P575.TP.indd 1 3/20/09 5:16:12 PMHANDBOOK OFPI AND PID CONTROLLERTUNING RULES3rd EditionThis page intentionally left blank This page intentionally left blankImperial College PressICPAidan ODwyerDublin Institute of Technology, IrelandP575.TP.indd 2 3/20/09 5:16:18 PMHANDBOOK OFPI AND PID CONTROLLERTUNING RULES3rd EditionBritish Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.Published byImperial College Press57 Shelton StreetCovent GardenLondon WC2H 9HEDistributed byWorld Scientific Publishing Co. Pte. Ltd.5 Toh Tuck Link, Singapore 596224USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601UK office: 57 Shelton Street, Covent Garden, London WC2H 9HEPrinted in Singapore.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 fromthe publisher.ISBN-13 978-1-84816-242-6ISBN-10 1-84816-242-1Typeset by Stallion PressEmail: enquiries@stallionpress.comAll rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic ormechanical, including photocopying, recording or any information storage and retrieval system now known or tobe invented, without written permission from the Publisher.Copyright 2009 by Imperial College PressHANDBOOK OF PI AND PID CONTROLLER TUNING RULES(3rd Edition)Steven - Hdbook of PI & PID (3rd).pmd 9/2/2009, 7:13 PM 1Dedication Once more, this book is dedicated with love to Angela, Catherine and Fiona and to my parents, Sean and Lillian. b720_FM 17-Mar-2009 FA b720_FM 17-Mar-2009 FAThis page intentionally left blank This page intentionally left blankvii Preface Proportional integral (PI) and proportional integral derivative (PID) controllers have been at the heart of control engineering practice for over seven decades. However, in spite of this, the PID controller has not received much attention from the academic research community until the past two decades, when work by K.J. strm, T. Hgglund and F.G. Shinskey, among others, 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 accessible, being scattered throughout the control literature; in addition, the notation used is not unified. The purpose of this book, now in its third edition, 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. In this edition, the structure of the book has been modified from the previous edition, with controller tuning rules for non-self-regulating process models being summarised in a different chapter from those for self-regulating process models. It is the authors 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 wish to thank the School of Electrical Engineering Systems, 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 years and to my family, Angela, Catherine and Fiona, for their love and understanding. Aidan ODwyer b720_FM 17-Mar-2009 FA b720_FM 17-Mar-2009 FAThis page intentionally left blank This page intentionally left blankix Contents Preface ........................................................................................................ vii 1. Introduction ........................................................................................................ 1 1.1 Preliminary Remarks ...................................................................................... 1 1.2 Structure of the Book...................................................................................... 2 2. Controller Architecture........................................................................................... 4 2.1 Introduction .................................................................................................. 4 2.2 Comments on the PID Controller Structures . ................................................ 11 2.3 Process Modelling .......................................................................................... 12 2.3.1 Self-regulating process models . ........................................................ 12 2.3.2 Non-self-regulating process models................................................... 14 2.4 Organisation of the Tuning Rules................................................................... 16 3. Controller Tuning Rules for Self-Regulating Process Models ............................... 18 3.1 Delay Model ................................................................................................. 18 3.1.1 Ideal PI controller Table 2 .............................................................. 18 3.1.2 Ideal PID controller Table 3 .......................................................... 23 3.1.3 Ideal controller in series with a first order lag Table 4 ................ 24 3.1.4 Classical controller Table 5 ........................................................... 25 3.1.5 Generalised classical controller Table 6 ........................................ 26 3.1.6 Two degree of freedom controller 1 Table 7 .................................. 27 3.2 Delay Model with a Zero ............................................................................... 28 3.2.1 Ideal PI controller Table 8 . ............................................................ 28 3.3 FOLPD Model ............................................................................................... 30 3.3.1 Ideal PI controller Table 9 .............................................................. 30 3.3.2 Ideal PID controller Table 10 ........................................................ 78 3.3.3 Ideal controller in series with a first order lag Table 11 ................ 118 3.3.4 Controller with filtered derivative Table 12.................................... 122 3.3.5 Classical controller Table 13 ........................................................ 134 3.3.6 Generalised classical controller Table 14 . ..................................... 149 3.3.7 Two degree of freedom controller 1 Table 15 ............................. 152 3.3.8 Two degree of freedom controller 2 Table 16 ............................. 168 3.3.9 Two degree of freedom controller 3 Table 17 ............................. 170 b720_FM 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules x 3.4 FOLPD Model with a Zero ............................................................................ 180 3.4.1 Ideal PI controller Table 18 ............................................................ 180 3.4.2 Ideal controller in series with a first order lag Table 19 ................ 182 3.5 SOSPD Model .............................................................................................. 183 3.5.1 Ideal PI controller Table 20 .......................................................... 183 3.5.2 Ideal PID controller Table 21 ........................................................ 206 3.5.3 Ideal controller in series with a first order lag Table 22 ................ 232 3.5.4 Controller with filtered derivative Table 23.................................... 236 3.5.5 Classical controller Table 24 ........................................................ 238 3.5.6 Generalised classical controller Table 25 . ..................................... 251 3.5.7 Two degree of freedom controller 1 Table 26 ............................. 253 3.5.8 Two degree of freedom controller 3 Table 27 ............................. 264 3.6 SOSPD Model with a Zero ............................................................................ 277 3.6.1 Ideal PI controller Table 28 ......................................................... 277 3.6.2 Ideal PID controller Table 29 ........................................................ 279 3.6.3 Ideal controller in series with a first order lag Table 30 ............... 282 3.6.4 Controller with filtered derivative Table 31.................................... 284 3.6.5 Classical controller Table 32 ......................................................... 286 3.6.6 Generalised classical controller Table 33 ....................................... 288 3.6.7 Two degree of freedom controller 1 Table 34 .............................. 289 3.6.8 Two degree of freedom controller 3 Table 35 .............................. 292 3.7 TOSPD Model .............................................................................................. 293 3.7.1 Ideal PI controller Table 36 . .......................................................... 293 3.7.2 Ideal PID controller Table 37 ......................................................... 296 3.7.3 Ideal controller in series with a first order lag Table 38 ................ 297 3.7.4 Controller with filtered derivative Table 39 .................................... 298 3.7.5 Two degree of freedom controller 1 Table 40 ................................. 299 3.7.6 Two degree of freedom controller 3 Table 41 . ............................... 302 3.8 Fifth Order System Plus Delay Model ........................................................... 303 3.8.1 Ideal PID controller Table 42 ......................................................... 303 3.8.2 Controller with filtered derivative Table 43 .................................... 305 3.8.3 Two degree of freedom controller 1 Table 44 . ............................... 308 3.9 General Model . .............................................................................................. 310 3.9.1 Ideal PI controller Table 45 ........................................................... 310 3.9.2 Ideal PID controller Table 46 ......................................................... 312 3.9.3 Ideal controller in series with a first order lag Table 47 ................ 315 3.9.4 Controller with filtered derivative Table 48 .................................. 316 3.9.5 Two degree of freedom controller 1 Table 49 . .............................. 317 3.10 Non-Model Specific ...................................................................................... 318 3.10.1 Ideal PI controller Table 50 ............................................................ 318 3.10.2 Ideal PID controller Table 51 ......................................................... 324 3.10.3 Ideal controller in series with a first order lag Table 52 . ................ 332 3.10.4 Controller with filtered derivative Table 53 ................................... 336 3.10.5 Classical controller Table 54 ......................................................... 341 3.10.6 Generalised classical controller Table 55 ...................................... 343 b720_FM 17-Mar-2009 FAContents xi 3.10.7 Two degree of freedom controller 1 Table 56 . .............................. 346 3.10.8 Two degree of freedom controller 3 Table 57 . .............................. 349 4. Controller Tuning Rules for Non-Self-Regulating Process Models ....................... 350 4.1 IPD Model ..................................................................................................... 350 4.1.1 Ideal PI controller Table 58 . .......................................................... 350 4.1.2 Ideal PID controller Table 59 ...................................................... 359 4.1.3 Ideal controller in series with a first order lag Table 60 ................ 364 4.1.4 Controller with filtered derivative Table 61 .................................. 366 4.1.5 Classical controller Table 62 ......................................................... 368 4.1.6 Generalised classical controller Table 63 ..................................... 371 4.1.7 Two degree of freedom controller 1 Table 64 . .............................. 372 4.1.8 Two degree of freedom controller 2 Table 65 . .............................. 378 4.1.9 Two degree of freedom controller 3 Table 66 . .............................. 381 4.2 IPD Model with a Zero ................................................................................. 383 4.2.1 Ideal PI controller Table 67 . .......................................................... 383 4.3 FOLIPD Model .............................................................................................. 385 4.3.1 Ideal PI controller Table 68 ......................................................... 385 4.3.2 Ideal PID controller Table 69 ........................................................ 388 4.3.3 Ideal controller in series with a first order lag Table 70 ................ 392 4.3.4 Controller with filtered derivative Table 71 ................................... 394 4.3.5 Classical controller Table 72 ......................................................... 395 4.3.6 Generalised classical controller Table 73 ...................................... 397 4.3.7 Two degree of freedom controller 1 Table 74 .............................. 399 4.3.8 Two degree of freedom controller 2 Table 75 .............................. 416 4.3.9 Two degree of freedom controller 3 Table 76 .............................. 418 4.4 FOLIPD Model with a Zero ........................................................................ 420 4.4.1 Ideal PID controller Table 77 . ....................................................... 420 4.4.2 Ideal controller in series with a first order lag Table 78 ............... 422 4.4.3 Classical controller Table 79 ......................................................... 423 4.5 PD I2Model..................................................................................................... 424 4.5.1 Ideal PID controller Table 80 . ....................................................... 424 4.5.2 Classical controller Table 81 ........................................................ 425 4.5.3 Two degree of freedom controller 1 Table 82 .............................. 426 4.5.4 Two degree of freedom controller 2 Table 83 ................................ 427 4.5.5 Two degree of freedom controller 3 Table 84 .............................. 429 4.6 SOSIPD Model ............................................................................................. 430 4.6.1 Ideal PI controller Table 85 . .......................................................... 430 4.6.2 Two degree of freedom controller 1 Table 86 . .............................. 431 4.7 SOSIPD Model with a Zero............................................................................ 436 4.7.1 Classical controller Table 87 ......................................................... 436 4.8 TOSIPD Model............................................................................................... 437 4.8.1 Two degree of freedom controller 1 Table 88 .............................. 437 b720_FM 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules xii 4.9 General Model with Integrator ....................................................................... 438 4.9.1 Ideal PI controller Table 89 . .......................................................... 438 4.9.2 Two degree of freedom controller 1 Table 90 ................................ 439 4.10 Unstable FOLPD Model ................................................................................ 440 4.10.1 Ideal PI controller Table 91 ............................................................ 440 4.10.2 Ideal PID controller Table 92 ........................................................ 447 4.10.3 Ideal controller in series with a first order lag Table 93 ............... 455 4.10.4 Classical controller Table 94 ....................................................... 458 4.10.5 Generalised classical controller Table 95 ....................................... 462 4.10.6 Two degree of freedom controller 1 Table 96 ................................ 463 4.10.7 Two degree of freedom controller 2 Table 97 .............................. 473 4.10.8 Two degree of freedom controller 3 Table 98 .............................. 475 4.11 Unstable FOLPD Model with a Zero ........................................................... 480 4.11.1 Ideal PI controller Table 99 ............................................................ 480 4.11.2 Ideal controller in series with a first order lag Table 100 ............. 481 4.11.3 Generalised classical controller Table 101 .................................... 483 4.11.4 Two degree of freedom controller 1 Table 102 ............................ 484 4.12 Unstable SOSPD Model (one unstable pole) ................................................ 486 4.12.1 Ideal PI controller Table 103 ........................................................ 486 4.12.2 Ideal PID controller Table 104 . ..................................................... 488 4.12.3 Ideal controller in series with a first order lag Table 105 .............. 490 4.12.4 Classical controller Table 106 ........................................................ 491 4.12.5 Two degree of freedom controller 1 Table 107 .............................. 497 4.12.6 Two degree of freedom controller 3 Table 108 .............................. 503 4.13 Unstable SOSPD Model (two unstable poles) ............................................... 506 4.13.1 Ideal PID controller Table 109 ...................................................... 506 4.13.2 Generalised classical controller Table 110 . ................................... 508 4.13.3 Two degree of freedom controller 2 Table 111 .............................. 509 4.14 Unstable SOSPD Model with a Zero.............................................................. 511 4.14.1 Ideal PI controller Table 112 . ........................................................ 511 4.14.2 Ideal controller in series with a first order lag Table 113 .............. 513 4.14.3 Generalised classical controller Table 114 ................................... 516 4.14.4 Two degree of freedom controller 1 Table 115 .............................. 518 4.14.5 Two degree of freedom controller 3 Table 116 .............................. 520 5. Performance and Robustness Issues in the Compensation of FOLPD Processes with PI and PID Controllers ................................................................... 521 5.1 Introduction ................................................................................................... 521 5.2 The Analytical Determination of Gain and Phase Margin ............................. 522 5.2.1 PI tuning formulae ............................................................................... 522 5.2.2 PID tuning formulae ........................................................................... 525 5.3 The Analytical Determination of Maximum Sensitivity ................................ 529 5.4 Simulation Results.......................................................................................... 529 b720_FM 17-Mar-2009 FAContents xiii 5.5 Design of Tuning Rules to Achieve Constant Gain and Phase Margins, for All Values of Delay................................................................................... 534 5.5.1 PI controller design.............................................................................. 534 5.5.1.1 Processes modelled in FOLPD form...................................... 534 5.5.1.2 Processes modelled in IPD form ........................................... 536 5.5.2 PID controller design .......................................................................... 539 5.5.2.1 Processes modelled in FOLPD form classical controller.... 539 5.5.2.2 Processes modelled in SOSPD form series controller......... 541 5.5.2.3 Processes modelled in SOSPD form with a negative zero classical controller ...................................................... 542 5.5.3 PD controller design . .......................................................................... 542 5.6 Conclusions . .................................................................................................. 543 Appendix 1 Glossary of Symbols and Abbreviations................................................. 544 Appendix 2 Some Further Details on Process Modelling........................................... 551 Bibliography ........................................................................................................ 565 Index ........................................................................................................ 599 b720_FM 17-Mar-2009 FA This page intentionally left blank This page intentionally left blank1 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 510% 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; strm and Hgglund, 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 (strm and Hgglund, 1995). 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 in 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 does not appear to have improved more recently, 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% b720_Chapter-01 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules 2of 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, pp. 12) 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 for a period of over seventy years. 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. 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 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 forgotten 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 in another. This chapter also details the process models used to define the controller tuning rules. Chapters 3 and 4 detail, in tabular form, PI and PID controller tuning rules (and their variations), as applied to a wide variety of self-regulating process models and non-self-regulating process models, respectively. One-hundred-and-sixteen such tables are provided altogether. To allow the reader to access data readily, the author has arranged that each table starts on its own page; 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, 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 b720_Chapter-01 17-Mar-2009 FAChapter 1: Introduction 3 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 used for the tuning rules; a glossary of the symbols used is provided in Appendix 1. Appendix 2 outlines the range of methods 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), strm and Hgglund (1995), (2006), 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. b720_Chapter-01 17-Mar-2009 FA4 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: ) s ( E ) s ( G ) s ( Uc= (2.1) with ) s Ts T11 ( K ) s ( Gdic c + + = (2.2) and with cK = proportional gain, iT = integral time constant and dT = derivative time constant. If =iT and 0 Td = (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 PID controller structure have been proposed (indeed, the PI controller structure is itself a subset of the PID controller structure). As Tan et al. (1999a) suggest, one 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. A substantial number of the variations in the controller structures used may be b720_Chapter-02 17-Mar-2009 FAChapter 2: Controller Architecture 5 summarised by controller structure supersets. In this book, nine such supersets are specified, which allows a sensible restriction on the number of tables that need to be detailed. The controller structures specified are detailed below. 1. The ideal PI controller structure: ||.|

