journal 2007 jcej jp

8/3/2019 Journal 2007 Jcej Jp

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Journal of Chemical Engineering of Japan, Vol. 40, No. 2, pp. 1451 63, 2007 Research Paper

Copyright 2007 The Society of Chemical Engineers, Japan 145

PID Control of Unstable Processes with Time Delay: A ComparativeStudy

Han-Qin ZHOU1,2

, Qing-Guo WANG1

and Leang-San SHIEH2

1Department of Electrical and Computer Engineering,

National University of Singapore, Singapore 1192602Department of Electrical and Computer Engineering,

University of Houston, Houston, TX 77204, U.S.A.

Keywords: Unstable Processes, Time Delay, PID Control, IMC, Nonlinear Control

In this paper, several linear PID control methods for unstable processes with time delays, which have

recently been reported in the literature, are studied in terms of a comprehensive set of control perform-

ance specifications. Their applicability and achievable performance are indicated with regard to differ-

ent cases of normalized time delay for ease of users choice of the methods. By analyzing these methods

pros and cons, a modified method which combines their respective strengths using simple linear time-

variant and nonlinear control strategies is obtained and demonstrated with performance enhancement.

Introduction

It is well-known that unstable dynamic systems

are inherently more difficult to control than their sta-

ble counterparts. For an unstable process, open-loop

control is impossible, and controller gain cannot be

tuned gradually from zero value. A sufficiently large

feedback gain must be used to stabilize the process

before addressing performance and robustness, and yet

the feedback loop may become unstable again if the

gain is too large. This stability range for the feedback

gain is limited. Moreover, as the time delay increases

relatively with the time constant, the range gets nar-

row and thus the system performance could be further

deteriorated. These and other difficulties have moti-

vated active research in control system design for un-

stable processes in recent years.

Design of PID controllers, the most commonly

used controllers in industrial process control, for un-

stable time-delay processes have been reported (DePaor

and OMalley, 1989; Venkatashankar and

Chindambaram, 1994; Shafiei and Shenton, 1994;Huang and Lin, 1995; Poulin and Pomerleau, 1996)

and surveyed by Chidambaram (1997). More recently,

Ho and Xu (1998) derived PID tuning formulas based

on gain and phase margin specifications. Visioli (2001)

proposed optimal PID parameters tuning in terms of

IAE, ISTEand ITSE specifications via genetic algo-

Received on November 29, 2004; accepted on May 28, 2005.

Correspondence concerning this article should be addressed to

Q.-G. Wang (E-mail address: [email protected]).

Partly presented at 5th Asian Control Conference, at Melbourne,

Australia, July 2023, 2004.

rithm. However, these two PID design methods exhibit

excessive overshoot and large setting time. To reduce

them, Park et al. (1998) and Majhi and Atherton

(2000a) developed PID-P and PI-PD strategies respec-

tively, both using inner feedback loops. Besides, Wang

and Cai (2002) employed the PID-P structure to de-

sign an equivalent two degree-of-freedom (DOF) sin-

gle loop PID control scheme in terms of gain and phase

margin specifications.

Owing to power and popularity of internal model

control (IMC) in process industry (Morari and Zafiriou,

1989), many efforts have been made to exploit the IMC

principle to design the equivalent feedback controllers

for unstable processes. Satisfactory results have been

obtained for SISO applications (Chen, 1988; Wang et

al., 2001). Rotstein and Lewin (1991) proposed explicit

PI and PID settings for first-order plus dead time

(FOPDT) and second-order plus dead time (SOPDT)

unstable processes. However, performance limitation

of implementing the IMC controller in an equivalent

feedback structure for unstable processes was not ad-

dressed. Lee et al. (2000) derived a set of PID tuningrules for FOPDT and SOPDT unstable processes, us-

ing Maclaurin series expansion to approximate the ideal

IMC controller with PID. Yang et al. (2002) developed

a unified framework for an IMC-based single loop con-

troller design method using frequency response fitting

for both high-order controllers as well as PID ones for

general processes, where stability is ensured. These two

methods give very good control performance with rela-

tively wide applicability.

The Smith predictor (Smith, 1959) has been

known as a very effective dead-time compensator,

which can eliminate the delay term from the closed-loop


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146 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

characteristic equation. Thus standard PI/PID control-

lers can be applied to the delay free systems. How-

ever, the Smith predictor will become internally un-

stable when applied to unstable processes (Wang et al.,

1999). Therefore, many modified Smith schemes have

been proposed to overcome this obstacle. DePaor

(1985) suggested a modified Smith predictor for time-

delayed unstable processes, by changing the denomi-nator of the process model from the original unstable

one to a stable one, but no time response is presented

in this work. Astrom et al. (1994) presented a modi-

fied Smith predictor for integrator plus dead-time

processes, which achieved fast setpoint response and

good disturbance rejection, by decoupling the setpoint

and load responses. Matausek and Micic (1996) con-

sidered the same problem and provided similar but

easier tuning rules. Majhi and Atherton (1999) pro-

posed another modified Smith predictor with good per-

formance particularly for integral unstable processes.

