Figures 2 to 4) and corresponding PID tuning rules for the classical controller structure (with a = 0.1) are used (Figures 5 to 7). In these results, Z-N refers to the process reaction curve method of Ziegler and Nichols (1942); W-W refers to the process reaction curve method of Witt and Waggoner (1990); IAE reg, ISE reg and ITAE reg refer to the tuning rules for regulator applications that minimise the integral of absolute error criterion, the integral of squared error criterion and the integral of time multiplied by absolute error criterion, respectively, as defined by Murrill (1967) for PI tuning rules and Kaya and Scheib (1988) for PID tuning rules based on the classical controller structure. Figures 8 to 15 show gain and phase margin comparisons between corresponding PI and PID controller tuning rules.

It is clear that the gain margin is generally less when the PID rather than the PI tuning rules are considered, over the ratios of time delay to

time constant taken; the difference between the phase margins is less clear cut. This suggests that these PID tuning rules should provide a greater degree of performance than the corresponding PI tuning rules, but may be less robust. Comparing the individual tuning rules, it is striking that the ISE based tuning rules have generally the smallest gain margin and have also a small phase margin, suggesting that this is a less robust tuning strategy. The results in Figures 4 and 7 confirm these comments.

Figure 2: Gain margin Figure 3: Phase margin Figure 4: Max. sensitivity

•Z-U | f 7= JAp ijegi : +4ltAJEreg

...

o - -°oo<

4 + , * 00°

' ' ' - - - * " 1 1 1 -

/ : t t f V i •... A// ; ' \ t ; V > ° fM--t°--i—1—r--;---:—

;>'! i \ i i - ! — ! — 1 — ! — t " - - \ - - - ; - - -

0.2 0.4 0.6 OB I 1.2 1.4 1.6 IB 2 0.2 0.4 0.6 OB I 12 1.4 1.6 IB 2 02 04 0.6 06 1 12 14 1.6 IB 2

Ratio of tm to T„ Ratio of tm to T„ Ratio of tm to T„

Figure 5: Gain margin Figure 6: Phase margin Figure 7: Max. sensitivity

1- 4 w-w M M : 4 4 iAri reg ; ; ; + = lTAEreg! i i +++++t

•^-lis^tegr-r^f-r--',i : i >*!*' i j^<\

"oofeooiooooo*"?000: : : i

60.. - : . . : . . j / t ; / ,

40 WA 3 0 ° P 4 ° ° J J

0 : - i - \ - f -: : \ ! '• '• *

0.2 0.4 0.6 0 6 1 1.2 1.4 1.6 18 2 0 0.2 04 0.6 0.8 1 1.2 14 1.6 1.8 02 0.4 06 OB 1 1.2 1.4 1.6 1.8 2

Ratio of xm to Tm Ratio of Tm to Tm Ratio of x_ to T„


No general conclusion can be reached as to the best tuning rule (as expected); it is interesting, though, that a full panorama of simulation results (O'Dwyer, 2000) show that many tuning rules may be applied at ratios of time delay to time constant greater than that normally recommended. One example may be seen in Figures 5 to 7, where the gain margin, phase margin and maximum sensitivity (associated with the use of the PID tuning rule for obtaining minimum IAE in the regulator mode) tends to level out when the ratio of time delay to time constant is greater than 1; normally, the tuning rule is used when the ratio is less than 1 (Murrill, 1967). On the other hand, it is clear from Figures 8 and 9 that there is a significant degradation of performance when using the PID tuning rule of Witt and Waggoner (1990) and the PI tuning rule of Ziegler and Nichols (1942) for large ratios of time delay to time constant, which is compatible with application experience.

The decision between the use of a PI and PID controller to compensate the process, depends on the ratio of time delay to time constant in the FOLPD model, together with the desired trade-off between performance and robustness, as expected. It turns out, however, that the analytical method explored allows the calculation of a far wider range of gain and phase margins for PI controllers; it is also true that stability tends to be assured when a PI controller is used (O'Dwyer, 2000). Thus, a cautious design approach is to use a PI controller, particularly at larger ratios of time delay to time constant.

Finally, the volume of tuning rules and data generated means that the use of an expert system to recommend a tuning rule based on user defined requirements is indicated; work is ongoing on such an implementation (Feeney and O'Dwyer, 2002).

Figure 8: Gain margin comparison Figure 9: Phase margin comparison

120

100

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Ratio of Tm to T^

0 .0.. .

0

o 7

oO - - . , - - - U -

: ov

: o : o : /

o •/ o /

5000 )000( >ooo6ooo(

v m v u * m

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Ratio of tm to Tm w m *•" m


I 1 1 1 1 1

: - ± IAE re|g PID i : o =j IAB reg PI: i

. _ _ „ . ; . - - - - : — - - ! : ; ; . . . „

: • : : : i Q o °

\—- — -t~ - : — - : - — :-0Q0-:—-

\ : : : ; 00° :

\ : : 0ov : : \ : j,o" : : :

: N»O° : : : ° O O O O ° ^ > ~ - ~ L _ ; ; ...j

; 1/ : /

/? i

960^

. i x ^ ' / ! ! o° ; | ;

A > ° °

r :

) 0o6^e <

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

R a t i o o f ! _ , t o T _ vm *" m

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Ratio of im to Tm * m m

2.5, ! 1 1 . 1 1 i 1 1 1 I 0 0 r

1SE reg JPIDJ 6 = ISEregPI

—-i—~i

i 1/ o0oO°0 :

,/ : o

•A~t~ : o : 'o ....

0? i o-°4"4- -

in00<

o?° 0° i

—4—-

- — -:- —

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Ratio of xm to Tm " m *•" m

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Ratio of tm to Tm m * v * m


L-.=i.iT^ | 0=1117

%fafif-

LEregJ m E reg PI

! _n°<

„<J 0

, 0 ° j

5U

> " i

c

/

, /

1 1 i i _ o O ° (

•A ! i ! '600<? - - - ; - - - - ! - - - - - . ;

pj* io0o^oofo°0(

0 0.2 0.4 0.6 0.6 1 1.2 1.4 1.6 1.8 2

Ratio of tm to T„ fcm v w m

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Ratio of xm to T„ v m ™ x m

5.5 Design of Tuning Rules to Achieve Constant Gain and Phase Margins, for all Values of Delay

Normally, the gain and phase margins of the compensated systems tend to increase as the time delay increases, supporting the common view that PI and PID controllers are less suitable for the control of dominant time delay processes. However, for the PI control of a FOLPD process model, O'Dwyer (2000) shows that the tuning rules proposed by Chien et al. (1952), Haalman (1965) and Pemberton (1972a), among others, facilitate the achievement of a constant gain and phase margin as the time delay of the process model varies. All of these tuning rules have the following structure: K c = a T m / K m t m , T i=T m . Following this observation, original approaches to the design of tuning rules for PI, PD and PID controllers are proposed, for a wide variety of process models, which allow constant gain and phase margins for the compensated system.

This work organised as follows. In Section 5.5.1, PI controller tuning rules are specified for processes modelled in FOLPD form and IPD form. In Section 5.5.2, PID controller tuning rules are described for processes modelled in FOLPD form, SOSPD form and SOSPD form with a negative zero. Section 5.5.3 deals with the design of PD controller tuning rules for the control of processes modelled in FOLIPD form.

5.5.1 PI controller design

5.5.1.1 Processes modelled in FOLPD form

For such processes and controllers, Equations (5.1) and (5.2) apply i.e.,

K e"STm

Gm(s) = -

with Gc (s) = K

l + sTm

1 1 +

From Section 5.2.1,

<t>m =71-0.571 +tan -1 WgTj -tan"1 cogTm -u)gTm (Equation 5.9)

with GO given by the solution of


K m K c J l + ©„2X2

Q)gTiA/l + cog2T,

and

2m 2 m

= 1 (Equation 5.10)

conT: l + co2T 2

Am = — ^ - , V V (Equation 5.11)

with cop given by the solution of

-0.5u + tan_1 c0pT; - tan - 1 copTm-copTm =-n (Equation 5.12)

If Kc and T; are designed as follows: aT

K c = — ^ - (5.43)

and T ;=Tm (5.44)

Then Equation (5.12) becomes

-0.5n-copxm =-7t (5.45)

i.e. cop = 7i/2xm (5.46)

Substituting Equation (5.46) into Equation (5.11) gives Am=7rTm/2KmKcTm (5.47)

Substituting Equation (5.44) into Equation (5.10) gives

KmK c /co gTm=l (5.48)

i.e. co g =K m K c /T m (5.49)

Substituting Equations (5.44) and (5.49) into Equation (5.9) gives

(|>m=0.57t-KmKcTm/Tm (5.50) Finally, substituting Equation (5.43) into Equation (5.47) gives

Am=7r/2a (5.51) and substituting Equation (5.43) into Equation (5.50) gives

(t>m =0.57t-a (5.52) Some typical tuning rules are shown in Table 186.


Table 186: Typical PI controller tuning rules - FOLPD process model

a

7l/3

7t/4

TC/6

Kc

1.047Tm/KmTm

0.785Tm/Kmxm

0.524Tm/KmTm

T Tm

T m

T

Am

1.5

2.0

3.0

*m

7t/6

7t/4

7t/3

These rules are also provided in Table 3, Chapter 3. It may also be demonstrated that, if Kc and Tj are designed as follows:

K„= m ,— u u (5.53) xm , / T 2 +4n2T

and T ^ T m (5.54)

where Ku and Tu are the ultimate gain and ultimate period, respectively, then the constant gain and phase margins provided in Equations (5.51) and (5.52) are obtained.

5.5.1.2 Processes modelled in IPD form

A similar analysis to that of Section 5.5.1.1 may be done for the design of PI controllers for processes modelled in IPD form. The process is modelled as follows:

Gm(s) = K m e- s ^ /s

Corresponding to Equations (5.4) and (5.9), the phase margin is

* , 71-0.571 +tan ' cOgTj-0.57t-cogxm

with cog given by the solution of

K Kcjl + V ? cog T,

If Kc and T; are designed as follows: a

= 1

K = • Kmxm

(5.55)

(5.56)

(5.57)

(5.58)

and

% = bxr

then it may be shown that

co„ = a a f ±——T-V; 2xm 2 b x „

a 2 b 2 + 4

0.5

(5.59)

(5.60)

and, substituting Equations (5.59) and (5.60) into Equation (5.56), -i0.5

<t>m=tan-1 0 . 5 a V +0.5abVa2b2+4

Corresponding to Equations (5.5) and (5.11), the gain margin,

A = -®p T i

KmKc>/l + a)p2Ti

2

0 .5a 2 +0 .5 -Va 2 b 2 +4

(5.61)

(5.62)

with co given by the solution of

-0.57i + tan cOpT;-0.57t-copTm

i.e. co is given by the solution of

tan' COpT; = copxm

(5.63)

(5.64)

An analytical solution to this equation is not possible, though if the - i ft i i

approximation tan a^T, « 0.571 (when coT: > 1) is used, P > 4 ( f l p T j | P . | >

simple calculations show that the following analytical solution for co

may be obtained:

%=" 0.5

0.57i + ,|0.257r — (5.65)

The inequality copTj > 1 may be shown to be equivalent to b > 1.273.

Substituting Equations (5.58), (5.59) and (5.65) into Equation (5.62), calculations show that:


_b_

4a A m = • (5.66)

1 + -

Some typical tuning rules are shown in Table 187.

Table 187: Typical PI controller tuning rules - IPD process model

Kc

0.558/Kmxm

0.484/Kmxm

0.458/KmTm

0-357/Kmtm

0.305/Kmxm

T;

l-4xm

1.55tra

3.35im

4-3xm

12.15tm

Am

1.5

2.0

3.0

4.0

5.0

*m

46.2°

45.5°

59.9°

60.0°

75.0°

It may also be shown that the maximum sensitivity is a constant if Kc

and Tj are specified according to Equations (5.58) and (5.59). The

maximum sensitivity may be calculated to be:

a2

M, l-2acos(co rxm)-2 2

-0.5

(5.67)

with corxm being a constant obtained numerically from the solution of

the equation (a /cor xm ) + cosa>rxm =sin(corxm)/corxm .

It may also be shown that, if Kc and Tj are designed as follows:

K c = a K u (5.68)

and Ti=bTu (5.69)

then the following constant gain and phase margins are determined:

A =

2b na

K i \n n 1 4 4b

2

1+- 2 2

an

7t

and <\>m = tan"

- + J - — 2 V 4 4b

T).5 27i2a2b2 +27tabV47i2a2b2 +4

- J0.1257i2a2 + - ^ - V 4 7 r 2 a 2 b 2 + 4

16b

(5.70)

(5.71)

5.5.2 PID controller design

5.5.2.1 Processes modelled in FOLPD form - classical controller 1

The classical PID controller is given by

Gc(s) = K(

f 1 Y l + sTd ^ 1 +

v T i s y 1 + saT, d J

and the process model is given by Kme-ST™

Gm(s) = -l + sT„

(Equation 5.19)

(Equation 5.2)

From Section 5.2.2,

<t>m = 0.57r + tan-1 co Tj + tan"1 co Td - tan"' co T - tan"1 co aTd - co x g m

with co given by the solution of

KmKcA/l + cog2T,.2A/l + co<)

2TH2

" ^ '% 1 d

cogTiA/l + cog2Tm

2^l + a2cog2T,

• = 1 2 „ 2rp 2

d

Also

A, ^pTj l(l + cop

2Tm2J(l + cop

2a2Td2)

KcKm \ (l + cop2

Ti2)(l + cop

2Td2)

(Equation 5.20)

(Equation 5.22)

(Equation 5.21)

with cop given by

+ tan_1 (OpTj -tan" ' copTm -tan"1 cop

(Equation 5.25)

0.571 + tan"1 copTd +tan cOpT; -tan"1 copTm -tan"1 copaTd -copxm =0

If K

and

then,

c, Tj and Td are designed

Kc =

T,- =

Td = Equation (5.25) becomes

0.5TI

as follows:

aTm

Kmxm

aTm

T ^m

- ® p ^ m = 0

(5.72)

(5.73)

(5.74)

(5.75)

i.e.

cop = V2xra (5.76)

and Equation (5.21) becomes Am=<xrcTm/2KmKcTm (5.77)

Equation (5.22) becomes K m K c / o g T m = l (5.78)

i.e. » g = K m K c / T m (5.79)

Finally, substituting Equations (5.73), (5.74) and (5.79) into Equation (5.20) gives

(t ,m=0.57i-KmKcxm /aTm (5.80)

Substituting Equation (5.72), (5.73), (5.74) and (5.76) into Equation (5.21) gives

Am=7ra/2a (5.81)

and substituting Equation (5.72) into Equation (5.80) gives <|>m= 0 .5TI-a /a (5.82)

This design reduces to the PI controller design when a = 1.


5.5.2.2 Processes modelled in SOSPD form - series controller

For such processes and controllers, K e"STm

Gm(s) = 2 (5.83) (l + sTml)(l + sTm2)

with

Gc(s) = Kc

Therefore, » T i s ,

(l + sTd) (5.84)

G m ( s ) G i ( s ) = i S i £ H ± S l ± ^ > (5.85, T|S(l + sTm,Xl + sTm2)

If Tj and Td are designed as follows:

Ti=Tm l (5.86)

and T d =T m 2 (5.87)

then, from Equation (5.85)

G m ( s ) G c ( s ) = K m ^ e (5.88)

which is equal to Gm(s)Gc(s) in Section 5.5.1.1 when T i=T m . Therefore, designing

aT K c = 2 - (5.89)

K m T m

will allow Am = n/2a and §m = 0.571-a , as before.

5.5.2.3 Processes modelled in SOSPD form with a negative zero classical controller 1

For such processes,

Kme- !T-(l + sTm3) Gm(s) = -^S 1 SLL (5.90)

(l + sTml)(l + sTm2) with Gc (s) given by Equation (5.19). Therefore,

0„(s)G c (s)=K A e""'( ' + S T^1 +;T 'X1 + ^ (5.9!)

If T, ,Td and a are designed as follows:

(5.92)

(5.93)

(5.94)

(5.95)

will allow Am =n/2a. and <j)m = 0.57t-a, as in Sections 5.5.1.1 and 5.5.2.2.

5.5.3 PD controller design

In this case, the process is modelled in FOLIPD form, i.e.

Kme-ST~

and

therefo ire, designing

Ti=Tm l

T<i = Tm2

« = Tm3/Tm2

K c = a T™ Km"1™

s(l + sTm)

with Gc(s) = Kc(l + Tds)

Therefore,

KmKce"STm(l + sTd) Gm(S)Gc(s)= m ' V 6j

S ( 1 + s T m )

Therefore, designing

K - a T m c —

K m T m

and T d = T m

(5.96)

(5.97)

(5.98)

(5.99)

(5.100)

will allow Am =7i/2aand <\>m = 0.57i-a , as in Sections 5.5.1.1, 5.5.2.2

and 5.5.2.3.


5.6 Conclusions

This chapter has considered the performance and robustness of a PI and PID controlled FOLPD process, with the parameters of the controllers determined by a variety of tuning rules. The original contributions of this work are as follows: (a) an expansion of the analytical approach of Ho et al. (1995b; 1996) to determine the approximate gain and phase margin analytically under all operating conditions (b) the analytical determination of the approximate maximum sensitivity of the compensated system and (c) the application of the algorithms using a wide variety of PI and PID tuning rules. The implementation of autotuning algorithms in commercial controllers means that choice of a suitable tuning rule is an important issue; the techniques discussed allow an analytical evaluation to be performed of candidate tuning rules.

Finally, the chapter discusses an original approach to design tuning rules for both PI and PID controllers, for a variety of delayed process models, with the objective of achieving constant gain and phase margins for all values of delay. In one of the cases discussed (PI control of an IPD process model), an analytical approximation is used in the development; this approximation may also be used to determine further tuning rules for other process models with delay, and for other PID controller structures.

