Adaptive Control 1

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Adaptive control of dominant time delay systemsvia polynomial identificationA. Besharati RadK.M. TsangW.L. Lo

Indexing terms : Dominant lime de lay, Adaptice control. Predictice PI c ontrol, Polynomial idenrifrcation

Abstract: An adaptive algorithm for controllingsystems with dominant time delay is presented.The algorithm is derived by integration of H ag -glund 's predictive P I (P IP) controller with theonline polynomial identification algorithm. Forthe purpose of identification and control, thesystem under control is modelled by a first-orderwith delay model an d its parameters, including thetime delay, are identified. The identified param-eters are used to tune the predictive PI controller.The performance of this algorithm is verified byan experimental study for a system with variabletime delay. It is also shown that this controllerperforms better than the popular Foxboro 'sEXACT controller for a dominant time delayprocess.

1 IntroductionControl of systems with large dead time imposes muchmore restriction on design methodology and conven-tional control algorithms m ay not be directly applicable.For example, i t is well kno wn tha t Zeigler-Nichols tunedPID controllers cannot give an acceptable performancefor this class of processes. To design an adaptive control-ler for a dominant time delay system, the parameters ofthe system, including the time delay, should be estimatedonline, this is if the delay is not known a priori. There aremany adaptive control algorithms derived specifically fortime delay systems [ I , 21 ; many of them, however, do notaddress the dominant time delay systems explicitly. Inthis paper, a new adaptive control algorithm is intro-duced which is designed particularly for dominant timedelay systems. The design is based on online tuningof th epredictive PI (P IP ) controller which was recently intro -duced by Hagglund [3,4]. T he purposeof the paper is toextend Hagglund 's PIP to adaptive capability . Thisadaptive controller can control high-order systems withdominant time delay. For the purpose of identificationand control, the system is modelled by a first-order tran s-fer function with time delay. The parameters of thismodel, including the time delay, are estimated via onlinepolynomial identification [ 51 . The controller is imple-mented in continuous-time [ 6 ] in contrast to conven-tional discrete-time approach. The proposed algorithmwill also take into account the case of variable time delay.r IEE . 1995Paper 1973D IC8).received 22nd February 1995The auth ors are with The Hong K ong Polytechnic University, Depart-ment of Electr ical Engneering . Hung Horn, Kowloon, Hon g KongI E E Pro . -C o nrro l Theo ry A p p l. , V o l . 142, N o . 5, September 1995

2 Controller designHagglund [3, 41 introduced a predictive PI controller(PI P) that is specially suited to systems with dominanttime delay. The controller is effectively a Smith predictor[7] whereby the structure of the model is fixed to a first-order with time delay. Note that a first-order model withdelay is sensible for dominant time delay systems. Hag-glund determined the parameters of the model (offline)from the process reaction curve and used them to calcu-late the controller parameters.

The high-order dominant delay process is approx-imated by

where m < n, L is the actual time delay, K is the staticgain, T is the apparent delay and T is the effective timeconsta nt. The system is labelled as a dom inant time delayprocess if T > 5s. Hagglund 's PIP has the followingst ructu re:

whereU([)= control signale ( t )= error signalK p = proportional gainTi = integral t ime constantT = model time delayp = differential operator d / d t

If the parameters of the model are known, the controllercan easily be tuned. The proportional gain and integraltime constant of the controller are selected based onmodel static gain an d effective time constant as1

K, = - an d 7 ; = r (3 )In process control, the param eters of the system and thusthe model change a nd therefore the controller needs con-stant retuning. Hagglund 's PIP is designed fo r a knownsystem a nd the first-order model is derived offline and a ssuch cannot cope with system parameter changes. More-over, this controller is essentially a Smith predictor a nd itis well known that a Smith predictor is sensitive to mis-match between the process and the model. This problemcan be solved if we somehow estimate online the modelparameters namely K , s a n d T f rom the inpu t u(t ) a n dthe ou tpu t y(f) of the original high-order system, we willbe able to update and hence retune the controller fromeqns. 2 and 3 as the system change.