Another paper of Majhi and Atherton (2000b) extended

it to unstable FOPDT processes. Kaya (2003) proposed

a more systematic tuning formula for this modified

Smith predictor structure (Majhi and Atherton, 2000b).

With so much work done, we are motivated to

conduct a comparative study to show a comprehensive

review and assessment of time-delayed unstable

processes control. Seven recent and representative

PID control methods are chosen in our investigation:

(A) Optimal PID Tuning Method (Visioli, 2001);

(B) PID-P Control Method (Parket al., 1998); (C) PI-

PD Control Method (Majhi and Atherton, 2000a);

(D) Gain and Phase Margin Method (Wang and Cai,

2002); (E) IMC-Maclaurin PID Tuning Method (Leeet al., 2000); (F) IMC-based Approximate PID Tuning

Method (Yang et al., 2002); (G) Modified Smith Pre-

dictor (Majhi and Atherton, 2000b). A brief summary

of each of them is given first, their performance is

evaluated against a set of most popular and important

specifications and is commented on, and their applica-

bility is concluded. It is observed from the compari-

son that the best achievable control performance among

the investigated methods has been very good already,

and further enhancement within these frameworks

could be quite difficult though it is not impossible.

However, these methods employ linear time invariant(LTI) controllers only. By using linear time variant

(LTV) and nonlinear components in the controller, the

best performance from LTI controllers can be further

improved by such modifications.

The rest of this paper is organized as follows. The

existing control schemes are reviewed in Section 1,

and their performance is evaluated in Section 2. In

Section 3 some new results on performance improve-

ment are presented.

1. Review of Control Methods

In this section, we will briefly review seven meth-

ods for control of time-delayed unstable processes. The

PID controller to be discussed below is represented in

the form of

G s KT s

T sc pi

d( ) = + +

( )1 1 1

1.1 Optimal PID tuning method

For the FOPDT unstable process

G sK

Tse

Lsp ( ) =

( )1

2

Visioli (2001) proposed three sets of PID auto-

tuning rules to minimize one of the following three

specifications, respectively:

ISE e t t = ( ) ( )

20 3d

ITSE te t t = ( ) ( )

20 4d

ISTE t e t t = ( ) ( )

2 20 5d

The optimal controller parameters are obtained by

means of genetic algorithms, which is well-known toprovide a global optimum in a stochastic framework,

in order to avoid local minima in the optimization pro-

cedure. The value ofKin the process transfer function

results in a simple scaling of the PID proportional gain

Kp, and thus is not required to address in the genetic

algorithm. Based on the optimal PID coefficients for

various values of process normalized dead time n

=

L/Tand time constant T, the tuning rules are obtained

by analytical interpolation. Each interpolation function

was selected manually and its parameters were deter-

mined again by genetic algorithms to minimize the sum

of the absolute values of the estimation errors. Two

sets of tuning formulas are displayed in Table 1, onefor setpoint response, while the other for disturbance

rejection.

It is noted that the configuration Figure 1 is only

of one degree-of-freedom (DOF), small overshoot and

fast settling-time can hardly be obtained at the same

time. The methods below are however all of two-DOF

structure (Figure 2), and have potential to perform

better than 1DOF counterparts. Therefore, for a fair

comparison of all the methods, a pre-filter is used and

tuned to best performance by trial and error when

Method A is implemented in the next section.


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VOL. 40 NO. 2 2007 147

1.2 PID-P control method

For an FOPDT unstable process, Parket al. (1998)

proposed the configuration in Figure 3, where a pro-

portional controller Giis used in the inner loop to sta-

bilize the unstable process, and the main PID control-

ler Go

is designed by viewing the inner closed-loop

system as a stable process G.With relay auto-tuning, the unstable process is

modelled by an FOPDT unstable process:

G s G s

K e

T s

L s

p mm

m

m

( ) ( ) = ( )

1 6

The P controller gain that can stabilize this unstable

FOPDT process is within the range:

kK

kK

T kmin max= < < + ( ) = ( )1 1

1 72

mci

mm u

where u

is the ultimate frequency. To have the opti-

mal gain margin, the controller gain derived by DePaor

and OMalley (1989) is used on the inner loop:

PID parameter ISE ITSE ISTE

Setpoint t racking

Kp 1 320 92

..

K

L

T

1 38

0 90.

.

K

L

T

1 35

0 95.

.

K

L

T

Ti

4 00

0 47

.

.L

T T

4 12

0 90

.

.L

T T

4 52

1 13

.

.L

T T

Td3 87 1 0 84

0 02

0 95

. .

.

.

TL

T

L

T

3 62 1 0 85

0 02

0 93

. .

.

.

TL

T

L

T

3 70 1 0 86

0 02

0 97

. .

.

.

TL

T

L

T

Disturbance rejection

Kp 1 371

.

K

L

T

1 37

1.

K

L

T

1 37

1.

K

L

T

Ti2 42

1 18

.