Appendix 1

Glossary of Symbols Used in the Book

a f l,a f2,b f] = Parameters of a filter in series with some PID controllers

a,,a2,a3,b[, b2 ,b3 = Parameters of a third order process model

a1,a2,a3,a4,a5,b1,b2,b3,b4,b5 = Parameters of a fifth order process model

A, =y,(oo), A2=y2(co), A3=y3(co), A4=y4(oo), A5=y5(oo)

Am = Gain margin

Ap = Peak output amplitude of limit cycle determined from relay

autotuning

b, c = Weighting factors in some PID controller structures

DR = Desired closed loop damping ratio

d(t) = Disturbance variable (time domain)

du/dt = Time derivative of the manipulated variable (time domain)

e(t) = Desired variable, r(t), minus controlled variable, y(t) (time domain) E(s) = Desired variable, R(s), minus controlled variable, Y(s) FOLPD model = First Order Lag Plus time Delay model FOLIPD model = First Order Lag plus Integral Plus time Delay model Fp,F;,Fd = Weights on the desired variable for one PID controller

structure

Gc (s) = PID controller transfer function

GCL(s) = Closed loop transfer function

GCL(jco) = Desired closed loop frequency response

G (s) = Ideal PID controller transfer function

485


Gcs(s) = Series PID controller transfer function

Gm(s) = Process model transfer function

Gp(s) = Process transfer function

Gp(jco) = Process transfer function at frequency co

Gp(jco) = Magnitude of Gp(jco)

G (jco,!,) = Magnitude of Gp(joo), at frequency co, corresponding to

phase lag of §

ZGp(jco)= Phase of Gp(jco)

+ h = Relay amplitude (relay autotuning)

IE = Integral of Error = (e(t)dt

IAE = Integral of Absolute Error = J~|e(t)|dt

IMC = Internal Model Controller IPD model = Integral Plus time Delay model

I2PD model = Integral Squared Plus time Delay model

ISE = Integral of Squared Error = f e2 (t)dt

ISTES = Integral of Squared Time multiplied by Error, all to be Squared

= | V e ( t ) ] 2 d t

ISTSE = Integral of Squared Time multiplied by Squared Error =

ft2e2(t)dt

ITSE = Integral of Time multiplied by Squared Error = \~ te2 (t)dt

ITAE = Integral of Time multiplied by Absolute Error = T t|e(t)|dt

Kc = Proportional gain of the controller

Kcp = Proportional gain of the ideal PID controller

Kcs = Proportional gain of the series PID controller

Appendix J: Glossary of Symbols Used in the Book 487

K f = Feedback gain used in non-interacting PID controller 10

2

" ' ' m l KH = 2 t m Km

m '-p 2 TTi i^ml^m (Hwang, 1995)

K 2Kmxm

xm- xmTm | 1 . 8 8 4 K c K u x m

9co„

2KmTm m m

( i m2 | 49Tm

2 | 7Tmxm 4 . 7 1 K u K c ( x m + 1 . 6 T j 1.775KC

2KU2

1324 324 162 81co„ 81a>/

(Hwang, 1995)

K H2 " 2K x 2

V T 2 ^ m x m T m , 1 . 8 8 4 K U K C T „

9co„ +-2 K m T m

-a

with

a :

and

T 4 , T " l , " ^ m ^ m ml T m * ml . ' ^ m l S mT m , ^ ^ m ' m l S m

i . H 1 1 1 a m' 324 81 9 81 9

0 . 4 7 1 K „ K „ a = •

a „

10TmJ | 4Tm,Tm(8^mTm+9Tml) 1.775Ku

2Kc2rm

2

81co„2 81 81

(Hwang, 1995)

Kj = Feedback gain term used in non-interacting PID controller 8, 11

K, = Integral-only controller gain (Pinnella et ah, 1986)

KL = Gain of a load disturbance process model

Km = Gain of the process model

K0 = Weighting parameter used in one PID controller structure

K = Gain of the process

Ku = Ultimate proportional gain

K^ = Proportional gain when Gc(jco)Gp(jco) has a phase lag of <|>

k u = Ultimate proportional gain estimate determined from relay autotuning


Kj = Feedback gain term used in one PI controller structure (labeled controller with proportional term acting on the output 2) and the PID non-interacting controller 12 structure

K25% = Proportional gain required to achieve a quarter decay ratio

Kx% = Proportional gain required to achieve a decay ratio of O.Olx

K)35o = Controller gain when Gc(jco)Gp(jco)has a phase lag of 135°

K15Q0 = Controller gain when Gc(jco)Gp(jco) has a phase lag of 150°

Ms = Closed loop sensitivity

Mmax = Maximum value of closed loop sensitivity, Ms

n = Order of a process model with a repeated pole N = Parameter that determines the amount of filtering on the derivative

term on some PID controller structures OS = Closed loop response overshoot PI controller = Proportional Integral controller PID controller = Proportional Integral Derivative controller r = Desired variable (time domain)

0.7071Mmax2-MmaxA/l-0.5Mmax

2

f ' ~ A * 2 i MmaX - 1 (Chen et al, 1999a)

R(s) = Desired variable (Laplace domain) s = Laplace variable SOSPD model = Second Order System Plus time Delay model SOSIPD model = Second Order System plus Integral Plus time Delay

model Tar = Average residence time = time taken for the open loop process step

response to reach 63% of its final value

Td = Derivative time of the controller

Tdi = Derivative feedback term used in non-interacting PID controller 8

Tdp = Derivative time of the ideal PID controller

Tds = Derivative time of the series PID controller

TCL = Desired closed loop system time constant

Appendix 1: Glossary of Symbols Used in the Book 489

TCL2 = Desired parameter of second order closed loop system response

TCL3 = Desired parameter of third order closed loop system response,

with a repeated pole

Tf = Time constant of the filter in series with some PI or PID controllers

T; = Integral time of the controller

Tip = Integral time of the ideal PID controller

Tis = Integral time of the series PID controller

TL = Time constant of a load disturbance FOLPD process model

Tm = Time constant of the process model

TmijT^.T^ = Time constants of second or third order process models, as

appropriate.

Tmi.Tmi,Tmi,T j = Time constants of a general process model

Tp = Time constant of the process

Tr = Parameter of the filter on the set-point in some PID controller

structures

TR = Closed loop rise time

Ts = Closed loop settling time

Tu = Ultimate period A

TU = Ultimate period estimate determined from relay autotuning

T25% = Period of the quarter decay ratio waveform, when the closed loop

system is under proportional control Tx%= Period of the waveform with a decay ratio of O.Olx, when the

closed loop system is under proportional control TOLPD model = Third Order Lag Plus time Delay model u(t) = Manipulated variable (time domain), u^ = Final value of the manipulated variable (time domain) U(s) = Manipulated variable (Laplace domain) V = Closed loop response overshoot (as a fraction of the controlled

variable final value)

x,, x2 , x3 , x4 , x5 = Coefficient values.

y = Controlled variable (time domain) y^ = Final value of the controlled variable (time domain)

Y(s) = Controlled variable (Laplace domain)

y,(t)= J^Km - ^ j d t , y2( t)= J(A, -y,(x))lx,

t t

y3(t)= J(A2-y2(x))iT, y4(0= J[A3-y3(T)]iT,

y5(t)=J[A4-y4(x)}iT

a , P, x = Weighting factors in some PI or PID controller structures 2 2

£ = 6 T m l +4^mTm ,xm+KHKmxm ( R w a n g ; 1 9 9 5 )

2Tml xmcoH

= 2Tmcou+KH1Kmxmcou-1.884KuKc ^

0.471KcKu<BH1xm 2 2

60 = 6 T " ^ + 4 T m 1 ^m+K u K m T m ( H w a n g ; 1 9 9 5 )

2 x m T ml fflu

C2 , 6Tm,2(Du +4Tml^mtm(0u + KH2Kmxm2cou -1.884KuKcxm

(0.471KuKcxm2+2xmTml

2cou)coH2

(Hwang, 1995) X = Parameter that determines robustness of compensated system. £, = Damping factor of the compensated system

^m = Damping factor of an underdamped process model

^ e s = Desired damping factor of an underdamped process model

£,m j , £m ; = Damping factors of a general underdamped process model

K = l/KmKu

<)) = Phase lag

Appendix 1: Glossary of Symbols Used in the Book 491

<j)c - Phase of the plant at the crossover frequency of the compensated

system, with a 'conservative' PI controller (Pecharroman and Pagola, 2000).

cpm = Phase margin

§a = Phase lag at an angular frequency of CO

xL = Time delay of a load disturbance process model

Tm = Time delay of the process model

co = Angular frequency

cobw = -3dB closed loop system bandwidth (Shi and Lee, 2002)

co_6dB= Frequency where closed loop system magnitude is -6dB (Shi and Lee, 2004)

coc = Maximum cut-off frequency

cod = Bandwidth of stochastic disturbance signal (Van der Grinten, 1963)

®CL = 2 " / T C L

cog = Specified gain crossover frequency (Shi and Lee, 2002)

coH = | 1 + K " K m (Hwang, 1995) 2 , 2 T m l x m ^ m | K H K m x I ]

TL i ml m ^ m .

1 + K H l K m (Hwang, 1995) xmTm | KH1KmTm

2 0.942KcKutm

3co„

C 0 „ T = l + KH2Km

[T 2 } 2^mxmTml | KH2Kmxm2 0.942KcKuxr

3 6 3cou

(Hwang, 1995)

coM = Frequency where the sensitivity function is maximized (Rotach,

1994)


con = Undamped natural frequency of the compensated system

A rh +0 Sir A (A — l) coD = A , p m '- (Hang e/a/. 1993a, 1993b)

lAm2-l>m

cor = Resonant frequency (Rotach, 1994)

cou = Ultimate frequency

co = Angular frequency at a phase lag of (j)

cog()0 = Angular frequency at a phase lag of 90°

co o = Angular frequency at a phase lag of 135° '135'

co o = Angular frequency at a phase lag of 150°

Appendix 2

Some Further Details on Process Modelling

Processes with time delay may be modelled in a variety of ways. The modelling strategy used will influence the value of the model parameters, which will in turn affect the controller values determined from the tuning rules. The modelling strategy used in association with each tuning rule, as described in the original papers, is indicated in the tables in Chapters 3 and 4. Some outline details of these modelling strategies are provided, together with the references that describe the modelling method in detail. The references are given in the bibliography. For all models, the label "Model: Method 1" indicates that the model method has not been defined or that the model parameters are assumed known. Of the 1134 tuning rules specified (442 PI controller tuning rules, 692 PID controller tuning rules), it is startling to relate that only 386 (or 34%) of tuning rules have been specified based on a defined process modelling method.

A2.1 FOLPD Model G m ( s )= K m C

A2.1.1 Parameters estimated from the open loop process step or impulse response

Method 2: Parameters estimated using a tangent and point method (Ziegler and Nichols, 1942).

Method 3: Km ,xm determined from the tangent and point method of

Ziegler and Nichols (1942); Tm determined at 60% of the

493


total process variable change (Fertik and Sharpe, 1979). Method4: Km ,xm assumed known; Tm estimated using a tangent

method (Wolfe, 1951). Method 5: Parameters estimated using a second tangent and point

method (Murrill, 1967). Method 6: Parameters estimated using a third tangent and point method

(Davydov et al, 1995). Method 7: xm and Tm estimated using the two-point method; Km

estimated from the open loop step response (ABB, 2001). Method 8: xm and Tm estimated from the process step response:

Tm=1.4( t 6 7 % - t 3 3 % ) , x m = t 6 7 % - l . l T m ; Km assumed known (Chen and Yang, 2000). '

Method 9: xm and Tm estimated from the process step response:

Tm = 1.245(t70O/o - t 3 3 % ) , xm = 1.498t33% -0.498t70% ; Km

assumed known (Viteckova et al., 2000b). 2

Method 10: Km estimated from the process step response; xm= time for

which the process variable does not change; Tm determined

at 63% of the total process variable change (Gerry, 1999).

Method 11: Km estimated from the process step response; xm = time for which the process variable has to change by 5% of its total value; Tm determined at 63% of the total process variable change (Kristiansson, 2003).

Method 12: Parameters estimated using a least squares method in the time domain (Cheng and Hung, 1985).

Method 13: Parameters estimated from linear regression equations in the time domain (Bi et al., 1999).

Method 14: Parameters estimated using the method of moments (Astrom and Hagglund, 1995).

Method 15: Parameters estimated from the process step response and its first time derivative (Tsang and Rad, 1995).

Method 16: Parameters estimated from the process step response using numerical integration procedures (Nishikawa et al., 1984).

' t67o/„, t33% are the times taken by the process variable to change by 67% and 33%,

respectively, of its total value. 2 170o/o is the time taken by the process variable to change by 70% of its total value.

Appendix 2: Some Further Details on Process Modelling 495

Method 17: Parameters estimated from the process impulse response (Peng and Wu, 2000).

Method 18: Parameters estimated using a "process setter" block (Saito et al, 1990).

Method 19: Parameters estimated from a number of process step response data values (Kraus, 1986).

Method 20: Parameters estimated in the sampled data domain using two process step tests (Pinnella et ah, 1986).

Method 21: A model is obtained using the tangent and point method of Ziegler and Nichols (1942); label the parameters Km , Tm

and xm. Subsequently, xm is estimated from the process

step response; then, a parameter labelled

xam=xm-im(Schaedel, 1997).

Method 22: A model is obtained using the tangent and point method of

Ziegler and Nichols (1942); label the parameters Km , Tm

and xm. Subsequently, xm is estimated from the process step response (Henry and Schaedel, 2005).

A2.1.2 Parameters estimated from the closed loop step response

Method 23: Km estimated from the open loop step response. T90% and

xm estimated from the closed loop step response under

proportional control (Astrom and Hagglund, 1988). Method 24: Parameters estimated using a method based on the closed

loop transient response to a step input under proportional control (Sain and Ozgen, 1992).

Method 25: Parameters estimated using a second method based on the closed loop transient response to a step input under proportional control (Hwang, 1993).

Method 26: Parameters estimated using a third method based on the closed loop transient response to a step input under proportional control (Chen, 1989; Taiwo, 1993).


Method 27: Parameters estimated from a step response autotuning experiment - Honeywell UDC 6000 controller (Astrom and Hagglund, 1995).

Method 28: Parameters estimated from the closed loop step response when process is in series with a PID controller (Morilla et al, 2000).

Method 29: Parameters estimated by modelling the closed loop response as a second order system (Ettaleb and Roche, 2000).

A2.1.3 Parameters estimated from frequency domain closed loop information

Method 30: Parameters estimated from two points, determined on process frequency response, using a relay and a relay in series with a delay (Tan et al., 1996).

Method 31: Tm and xm estimated from the ultimate gain and period, determined using a relay in series with the process in closed loop; Km assumed known (Hang and Cao, 1996).

Method 32: Tm and xm estimated from the ultimate gain and period, determined using a relay in series with the process in closed loop; Km estimated from the process step response (Hang et al, 1993b).

Method 33: Tm and xm estimated from the ultimate gain and period, determined using the ultimate cycle method of Ziegler and Nichols (1942); Km estimated from the process step response (Hang et al., 1993b).

Method 34: Tm and xm estimated from relay autotuning method (Lee

and Sung, 1993); Km estimated from the closed loop process step response under proportional control (Chun et al, 1999).

Method 35: Parameters estimated in the frequency domain from the data determined using a relay in series with the closed loop system in a master feedback loop (Hwang, 1995).


Method 36: Ku and Tu estimated from relay autotuning method; xm

estimated from the open loop process step response, using a tangent and point method (Wojsznis et al, 1999).

Method 37: Parameters estimated by including a dynamic compensator outside or inside an (ideal) relay feedback loop (Huang and Jeng, 2003).

Method 38: Parameters estimated from measurements performed on the manipulated and controlled variables when a relay with hysteresis is introduced in place of the controller (Zhang et al., 1996).

Method 39: Non-gain parameters estimated using a relay, with hysteresis, in series with the process in closed loop; Km is estimated with the aid of a small step signal added to the reference (Potocnik et al., 2001).

Method 40: Parameters estimated using a relay in series with the process in closed loop (Peric et al, 1997).

Method 41: Parameters estimated using an iterative method, based on data from a relay experiment (Leva and Colombo, 2004).

Method 42: Km estimated from relay autotuning method; xm , Tm

estimated using a least squares algorithm based on the data recorded from the relay autotuning method, with the aid of neural networks (Huang et al., 2005).

A2.1.4 Other methods

Method 43: Parameter estimates back-calculated from a discrete time identification method (Ferretti et al., 1991).

Method 44: Parameter estimates determined graphically from a known higher order process (McMillan, 1984).

Method 45: Parameter estimates determined from a known higher order process.

Method 46: The parameters of a second order model plus time delay are estimated using a system identification approach in discrete time; the parameters of a FOLPD model are subsequently determined using standard equations (Ou and Chen, 1995).


Method 47: Parameter estimates back-calculated from a discrete time identification non-linear regression method (Gallier and Otto, 1968).

Table 188 summaries the most common FOLPD process modelling methods.

Table 188: The most common FOLPD process modelling methods

Model Method Method 1 Method 2 Method 5 Method 31

PI 112 18 12 6

PID 147 36 26 6

Total 259 54 38 12

It is interesting that in 57% of tuning rules based on the FOLPD process model (252 out of 443), the modelling method has not been specified or the model parameters are known a priori.

A2.2 FOLPD Model with a Negative Zero Gm (s) = - - - -^+ sTm2 ^ l + sT

ml

Method 2: Non-delay parameters are estimated in the discrete time domain using a least squares approach; time delay assumed known (Chang et al, 1997).

A2.3 Non-Model Specific

Modelling methods have not been specified in the tables in most cases, as typically the tuning rules are based on Ku and Tu. Modelling methods have been specified whenever relevant data have been estimated using a relay method. Method 2: Ku and Tu estimated from an experiment using a relay in

series with the process in closed loop (Jones et al., 1997). Method 3: Ku and Tu estimated with the assistance of a relay (Lloyd,

1994).


Method 4: co90„ and Ap estimated using a relay in series with an

integrator (Tang et al., 2002).

A2.4 IPD Model Gm (s) = Km&

s

A2.4.1 Parameters estimated from the open loop process step response

Method 2: xm assumed known; Km estimated from the slope at start of the process step response (Ziegler and Nichols, 1942).

Method 3: Km ,xm estimated from the process step response (Hay, 1998).

Method 4: Km ,xm estimated from the process step response using a

tangent and point method (Bunzemeier, 1998).


Method 5: Parameters estimated from the servo or regulator closed loop transient response, under PI control (Rotach, 1995).

Method 6: Parameters estimated from the servo closed loop transient response under proportional control (Srividya and Chidambaram, 1997).

Method 7: Parameters estimated from the closed loop response, under the control of an on-off controller (Zou et al., 1997).

Method 8: Parameters estimated from the closed loop response, under the control of an on-off controller (Zou and Brigham, 1998).


Method 9: Parameters estimated from Ku and Tu values, determined from an experiment using a relay in series with the process in closed loop (Tyreus and Luyben, 1992).

Method 10: Parameters estimated from values determined from an


experiment using an amplitude dependent gain in series with the process in closed loop (Pecharroman and Pagola, 1999).

The most common IPD modeling method is the one where the model parameters are known a-priori or the modeling method is unspecified (i.e. Model: Method 1); 46 PI controller tuning rules, and 47 PID controller tuning rules have been specified using the method. Altogether 76% of tuning rules based on the IPD process model (93 out of 122) have been specified using the method.

A2.5 FOLIPD Model Gm(s) Kme"

3(l + sTm)

Method 2: Parameters estimated from the open loop process step response and its first and second time derivatives (Tsang andRad, 1995).

Method 3: Parameters estimated using the method of moments (Astrom andHagglund, 1995).

Method 4: Parameters estimated from values determined from an experiment using an amplitude dependent gain in series with the process in closed loop (Pecharroman and Pagola, 1999).

Method 5: Parameters estimated from the open loop response of the process to a pulse signal (Tachibana, 1984).

Method 6: Parameters estimated from the waveform obtained by introducing a single symmetrical relay in series with the process in closed loop (Majhi and Mahanta, 2001).

Method 7: Km estimated from the response of the process to a square

wave pulse; other process parameters estimated from the wavefrom obtained by introducing a relay in series with the process in closed loop (Wang et ah, 2001a).

Method 8: Parameters estimated using a relay in series with the process in closed loop (Peric et ah, 1997).

The most common FOLIPD modeling method is the one where the model parameters are known a-priori or the modeling method is


unspecified (i.e. Model: Method 1); 25 PI controller tuning rules, and 49 PID controller tuning rules have been specified using the method. Altogether 85% of tuning rules based on the FOLIPD process model (74 out of 87) have been specified using the method.

A2.6 SOSPD Model Gm(s) = K e~

Tml2s2

+2^mTmls + l or

K e"STm

(l + Tmls)(l + Tm2s)

A2.6.1 Parameters estimated from the open loop process step response

Method 2:

Method 3:

Method 4:

Method 5:

Method 6:

Km , Tml and xm are determined from the tangent and point method of Ziegler and Nichols (1942) (Shinskey, 1988, page 151); Tm2 assumed known.

FOLPD model parameters estimated using a tangent and point method (Ziegler and Nichols, 1942); corresponding SOSPD parameters subsequently deduced (Auslander et al., 1975). Tml = Tm2. xm, Tml estimated from process step response:

Tm l=0.794(t7 0 %- t3 3 %) , Tm=1.937t33%-0.937t70%. Km

assumed known (Viteckova et ah, 2000b). Tml = Tm2. Km , xm and Tml estimated from the process

step response; xm = time for which the process variable does

not change; Tml determined at 73% of the total process variable change (Pomerleau and Poulin, 2004). Parameters estimated from the underdamped or overdamped transient response in open loop to a step input (Jahanmiri and Fallahi, 1997).