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3 Adaptive PIP controllerTo design an adaptive controller around the predictivePI controller, the parameters of the system including theunknown time delay should be identified recursively. Inthis Section, we first present the identification algorithmand then outline the design procedure for the adaptivePIP controller.3.1 Parameter estimationThe, possibly, high-order do min ant delay process is mod -elled by eqn. 1. However, the adaptive controller requiressimultaneous estimation of time delay ( T ) , gain ( K ) andtime constant (5). Standard least squares cannot bedirectly applied since the presence of unknown delayimposes a nonlinear in the parameter configuration. Letus define the parameter vector O ( t ) = [ T , aO, al]T wherea, = l / K a n d a , = T/K . Now , we can use Polynomialidentification method [ S I which extends the continuous-least squares criterion to polynomial in the parametersystem; i.e. the parameters o f a general nth order systemwith unknow n time delay can be identified. This isachieved by minimising the following cost function:

~ [ e ( t ) ,] = + e - @ ' [ d ( t ) d o l T ~ o [ d ( t ) o ]+ l e - 8 ( r - r ) t 2 ( t ,) dr (4)

where e ( t ) is the parameter vector, and e(t , T) is theestimation erro r defined asqt, T) = d Y ( T ) , U(T), &t) l ( 5 )

In eqn. 4, P is the nonnegative scalar forgetting factor:an d So is the positive definite matrix initial cost weight-ing.Theorem: If the optimal estimate of d of 0 exists and isunique for every t , then it satisfies the following system ofdifferential equation s:

and for i > 1, J i [e ( t ) , ] is given by differential equation

where

? J [ d ( t ) , t ]iiei [ d ( t ) , ] =

p ( t ) = J , [ d ( t ) , t ] - ( 9 440) eo J,(o) = so (10)

The only nonzero initial conditions are

Proof: The proof is obtained by repeated differentiationof the cost function eqn. 4.The general algori thm con-siders a transfer function with both numerator anddenominator as well as time delay which makes it not434

feasible for single processor implementation. However,since eqn. 1 can be represented as a system with nonumerator dynamics, the identification is very muchsimplified and is implemented by first partitioning theparameter vector O ( t ) as0 = [ z ] where A = [ao, U , ] ~ (11)

Let us define hi(t) , i ( t) n d Y ( t ) sl%(t) d'u(t - T )q i ( t ) = 7 7 = ( - 1 y + 10' ~ dt' fo r i 2 1

an dY( t )=- ["- ( t ) T

?AIn the above equations, u ( t ) is the control signal, i is thetruncation order an d is selected by the user and y ( t ) s thesystem output. Similarly, H , ( t ) s partitioned as

Differentiating once again with respect to B(t) gives

For i 2 2, the only component of Hi([) hat is not zero isthat corresponding to T and is given by

Thus, the differential equation for updating the cost (eqn.8) becomes

3.2 Design of the adaptive controllerA step by step procedure for the design of the adaptivePIP controller is as follows:Step I : The high-order system is modelled by a first-order with time delay (eqn. I), the initial parametervector [6(0)], nitial cost [ J 2 ( 0 ) ] , he forgetting factor (P ) ,the truncation order of estimation ( i ) are chosen. Basedon the a priori knowledge of the system bandwidth, thestate variable filter 161 is selected.Step 2: The parameters of this model are identifiedfrom the input and output of the original high-ordersystem (eqns. 6 and 12-17).Step 3 : The parameters of th e PIP are updated (eqn. 2and 3).Step 4 : Go back to step 2.4 Experimental studiesIn this Section we demonstrate the performance of theadaptive algorithm proposed in this paper on two realsystems. The first system is a flow control problem andthe second one is a temperature control problem. Sincethe algorithm reported in this paper is applicable tosystems with dominant time delay, we added artificial