.L

TT

3 76

1 39

.

.L

TT

4 68

1 52

.

.L

TT

Td 0 60.L

TT

0 55.

L

TT

0 50.

L

TT

Table 1 Optimal PID tuning formulas of Method A

Fig. 1 Unity feedback system with PID controller

Fig. 2 Two-degree-of-freedom unity feedback control sys-

tem

Fig. 3 Double-loop control scheme


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k k kci = ( )min max 8

As a result, the closed-loop transfer function of the

inner feedback loop becomes

H sk e

T s k k e

L s

L s( ) = +( )

m

m m ci

m

m19

which can be approximated by a stable SOPDT sys-

tem

G ske

s s

s

( ) =+ +

( )

2 2 110

by either (i) a model reduction technique, or (ii) Taylor

series expansion. Their study of two methods shows

that the model reduction technique is superior to the

Taylor series expansion regarding the system perform-

ance. However, the Taylor series expansion is much

easier to carry out.

Since the unstable process has been stabilized by

the proportional controller on the inner loop, the pri-

mary PID controller then acts for a stable plant, G(s).With k

1,

1,

1, and

1in Eq. (10) available, the tuning

rules proposed by Sung et al. (1996) in terms ofITAE

criteria are used to obtain PID controller settings, as

shown in Table 2.

Essentially, this PID-P scheme is equivalent to a

two-degree-of-freedom controller. Therefore,

stabilization and performance problems can be treated

separately and better performance than 1DOF control

can be expected. However, the normalized dead-time

of the process should be no larger than 0.693, which is

the limitation imposed by the normal relay feedback

identification. Robustness is not analyzed in this work.1.3 PI-PD control method

For an FOPTD unstable process, Majhi and

Atherton (2000a) presented another double-loop PID

control. The structure is similar to that of Method B as

shown in Figure 3. But now Go

is a PI controller, while

Gion the inner loop is a PD controller.

The unstable FOPDT process is described by

G s G sK e

T s

Ke

s

L s s

p mm

m

m n

( ) ( ) =

( )

1 111

Setpoint response

kKp = + +

0 04 0 333 0 949 0 9

0 983

. . . , .

.

kKp = + + >

0 544 0 308 1 408 0 9

0 832

. . . , .

.

Ti

= +

2 055 0 072 1. . ,

Ti

= + >

1 768 0 329 1. . ,

Td

= +( )

( )[ ]1

0 870 55 1 683

1 061 09

exp.

. .

..

Disturbance rejection

kKp = + +

0 67 0 0297 2 189 0 9

2 001 0 766

. . . , .

. .

kKp = + + >

0 365 0 26 1 4 2 189 0 9

2 0 766

. . . . , .

.

Ti

=


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VOL. 40 NO. 2 2007 149

where n =Lm/Tm is the normalized dead-time. A directrelay feedback identification is appplied to the plant

to obtain the parametersLm, T

mand K

mof Eq. (11). For

processes with n

< 0.693, the normal relay feedback

can be used. However, ifn

is large, i.e., n

> 0.693,

the limit cycle does not exist in the normal relay feed-

back, which explains why the Method B is only appli-

cable for processes with normalized dead-time less than

0.693. Thus an additional inner loop P controller has

to be added to overcome this problem, by which the

range of normalized dead time for the existing limit

cycle is extended to n

< 1. As a result, the method will

be applicable to control the FOPDT unstable processwith 0 <

n< 1.

The inner controller, Gi(s), is to stabilize the un-

stable process and takes a PD form:

G s KT

Ts K T si f

d

if f( ) = +

= +( ) ( )1 1 12

To approximate the resultant closed-loop transfer func-

tion by a stable FOPDT process after Pade approxima-

tion of esn , T

iis set toL/2. K

fis given by 1/K 2 n

as in Eq. (8) with the optimal gain margin. The main

controller is of the PI form:

G s KT s

o pi

( ) = +

( )11

13

and is tuned for satisfactory setpoint tracking. The in-tegral square time error optimization criterion, ISTE,

is used to design the PI controller and tuning formulas

for this PI-PD controller are shown in Table 3. Ro-

bustness of the control method is examined for

perturbations in process time delay.

1.4 Gain and phase margin PID tuning method

For the unstable FOPDT process, Wang and Cai

(2002) employed the same control configuration of

Method B as in Figure 3, where Go(s) is the primary

PID controller, and Klis the proportional controller on

the inner loop. But they designed the primary PID con-

troller based on gain and phase margin specifications.Such a double-loop configuration can be implemented

in an equivalent single-loop PID feedback system with

a setpoint weighting in Figure 4, where Kp, T

i, and T

d

are PID settings and b is the setpoint weighting.

With the P controller in the inner loop, the inner

closed-loop transfer function Gl(s) is

G ll

sKe

Ts KK e

Ls

Ls( ) = +( )

114

By its Taylor series expansion and truncation of the

time delay term in the denominator:

e Ls L sLs + ( )1 0 5 152 2.

