a new smith predictor and controller for control of processes

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ISATRANSACTIONS®

ISA Transactions 42~2003! 101–110

A new Smith predictor and controller for control of processewith long dead time

Ibrahim Kaya*Engineering Faculty, Electrical and Electronics Dept., Inonu University, 44069, Malatya, Turkey

~Received 12 July 2001; accepted 13 January 2002!

Abstract

Good control of processes with long dead time is often achieved using a Smith predictor configuration. TypPI or PID controller is used; however, it is shown in this paper that for some situations improved set poidisturbance responses can be obtained if a PI-PD controller is used. Several methods are possible for seleparameters of the PI-PD controller but when the plant transfer function has no zeros, the use of the standaprovides a simple algebraic approach, and also reveals why difficulties may be encountered if a PID controllerSome examples are given to show the value of the approach presented. © 2003 ISA—The Instrumentation,and Automation Society.

Keywords: Predictive control; Integral performance indices; Delay compensation; PID controllers; Disturbance rejection

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1. Introduction

Plants with long time delays can often not bcontrolled effectively using a simple PID controler. The main reason for this is that the additionphase lag contributed by the time delay tendsdestabilize the closed loop system. The stabiproblem can be solved by decreasing the contler gain. However, in this case the responsetained is very sluggish.

The Smith predictor, shown in Fig. 1, is weknown as an effective dead-time compensatora stable process with long time delays@1#. Theclosed loop transfer function for the system of F1 is given by

T~s!5C~s!

R~s!5

Gc~s!G~s!e2us

11Gc~s!@Gm~s!1Ge~s!#,

~1!

*Fax: 190 422 3401046.E-mail address:[email protected]

0019-0578/2003/$ - see front matter © 2003 ISA—The Instru

where Ge(s)5G(s)e2us2Gm(s)e2ums.Gm(s)e2ums, G(s)e2us, and Gc(s) are, respec-tively, the plant’s dynamic model and the transffunctions of the plant and the controller, whichusually a PI or a PID controller.

The stability of the Smith predictor is affecteby the accuracy with which the model representhe plant. Based on the assumption that the moused matches perfectly the plant dynamiGe(s)50, and the closed loop transfer function oEq. ~1! reduces to

T0~s!5Gc~s!Gm~s!e2ums

11Gc~s!Gm~s!. ~2!

According to Eq.~2!, the parameters of the primary controller,Gc(s), which is typically taken asPI or PID, may be determined using a modelthe delay-free part of the plant.

Many possible approaches for determiningtuning the parameters of an appropriate controlGc(s), have been given in the literature and rececontributions include Refs.@2–4#. An interesting

mentation, Systems, and Automation Society.

102 Ibrahim Kaya / ISA Transactions 42 (2003) 101–110

Fig. 1. The Smith predictor control scheme.

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result has also been provided recently by Mtausek and Micic @5#, who have shown that byadding an additional controller the disturbancesponse of the Smith predictor can be controllindependently of the set point response. The regiven by Matausˇek and Micic @5# was later im-proved by the same authors@6# by adding a PDcontroller so that a better disturbance responsebe achieved.

In this paper a new control strategy for thSmith predictor is presented that replaces the cventional controller by a PI-PD structure, a cotroller that has been shown to significantly outpform a PID controller in some standard singlinput–single-output systems@7#. The disturbancerejection controller suggested by Matausˇek andMicic @6# is also included. It is also shown thawhen the plant transfer function is of relativelow order, the parameters of the PI-PD controlcan be suitably chosen using the standard fomethod, which is a simple algebraic approachcontrol system design.

The paper is organized as follows. In Sectiondesign using integral performance criteriabriefly reviewed and the standard forms for intgral squared time error~ISTE! are given. Section 3presents the new PI-PD Smith predictor controland then outlines the design procedure for sevesimple process plant transfer function models. Eamples are given in Section 4, which by compason with previous studies clearly show the advatages of the new design approach presenFinally, conclusions are given in Section 5.