\|+ =s T11 K ) s ( Gic c (2.3) 2. The ideal PID controller structure: ||.|

\|+ + = s Ts T11 K ) s ( Gdic c (2.4) This controller structure 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). A variation of the controller is labelled the parallel controller structure (McMillan, 1994). s Ts T1K ) s ( Gdic c + + = (2.5) This variation has also been labelled the ideal parallel, non-interacting, independent or gain independent algorithm. The ideal PID 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 9030 and 9070 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 (strm and Hgglund, 1988) (g) Yokogawa Field Control Station (FCS) PID algorithm (EZYtune, 2003). b720_Chapter-02 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules 63. Ideal controller in series with a first order lag: 1 s T1s Ts T11 K ) s ( Gfdic c+||.|

\|+ + = (2.6) 4. Controller with filtered derivative: ||||.|

\|++ + =NTs 1s Ts T11 K ) s ( Gddic c (2.7) 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 30 N 2 (McMillan, 1994; EZYtune, 2003) (g) Siemens Teleperm/PSC7 ContC/PCS7 CTRL PID products with N = 10 (ISMC, 1999) and the S7 FB41 CONT_C PID product (EZYtune, 2003). 5. Classical controller: 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). NTs 1sT 1s T11 K ) s ( Gddic c++||.|

\|+ = (2.8) b720_Chapter-02 17-Mar-2009 FAChapter 2: Controller Architecture 7 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 10 N 33 . 3 (McMillan, 1994) (c) Foxboro EXACT Model 761 product with N = 10 (McMillan, 1994) (d) Honeywell UDC6000 product with N = 8 (strm and Hgglund, 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). A subset of the classical PID controller is the so-called series controller structure, also labelled the interacting controller or the analog algorithm (McMillan, 1994) or the dependent controller (EZYtune, 2003). ( )dic csT 1s T11 K ) s ( G +||.|

\|+ = (2.9) The structure is used in the following products: (a) Turnbull TCS6000 series product (McMillan, 1994) (b) Alfa-Laval Automation ECA400 product (strm and Hgglund, 1995) (c) Foxboro EXACT 760/761 product (strm and Hgglund, 1995). A further related structure is one labelled the interacting controller (Fertik, 1975). ||||.|

\|++||.|

\|+ =NTs 1s T1s T11 K ) s ( Gddic c (2.10) 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). b720_Chapter-02 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules 86. Generalised classical controller: ||.|

\|+ ++ +||||.|

\|++ + =22 f 1 f22 f 1 f 0 fddic cs a s a 1s b s b bsNT1s Ts T11 K ) s ( G (2.11) 7. Two degree of freedom controller 1: [ ] [ ]) s ( RsNT1s T 1s T11 K ) s ( Uddic||||.|

\|+ + + = ) s ( YsNT1s Ts T11 Kddic||||.|

\|++ + (2.12) 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 (Mizutani and Hiroi, 1991). This structure is used in the following products: (a) Bailey Net 90 PID PV and SP product (McMillan, 1994) (b) Yokogawa SLPC products with 1 = , 1 = , 10 N = (McMillan, 1994) (c) Omron E5CK digital controller with 1 = and 3 N = (ISMC, 1999). Notable subsets of this controller structure are: ) s ( YsNT1s Ts T11 K ) s ( Rs T11 K ) s ( Uddicic||||.|

\|++ + ||.|

\|+ = (2.13) which 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). b720_Chapter-02 17-Mar-2009 FAChapter 2: Controller Architecture 9 ) s ( Y s Ts T11 K ) s ( Rs T11 K ) s ( Udicic ||.|

\|+ + ||.|

\|+ = (2.14) Also labelled the PI+D controller structure (Chen, 1996) or the dependent, ideal, non-interacting controller structure (Cooper, 2006a), it 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). ) s ( Y s Ts T11 K ) s ( Rs T1K ) s ( Udicic ||.|

\|+ + ||.|

\|= (2.15) which is used in the following products: (a) Toshiba AdTune TOSDIC 211D8 product (Shigemasa et al., 1987) (b) Honeywell TDC3000 Process Manager Type C non-interactive mode product (ISMC, 1999). 8. Two degree of freedom controller 2: ( ) ) s ( Y K ) s ( Y ) s ( R ) s ( Fs a 1s b 1sNT1s Ts T11 K ) s ( U01 f1 fddic ++||||.|

