paper a process-model control for linear systems with

8/6/2019 PAPER a Process-Model Control for Linear Systems With

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-26,NO . 6, DECEMBER 1981261Albert Benveniste was born in Paris, France, in1949. H e graduated from the Ecole des M ines deParis in 1971.From 1971 to 1973 he was with the CentredAutomatique de 1Ecole des Mines, Fontaine-hleau. From 1974 to 1976 he was with the IRIA ,Rocqu encourt. Since 1976 he has been with theIRISA, Rennes. and is a consultant at the CNET.National Research Center for Telecommunica-tions.Afterome work on probability theory(Markov processes and ergodic theory) for his These dEtat, Paris, 1975,his interests moved towards the area of applied mathematics. automaticcontrol, identification and recursive algorithms. signal processing, speechand image coding, and data communication systems.

Christianhaure was born in Laardrieux,France, in 1954. He was graduated from theInstitutNational des Sciences App liquk s ofRennes in 1977.From 1977 to 1979 he was taking his These deDocteur-Ingknieur in speech coding. He is cur-rently with Thomson-CSF Brest, France, wherehe is interested in radar.

Short PapersA Process-ModelControl for Linear SystemswithDelay

KEIJI WATANABE AND MASAMI I T0Abstrucf-This paper is concerned with linear system s containing timedelay in control. A process-model control which utilize s amathematicalmodel of the process in the minor feedback loop around the conventional

controller t o overcome delay is developed. A new process-model controlsystem vhich can ield zero teadystate error nddesiredransientres ponse s to step disturbances and arbitrary initial conditions is proposed.The remarkable feature of the proposed system is that the states of themodel and the integrator are unobservable in the output of the model. It isshown that th is structural property is the key by which the major problemsof process-model control can be solved.

I. INTRODUCTIONTime delay occurs frequently in process control problems. Compared toprocesses without delay, the presence of delay in processes greatly com-plicates the analytical aspects of control system design and makes satis-factory control more difficult to achieve.Smith [l ] proposed a delay compensation technique wbich utilizes amathem atical model of the process in the minor feedback loop around theconventional controller. This technique became known as the Smithpredictor method [2]. The main advantage of the Smith predictor methodis that time delay is eliminated from the characteristic equation of th eclosed loop system. Thus, the design problem for the process with delaycan be converted to the one without delay.On the other hand, a n alternative design approach for systems withdelay has b een d iscussed using state equations [4]-[9]. F uller [4], Mee [5],Kleinman [6], and Koivo er al. [7] investigated the optimal regulator

recommended by W . E Schmitendorf, Chairman of the Optimal Systems Committee.Yamagata University. Jonan. Yonezawa-shi. Yamagata 992, JapanUniversity, Furo-cho. Chikusa-ku. Nagoya 4 6 4 , Japan.

ManuscriptreceivedJuly 16, 1979; revised May 5. 1980 and December 9. 1980. PaperK . Watanak is with the Department of Uectronic Engineering, Faculty of Engineering.M . Ito iswith he Automatic ControlLaboratory.Faculty of Engineering. Nagoya

problem with time delays. Lewis [8] and Manitius et al . [9] considered thefinite spectrum assignment problem of delay systems. The resultingcontrol law con tains certain functionals of the present state of the delayelement. To construct the control systems, it is necessary to continuallymeasure the present s tate of the delay element or measure the functionals.Various methods of the implementation of the functionals are suggestedin [4]. One of them is to use a certain m odel of the process. By passing thecontrol signal through the model, the functionals can be generated w ithoutthe mea surement of the state of the d elay element [4]-[6], [lo ]. Theresulting control systems are similar to the Smith predictor co ntrol one.Many papers on the control of the time delay system which have beenpublished embrace Smiths principle explicitly or implicitly [IS], [19]. Inthis paper, the Smith predictor method and the state space approacheswith the implementation of the functionals by a process model are saidgenerically to be process-model controls [3].Am ong process-mo del contro l systems -which have been reported, thereexist significant differences of characteristics [ I 11, [13], [21]. Do nog hue[ 1 I] , [ 131 compared the characteristics of two control systems resultingfrom application of the Sm ith predictor m ethod an d the optimal designapproach o a first-order lag process with delay, and showed that heSmith predictor control system can yield zero steady-state e rror to stepdisturbances, whereas the optimal system has nonzero steady-state errorto the same disturbance even though the controller contains an integrator.Cook and Price [121 had objection to Donoghues results and p ointed outthat, if the disturbances are steps of random magnitudes, then the Smithpredictor is optimal. However, Cook and Price did not consider thetransient responses. Considering them, one may see that the Smithpredictor suffers some shortcomings. In particular, if the process has polesnear the origin in the left half plane, then the responses may be sluggishenough to be unacceptable. Many improvements have been tried [19], hutthe problem has not been solved to a satisfactory level within theframework of feedback. Consequently, disturbance attenuation has beenstill one of the major problems of process-model control.

Furthermo re, there exist differences of characteristics among som e statespace approaches. The control systems presented by Kleinman [6] andMarshall et a/. [lo] can yield desired transient responses to arbitraryinitial conditions of the system, but those of Fuller [4] and D onogh ue [1 ]0018-9286/81/1200-1261$00.75 01981 IEEE


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I262 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-26,NO. , DECEMBER 1981cannot. The mechanism which causes these differences has not beenclarified.

This paper is concerned w ith process-model control for linear systemscontaining delay in control subject to step disturbances. The concept ofthe error system [14], [I51 is used because of its advantages for hetreatment of disturbances and initial conditions. A new process-modelcontrol system, which can yield zero steady-state error and desiredtransient responses to step disturbances and arbitrary initialconditions, incontrast to the other xisting ones, is proposed. The remarkable feature ofthe proposed system is that the states of the delay-free part of the modeland the integrator are unobservable in the output of the model. Thisstructu ral property is the key by which the major problems of process-model control can be solved. This is illustrated by comparing the pro-posed system with the others.