Equation (14) becomes

G l pl l l

s G sKe

KK L s T KK L s KK

Ls

( ) ( ) =+ ( ) +

( )

0 5 1

16

2 2.

Table 3 Tuning rule for PI-PD controller of Method

C

Apeak

, h and =Apeak

/(kmh) are peak output amplitude,

relay amplitude and normalized peak output respec-

tively.

Fig. 4 PID controller with setpoint weighting

0 < 0.693

K Km p

= +

+

( )[ ]( )

0 8011 1 0 9358 1

1

. . ln

T

T

m

i= + + +

( ) ( )[ ]2

0 1227 1 4550 1 1 2711 1

1

2

tanh

. . ln . ln

T

T

d

m

=+( )

ln

tanh

1

41

Kf

=+( )

2

1ln

0.693 < < 1

K Km p =+

+

+

+

( )0 8011 0 99460 0682

0 7497 0 9946

0 0682

. .

.

. .

.

T

T

m

i =

+ + +

+ + +

3 2

3 2

5 2158 4 481 0 2817

0 0145 0 5773 2 6554 0 3488

. . .

. . . .

T

T

d

m

=+

+

( )0 0237 34 53384 1530

. .

.

Kf =

+

+

( )2 0 99460 0682

.

.


6/19


To make Gp(s) stable, it follows from the Routh

Hurwits criterion that

KK

KT

LKKmin max= < < = ( )

117l

Again, to have the optimal gain margin, the P control-

ler gain is chosen as

K K KK

T

Ll = = ( )mi n ma x

118

Substituting it into Eq. (16) yields

( ) =+ +

( )

G se

as bs c

Ls

p 219

where a = 0.5L/K TL , b = 1/K(T TL ) and c =

1/K( T L 1 ).The transfer function of the PID controller is re-

written as

G s kAs Bs C

sc ( ) =

+ +

( )

2

20

whereA = Kd/k,B = K

p/kand C= K

i/k. The controller

zeros are chosen to cancel the poles of model Gp(s),

i.e.,A = a,B = b and C= c. This leads to

( ) ( ) = ( )

G s G s k e

s

Ls

p c 21

kis to be determined based on gain and phase margin

specifications. By assigning a gain margin ofAm

= 3

and a phase margin ofm

= 60, one has

kA L

= = ( )

2 622

mL

Hence, the tuning formulas are given as follows:

KK

T

L K

T

L

T

Lp = +

( )

1

623

TK

KL

T

L

i

p=

( )

61

24

TK K

TLdp

= ( )1

1225

b

L

T

L

T

=

+

( )

1

16

126

For implementation, the 2DOF control system in

Figure 4 is applied with all parameters set according

to Eqs. (23)(26), which is simpler and more straight-

forward than Figure 3. However, this method is lim-

ited into FOPDT unstable processes with the normal-

ized dead-timeL/Tless than 1, as indicated by the in-

equality of Eq. (17).

1.5 IMC-Maclaurin PID tuning method

For both FOPDT and SOPDT unstable processes,Lee et al. (2000) proposed explicit PID tuning rules

based on the internal model control (IMC).

The IMC scheme is shown in Figure 5 and the

closed-loop transfer functions are

HGq

q G GH

Gq G

q G Gyr yd

D=

+ ( )=

( )+ ( )11

1,

where G is the plant, G is the model, and q is the IMC

controller. They reduce to

H Gq H Gq Gyr yd D= = ( ), 1

in the case of perfect model-plant match, i.e., G = G .

The IMC system can be put into the equivalent con-

ventional single loop control scheme in Figure 2 with

HG G

G GH

G

G Gyr

c

cyd

D

c

=+

=+1 1

,

where

Fig. 5 IMC control system


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VOL. 40 NO. 2 2007 151

Gq

Gqc = 1

Write an unstable process model as G(s) = PM

(s)PA(s),

where PM

(s) is the invertible portion, while PA(s) is

the non-invertible part, i.e., RHP zeros and time delay.

Ifq has zeros to cancel the unstable poles ofG, and if

(1 G q) has zeros to cancel the unstable poles ofGD,

the closed-loop responses under both a setpoint change

and a load disturbance will be stable. To satisfy the

above conditions, the IMC controller is set as q =

PM

1(s)f. Here,f=fsf

dis comprised of two parts:f

s=

1/(s + 1)n is to make the controller proper by choos-ing a suitable n; andf

d= (

im=1 is

i + 1)/(s + 1)m is tocancel the unstable/stable poles near the zeros ofG

D,

in which iis to be determined to cancel the m unsta-

ble poles. Thus,fis the IMC filter with an adjustable

time constant , and the IMC controller is

q P s f P s

s

s

sn

ii

i

m

m= ( ) =

( )

+( )

+

+( )( )

=

MM1

11

1

1

127

It follows that

H GqP s

s

s

sn

ii

i

m

myrA= =

( )

+( )

+

+( )( )=

1

1

1281

H Gq GP s

s

s

sG

n

ii

i

m

myd DA

D= ( ) = ( )