\|++ + =, 45 f34 f23 f 2 f45 f34 f23 f 2 fs a s a s a s a 1s b s b s b s b 1) s ( F+ + + ++ + + += (2.16) b720_Chapter-02 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules 109. Two degree of freedom controller 3: ) s ( Y ) s ( F ) s ( R ) s ( F ) s ( U2 1 = , [ ] [ ] ( )||||.|

\|+ +++ + =sNT1s T 1s T 1 s T11 K ) s ( Fddi ic 122 f 1 f1 fs a s a 1s b 1+ ++, [ ] [ ] [ ]||||.|

\|+ + + =sNT1s T 1s T11 K ) s ( Fddic 224 f 3 f2 fs a s a 1s b 1+ ++ ( )s a 1s b b K5 f4 f 3 f 0++ (2.17) Notable subsets of this controller structure are: ( ) ||.|

\|((

++ ||.|

\|+ = ) s ( Ys N T 1s T1 ) s ( Rs T11 K ) s ( Uddic (2.18) Also labelled the reset-feedback controller structure (Huang et al., 1996), it is used in the following products: (a) Bailey Fisher and Porter 53SL6000 and 53MC5000 products (ISMC, 1999) (b) Moore Model 352 Single-Loop Controller product (Wade, 1994). ( ) ||.|

\|++||.|

\|+ = ) s ( Ys N T 1s T 1) s ( Rs T11 K ) s ( Uddic (2.19) Also labelled the industrial controller structure (Kaya and Scheib, 1988), it is used in the following products: (a) Fisher-Rosemount Provox product with 8 N = (ISMC, 1999; McMillan, 1994) (b) Foxboro Model 761 product with 10 N = (McMillan, 1994) (c) Fischer-Porter Micro DCI product (McMillan, 1994) b720_Chapter-02 17-Mar-2009 FAChapter 2: Controller Architecture 11 (d) Moore Products Type 352 controller with 30 N 1 (McMillan, 1994) (e) SATT Instruments EAC400 product with 33 . 8 N = (McMillan, 1994) (f) Taylor Mod 30 ESPO product with 7 . 16 N = (McMillan, 1994) (g) Honeywell TDC3000 Process Manager Type B, interactive mode product with 10 N = (ISMC, 1999). ) s ( YNsT1sT 1s T11 K ) s ( Rs TK) s ( Uddicic||||.|

\|++||.|

\|+ = (2.20) which 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). 2.2 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 2.4 and 2.5) 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 (2.22) and the series PID controller is given in equation (2.23). ||.|

\|+ + = s Ts T11 K ) s ( Gdpipcp cp (2.22) ( )dsiscs cssT 1s T11 K ) s ( G +||.|

\|+ = (2.23) Then, it may be shown that csisdscpKTT1 K ||.|

\|+ = , ( )ds is ipT T T + = , dsds isisdpTT TTT ||.|

\|+= . b720_Chapter-02 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules 12Similarly, it may be shown that, provided dp ipT 4 T > , ||.|