11. SYSTEM ESCRIFTIONIn this paper, we consider a single input-single output process withdelay in control described by

i ( t ) = A x ( t ) + B u ( t - L )r ( t ) = C x ( t )x ( 0 ) = x o , u ( T ) = u ~ ( T ) - L ~ T < O (1)

where u is the input, x is the nX 1 state v ariable, y is the output, L is th edelay, and uo is the initial function of the input. A , B , an d C are constantmatrices of appropriate dimensions for the process. It is assumed that( A ,B ) s controllable, ( C ,A ) is observable, and

The last assumption is needed when an integrator is tandemly connectedto the process in order to make the steady-state error zero. If (2) issatisfied, then the augmented process with the integrator is observable.111. PROCESS-MODELONTROL

Some existing process-model control systems are considered in thissection to state the problems.The S mith predictor control system for ( 1 ) is shown in Fig. 1. The blockG(s)e -ZL. here G ( s ) = C ( s l - A ) - ' B , is the process. The block G,(s) isa conventional controller and the feedback element G ( s ) - G(s)e-'L is aprocess model. The transfer function from the reference input r to th eoutput y is given by

y ( s ) - G , ( s ) G ( s ) e - ' Lr ( s ) l + C , ( s ) G ( s ) ' (3 )

It should be noted that time delay is eliminated from the characteristicequation of the closed loop system. The design problem for the processwith delay can be converted to the one without delay. Thus. the controllerG,(s) can be designed without considering delay. This is the majoradvantage of the Smith predictor control. However, the Smith predictorcontrol canno t handle the disturbances and nonzero initial conditions. Inparticular. if the proc ess has poles in the left half plane n ear the orig in.then the responses to disturbances and n onzero initial conditions may besluggish enough to be unacceptable, as seen later.Fuller [4] nvestigated an optimal regulator problem for processes withdelay in control. Applying the result to the process ( 1 ) . the control law isgiven by

--

u ( f ) = / x ( r + L ) . (4 )Since x ( + L ) s expanded into

x ( t + L ) = e A L x ( t ) + / 'A " - " i ? u ( T ) d T (5)the control law contains the present state of the delay element. Toconstruct the control system, it is necessary to continually measure thepresent state of the delay element. How ever, the measurem ent of the

I - L

Fig. 1. Smith predictor control system

II I'Fig. 2. Optimal o n t r o l system proposed by Fuller.

T-- s + l l IFig. 3. Optimal ontrol system proposed by Donoghue

initial condition of the delay element is not feasible and it is necessary toadopt suboptimal control. One method of suboptimal control is to use aprocess model as follows.The second term of the right of (5) can be written

- e a L j d - L e a c ' - L - T ) Bu ( r ) d ~ . 6)The first integral on he right of (6) is the convolution of u with theimpulse response of the system with the transfer function ( ~ I - A ) - ' B .Thus, the first te rn on the right of (6) can be generated by passing thesignal u through the system. In the same manner, the second term on theright of (6) can be generated by passing u through the system with thetransfer function eAL(sJ -A ) - 'Be - 'L . Hence, Fu ller presented a controlsystem shown in Fig. 2. An important result which emerges is that thecontrol law can be generated without the m easurement of the state of thedelay element by using the process model.

Donoghue [ 1 ] tried to extend Fuller's result to a servomechanismproblem in which a step reference input is taken into account. Theresulting system for a process with transfer function Be - ' L / ( s + a ) isshown in Fig. 3 . This system can yield zero steady-state error o stepreference inputs, b ut it has onzero steady-state error to step disturbanceseven though the controller contains an integrator. Donoghue suggestedthat if the zero steady-state error to step disturbance s is desired in theoptimal system, then additional feedback is required which tries toestimate the disturbances.Cook and Price [121 made objections to Donoghue's conclusion andpointed out that the optimal control system for processes with delay isoutlined by Kleinman [ 6 ] .Moreover, Cook and Price generalized this abit, as shown in Fig. 4, where if the disturbances are white noises thenM = e F u L and if the disturbances are steps of random magnitudes then


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IEEE TRANSACTIONS ON AUTOMATIC C O N T R O L ,OL. AC-26,NO. 6, DECEMBER 1981263Id

U -3L y ,s + aprocess

Fi g 4. Optimal control system generalized by Cook an d Price. If M = , then the figureis equivalent to the Smith predictor conuoL IfM = e- 'L , then the figure is equivalent tothat of Marshall ef or.)

Fig. 5. Process-model version of state feedback control system by Manitius nd Olbrot.

M= . If M = e - a L , then the figure is basically the same as the system ofMarshall et nl . [lo]. Cook and Price concluded that the Smith predictorcon trol is optimal if the disturb ances are steps of random mag nitude s.However, they did not take the transientbehavior into account. Consider-ing it, the Smith predictor control suffers some shortcomings, as men-tioned above. Moreover, if the process is astat ic with in tegra l prop erty,then the Smith predictor control cannot yield zero steady-state error tostep disturbances as pointed out later. Therefore, the conclusion of Cookand Price is not valid in these circumstances.On th eotherhand,Manitius ndOlbrot [9] investigated initespectrum assignment problem for the system with delay in control andstate. Applying the result to (l ), the control law is given by

u ( t ) = j x ( t ) + f r eA( ' - L - ' )Bu (T ) d T . (7)The ntegral erm on the right of (7)can be ealized bya n u m e n dcomputation.The esulting system has desired responses to arbitraryinitial conditions [9]. On the other hand, letz ( t ) be

1- L

Differentiating z ( t ) , the following model which generates the integralterm on the right of (7 ) is obtained 19, eq. (2.18)]:i ( t > = A z ( t ) + e - A L B u ( t ) - B u ( r - L ) . (9 )

The resulting control system is shown in Fig. 5 . However, if the initialcon ditio n of th e model is not zero, th en the system cann ot yield desiredresponses as seen later. The systems presented by Smith, Fuller,an dDono ghue are n the same condition.Collecting the esults, it an be concluded that the xisting process-modelcontrol systems suffer some shortcomings in the presence of disturbancesand certain initial cond itions. The m echanism which causes the shortcom -ings has not been clarified.