Method 7: Parameters estimated from a step response autotuning experiment - Honeywell UDC 6000 controller (Astrom and Hagglund, 1995).


Method 8: Tm and xm estimated In this method, Tml = Tm2 = T,

from Ku , Tu determined using a relay autotuning method;

Km estimated from the process step response (Hang et al., 1993a).

Method 9: Parameters estimated using a two-stage identification procedure involving (a) placing a relay and (b) placing a proportional controller, in series with the process in closed loop (Sung et al, 1996).

Method 10: Parameters estimated in the frequency domain from the data determined using a relay in series with the closed loop system in a master feedback loop (Hwang, 1995).


Method 12: Parameters estimated from data obtained when the process phase lag is -90° and -180°, respectively (Wang et al., 1999).

Method 13: Model parameters estimated by including a dynamic compensator outside or inside an (ideal) relay feedback loop (Huang and Jeng, 2003).

Method 14: Km estimated from a relay autotuning method; xm, Tml and

2;m estimated using a least squares algorithm based on the

data recorded from the relay autotuning method, with the aid of neural networks (Huang et al, 2005).


A2.6.4 Other methods

Method 15: Parameter estimates back-calculated from a discrete time identification method (Ferretti et al., 1991).

Method 16: Parameter estimates back-calculated from a second discrete time identification method (Wang and Clements, 1995).

Method 17: Parameter estimates back-calculated from a third discrete time identification method (Lopez et al, 1969).

Method 18: Model parameters estimated assuming higher order process parameters are known.

Method 19: Parameter estimates back-calculated from a discrete time identification non-linear regression method (Gallier and Otto, 1968).

The most common SOSPD modeling method is the one where the model parameters are known a-priori or the modeling method is unspecified (i.e. Model: Method 1); 15 PI controller tuning rules, and 93 PID controller tuning rules have been specified using the method. Altogether 68% of tuning rules based on the SOSPD process model (108 out of 160) have been specified using the method.

A2.7 SOSIPD model - Repeated Pole G_(s) = —^ -r s(l + sTm)2



A2.8 SOSPD Model with a Positive Zero G_(s) - — •"-" s T m 3 ^ (l + sTmlXl + sTm2)

Method 2: Parameters estimated from a closed loop step response test using a least squares approach (Wang et al., 2001a).

Method 3: Tml = Tm2. Km , xm, Tml and Tm3 estimated from the process step response; the latter three parameters are estimated using a tabular approach (Pomerleau and Poulin, 2004).

A2.9 SOSPD model with a Negative Zero Gm (s) - K m '* + sTm3 ^ (l + sTmlXl + sTm2)

Method 2: Parameters estimated from a closed loop step response test using a least squares approach (Wang et al, 2001a).

Method 3: Tml = Tm2. Km , xm, Tml and Tm3 estimated from the process step response; the latter three parameters are estimated using a tabular approach (Pomerleau and Poulin, 2004).

A2.10 Unstable FOLPD Model Gm (s) = K™e

Tms-1

Method 2: Parameters estimated by least squares fitting from the open loop frequency response of the unstable process; this is done by determining the closed loop magnitude and phase values of the (stable) closed loop system and using the Nichols chart to determine the open loop response (Huang and Lin, 1995; Deshpande, 1980).

Method 3: Parameters estimated using a relay feedback approach (Majhi and Atherton, 2000).


Method 4: Parameters estimated using a biased relay feedback test (Huang and Chen, 1999).

K e~STm

A2.ll Unstable SOSPD Model Gm(s) = -, - ^ r or (TmlS-ljO+sTmJ

K e - S T „

(Tm ls-l)(Tm 2s-l)

Method 2: Parameters estimated by least squares fitting from the open loop frequency response of the unstable process; this is done by determining the closed loop magnitude and phase values of the (stable) closed loop system and using the Nichols chart to determine the open loop response (Huang and Lin, 1995; Deshpande, 1980).

Method 3: Parameters estimated by minimising the difference in the frequency responses between the high order process and the model, up to the ultimate frequency. The gain and delay are estimated from analytical equations, with the other parameters estimated using a least squares method (Kwak et al, 2000).

Method 4: Parameters estimated using a biased relay feedback test (Huang and Chen, 1999).

K e""" A2.12 General Model with a Repeated Pole Gm(s)

0 + sTm)n

Method 2: A FOLPD model is obtained using the tangent and point method of Ziegler and Nichols (1942); label the parameters

Km , Tm and xm . Then, n = 10-rL + 1 . Subsequently, xm is

estimated from the open loop step response, and Tar is estimated using "known methods" (Schaedel (1997)).

http://A2.ll

Bibliography

ABB (1996). Operating Guide for Commander 300/310, Section 7. ABB (2001). Instruction manual for 53SL6000 [Online]. Document: PN24991.pdf.

Available at www.abb.com [accessed 1 September 2004]. Abbas, A. (1997). A new set of controller tuning relations, ISA Transactions, 36, pp.

183-187. Aikman, A.R. (1950). The frequency response approach to automatic control problems,

Transactions of the Society of Instrument Technology, pp. 2-16. Alvarez-Ramirez, J., Morales, A. and Cervantes, I. (1998). Robust proportional-integral

control, Industrial Engineering Chemistry Research, 37, pp. 4740^747. Andersson, M. (2000). A MATLAB tool for rapid process identification and PID design,

M.Sc. thesis, Department of Automatic Control, Lund Institute of Technology, Lund, Sweden.

Ang, K.H., Chong, G. and Li, Y. (2005). PID control system analysis, design and technology, IEEE Transactions on Control Systems Technology, 13(4), pp. 559-576.

Argelaguet, R., Pons, M., Quevedo, J. and Aguilar, J. (2000). A new tuning of PID controllers based on LQR optimization, Preprints of Proceedings of PID '00: IF AC Workshop on Digital Control, pp. 303-308.

Arvanitis, K.G., Akritidis, C.B., Pasgianos, G.D. and Sigrimis, N.A. (2000). Controller tuning for second order dead-time fertigation mixing process, Proceedings of EurAgEng Conference on Agricultural Engineering, Paper No. OO-AE-011.

Arvanitis, K.G., Syrkos, G., Stellas, I.Z. and Sigrimis, N.A. (2003a). Controller tuning for integrating processes with time delay. Part I: IPDT processes and the pseudo-derivative feedback control configuration, Proceedings of ll'h Mediterranean Conference on Control and Automation, Paper No. T7-040.

Arvanitis, K.G., Syrkos, G., Stellas, I.Z. and Sigrimis, N.A. (2003b). Controller tuning for integrating processes with time delay. Part III: The case of first order plus integral plus dead-time processes, Proceedings of 11' Mediterranean Conference on Control and Automation, Paper No. T7-042.

Astrom, K.J. (1982). Ziegler-Nichols auto-tuner, Department of Automatic Control, Lund Institute of Technology, Lund, Sweden, Report TFRT-3167.

507

http://www.abb.com


Astrom, K.J. and Hagglund, T. (1984). Automatic tuning of simple regulators with specifications on phase and amplitude margins, Automatical 20, pp. 645-651.

Astrom, K.J. and Hagglund, T. (1988). Automatic tuning of PID controllers (Instrument Society of America, Research Triangle Park, North Carolina, U.S.A.).

Astrom, K.J. and Hagglund, T. (1995). PID Controllers: Theory, Design and Tuning (Instrument Society of America, Research Triangle Park, North Carolina, U.S.A., 2nd

Edition). Astrom, K.J. and Hagglund, T. (1996) in The Control Handbook, Editor W.S. Levine,

(CRC/IEEE Press), pp. 198-209. Astrom, K.J. and Hagglund, T. (2000). The future of PID control, Preprints of

Proceedings of PID '00: IF AC Workshop on Digital Control, pp. 19-30. Astrom, K.J. and Hagglund, T. (2004). Revisiting the Ziegler-Nichols step response

method for PID control, Journal of Process Control, 14, pp. 635-650. Atkinson, P. and Davey, R.L. (1968). A theoretical approach to the tuning of pneumatic

three-term controllers, Control, March, pp. 238-242. Auslander, D.M., Takahashi, Y. and Tomizuka, M. (1975). The next generation of single

loop controllers: hardware and algorithms for the discrete/decimal process controller, Transactions of the ASME: Journal of Dynamic Systems, Measurement and Control, September, pp. 280-282.

Bain, D.M. and Martin, G.D. (1983). Simple PID tuning and PID closed-loop simulation, Proceedings of the American Control Conference, pp. 338-342.

Bateson, N. (2002). Introduction to Control System Technology (Prentice-Hall Inc.). Bekker, J.E., Meckl, P.H. and Hittle, D.C. (1991). A tuning method for first-order

processes with PI controllers, ASHRAE Transactions, 97(2), pp. 19-23. Belanger, P.W. and Luyben, W.L. (1997). Design of low-frequency compensators for

improvement of plantwide regulatory performances, Industrial Engineering Chemistry Research, 36, pp. 5339-5347.

Bequette, B.W. (2003). Process Control: Modeling, Design and Simulation (Pearson Education, Inc.).

Bi, Q., Cai, W.-J., Lee, E.-L., Wang, Q.-G., Hang, C.-C. and Zhang, Y. (1999). Robust identification of first-order plus dead-time model from step response, Control Engineering Practice, 7, pp. 71-77.

Bi, Q., Cai, W.-J., Wang, Q.-G., Hang, C.-C, Lee, E.-L., Sun, Y., Liu, K.-D., Zhang, Y. and Zou, B. (2000). Advanced controller auto-tuning and its application in HVAC systems, Control Engineering Practice, 8, pp. 633-644.

Bialkowski, W.L. (1996) in The Control Handbook, Editor: W.S. Levine (CRC/IEEE Press), pp. 1219-1242.

Blickley, G.J. (1990). Modern control started with Ziegler-Nichols tuning, Control Engineering, 2 October, pp. 11-17.

Boe, E. and Chang, H.-C. (1988). Dynamics and tuning of systems with large delay,

Proceedings of the American Control Conference, pp. 1572-1578.

Bibliography 509

Bohl, A.H. and McAvoy, T.J. (1976a). Linear feedback vs. time optimal control. I. The servo problem, Industrial Engineering Chemistry Process Design and Development, 15, pp. 24-30.

Bohl, A.H. and McAvoy, T.J. (1976b). Linear feedback vs. time optimal control. II. The regulator problem, Industrial Engineering Chemistry Process Design and Development, 15, pp. 30-33.

Borresen, B.A. and Grindal, A. (1990). Controllability - back to basics, ASHRAE Transactions - Research, pp. 817-819.

Brambilla, A., Chen, S. and Scali, C. (1990). Robust tuning of conventional controllers, Hydrocarbon Processing, November, pp. 53-58.

Branica, I., Petrovic, I and Peric, N. (2002). Toolkit for PID dominant pole design, Proceedings of the 9th IEEE Conference on Electronics, Circuits and Systems, 3, pp. 1247-1250.

Bryant, G.F., Iskenderoglu, E.F. and McClure, C.H. (1973). Design of controllers for time delay systems, in Automation of Tandem Mills, Editor: G.F. Bryant (The Iron and Steel Institute, London), pp. 81-106.

Buckley, P.S. (1964). Techniques in Process Control (John Wiley and Sons). Bunzemeier, A. (1998). Ein vorschlag zur regelung integral wirkender prozesse mit

eingangsstorung, Automatisierungstechnische Praxis, 40, pp. 26-35 (in German). Calcev, G. and Gorez, R. (1995). Iterative techniques for PID controller tuning,

Proceedings of the 34,h Conference on Decision and Control (New Orleans, LA., U.S.A.), pp. 3209-3210.

Callendar, A., Hartree, D.R. and Porter, A. (1935/6). Time-lag in a control system, Philosophical Transactions of the Royal Society of London Series A, 235, pp. 415-444.

Camacho, O.E., Smith, C. and Chacon, E. (1997). Toward an implementation of sliding mode control to chemical processes, Proceedings of the IEEE International

'Symposium on Industrial Electronics (Guimaraes, Portugal), 3, pp. 1101-1105. Carr, D. (1986). AN-CNTL-13: PID control and controller tuning techniques [Online].

Available at http://www.eurotherm.com/training/tutorial/instrumentation/anl3 2.doc [accessed 3 September 2004].

Chandrashekar, R., Sree, R.P. and Chidambaram, M. (2002). Design of PI/PID controllers for unstable systems with time delay by synthesis method, Indian Chemical Engineer Section A, 44(2), pp. 82-88.

Chang, D.-M., Yu, C.-C. and Chien, I.-L. (1997). Identification and control of an overshoot lead-lag plant, Journal of the Chinese Institute of Chemical Engineers, 28, pp. 79-89.

Chao, Y.-C, Lin, H.-S., Guu, Y.-W. and Chang, Y.-H. (1989). Optimal tuning of a practical PID controller for second order processes with delay, Journal of the Chinese Institute of Chemical Engineers, 20, pp. 7-15.

Chau, P.C. (2002). Process control - a first course with MATLAB (Cambridge University Press).

http://www.eurotherm.com/training/tutorial/instrumentation/anl3


Chen, C.-L. (1989). A simple method for on-line identification and controller tuning AIChE Journal, 35, pp. 2037-2039.

Chen, C.-L., Hsu, S.-H. and Huang, H.-P. (1999a). Tuning PI/PD controllers based on gain/phase margins and maximum closed loop magnitude, Journal of the Chinese Institute of Chemical Engineers, 30, pp. 23-29.

Chen, C.-L., Huang, H.-P. and Hsieh, C.-T. (1999b). Tuning of PI/PID controllers based on specification of maximum closed-loop amplitude ratio, Journal of Chemical Engineering of Japan, 32, pp. 783-788.

Chen, C.-L., Huang, H.-P. and Lo, H.-C. (1997). Tuning of PID controllers for self-regulating processes, Journal of the Chinese Institute of Chemical Engineers, 28, pp. 313-327.

Chen, C.-L. and Yang, S.-F. (2000). PI tuning based on peak amplitude ratio, Preprints of Proceedings of PID '00: IF AC Workshop on digital control (Terrassa, Spain), pp. 195— 198.

Chen, C.-L., Yang, S.-F. and Wang, T.-C. (2001). Tuning of the blending PID controllers based on specification of maximum closed-loop amplitude ratio, Journal of the Chinese Institute of Chemical Engineers, 32, pp. 565-570.

Chen, D. and Seborg, D.E. (2002). PI/PID controller design based on direct synthesis and disturbance rejection, Industrial Engineering Chemistry Research, 41, pp. 4807-4822.

Chen, G. (1996). Conventional and fuzzy PID controllers: an overview, International Journal of Intelligent Control and Systems, 1, pp. 235-246.

Cheng, G.S. and Hung, J.C. (1985). A least-squares based self-tuning of PID controller, Proceedings of the IEEE South East Conference (Raleigh, North Carolina, U.S.A.), pp. 325-332.

Cheng, Y.-C. and Yu, C.-C. (2000). Nonlinear process control using multiple models: relay feedback approach, Industrial Engineering Chemistry Research, 39, pp. 420-431.

Chesmond, C.J. (1982). Control System Technology (Edward Arnold). Chidambaram, M. (1994). Design of PI controllers for integrator/dead-time processes,

Hungarian Journal of Industrial Chemistry, 22, pp. 37-39. Chidambaram, M. (1995a). Design formulae for PID controllers, Indian Chemical

Engineer Section A, 37(3), pp. 90-94. Chidambaram, M. (1995b). Design of PI and PID controllers for an unstable first-order

plus time delay system, Hungarian Journal of Industrial Chemistry, 23, pp. 123-127. Chidambaram, M. (1997). Control of unstable systems: a review, Journal of Energy,

Heat and Mass Transfer, 19, pp. 49-56. Chidambaram, M. (2000a). Set point weighted PI/PID controllers for stable systems,

Chemical Engineering Communications, 179, pp. 1-13. Chidambaram, M. (2000b). Set-point weighted PI/PID controllers for integrating plus

dead-time processes, Proceedings of the National Symposium on Intelligent Measurement and Control (Chennai, India), pp. 324-331.

Chidambaram, M. (2000c). Set-point weighted PI/PID controllers for unstable first-order plus time delay systems, Proceedings of the International Conference on

Bibliography 511

Communications, Control and Signal Processing (Bangalore, India), pp. 173-177. Chidambaram, M. and Kaylan, V.S. (1996). Robust control of unstable second order plus

time delay systems, Proceedings of the International Conference on Advances in Chemical Engineering (ICAChE-96) (Chennai, India: Allied Publishers, New Delhi, India), pp. 277-280.

Chidambaram, M. and Sree, R.P. (2003). A simple method of tuning PID controllers for integrator/dead-time processes, Computers and Chemical Engineering, 27, pp. 211-215.

Chidambaram, M., Sree, R.P. and Srinivas, M.N. (2005). Reply to the comments by Dr. A. Abbas on "A simple method of tuning PID controllers for stable and unstable FOPTD systems" [Comp. Chem. Engineering V28 (2004) 2201-2218], Computers and Chemical Engineering, 29, pp. 1155.

Chien, I.-L. (1988). IMC-PID controller design - an extension, Proceedings of the IFAC Adaptive Control of Chemical Processes Conference (Copenhagen, Denmark), pp. 147-152.

Chien, I.-L. and Fruehauf, P.S. (1990). Consider IMC tuning to improve controller performance, Chemical Engineering Progress, October, pp. 33—41.

Chien, I.-L., Huang, H.-P. and Yang, J.-C. (1999). A simple multiloop tuning method for PID controllers with no proportional kick, Industrial Engineering Chemistry Research 38, pp. 1456-1468.

Chien, K.L., Hrones, J.A. and Reswick, J.B. (1952). On the automatic control of generalised passive systems, Transactions oftheASME, February, pp. 175-185.

Chiu, K.C., Corripio, A.B. and Smith, C.L. (1973). Digital controller algorithms. Part III. Tuning PI and PID controllers, Instruments and Control Systems, December, pp. 4 1 -43.

Chun, D., Choi, J.Y. and Lee, J. (1999). Parallel compensation with a secondary measurement, Industrial Engineering Chemistry Research, 38, pp. 1575-1579.

Cluett, W.R. and Wang, L. (1997). New tuning rules for PID control, Pulp and Paper Canada, 3(6), pp. 52-55.

Cohen, G.H. and Coon, G.A. (1953). Theoretical considerations of retarded control, Transactions oftheASME, May, pp. 827-834.

Coon, G.A. (1956). How to find controller settings from process characteristics, Control Engineering, 3(May), pp. 66-76.

Coon, G.A. (1964). Control charts for proportional action, ISA Journal, ll(November), pp. 81-82.

Connell, B. (1996). Process instrumentation applications manual (McGraw-Hill). ControlSoft Inc. (2005). PID loop tuning pocket guide (Version 2.2, DS405-02/05)

[Online]. Available at http://www.controlsoftinc.com [accessed 30 June 2005]. Corripio, A.B. (1990). Tuning of industrial control systems (Instrument Society of

America, Research Triangle Park, North Carolina, U.S.A.). Cox, C.S., Arden, W.J.B. and Doonan, A.F. (1994). CAD Software facilities tuning of

traditional and predictive control strategies, Proceedings of the ISA Advances in

http://www.controlsoftinc.com


Instrumentation and Control Conference (Anaheim, U.S.A.), 49(2), pp. 241-250. Cox, C.S., Daniel, P.R. and Lowdon, A. (1997). Quicktune: a reliable automatic strategy

for determining PI and PPI controller parameters using a FOLPD model, Control Engineering Practice, 5, pp. 1463-1472.