\| + =ipdpcp csTT4 1 1 K 5 . 0 K , ||.|

\| + =ipdpip isTT4 1 1 T 5 . 0 T , ||.|

\| =ipdpdp dsTT4 1 1 T 5 . 0 T . strm and Hgglund (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 isT and dsT . ODwyer (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, it is straightforward to show that controller structures (2.6), (2.7) and (2.8) are subsets of a more general controller structure, and controller parameters may be transformed readily from one structure to another. 2.3 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. Process models may be classified as self-regulating or non-self-regulating, and the models that fall into these categories are now detailed. 2.3.1 Self-regulating process models 1. Delay model: msm me K ) s ( G = 2. Delay model with a zero: ms3 m m me ) s T 1 ( K ) s ( G + = or ms4 m m me ) s T 1 ( K ) s ( G = b720_Chapter-02 17-Mar-2009 FAChapter 2: Controller Architecture 13 3. First order lag plus time delay (FOLPD) model: msmmsT 1e K) s ( Gm+= 4. FOLPD model with a zero: 1 ms3 m mmsT 1e ) sT 1 ( K) s ( Gm++= or 1 ms4 m mmsT 1e ) sT 1 ( K) s ( Gm+= 5. Second order system plus time delay (SOSPD) model: = ) s ( Gm1 s T 2 s Te K1 m m2 21 msmm+ + or ( )( ) s T 1 s T 1e K) s ( G2 m 1 msmmm+ += 6. SOSPD model with a zero: 1 s T 2 s Te ) sT 1 ( K) s ( G1 m m2 21 ms3 m mmm+ ++= or ) sT 1 )( sT 1 (e ) sT 1 ( K) s ( G2 m 1 ms3 m mmm+ ++= ( )( )( )2 m 1 ms4 m mmsT 1 sT 1e sT 1 K) s ( Gm+ += or 1 s T 2 s Te ) sT 1 ( K) s ( G1 m m2 21 ms4 m mmm+ += 7. Third order system plus time delay (TOSPD) model: ( )3322 1s 3322 1 mms a s a s a 1e ) s b s b s b 1 ( K) s ( Gm+ + ++ + += or = ) s ( Gm( )( )( )3 m 2 m 1 msmsT 1 sT 1 sT 1e Km+ + + or ( )( ) 1 s T 1 s T 2 s Te ) 1 s T ( K) s ( G2 m 1 m m2 21 ms3 m mmm+ + ++= 8. Fifth order system plus delay model: ( )55443322 1s 5s443322 1 mms a s a s a s a s a 1e ) s b s b s b s b s b 1 ( K) s ( Gm+ + + + ++ + + + += b720_Chapter-02 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules 149. General model. 10. Non-model specific. Corresponding tuning rules may also apply to non-self-regulating processes. 2.3.2 Non-self-regulating process models 1. Integral plus time delay (IPD) model: se K) s ( Gmsmm = 2. IPD model with a zero: ( )se sT 1 K) s ( Gms3 m mm += or ( )se sT 1 K) s ( Gms4 m mm = 3. First order lag plus integral plus time delay (FOLIPD) model: ( )msmmsT 1 se K) s ( Gm+= 4. FOLIPD model with a zero: ( )( )1 ms3 m mmsT 1 se sT 1 K) s ( Gm++= or ( )( )1 ms4 m mmsT 1 se sT 1 K) s ( Gm+= 5. Integral squared plus time delay ( PD I2) model: = ) s ( Gm2smse Km 6. Second order system plus integral plus time delay (SOSIPD) model: = ) s ( Gm( )21 msmsT 1 se Km+ or ) 1 s T 2 s T ( se K) s ( G1 m m2 21 msmmm+ += 7. SOSIPD model with a zero: ( )( )( )2 m 1 ms3 m mmsT 1 sT 1 se sT 1 K) s ( Gm+ ++= 8. Third order system plus integral plus time delay (TOSIPD) model: ( )( )( )3 m 2 m 1 msmmsT 1 sT 1 sT 1 se K) s ( Gm+ + += or ( )31 msmmsT 1 se K) s ( Gm+= b720_Chapter-02 17-Mar-2009 FAChapter 2: Controller Architecture 15 9. General model with integrator: ( )n1 msmmsT 1 se K) s ( Gm+= or ( ) ( )( ) ( )m4 d 4 32 n 2 1si ii m i m2 2i m i mii m i m2 2i mii mmme1 s T 2 s T 1 s T1 s T 2 s T 1 s TsK) s ( G + + ++ + += 10. Unstable FOLPD model: = ) s ( Gm1 s Te Kmsmm 11. Unstable FOLPD model with a zero: ( )1 s Te sT 1 K) s ( G1 ms3 m mmm+= or = ) s ( Gm1 s Te ) sT 1 ( Kms4 m mm 12. Unstable SOSPD model (one unstable pole): = ) s ( Gm( )( )2 m 1 msmsT 1 1 s Te Km+ 13. Unstable SOSPD model (two unstable poles): = ) s ( Gm( )( ) 1 s T 1 s Te K2 m 1 msmm 14. Unstable SOSPD model with a zero: = ) s ( Gm ( )1 s T 2 s Te s T 1 K1 m m2 21 ms3 m mm+ + or ( )1 s T 2 s Te s T 1 K1 m m2 21 ms3 m mm+ + + or = ) s ( Gm1 s T 2 s Te ) sT 1 ( K1 m m2 21 ms4 m mm+ b720_Chapter-02 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules 16Table 1 shows the number of tuning rules defined for each PI/PID controller structure and types of process model. The following data is key to the process model type: Model 1: Stable FOLPD model (with or without a zero) Model 2: Stable SOSPD model (with or without a zero) Model 3: Other stable models Model 4: Non-model specific Model 5: Models with an integrator Model 6: Open loop unstable models Table 1: PI/PID controller structure and tuning rules a summary Process model Controller Equation 1 2 3 4 5 6 Total (2.3) 261 63 58 59 90 32 563 (2.4) 140 82 20 63 66 35 406 (2.6) 20 23 3 9 16 17 88 (2.7) 36 9 5 11 6 0 67 (2.8) 74 53 1 12 26 17 183 (2.11) 7 7 1 4 5 6 30 (2.12) 74 36 14 9 106 37 276 (2.16) 7 3 0 0 16 8 34 (2.17) 28 15 1 2 8 30 84 Total 649 291 103 169 339 182 1731 Table 1 shows that for the tuning rules defined, 60% are based on a self-regulating (or stable) model, 10% are non-model specific, with the remaining 30% based on a non-self-regulating model. 2.4 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. b720_Chapter-02 17-Mar-2009 FAChapter 2: Controller Architecture 17 (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 into 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. All symbols used in the tables are defined in Appendix 1. b720_Chapter-02 17-Mar-2009 FA 18 Chapter 3 Controller Tuning Rules for Self-Regulating Process Models 3.1 Delay Model msm me K ) s ( G = 3.1.1 Ideal PI controller + =s T11 K ) s ( Gic c Table 2: PI controller tuning rules msm me K ) s ( G = Rule cK iT Comment Process reaction 1m mK 568 . 0 m64 . 3 2m mK 65 . 0 m6 . 2 3m mK 79 . 0 m95 . 3 Callender et al. (1935/6). Model: Method 1 4m mK 95 . 0 m3 . 3 Representative results deduced from graphs. 1 Decay ratio = 0.015; period of decaying oscillation = m5 . 10 . 2 Decay ratio = 0.1; period of decaying oscillation = m8 . 3 Decay ratio = 0.12; period of decaying oscillation = m28 . 6 . 4 Decay ratio = 0.33; period of decaying oscillation = m56 . 5 . +s T11 KicR(s) +E(s) Y(s) U(s)msme K b720_Chapter-03 17-Mar-2009 FAChapter 3: Controller Tuning Rules for Self-Regulating Process Models 19 Rule cK iT Comment m mK2 . 0 m3 . 0 Decay ratio is as small as possible. Two constraints method Wolfe (1951). Model: Method 1 Minimum error integral (regulator mode). Minimum performance index: regulator tuning undefined m5 . 1 s T 1 ) s ( Gi c = Minimum ISE Haalman (1965). Model: Method 1 9 . 1 Ms = ; 36 . 2 Am = ; 0m50 = . Minimum error: step load change Gerry (1998). mK3 . 0 m42 . 0 Model: Method 1 Minimum IAE Gerry and Hansen (1987). mK345 . 0 m455 . 0 Model: Method 1 Minimum IAE Shinskey (1988), p. 123. mK43 . 0 m5 . 0 Model: Method 1 mK 4 . 0 m5 . 0 p. 67. Minimum IAE Shinskey (1994). Model: Method 1 undefined m mK 6 . 1 s T 1 ) s ( Gi c = ; p. 63. Minimum IAE Shinskey (1988), p. 148. mK4255 . 0 uT 25 . 0 Model: Method 1 Minimum IAE Shinskey (1996), p. 167. uK 4 . 0 uT 25 . 0 Model: Method 1 Minimum performance index: servo tuning mK 36 . 0 m47 . 0 Model: Method 1 Minimum IAE Huang and Jeng (2002). Minimum IAE = m377 . 1 . 2 Am = ; 0m60 = . undefined m33 . 1 s T 1 ) s ( Gi c = ; Minimum ISE = m53 . 1 ; 1 Km = . Minimum ISE Keviczky and Cski (1973). Model: Method 1 0.5 m635 . 0 1 Km = Minimum performance index: other tuning Fertik (1975). Model: Method 1 mK 35 . 0 m4 . 0 i cT , K values deduced from graph. strm and Hgglund (2000). Model: Method 1 mK25 . 0 m35 . 0 i cT K maximised subject to 2 Mmax = . b720_Chapter-03 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules 20 Rule cK iT Comment Skogestad (2001). Model: Method 1 mK 16 . 0 m340 . 0 4 . 1 Mmax = Skogestad (2003). Model: Method 1 mK 200 . 0 m318 . 0 6 . 1 Mmax = Skogestad (2001). Model: Method 1 mK 26 . 0 m306 . 0 0 . 2 Mmax = m 1K x m 2x Coefficient values: 1x 2x maxM 1x 2x maxM 0.057 0.400 1.1 0.211 0.342 1.6 0.103 0.389 1.2 0.227 0.334 1.7 0.139 0.376 1.3 0.241 0.326 1.8 0.168 0.363 1.4 0.254 0.320 1.9 strm and Hgglund (2004). Model: Method 1 0.191 0.352 1.5 0.264 0.314 2.0 Direct synthesis: time domain criteria mK 5 . 0 m5 . 0 Step disturbance. Van der Grinten (1963). Model: Method 1 1cK iT Stochastic disturbance. undefined m mK 70 . 2 Critically damped dominant pole. Bryant et al. (1973). Model: Method 1; s T 1 ) s ( Gi c = undefined m mK 50 . 2 Damping factor (dominant pole) = 0.6. undefined m 2x s T 1 ) s ( Gi c = Coefficient values: 2x OS 2x OS 2x OS 2x OS 0.64 105% 1.00 50% 1.5 17% 2 4% 1x m2 Coefficient values: 1x OS 1x OS 1x OS 1x OS Keviczky and Cski (1973). Model: Method 1; representative coefficient values estimated from graphs; 1 Km = . 0.68 5% 0.72 10% 0.75 15% 0.78 20% Kuwata (1987). 2cK iT Model: Method 1 Nomura et al. (1993). mK 223607 . 0 m309017 . 0 Model: Method 1 1 ( )m dm de 5 . 0 1KeKmc = , ( )m dm de 5 . 0 1eTmi = . 2 ( )m mmc267 . 0 1 1 K4 . 0K = , m m m i25 . 0 267 . 0 1 25 . 1 T = . b720_Chapter-03 17-Mar-2009 FAChapter 3: Controller Tuning Rules for Self-Regulating Process Models 21 Rule cK iT Comment undefined m mK 2 OS = 10% Also given by Schlegel (1998); 1 Ms = . undefined m mK 6667 . 2 OS = 0% Kamimura et al. (1994), Manum (2005). Model: Method 1; s T 1 ) s ( Gi c = . undefined m mK 3529 . 2 Quick servo response; negligible overshoot. mK 305 . 0 m279 . 0 3 . 0 = mK 273 . 0 m285 . 0 4 . 0 = mK 218 . 0 m288 . 0 6 . 0 = mK 174 . 0 m276 . 0 8 . 0 = strm and Hgglund (1995), pp. 187189. Model: Method 1 mK 154 . 0 m265 . 0 9 . 0 = mK 388 . 0 m258 . 0 1 . 0 = mK 343 . 0 m270 . 0 2 . 0 = mK 244 . 0 m288 . 0 5 . 0 = mK 195 . 0 m284 . 0 707 . 0 = strm and Hgglund (1995), pp. 187189, strm and Hgglund (2006), pp. 185186. Model: Method 1 mK 135 . 0 m250 . 0 0 . 1 = Hansen (2000). mK 2 . 0 m3 . 0 Model: Method 1 Direct synthesis: frequency domain criteria undefined m mK 45 . 1 s T 1 ) s ( Gi c = mK 075 . 0 m1 . 0 mK 53 . 0 m Buckley (1964). Model: Method 1; dB 2 Mmax = . mK 55 . 0 m59 . 1 Representative results. Schlegel (1998). Model: Method 1 mK 318 . 0 m405 . 0 1 Ms = mK 15 . 0 m35 . 0 Hgglund and strm (2004). Model: Method 1 uK 15 . 0 uT 17 . 0 4 . 1 Ms = strm and Hgglund (2006), p. 243. mK 1677 . 0 m363 . 0 Model: Method 1; 4 . 1 M Mt s = = . Robust Bequette (2003), p. 300. ( )m mm2 K + m5 . 0 Model: Method 1; m > (Yu, 2006, p. 19). Skogestad (2003). Model: Method 1 mK 294 . 0 m5 . 0 IMC based rule. b720_Chapter-03 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules 22 Rule cK iT Comment strm and Hgglund (2006), p. 189. 3 + mmm5 . 05 . 0K1 m5 . 0 Model: Method 1 Urrea et al. (2006). Model: Method 1 undefined m m 1K x s T 1 ) s ( Gi c = ; 2 x 5 . 11 . Other tuning Skogestad (2003). mK 25 . 0 m333 . 0 Model: Method 1 McMillan (2005), p. 35. mK 25 . 0 m25 . 0 Model: Method 1 Ultimate cycle Yu (2006), p. 18. uK 30 . 0 uT 25 . 0 Model: Method 1 3 mT = (aggressive tuning). mT 3 = (robust tuning). b720_Chapter-03 17-Mar-2009 FAChapter 3: Controller Tuning Rules for Self-Regulating Process Models 23 3.1.2 Ideal PID controller + + = s Ts T11 K ) s ( Gdic c Table 3: PID controller tuning rules delay model msm me K ) s ( G = Rule cK iT dT Comment Process reaction 1m mK 749 . 0 m734 . 2 m303 . 0 Callender et al. (1935/6). Model: Method 12m mK 24 . 1 m31 . 1 m303 . 0 Representative results deduced from graphs. Direct synthesis: time domain criteria Ream (1954). Model: Method 1mK 61 . 0 m491 . 0 m015 . 0 Decay ratio % 33 . mK 230 . 0 m315 . 0 m0086 . 0 Kitamori. mK 2269 . 0 m3109 . 0 m0264 . 0 Butterworth. mK 2635 . 0 m3610 . 0 m1911 . 0 Minimum ITAE. Nomura et al. (1993). Model: Method 1mK 3548 . 0 m4860 . 0 m2735 . 0 Binomial. Robust Gong et al. (1998). mK 511 . 0 m511 . 0 m1247 . 0 Model: Method 1 mK 40 . 0 m46 . 0 m149 . 0 Regulator. mK 29 . 0 m40 . 0 m096 . 0 Servo; % 10 overshoot. Wang (2003), pp. 1517. Model: Method 1Deduced from graphs; robust to % 25 uncertainty in process model parameters. 1 Decay ratio = 0.015; period of decaying oscillation = m5 . 10 . 2 Decay ratio = 0.12; period of decaying oscillation = m28 . 6 . + + s Ts T11 KdicR(s) +E(s) Y(s)U(s)msme K b720_Chapter-03 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules 24 3.1.3 Ideal controller in series with a first order lag 1 s T1s Ts T11 K ) s ( Gfdic c++ + = Table 4: PID controller tuning rules delay model msm me K ) s ( G = Rule cK iT dT Comment Robust ( )m mm4 K + m5 . 0 m167 . 0 =fT m2m24167 . 0 2 + Bequette (2003), p. 300. Model: Method 1m7 . 1 > (Yu, 2006, p. 19). + + s Ts T11 KdicR(s) +E(s) Y(s) U(s)s T 11f+msme K b720_Chapter-03 17-Mar-2009 FAChapter 3: Controller Tuning Rules for Self-Regulating Process Models 25 3.1.4 Classical controller +++ =Ns T1s T 1s T11 K ) s ( Gddic c Table 5: PID controller tuning rules delay model msm me K ) s ( G = Rule cK iT dT Comment Process reaction 1m mK70 . 0 m66 . 2 m Hartree et al. (1937). Model: Method 1 2m mK78 . 0 m97 . 2 m5 . 0 Representative results. 1 N = 2. Decay ratio = 0.15; period of decaying oscillation = m49 . 4 . 2 N = 2. Decay ratio = 0.042; period of decaying oscillation = m03 . 5 . +++sNT1s T 1s T11 KddicR(s) +E(s) Y(s)U(s)msme K b720_Chapter-03 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules 26 3.1.5 Generalised classical controller + ++ +++ + =22 f 1 f22 f 1 f 0 fddic cs a s a 1s b s b bsNT1s Ts T11 K ) s ( G Table 6: PID controller tuning rules delay model msme K Rule cK iT dT Comment Ultimate cycle uK 30 . 0 uT 23 . 0 0 Yu (2006), p. 18. Model: Method 10 b0 f = , u 1 fT 12 . 0 b = , 0 b2 f = ,u 1 fT 012 . 0 a = , 0 a2 f = . msme K +E(s) Y(s) U(s)22 f 1 f22 f 1 f 0 fddics a s a 1s b s b bsNT1s Ts T11 K+ ++ +++ +R(s) b720_Chapter-03 17-Mar-2009 FAChapter 3: Controller Tuning Rules for Self-Regulating Process Models 27 3.1.6 Two degree of freedom controller 1 ) s ( RsNT1s TK ) s ( EsNT1s Ts T11 K ) s ( Uddcddic++ ++ + = Table 7: PID controller tuning rules delay model msm me K ) s ( G = Rule cK iT dT Comment Direct synthesis: time domain criteria 1cK m278 . 0 Kuwata (1987). Model: Method 1mK 2 . 0 167 . 4 667 . 1m 0 1 = mK 1 . 0 m278 . 0 Nomura et al. (1993). Model: Method 1mK 1 5 . 2 667 . 1m 0 1 = ; also given by Kuwata (1987). 1 ( )m mmc267 . 0 1 1 K8 . 0K = . ++ +sNT1s Ts T11 KddicR(s) +E(s) Y(s)U(s)cddKsNT1s T++ +msme K b720_Chapter-03 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules 28 3.2 Delay Model with a Zero ms3 m m me ) s T 1 ( K ) s ( G + = or ms4 m m me ) s T 1 ( K ) s ( G = 3.2.1 Ideal PI controller + =s T11 K ) s ( Gic c Table 8: PI controller tuning rules delay model with a zero ms3 m m me ) s T 1 ( K ) s ( G + = or ms4 m m me ) s T 1 ( K ) s ( G = Rule cK iT Comment Direct synthesis: time domain criteria 1iT Smaller 4 m mT . Chidambaram (2002), p. 157. Model: Method 1; s T 1 ) s ( Gi c = . undefined 2iT Larger 4 m mT . 1 ( )2m 4 m m iT 142 . 3 1 637 . 0 T + = . 2 ( )2m 4 m m iT 712 . 4 1 424 . 0 T + = . +s T11 KicR(s) +E(s) Y(s) U(s)Modelb720_Chapter-03 17-Mar-2009 FAChapter 3: Controller Tuning Rules for Self-Regulating Process Models 29 Rule cK iT Comment Direct synthesis: frequency domain criteria 3iT 1 T3 m m < 4iT 1 T3 m m > ( )3 m m mT K 5 . 1 5iT 1 T 03 m m < < Jyothi et al. (2001). Model: Method 1; s T 1 ) s ( Gi c = . undefined 6iT 10 T 03 m m < < 4 m mT K 2 1 T4 m m < 7iT 1 T4 m m > ( )4 m m mT K 5 . 1 + 8iT 1 T 04 m m < < Jyothi et al. (2001). Model: Method 1; s T 1 ) s ( Gi c = . undefined 9iT 10 T 04 m m < < 3 ( )2m 3 m m m iT 870 . 9 1 K 64 . 0 T + = . 4 ( )2m 3 m m m iT 467 . 2 1 K 27 . 1 T + = . 5 3 m mT 419692 . 03 m m ie T K 0309 . 2 T = . 6 ( ) [ ] 962232 . 0 T ln 706636 . 0 expT KT3 m m3 m mi = . 7 ( )2m 4 m m m iT 467 . 2 1 K 27 . 1 T + = . 8 ( ) ( )24 m m 4 m m4 m miT 03091 . 0 T 001945 . 0 5 . 0T KT += . 9 ( ) ( ) ( )34 m m24 m m 4 m m4 m miT 0004883 . 0 T 0141 . 0 T 1452 . 0 6178 . 0T KT + = . b720_Chapter-03 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules 30 3.3 FOLPD Model msmmsT 1e K) s ( Gm+= 3.3.1 Ideal PI controller ||.|