N. A N E W PROCESS-MODEL CONTROLIn this section, wewill propo se a new process-model control systemwhich can yield zero steady-state error and desired transient response tostep d isturbances and arbitrary initial conditions.We first suppose that both a reference input and a disturbance are zeroand consider a regulator problem for (1). The state at +L for (1 ) is givenby (5). Le t X( t ) be X( t) = x ( r +L ) , then (1 ) can be rewritten as follows:

process-modelI

Fi g 6. A new process-model control system or the regulator problem.

The time delay is expressed as an integral-termof the output equation in(IO), whereas it is denoted by u ( r - L ) in ( I ) . Lety( r ) be

J ( t ) = y ( t ) + C e - A L ] f eA ( ' - ' )Bu ( 7 ) dT . ( 11 )1- L

Then (10) becomesx ' ( t ) = A x ( t ) + B u ( r )v ( r ) = C e - A L x ( r ) . (12)

Since ( A , B ) is controllable and ( Ce -AL ,A ) is observable, there exists acontrol inputu( t ) such that X( t )+0 as t+ 03.If X( t )+0 as +cg , he nu ( t ) +O and, therefore, x ( t ) -O , as seen in ( 5 ) . Consequently, the regula-tor problem for (1) is converted to the one for (12), which is a delay-freeproblem.To construct hecontrol system, it is necessary to mplement heintegral term in (1 1). This integral term can be generatedby the followingmodel of the process:

An important point tobe noted is that he nfluence of the nitialcon ditio n of the model is canceled inyL( ) at r 2 , as seen in (14). That isto say, the stat e of the delay-free part of the model is unobservable inyLat e L , where he term "unobservable" is used to show that the corre-sponding outputmatrix is zero. The resulting contro l system is shown inFig. 6 .Next, let us extend the ab ove result to the following servomecha nismproblem with disturbance. The process is subject to a step disturbance das follows:

where d is given by

The ou tput of the process is required to follow the step reference input rwith zero steady -state error, where r is given byi( ) =0, r ( 0 ) =ro. (17)

To make steady-state error zero,we consider the following integ rator:


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I264 IEEE TWNSACTIONS ON AUTOMATIC CONTROL,OL. AC-26, NO . 6, DECEMBER 1981

where e ( ? ) s the output error given bye ( ? ) = . ( ? ) - y ( t ) .

It is seen from (2) that there exists a matrix T which satisfies

Le t [ ( t ) e

Then we have

where

It shou ld be noted that the servomechanism problem can be conv erted toa regulator problem. Furthermore, (23) can be reduced to the delay-freeprocess of the form

1 ( t ) = a S ( t >+ i o ( t >g ( r > = - t e - " g ( r ) (24)in the same manner as (12) where i(r) s

$(I) = t ( t + L )=eA^ , [ ( t ) + j r ei ( l -T )Bo (7 ( 25 )r - L

an d e ( ? ) s?(r)=e(t>-&" A ( r - r ) & 7- 41 - ( ) d T . (26)

Since ( A ,k ) s con trollable ant (tCAL,) is observable, there exists acontrol nput u ( t ) such that [ ( t ) - O as t-cc, in the sense of poleassignment of the closed loop system. It follows from (24) an d (26) thate ( t ) 0 as r+w in the same condition. The integral term on the right of(26) can be implemented by the following model in the same mann er as(13):

The state of the delay-free part of (27) is also unobservable in y, at r > L ,Le t 2; =[x;. x , ] an d x , is the state of the integrator. Since

I

pro c e ~ s - rmode l (30)I I( b)

Fig. 7. A new process-model control system for the sewomechanism problem. (a) Thesystem derived from that of Fig 6 and @) the reduced system.

= C e - d L x p ( t ) - C j L e - A ' B d T x , ( t ) (28)

t ~ . , ( t - L )=Cx , ( r -L ) (29)the resulting control system is represented by block diagram as shown inFig. 7 (a).Furthermore, reducing two integrators to one, then the system of Fig. 7(a) can be converted to that of Fig. 7 @) where the reduced process modelis given by

an d

ip() =Axp( ) +Bu( t )~ ~ ( f ) = C e - ~ ~ ~ ~ ( t ) - C x ~ ( f - L ) - lCe-" ' Bd . ru ( t ) . (30)This system is the new process-model control system which we intend topropose in the present paper. The structural properties of the systemshown in Fig. 7 (a) remain unchanged in the system of F ig. 7 (b). That isto say, in Fig. 7 (b), x p and the state of the integrator are unobservable inY, at ra L . The direct path from u to . y L , that is, the third term on theoutput equation of (30). is needed to make the state of the integratorunobservable in the output of the model. This structural property is thekey by which the major problems of process-model control systems can besolved. This can be clarified via comparison of the characteristics ofprocess-model control systems.

0

v. COMPARISON O F CHARACTERISTICSOF%ME ROCESS-MODELCONTROLYSTEMSA. Response to a Srep Reference Inpui

Generally, the process-model control systems are represented by ablock diagram as shown in Fig. 8, where G ( s ) = C ( s l - A ) - ' B , G , ( s ) =C , ( s I - A ) - ' B + D an d M is the constant number. Moreover, C , is the1X n onstant matrix and D s the constant number. I t is a s s u m e d hat thecontroller G , ( s ) contains an integrator, (C,,) is observable, and


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IEEE TRANSACTIONSON AUTOMAT IC CONTROL, VOL. AC-26, NO. 6 , DECEMBER 1981 1265The transfer function from theeference inpu t r to the output y is given

by

Since (C,, A ) is observablean d ( A , B ) is controllable, zeros of 1 +G,(s)G,(s) can ake preassigned values in the left half plane. The steady-state behavior is as follows,Proposition V.1: In the process-model control system of Fig. 8, let theinitial conditionof the system be zero, the disturbanced be zero and zerosof I + G c ( s ) G , ( s ) e in the left half plane. The steady-state error to thestep reference input is zero if and only if

(33)Proof Since G,(s)G(s) an d G,(s)G,(s) contain poles at s=O, theoutput error at t -+ 03 is given by

(34)where r ( s ) = 1,s. Therefore, the proposition is true.