+( )

+

+( )

( )

=

1 11

1

1

29

1

The term (im=1 is

i + 1) in Hyr

causes an overshoot in

the closed-loop response for setpoint changes. This

problem could be solved by adding a pre-filter FR

(s) =

1/( im=1 is

i + 1). The equivalent conventional feedback

controller from this IMC controller is

Gq

Gq

P s

s

s

s

P s

s

s

s

n

ii

i

m

m

n

ii

i

m

m

c

M

A

=

=

( )

+( )

+

+( )

( )

+( )

+

+( )

( )

=

=

11

1

1

11

1

1

30

11

1

Gc(s) is further approximated by a PID controller with

its settings taken as the first three terms of its Maclaurin

series expansion in s:

G ss

f f sf

sc ( ) = ( ) + ( ) +( )

+

( )1

0 00

2312

!K

The tuning formulas for first-order and second-order

unstable time-delayed processes are given in Table 4.

The limitation is due to the stability constraint: for the

FOPDT unstable process, no stabilizing PID param-

eters can be found unless its normalized dead-time

L/T 2. Robustness is fully analyzed for parametricuncertainty.

1.6 IMC-based approximate PID tuning method

For quite general unstable processes, Yang et al.

(2002) developed a frequency response fitting approach

to derive feedback controllers from an IMC one with

stability. It has options to choose between PID and

high-order forms. The ideal controller is first formu-

lated according to the standard IMC design. Stability

is ensured when converting to the single-loop control-

ler. It is shown that high-order controllers become nec-

essary for high performance for processes of essential

high orders, where PID controllers become inappro-

priate. For simple processes, model reduction is ex-

ploited to approximate the ideal IMC equivalent feed-

back controller in Eq. (30) by a standard PID control-

ler.

G s K T s T s K K s K sC,PID p

id p

id( ) = + +

= + + ( )11 1

32

Given the desired closed-loop bandwidth b, the

standard non-negative least squares method is used to

find the optimal PID parameters {Kp, K

i, K

d} to mini-

mize the criterion

EG j G j

G j=

( ) ( )( )

( )( )max ,

033

b

C,PID C

C

where is the desired accuracy of the PID approxima-tion to the IMC controller. The default is set as 5%.Once this criterion is satisfied, the controller design

procedure is completed. Similar to Lee et al. (2000)s

method, a pre-filter is added to eliminate the overshoot.

1.7 Modified Smith predictor control method

For an FOPDT unstable process Gp(s) in Eq. (2),

Majhi and Atherton (2000b) presented a modified

Smith predictor control scheme depicted in Figure 6,

where three controllers serve different objectives. Gc1

in the inner loop is to stabilize the integrating and un-

stable process. The other two controllers, Gc

and Gc2

are used for set-point tracking and disturbance rejec-


8/19


tion, respectively, by regarding the inner loop as an

open-loop stable process. This structure is similar to

that of Matausek and Micic (1996) when Gc1

= 0 and is

equivalent to the standard Smith predictor when Gc1

=

Gc2

= 0.

Suppose that the method perfectly matches the

process dynamics, i.e., Gm(s) e

L s m= Gp(s) with Gm(s)= K

m/T

ms 1. The closed-loop setpoint response and

disturbance response are given by

Y sG G e

G G GY s e

L sL s

rc m

m c c1r

m

m

1+( ) =

+( )= ( ) ( )

34

Y sG e

G G G

G G G G G e

G G e

Y s e

L s L s

L s

L s

lm

m c c1

m c c1 c m

m c2

l

m m

m

m

1 +

1 +( ) =

+( )

+( )

+

= ( ) ( )

1

35

Since the time delay term is eliminated from the

denominator of the setpoint response transfer function

Yr(s), the G

ccan be taken as a PI controller G

c(s) =

Kp(1 + 1/T

is) for pole placement on the delay free sys-

tem. The controller Gc1

is chosen in the PD form: Gc1

(s)

= Kf(1 + T

fs). The proportional controller G

c2= K

d,

whose another job is to reject unwanted load distur-

bances, is designed to stabilize the second part of the

characteristic equation of Eq. (35), and a possible

choice is Kd

= 1/Km

T Lm m under the constraintLm/Tm< 1, as suggested by DePaor and OMalley (1989).

Hence, it follows that Yr(s) and Yl

(s) in Eqs. (34)and (35) become

( ) =+( )[ ] +

=+

( )Y sT T K K s s

r

m f m p

1

2 1

1

136

( ) =+ ( )

+( ) +( )( )

Y s

K s e

s s K K e

L s

L sl

m

m d

m

m

1

1 137

where is the tuning parameter. Taking = Tm

+ 2Tf

results in the following controller parameters:

Table4

IMC-PIDtuningrulesofMethodE

Fig. 6 Modified Smith predictor control system


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VOL. 40 NO. 2 2007 153

KK

pm

= ( )1

38

T T Ti m f= + ( )2 39

KK

fm

= ( )2 40

TT

fm=

( )

241

KK

T

Ld

m

m

m

= ( )1

42

Kaya (2003) proposed an alternative formula for tun-

ing , from the user-specified settling time and assumesthat the settling time t

sis proportional to the time con-

stant , i.e.,

t ks = ( ) 43

In the coefficient diagram method (CDM)

(Hamamci et al., 2001), k is chosen between 2.5 and

3.0 to perform well for processes with large time con-

stants, an integrator or unstable pole. They suggested

k= 2.5, or

= ( )ts2 544

.