Davydov, N.I., Idzon, O.M. and Simonova, O.V. (1995). Determining the parameters of PID-controller settings using the transient response of the controlled plant, Thermal Engineering (Russia), 42, pp. 801-807.

De Oliveira, R., Correa, R.G. and Kwong, W.H. (1995). An IMC-PID tuning procedure based on the integral squared error (ISE) criterion: a guide tour to understand its features, Proceedings of the IF AC Workshop on Control Education and Technology Transfer Issues (Curitiba, Brazil), pp. 87-91.

De Paor, A.M. (1993). A fiftieth anniversary celebration of the Ziegler-Nichols PID controller, InternationalJournal of Electrical Engineering Education, 30, pp. 303-316.

De Paor, A.M. and O' Malley, M. (1989). Controllers of Ziegler-Nichols type for unstable processes with time delay, InternationalJournal of Control, 49, pp. 1273-1284.

Deshpande, P.B. (1980). Process identification of open-loop unstable systems, AIChE Journal, 26, pp. 305-308.

Devanathan, R. (1991). An analysis of minimum integrated error solution with application to self-tuning controller, Journal of Electrical and Electronics Engineering, Australia, 11, pp. 172-177.

Dutton, K., Thompson, S. and Barraclough, B. (1997). The art of control engineering (Addison-Wesley Longman).

ECOSSE Team (1996a). The ECOSSE Control Hypercourse [Online]. Available at http://eweb.chemeng.ed.ac.uk/courses/control/course/map/controllers/correlations.html [accessed 24 May 2005].

ECOSSE Team (1996b). The ECOSSE Control Hypercourse [Online]. Available at http://eweb.chemeng.ed.ac.uk/courses/control/course/inap/controllers/damped.html [accessed 24 May 2005].

Edgar, T.F., Smith, C.L., Shinskey, F.G., Gassman, G.W., Schafbuch, P.J., McAvoy, T.J. and Seborg, D.E. (1997). Process Control, in Perrys Chemical Engineers Handbook (McGraw-Hill International Edition, Chemical Engineering Series), pp. 8-1 to 8-84.

Ender, D.B. (1993). Process control performance: not at good as you think, Control Engineering, September, pp. 180-190.

Ettaleb, L. and Roche, A. (2000). On-line tuning of mal-functioning control loops, Proceedings of Control Systems 2000 (Victoria, British Columbia, Canada), pp. 139— 144.

EZYtune software package [Online]. Available at http://www.unac.com.au/ezytune [accessed 28 June 2003].

Farrington, G.H. (1950). Communications on "The practical application of frequency response analysis to automatic process control", Proceedings of the Institution of Mechanical Engineers (London), 162, pp. 346-347.

Feeney, M. and O'Dwyer, A. (2002). Towards an expert system approach for PI

http://eweb.chemeng.ed.ac.uk/courses/control/course/map/controllers/correlations.html

http://eweb.chemeng.ed.ac.uk/courses/control/course/inap/controllers/damped.html

http://www.unac

http://com.au/ezytune

Bibliography 513

controller design of delayed processes, Proceedings of the Irish Signals and Systems Conference (University College Cork, Ireland), pp. 175-180.

Ferretti, G., Maffezzoni, C. and Scattolini, R. (1991). Recursive estimation of time delay in sampled systems, Automatica, 27, pp. 653-661.

Fertik, H.A. (1975). Tuning controllers for noisy processes, ISA Transactions, 14, pp. 292-304.

Fertik, H.A. and Sharpe, R. (1979). Optimizing the computer control of breakpoint chlorination, Advances in Instrumentation: Proceedings of the ISA Conference and Exhibit (Chicago, U.S.A.), 34(1), pp. 373-386.

Ford, R.L. (1953). The determination of the optimum process-controller settings and their confirmation by means of an electronic simulator, Proceedings of the IEE Part 2, 101 (April), pp. 141-155 and pp. 173-177.

Foxboro (1994). I/A/ Series EXACT Multivariate Control Product Specifications (PSS 21S-3A2 B3, Foxboro Company, U.S.A.).

Frank, P.M. and Lenz, R. (1969). Entwurf erweiterter Pl-regler fur totzeitstrecken mit verzogerung erster ordnung, Elektrotechnische Zeitschrift ETZ A, 90(3), pp. 57-63 (in German).

Friman, M. and Waller, K.V. (1997). A two channel relay for autotuning, Industrial Engineering Chemistry Research, 36, pp. 2662-2671.

Fruehauf, P.S., Chien, I.-L. and Lauritsen, M.D. (1993). Simplified IMC-PID tuning rules, Proceedings of ISA/93 Advances in Instrumentation and Control Conference (Chicago, U.S.A.), 48, pp. 1745-1766.

Gaikward, R. and Chidambaram, M. (2000). Design of PID controllers for unstable systems, Proceedings of the National Symposium on Intelligent Measurement and Control (Chennai, India), pp. 332-339.

Gallier, P.W. and Otto, R.E. (1968). Self-tuning computer adapts DDC algorithms, Instrumentation Technology, February, pp. 65-70.

Garcia, R.F. and Castelo, F.J.P. (2000). A complement to autotuning methods on PID controllers, Preprints of Proceedings of PID '00: IFAC Workshop on digital control (Terrassa, Spain), pp. 101-104.

Geng, G. and Geary, G.M. (1993). On performance and tuning of PID controllers in HVAC systems, Proceedings of the 2nd IEEE Conference on Control Applications (Vancouver, B.C., Canada), pp. 819-824.

Gerry, J.P. (1998). How to control processes with large dead times [Online]. Available at http://www.expertune.com/artdt.html [accessed 27 June 2005].

Gerry, J.P. (1999). Tuning process controllers starts in manual [Online]. Available at http://www.expertune.com/ArtlnTechMay99.html [accessed 27 June 2005].

Gerry, J.P. (2003). How to control a process with long dead time [Online]. Available at http://www.expertune.com/learncast.html [accessed 27 June 2005].

Gerry, J.P. and Hansen, P.D. (1987). Choosing the right controller, Chemical Engineering, May 25, pp. 65-68.

http://www.expertune.com/artdt.html

http://www.expertune.com/ArtlnTechMay99.html

http://www.expertune.com/learncast.html


Gong, X., Gao, J. and Zhou, C. (1996). Extension of IMC tuning to improve controller performance, Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, pp. 1770-1775.

Goodwin, G.C., Graebe, S.F. and Salgado, M.E. (2001). Control system design (Prentice Hall).

Gorecki, H., Fuska, S., Grabowski, P. and Korytowski, A. (1989). Analysis and synthesis of time delay systems (John Wiley and Sons, New York).

Gorez, R. (2003). New design relations for 2-DOF PID-like control systems, Automatica, 39, pp. 901-908.

Gorez, R. and Klan, P. (2000). Nonmodel-based explicit design relations for PID controllers, Preprints of Proceedings of PID '00: IFAC Workshop on digital control (Terrassa, Spain), pp. 141-148.

Haalman, A. (1965). Adjusting controllers for a deadtime process, Control Engineering, July, pp. 71-73.

Haeri, M. (2002). Tuning rules for the PID controller using a DMC strategy, Asian Journal of Control, 4, pp. 410^117.

Haeri, M. (2005). PI design based on DMC strategy, Transactions of Institute of Measurement and Control, 27, pp. 21-36.

Hagglund, T. and Astrom, K.J. (1991). Industrial adaptive controllers based on frequency response techniques, Automatica, 27, pp. 599-609.

Hagglund, T. and Astrom, K.J. (2002). Revisiting the Ziegler-Nichols tuning rules for PI control, Asian Journal of Control, 4, pp. 364-380.

Hang, C.C. and Astrom, K.J. (1988a). Refinements of the Ziegler-Nichols tuning formuale for PID auto-tuners, Proceedings of the ISA/88 International Conference and Exhibition. Advances in Instrumentation, 43, pp. 1021-1030.

Hang, C.C. and Astrom, K.J. (1988b). Practical aspects of PID auto-tuners based on relay feedback, Proceedings of the IFAC Adaptive control of Chemical Processes Conference, (Copenhagen, Denmark), pp. 153-158.

Hang, C.C, Astrom, K.J. and Ho, W.K. (1991). Refinements of the Ziegler-Nichols tuning formula, IEE Proceedings, Part D, 138, pp. 111-118.

Hang, C.C. and Cao, L. (1996). Improvement of transient response by means of variable set-point weighting, IEEE Transactions on Industrial Electronics, 43, pp. 477-^184.

Hang, C.C, Ho, W.K. and Cao, L.S. (1993a). A comparison of two design methods for PID controllers, Proceedings of the ISA/93 Advances in Instrumentation and Control Conference (Chicago, U.S.A.), 48, pp. 959-967.

Hang, C.C, Ho, W.H. and Cao, L.S. (1994). A comparison of two design methods for PID controllers, ISA Transactions, 33, pp. 147-151.

Hang, C.C, Lee, T.H. and Ho, W.K. (1993b). Adaptive Control (Instrument Society of America, Research Triangle Park, North Carolina, U.S.A.).

Hansen, P.D. (1998). Controller structure and tuning for unmeasured load disturbance, Proceedings of the American Control Conference (Philadelphia, Pennsylvania, U.S.A.), l,pp. 131-136.

Bibliography 515

Hansen, P.D. (2000). Robust adaptive PID controller tuning for unmeasured load rejection, Preprints of Proceedings of PID '00: IF AC Workshop on digital control, (Terrassa, Spain), pp. 487-^194.

Harriott, P. (1964). Process Control (McGraw-Hill, New York). Harris, T.J. and Tyreus, B.D. (1987). Comments on Internal Model Control. 4. PID

controller design, Industrial Engineering Chemistry Research, 26, pp. 2161-2162. Harrold, D. (1999). Process controller tuning guidelines [Online]. Available at

http://www.manufacturing.net/ctl/index.asp?lavout=articlePrint&articleID=CAl 88338 [accessed 2 May 2003].

Hartree, D.R., Porter, A., Callender, A. and Stevenson, A.B. (1937). Time-lag in a control system - II, Proceedings of the Royal Society of London, 161(A), pp. 460-476.

Hassan, G.A. (1993). Computer-aided tuning of analog and digital controllers, Control and Computers, 21(1), pp. 1-6.

Hay, J. (1998). Regeltechniek 1 (Die Keure n.v., Brugge, Belgium; in Flemish). Hazebroek, P. and Van der Waerden, B.L. (1950). The optimum tuning of regulators,

Transactions oftheASME, April, pp. 317-322. Heck, E. (1969). Stetige regler an regelstrecken - bestehend aus verzogerungsgliedern

erster ordnung, Heizung-Lueftung-Haustechnik, 20(9), September, pp. 333-337 (in German).

Henry, J. and Schaedel, H.M. (2005). International co-operation in control engineering education using online experiments, European Journal of Engineering Education, 30(2), pp. 265-274.

Hiroi, K. and Terauchi, Y. (1986). Two degrees of freedom algorithm, Proceedings of the ISA/86 International Conference and Exhibition. Advances in Instrumentation, pp. 789-796.

Ho, W.K., Gan, O.P., Tay, E.B. and Ang, E.L. (1996). Performance and gain and phase margins of well-known PID tuning formulas, IEEE Transactions on Control Systems Technology, 4, pp. 473^477.

Ho, W.K., Hang, C.C. and Cao, L.S. (1995a). Tuning of PID controllers based on gain and phase margin specifications, Automatica, 31, pp. 497-502.

Ho, W.K., Hang, C.C. and Zhou, J.H. (1995b). Performance and gain and phase margins of well-known PI tuning formulas, IEEE Transactions on Control Systems Technology, 3, pp. 245-248.

Ho, W.K., Hang, C.C. and Zhou, J. (1997). Self-tuning PID control of a plant with under-damped response with specifications on gain and phase margins, IEEE Transactions on Control Systems Technology, 5, pp. 446—452.

Ho, W.K., Hang, C.C, Zhou, J.H. and Yip, C.K. (1994). Adaptive PID control of a process with underdamped response, Proceedings of the Asian Control Conference (Tokyo, Japan), pp. 335-338.

Ho, W.K., Lee, T.H., Han, H.P. and Hong, Y. (2001). Self-tuning IMC-PID control with interval gain and phase margins assignment, IEEE Transactions on Control Systems Technology, 9, pp. 535-541.

http://www.manufacturing.net/ctl/index.asp?lavout=articlePrint&articleID=CAl


Ho, W.K., Lim, K.W., Hang, C.C. and Ni, L.Y. (1999). Getting more phase margin and performance out of PID controllers, Automatica, 35, pp. 1579-1585.

Ho, W.K., Lim, K.W. and Xu, W. (1998). Optimal gain and phase margin tuning for PID controllers, Automatica, 34, pp. 1009-1014.

Ho, W.K. and Xu, W. (1998). PID tuning for unstable processes based on gain and phase-margin specifications, IEE Proceedings - Control Theory and Applications, 145, pp. 392-396.

Horn, I.G., Arulandu, J.R., Gombas, C.J., VanAntwerp, J.G. and Braatz, R.D. (1996). Improved filter design in internal model control, Industrial Engineering Chemistry Research, 35, pp. 3437-3441.

Hougen, J.O. (1979). Measurement and Control Applications (Instrument Society of America, Research Triangle Park, North Carolina, U.S.A.).

Hougen, J.O. (1988). A software program for process controller parameter selection, Proceedings of the ISA/88 International Conference and Exhibition. Advances in Instrumentation, 43(1), pp. 441-456.

Huang, C.-T. (2002). Private communication to Mr. T. Kealy, DIT. Huang, C.-T., Chou, C.-J. and Wang, J.-L. (1996). Tuning of PID controllers based on

the second order model by calculation, Journal of the Chinese Institute of Chemical Engineers, 27, pp. 107-120.

Huang, C.-T. and Lin, Y.-S. (1995). Tuning PID controller for open-loop unstable processes with time delay, Chemical Engineering Communications, 133, pp. 11-30.

Huang, H.-P. and Chao, Y.-C. (1982). Optimal tuning of a practical digital PID controller, Chemical Engineering Communications, 18, pp. 51-61.

Huang, H.-P. and Chen C.-C. (1997). Control-system synthesis for open-loop unstable process with time delay, IEE Proceedings - Control Theory and Applications, 144, pp. 334-346.

Huang, H.-P. and Chen, C.-C. (1999). Auto-tuning of PID controllers for second-order unstable process having dead time, Journal of Chemical Engineering of Japan, 32, pp. 486-497.

Huang, H.-P., Chien, I.-L., Lee, Y.-C. and Wang, G.-B. (1998). A simple method for tuning cascade control systems, Chemical Engineering Communications, 165, pp. 89-121.

Huang, H.-P. and Jeng, J.-C. (2002). Monitoring and assessment of control performance for single loop systems, Industrial Engineering Chemistry Research, 41, pp. 1297-1309.

Huang, H.-P. and Jeng, J.-C. (2003). Identification for monitoring and autotuning of PID controllers, Journal of Chemical Engineering of Japan, 36, pp. 284-296.

Huang, H.-P., Jeng, J.-C. and Luo, K.-Y. (2005). Auto-tune system using single-run relay feedback test and model-based controller design, Journal of Process Control, 15, pp. 713-727.

Bibliography 517

Huang, H.-P., Lee, M.-W. and Chen, C.-L. (2000). Inverse-based design for a modified PID controller, Journal of the Chinese Institute of Chemical Engineers, 31, pp. 225-236.

Huang, H.-P., Roan, M.-L. and Jeng, J.-C. (2002). On-line adaptive tuning for PID controllers, IEE Proceedings - Control Theory and Applications, 149, pp. 60-67.

Huba, M. and Zakova, K. (2003). Contribution to the theoretical analysis of the Ziegler-Nichols method, Journal of Electrical Engineering, Slovakia, 54(7-8), pp. 188-194.

Hwang, S.-H. (1993). Adaptive dominant pole design of PID controllers based on a single closed-loop test, Chemical Engineering Communications, 124, pp. 131-152.

Hwang, S.-H. (1995). Closed-loop automatic tuning of single-input-single-output systems, Industrial Engineering Chemistry Research, 34, pp. 2406-2417.

Hwang, S.-H. and Chang, H.-C. (1987). A theoretical examination of closed-loop properties and tuning methods of single-loop PI controllers, Chemical Engineering Science, 42, pp. 2395-2415.

Hwang, S.-H. and Fang, S.-M. (1995). Closed-loop tuning method based on dominant pole placement, Chemical Engineering Communications, 136, pp. 45-66.

Isaksson, A.J. and Graebe, S.F. (1999). Analytical PID parameter expressions for higher order systems, Automatica, 35, pp. 1121-1130.

Isermann, R. (1989). Digital Control Systems Volume 1. Fundamentals, Deterministic Control (2nd Revised Edition, Springer-Verlag).

ISMC (1999). RAPID: Robust Advanced PID Control Manual (Intelligent System Modeling and Control nv, Belgium).

Jahanmiri, A. and Fallahi, H.R. (1997). New methods for process identification and design of feedback controller, Transactions of the Institute of Chemical Engineers, 75(A), pp. 519-522.

Jhunjhunwala, M. and Chidambaram, M. (2001). PID controller tuning for unstable systems by optimization method, Chemical Engineering Communications, 185, pp. 91-113.

Jones, K.O., Williams, D. and Montgomery, P. A. (1997). On-line application of an auto-tuning PID controller for dissolved oxygen concentration in a fermentation process, Transactions of the Institute of Measurement and Control, 19, pp. 253-262.

Juang, W.-S. and Wang, F.-S. (1995). Design of PID controller by concept of Dahlin's Law, Journal of the Chinese Institute of Chemical Engineers, 26, pp. 133-136.

Kamimura, K., Yamada, A., Matsuba, T., Kimbara, A., Kurosu, S. and Kasahara, M. (1994). CAT (Computer-aided tuning) software for PID controllers, ASHRAE Transactions: Research, 100(1), pp. 180-190.

Karaboga, D. and Kalinli, A. (1996). Tuning PID controller parameters using Tabu search algorithm, Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, pp. 134-136.

Kasahara, M., Matsuba, T., Kuzuu, Y., Yamazaki, T., Hashimoto, Y., Kamimura, K. and Kurosu, S. (1999). Design and tuning of robust PID controller for HVAC systems, ASHRAE Transactions: Research, 105(2), pp. 154-166.


Kasahara, M., Yamazaki, T., Kuzuu, Y., Hashimoto, Y„ Kamimura, K., Matsuba, T. and Kurosu, S. (2001). Stability analysis and tuning of PID controller in VAV systems, ASHRAE Transactions: Research, 107, pp. 285-296, 2001.

Kasahara, M, Matsuba, T., Murasawa, I., Hashimoto, Y., Kamimura, K., Kimbara, A. and Kurosu, S. (1997). A tuning method of two degrees of freedom PID controller, ASHRAE Transactions: Research, 103, pp. 278-289.

Kaya, A. and Scheib, T.J. (1988). Tuning of PID controls of different structures, Control

Engineering, July, pp. 62-65. Kaya, I. (2003). A PI-PD controller design for control of unstable and integrating

processes, ISA Transactions, 42, pp. 111-121. Kaya, I. and Atherton, D.P. (1999). A PI-PD controller design for integrating processes,

Proceedings of the American Control Conference (San Diego, U.S.A.), pp. 258-262. Kaya, I., Tan, N. and Atherton, D.P. (2003). A simple procedure for improving

performance of PID controllers, Proceedings of the IEEE Conference on Control Applications (U.S.A.), pp. 882-885.

Keane, M.A., Yu, J. and Koza, J.R. (2003). Automatic synthesis of both the topology and tuning of a common parameterized controller for two families of plants using genetic programming, Proceedings of the Genetic and Evolutionary Computation Conference (Las Vegas, Nevada, U.S.A.), pp. 496-504.