Remark V.1: In the proposed system, C,(s) isG,(s)=Ce-AL(sl-A)-B-JLCe-rBd7.

If the process does not have poles at s=O, that is to say, A is nonsingular,then

On the other hand, if A =0, B = 1, an d C= 1, then

This argument can beextended to the other processes with integralproperties. The proposed system satisfies (33).Remark V.2: In he systems of Smith, Donoghue, and Cook et d . ,G,(s)=G(s). Then (33) is satisfied.

B. Disturbance AttenuationThe behavior of the process-model control system subject to dis-turbances is as follows.Proposirion V.2: In the system of Fig. 8, let the initial conditio n of thesystem bezeroand he reference input be zero. To yield he desiredtransient responseto the disturbance afterL, t is necessary and sufficientthat poles of G , ( s ) - M G ( s ) e p S Lre canceled with its zeros.Proof: In the system of Fig. 8, the transfer unction from thedisturbance d to the output y is given by

Poles of CJs) are zeros of I + G & s ) G , ( s ) and oles of G,(s)-M c ( ~ ) e - ~ ,aken together.Zeros of 1+G,(s)G,(s) can ake preassignedvalues in the eft half plane, but poles of G , ( s ) - M G ( s ) e - s L cannot.Hence, it is necessary that poles of G , ( s ) - M G ( s ) e - L are canceled withits zeros. Co nversely, if poles of G, ( s ) - M G ( s ) e - E L re canceled with its

G(s)a-sLprocess

Fig 8. General form of process-model control system using ransfer function notation.

zeros, then the impulse responsef G , ( s ) - M G ( s ) e - L is eliminated afterL . This implies that the response to the disturbance can be ad justed bythe controller after L , as seen in (37). The proof is complete.Proposition V.3: In the system of Fig. 8, let the initial condition of thesystem be zero, (33) be satisfied, and zerosf 1+G&s)G,(s) be in the lefthalf plane. The process-model control system has zero steady-state errorto he tepdisturb ance if andonly if (a) he mpulse esponse ofG , ( s ) - M G ( s ) e - Lbecomes zero as t -+m and (b)

Proof The output y at t+ m is given by

r - m s- 0= lim { G,(s) M G ( s ) e-,)s- 0 (39)

where d(s)= 1,s. Equation (39) implies that the proposition is true.Corollary V.1: The condition given by (38) is equivalent to the one thatthe state of the integrator in the controller s unobservable in y L .This can be proved as follows. If (38) holds, then he process model has,at least, a zero at s=O . Hence, the state of the integrator is unobservablein y,. Conve rsely, if the state of the integrator is unobse rvable in ,, thenthe state does not affect yL since unobservable means that the outp utmatrix corresponding to the state is zero. Equation (38) holds.Corollary V.2: When the process is asymptotically stable, he necessaryand sufficient condition for yielding zero steady-state error to step dis-

turbances is tha t (38) is satisfied.Remark V.3: In the proposed system, the states of the delay-free partof the model and the integrator are unobservable n he outp ut of themodel at t 2 . These imply that poles of G , ( s ) - M G ( s ) e - L are canceledwith its zeros and (38) is satisfied. There fore, th e erro rdecay can be ma deas fast as desired.Remark V.4: In the Don oghue system shown in Fig. 3, the ransferfunctio n of the model i s given by

Clearly, the pole of he mode l is canceled with the zero. However, thetransfer function as s+0 becomes

Equation (38) is not satisfied.Donoghues system has nonzero steady-stateerror to step disturbance even though the controller contains an integra-tor. The system of Fig. 4, where M = e - a L , is in the same condition.Remark V.5: In the Smith predictor control system, the transfer func-tion of the model isgiven by G , ( s ) - M C ( s ) e - L = G ( s ) - G ( s ) e - L ,where G ( s ) = C ( s l - A ) - B . If the process is asymptotically stable, thenthe system ha s 7.ero steady-state error because the impu lse response ofG ( s ) - G ( s ) e - L becomes zero at t-03 an d (38) is satisfied. However,poles of G ( s ) - G(s)ePELre not canceled with its zeros except for theprocess with n = 1 an d A= O. The Smith predictor control cannot yielddesired transient response to disturbances. If the process has poles nearthe origin in he lefthalf plane , hen he response may be sluggish.Moreover, it should be noted that if the process is astatic with integralproperties, then the Smith predictor control system has nonze ro steady-state error. For example, consider a process given by


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1266 IEEE TRANSACTIONS ON AUTOMATIC COSTROL, VOL. AC-26.KO. 6. DECEhIBER 1981G ( s ) e - I L = l e - s ,

S

The transfer function of the Smith's model as s - 0 is(43)

Equation (38) is not satisfied. Hence, the Smith predictor control systemfor astatic process with integral properties cannot yield zero steady-stateerror to step disturbances even though the controller contains an integra-tor. The system of Fig. 4, where M= l, is in the same condition.The systems presented by Kleinman [6] and Marshall et al . [101 cann othandle step disturbance because the step disturbances are not taken intoaccount in the problem formulation.To understand better the differences of characteristics. we will presentnumerical examples.Example V.1 (Comparison of responses to a step disturbance):Consider a process described by

1-e-' .s+0.2The responses of some process-model control systems to a step dis-turbance are calculated numerically, as shown in Fig. 9. where the initialconditions of the process and the mod el are zero, and wh ere poles of theclosed-loop system are - .8, - .82, and - .84. It is een that theresponse of the Smith predictor control system is sluggish and D onogh ue'ssystem has nonzero steady-state error. Only the proposed system canmake the error zero as fas t as desired.Example V.2 (The responses of astatic process):Consider an astatic process given by

- e - 2 ss(s+ 1) (45)The response of the Smith predictor control system and the proposed oneare calculated numerically as show n with full lines in Fig. 10, wshere zerosof 1 +G,(s)G,(s) are -0.3. -0.3, -0.4, -0.4, and - 1.0. It is seen thatthe proposed system can yield zero steady-state error whereas the Smithpredictor control system cannot.C. Response to .Vonzero Initial Condition