\|+ =s T11 K ) s ( Gic c Table 9: PI controller tuning rules FOLPD model msmmsT 1e K) s ( Gm+= Rule cK iT Comment Process reaction 1m mK 568 . 0 m64 . 3 Callender et al. (1935/6). Model: Method 1 2m mK 690 . 0 m45 . 2 3 . 0Tmm = Ziegler and Nichols (1942). Model: Method 2 m mmKT 9 . 0 m33 . 3 Quarter decay ratio; 1 Tm m . m m m 1K T x m 2x Coefficient values: mmT 1x 2x mmT 1x 2x mmT 1x 2x 0.2 0.68 7.14 1.1 0.90 1.49 2.0 1.20 1.00 0.3 0.70 4.76 1.2 0.93 1.41 2.2 1.28 0.95 0.4 0.72 3.70 1.3 0.96 1.32 2.4 1.36 0.91 0.5 0.74 3.03 1.4 0.99 1.25 2.6 1.45 0.88 0.6 0.76 2.50 1.5 1.02 1.19 2.8 1.53 0.85 0.7 0.79 2.17 1.6 1.06 1.14 3.0 1.62 0.83 0.8 0.81 1.92 1.7 1.09 1.10 3.2 1.71 0.81 0.9 0.84 1.75 1.8 1.13 1.06 3.4 1.81 0.80 Hazebroek and Van der Waerden (1950). Model: Method 2 1.0 0.87 1.61 1.9 1.17 1.03 1 Decay ratio = 0.015; period of decaying oscillation = m5 . 10 . 2 Decay ratio = 0.043; period of decaying oscillation = m28 . 6 . ||.|

\|+s T11 KicR(s) +E(s) Y(s) U(s)msmsT 1e Km+ b720_Chapter-03 17-Mar-2009 FAChapter 3: Controller Tuning Rules for Self-Regulating Process Models 31 Rule cK iT Comment Hazebroek and Van der Waerden (1950) continued. ||.|

\|+1T5 . 0KTmmm mmm m2mT 2 . 1 6 . 1 5 . 3 Tm m > m66 . 1 m mT >> Oppelt (1951). Model: Method 2 3cK m32 . 3 m mT m m mK T 00 . 1 m0 . 3 2 . 0 Tm m = m m mK T 04 . 1 m25 . 2 5 . 0 Tm m = m m mK T 08 . 1 m45 . 1 1 Tm m = Minimum IAE Shinskey (1988), p. 123. Model: Method 1 m m mK T 39 . 1 m 2 Tm m = 6 256 . 1mmT346 . 0 214 . 0mLmcT TTK685 . 0Kmm||.|

\| ||.|

\|= , 123 . 1mm55 . 0T977 . 1mL miT TT214 . 0TTmm||.|

\| ||.|

\|=, 16 . 0T641 . 2TTmmmL+ . 7 041 . 1mmT159 . 0 099 . 0mLmcT TTK874 . 0Kmm+ ||.|

\| ||.|

\|= , 876 . 0mm067 . 0T515 . 4mL miT TT415 . 0TTmm||.|

\| ||.|

\|=+, 1TT16 . 0T641 . 2mLmm +. 8 055 . 1mmT384 . 0 015 . 0mLmcT TTK871 . 0Kmm+ ||.|