Keviczky, L. and Csaki, F. (1973). Design of control systems with dead time in the time domain, Acta Technica Academiae Scientiarum Hungaricae, 74(1-2), pp. 63-84.

Khan, B.Z. and Lehman, B. (1996). Setpoint PI controllers for systems with large normalised dead time, IEEE Transactions on Control Systems Technology, 4, pp. 459-466.

Khodabakhshian, A. and Golbon, N. (2004). Unified PID design for load frequency control, Proceedings of the IEEE International Conference on Control Applications (Taipei, Taiwan), pp. 1627-1632.

Kim, I.-H., Fok, S., Fregene, K., Lee, D.-H., Oh, T.-S. and Wang, D.W.L. (2004). Neural-network bases system identification and controller synthesis for an industrial sewing machine, International Journal of Control, Automation and Systems, 2(1), pp. 83-91.

Kinney, T.B. (1983). Tuning process controllers, Chemical Engineering, September 19, pp. 67-72.

Klein, M., Walter, H. and Pandit, M. (1992). Digitaler Pl-regler: neue einstellregeln mit hilfe der streckensprungantwort (Digital Pi-control: new tuning rules based on step response identification), at-Automatisierungstechnik, 40(8), pp. 291-299 (in German).

Koivo, H.N. and Tanttu, J.T. (1991). Tuning of PID controllers: survey of SISO and MIMO techniques, Proceedings of the IF AC Intelligent Tuning and Adaptive Control Symposium (Singapore), pp. 75-80.

Kookos, I.K., Lygeros, A.I. and Arvanitis, K.G. (1999). On-line PI controller tuning for integrator/dead time processes, European Journal of Control, 5, pp. 19-31.

Bibliography 519

Kraus, T.W., Pattern-recognizing self-tuning controller, U.S. Patent Number 4,602,326 (1986).

Kristiansson, B. (2003). PID controllers design and evaluation, Ph.D. thesis, Chalmers University of Technology, Sweden.

Kristiansson, B. and Lennartson, B. (1998a). Robust design of PID controllers including auto-tuning rules, Proceedings of the American Control Conference, 5, pp. 3131-3132.

Kristiansson, B. and Lennartson, B. (1998b). Optimal PID controllers for unstable and resonant plants, Proceedings of the Conference on Decision and Control (Tampa, Florida, U.S.A.), pp. 4380-4381.

Kristiansson, B. and Lennartson, B. (2000). Near optimal tuning rules for PI and PID controllers, Preprints of Proceedings of PID '00: IFAC Workshop on digital control (Terrassa, Spain), pp. 369-374.

Kristiansson, B. and Lennartson, B. (2002). Robust and optimal tuning of PI and PID controllers, IEE Proceedings - Control Theory and Applications, 149, pp. 17-25.

Kristiansson, B., Lennartson, B. and Fransson, C.-M. (2000). From PI to H„ control in a unified framework, Proceedings of the 39th IEEE Conference on Decision and Control (Sydney, Australia), pp. 2740-2745.

Kuhn, U. (1995). Ein praxisnahe einstellregel fur PID-regler: die T-summen-regel, atp -Automatisierungstechnische Praxis, 37(5), pp. 10-16 (in German).

Kwak, H.J., Sung, S.W. and Lee, I.-B. (2000). Stabilizabihty conditions and controller design for unstable processes, Transactions of the Institute of Chemical Engineers, 78(A), pp. 549-556.

Landau, I.D. and Voda, A. (1992). An analytical method for the auto-calibration of PID controllers, Proceedings of the 31st Conference on Decision and Control (Tucson, Arizona, U.S.A.), pp. 3237-3242.

Lee, J. and Edgar, T.F. (2002). Improved PI controller with delayed or filtered integral mode, AIChE Journal, 48, pp. 2844-2850.

Lee, J. and Sung, S.W. (1993). Comparison of two identification methods for PID controller tuning, AIChE Journal, 39, pp. 695-697.

Lee, W.S. and Shi, J. (2002). Modified IMC-PID controllers and generalised PID controllers for first-order plus dead-time processes, Proceedings of the Seventh International Conference on Control, Automation, Robotics and Vision (Singapore), pp. 898-903.

Lee, Y., Lee, J. and Park, S. (2000). PID tuning for integrating and unstable processes with time delay, Chemical Engineering Science, 55, pp. 3481-3493.

Lee, Y., Park, S., Lee, M. and Brosilow, C. (1998). PID controller tuning to obtain desired closed-loop responses for SI/SO systems, AIChE Journal, 44, pp. 106-115.

Lelic, M. and Gajic, Z. (2000). A reference guide to PID controllers in the nineties, Preprints of Proceedings of PID '00: IFAC Workshop on digital control (Terrassa, Spain), pp. 73-82.


Lennartson, B. and Kristiansson, B. (1997). Pass band and high frequency robustness for PID control, Proceedings of the 36th IEEE Conference on Decision and Control, (San Diego, California, U.S.A.), pp. 2666-2671.

Leonard, F. (1994). Optimum PIDS controllers, an alternative for unstable delayed systems, Proceedings of the IEEE Conference on Control Applications (Glasgow, U.K.), pp. 1207-1210.

Leva, A. (1993). PID autotuning algorithm based on relay feedback, IEE Proceedings, PartD, 140, pp. 328-338.

Leva, A. (2001). Model-based tuning: the very basics and some useful techniques, Journal A: Benelux quarterly journal on Automatic Control, 42(3).

Leva, A. (2005). Model-based proportional-integral-derivative autotuning improved with relay feedback identification, IEE Proceedings - Control Theory and Applications, 152, pp. 247-256.

Leva, A. and Colombo, A.M. (2000). Estimating model mismatch overbounds for the robust autotuning of industrial regulators, Automatica, 26, pp. 1855-1861.

Leva, A. and Colombo, A.M. (2001). Implementation of a robust PID autotuner in a control design environment, Transactions of the Institute of Measurement and Control, 23, pp. 1-20.

Leva, A. and Colombo, A.M. (2004). On the IMC-based synthesis of the feedback block of ISA-PID regulators, Transactions of the Institute of Measurement and Control, 26, pp. 417^140.

Leva, A., Cox, C. and Ruano, A. (2003). IFAC Professional Brief - Hands-on PID autotuning: a guide to better utilization [Online]. Available at http://www.ifac-control.org/publications/pbriefs/PB Final LevaCoxRuano.pdf [Accessed 28 June 2005].

Leva, A., Maffezzoni, C. and Scattolini, R. (1994). Self-tuning PI-PID regulators for stable systems with varying delay, Automatica, 30, pp. 1171-1183.

Li, Y.-N., Zhang, W.-D. and Liu, T. (2004). New design method of PID controller for inverse response processes with time delay, Shanghai Jiaotong Daxue Xuebao/Journal of Shanghai Jiaotong University, 38(4), pp. 506-509 (in Chinese).

Li, Z., Su., X. and Lin, P. (1994). A practical algorithm for PID auto-tuning, Advances in Modelling and Analysis C (ASME Press), 40(2), pp. 17-27.

Lim, CM., Tan, W.C. and Lee, T.S. (1985). Tuning of PID controllers for first and second-order lag processes with dead time, International Journal of Electrical Engineering Education, 22, pp. 345-353.

Liptak, B. (1970, Editor), Instrument Engineers Handbook Volume II: Process Control (Chilton Book Co., Philadelphia, U.S.A.)

Liptak, B. (2001). Controller tuning II: Problems and Methods [Online]. Available at http://www.controlmagazine.com [accessed 21 November 2003].

Liu, T., Gu, D. and Zhang, W. (2003). A H infinity design method of PID controller for

second-order processes with integrator and time delay, Proceedings of the 42nd IEEE Conference on Decision and Control (Maui, Hawaii, U.S.A.), pp. 6044-6049.

http://www.ifac-

http://control.org/publications/pbriefs/PB

http://www.controlmagazine.com

Bibliography 521

Lloyd, S.G. (1994). Tuning arrangements for turning the control parameters of a controller, U.S. Patent Number 5,283,729.

Lopez, A.M., Smith, C.L. and Murrill, P.W. (1969). An advanced tuning method, British Chemical Engineering, 14, pp. 1553-1555.

Loron, L. (1997). Tuning of PID controllers by the non-symmetrical optimum method, Automatica, 33, pp. 103-107.

Luo, K.-N., Kuo, C.-Y. and Sheu, L.-T. (1996). A novel method for fuzzy self-tuning PID controllers, Proceedings of the Asian Fuzzy Systems Symposium, pp. 194-199.

Luyben, W.L. (1996). Tuning proportional-integral-derivative controllers for integrator/deadtime processes, Industrial Engineering Chemistry Research, 35, pp. 3480-3483.

Luyben, W.L. (1998). Tuning temperature controllers on openloop unstable reactors, Industrial Engineering Chemistry Research, 37, pp. 4322^1331.

Luyben, W.L. (2000). Tuning proportional-integral controllers for processes with both inverse response and deadtime, Industrial Engineering Chemistry Research, 39, pp. 973-976.

Luyben, W.L. (2001). Effect of derivative algorithm and tuning selection on the PID control of dead-time processes, Industrial Engineering Chemistry Research, 40, pp. 3605-3611.

Luyben, W.L. and Luyben, M.L. (1997). Essentials of process control (McGraw-Hill International Edition, Singapore).

MacLennan, G.D.S. (1950). Communications on "The practical application of frequency response analysis to automatic process control", Proceedings of the Institution of Mechanical Engineers (London), 162, pp. 347-348.

Maffezzoni, C. and Rocco, P. (1997), Robust tuning of PID regulators based on step-response identification, European Journal of Control, 3, pp. 125-136.

Majhi, S. and Atherton, D.P. (2000). Online tuning of controllers for an unstable FOPDT process, IEE Proceedings - Control Theory and Applications, 147, pp. 421-427.

Majhi, S. and Litz, L. (2003). On-line tuning of PID controllers, Proceedings of the American Control Conference (Denver, Colorado, U.S.A.), pp. 5003-5004.

Majhi, S. and Mahanta, C. (2001). Tuning of controllers for integrating time delay processes, Proceedings of the IEEE Region 10 Int. Conference on Electrical and Electronic Technology (TENCON 2001), 1, pp. 317-320.

Mann, G.K.I., Hu, B.-G. and Gosine, R.G. (2001). Time-domain based design and analysis of new PID tuning rules, IEE Proceedings - Control Theory and Applications, 148, pp. 251-261.

Mantz, R.J. and Tacconi, E.J. (1989). Complementary rules to Ziegler and Nichols' rules for a regulating and tracking controller, International Journal of Control, 49, pp. 1465-1471.

Marchetti, G. and Scali, C. (2000). Use of modified relay techniques for the design of model-based controllers for chemical processes, Industrial Engineering Chemistry Research, 39, pp. 3325-3334.


Marlin, T.E. (1995). Process Control (Mc-Graw Hill Inc., New York). Matausek, M.R. and Kvascev, G.S. (2003), A unified step response procedure for

autotuning of PI controller and Smith predictor for stable processes, Journal of Process Control, 13, pp. 787-800.

Matsuba, T., Kasahara, M., Murasawa, I., Hashimoto, Y., Kamimura, K., Kimbara, A. and Kurosu, S. (1998). Stability limit of room air temperature of a VAV system, ASHRAE Transactions, 104(2), pp. 257-265.

McAnany, D.E. (1993). A pole placement technique for optimum PID control parameters, Proceedings of the ISA Advances in Instrumentation and Control Conference (Chicago, U.S.A.), 48, pp. 1775-1782.

McAvoy, T.J. and Johnson, E.F. (1967). Quality of control problem for dead-time plants, Industrial and Engineering Chemistry Process Design and Development, 6, pp. 440-446.

McMillan, G.K. (1984). Control loop performance, Proceedings of the ISA/84 International Conference and Exhibition (Houston, Texas, U.S.A.), 39, pp. 589-603.

McMillan, G.K. (1994). Tuning and control loop performance - a practitioner's guide (Instrument Society of America, Research Triangle Park, North Carolina, U.S.A., 3rd

Edition). Mizutani, M. and Hiroi, K. (1991). New two degrees of freedom PID algorithm with 1st

lag mean (super 2dof PID algorithm), Proceedings of the ISA Advances in Instrumentation and Control, 46(2), pp. 1125-1132.

Mollenkamp, R.A., Smith, C.L. and Corripio, A.B. (1973). Using models to tune industrial controllers, Instruments and Control Systems, September, pp. 46-47.

Morari, M. and Zafiriou, E. (1989). Robust process control (Prentice-Hall Inc., Englewood Cliffs, New Jersey, U.S.A.).

Morilla, F., Gonzalez, A. and Duro, N. (2000). Auto-tuning PID controllers in terms of relative damping, Preprints of Proceedings of PID '00: IF AC Workshop on digital control (Terrassa, Spain), pp. 161-166.

Moros, R. (1999). Strecke mit Ausgleich hoherer Ordung [Online]. Available at http://techni.tachemie.uni-leipzig.de/reg/regeintn.html [accessed 28 January 2005] (in German).

Murrill, P.W. (1967). Automatic control of processes (International Textbook Co.). NI Labview (2001). PID control toolkit user manual [Online]. Available at

http://www.ni.com/pdf/manuals/322192a.pdf [accessed 3 September 2004]. Nishikawa, Y., Sannomiya, N., Ohta, T. and Tanaka, H. (1984). A method for auto

tuning of PID control parameters, Automatica, 20, pp. 321-332. Normey-Rico, J.E., Alcala, I., Gomez-Ortega, J. and Camacho, E.F. (2000). Robust PID

tuning application to a mobile robot path tracking problem, Preprints of Proceedings of PID '00: IFAC Workshop on digital control (Terrassa, Spain), pp. 648-653.

Normey-Rico, J.E., Alcala, I., Gomez-Ortega, J. and Camacho, E.F. (2001). Mobile robot path tracking using a robust PID controller, Control Engineering Practice, 9, pp. 1209-1214.

http://techni.tachemie.uni-leipzig.de/reg/regeintn.html

http://www.ni.com/pdf/manuals/322192a.pdf

Bibliography 523

O'Connor, G.E. and Denn, M.M. (1972). Three mode control as an optimal control, Chemical Engineering Science, 27, pp. 121-127.

O'Dwyer, A. (1998). Performance and robustness in the compensation of FOLPD processes with PI and PID controllers, Proceedings of the Irish Signals and Systems Conference (Dublin Institute of Technology, Ireland), pp. 227-234.

O'Dwyer, A. (2000). Performance and robustness of PI and PID tuning rules, Technical Report AOD-00-12 (Dublin Institute of Technology, Ireland).

O'Dwyer, A. (2001a). PI and PID controller tuning rule design for processes with delay, to achieve constant gain and phase margins for all values of delay, Proceedings of the Irish Signals and Systems Conference (N.U.I. Maynooth, Ireland), pp. 96-100.

O'Dwyer, A. (2001b). Series PID controller tuning rules, Technical Report AOD-01-13 (Dublin Institute of Technology, Ireland).

Ogawa, M. and Katayama, T. (2001). A robust tuning method for I-PD controller incorporating a constraint on manipulated variable, Transactions of the Society of Instrument and Control Engineers, Japan, E-l(l), pp. 265-273.

Ogawa, S. (1995). PI controller tuning for robust performance, Proceedings of the 4th

IEEE Conference on Control Applications, pp. 101-106. Oppelt, W. (1951). Einige Faustformeln zur Einstellung von Regelvorgangen, Chemie

Ingenieur Technik, 23(8), pp. 190-193 (in German). Ou, W.-H. and Chen, Y.-W. (1995). Adaptive actual PID control with an adjustable

identification interval, Chemical Engineering Communications, 134, pp. 93-105. Oubrahim, R. and Leonard, F. (1998). PID tuning by a composed structure, Proceedings

of the UKACC International Conference on Control (Swansea, Wales, U.K.), 2, pp. 1333-1338.

Ozawa, K., Noda, Y., Yamazaki, T., Kamimura, K. and Kurosu, S. (2003). A tuning method for PID controller using optimisation subject to constraints on control input, ASHRAE Transactions: Research, 109, pp. 79-88.

Panda, R.C., Yu, C.-C. and Huang, H.-P. (2004). PID tuning rules for SOPDT systems: review and some new results, ISA Transactions, 43, pp. 283-295.

Paraskevopoulos, P.N., Pasgianos, G.D. and Arvanitis, K.G. (2004). New tuning and identification methods for unstable first order plus dead-time processes based on pseudoderivative feedback control, IEEE Transactions on Control Systems Technology, 12, pp. 455^164.

Park, J.H., Sung, S.W. and Lee, I.-B. (1998). An enhanced PID control strategy for unstable processes, Automatica, 34, pp. 751-756.

Parr, E.A. (1989). Industrial Control Handbook, Vol. 3 (BSP Professional Books, Oxford, U.K.).

Pecharroman, R.R. (2000). Private communication (9 May). Pecharroman, R.R. and Pagola, F.L. (1999). Improved identification for PID controllers

auto-tuning, Proceedings of the 5th European Control Conference (Karlsruhe, Germany), Paper F453.


Pecharroman, R.R. and Pagola, F.L. (2000). Control design for PID controllers auto-tuning based on improved identification, Preprints of Proceedings of PID '00: IF AC Workshop on digital control (Terrassa, Spain), pp. 89-94.

Pemberton, T.J. (1972a). PID: The logical control algorithm, Control Engineering, 19(5), pp. 66-67.

Pemberton, T.J. (1972b). PID: The logical control algorithm II, Control Engineering, 19(7), pp. 61-63.

Peng, H. and Wu, S.-C. (2000). Journal of Cent. South Univ. Technol, (China), 7, pp. 165-169.

Penner, A. (1988). Tuning rules for a PI controller, Proceedings of the ISA International Conference and Exhibition (Houston, Texas, U.S.A.), 43(3), pp. 1037-1051.

Peric, N., Petrovic, I. and Branica, I. (1997). A method of PID controller autotuning, Proceedings of the IFAC Conference on Control of Industrial Systems (Belfort, France), pp. 597-602.

Pessen, D.W. (1953). Optimum three-mode controller settings for automatic start-up, Transactions of the ASME, July, pp. 843-849.

Pessen, D.W. (1954). How to "tune in" a three-mode controller, Instrumentation, 7(3), pp. 29-32.

Pessen, D.W. (1994). A new look at PID-controller tuning, Transactions of the ASME. Journal of Dynamic Systems, Measurement and Control, 116, pp. 553-557.

Pettit, J.W. and Carr, D.M. (1987). Self-tuning controller, US Patent No. 4,669,040. Pinnella, M.J., Wechselberger, E., Hittle, D.C. and Pedersen, CO. (1986). Self-tuning

digital integral control, ASHRAE Transactions, 92(2B), pp. 202-209. Polonyi, M.J.G. (1989). PID controller tuning using standard form optimization, Control

Engineering, March, pp. 102-106. Pomerleau, A. and Poulin, E. (2004). Manipulated variable based PI tuning and detection

of poor settings: an industrial experience, ISA Transactions, 43, pp. 445^157. Potocnik, B., Skrjanc, I., Matko, D. and Zupancic, B. (2001). Samonastavljivi PI

regulator na podlagi metode z relejskim preskusom, Elektrotehniski vestnik (Electrotechnical Review) Ljubljana Slovenija, 68, pp. 115-122 (in Slovenian).

Poulin, E., Pomerleau, A. Desbiens, A. and Hodouin, D. (1996). Development and evaluation of an auto-tuning and adaptive PID controller, Automatica, 32, pp. 71-82.

Poulin, E. and Pomerleau. A. (1996). PID tuning for integrating and unstable processes, IEE Proceedings - Control Theory and Applications, 143, pp. 429^135.

Poulin, E. and Pomerleau. A. (1997). Unified PID design method based on a maximum peak resonance specification, IEE Proceedings - Control Theory and Applications, 144, pp. 566-574.