2. Integral performance criteria

In this section a very brief review of integraperformance criteria is given, since the approa

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can be used to produce standard forms used forcontroller designs used in the next section. Tuse of integral performance indices for contrsystem design is well known. Many textbooksuch as Refs.@8# and @9#, include short sectionsdevoted to the procedure. When integral perfmance criteria were first suggested in the ea1950s, digital computers were in their infancy aevaluations could take a long computation time

For linear systems, the ISTE can be evaluaefficiently on digital computers using ths-domain approach with A˚ strom’s recursive algo-rithm @10#. Thus for

J05E0

`

@e2~ t !#dt, ~3!

the s-domain solution is given by

J051

2p j E0

`E~s!E~2s!ds, ~4!

whereE(s)5B(s)/A(s) andA andB are polyno-mials with real coefficients, given by

A~s!5a0sm1a1sm211¯1am21s1am ,

B~s!5b1sm211¯1bm21s1bm .

Criteria of the formJn5*0`(tne)2dt can also be

evaluated @11# using this approach, sincL@ t f (t)#5(2d/ds)F(s), where L denotes theLaplace transform andL@ f (t)#5F(s).

Using this approach, it is possible to obtain toptimal parameters of a closed loop transfer funtion that will provide a minimum value of theISTE. Tables of such all pole transfer functionwere given many years ago@12# but are of littleuse in design, because even with an all-pole pltransfer function the addition of a typical contro

103Ibrahim Kaya / ISA Transactions 42 (2003) 101–110

Fig. 2. Optimum values ofd1 .

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ler produces a closed loop transfer function withzero. Results with a single zero were also giventhat the feedback loop would follow a ramp inpwith zero steady-state error, but these expressare not appropriate for step response design. Fclosed loop transfer function with one zero iteasy to present results for these optimum transfunctions as the position of the zero varies.

Assuming a plant transfer function with no zeand a controller with a zero, then a closed lotransfer function,T1 j , of the form

T1 j5c1s11

sj1dj 21sj 211¯1d1s11~5!

is obtained, where the subscript 1 inT1 j indicatesa zero in the numerator of the standard form athe subscriptj indicates the order of the denomnator. Also, for a unit step set point, the errorobtained as

E1 j5sj 211dj 21sj 221¯1~d12c1!

sj1dj 21sj 211¯1d1s11. ~6!

Minimizing E1 j for the ISTE, the optimum valueof the d’s as functions ofc1 are shown in Figs. 2and 3 forT12(s) andT13(s), respectively.

Fig. 4 shows howJ1 ~the minimum value for theISTE criterion! varies asc1 increases. The figureillustrates that asc1 increases the step response

s

r

the closed loop improves. However, it is also sefrom the figure that any further increase inc1above the value of 4 or 5 has a negligible improvment in the response. Also the step responsesthe J1 criterion for a few differentc1 values areshown in Fig. 5. It is seen that asc1 increases thestep responses are faster.

3. Controller design

The proposed Smith predictor is shown in Fi6, whereGc1 is a PI controller,Gc2 is a PD or Pcontroller, where it is appropriate, andGd is thedisturbance controller introduced in Ref.@6#. Gc2is used to stabilize an unstable or integrating pcess and modify the pole locations for a stabprocess. The other two controllers,Gc1 and Gd ,are used to take care of servotracking and regutory control, respectively. WhenGc25Gd50,then the standard Smith predictor is obtained.

Assuming exact matching between the proceand the model parameters, then the set pointdisturbance responses are given by

C~s!5Tr~s!R~s!1Td~s!D~s!, ~7!

where


Fig. 3. Optimum values ofd1 andd2 .

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Tr~s!5GmGc1e2ums

11Gm~Gc11Gc2!, ~8!

Td~s!5

Gm[11Gm(Gc21Gc1

2Gc1e2ums)]e2ums

@11Gm~Gc11Gc2!#@11GdGme2ums#.