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time delay to outp uts of both systems. For flow control, afixed delay and for temperature control, a variable delaywas added to their respective outputs. Furthermore, toassess the performance of the controller we have com-pared it with a popular commercial PID self-tuner, i.e.the Foxboro EXACT.4.1 Flow controlFlow control is a typical process control situation infood, drink and petrochemical industries. A laboratory-scale process control unit (PCU) from Bytronic [SI wa sused to dem onstrate the performance of the algorithm ona real system. The system rig consisted of a sump, apump, a manual/computer control diverter valve, acooler and a drain valve. The sump water was pumpedthrough the pipeline, the cooler and the manual flowcontrol valve to the process tank. The flow rate wasmeasured by an impeller type flow meter located near theprocess tank. The water was fed back to the sump via thedrain valve, thus completing the flow cycle. The objectivewas to control the f low rate by manipulat ing the pumpvoltage. Fig. 1 shows the PCU layout for the flow control

strrrerprocess

I

s u m p pumpF lg 1menf BItronir process control lajour for Jon rate control erperr-

experiment. From omine modelling, the model of the flowrate control system was found to be

where U s ) and T/,(s) were the Laplace transform of theflow rate and pump drive voltage, respectively. This wasclearly not a dominant time delay system. To show theperformance of the proposed algorithm for a dominanttime delay system, extra time delay of 7.8 s was added tothe system output signal (flow rate) to make the overallsystem time delay to 8 s which was about six times thesystem time constant. Therefore, the overall system trans-fer function was

The flow rate was measured every 0.05 s and was delayedfor 7.8 s before being used to calculate the updatedcontro l signal (i.e., the p um p drive voltage). Thre e param -eters were estimated. The estimator initial parametersand the initial cost, the forgetting factor and the estima-tion truncation order were selected as O(0)= [6 , I , l],12(0) 0.011, where Z3 is a 3 x 3 identity matrix, for-getting factor b = 0.05 and the identification truncationorder i = 4, respectively. The design was implemented incontinuous-time and the state variable filter of the formI E E Proc.-Conrrol Theo ry Appl. , Vo l . / 4 2 , N o . 5 , September 1995

1/(0.5s+ 1 )4 was applied to the system input and outputfor generating filtered high-order derivatives of input andoutput which are needed for parameter estimation (eqns.12 a n d 13). The set-point was a square-wave with anampli tude of 0.15 and a period of 80 s which was super-imposed on a DC offset level of 0.75. Fig. 2 shows the

O 00 50 100 150 2 00t1me.s

Li

010 50 100 150 200t1me.sb

-0 50 100 150 200t1me.sCFig. 2

(I Set point and output flow rale IF ) against timeh Pump drive voltage (V I against timec Estimated parameters ( T d .K . r ) against time

Adaptiae.Pow control oJByfron ic process control unit

overall experimental results. In particular, Fig. 2a showsthe set-point, output pair, Fig. 26 shows the controlsignal (pump driving voltage) and the estimated param-eters are shown in Fig. 2c. As shown in this figure, theadaptive PIP controller shows a very satisfactory per-formance.Next, to compare the proposed controller with anindustrial controller, an experiment for Foxboro EXACTself-tuning controller was also carried out. The FoxboroEXACT self-tuning controller uses a pattern recognitiontechnique to tune the controller. By continously monitor-ing the behaviour pattern of the error signal, the control-ler settings will be adjusted in order to match theuser-specified dam ping an d overshoot co nstrain ts when aseries of set point changes or load disturbance changes isapplied to the process.

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The initial estimates of the controller parameters weredetermined using the prerune function. The MAIN N B(no ise band) MAIN cornax(maximum wait t ime) andMAIN DFCT (derivative factor) were determined auto-matically through this pretune function. The MAIND M P (damping facto r) and the MAIN OVR (overshoo t)were both selected at the default value of 0.25. Theprocess was then subjected to a series of set-pointchanges (three or more set point changes), hence theEXACT self-tuning algorithm obtained the opimal con-troller parameters. This industrial PID controller imple-ments a series PID control algorithm which has thefollowing form :

where U ( s ) and E ( s ) are the Laplace transform of th econtrol and the error signals, respectively. After a seriesof set-point changes, the EXACT tuning algorithm con-verged a nd the final settings were obtaine d. The EXACTcontroller performance after tuning period (abo ut 300 s)is shown in Fig. 3. In particular, Fig. 3a shows the set: 4 ,' 2.