Poulin, E. and Pomerleau, A. (1999). PI settings for integrating processes based on ultimate cycle information, IEEE Transactions on Control Systems Technology, 7, pp. 509-511.

Bibliography 525

Pramod, S. and Chidambaram, M. (2000). Closed loop identification of transfer function model for unstable bioreactors for tuning PID controllers, Bioprocess and Biosystems Engineering, 22, pp. 185-188.

Prashanti, G. and Chidambaram, M. (2000). Set-point weighted PID controllers for unstable systems, Journal of the Franklin Institute, 337, pp. 201-215.

Pulkkinen, J., Koivo, H.N. and Makela, K. (1993). Tuning of a robust PID controller -application to heating process in extruder, Proceedings of the 2nd IEEE Conference on Control Applications (Vancouver, B.C., Canada), pp. 811-816.

Reswick, J.B. (1956). Disturbance-response feedback - a new control concept, Transactions oftheASME, January, pp. 153-162.

Rice, R. and Cooper, D.J. (2002), Design and tuning of PID controllers for integrating (non self-regulating) processes, Proceedings of the ISA Annual Meeting (Chicago, IL, U.S.A.), 424, Paper P057.

Rivera, D.E. and Jun, K.S. (2000). An integrated identification and control design methodology for multivariable process system applications, IEEE Control Systems Magazine, June, pp. 25-37.

Rivera, D.E., Morari, M. and Skogestad, S. (1986). Internal Model Control. 4. PID controller design, Industrial and Engineering Chemistry Process Design and Development, 25, pp. 252-265.

Robbins, L.A. (2002). Tune control loops for minimum variability, Chemical Engineering Progress, January, pp. 68-70.

Rotach, Y. Va. (1994). Calculation of the robust settings of automatic controllers, Thermal Engineering (Russia), 41, pp. 764-769.

Rotach, V. Ya. (1995). Automatic tuning of PID-controllers - expert and formal methods, Thermal Engineering (Russia), 42, pp. 794-800.

Rotstein, G.E. and Lewin, D.E. (1991). Simple PI and PID tuning for open-loop unstable systems, Industrial Engineering Chemistry Research, 30, pp. 1864-1869.

Rovira, A.A., Murrill, P.W. and Smith, C.L. (1969). Tuning controllers for setpoint changes, Instruments and Control Systems, 42 (December), pp. 67-69.

Rutherford, C.I. (1950). The practical application of frequency response analysis to automatic process control, Proceedings of the Institution of Mechanical Engineers (London), 162, pp. 334-354.

Sadeghi, J. and Tych, W. (2003). Deriving new robust adjustment parameters for PID controllers using scale-down and scale-up techniques with a new optimization method, Proceedings oflCSE: 16th Conference on Systems Engineering (Coventry, U.K.), pp. 608-613.

Sain, S.G. and Ozgen, C. (1992). Identification and tuning of processes with large deadtime, Control and Computers, 20(3), pp. 73-78.

Saito, T., Kawakami, J., Takahashi, S., Suehiro, T., Matsumoto, H. and Tachibana, K. (1990). PID controller system, US Patent No. 4,903,192.

Sakai, Y., Nakai, Y., Miyabe, A. and Kawano, T. (1989). Self-tuning controller, US Patent 4,881,160.


Schaedel, H.M. (1997). A new method of direct PID controller design based on the principle of cascaded damping ratios, Proceedings of the European Control Conference (Brussels, Belgium), Paper WE-A H4.

Schneider, D.M. (1988). Control of processes with time delays, IEEE Transactions on Industry Applications, 24, pp. 186-191.

Seki, H., Ogawa, M. and Ohshima, M. (2000). Retuning PID temperature controller for an unstable gas-phase polyolefin reactor, Preprints of Proceedings of PID '00: IF AC Workshop on digital control (Terrassa, Spain), pp. 473^178.

Setiawan, A., Albright, L.D. and Phelan, R.M. (2000). Application of pseudo-derivative-feedback algorithm in greenhouse air temperature control, Computers and Electronics in Agriculture, 26, pp. 283-302.

Shen, J.-C. (1999). Tuning PID controller for a plant with under-damped response, Proceedings of the IEEE International Conference on Control Applications (Hawaii, U.S.A.), pp. 115-120.

Shen, J.-C. (2000). New tuning method for PID control of a plant with underdamped response, Asian Journal of Control, 2(1), pp. 31-41.

Shen, J.-C. (2002). New tuning method for PID controller, ISA Transactions, 41, pp. 473-484.

Shi, J. and Lee, W.S. (2002). IMC-PID controllers for first-order plus dead-time processes: a simple design with guaranteed phase margin, Proceedings of IEEE Region 10 Conference on Computers, Communications, Control and Power Engineering (Beijing, China), pp. 1397-1400.

Shi, J. and Lee, W.S. (2004). Set point response and disturbance rejection tradeoff for second-order plus dead time processes, Proceedings of the 5lh Asian Control Conference, (Melbourne, Australia).

Shigemasa, T., lino, Y. and Kanda, M. (1987). Two degrees of freedom PID auto-tuning controller, Proceedings of the ISA International Conference and Exhibition. Advances in Instrumentation and Control, pp. 703-711.

Shin, C.-H., Yoon, M.-H. and Park, I.-S. (1997). Automatic tuning algorithm of the PID controller using two Nyquist points identification, Proceedings of the Society of Instrument and Control Engineers annual conference (Tokyo, Japan), pp. 1225-1228.

Shinskey, F.G. (1988). Process Control Systems - Application, Design and Tuning (McGraw-Hill Inc., New York, 3rd Edition).

Shinskey, F.G. (1994). Feedback controllers for the process industries (McGraw-Hill Inc., New York).

Shinskey, F.G. (1996). Process Control Systems - Application, Design and Tuning (McGraw-Hill Inc., New York, 4th Edition).

Shinskey, F.G. (2000). PID-deadtime control of distributed processes, Preprints of Proceedings of PID '00: IFAC Workshop on digital control (Terrassa, Spain), pp. 14-18.

Shinskey, F.G. (2001). PID-deadtime control of distributed processes, Control Engineering Practice, 9, pp. 1177-1183.

Bibliography 527

Sklaroff, M. (1992). Adaptive controller in a process control system and a method therefor, US Patent 5,170,341.

Skoczowski, S. and Tarasiejski, L (1996). Tuning of PID controllers based on gain and phase margin specifications using Strejc's process model with time delay, Proceedings of the 3rd International Symposium on Methods and Models in Automation and Robotics (MMAR '96), (Miedzyzdroje, Poland), pp. 765-770.

Skogestad, S. (2001). Probably the best simple PID tuning rules in the world, AIChE

Annual Meeting (Reno, Nevada, U.S.A.) [Online]. Available at http://www.nt.ntnu.no/users/skoge/publications/2001/skogestad reno/tuningpaper.pdf [accessed 24 September 2004].

Skogestad, S. (2003). Simple analytic rules for model reduction and PID controller tuning, Journal of Process Control, 13, pp. 291-309.

Skogestad, S. (2004a). Lower limit on controller gain for acceptable disturbance rejection, Proceedings of the International Symposium on Advanced Control of Chemical Processes (Hong Kong, China).

Skogestad, S. (2004b). Simple analytic rules for model reduction and PID controller tuning, Modeling, Identification and Control, 25, pp. 85-120.

Smith, C.A. and Corripio, A.B. (1997). Principles and practice of automatic process control (John Wiley and Sons, New York, 2nd Edition).

Smith, C.L. (2002). Intelligently tune PI controllers, Chemical Engineering, August, pp. 169-177.

Smith, C.L. (2003). Intelligently tune PID controllers, Chemical Engineering, January, pp. 56-62.

Smith, C.L., Corripio, A.B. and Martin, J. (1975). Controller tuning from simple process models, Instrumentation Technology, December, pp. 39-44.

Smith, L. (1998). A modified Smith predictor for extruded diameter control, InstMC Mini Symposium in UKACC International Conference on Control (Swansea, U.K.), Lecture 5.

Somani, M.K., Kothare, M.V. and Chidambaram, M. (1992). Design formulae for PI controllers, Hungarian Journal of Industrial Chemistry, 20, pp. 205-211.

Sree, R.P. and Chidambaram, M. (2003a). Simple method of tuning PI controllers for stable inverse response systems, Journal of the Indian Institute of Science, 83 (May-August), pp. 73-85.

Sree, R.P. and Chidambaram, M. (2003b). Control of unstable bioreactor with dominant unstable zero, Chemical and Biochemical Engineering Quarterly, 17(2), pp. 139-145.

Sree, R.P. and Chidambaram, M. (2003c). A simple method of tuning PI controllers for unstable systems with a zero, Chemical and Biochemical Engineering Quarterly, 17(3), pp. 207-212.

Sree, R.P. and Chidambaram, M. (2004). Control of unstable reactor with an unstable zero, Indian Chemical Engineer Section A, 46(1), pp. 21-26.

Sree, R.P. and Chidambaram, M. (2005a). Set point weighted PID controllers for unstable systems, Chemical Engineering Communications, 192, pp. 1-13.

http://www.nt.ntnu.no/users/skoge/publications/2001/skogestad


Sree, R.P. and Chidambaram, M. (2005b). A simple and robust method of tuning PID controllers for integrator/dead time processes, Journal of Chemical Engineering of Japan, 38(2), pp. 113-119.

Sree, R.P., Srinivas, M.N. and Chidambaram, M. (2004). A simple method of tuning PID controllers for stable and unstable FOPTD systems, Computers and Chemical Engineering, 28, pp. 2201-2218.

Srinivas, M.N. and Chidambaram, M. (2001). Set-point weighted PID controllers, Proceedings of the International Conference on Quality, Reliability and Control (ICQRC-2001), (Mumbai, India), pp. C22-1-C22-5.

Srividya, R. and Chidambaram, M. (1997). On-line controllers tuning for integrator plus delay systems, Process Control and Quality, 9, pp. 59-66.

St. Clair, D.W. (1997). Controller tuning and control loop performance (Straight Line Control Co. Inc., Newark, Delaware, U.S.A., 2nd Edition).

St. Clair, D.W. (2000). The PID algorithm [Online]. Available at http://members.aol.com/pidcontrol/pid_algorithm.html [accessed 28 June 2005].

Streeter, M.J., Keane, M.A. and Koza, J.R. (2003). Automatic synthesis using genetic programming of improved PID tuning rules, Preprint of Proceedings of Intelligent Control Systems and Signal Processing Conference (Ed: A.D. Ruano), pp. 494-499.

Sung, S.W. O., J., Lee, I.-B., Lee, J. and Yi, S.-H. (1996). Automatic tuning of PID controller using second-order plus time delay model, Journal of Chemical Engineering of Japan, 29, pp. 991-999.

Suyama, K. (1992). A simple design method for sampled-data PID control systems with adequate step responses, Proceedings of the International Conference on Industrial Electronics, Control, Instrumentation and Automation, pp. 1117-1122.

Suyama, K. (1993). A simple design method for sampled-data I-PD control systems, Proceedings of the Annual Conference of the IEEE Industrial Electronics Society (HI, U.S.A.), pp. 2293-2298.

Syrcos, G. and Kookos, I. K. (2005). PID controller tuning using mathematical programming, Chemical Engineering and Processing, 44, pp. 41-49.

Tachibana, Y. (1984). Identification of a system with dead time and its application to auto tuner, Electrical Engineering in Japan, 104, pp. 128-137.

Taguchi, H. and Araki, M. (2000). Two-degree-of-freedom PID controllers - their functions and optimal tuning, Preprints of Proceedings of PID '00: IF AC Workshop on digital control (Terrassa, Spain), pp. 95-100.

Taiwo, O. (1993). Comparison of four methods of on-line identification and controller tuning, IEE Proceedings, Part D, 140, pp. 323-327.

Tan, K.K., Lee, T.H. and Wang, Q.G. (1996). Enhanced automatic tuning procedure for process control of PI/PID controller, AIChE Journal, 42, pp. 2555-2562.

Tan, K.K., Lee, T.H. and Jiang, X. (2001). On-line relay identification, assessment and tuning of PID controller, Journal of Process Control, 11, pp. 483-496.

Tan, K.K., Wang, Q.-G., Hang, C.C. and Hagglund, T.J. (1999a). Advances in PID control (Advances in Industrial Control Series, Springer-Verlag London).

http://members.aol.com/pidcontrol/pid_algorithm.html

Bibliography 529

Tan, W., Liu, J. and Sun, W. (1998a). PID tuning for integrating processes, Proceedings of the IEEE International Conference on Control Applications (Trieste, Italy), 2, pp. 873-876.

Tan, W., Liu, K. and Tarn, P.K.S. (1998b). PID tuning based on loop-shaping H„ control, IEE Proceedings - Control Theory and Applications, 145(6), pp. 485^90.

Tan, W., Yuan, Y. and Niu, Y. (1999b). Tuning of PID controller for unstable process, Proceedings of the IEEE International Conference on Control Applications (Hawaii, U.S.A.), pp. 121-124.

Tang, W., Shi, S. and Wang, M. (2002). Autotuning PID control for large time-delay processes and its application to paper basis weight control, Industrial and Engineering Chemistry Research, 41, pp. 4318-4327.

Tavakoli, S. and Fleming, P. (2003). Optimal tuning of PI controllers for first order plus dead time/long dead time models using dimensional analysis, Proceedings of the European Control Conference (Cambridge, UK).

Tavakoli, S. and Tavakoli, M. (2003). Optimal tuning of PID controllers for first order plus time delay models using dimensional analysis, Proceedings of the 4th IEEE International Conference on Control and Automation (Canada), pp. 942-946.

Thomasson, F.Y. (1997). Tuning guide for basic control loops, Proceedings of the Process Control, Electrical and Information (TAAPI) Conference, pp. 137-148.

Tinham, B. (1989). Tuning PID controllers, Control and Instrumentation, September, pp. 79-83.

Tsang, K.M. and Rad, A. B. (1995). A new approach to auto-tuning of PID controllers, InternationalJournal of Systems Science, 26(3), pp. 639-658.

Tsang, K.M., Rad, A.B. and To, F.W. (1993). Online tuning of PID controllers using delayed state variable filters, Proceedings of the IEEE Region 10 Conference on Computer, Communication, Control and Power Engineering (Beijing, China), 4, pp. 415^19.

Tyreus, B.D. and Luyben, W.L. (1992). Tuning PI controllers for integrator/dead time processes, Industrial Engineering Chemistry Research, 31, pp. 2625-2628.

Valentine, C.C. and Chidambaram, M. (1997a). Robust PI and PID control of stable first order plus time delay systems, Indian Chemical Engineer Section A, 39(1), pp. 9-14.

Valentine, C.C. and Chidambaram, M. (1997b). PID control of unstable time delay systems, Chemical Engineering Communications, 162, pp. 63-74.

Valentine, C.C. and Chidambaram, M. (1998). Robust control of unstable first order plus time delay systems, Indian Chemical Engineer Section A, 40(1), pp. 19-23.

Van der Grinten, P.M.E.M. (1963). Finding optimum controller settings, Control Engineering, December, pp. 51-56.

VanDoren, V.J. (1998). Ziegler-Nichols methods facilitate loop tuning, Control Engineering, December.

Van Overschee, P. and De Moor, B. (2000). RaPID: the end of heuristic PID tuning, Preprints of Proceedings of PID '00: IF AC Workshop on digital control (Terrassa, Spain), pp. 687-692.


Venkatashankar, V. and Chidambaram, M. (1994). Design of P and PI controllers for unstable first order plus time delay systems, International Journal of Control, 60, pp. 137-144.

Visioli, A. (2001). Tuning of PID controllers with fuzzy logic, IEE Proceedings -Control Theory and Applications, 148(1), pp. 180-184.

Viteckova, M. (1999). Sefizeni cislicovych I analogovych regulatorfl pro regulovane soustavy s dopravnim zpozdSnim (Digital and analog controller tuning for processes with time delay), Automatizace, 42(2), pp. 106-111 (in Czech).

Viteckova, M. and Vitecek, A. (2002). Analytical controller tuning method for proportional non-oscillatory plants with time delay, Proceedings of the International Carpathian Control Conference (Malenovice, Czech Republic), pp. 297-302.

Viteckova, M. and Vitecek, A. (2003). Analytical digital and analog controller tuning method for proportional non-oscillatory plants with time delay, Proceedings of the 2nd

IFAC Conference on Control Systems Design (Bratislava, Slovak Republic), pp. 59-64.

Viteckova, M., Vitecek, A. and Smutny, L. (2000a). Controller tuning for controlled plants with time delay, Preprints of Proceedings of PID '00: IFAC Workshop on digital control (Terrassa, Spain), pp. 283-288.

Viteckova, V., Vitecek, A. and Smutny, L. (2000b). Simple PI and PID controllers tuning for monotone self-regulating plants, Preprints of Proceedings of PID '00: IFAC Workshop on digital control (Terrassa, Spain), pp. 289-294.

Voda, A. and Landau, I.D. (1995). The autocalibration of PI controllers based on two frequency measurements, International Journal of Adaptive Control and Signal Processing, 9, pp. 395-421.

Vrancic, D. (1996). Design of anti-windup and bumpless transfer protection. Part II: PID controller tuning by multiple integration method, PhD thesis (University of Ljubljana, J. Stefan Institute, Ljubljana, Slovenia).

Vrancic, D., Kocijan, J. and Strmcnik, S. (2004a). Simplified disturbance rejection tuning method for PID controllers, Proceedings of the 5' Asian Control Conference (Melbourne, Australia).

Vrancic, D., Kristiansson, B. and Strmcnik, S. (2004b). Reduced MO tuning method for PID controllers, Proceedings of the 5th Asian Control Conference (Melbourne, Australia).

Vrancic, D. and Lumbar, S. (2004). Improving PID controller disturbance rejection by means of magnitude optimum, Report DP-8955 (J. Stefan Institute, Ljubljana, Slovenia).

Vrancic, D., Peng, Y., Strmcnik, S. and Hanus, R. (1996). A new tuning method for PI controllers based on a process step response, Proceedings of CESA '96 IMACS MultiConference Symposium on Control, Optimisation and Supervision (Lille, France), 2, pp. 790-794.

Vrancic, D., Peng, Y., Strmcnik, S. and Juricic, D. (1999). A multiple integration tuning

method for filtered PID controller, Preprints of Proceedings of IFAC I4'h World

Bibliography 531

Congress (Beijing, China), Paper 3b-02-3. Vrancic, D., Strmcnik, S. and Kocijan, J. (2004c). Improving disturbance rejection of PI

controllers by means of the magnitude optimum method, ISA Transactions, 43, pp. 73-84.

Wade, H.L. (1994). Regulatory and advanced regulatory control: system development (ISA).

Wang, F.-S., Juang, W.-S. and Chan, C.-T. (1995a). Optimal tuning of PID controllers for single and cascade control loops, Chemical Engineering Communications, 132, pp. 15-34.

Wang, H. and Jin, X. (2004). Direct synthesis approach of PID controller for second-order delayed unstable processes, Proceedings of the 5th World Congress on Intelligent Control and Automation (Hangzhou, P.R. China), pp. 19-23.

Wang, L., Barnes, T.J.D. and Cluett, W.R. (1995b). New frequency-domain design method for PID controllers, IEE Proceedings - Control Theory and Applications, 142, pp. 265-271.

Wang, L. and Cluett, W.R. (1997). Tuning PID controllers for integrating processes, IEE Proceedings - Control Theory and Applications, 144(5), pp. 385-392.

Wang, L. and Cluett, W.R. (2000). From plant data to process control (Taylor and Francis Ltd.).

Wang, Q.-G., Lee, T.-H., Fung, H.-W., Bi, Q. and Zhang, Y. (1999). PID tuning for improved performance, IEEE Transactions on Control Systems Technology, 7(4), pp. 457^165.