When the reference input an d the disturbance are zero. the beha tiors ofthe process-model control systems with nonzero initial conditions areshown as follows.First. we suppose that the initial condition of the mod el is zero. but thatof the p rocess is not zero . In this case, the behavior of the system issimilar to the case w here the process is subject to disturbances. Therefore,to yield desired transient response to nonzero initial condition of theprocess after L , it is necessary and sufficient that poles of G , ( s ) -~ b f G ( r ) e - ' ~re canceled \pith its zeros.Second , we consider the case where the initial condition of the pro cessan d that of the model are not zero. In this case. the cancelation of poles ofG l ( s ) - M C ( s ) e - " - with its zeros does not imply that the system hasdesired response. In the process-model control system of Fig.8, theprocess and the model are connected in parallel to the control input u an dpoles of the model are equivalent to those of the process. Hence, bothresponse of the process and the model cannot be controlled simulta-neously by the control input u. That is to say. there exist dynamicalmodes which are not controllable by u. The primary concern is thebehavio r of the process. If the state of the delay-free part of the m odel isunobservable in yL , hen contains only the state of the process. The stateof the process is estimated from e and so the controller can yield thecontrol signal u . at least, to adjust the respo nse of the process desirably.However, if the state of the delay-free part of the model is not unobserva-ble in .vL. then the response of the model to its nonzero initial conditionaffects P.The state of th e process cann ot be estimated exactly from e sincethe process and the m odel are connected in parallel to 2 and poles of themodel are equivalent to those of the process. The controller cannotgenerate the control signal u to adjust the response of the process asdesired. Consequently, the following proposition is obtained.

Fig. 9. Responses of (a) theproposed system. (b) theSmithpredictor ontrol vstem(Cook's systemwhere .M= I ) . and (c) Donoghue's srstem (Cook's systemwhere- .M=.-OZ ) to the step disturbance.

? . O I0 2 0 $0 t

Fig. 10 Responses of (a) theproposed system. (b) theSmithpredictorcontrol ystem,and (c ) the approximate process-model control system for the astatic process with thestep disturbance.

Proposition V.4: When the reference input an d the dwu rba nc e are zeroin the process-m odel control sy stem of Fig . 8. the response of the processto the arbitraty initial condition of the system can be adjusted as desiredif and only if the state of the delay-free pa rt of the mo del is unob servablein ? v L .

Remark V.6: The proposed system satisfies this condition as men-tioned above an d so the response of the process can be adjusted desirably.Remark V.7: In the Smith predictor control system. the transfer func-tion of the model is given by C ( ~ ~ - A ) - ' B - C ( S I - . ~ ) - ' B ~ ~ ' ~ .oles ofthe model are not canceled with its zeros except for the process with n= 1and A =O . Hence. the Smith predictor control system cannot yield desiredresponse to a nonzero initial condition of the process even though theinitial condition of the model is zero.Remark V.8: In the systems presented by Fuller [4] and Donoghue

[ I I]. poles of the transfer function of th e model are canceled with itszeros. as seen in (40). but the state of the delay-free part of the model isnot unobservable inyL.Tlus is shown as follows. In the system of Fig s. 2an d 3, the process model consists of hvo dynamical portions and so theresponse to the initial condition is not canceled with its delayed version.This implies that the systems do not satisfy the unobservability of thestate of the de lay-free part of the mod el. Therefore. if the initial conditionof the m odel is zero, then the control systems can yield desired responsesto the initial co nd tio n of the process, but if the initial condition of themodel is not zero, then the control system cannot have desired responses.

Remark V.9: In the systems proposed by Kleinman [6]. Marshall et al.[IO]. and Cook et 01. [ 121. the response of the model to its initial conditionis canceled with the delayed version, as seen in Fig. 4. where M = e - " L .This implies that the state of the delay-free part of the mo del is unob-servable in y L after L . These systems satisfy the condition of the Prop osi-tion V.4.Remark V.10: If the integral term on the right of (7) is realized by anumerical computation, then poles of the closed loop system are onlyzeros of de t(sl -A -eP AL Bf) and the system can yield desired response


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IEEE TRANSACTIONS ON AUTOMATIC COhmOL, VOL. AC-26,NO. 6 , DECEhlBER 1981 1267

10k t

0 '. 5 100.0- _ _ - - - tb.---.s(t)*op>; l'o0.0 - - - - - - - - - - - - - _ _ _ _ _ _ _ _ _ _ _ _ _

t

1 O k

Fig. 1 I. Response of (a) the proposed system, (b) the Smith predictor control system, (c )Donoghue's system. (d) Cook's system where M = e - 0 . 2 (the systems of Kleinman andMarshall et d ) , nd (e) process-model version of Manitius el nl . to nonzero initialcondition of the process. The full limes show the case where the initial condition of themodel is zero an d the dotted lines show the m e vhere the initial condition of the modelis not zero.

to arbitrary initial conditions [9]. On the other hand, if the integral term isrealized by the process model, as shown in Fig. 5, then additional polesare introduced in the closed loop system as suggested in [9]. The statecorresponding to the additional poles, that is, the state of the model, isobservable in z as seen in Fig. 5. The condition of Proposition V.4 is notsatisfied. However, the transfer function of the model is given by ( s l -A ) - ' e - A L B - ( s l - A ) - ' B e - S L from (9) and the impulse response of themodel is eliminated after L. This implies that poles of the model arecanceled with its zeros. Therefore, the behavior of the system shown inFig. 5 is similar to that of the system by Fuller and Donoghue.To understand better he above differences, numerical examples arepresented.