\| ||.|

\|= , 867 . 0mm213 . 0T217 . 0mL miT TT444 . 0TTmm||.|

\| ||.|

\|=, 3TT1mL < . 9 451 . 1mmT218 . 0mLmcT TTK513 . 0Kmm||.|

\| ||.|

\|= , 56 . 0mm084 . 0T003 . 0mL miT TT670 . 0TTmm||.|

\| ||.|

\|=. b720_Chapter-03 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules 36 Rule cK iT Comment m 1K x m 2T x Load disturbance model = LsT 11+. Minimum IAE Hill and Venable (1989). Model: Method 1 1x values (this page) and 2x values (next page) deduced from graphs; robust to % 30 process model parameter changes. m mT 0.05 0.07 0.1 0.15 0.2 0.3 0.4 0.5 0.75 1.0 =m LT T 0.05 14.0 9.8 6.2 4.0 3.0 2.08 1.60 1.32 0.95 0.78 =m LT T 0.1 16.8 10.3 7.2 4.3 3.2 2.14 1.63 1.33 0.97 0.77 =m LT T 0.3 17.6 11.8 8.8 5.4 4.0 2.48 1.83 1.46 1.00 0.80 =m LT T 0.5 17.0 11.4 8.6 5.6 4.2 2.67 1.96 1.58 1.06 0.84 =m LT T 0.7 16.4 11.0 8.2 5.4 4.1 2.74 2.04 1.64 1.11 0.88 =m LT T 1.0 15.7 10.5 7.8 5.2 4.0 2.70 2.07 1.68 1.17 0.92 =m LT T 1.2 15.7 10.5 7.8 5.1 4.0 2.69 2.05 1.68 1.18 0.94 =m LT T 1.5 15.8 10.5 7.8 5.1 3.9 2.68 2.01 1.68 1.18 0.95 =m LT T 2.0 15.8 10.5 7.6 5.1 3.9 2.65 2.02 1.65 1.18 0.95 =m LT T 2.5 15.9 10.5 7.8 5.1 3.9 2.64 1.95 1.63 1.17 0.95 =m LT T 3.0 15.8 10.5 7.8 5.1 3.9 2.62 2.01 1.62 1.16 0.94 =m LT T 3.5 15.8 10.4 7.8 5.1 3.9 2.59 1.98 1.61 1.15 0.93 =m LT T 4.0 15.7 10.5 7.8 5.1 3.9 2.60 2.00 1.63 1.13 0.91 =m LT T 4.5 15.6 10.4 7.8 5.1 3.8 2.54 1.98 1.60 1.13 0.92 =m LT T 5.0 15.7 10.3 7.8 5.1 3.8 2.56 1.98 1.61 1.15 0.93 =m LT T 5.5 15.5 10.3 7.8 5.1 3.9 2.55 1.98 1.62 1.17 0.92 =m LT T 6.9 15.5 10.3 7.6 5.1 3.9 2.52 1.98 1.60 1.17 0.92 =m LT T 6.5 15.5 10.3 7.8 5.2 4.0 2.60 2.00 1.62 1.18 0.93 =m LT T 7.0 15.6 10.3 7.8 5.2 3.9 2.56 2.00 1.62 1.18 0.92 b720_Chapter-03 17-Mar-2009 FAChapter 3: Controller Tuning Rules for Self-Regulating Process Models 37 Rule cK iT Comment m 1K x m 2T x Load disturbance model = LsT 11+. Minimum IAE Hill and Venable (1989). Model: Method 1 1x values (previous page) and 2x values (this page) deduced from graphs; robust to % 30 process model parameter changes. m mT 0.05 0.07 0.1 0.15 0.2 0.3 0.4 0.5 0.75 1.0 =m LT T 0.05 1.0 1.0 1.0 1.0 1.0 0.92 0.90 0.88 0.80 0.74 =m LT T 0.1 1.0 1.0 1.0 1.0 1.0 0.92 0.90 0.88 0.80 0.74 =m LT T 0.3 3.4 1.8 1.3 1.0 1.0 0.93 0.88 0.86 0.79 0.73 =m LT T 0.5 4.4 2.8 2.0 1.7 1.2 1.00 0.93 0.87 0.78 0.72 =m LT T 0.7 4.8 3.3 2.5 1.7 1.4 1.10 1.00 0.90 0.79 0.72 =m LT T 1.0 5.4 3.8 3.0 2.2 1.7 1.32 1.11 0.99 0.82 0.74 =m LT T 1.2 5.6 4.0 3.2 2.3 1.8 1.42 1.19 1.06 0.85 0.76 =m LT T 1.5 6.0 4.2 3.4 2.5 2.0 1.54 1.32 1.13 0.91 0.80 =m LT T 2.0 6.3 4.6 3.7 2.6 2.2 1.68 1.42 1.26 1.00 0.86 =m LT T 2.5 6.7 4.7 3.8 2.7 2.3 1.76 1.53 1.32 1.06 0.90 =m LT T 3.0 6.9 4.9 3.9 2.8 2.3 1.84 1.55 1.38 1.11 0.96 =m LT T 3.5 7.1 5.1 4.0 3.0 2.4 1.91 1.62 1.43 1.16 0.99 =m LT T 4.0 7.3 5.2 4.1 3.0 2.5 1.93 1.63 1.46 1.20 1.04 =m LT T 4.5 7.4 5.3 4.2 3.1 2.5 2.01 1.68 1.50 1.21 1.05 =m LT T 5.0 7.5 5.4 4.3 3.1 2.5 2.02 1.70 1.51 1.23 1.06 =m LT T 5.5 7.6 5.4 4.2 3.2 2.6 2.04 1.72 1.52 1.23 1.09 =m LT T 6.0 7.7 5.4 4.4 3.2 2.6 2.08 1.73 1.56 1.25 1.10 =m LT T 6.5 7.9 5.5 4.3 3.2 2.6 2.05 1.74 1.56 1.26 1.10 =m LT T 7.0 7.9 5.5 4.5 3.2 2.7 2.10 1.75 1.57 1.27 1.12 b720_Chapter-03 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules 38 Rule cK iT Comment mK 4 . 1 m24 . 0 11 . 0 Tm m = mK 8 . 1 m52 . 0 25 . 0 Tm m = mK 4 . 1 m75 . 0 43 . 0 Tm m = mK 0 . 1 m68 . 0 67 . 0 Tm m = mK 8 . 0 m71 . 0 0 . 1 Tm m = mK 55 . 0 m60 . 0 50 . 1 Tm m = mK 45 . 0 m54 . 0 33 . 2 Tm m = Minimum IAE Marlin (1995), pp. 301307. Model: Method 1; cK ,iT deduced from graphs; robust to % 25 process model parameter changes. mK 35 . 0 m49 . 0 0 . 4 Tm m = Minimum IAE Huang et al. (1996). Model: Method 1 10cK iT 1T1 . 0mm m m mK T 95 . 0 m4 . 3 1 . 0 Tm m = Minimum IAE Shinskey (1996), p. 38. Model: Method 1 m m mK T 95 . 0 m9 . 2 2 . 0 Tm m = mK 4 . 0 m5 . 0 Time delay dominant. Minimum IAE Edgar et al. (1997), pp. 8-14, 8-15. Model: Method 1 m m mK T 94 . 0 m4 Time constant dominant. Minimum IAE Edgar et al. (1997), p. 8-15. m m mK T 67 . 0 m5 . 3 Model: Method 2 10(((

||.|

\| ||.|

\| ++ = 063 . 0mm9077 . 0mmmmmcT2132 . 5T8196 . 0T6198 . 4 4884 . 6K1K (((

||.|

\| +mmT5961 . 0mmme 7241 . 0T2712 . 7K1, (((

||.|

\| ||.|

\| +||.|

\| + =4mm3mm2mmmmm iT7619 . 10T4348 . 9T4789 . 6T9574 . 3 0064 . 0 T T (((

||.|

\| ||.|

\| +6mm5mmmT2236 . 2T5146 . 7 T . b720_Chapter-03 17-Mar-2009 FAChapter 3: Controller Tuning Rules for Self-Regulating Process Models 39 Rule cK iT Comment 11cK iT 2 . 1 T 1 . 0m m Minimum IAE Arrieta Orozco (2003). Model: Method 10 Also given by Arrieta Orozco and Alfaro Ruiz (2003). Minimum IAE Shinskey (2003). m m mK T 74 . 0 m06 . 4 Model: Method 2 u 1K x u 2T x 1x 2x m mT 1x 2x m mT 0.55 0.91 0.1 0.43 0.45 1.0 0.52 0.81 0.2 0.42 0.33 2.0 0.50 0.64 0.5 uK 5 . 0 u 2T x 2x m mT 2x m mT 2x m mT 0.80 0.1 0.64 0.5 0.43 2.0 Minimum IAE Harriott (1988). Model: Method 1; coefficients of i cT , K deduced from graphs. 0.77 0.2 0.53 1.0 uK 58 . 0 uT 81 . 0 2 . 0 Tm m = uK 54 . 0 uT 66 . 0 5 . 0 Tm m = uK 48 . 0 uT 47 . 0 1 Tm m = Minimum IAE Shinskey (1988), p. 148. Model: Method 1 uK 46 . 0 uT 37 . 0 2 Tm m = Minimum IAE Shinskey (1994), p. 167. ( )m uuT 35 . 0 05 . 3K 12iT Model: Method 1 Minimum IAE Shinskey (1996), p. 121. uK 55 . 0 uT 78 . 0 Model: Method 1; 2 . 0 Tm m = . Minimum ISE Hazebroek and Van der Waerden (1950). 13cK mT 43 . 1 Model: Method 2; 2 . 0 Tm m < . 11 (((

||.|

\|+ =1251 . 1mmmcT6494 . 0 4485 . 0K1K , (((

||.|

\| + =4749 . 0mmm iT8205 . 1 2551 . 0 T T . 12 ||.|

\|((

+ =2mumuu iT172 . 0T855 . 0 87 . 0 T T . 13 ||.|

\| +=mmm mmcT3 . 0 74 . 0KTK . b720_Chapter-03 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules 40 Rule cK iT Comment m m m 1K T x m 2x Coefficient values: mmT 1x 2x mmT 1x 2x mmT 1x 2x 0.2 0.80 7.14 0.7 0.96 2.44 2.0 1.46 1.18 0.3 0.83 5.00 1.0 1.07 1.85 3.0 1.89 0.95 Hazebroek and Van der Waerden (1950) continued. 0.5 0.89 3.23 1.5 1.26 1.41 5.0 2.75 0.81 Minimum ISE Haalman (1965). Model: Method 1 m mmKT 67 . 0 mT maxM = 1.9; mA = 2.36; m = 050 . Minimum ISE Murrill (1967), pp. 358363. Model: Method 5 959 . 0mmmTK305 . 1||.|

\| 739 . 0mm mT 492 . 0T||.|

\| 0 . 1T1 . 0mm ||.|

\|+mm2 1mTx xK1||.|

\|+mm2 12mTx xx Representative coefficient values (deduced from graph): m mT 1x 2x m mT 1x 2x 0.2 0.53 0.82 1.0 0.63 1.12 0.4 0.58 0.88 1.2 0.63 1.17 0.6 0.61 0.97 1.4 0.62 1.20 Minimum ISE Frank and Lenz (1969). Model: Method 1 0.8 0.62 1.05 Minimum ISE Yu (1988). Model: Method 1 14cK iT 35 . 0Tmm 14 Load disturbance model = LsLsT 1e KL+ . 214 . 1mmT205 . 0 181 . 0mLmcT TTK921 . 0Kmm||.|

\| ||.|

\|= , 639 . 0mm49 . 0T954 . 0mL miT TT430 . 0TTmm||.|

\| ||.|

\|=, 077 . 0T310 . 2TTmmmL+ . b720_Chapter-03 17-Mar-2009 FAChapter 3: Controller Tuning Rules for Self-Regulating Process Models 41 Rule cK iT Comment 15cK iT 16cK iT 35 . 0Tmm Minimum ISE Yu (1988) continued. 17cK iT 35 . 0 Tm m > 945 . 0mmmTK279 . 1||.|

\| 586 . 0mm mT 535 . 0T||.|

\| 0 . 1T1 . 0mm 675 . 0mmmTK346 . 1||.|

\| 438 . 0mm mT 552 . 0T||.|

\| 0 . 2T1 . 1mm Minimum ISE Zhuang (1992), Zhuang and Atherton (1993). Model: Method 1 or Method 5 or Method 6 or Method 15. Minimum ITAE Murrill (1967), pp. 358363. Model: Method 5 977 . 0mmmTK859 . 0||.|

\| 680 . 0mm mT 674 . 0T||.|

\| 0 . 1T1 . 0mm ||.|

\|+mm2 1mTx xK1||.|

\|+mm2 12mTx xx Representative coefficient values (deduced from graph): m mT 1x 2x m mT 1x 2x 0.2 0.32 0.77 1.0 0.38 1.00 0.4 0.35 0.80 1.2 0.39 1.05 0.6 0.37 0.86 1.4 0.39 1.10 Minimum ITAE Frank and Lenz (1969). Model: Method 1 0.8 0.38 0.93 15 014 . 1mmT344 . 0 045 . 0mLmcT TTK157 . 1Kmm+ ||.|