This modified tuning rule looks systematic, compared

with Majhi and Atherton (2000b), where is chosenarbitrarily or equal to the estimated dead time.

2. Performance Comparison

To assess the performance of seven methods men-

tioned in the preceding section, the FOPDT unstable

process is used for simulation study as it is most popu-

lar. Three cases of different ranges of normalized dead-

time are considered: (i) 0 < L/T< 0.693, (ii) 0.693

L/T < 1, and (iii) 1 L/T < 2. Controller settingsused are obtained strictly following respective meth-

ods, and when there are some design parameters which

are not specified by certain methods, the optimum val-

ues are found by trial and error and applied. There-

fore, the true or best performance of seven methods

are compared. Performance is evaluated in terms of its

applicability, time domain specifications on set-point

response and disturbance rejection, control action size

and system robustness. The time domain specifications

under consideration are listed as follows:

(1) Rise t ime trof the set-point response: the time for

the step response to rise from 10% to 90% of its

steady state value.

(2) Settling-time tsof the set-point response: the time

for the step response to stay within 2% of its steady

state value.

(3) OvershootMp

of the set-point response: the ratio

of the difference between the first peak and thesteady state value to the steady state value of the

step response.

(4) Integral absolute error of the set-point response:

IAE = 0 |e(t)|dt.

(5) Integral square error of the set-point response: ISE

= 0 [e(t)]2dt.

(6) Recovery time tR

of disturbance response: the time

for the response to stay within 2% of its steady

state.

(7) Maximum error emax

of disturbance response: the

maximum absolute error during the response.

Control signals are observed in simulation and

commented. Maximum control signals should be of

similar magnitudes for judgement of controller designs.

Robustness is evaluated by varying some process pa-

rameters.

2.1 Small normalized dead-time: 0 < L/T< 0.693

Consider the following FOPDT unstable process

G se

s

s

p ( ) = ( )

4

4 145

2

which has been studied extensively in the literature.One seesL = 2, T= 4, K= 4, and the normalized dead-

time L/T = 0.5. A step change is introduced in the

setpoint at t= 0 with a unity magnitude and in the dis-

turbance at t= 75 with a magnitude of0.1. To have a

fair comparison, a pre-filter is added in Method A to

reduce the excessive overshoots.

The output time responses of Methods A to G are

shown in Figures 79. The performance specifications

are listed in Table 5.

Let us first look at the control signal u(t). Figure

10 exhibits input time responses of all seven methods.

It is seen that in this scenario the sizes of control ac-

tions are similar. In the case of stable processes, a bigcontrol action, when properly desiged, could poten-

tially give better output response. It seems that the case

becomes much complicated when an unstable process

is considered. As it is known to all, feedback system is

only conditionally stable for unstable processes. It is

guessed that control action is already limited due to

stabilization requirement and leaves less room to ma-

nipulate for performance than the stable process case.

This argument is more appealing for other two cases

of median and large normalized dead-time as they are

more difficult to stabilize than the small case. Therefore,


10/19


we will drop this factor of control action from now on

and focus on the system performance and robustness.

In the sense of the setpoint response, we can eas-

ily see that Method G, the modified Smith predictor

control, gives the best performance: fastest settling,

no overshoot and lowest IAE/ISE. The merit of Smith

predictor is thus obvious: the time delay term is elimi-

nated from the characteristic equation of the setpoint

transfer function, and thus closed-loop time constant

can be taken as the design parameter. Consequently,

settling time is under control. Moreover, there is no

closed-loop zero introduced by the PI controller on the

forward transfer function, and thus no overshoot is to

be taken care of. However, for the disturbance rejec-

tion, the performance is not satisfactory compared with

other methods. Note also that the control system of

Method G is the most complicated, too, in which there

are three controllers to be designed.

Methods E and F, which are both IMC-based, give

the second best setpoint responses: short settling time,

Fig. 7 System response of Method A for small normalized dead-time

Fig. 8 System response of Methods B, C and D for small normalized dead-time

Fig. 9 System response of Methods E, F and G for small normalized dead-time


11/19

VOL. 40 NO. 2 2007 155

very small or even no overshoots and small IAE/ISE

as well. The recovery times of Methods E and F from

the disturbance are the smallest. IMC based design is

essentially more complicated than the traditional PID

design, and the PID settings obtained by approximat-

ing the ideal IMC controller are superior to those from

traditional PID tuning rules. More desirable closed-

loop responses can be expected. Since Method E gives

an explicit tuning rule, it is more convenient for prac-tical implementation as a PID auto-tuning strategy.