~9!

The transfer function for the set point respongiven by Eq.~8!, reveals that the parameters of tmain controllers,Gc1 and Gc2 , may be deter-mined using a model of the delay-free part of tplant. Also it is seen that only the disturbancesponse is affected by the controllerGd . It hasbeen shown@3# that the original Smith predictogives a steady-state error under disturbancesintegrating processes. That is why the use ofcontrollerGd was proposed in Ref.@6# to improvethe disturbance rejection of integrating processHere, it has been adopted in the proposed methagain primarily to improve disturbance rejectiofor integrating processes.

The proposed PI-PD Smith predictor contrstructure gives superior performance over classPI or PID Smith predictor control configuratiofor both the set point response and disturbance

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jection. The superior performance of the proposSmith predictor is more evident when the procehas a large time constant, with or without an intgrator, and for processes with ‘‘poorly’’ locatepoles, i.e., lightly damped. This is illustrated latby examples. However, the proposed Smith pdictor configuration still suffers from a mismatcbetween the actual process and model dynamwhich is a case also for classical PI~D! Smith pre-dictor scheme.

In the following, it is shown how standard formare used in order to obtain the controller paraeters for several specific plants.

Case 1: If the process can be modeled by a firsorder plus dead-time transfer function given by

Gm~s!5Km

s1ae2ums ~10!

then the controllers are given by

Gc1~s!5KcS 111

TisD ~11!

and

Gc2~s!5K f . ~12!


Fig. 4. The optimum values ofJ1 for increasingc1 .

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Using the delay-free part of Eq.~8!, then the re-sulting closed loop transfer function,T(s), isgiven by

T~s!5KmKc~Tis11!

Tis21~a1Kc1K f !KmTis1KmKc

.

~13!

Rearranging Eq.~13!,

T~sn!5c1sn11

sn21d1sn11

~14!

is obtained, where the normalized Laplace coplex variable sn5s(Ti /KKc)

1/25s/a, whichmeans the response of the system will be fasthan the normalized response by a factor ofa,where

a5~KmKc /Ti !1/2, ~15!

andc1 andd1 in Eq. ~14! are given by

c15aTi , ~16!

d15a1~Kc1K f !Km

a. ~17!

In principle the time scale,a, can be selected bthe choice ofKc , c1 by the choice ofTi andd1 by

the choice ofK f . In practice,Kc will be con-strained, possibly to limit the initial value of thcontrol effort, so that the choice ofKc andTi mayinvolve a tradeoff between the values chosen foaand c1 . The controllerGd is not needed in thiscase as the plant is a stable one.

Case 2: If the process can be modeled bysecond-order plus dead-time transfer function wcomplex poles, which is given by

G~s!5Km

s21as1be2ums, ~18!

then the controller,Gc1 , in the forward loop isstill a PI controller given by Eq.~11!. However,the controller,Gc2 , in the inner feedback loop isnow a PD controller, given by

Gc25Tds1K f . ~19!

Following the same procedure as in case 1,closed loop transfer function can be put in the nmalized form

T~sn!5c1sn11

s31d2sn21d1sn11

, ~20!

where


Fig. 5. Step responses forT13(s).

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srdq.

c15aTi , ~21!

d25a1KmTd

a, ~22!

d15b1~Kc1K f !Km

a2 , ~23!

and

a5~KmKc /Ti !1/3. ~24!

In this casea is again selected by the choice ofKc

andc1 by the choice ofTi , d1 by the choice ofTd

andd2 by the choice ofK f . Also the choice ofKc

andTi may involve a trade-off between the valu

chosen fora and c1 . Again the controllerGd isnot needed because the plant is a stable one.

Case 3: An integrating process plus dead timwith the following transfer function is considered

G~s!5Km

se2ums. ~25!

This is a special case of Eq.~10! with the constanta equal to zero. In this case the controllersGc1andGc2 are again given by the transfer functionof case 1. It is easy to show that the standaclosed loop transfer function is again given by E

Fig. 6. The proposed Smith predictor control scheme.