\| ||.|

\|= , 899 . 0mm292 . 0T532 . 2mL miT TT359 . 0TTmm||.|

\| ||.|

\|=, 1TT077 . 0T310 . 2mLmm +. 16 047 . 1mmT234 . 0 065 . 0mLmcT TTK07 . 1Kmm+ ||.|

\| ||.|

\|= , 898 . 0mm094 . 0T112 . 1mL miT TT347 . 0TTmm||.|

\| ||.|

\|=, 3TT1mL < . 17 889 . 0mmT067 . 0 04 . 0mLmcT TTK289 . 1Kmm+||.|

\| ||.|

\|= , 46 . 0mm44 . 0T372 . 0mL miT TT596 . 0TTmm||.|

\| ||.|

\|=. b720_Chapter-03 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules 42 Rule cK iT Comment m 1K x ( )m m 2T x + mmT 1x 2x mmT 1x 2x mmT 1x 2x 0.11 5.0 0.53 0.67 0.94 0.76 2.33 0.47 0.59 0.25 2.6 0.74 1.00 0.66 0.71 4.00 0.42 0.52 Minimum ITAE with OS % 8 Fertik (1975). Model: Method 3; 1x ,2x deduced from graphs. 0.43 1.45 0.80 1.50 0.54 0.65 9.00 0.375 0.47 18cK iT 19cK iT 20cK iT Model: Method 1; 35 . 0Tmm. Minimum ITAE Yu (1988). Load disturbance model = LsLsT 1e KL+ . 21cK iT 35 . 0 Tm m > 18 341 . 1mmT254 . 0 272 . 0mLmcT TTK598 . 0Kmm||.|

\| ||.|

\|= , 196 . 0mm112 . 0T304 . 0mL miT TT805 . 0TTmm||.|

\| ||.|

\|=, 112 . 0T385 . 2TTmmmL+ . 19 055 . 1mmT945 . 1 011 . 0mLmcT TTK735 . 0Kmm ||.|

\| ||.|

\|= , 901 . 0mm241 . 0T809 . 5mL miT TT425 . 0TTmm||.|

\| ||.|

\|=+, 1TT112 . 0T385 . 2mLmm +. 20 042 . 1mmT154 . 0 084 . 0mLmcT TTK787 . 0Kmm+||.|

\| ||.|

\|= , 901 . 0mm365 . 0T148 . 0mL miT TT431 . 0TTmm||.|

\| ||.|

\|=, 3TT1mL < . 21 909 . 0mmT057 . 0 172 . 0mLmcT TTK878 . 0Kmm||.|

\| ||.|

\|= , 489 . 0mm257 . 0T228 . 0mL miT TT794 . 0TTmm||.|

\| ||.|

\|=. b720_Chapter-03 17-Mar-2009 FAChapter 3: Controller Tuning Rules for Self-Regulating Process Models 43 Rule cK iT Comment 22cK iT 2 . 1T1 . 0mm Minimum ITAE Arrieta Orozco (2003). Model: Method 10. Also given by Arrieta Orozco and Alfaro Ruiz (2003). Minimum ITAE Arrieta Orozco (2006), pp. 9092. 23cK iT 4 A 2m ; Model: Method 10 ||.|

\|+mm2 1mTx xK1||.|

\|+mm2 12mTx xx Representative coefficient values (deduced from graph): m mT 1x 2x m mT 1x 2x 0.2 0.46 0.84 1.0 0.55 1.09 0.4 0.50 0.89 1.2 0.54 1.15 0.6 0.53 0.97 1.4 0.54 1.20 Minimum ITSE Frank and Lenz (1969). Model: Method 1 0.8 0.55 1.04 22 (((

||.|

\|+ =1055 . 1mmmcT6470 . 0 2607 . 0K1K , (((

||.|

\| + =1789 . 0mmm iT9191 . 2 5926 . 1 T T . 23 ((((

||.|

\| + + = + 2m mA 0389 . 0 A 2865 . 0 5345 . 1mm12m mmcTx A 0764 . 0 A 6179 . 0 2585 . 1K1K , 2m m 1A 0127 . 0 A 1783 . 0 8653 . 0 x + = , 0 . 2T1 . 0mm ; ((((

||.|

\| + + =+ 2m mA 6454 . 0 A 4925 . 4 7448 . 7mm22m m m iTx A 7267 . 2 A 4081 . 17 5244 . 24 T T , 2m m 2A 6258 . 2 A 5760 . 16 8710 . 21 x + = , 0 . 1T1 . 0mm < ; ((((

||.|

\| + + = 2m mA 0110 . 0 A 2639 . 0 3703 . 1mm22m m m iTx A 0272 . 0 A 2739 . 0 1097 . 1 T T , 2m m 2A 0417 . 0 A 3504 . 0 1821 . 1 x + = , 0 . 2T0 . 1mm . b720_Chapter-03 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules 44 Rule cK iT Comment 957 . 0mmmTK015 . 1||.|

\| 552 . 0mm mT 667 . 0T||.|

\| 0 . 1T1 . 0mm 673 . 0mmmTK065 . 1||.|

\| 427 . 0mm mT 687 . 0T||.|

\| 0 . 2T1 . 1mm Minimum ISTSE Zhuang (1992), Zhuang and Atherton (1993). Model: Method 1 or Method 5 or Method 6 or Method 15. 24cK iT Model: Method 1 Minimum ISTSE Zhuang (1992), Zhuang and Atherton (1993). 25cK iT Model: Method 37 953 . 0mmmTK021 . 1||.|

\| 546 . 0mm mT 629 . 0T||.|

\| 0 . 1T1 . 0mm 648 . 0mmmTK076 . 1||.|

\| 442 . 0mm mT 650 . 0T||.|

\| 0 . 2T1 . 1mm Minimum ISTES Zhuang (1992), Zhuang and Atherton (1993). Model: Method 1 or Method 5 or Method 6 or Method 15. Nearly minimum IAE, ISE, ITAE Hwang (1995). Model: Method 44 26( )u 1K 1 u u 2 cK K Decay ratio = 0.15; 0 . 2T1 . 0mm . Nearly minimum IAE and ITAE Hwang and Fang (1995). Model: Method 30 27cK iT Decay ratio = 0.12; 0 . 2T1 . 0mm . 24 uu mu mcK097 . 2 K K 249 . 3244 . 0 K K 892 . 1K ||.|

\|++= , uu mu miT2736 . 1 K K 7229 . 0227 . 0 K K 706 . 0T ||.|

\|+= , 0 . 2T1 . 0mm . 25 uumumcK06 . 1 K K 848 . 5610 . 2 K K 126 . 4K |||.|

\|= , uumumiT036 . 5 K K 539 . 5926 . 2 K K 352 . 5T |||.|

\|+= , 0 . 2T1 . 0mm . 26 ( )) K K 1 ( K K068 . 0 482 . 0 1 14 . 1m u m u2m2u m u1+ + = , ( )) K K 1 ( K K367 . 0 1 . 2 1 0694 . 0m u m u2m2u m u2+ + = . 27 u2m2mmmcKT0254 . 0T0521 . 0 515 . 0 K ||.|

\| + = , ||.|

\| +=2m2mmmu u ciT0141 . 0T0918 . 0 0877 . 0) K / K (T b720_Chapter-03 17-Mar-2009 FAChapter 3: Controller Tuning Rules for Self-Regulating Process Models 45 Rule cK iT Comment Minimum error: step load change Gerry (2003). mK 3 . 0 m42 . 0 Model: Method 1; 5 Tm m . m 1K x ( )m m 2T x + Coefficient values: m mT 1x 2x m mT 1x 2x 0.11 1.1 0.23 0.11 5.8 0.5 0.25 1.8 0.23 0.25 3.1 0.6 0.43 1.1 0.72 0.43 2.1 0.7 0.67 1.0 0.72 0.67 1.7 0.8 1.0 0.8 0.70 1.0 0.91 0.9 1.50 0.59 0.67 2.33 0.42 0.60 Minimum IAE or ISE ECOSSE Team (1996a). Model: Method 1; coefficients of cK ,iT deduced from graphs. 4.0 0.32 0.53 28cK u 2T x 2x coefficient values (deduced from graphs): m mT 2x m mT 2x m mT 2x 0.2 0.35 0.8 0.23 1.4 0.18 0.4 0.29 1.0 0.21 1.6 0.18 Minimum IE Devanathan (1991). Model: Method 1 0.6 0.26 1.2 0.19 1.8 0.17 Minimum performance index: servo tuning m 1K x ( )m m 2T x + Representative coefficient values (deduced from graphs): m mT 1x 2x m mT 1x 2x 0.053 12 0.95 0.67 1.15 0.75 0.11 6.4 0.91 1.50 0.65 0.66 Minimum IAE Gallier and Otto (1968). Model: Method 58 0.25 2.9 0.84 4.0 0.45 0.55 Minimum IAE Rovira et al. (1969). Model: Method 5 861 . 0mmmTK758 . 0||.|

\| mmmT323 . 0 020 . 1T 0 . 1T1 . 0mm 28 (((

||.|

\| (((

||.|

\| =um 1mRR1cTT 28 . 6tan cos KD16 . 0D 57 . 1 tan cosK . b720_Chapter-03 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules 46 Rule cK iT Comment m 1K x m 2T x 1x 2x m mT 1x 2x m mT 2.7 0.9 0.2 0.9 1.3 0.8 1.5 1.1 0.4 0.8 1.4 1.0 Minimum IAE Murata and Sagara (1977). Model: Method 2; 2 1x , x deduced from graphs. 1.1 1.2 0.6 Minimum IAE Bain and Martin (1983). Model: Method 1 ((