U- '3 4 !2 2ji

2 r~. ? L

3 3;

1 2-&' I I-

-0 B O 123 :60. rne.55Fig. 3o O ~ i p ~ ind sei poini flou raie iFI against timeh P u m p dri\e b o l t a g e I I I against time

Conrrol 0: B r r r o n r c process control unir h i Fo r h o r o E X . 4 C T

point and the output and Fig. 3b shows the controlact ion . The PID parameters ob tained f rom FoxboroEXACT controller were K , = 0.3125 (proportionalband = 3 2 0 % ) , 7; = 2.4 and & = 0 . 6 (M A IN P = 320,M A I N I = 0 . 4 m i n a n d M A I N D =0 .01 min). Theclosed-loop response after 300 s for the proposed control-ler PIP and the EXACT controller are shown in Fig. 4. I tis apparent that the proposed controller performs betterthan the EXACT controller for this dominant time delayprocess.4.2 TemperarurecontrolA popular piece of control engineering laboratoryequipment, the feedback process trainer PT326 [SI, wasused to dem onstrate the performance of the adaptiv e PIP47 6

controller on a variable time delay system. The systemconsisted of an adjustable air blower, a length of tubeand a heater . The ai r was d rawn th rough the th ro t t le

:;i , , , , , , , ,0 40 80 12 0 160

time,s0

a

'> O 8 84 -

0 40 80 120 160time. sb

0 20

Fig. 4o Set p i n 1 and oulput Row rate(F ) gainst timeh Pump drive voltage ( V )against time

Compnrison of the proposed algorithm with E X A C T

PIPIFoxboro

~~

opening by the centrifugal blower. It was then warmedup by the heater as it passed throu gh the tube length an dsent back to the atmosphere. The system input was theheater drive voltage and with its output the air tem-perature which was measured by a thermistor fitted tothe end of the tube. The schematic diagram of the heateris shown in Fig. 5.

adjustablea ir f low

controllermeasurementI t1,

txternalac tuot on slgna IFig. 5 Block dingram fo r the feedhnck process trainer (P T 326)

The adjustable air blower was set at 40" an d w asmaintained at this angle for the durat ion of the experi-ment. An open loop test was then carried out on thesystem and from the process reaction curve, a first-orderI E E Pro?.-Control Theorv A p p l , V o l . 142,N o 5 , September IYYS


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model was deduced as

Since the time delay of the system was fixed and verysmall as compared to its t ime constant, a variable delaywas added to the measured system output. Therefore, thesystem transfer function wasV ( s ) l . l e -L sV,(s ) 0.5s + 1G,(s) =-- (22)L = 2.0L = 2.5L = 3.0 fo r 0 < t < 60fo r 0 < t < 12 0fo r 0 < t < 180A state variable filter of the form 1/(0.6s+ wasapplied to the system input and output for polynomialidentification. Th e estima tor initial parameters, the initialcost, the forgetting factor and the estimator truncationorder were selected as O(0)= Cl.0, 0.5, O.5lr , J z ( 0 )=1 x 10-513 where 1, is a 3 x 3 identity matrix, = 0.05a n d i = 4, espectively. The set-point was a square-wavewith amp litude of 1.0 and per iod o f 40 s superimposed onDC offset level 4.0. Fig. 6a-c show s the overall results. In

>L 10," 8-

x olL.._---,* o 30 60 90 120 150 180t 1 m e . sa

o h - ,30 60 90 120 150 180h m e , sb

30 60 90 120 150 180t ime.5

C

Fig. 6n Set po in t and output aga ins t t imeh Contro l ac t ion agamr t t imec Estimated param ete r s agamr t t ime