Example V.3 (Comparison of responses to initial conditionsj:Consider the process given by (44).When the initial condition of themodel is zero, the responses to the nonzero initial condition of the processare calculated numerically, as shown with the full lines in Fig. 11. Whenthe initial condition of the mod el is not zero, the responses are shown withthe dotted lines in Fig. 11.VI. STABILIZATIONF U N S T ~ L EROCESS

As mentioned above, the process-model control system has uncontrolla-ble modes, since the process and the model are connected in parallel tothe control input u and poles of the model are equivalent to those of theprocess. The uncontrollable modes are governed by the same poles as theprocess. Therefore, if the process has poles in the right half plane. then theprocess-model control system is unstable. In this section, we ad 1 considera way to stabilize the process-model control system for unstable processesby using a certain approximate model. Moreover, the results derived herear e also concerned with the discussion about the behavior of the systemunder the mismatch between the process and the m odel.In the proposed system, the transfer function of the model is described

byGl(s)-MG(s)e-SL=Ce-"L(sl-A)-'B-lLCe-ATBdT

- C ( s l - A ) - ' B e - ' L . (46)

Then, the impulse response sf the model is given by( t = O )

~ - ' [ G , ( s ) - M C ( s ) e - S L ] = ' ( O < t < L )(LG t ) .

(47)Since the impulse response maintains zero after L, we can consider astable system described by C s e - A ~ L ( s I - A , ) - ' B , , he impulse response ofwhich is approximate to that of the unstable model in the interval (0,L).If such a system is obtained, then the following approximate model isconstructed:

where the state of the delay-free part of the model is intended to beunobservable in y L a t r > L. Hence, the impulse response of the model iseliminated after L an d is approximate to that of (47). To yield zerosteady-state error zero, a term corresponding to the third one in (30),which makes the state of the integratorunobservable iny L, must be addedto (48). The resulting approximate model is given by

Replacing (30) with (49) in the system of Fig. 7@), theapproximateprocess-model control system is obtained.The input-output transfer function of the appro xima te process-modelcontrol system is given byCr(s)=-y ( s ) - G , ( s ) G ( s ) e - S Lr ( s ) l +G , ( s ) (G , ( s )+~ ( s ) }

Since c ( s ) is the Laplace transformation of the difference in the impulseresponses of the models and zeros of 1 +G,(s)G,(s) that are in the lefthalf plane, the approximate process-model control system is stable if thedifference in the impulse responses is small enough to leave zeros of1 +G,(s)(G,(s)+e(s)) in the left half plane. The similar argument on thestability is seen in [9], [16]. an d [17].When the approximate process-model control system is stable, thesteady-state and transient behaviors are as follows. Since G , ( S ) - G ( S ) ~ - ' ~in (51) represents the transfer function of the proposed model and it has azero at s=O, we have

lim { G , ( ~ ) - G ( s ) ~ - ' ~ }O . ( 56 )Moreover, {GSl(s)-G,(s)e-'L) is the transfer function of the approxi-mate model which has the same structure as the proposed one. Hence,

s - 0

lim { G , , ( s ) - G , ( s ) e - S L } = O . ( 57 )s -0


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1268 IEEE TRANSACTIONS ON AUTOMATIC COhTROL, VOL. AC-26,NO. 6 , DECEBER 1981Substituting (56) and (57) into ( 5 l), we have

s -0l i m ( S ) = O .

Considering (58) and (33), the following equation holds:limsc,(s)-= l i m G , ( s ) = ls- 0 s s- 0 (59)

The output of the system can follow the step reference input ai th zerosteady-state error. If the magnitude of E ( S ) is small, then the transientresponse may be approximate to that of the system of Fig. 7(b), as seen bycomparing (50) with (32).On the otherhand, the behavior of the approximate process-modelcontrol system to disturbances is shown as follows. The disturbanceoutput transfer function of the app roximate system is given by

Substituting (57). 58). and (59) into (60). and considering that G X s )contains an integrator, we have

The stead y-state error to step disturbances is also zero. If the m agnitudeof E ( S ) is small. then the transient response to the disturbance is alsoapprox imate t o that of th e system of Fig . 7(b), since the impulse responseof G S l ( s ) - G , ( s ) e - Ls eliminated after L .

Example V I . l (Response of the approximate process - model control sys-tem):Consider the process given by (45), which is not asymptotically stable.Let the exact model be replaced with the approximate one of the form0.48-6.17s( S+O. l ) ( S+l ) +4 .18- 0 .9 e - 2 s( s + O . l ) ( s + l ) ( 6 2 )

in the proposed system, as shown in Fig. 7@). The resp onse of th eresulting approximate p rocess-model control system is simulated digitally,as shown with the dotted line in Fig. IO.Remark VI.1: An important result which emerges is that the unob-sew ability of the tates of the delay-free part of the model and theintegrator in y L also plays a significant role in order to yield zerosteady-state error an d desired transient responses even though mismatchexists between the process and the model. Moreover. it should be em-phasized tha t the state s of the delay-free pa rt of the model and heintegrator are made unobservable in yL in presence of mismatch. This isobvious from the following fact. The observability of these states in y L

depends only on the structure of the model. The structure of the ap-proximate model is intended to be identical to that of the exact one.Hence. the unobsewability is satisfied. This implies that the unobserv-ability can be satisfied without the exact knowledge of the process.Therefore, in practical applications with inevitable mismatch between theprocess and the model, one can make the states of the delay-free part ofthe model and the integrator unobservable in the output of the model. Ifthe mismatch is small. then the process-model control system has zerosteady-state error and an yield approxim ately desired transient re-sponses.121. CONCLUSIONS

1) The necessary and sufficient condition for yielding desired transientresponse to disturbances at r > L in the process-model control system isthat poles of the m odel are canceled with its zeros. The Sm ith predictorcontrol cann ot satisfy this condition.2) To make the steady-state rror to stepdisturbances zero, it isnecessary that the state of the integrator in the controller is unobservable

in the output of the model. If the process is asymptotically stable, then thecondition is also the sufficient one. The Smith predictor for astaticprocesses with integral properties and Donoghues system cannot satisfythis condition. Moreover, the systems proposed by Kleinman an d M arshallet al . cannot handle step disturbances.3) The necessary and sufficient condition for yielding desired transientresponses to arbitrary initial conditions is that the state of the delay-freepar t of the model is unobservable in the output of the model. The systemsproposed by Kleinman and Marshall er a/ . can satisfy this condition, bu tthose of Fuller and Donoghue cannot.4) The remarkable feature of the proposed system is that the states ofthe delay-free part of the m odel and the integrator are unobservable in theoutput of the model at t S L . All conditions pointed out above can besatisfied. Consequently, the proposed system can yield zero steady-stateerror and desired transient responses to step disturbances and arbitraryinitial conditions.