+mmmT3 . 0 3 . 0K1 m m4 . 0 T + 2 . 0Tmm> mK 4 . 1 m72 . 0 11 . 0 Tm m = Minimum IAE Marlin (1995), pp. 301307. Model: Method 1 cK , iT deduced from graphs; robust to % 25 process model parameter changes. Minimum IAE Huang et al. (1996). Model: Method 1 29cK iT 1T1 . 0mm Minimum IAE Smith and Corripio (1997), p. 345. m mmKT 6 . 0 mT Model: Method 1; 5 . 1 T 1 . 0m m . Minimum IAE Huang and Jeng (2002). Model: Method 1 m mmKT 59 . 0 mT Minimum IAE = m1038 . 2 ; 2 . 0 Tm m < . 29(((

||.|

\| +||.|

\| + = 5959 . 3mm0169 . 1mmmmmcT1075 . 1T3053 . 0T0916 . 9 0454 . 13K1K (((

+||.|

\| +mmT6843 . 3mmme 8259 . 4T2927 . 2K1, (((

||.|

\| +||.|

\| ||.|

\| + =4mm3mm2mmmmm iT2567 . 8T4538 . 7T4651 . 3T2492 . 0 9771 . 0 T T (((

||.|

\| +||.|

\| +6mm5mmmT1496 . 1T7536 . 4 T . b720_Chapter-03 17-Mar-2009 FAChapter 3: Controller Tuning Rules for Self-Regulating Process Models 47 Rule cK iT Comment 30cK iT 2 . 1T1 . 0mm Minimum IAE Arrieta Orozco (2003). Model: Method 10 Also given by Arrieta Orozco and Alfaro Ruiz (2003). 31cK iT 10T1 . 0mm Minimum IAE Tavakoli and Fleming (2003). Model: Method 1 Minimum dB 6 Am = ; minimum 0m60 = . Minimum IAE Tavakoli et al. (2006). 32cK iT Model: Method 42; 3 Am ;0m60 . uK 5 . 0 u 2T x Model: Method 1 2x m mT 2x m mT 2x m mT 1.61 0.2 0.61 1.0 0.48 2.0 Minimum IAE Harriott (1988). iT coefficients deduced from graph. 0.94 0.5 30(((

||.|

\|+ =2099 . 1mmmcT5305 . 0 2438 . 0K1K , (((

||.|

\| + =8714 . 0mmm iT4337 . 0 9377 . 0 T T . 31 ((

+= 3047 . 0T4849 . 0K1Kmmmc, ((

+= 9581 . 0T4262 . 0 T Tmmm i. 32 ((

+= 071 . 0T5 . 0K1Kmmmc, [ ]m m m i9 , 143 . 0 T min T + = . b720_Chapter-03 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules 48 Rule cK iT Comment Small IAE Hwang (1995). Model: Method 44 33( )u 1K 1 u u 2 cK K Decay ratio = 0.1; 0 . 2T1 . 0mm . 892 . 0mmmTK980 . 0||.|

\| mmmT155 . 0 690 . 0T 0 . 1T1 . 0mm Minimum ISE Zhuang (1992), Zhuang and Atherton (1993). Model: Method 1 or Method 5 or Method 6 or Method 15. 560 . 0mmmTK072 . 1||.|

\| mmmT114 . 0 648 . 0T 0 . 2T1 . 1mm 34cK iT 2 . 0 T 01 . 0m m Minimum ISE Khan and Lehman (1996). Model: Method 1 35cK iT 20 T 2 . 0m m Minimum ITAE Rovira et al. (1969). Model: Method 5 916 . 0mmmTK586 . 0||.|

\| mmmT165 . 0 030 . 1T 0 . 1T1 . 0mm 33 ( )) K K 1 ( K K0667 . 0 466 . 0 1 07 . 1m u m u2m2u m u1+ + = , ( )) K K 1 ( K K338 . 0 21 . 2 1 0328 . 0m u m u2m2u m u2+ + = , r = 0.5; ( )) K K 1 ( K K0657 . 0 467 . 0 1 11 . 1m u m u2m2u m u1+ + = , ( )) K K 1 ( K K333 . 0 07 . 2 1 0477 . 0m u m u2m2u m u2+ + = , r = 0.75; ( )) K K 1 ( K K0647 . 0 466 . 0 1 14 . 1m u m u2m2u m u1+ + = , ( )) K K 1 ( K K323 . 0 97 . 1 1 0609 . 0m u m u2m2u m u2+ + = , r = 1.0; r = parameter related to the position of the dominant real pole. 34mmm mcKTT3185 . 0 7388 . 0K ||.|

\|+= ,iT = ((

+ +m mm mm5291 . 0 T 0003082 . 03185 . 0 T 7388 . 0. 35mmm mm mcKTT255 . 0T511 . 0 808 . 0K ||.|

\| += , iT =(((

+ +m m m mm m m mmT 381 . 0 846 . 0 T 095 . 0T 255 . 0 511 . 0 T 808 . 0. b720_Chapter-03 17-Mar-2009 FAChapter 3: Controller Tuning Rules for Self-Regulating Process Models 49 Rule cK iT Comment m 1K x ( )m m 2T x + mmT 1x 2x mmT 1x 2x mmT 1x 2x 0.11 3.05 0.95 0.67 0.71 0.76 2.33 0.42 0.59 0.25 1.70 0.88 1.00 0.55 0.71 4.00 0.39 0.52 Minimum ITAE with OS % 8 Fertik (1975). Model: Method 3; 1x ,2x deduced from graphs. 0.43 1.05 0.82 1.50 0.47 0.65 9.00 0.365 0.47 36cK iT 2 . 1T1 . 0mm Minimum ITAE Arrieta Orozco (2003). Model: Method 10 Also given by Arrieta Orozco and Alfaro Ruiz (2003). Minimum ITAE Arrieta Orozco (2006), pp. 9092. 37cK iT 4 A 2m ; Model: Method 10 Minimum ITAE Barber (2006). 38cK mmT9707 . 0e 9894 . 0 Model: Method 1 36(((

||.|

\|+ =0382 . 1mmmcT5131 . 0 1440 . 0K1K , (((

||.|

\| + =5426 . 1mmm iT2073 . 0 9953 . 0 T T . 37((((

||.|

\| + + =+ + 2m mA 0047 . 0 A 0048 . 0 0871 . 1mm12m mmcTx A 0496 . 0 A 4436 . 0 9639 . 0K1K , 2m m 1A 0509 . 0 A 4355 . 0 3296 . 1 x + = , 0 . 2 T 1 . 0m m ; ((((

||.|

\| + + =+ 2m mA 4764 . 0 A 4585 . 3 1845 . 6mm22m m m iTx A 1692 . 0 A 2511 . 1 7836 . 2 T T , 2m m 2A 0083 . 0 A 1193 . 0 8298 . 0 x + = , 0 . 1 T 1 . 0m m < ; ((((

||.|

\| + + = 2m mA 0447 . 0 A 2509 . 0 6705 . 1mm22m m m iTx A 0311 . 0 A 5089 . 0 0174 . 2 T T , 2m m 2A 0889 . 0 A 4874 . 0 0065 . 1 x + = , 0 . 2 T 0 . 1m m . 38 (((

++||.|

\| =00911 . 0T0214 . 2T5853 . 1 K1Kmm2mmmc. b720_Chapter-03 17-Mar-2009 FAHandbook of PI and PID Controller Tuning Rules 50 Rule cK iT Comment m m mK T 586 . 0 mT Model: Method 1 Smith (2002). Approximately minimum ITAE. Minimum ISTSE Fukura and Tamura (1983). ((

+2 . 0T56 . 0K1mmm m m36 . 0 T + Model: Method 1; 1 T 05 . 0m m . 921 . 0mmmTK712 . 0||.|

\| m mmT 247 . 0 968 . 0T 0 . 1T1 . 0mm Minimum ISTSE Zhuang (1992), Zhuang and Atherton (1993). Model: Method 1 or Method 5 or Method 6 or Method 15. 559 . 0mmmTK786 . 0||.|

\| m mmT 158 . 0 883 . 0T 0 . 2T1 . 1mm Minimum ISTSE Zhuang (1992). 39cK iT Model: Method 1; 0 . 2 T 1 . 0m m . uK 361 . 0 40iT Model: Method 1 Minimum ISTSE Zhuang (1992), Zhuang and Atherton (1993). 41cK iT Model: Method 37 921 . 0mmmTK712 . 0||.|

\| m mmT 182 . 0 694 . 0T 712 . 0 0 . 1T1 . 0mm 543 . 0mmmTK780 . 0||.|

\| m mmT 120 . 0 683 . 0T 780 . 0 0 . 2T1 . 1mm Minimum ISTSE Majhi (2005). Model: Method 52 331 . 0mmmTK673 . 0||.|

\| m mmT 066 . 0 511 . 0T 673 . 0 5 . 3T1 . 2mm 39 uu mu mcKK K 432 . 0 119 . 12K K 148 . 0 264 . 4K ||.|

\|= , ( )u u m iT 1 K K 935 . 1 083 . 0 T + = . 40 ( )u u m iT 1 K K 935 . 1 083 . 0 T + = , 0 . 2T1 . 0mm . 41 uumumcK606 . 0 K K 341 . 3177 . 0 K K 506 . 1K |||.|

\|+= , u um iT 1 K K 616 . 3 055 . 0 T ||.|

\| + = , 0 . 2T1 . 0mm . b720_Chapter-03 17-Mar-2009 FAChapter 3: Controller Tuning Rules for Self-Regulating Process Models 51 Rule cK iT Comment 951 . 0mmmTK569 . 0||.|

\| mmmT179 . 0 023 . 1T 0 . 1T1 . 0mm Minimum ISTES Zhuang (1992), Zhuang and Atherton (1993). Model: Method 1 or Method 5 or Method 6 or Method 15. 583 . 0mmmTK628 . 0||.|

\| mmmT167 . 0 007 . 1T 0 . 2T1 . 1mm Nearly minimum IAE and ITAE Hwang and Fang (1995). 42cK iT Model: Method 30; 0 . 2 T 1 . 0m m ; decay ratio = 0.03. Minimum performance index: other tuning Simultaneous servo/regulator tuning Hwang and Fang (1995). 43cK iT Model: Method 30; 0 . 2 T 1 . 0m m ; decay ratio = 0.05. m 1K x mT Coefficient values: m mT 1x b m mT 1x b 0.1 750.0 0.0001 0.2 3.3675 0.008 0.1 17.487 0.04 0.2 2.2946 0.04 0.1 4.8990 0.2 0.2 1.7146 0.2 0.1 2.1213 1 0.2 1.1313 1 0.2 0.8764 5 0.4 1.6837 0.008 0.5 150.01 0.0001