Adaprire ronrrol uJPT326 tiirh iuriahle de lay

particular, Fig. 6u shows the setpoint, output pair, Fig. 6hshows the control signal and Fig. 6c shows the estimatedparameters. As shown in this figure, the adaptive PIPcontroller shows excellent performance under variabletime delay condition. The polynomial identification canI E E P r oc . - Con tr o l Theory .4ppl . . V o l . 142. N o. 5 , September 1995

track the variable time delay and P IP can give a verygood contro l for this system. The model of the heater wasalso used to compare simulation results with the actualexperimental results. Fig. 7a-c shows the simulationresults an d co mpa re very well with those of Fig. 6a-c.>

t imesa

> l o rP 81c' '

0 30 60 90 120 150 180t ime. sb

I I-0 60 90 120 150 180o!0t ime, sCFig. 7dela!

U Set po in t and ou tput aga ins t t imeh Contro l action against timec Estimated paramete r s aga ins t t ime

Adaprirr control of sirnuluted model of PT326 with variable

5 ConclusionsA new adaptiv e controller is proposed for the specialclass of systems with dominant time delay. This class ofsystems are known to be difficult to contro l and there a refew algorithms which can successfully control this type ofprocesses. However, for the purpose of identification andcontrol, dominant time delay systems can be representedby a first-order with time delay model. The param eters ofthis model are recursively identified by the polynomialidentification an d are used to tune Hagglund 's PI P con-troller. Experimental studies were carried out to show th eperformance of algorithm under noisy environment,parameter var ia t ions and inpu t d is tu rbances . The con-troller performance deteriorates as the ratio of the timedelay to the effective time constant of the systemdecreases. Therefore, i t can be concluded that for thedominant time delay systems, this approach can be veryuseful and lead to a much improved performance com-pared to the Zeigler and Nichols tuned PID controllers.One of the main attractive properties of the proposedcontroller is its ability to track a variable time delay.These claims have been verified by simulation as well as

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experimental studies on two laboratory-scale realsystems.6 ReferencesI CLARKE, D.W.. MOHTADI, C. , and TUFFS, P.S.: 'Generalisedpredictive control. Pa rt I : The basic aleori thm'. Auromarica. 1987. 23.( 2 ) pp 137-1482 DUM ONT. G A ELNAGG AR, A and ELSHAFEI, A 'Adap t ivepredictive control of systems with varying time delay', Int . J. Adapt ioeConrrol & Signal Processing, 1993, 7 , pp. 91-1013 H A G G L U N D , T.: 'A de ad time com pensatin g three term controller'.IFAC Identificat ion and system parameter est imation, Pergamonpress plc, Budapest. Hungary, 1991,pp . 1167-117?

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4 H A G G L U N D , T. : 'A predictive PI control ler for processes with longdead times'. IEEE Control Systems, 1992, 12, ( I ) , pp. 57-605 GAWTHROP, P.J ., NIHTILA, M.T., and BESHARATI RAD, A.:'Recursive parameter estimation of continuous systems withunknown time delay', C o nt ro l , Theo ry & A d u. Tec hno / . , 1989, 5, (3)pp. 227-2486 G A W T H R O P , P.J.: 'Continuous self-tuning control'. Vol. I , 'Design'(Research Studies Press, UK, 1987)7 SM ITH , O.J .M.: 'Closer control of loops with dead time', C hem.Engng, 1957,53, 5 ) . pp. 217-2198 B Y T R O N I C ( U K) : Documentat ion for the Bytronic process controlunit' (Bytronic, UK , 1994)9 FEEDBA CK INSTRUM ENTS LTD.: 'Manual fo r PT326 '(Feedback Inst. Ltd.. U K , 1982)

IE E Pro c . -C o nlro l The o ry A p p l . , Vo l . 142, N o . 5 , September I995

Adaptive Control 1

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