5 ) If th e process has poles in the right half plane, then the process-modelcontrol system is unstable. If the impulse response of the model iseliminated after L , like the proposed one. then the system can bestabilized by using a certain stable model. the imp ulse response of whichis approximate to that of the exact model.6) In practical applications, there exists inevitable mismatch betw eenthe process and the model. In spite of this, the unobservability of thestates of the delay-free part of the model and the integrator in the outputof the mo del can be satisfied because the observability of these statesdepends only on the structure of the model. Therefore, the unobserva-bhty can be satisfied without the exact knowledge of the process. If themisma tch is small enough to mainta in the stability of the closed loopsystem. then the system has zero steady-state error and can yield ap-proximately the desired transient responses.This paper restricts attention to single input-single output systems forthe sake of clarity of ex position. The results obtained h ere should beextended to process-model control for multivariable processes with m ulti-ple delays. A process-model control for m ultivariable discrete system isshown in [22].In this paper, the process-model control is developed within the frame-work of feedback. The improvement of dlsturbance attenuation usingfeedforward is referred to in [20].

REFERENCES0. J. M. Smith. A controller to overwme dead time. IS 4 J . . vol. 6, no. 2. pp.28-33. 1959.P. S. Buckley. Automatic control of processes u.ith dead time. Proc Frrsr Int.Congress IFA C. 1960. pp. 33-40.S. Saa-ano, Analog study of process-model control systems. J . Sor . Insrrm. Conrr .A T Fuller. Optimal nonlinear control of systems with pure delay. Inr. J . Conrr.,E ng . . Japu n . vol. I . no. 3. pp. 198-203. 1962.vol. 8. no. 2. pp 145-168. 1968Inr. J . Conrr. . vol. 18 .no . 6.pp. 1151-116R. 1973D. H. Mee. An extension of redictor control fo r systemsuith control tlme-delays.

nolse. I E E E Trum Auronmr. Comr. vo l AC-14. pp. 524-527. Oct. 1969.D. L Kleinman. Optimal control of linear systems w th time-delay and observation

state and control variables andquadratic cost. Auromortcu. vol. 8. no. 2. pp.H. h. Kolvo and E. B. Lee. Controller synthesis for linear systems ui th retarded203-208.1972.R. M. euis. Control-delayed system properties via anordinary model. Inr. J.Conrr.. vol. 30. no 3. pp. 477-490. 1979.A. Maninus and A W. lbrot. Finite spectrum assignment problem for systemsu,ith delays. I E E E T r a m Auronlur. Conrr . . rol. AC-24. pp 541-553, Aug. 1979J. E Marshall. B. Ireland. and B. Garland. Comments on an extension of predictorcontrol for systems with control time-delays. In r J . Conrr.. rol. 26, no. 6. pp.981-982.1977.J. F. Donoghue. A comparison of the Smth predictor and optlmal design ap-proaches fo r systems uirh delay in he control. I E E E T r a m Ind Elerrron. Conrr.lmrrurn , bel. IECI-24. pp. 109-117. Feb 1977.G Cook and X. rice. Comments on A comparison of the Smith predictor andoptimal deslgn approaches for systems with delay in the control. I E E E Trans. Ind.J . F. Doooghue. Fu rther mmments on A comparison of the Smith predictor andElecrron Conrr. Imrun. . vol. IECI-25, pp 180-181. May 1978optimal design approaches for sy ste m ui th delay in the control. I E E E Trans. Ind.S. Bhattachanya and J. B. Pearson. On the h e a r servomechanism problem, Inr. J .Elecrron. Conrr. Im rm . . vol. IECI-25, pp. 379-380. Nov. 1978Conrr..vol. 12. no. 5 . pp. 795-806. 1970.

J u p a n . %,ol 92s. no. IO. pp. 369-375. 1972S . Hosoe and M. to. On the error systems in linear sen0 problems. T r a m . I E EA. C Ioannides. G. I. Rogers. and V . Latham. Stabilln, limits of a Smith controllerIn simple systems containing a time delay. Inr . J . Conrr..vol. 29. no. 4. pp. 557-563,B. Garland and J E. Marshall. Sensitlrit) considerations of Smiths method for1979.time-delay s>-stem: Elerrron I x r r . . vol. IO . no. 15. pp 308-309.1974.


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0018-9286/81/1200-1269$00.75 Q1981 IEE E

[IS] B. Garland and J. E. Marshall, "A shor t bibliography on Smith's method," presentedat I E E Colloquium Time Delay Syst. Contr.-The Status of Smith's Method, London,

[I91 J. E Marshall. "Explorations of Smith's principle," IEE Colloquium TimeDelay1979.

[20] C. C. Prasad and P. R. Krishnaswamy. "Control of pure time delay processes,"Syst. Contr.-The Status of Smith's Method, London, 1979.

[21] C. B. G. Meyer, R. K . W o o d , and D. E Seborg, "Experimental exvaluation ofChem. Eng. Sa.. ol. 30, no. 2, pp. 207-215, 1975.analytical and Smith predictors for distillation column control." Amer. Insf.Chem.J. , voL 25, no. . pp. 24-32. 1979.[22] K Watanabe and M. Ito, "Process-model control for multivariable systems with timedelays in inputs and outputs." Trans. I E E Japan, vol. 99-c, no. 7, pp. 147-154, 1979.