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~14! and also the constantsc1 anda are given bythe same equations as in case 1. However,d1 isnow given by

d15Km~Kc1K f !

a. ~26!

The choice ofa, c1, andd1 is made according tothe discussions in case 1.

The controllerGd

Gd5K0~11T0s! ~27!

is necessary for a satisfactory load disturbancejection. Gd is designed using the approach givin Ref. @6# where the relationT05aum has beenused to obtain

K05p/22Fpm

KmumA~12a!21~p/22Fpm!2a2

~28!

for a specific phase marginFpm. The best resultscan be obtained@6# by a50.4 andFpm564°.

Case 4: A first-order plus integrator plus deatime with the transfer function

G~s!5Km

s~s1a!e2ums ~29!

is a special case of 2 withb50. Thus the Eqs.~21!–~24! are used withb50 in Eq. ~23!.

The choice ofa, c1 , d1 , andd2 is made accord-ing to the discussions in case 2. The controlGd , given by Eq. ~27!, may again be used fodisturbance rejection. However, it should be nticed that in this caseum used in Eq.~28! must bereplaced byum1Ts where Ts is the sum of thetime constants.

4. Examples

In this section several examples are given tolustrate the use of the method. The method is copared with the methods of both Ha¨gglund and Ma-tausek for processes that can be modeled byfirst-order plus dead-time and integrator plus detime transfer functions, respectively. Also, thcontrol of a second order plus dead-time transfunction with complex poles and a long dead timis given using the proposed method and it is copared with a Smith predictor using a PID controler as the main controller.

-

Example 1. A FOPDT transfer function given bye210s/(10s11) is considered. It should be noticethat the time constant is quite large.Kc was lim-ited to 1.00 andTi was chosen as 0.25, leadinga50.6324andc150.1581.For the ISTE integralperformance criteriond151.3563, which givesthe controller parameterK f56.5772.The resultsare compared with the initial tuning for a PI controller suggested by Ha¨gglund for whichKc51and Ti510. The response of the system to thproposed method and Ha¨gglund’s method, whenthe starting tuning parameters are used, is givenFig. 7. At time t560 sa disturbance with magnitude of20.1 is introduced to the system. The prposed method gives better results for the set poresponse, although there is not a significant diffence in disturbance rejection. The main reasonproposed method gives better results is becadue to its large-time constant its responsesembles more an integrating process rather thanonintegrating one.

Example 2. A SOPDT transfer function given bye215s/(s210.2s11) is considered.Kc was againlimited to 1.00, with the choice ofTi50.50.Thesetwo values givea51.26andc150.63.The ISTEoptimum values ofd2 and d1 are, from Fig. 3,1.644 and 2.163, respectively. These values canachieved with 1.8714 and 1.4334 forTd and K f

respectively, from Eqs.~21! and ~22!. The re-sponse to set point change is given in Fig. 8 for tPI-PD controller and also shown for comparisoare the results for a PID controller with paramete1.00, 0.198, and 5.065 for the gain, integral aderivative time constants, respectively, whicwere obtained first by limitingKc51.00, then byusing an ISTE optimization program to get thother two parameters. The proposed methodseen to be superior to using a PID controller, sinthe plant’s complex poles are near to the imanary axis.

Example 3. An integrating process givenby e25s/s(s11)(0.5s11)(0.2s11)(0.1s11),which was given in Ref.@6# is considered. As inRef. @6# K050.08andT052.72was obtained fora50.4 and Fpm564°. The proportional onlycontroller for Matausˇek and Micic @6# was takenas 0.56. Using the identification method givenKaya @13# first a proper model Gm(s)5e25.6s/s(1.205s11) was obtained. Then, limit-ing Kc to 1.00 and choosingTi as 0.5 and follow-


Fig. 7. Step responses for example 1.