Wang, Q.-G., Zhang, Y. and Guo, X. (2000b). Robust closed-loop controller auto-tuning, Proceedings of the 15'h IEEE International Symposium on Intelligent Control (Rio, Patras, Greece), pp. 133-138.

Wang, Q.-G., Zhang, Y. and Guo, X. (2001a). Robust closed-loop identification with application to auto-tuning, Journal of Process Control, 11, pp. 519-530.

Wang, T.-S. and Clements, W.C. (1995). Adaptive multivariable PID control of a distillation column with unknown and varying dead time, Chemical Engineering Communications, 132, pp. 1-13.

Wang, Y., Schinkel, M, Schmitt-Hartmann, T. and Hunt, K.J. (2002). PID and PID-like controller design by pole assignment within D-stable regions, Asian Journal of Control, 4(4), pp. 423^132.

Wang, Y.-G. and Cai, W.-J. (2001). PID tuning for integrating processes with sensitivity specification, Proceedings of the IEEE Conference on Decision and Control (Orlando, Florida, U.S.A.), pp. 4087^091.

Wang, Y.-G. and Cai, W.-J. (2002). Advanced proportional-integral-derivative tuning for integrating and unstable processes with gain and phase margin specifications, Industrial Engineering Chemistry Research, 41, pp. 2910-2914.

Wang, Y.-G., Cai, W.-J. and Shi, Z.-G. (2001b). PID autotuning for integrating processes with specifications on gain and phase margins, Proceedings of the American Control Conference (Arlington, VA, U.S.A.), pp. 2181-2185.


Wang, Y.-G. and Shao, H.-H. (1999). PID autotuner based on gain- and phase-margin specification, Industrial Engineering Chemistry Research, 38, pp. 3007-3012.

Wang, Y.-G. and Shao, H.-H. (2000a). Optimal tuning for PI controller, Automatica, 36, pp. 147-152.

Wang, Y.-G. and Shao, H.-H. (2000b). PID auto-tuner based on sensitivity specification, Transactions of the Institute of Chemical Engineers, 78(A), pp. 312-316.

Wang, Y.-G., Shao, H.-H. and Wang, J. (2000a). PI tuning for processes with large dead time, Proceedings of the American Control Conference (Chicago, U.S.A.), pp. 4274-4278.

Wills, D.M. (1962a). Tuning maps for three-mode controllers, Control Engineering, April, pp. 104-108.

Wills, D.M. (1962b). A guide to controller tuning, Control Engineering, August, pp. 93-95.

Wilton, S.R. (1999). Controller tuning, ISA Transactions, 38, pp. 157-170. Witt, S.D. and Waggoner, R.C. (1990). Tuning parameters for non-PID three-mode

controllers, Hydrocarbon Processing, June, pp. 74-78. Wojsznis, W.K. and Blevins, T.L. (1995). System and method for automatically tuning a

process controller, U.S. Patent Number 5,453,925. Wojsznis, W.K., Blevins, T.L. and Thiele, D. (1999). Neural network assisted control

loop tuner, Proceedings of the IEEE International Conference on Control Applications (HI, U.S.A.), 1, pp. 427-431.

Wolfe, W.A. (1951). Controller settings for optimum control, Transactions of the ASME, May, pp. 413-418.

Xu, J. and Shao, H. (2003a). Advanced PID tuning for integrating processes with a new robustness specification, Proceedings of the American Control Conference (Denver, CO., U.S.A.), pp. 3961-3966.

Xu, J. and Shao, H. (2003b). Advanced PID tuning for unstable processes based on a new robustness specification, Proceedings of the American Control Conference (Denver, CO., U.S.A.), 368-372.

Xu, J. and Shao, H. (2003c). A novel method of PID tuning for integrating processes, Proceedings of the 42nd IEEE Conference on Decision and Control (Maui, Hawaii, U.S.A.), pp. 139-142.

Xu, J.-H., Sun, R. and Shao, H.-H. (2004). PI controller tuning for large dead-time processes, Kongzhi Yu Juece/Control and Decision, 19(1), pp. 99-101 (in Chinese).

Yang, J. C.-Y. and Clarke, D.W. (1996). Control using self-validating sensors, Transactions of the Institute of Measurement and Control, 18, pp. 15-23.

Yi, C. and De Moor, B.L.R. (1994). Robustness analysis and control system design for a hydraulic servo system, IEEE Transactions on Control Systems Technology, 2(3), pp. 183-197.

Young, A.J. (1955). An introduction to process control system design (Longman, Green and Company, London).

Bibliography 533

Yu, C.-C. (1999). Autotuning of PID controllers (Advances in Industrial Control Series, Springer-Verlag London Ltd.).

Yu, S. W. (1988). Optimal PI tuning for load disturbances, Journal of the Chinese Institute of Chemical Engineers, 19(6), pp. 349-357.

Zhang, D. (1994). Process dynamics and controller selection of DCS, Proceedings of the ISA Advances in Instrumentation and Control Conference (Anaheim, U.S.A), 49(2), pp. 231-240.

Zhang, W., Xi, Y., Yang, G. and Xu, X. (2002). Design PID controllers for desired time-domain or frequency-domain response, ISA Transactions, 41, pp. 511-520.

Zhang, W. and Xu, X. (2002). H,, PID controller design for runaway processes with

time delay, ISA Transactions, 41, pp. 317-322. Zhang, W., Xu, X. and Sun, Y. (1999). Quantitative performance design for integrating

processes with time delay, Automatica, 35, pp. 719-723. Zhang, W.D., Sun, Y.X. and Xu, X.M. (1996). Modified PID controller based on HM

theory, Proceedings of the IEEE International Conference on Industrial Technology, pp. 9-12.

Zhong, Q.-C. and Li, H.-X. (2002). 2-degree-of-freedom proportional-integral-derivative-type controller incorporating the Smith principle for processes with dead time,

Industrial Engineering Chemistry Research, 41, pp. 2448-2454. Zhuang, M. and Atherton, D.P. (1993). Automatic tuning of optimum PID controllers,

IEE Proceedings, Part D, 140, pp. 216-224. Ziegler, J.G. and Nichols, N.B. (1942). Optimum settings for automatic controllers,

Transactions oftheASME, November, pp. 759-768. Zou, H., Hedstrom, K.P., Warrior, J. and Hays, C.L. (1997). Field based process control

system with auto-tuning, U.S. Patent Number 5,691,896 . Zou, H. and Brigham, S.E. (1998). Process control system with asymptotic auto-tuning,

U.S. Patent Number 5,818,714.

Index

ABB (1996), 237 ABB (2001), 15, 29,42, 71, 224,225 Abbas (1997), 47, 169 Aikman(1950), 72 Alfa-Laval, 12, 73 Allen Bradley, 7, 15 Alvarez-Ramirez et al. (1998), 81 Andersson (2000), 58, 81 Argelaguet et al. (2000), 229 Arvanitis et al. (2000), 327, 332, 341,

371 Arvanitis et al. (2003a), 85, 86, 274,

275 Arvanitis et al. (2003b), 94-100, 298-

302 Astrom (1982), 73,236 Astrom and Hagglund (1984), 236 Astrom and Hagglund (1988), 155, 236 Astrom and Hagglund (1995), 45, 64,

70, 73, 75, 76, 93, 185, 236, 245, 259, 284, 349

Astrom and Hagglund (2000), 150 Astrom and Hagglund (2004), 150,

167-168, 223, 260, 272 Atkinson and Davey (1968), 235 Auslandere/a/. (1975), 309

Bailey, 7, 9, 12, 13 Bailey Fisher and Porter, 13 Bain and Martin (1983), 38, 43 Bateson (2002), 73, 238, 258 Bekkeretal. (1991), 45 Belanger and Luyben (1997), 88, 267 Bequette (2003), 151, 181, 450

Bialkowski (1996), 58, 81 Bi et al. (1999), 47 Bi et al. (2000), 47, 327 Blickley (1990), 235 Boe and Chang (1988), 60 Bohl and McAvoy (1976a), 372, 373 Bohl and McAvoy (1976b), 311, 330,

352,353 Borresen and Grindal (1990), 29, 155 Brambilla et al. (1990), 58, 118, 178,

328 Branica et al. (2002), 176 Bryant et al. (1973), 42, 114-115, 151 Buckley (1964), 151 Bunzemeier(1998), 77, 266

Calcev and Gorez (1995), 71, 236 Callender et al. (1935/6), 26, 149, 154,

449 Camachoefa/. (1997), 169 Carr (1986), 235 Chandrashekar et al. (2002), 133, 136,

408,409,417,418 Changed al. (1997), 69 Chao et al. (1989), 106, 107, 111, 342,

344-349 Chau (2002), 237 Chen and Seborg (2002), 48, 81, 172,

261,280,327-328 Chen and Yang (2000), 56 Chen etal. (1997), 59 Chen et al. (1999a), 51, 80, 261, 280


Chen etal. (1999b), 51,324 Chen et al. (2001), 193 Cheng and Hung (1985), 157, 164 Cheng and Yu (2000), 80 Chesmond(1982), 73, 238 Chidambaram (1994), 79 Chidambaram (1995a), 175-176 Chidambaram (1995b), 131,404 Chidambaram (1997), 131, 140, 405 Chidambaram (2000a), 63, 226 Chidambaram (2000b), 84, 271 Chidambaram (2000c), 132, 135, 416 Chidambaram and Kalyan (1996), 433 Chidambaram and Sree (2003), 79,

261,271 Chidambaram et al. (2005), 49 Chien (1988), 58, 81, 190, 206, 211,

264, 267, 269, 285, 291, 294, 386, 387, 389, 393, 394, 395

Chien and Fruehauf (1990), 81, 206, 211, 264, 267, 269, 285, 291, 294, 386, 387, 389, 393, 394, 395

Chien era/. (1952), 27, 155 Chien et al. (1999), 65, 86, 229, 275 Chiue/a/. (1973), 42, 326 Chun etal. (1999), 59 Cluett and Wang (1997), 54-55, 78,

170-171,260 Cohen and Coon (1953), 28, 155 Concept, 9 Connell(1996), 156 ControlSoft Inc. (2005), 156 Coon (1956), 76, 89 Coon (1964), 76, 89 Corripio (1990), 236, 253 Cox etal. (1994), 73 Cox etal. (1997), 55

Davydov et al. (1995), 46, 188 De Paor (1993), 236 De Paor and O'Malley (1989), 131,404 Devanathan(1991), 37, 161 Direct synthesis, 24-25, 42-58, 61, 62,

63-64, 65, 67, 68, 71-72, 75, 78-81, 84, 86, 90-91, 93, 100, 114-118,

123-124, 125, 126-127, 129, 131-133, 135-136, 138-139, 141, 142, 143-144, 146, 147, 148, 151, 152, 153, 168-178, 181, 183-184, 185, 186, 188-189, 193, 203-205, 208, 210, 212, 219-220, 225, 226, 227-229, 230, 242, 245, 246-251, 260-261, 263, 264, 265, 268, 271, 275, 277, 279-280, 284, 285, 291, 292, 293, 296, 302, 303, 304, 306, 307, 308, 321-328, 332, 338, 341, 349-350, 351, 353, 354, 367, 370, 371, 372, 378, 379, 380, 385, 387, 388, 390, 391, 392, 394, 396, 397, 399, 400, 404-407, 408-409, 410-413, 415, 416^119, 427, 433, 439^141, 442, 445^146, 447^148, 452, 453, 454, 455^456, 457^459, 460-461 Frequency domain criteria, 49-58,

64, 79-81, 124, 152, 174-178, 181, 183-184, 193, 208, 268, 292, 321-324, 332, 399, 400, 433, 453

Time domain criteria, 42-49, 61, 62, 63-64, 65, 67, 68, 78-79, 123, 141, 142, 143-144, 147, 148, 168-173, 227-229, 230, 265, 306, 325-328, 332, 372, 380, 385, 415, 433, 445-446, 447-448

Dmonetal. (1997), 240

ECOSSE Team (1996a), 37 ECOSSE Team (1996b), 73, 238 Edgar et al. (1997), 31, 71, 197, 224,

253, 257 Ettaleb and Roche (2000), 47 expert system, 472

Farrington (1950), 235 Fanuc, 7 Fertik(1975), 150 Fertik and Sharpe (1979), 29 Fischer and Porter, 9, 12, 17 Fisher-Rosemount, 17 Ford (1953), 259 Foxboro, 9, 11, 12, 17, 209

Index 537

Frank and Lenz (1969), 30, 33, 34, 35 Friman and Waller (1997), 52, 71, 178 Fruehauf etal. (1993), 46, 78, 169

Gaikward and Chidambaram (2000), 132,406

Gain margin, analytical calculation, 463-469

Gain margin, figures, 471, 473-474 Gain and phase margins, constant, 475-

483 PI controller, FOLPD model, 475^77 PI controller, IPD model, 477-480 PD controller, FOLIPD model, 483 PID controller, FOLPD model,

classical controller 1, 480-481 PID controller, SOSPD model,

series controller, 482 PID controller, SOSPD model with a

negative zero, classical controller 1, 482-483

Gallier and Otto (1968), 38, 108, 162, 315

Garcia and Castelo (2000), 239 Genesis, 15 Geng and Geary (1993), 60, 179 Gerry (1988), 149, 159 Gerry (1999), 58, 178 Gerry (2003), 37 Gerry and Hansen (1987), 149 Gong et al. (1996), 190 Goodwin etal. (2001), 189 Gorecki et al. (1989), 45, 49, 168, 174 Gorez (2003), 48, 173,185 Gorez and Klan (2000), 326,454

Haalman (1965), 33, 77, 149, 279, 352 Haeri (2002), 225 Haeri (2005), 49 Hagglund and Astrom (1991), 73 Hagglund and Astrom (2002), 56, 64,

84, 136 Hang and Astrom (1988a), 220-221 Hang and Astrom (1988b), 236 Hang and Cao (1996), 221

Hanged/. (1991), 45, 221 Hang et al. (1993a), 50, 321, 332 Hang et al. (1993b), 50, 70, 211, 255 Hange/a/. (1994), 338 Hansen (1998), 370 Hansen (2000), 151, 271, 379 Harriott (1964), 238 Harris and Tyreus (1987), 206, 234 Harrold (1999), 195,253 Hartmann and Braun, 9 Hartreeera/. (1937), 451 Hassan (1993), 106,312 Hay (1998), 29, 73, 76-77, 156, 238,

259 Hazebroek and Van der Waerden

(1950), 27, 32-33, 77 Heck (1969), 28-29 Henry and Schaedel (2005), 52-53 Hiroi and Terauchi (1986), 217, 222 Ho and Xu (1998), 132,433 Ho era/. (1994), 321 Ho etal. (1995a), 321-322 Ho etal. (1997), 322 Wo etal. (1998), 158, 163 Wo etal. (1999), 51, 158, 163 no etal. (2001), 234 Honeywell, 7, 11, 15, 16, 18, 45, 349 Horned/. (1996), 181, 184 Hougen (1979), 49, 79, 90, 115, 126,

208,268,292,351 Hougen (1988), 116,268,351 Huang and Chao (1982), 195, 197, 198,

199,201,203 Huang and Chen (1997), 415, 433 Huang and Chen (1999), 415, 433 Huang and Jeng (2002), 38, 150 Huang and Jeng (2003), 167, 320-321 Huang and Lin (1995), 420, 434^138 Huang et al. (1996), 32, 38, 103-104,

108-110,355-362 Huang e^a/. (1998), 58, 329 Huang et al. (2000), 212, 219, 354,

367,390,391,396,397 Huang et al. (2002), 182


Huang et al. (2005), 58, 60, 205, 207, 332,335

Huba and Zakova (2003), 79, 91 Hwang (1995), 36, 39, 107, 112-113,

159-160, 165-166, 314-315, 316-319

Hwang and Chang (1987), 70 Hwang and Fang (1995), 37, 41, 160,

166, 167

Intellution, 7 Isaksson and Graebe (1999), 59

Jahanmiri and Fallahi (1997), 336 Jhunjhunwala and Chidambaram

(2001), 135,404,413 Jonese/a/. (1997), 73,239 Juang and Wang (1995), 169

Kamimura et al. (1994), 46, 151 Karaboga and Kalinli (1996), 237 Kasahara e? a/. (1997), 219 Kasahara et al. (1999), 190-191 Kasaharae/o/. (2001), 229 Kaya (2003), 277 Kaya and Atherton (1999), 277 Kaya and Scheib (1988), 196, 197, 198,

199,201,215-216,233 Keane et al. (2000), 320 Keviczky and Csaki (1973), 43, 111,

150, 151 Khan and Lehman (1996), 39-40, 45,

47 Khodabakhshian and Golbon (2004),

124 Kim et al. (2004), 219-220 Kinney (1983), 253 Klein etal. (1992), 29 Kookos etal. (1999), 79 Kraus (1986), 209 Kristiansson (2003), 57, 72, 119, 183-

184,241,244,330,334,337 Kristiansson and Lennartson (1998a),

248

Kristiansson and Lennartson (1998b), 248

Kristiansson and Lennartson (2000), 72, 249-250

Kristiansson and Lennartson (2002), 72,250-251,264,285

Kristiansson et al. (2000), 243 Kuhn(1995),46, 188 Kwake/a/. (2000), 439-141

Landau and Voda (1992), 330-331 Lee and Edgar (2002), 66, 87, 101, 140,

182, 231, 232, 276, 278, 303, 305, 333, 422,425,426

Lee and Shi (2002), 62, 206 Lee et al. (1998), 59, 178, 328, 329 Lee et al. (2000), 407, 413, 428, 431,

443, 444 Leeds and Northup, 7 Lennartson and Kristiansson (1997),

248 Leonard (1994), 260 Leva (1993), 73, 246 Leva (2001), 53, 58, 178, 192 Leva (2005), 220 Leva and Colombo (2000), 191 Leva and Colombo (2004), 220 Leva et al. (1994), 57, 322-323 Leva etal. (2003), 57, 118, 192, 324 Li etal. (1994), 177 Li et al. (2004), 385 Lime/a/. (1985), 61,332 Liptak (1970), 238 Liptak(2001), 30, 156 Liu et al. (2003), 286, 287-288, 374-

375,376-377 Lloyd (1994), 239 Lopez et al. (1969), 106,310 Loron(1997), 153 Luoetal. (1996), 237 Luyben (1996), 267 Luyben(1998), 132 Luyben (2000), 124, 181 Luyben and Luyben (1997), 237

Index 539

MacLellan(1950), 72, 238 Maffezzoni and Rocco (1997), 190 Majhi and Atherton (2000), 130, 423-

424 Majhi and Mahanta (2001), 304 Majhi and Litz (2003), 353 Manned/. (2001), 44, 172 Mantz and Tacconi (1989), 252 Marchetti and Scali (2000), 329 Marlin (1995), 31,38,157, 162 Matausek and Kvascev (2003), 57 Matsuba et al. (1998), 60, 179 Maximum sensitivity, analytical

calculation, 469^170 Maximum sensitivity, figures, 471 Maximum sensitivity, constant, 479-

480 McAnany(1993), 45 McAvoy and Johnson (1967), 105, 235,

343 McMillan (1984), 91, 146, 179, 281,

429 McMillan (1994), 29, 60, 70, 73, 236,

238 Mizutani and Hiroi (1991), 14, 15 Modicon, 9 Modcomp, 15 Mollenkamp et al. (1973), 42, 326 Moore, 13, 17 Morari and Zafiriou (1989), 190 Morilla et al. (2000), 171 Moras (1999), 27, 156 Murrill (1967), 28, 30, 33, 34, 155,

157, 158

NI Labview (2001), 74, 240 Normey-Rico et al. (2000), 186, 263 Normey-Rico et al. (2001), 265