Among the remaining four traditional PID con-

trol schemes, Method C provides the highest perform-

ance, although not as good as the former three. Except

for some oscillatory behaviors, it generates very low

overshoots but a fast rise-time. Method A has very sim-

ple tuning rules in line with different optimization

specifications. However, given a properly tuned pre-

filter, these simple tuning formulas gives PID settings

with relatively good system responses, especially, in

the optimized specifications they aim to achieve, for

example, ISE specification of Method A (ISTE setting)

is the lowest among all the methods. Also derived from

the double loop PID-P structure, Method D gives a

reasonable output response, which is similar but bet-

ter than Method B. In fact, Method B doesnt have a

very good performance in the comparison, but it pio-

neered in the double-loop PID controller design for

unstable plants.

The robustness performances of Methods A to G

are addressed by assuming a mismatch of 10% onthree parameters, K, T and L, of the process model,

respectively. The results are presented in Figures

1113 , from which it can be concluded that all the

Table5

Controlpe

rformanceforsmallnormalizeddeadtime

Fig. 10 Control signals of investigated methods for smallnormalized dead-time


12/19


investigated methods are able to maintain stability and

satisfactory performance in small modelling mismatch.

However, in the case of +10% perturbation, Methods

A and C present evident oscillating behaviors. Over-

all, the IMC-based Methods E and F and the modified

Smith predictor Method G still outperform the others

under parametric uncertainty, as shown by specifica-

tions in Table 6.

2.2 Medium normalized dead-time: 0.693 L/T< 1

Consider now the time-delayed unstable process

G se

s

s

p ( ) = ( )

1 2

1 5 146

.

.

with the normalized dead-time ofL/T= 0.8. Method B

is excluded since it has been stated in Parket al. (1998)

that is only designed for 0 < L/T < 0.693. The same

Fig. 11 Robustness analysis of Method A

Fig. 12 Robustness analysis of Methods B, C and D


13/19

VOL. 40 NO. 2 2007 157

Mp [%] tr ts IAE ISE

Method A (ISE) nominal 17.27 5.32 29.26 5.75 1.35

10% mismatch 20.83 5.06 26.45 6.04 1.63

+10% mismatch 18.82 3.25 33.56 5.46 1.24

Method A (ISTE) nominal 6.92 3.28 14.58 4.34 0.95

10% mismatch 10.60 4.13 16.72 4.49 1.05+10% mismatch 8.65 2.91 26.81 4.21 0.99

Method A (ITSE) nominal 11.05 3.56 19.16 4.58 1.00

10% mismatch 13.02 4.66 18.48 4.75 1.14

+10% mismatch 11.52 2.99 28.28 4.41 0.99

Method B nominal 42.47 4.10 50.13 10.16 5.49

10% mismatch 52.78 4.14 67.50 12.85 6.32

+10% mismatch 36.65 4.01 37.82 8.92 5.25

Method C nominal 10.81 2.68 15.62 4.45 3.88

10% mismatch 6.67 2.92 15.02 4.30 3.24

+10% mismatch 20.60 2.56 33.74 6.14 3.86

Method D nominal 25.28 4.33 27.20 7.28 4.6110% mismatch 35.14 4.34 41.25 8.76 4.88

+10% mismatch 21.19 4.26 30.34 7.09 4.69

Method E nominal 0 7.44 13.48 5.79 1.09

10% mismatch 3.03 6.42 18.91 5.93 1.32

+10% mismatch 0.45 9.74 21.20 5.23 0.98

Method F nominal 1.30 6.83 12.01 5.02 0.95

10% mismatch 3.52 5.79 17.66 5.18 1.10

+10% mismatch 3.84 3.92 23.65 4.69 0.91

Method G nominal 0 4.39 9.83 2.01 1.01

10% mismatch 0 3.58 20.73 2.43 1.03

+10% mismatch 8.23 5.17 19.75 2.43 1.02

Table 6 Performance robustness

Fig. 13 Robustness analysis of Methods E, F and G


14/19


external signals as before are introduced to the system

to generate time responses for Methods A, C, D, E, F

and G, as shown in Figures 1416. The performance

specifications are listed in Table 7.

Note that Method G gives almost unchanged

setpoint specifications. It is because in this example,

the design parameter , closed-loop time constant, re-mains the same as in the first case. Although it still

performs the best in setpoint performance, Method G

becomes less effective for disturbance rejection for

larger normalized dead-time. It can be observed from

Eq. (35) that the disturbance transfer function of the

modified Smith predictor is not delay-free, which is

the reason why its disturbance rejection is much worse

than its setpoint response.

The IMC-based Methods E and F have no over-

shoot, very short settling time and recovery time for

the process Eq. (46). They are superior to all the other

Fig. 14 System responses of Method A for median normalized dead-time

Fig. 15 System responses of Methods C, D and E for median normalized dead-time

Fig. 16 System responses of Methods F and G for median normalized dead-time


15/19

VOL. 40 NO. 2 2007 159

methods in overall behaviors.