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ing the same procedure as before,a51.184, c150.592, d251.629, d152.149, giving K f

52.629 and Td51.323, were obtained. The responses to a unit set point and a disturbanchange, which is of magnitude20.1 at t540 s,are given in Fig. 9. In this case only a small im

provement over Matausˇek’s approach is obtainedHowever, in the following example it is showthat the proposed method can be more advageous if the process has a large time constawhich means that the process almost has a douintegrator action.




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Example 4. ConsiderG(s)5e26.7s/s(10s11),where the plant has both an integrator and a retively large time constant. WithKc50.50 and Ti

52.00 and again using the ISTE standard forapproach, one obtainsTd53.757and K f51.336.K050.031 and T056.68 were obtained using

equations given in Section 3 fora50.4 andFpm

564°. The Matausˇek and Micicmethod has thesameK0 andT0 values and a main controller gaiof 0.1. The responses of the system to a unitpoint and20.1 unit disturbance att5100 s areshown in Fig. 10. The far superior response of t


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PI-PD controlled system is now clearly evident.

5. Conclusions

A new approach to control processes, both ingrating and nonintegrating, with long dead timbased on a Smith predictor concept and a PI-controller has been introduced. It is shown thatproposed method can be advantageous whenprocess has a large time constant, with or withan integrator, and for processes with ‘‘poorly’’ located poles, i.e., lightly damped. Several produres for obtaining the parameters of the PI-Pcontrollers are possible, but one of the simplapproaches, which is used in this paper, is to eploy ISTE standard forms as this enables thesign to be completed using simple algebra.

References

@1# Smith, O. J., A controller to overcome dead time. ISJ. 6, 28–33~1959!.

@2# Hagglund, T., A predictive PI controller for processewith long dead-time delay. IEEE Control Syst. Ma12, 57–60~1992!.

@3# Watanabe, K. and Ito, M., A process-model control flinear systems with delay. IEEE Trans. Autom. ContrAC-26, 1261–1266~1981!.

@4# Astrom, K. J., Hang, C. C., and Lim, B. C., A newSmith predictor for controlling a process with an intgrator and long dead time. IEEE Trans. Autom. Cotrol 39, 343–345~1999!.

@5# Matausek, M. R. and Micic, A. D., A modified Smithpredictor for controlling a process with an integratand long dead time. IEEE Trans. Autom. Control41,1199–11203~1996!.

@6# Matausek, M. R. and Micic, A. D., On the modified

e

Smith predictor for controlling a process with an intgrator and long dead time. IEEE Trans. Autom. Cotrol 44, 1603–1606~1999!.

@7# Atherton, D. P. and Boz, A. F., Using standard formfor controller design. Proceedings of Control’98, Setember 1998, Sweansea, UK.

@8# Dorf, R. C. and Bishop, R. H., Modern Control Systems. Addison-Wesley, Reading, 1995.

@9# Chen, C. T., Analog and Digital Control System Dsign. Saunders College, 1993.

@10# Astrom, K. J., Introduction to Stochastic ControTheory. Academic, New York, 1970.

@11# Zhuang, M. and Atherton, D. P., Tuning PID controlers with integral performance criteria. In Matlab Tooboxes and Applications, Peter Peregrinus, Chap. 8131–144, 1993.

@12# Graham, D. and Lathrop, R. C., The synthesis of otimum response: criteria and standard forms, II. TraAIME 72, 273–288~1953!.

@13# Kaya, I., Relay feedback identification and modbased controller design. D.Phil. thesis, UniversitySussex, U.K.~1999!.

Ibrahim Kaya was born inDiyarbakir, Turkey, on 17 Sep-tember 1971. He received aB.Sc. degree from GaziantepUniversity, Turkey and a D.Phil. degree from University ofSussex, England. He is currently with Inonu University,Electrical & Electronics De-partment. He is interested inIdentification, Autotuning,Computer-Aided Control Sys-tem Design, and User InterfaceToolkits.

a new smith predictor and controller for control of processes

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