O'Connor and Denn (1972), 161 O'Dwyer (2001a), 51, 80-81, 204, 280,

353, 387 O'Dwyer (2001b), 209-210, 254 Ogawa(1995), 59, 81 Ogawa and Katayama (2001), 230

Omron, 13 Oppelt(1951),27 Other tuning rules, 72-74, 82, 151,

237-240, 258 Oubrahim and Leonard (1998), 289,

340 Ou and Chen (1995), 199, 203 Ozawa et al. (2003), 229

Panda et al. (2004), 119, 329, 332, 334, 350

Paraskevopoulos et al. (2004), 137-139 Parked/ . (1998), 421 Parr (1989), 70, 73, 155, 235, 238 Pecharroman (2000), 84, 92, 121, 122,

365-366 Pecharroman and Pagola (2000), 121,

270, 295, 365, 381-382 Pemberton (1972a), 30, 325 Pemberton (1972b), 325, 326 Peng and Wu (2000), 157 Penner(1988), 82 Performance index minimisation, 24,

30-42, 63, 65, 77-78, 84, 85, 89-90, 92, 94-100, 102-113, 120-121, 122, 128, 130, 137-138, 145, 149-150, 157-168, 187, 195-203, 213-214, 215-216, 218, 223, 224-225, 229, 233, 241, 243-244, 253, 256, 257, 259-260, 266-267, 270-271, 272, 273, 274-275, 279, 290-291, 295-296, 297, 298-302, 310-321, 342-349, 352, 355-362, 363-367, 368-369, 381-382, 383-384, 398, 401-402, 403^104, 414, 420, 423^124, 432,434-438

Regulator tuning, 30-37, 77, 85, 89-90, 94-97, 102-107, 137, 145, 149-150, 157-161, 195-199, 215-216, 224, 233, 253, 257, 259, 266-267, 273, 274, 279, 290-291, 297, 298-299, 310-315, 342-346, 352, 355-358, 363-365, 368-369, 403, 414,420,432,434^35 Minimum error, 37, 149, 159


Minimum IAE, 30-32, 77, 89, 102-105, 149-150, 157, 195-197, 215, 224, 233, 253, 257, 266-267, 273, 290, 297, 310, 342-343, 355-358, 368-369, 414, 420, 434^435

Minimum IE, 37, 161, 363-365 Minimum ISE, 32-34, 77, 85, 94,

96, 105, 137, 149, 158, 197,215, 233, 259, 274, 279, 298, 299, 343-344, 352, 403

Minimum ISTES, 36, 159 Minimum ISTSE, 36, 158-159,

259, 403 Minimum ITAE, 34-35, 77, 90,

106, 145, 158, 198-199, 216, 233, 291, 310-313, 344-345, 352,432

Minimum ITSE, 35, 107, 199, 259, 345-346, 403

Minimum IAE or ISE, 37 Minimum performance index -

other, 85, 95, 96-97, 161, 274, 298-299

Modified minimum IAE, 157 Nearly minimum IAE and ITAE,

37, 160 Nearly minimum IAE, ISE and

ITAE, 36, 107, 159-160, 314-315

Servo tuning, 38^11, 85, 97-100, 108-113, 130, 137, 150, 162-166, 187, 199-203, 213-214, 216, 229, 233, 260, 274-275, 300-302, 315-319, 346-349, 359-362, 383-384, 403^104, 420, 423^24, 436^38 Approximately minimum ITAE, 40 Minimum IAE, 38-39, 108-110,

150, 162, 187, 199-200, 216, 233, 315, 346, 359-362, 383, 420, 436^138

Minimum ISE, 39-^10, 85, 97, 99, 111, 137, 150, 162-163, 187, 201, 213, 216, 229, 233, 260, 274,300,301,347,384,403

Minimum ISTES, 40, 165, 214 Minimum ISTSE, 40, 164-165,

213-214,260,404 Minimum ITAE, 40, 111, 163-164,

187, 201-203, 216, 233, 315-316,347-348,384

Minimum ITSE, 111, 130, 203, 260, 348-349, 403, 423^124

Minimum performance index -other, 85, 98, 99-100, 274-275, 300-302

Modified minimum ITAE, 164 Nearly minimum IAE and ITAE,

41, 166 Nearly minimum IAE, ISE and

ITAE, 112-113, 165-166, 316-319

Small IAE, 39 Other tuning, 41^12, 63, 65, 77-78,

84, 92, 120-121, 122, 128, 138, 150, 167-168, 218, 223, 224-225, 241, 243-244, 256, 260, 270-271, 272, 295-296, 319-321, 365-367, 381-382,398,401-402,404

Peric et al. (1997), 60, 91, 179, 281 Pessen (1953), 254 Pessen (1954), 254 Pessen (1994), 70, 157,210 Petitt and Carr (1987), 235 Phase margin, analytical calculation,

463-469 Phase margin, figures, 471, 473^174 PI controller structures

Controller with a double integral term, 6-7, 22, 88

Controller with proportional term acting on the output 1, 6, 22, 65, 85-86,94-100,137-139

Controller with proportional term acting on the output 2, 6, 22, 66, 87, 101, 140

Controller with set-point weighting, 6, 22, 63-64, 75, 84, 92-93, 120-121, 122, 128-129, 135-136, 143-144, 148

541

Ideal controller, 6, 22, 26-60, 67, 70-74, 76-82, 89-91, 102-119, 123-124, 126-127, 130-134, 141, 145-146, 147, 149-151, 152, 153

Ideal controller in series with a first order lag, 6, 22, 61, 68, 69, 83, 125, 142

Ideal controller in series with a second order filter, 6, 22, 62

PID controller structures Alternative controller 1,18, 24, 307 Alternative controller 2, 18, 24, 308 Alternative controller 3, 18, 24 Alternative controller 4, 18, 24, 372-

373 Blending controller, 11, 23, 193 Classical controller 1, 11, 23, 194-

207, 253, 266-267, 290-291, 342-350,387,394,414,432,451

Classical controller 2, 12, 23, 208, 268,292,351

Classical controller 4, 12, 23, 211, 255, 269, 294, 389, 395

Controller with filtered derivative, 8-9, 23, 187-192, 246-251, 264, 285, 338,386,393,400,457-459

Controller with filtered derivative and dynamics on the controlled variable, 10,23,265

Controller with filtered derivative in series with a second order filter, 9, 23, 339

Controller with filtered derivative with setpoint weighting 1, 9, 23, 286

Controller with filtered derivative with setpoint weighting 2, 9-10, 23,374-375

Controller with filtered derivative with setpoint weighting 3, 10, 23, 287-288

Controller with filtered derivative with setpoint weighting 4, 10, 23, 376-377

Ideal controller, 5, 7, 19, 23, 24, 154-180, 235-240, 259-261, 279-281, 309-331, 383-384, 392, 398, 403-407, 427-429, 442^143, 449, 452, 454, 455-456

Ideal controller in series with a first order filter, 8, 23, 336

Ideal controller in series with a first order lag, 8, 23, 181-182, 241-242, 262, 282-283, 332-335, 385, 399, 408^109, 430, 445-446, 450, 453

Ideal controller in series with a second order filter, 8, 23, 183-184, 243-244, 337

Ideal controller with first order filter and dynamics on the controlled variable, 8, 23

Ideal controller with first order filter and setpoint weighting 1, 8, 23, 186

Ideal controller with first order filter and setpoint weighting 2, 8, 23, 263

Ideal controller with setpoint weighting 1,10, 23, 252, 289, 340, 410-413,431,444

Ideal controller with setpoint weighting 2, 10, 23, 341

Ideal controller with weighted proportional term, 8, 23, 185, 245, 284

Industrial controller, 17-18, 24, 233-234, 306, 380

Non-interacting controller 1, 12-13, 23,212,354-362,390,396

Non-interacting controller 2a, 13, 23, 213-214

Non-interacting controller 2b, 13, 23, 215-216

Non-interacting controller 3, 15, 23, 420, 434^138

Non-interacting controller 4, 15, 23, 224-225, 257, 273, 297, 368-369

Non-interacting controller 5, 16, 23, 226, 370


Non-interacting controller 6 (I-PD controller), 16, 23, 227-229, 274-275, 298-302, 371

Non-interacting controller 7, 16, 23, 230,460^161

Non-interacting controller 8, 16, 23, 276,303,421-422,439^141

Non-interacting controller 9, 16, 23, 258

Non-interacting controller 10, 16, 24, 277, 423^124

Non-interacting controller 11, 16-17, 24,231,304

Non-interacting controller 12, 17, 24, 232, 278, 305, 425^126

Non-interacting controller 13, 17 Non-interacting controller based on

the two degree of freedom structure 1, 13, 23, 217-221, 256, 270-271, 295-296, 363-367, 379, 381-382, 391, 397, 401^02, 416—419, 447-448

Non-interacting controller based on the two degree of freedom structure 2, 13-14, 23, 222

Non-interacting controller based on the two degree of freedom structure 3, 14, 23, 223, 272

PD-I-PD (only PD is DOF) incomplete 2DOF algorithm, 14

PI-PID (only PI is DOF) incomplete 2DOF algorithm, 14

Series controller (classical controller 3), 12, 19, 23, 209-210, 254, 293, 352-353,378,388,415,433

Super (or complete) 2DOF PID algorithm, 15

Pinnella e« a/. (1986), 44 Polonyi (1989), 398 Pomerleau and Poulin (2004), 118, 123,

125 Potocnik etal. (2001), 53-54 Poulin and Pomerleau (1996), 77, 90,

145,291,432 Poulin and Pomerleau (1997), 291, 432

Poulin and Pomerleau (1999), 78, 90 Poulin et al. (1996), 117, 146, 291,

350, 387, 394 Pramod and Chidambaram (2000), 406 Prashanti and Chidambaram (2000),

410,411,412-413 Process modelling

Delay model, 21, 149-151, 449^51 Fifth order system plus delay model,

21,455-461 FOLIPD model, 20, 22, 23, 24, 89-

101,279-308,500-501 FOLPD model, 20, 22, 23, 24, 26-66,

154-234,493^98 FOLPD model with a negative zero,

20, 69,498 FOLPD model with a positive zero,

20, 67-68 General model with a repeated pole,

21, 152,452-453,505 General model with integrator, 22,

153 General stable non-oscillating model

with a time delay, 22, 454 IPD model, 20, 22, 23, 24, 76-88,

259-278, 499-500 I2PD model, 20, 374-380

Non-model specific, 20, 22, 23, 24, 70-75, 235-258, 498-499

SOSIPD model, 20, 122, 381-382, 503

SOSPD model, 20, 22, 23, 24, 102-121, 309-373, 501-503

SOSPD model with a negative zero, 21,125,392-397,504

SOSPD model with a positive zero, 21, 123-124,383-391,504

Third order system plus time delay model, 21, 126-129, 398^102

Unstable FOLPD model, 21, 130-140, 403^126, 504-505

Unstable FOLPD model with a positive zero, 21, 141-144

Index 543

Unstable SOSPD model (one unstable pole), 21, 145-146, 427-441,505

Unstable SOSPD model with a positive zero, 21, 147-148, 445-448

Unstable SOSPD model (two unstable poles), 21, 442^144, 505

Process reaction curve methods, 24, 26-30, 76-77, 89, 149, 154-156, 194-195, 209-210, 211, 217, 222, 224,259,266,309,449,451

Pulkkinene/a/. (1993), 59

Reswick(1956),28 Rice and Cooper (2002), 262, 267 Rivera etal. (1986), 58, 178 Rivera and Jun (2000), 61, 83, 280,

283, 329, 333 Robbins (2002), 71,237 Robust, 25, 58-59, 61, 62, 66, 69, 81-

82, 83, 86, 87, 101, 118-119, 133-134, 140, 151, 178, 181-182, 184, 190-192, 206, 211, 220, 231, 232, 234, 261, 262, 264, 267, 269, 276, 278, 280, 282-283, 285, 286, 287-288, 291, 294, 303, 305, 328-330, 332-334, 336, 337, 338, 339, 374-375, 376-377, 386, 387, 389, 393, 394, 395, 407, 413, 421-422, 425-426,428,430,431,444,450

Rotach (1994), 238-239 Rotach(1995), 78, 261 Rotstein and Lewin (1991), 133-134,

428 Rovira et al. (1969), 38,40,162, 163 Rutherford (1950), 72,237

Sadeghi and Tych (2003), 162, 163, 164

Sain and Ozgen (1992), 156 Saitoh a/. (1990), 169 Sakaie/a/. (1989), 29 S ATT Instruments, 17

Schaedel (1997), 52, 118, 152, 181, 189,204,350,399,400,453

Schneider (1988), 45 Seki et al. (2000), 236 Setiawan et al. (2000), 65 Siemens, 9 Shen (1999), 363 Shen (2000), 364, 365 Shen (2002), 218, 256 Shi and Lee (2002), 206 Shi and Lee (2004), 339 Shigemasa et al. (1987), 227-228 Shines / . (1997), 240 Shinskey (1988), 31, 32, 77, 150, 196,

197, 224, 266, 273, 310, 342, 414 Shinskey (1994), 32, 77, 89, 102, 150,

197, 225, 266, 273, 290, 297, 343, 369

Shinskey (1996), 31, 32, 77, 102, 105, 150, 197, 224, 267, 273, 343, 368, 369

Shinskey (2000), 29, 195 Shinskey (2001), 29 Sklaroff(1992),45, 349 Skoczowski and Tarasiejski (1996),

452 Skogestad (2001), 77, 78, 150 Skogestad (2003), 48, 78, 79, 150, 151,

293,353,378 Skogestad (2004a), 82 Skogestad (2004b), 79, 306, 327, 380 Smith (1998), 55 Smith (2002), 40, 58, 59, 81,82 Smith (2003), 164, 237, 330 Smith and Corripio (1997), 30, 38, 43,

197, 200, 203 Smith et al. (1975), 43, 326, 349 Somani et al. (1992), 50, 116-117, 174 Square D, 15 Sree and Chidambaram (2003a), 67, 68 Sree and Chidambaram (2003b), 142,

143, 144 Sree and Chidambaram (2003c), 141,

144


Sree and Chidambaram (2004), 147, 148, 445^46, 447-448

Sree and Chidambaram (2005a), 419 Sree and Chidambaram (2005b), 261 Sree et al. (2004), 49, 133, 173, 406-

407 Srividya and Chidambaram (1997), 79 Srinivas and Chidambaram (2001), 64,

219,417 St. Clair (1997), 29, 195, 253 Streeter et al. (2003), 245 Sung era/. (1996), 313, 316 Suyama(1992),43, 169,326 Suyama (1993), 227-228 Syrcos and Kookos (2005), 161

Tachibana (1984), 279 Taguchi and Araki (2000), 63, 84, 92,

120-121, 122, 128, 218, 271, 296, 366-367, 382, 401-402

Tan etal. (1996), 58, 117, 177 Tan et al. (1998a), 282-283 Tan et al. (1998b), 83, 182, 282 Tan et al. (1999a), 209, 237, 254 Tan era/. (1999b), 407 Tan era/. (1999c), 134 Tan era/. (2001), 237, 254 Tang et al. (2002), 74, 240 Tavakoli and Fleming (2003), 39 Tavakoli and Tavakoli (2003), 187 Taylor, 17 Thomasson (1997), 58, 59, 81 Tinham (1989), 235 Toshiba, 11, 16 Tsang and Rad (1995), 203, 307 Tsang et al. (1993), 62, 203, 210, 307,

308 Turnbull, 12 Tyreus and Luyben (1992), 78

Ultimate cycle, 25, 60, 70-71, 88, 91, 124, 146, 179-180, 207, 210, 220-221, 235-237, 252, 253, 254, 255, 257, 267, 281, 289, 330-331, 335, 340, 353, 373, 429

Valentine and Chidambaram (1997a), 47, 171

Valentine and Chidambaram (1997b), 405

Valentine and Chidambaram (1998), 131

VanDoren (1998), 224, 257 Van der Grinten (1963), 151, 168, 279,

325 Venkatashankar and Chidambaram

(1994), 131 Visioli (2001), 259, 260, 403, 404 Viteckova (1999), 43, 79, 280, 326-

327 Viteckova and Vitecek (2002), 44, 173 Viteckova and Vitecek (2003), 44, 173 Viteckova et al. (2000a), 43, 79, 280,

326-327 Viteckova et al. (2000b), 43, 327 Voda and Landau (1995), 43, 51-52 Vrancic (1996), 71, 75, 127, 129, 247,

457 Vrancic and Lumbar (2004), 459, 460-

461 Vrancic era/. (1996), 71, 127 Vrancic et al. (1999), 247, 455^56,

458 Vrancic et al. (2004a), 127, 459, 460-

461 Vrancic et al. (2004b), 127

Wade (1994), 73, 238 Wang and Clements (1995), 326 Wang and Cai (2001), 296 Wang and Cai (2002), 296, 418 Wang and Cluett (1997), 78, 260 Wang and Cluett (2000), 54-55, 170-

171 Wang and Jin (2004), 427, 442 Wang and Shao (1999), 323 Wang and Shao (2000a), 56 Wang and Shao (2000b), 324 Wang era/. (1995a), 162, 164, 383, 384 Wang era/. (1995b), 71,242 Wang era/. (1999), 323

Index 545

Wang et al. (2000a), 55 Wang et al. (2000b), 384, 392 Wang et al. (2001a), 384, 392 Wang et al. (2001b), 303 Wang et al. (2002), 204-205 Wills (1962a), 315, 325, 330 Wills (1962b), 310, 315 Wilton (1999), 41-42, 319-320 Witt and Waggoner (1990), 194-195,

196, 198,200,202 Wojsznis and Blevins (1995), 179 Wojsznis et al. (1999), 179-180, 237 Wolfe (1951), 28, 76, 149

Xu and Shao (2003a), 296 Xu and Shao (2003b), 419 Xu and Shao (2003c), 296 Xu et al (2004), 58

Yokogawa, 8, 11, 13 Young (1955), 72 Yu (1988), 30-31, 33-34, 35 Yu (1999), 237

Zhang (1994), 46, 388, 394 Zhang and Xu (2002), 139, 430 Zhang era/. (1996), 206 Zhang etal. (1999), 81,282 Zhang et al. (2002), 182, 192, 206, 333 Zhong and Li (2002), 59, 81, 338 Zhuang and Atherton (1993), 33, 36,

39,40, 158, 159, 163, 164-165, 176-177,213-214

Ziegler and Nichols (1942), 27, 70, 76, 154, 235

Zou and Brigham (1998), 261 Zou etal. (1997), 178,261

Yang and Clarke (1996), 321, 353 Yi and De Moor (1994), 246

Handbook of

Controller Timing Rules

2nd Edition

. .ic vast majority ol automatic controllers used to compensate industrial processes arc ol I'l or I'll) type. This hook comprehensively compiles, using a unified notation, tuning rules for these controllers proposed over the last sewn decades i 1935-2005). The tuning rides are carefully categorized and application information about each rule is given. The book ''•"•"sses controller architecture and process modeling

;. as well as the performance and robustness of loops compensated with l'l or I'll) controllers. This unique publication brings together in an easv-to-u.se format material previously published in a large number ol papers and hooks.

This wholly revised second edition extends the presentation ol l'l and I ' l l ) controller tuning rules, lor single variable processes with time delays, to include additional rules compiled since the first edition was published in 2008.

Key Features • Addresses the needs of a niche market where no

comparable book is available

A comprehensive compilation of PI and FID controller tuning rules

Makes t h e tun ing rules easily accessible to researchers and practi t ioners through unified notation

Highlights the marked increase in the number of tuning rules compiled, from 600 in the first edition to 1,134 in this second edition

Imperial College Press

P424 he ISBN 1-86094 <>?'/ 1

www.icpress.co.uk

http://easv-to-u.se

http://www.icpress.co.uk

control handbook

Documents