Methods C and A are also working pretty well in

this range of normalized dead-time, especially with

regard to the disturbance rejection. And it is worth

mentioning that Method A outperforms other methods

at IAE and ISE indices. Method D gives the poorest

performance in this scenario. This may be due to the

choice of the setpoint weight value b provided byEq. (26) and the Taylor series approximation ofeLs.

However, if the process to be controlled is only of an

insignificant time delay, Method D will become a con-

venient design approach.

2.3 Large normalized dead-time: 1 L/T< 2Finally consider this process:

G se

s

s

p ( ) = ( )

1 5

147

.

with the normalized dead-time ofL/T= 1.5. Now, only

Methods E and F remain applicable to the process ofthe ratio L/T 1. The system responses are shown inFigure 17. The performance indices are listed in Ta-

ble 8. Note that in this case, Method E yields more

oscillatory behaviors than Method F does.

It can be concluded from the above three case stud-

ies that the following overall ranking in terms of gen-

eral applicabilities and achievable control perform-

ance: (1) Method F, (2) Method E, (3) Method G, (4)

Method C, (5) Method A, (6) Method D, (7) Method

B.

It has been reported in the literature that properly

tuned P/PI controllers can stabilize the FOPDT unsta-ble process if and only if its normalized dead timeL/T

1, and that the PD/PID controller can relax the con-straint to L/T 2, as the D portion contributes phaselead in the control system. Therefore, for those feed-

back configurations with PD/PID controllers, like

Method C, they have potential to be extended to the

case of 0 L/T 2, but need some more future re-search. And similar extension can also be made for the

modified Smith predictor structure of Method G, by

changing the controller Gc2

into the PD form to increase

its applicability. Improvements on disturbance attenu-

ation are also expected in Method G, while retaining

the merit of pole placement in its closed-loop setpointtransfer function, Eq. (34).

3. Nonlinear PID Control

It can be seen from comparative studies in the pre-

ceding section that the best achievable performance

among seven studied methods has been impressively

good. It would be difficult to devise a new and better

control scheme within their frameworks. However, we

noticed that all these methods use linear time invari-

ant (LTI) controllers only. As there are many nonlinear

Table7

Controlper

formanceformediumnormalizeddead

time


16/19


17/19

VOL. 40 NO. 2 2007 161

Thus kmax

= k0

+ k1

when e(t) = , and kmin

= k0

when

e(t) = 0. k2

defines the change rate in the range from

kmin

to kmax

together with e(t). k(t) can also be of the

sigmoidal function:

k t k k k e t

( ) = ++ ( )( )

( )0 12

2

11 51

exp

where kmin

= k0k

1when e(t) = , k

max= k

0+ k

1when

e(t) = + and k(t) = k0

when e(t) = 0. Let us implement

the above three NPID control strategies in Method B

and repeat simulation example Eq. (45) to validate them

in controlling unstable processes. The NPID settings

are listed in Table 9, which are tuned by trial and er-

ror. From Table 9 and Figure 18, it is obvious that the

system performance has been improved greatly by

NPID controllers, especially the setpoint response,

which indicates the usefulness of NPID control.

It is however noted that the above modifications

of Method B yet could not achieve performance as good

as the best available ones in Table 5, say those of

Method F. Hence, it is logical to incorporate nonlinear

PID control into the best linear controller. As Method

F gives an overall most satisfactory control, we use its

controller settings and modify the proportional gain

into a nonlinear component, while the linear integral

and derivative gains are retained. Therefore,

u t k k t e t k e t t k e t t

( ) = ( ) ( ) + ( ) + ( ) ( )p i dd 0 52

where kp, k

i, k

dare the fixed gains provided by Method

F, and k(t) is in the form of our modified sigmoidal

function:

k t k k k e t k e t

( ) = +( )( ) + ( )( )

( )0 12 3

21 53

exp exp

The simulation examples, Eqs. (45), (46) and (47), have

been repeated to test such a nonlinear control strategy.

The parameters in the nonlinear equation, Eq. (53), are

determined by trial and error. The system responses

are shown in plots A, B and C of Figure 19, respec-

tively. The performance specifications and controller

Table 9 Control performance of nonlinear modification of Method B

Fig. 18 Nonlinear modification of Method B


18/19


settings are listed in Table 10. From simulation results,we can see that performance improvement is achieved

from the nonlinear modification on the original con-

troller settings of Method F, although not very signifi-

cant. It is also worth noting that disturbance rejection

is not as good as that provided by linear controllers.

We also noticed that the setpoint response of

Method G is the best among all the investigated linear

methods. Due to the merit of Smith predictor control

structure, there is no RHP zero introduced in the

setpoint transfer function. Therefore, we try to further

accelerate the step response by using LTV setpoint

weighting. The time varying gain of the pre-filter is

chosen as follows:

f t M t

T M T t r ( ) = +

( ) ( )

1 54sgn

where

sgn,

,t

t

t( ) =

journal 2007 jcej jp

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