Global Convergence for Adaptive One-Step-AheadOptimalControllers Basedon Input Matching

Abstraet-This paper establishes global convergence for adaptive one-stega head optimal controllers applied to a class of linear discrete timesingle-input single-output systems. The class of systems includes all stablesystem s whether heyareminimumphase or not, all minimumphasesyst em s whether they are stable or not, and some unstable nonminimmphase systems. The key substantive assumption is that the one-step-aheadoptimal controller designed using the true system parameters eads to astable closed-loop system. Subject to this natural restriction, t is shownthat a simple adaptive control algorithm based n input matching is globallyconl'ergent in the sen se that the system inputsand outputs remain boundedfor all lime and the input converges to the one-step ahead optimal input.Both deterministic and stochastic cases are treated.

I. INTRODUCTIONRecently, global convergence has been established [1]-[7 or a c lass ofadaptive control algorithms applied to single-input single-output andmultiinput m ultioutput discrete time systems. A key assumption in theseresults is that the system has a stable inverse. In the single-input single-outp ut case, when the transfer function is q-dB(q- l ) /A(q- ' ) , the stablyinvertible assumption is equivalent to the assumption that B ( q - I ) is astable polynomial. To date, n o global convergence results are available foradaptive control algorithms applied to systems that do not satisfy thisassumption.One technique that has been suggested [8]-[IO] for extending t he classof systems to which adaptive control applies is to consider inputs that

mi n i m i z e single stage optimal cost functions of the formJ = ~ ( y ( r ) - y * ( t ) )2 x, u ( r - d )2 (1.1)

where { ~ ( t ) ) , y ( t ) ) , { y * ( t ) } denote he system input,output anddesired output, respectively.An advantage of the use of this cost functionis that large input signals are penalized. This reduces the sizes of the inpu tvariations even in the minimum phase case [9], [IO]. Apart from any otherconsideration, the usual minimum variance adaptive controllers [3], [6],[ 1 I] , correspond to a special case of (1 ) in which A is a priori fixed a t zero.For simplicity, we will first treat the case when d = 1. We will note thecorresponding results for d > 1. For d = 1, the system model isq-'B(q-')/A(q-'), where B(q-')=bo+bl,+ . . . +b,q-"; b,PO an dA ( ~ - ' ) = I + n , q - l + . . . +a,q-". Therefore, the optimal single stagecontrol, u * ( f - l), is readily seen to satisfy [12]G. Kreisselmeier. Chairman of the Adaptive, Learning Systems. Pattern RecognitionManuscript received May 14. 1980; revised Fe bma y 16. 1981. Paper recommended byCommittee. This work was supported by the Australian Research Grant s Committee andthe U.S. National Science Foundation under Grant ECS-79-18577.Engineering. University of Newcastle. New South Wales 2308. AustraliaG. C. Goodwin and K. S . Sin are with the Department of Electrical and Computer

Institute and State University, Blacksburg.VA 24061.C. R.Johnson. r.. is with the Department of Electrical Engineering. Virginia Polytechnic

b o [ ( l - A ( q - 1 ) ) y ( t ) + B ( q - ' ) ~ * ( ~ - l ) - ~ * ( t ) ] + h u * ( t - 1 ) = 0 .( 14

This leads to a closed-loop system described by( b o B ( 4 - ' ) + x A ( 4 - ' ) ) r c t ) = b , B ( q - ' ) y * ( t ) (1.3)

an d(b,B(q-')+hA(q-l))u(t)=boA(q-l)y*(t+l). (1.4)

Stability of the closed-loop system therefore depends up on the stability of[bo B(q - ')+ hA (q- ') ] . We shall be interested in adaptive control whenthe system parameters are initially unknown. However, to make thissensible, we shall introduce the natural requirement that minimization of(1.1) would lead to a stable closed -loop system if the parameters wereknown. A simple root locus argument [9], [12], using the polynomial[b,B(q- ')fhA(q- ')] shows that a value of h can always be foundwhich guarantees closed-loop stability provided the system is either o pen-loop table or minimum phase. Certain unstable-nonminimum phasesystems are also included 191, [12], [13].We will first treat the case of deterministic systems. Subsequently, wewill extend the results to stochastic system s where we will use the costfunction

J ' = T E { ( y ( t ) - y * ( f ) ) Z I F - l } + 2 1 1 ( f - 1 ) 2x (1.5)where denotes the sigma algebra generated by data up to andincluding time t- 1. Throughout we will assume that t is single-inputsingle-output system. However, the extension to multiinput multioutputsystems is possible along the line of the stably invertible case [7]. We w d lalso restrict our attentio n to algorithms having scalar gain sequences, e.g.,projection algorithms for deterministic systems and stochastic approxima-tion for stochastic systems. The extension to least squa r e s based algo-rithms is immed iate along the lines of the corresponding results for stablyinvertible systems [3], [ 141.

The convergence theory presented here generalizes and makes rigorousthe previous analysis presented by Johnson and Tse in [ I O ] for the noisefree case. Convergence was not fully established in [ IO ] because it wasimplicitly assumed that the algorithm lead to a stable closed-loop system(see [lo, Theorem A.21). Here we establish this crucial property of thealgorithm using the key technical device of [3], [6].11. INPUT MATCHINGLGORITHM FOR DETERMINISTTCYSTEMS

Here we consider the class of algorithms where the leading coefficientin the B( q-I) polynomial is effectively held a t an a priori estimate. Thisclass of algorithms has been termed input matching for reasons that willbecome ap parent presently. The stably invertible case has been shown tobe globally convergent in [3] (as rojection algorithm 11).The system model that we are concerned with here isA ( q - ' ) y ( t ) = B ( q - l ) u ( r - l ) (2 . 1 )

whereA(q- ' )=I+a , q - '+ " . t a " q - " (2.2)B ( q - l ) = b o + b I q - l + . . . +b,,q-"', bo#O. (2.3)

The one-step-ahead optimal nput u * ( t - l), minimizing (1.1) given thepast possibly suboptimal input and ou tput record then satisfiesb,[(l-A(q-~))y(r)+bou*(r-l)tq(B(q-')-bo)u(t-2)-y*(r)]

+Xu*( t - l ) =O.2 .4)Factoring ( b i i - A) from the left of (2.4) gives

paper a process-model control for linear systems with

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