Adaptive control of dominant time delay systems via polynomial identification

A. Besharati Rad K.M. Tsang W.L. Lo

Indexing terms: Dominant lime delay, Adaptice control. Predictice PI control, Polynomial idenrifrcation

Abstract: An adaptive algorithm for controlling systems with dominant time delay is presented. The algorithm is derived by integration of Hag- glund's predictive PI (PIP) controller with the online polynomial identification algorithm. For the purpose of identification and control, the system under control is modelled by a first-order with delay model and its parameters, including the time delay, are identified. The identified param- eters are used to tune the predictive PI controller. The performance of this algorithm is verified by an experimental study for a system with variable time delay. It is also shown that this controller performs better than the popular Foxboro's EXACT controller for a dominant time delay process.

1 Introduction

Control of systems with large dead time imposes much more restriction on design methodology and conven- tional control algorithms may not be directly applicable. For example, i t is well known that Zeigler-Nichols tuned PID controllers cannot give an acceptable performance for this class of processes. To design an adaptive control- ler for a dominant time delay system, the parameters of the system, including the time delay, should be estimated online, this is if the delay is not known a priori. There are many adaptive control algorithms derived specifically for time delay systems [ I , 21; many of them, however, do not address the dominant time delay systems explicitly. In this paper, a new adaptive control algorithm is intro- duced which is designed particularly for dominant time delay systems. The design is based on online tuning of the predictive PI (PIP) controller which was recently intro- duced by Hagglund [3,4]. The purpose of the paper is to extend Hagglund's PIP to adaptive capability. This adaptive controller can control high-order systems with dominant time delay. For the purpose of identification and control, the system is modelled by a first-order trans- fer function with time delay. The parameters of this model, including the time delay, are estimated via online polynomial identification [ 151. The controller is imple- mented in continuous-time [ 6 ] in contrast to conven- tional discrete-time approach. The proposed algorithm will also take into account the case of variable time delay.

r IEE. 1995 Paper 1973D IC8). received 22nd February 1995 The authors are with The Hong Kong Polytechnic University, Depart- ment of Electrical Engneering. Hung Horn, Kowloon, Hong Kong

I E E Pro.-Conrrol Theory Appl., Vol . 142, No. 5, September 1995

2 Controller design

Hagglund [3, 41 introduced a predictive PI controller (PIP) that is specially suited to systems with dominant time 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 with delay 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 static gain, T is the apparent delay and T is the effective time constant. The system is labelled as a dominant time delay process if T > 5s. Hagglund's PIP has the following structure:

where U([) = control signal e( t ) = error signal K p = proportional gain Ti = integral time constant T = model time delay p = differential operator d /d t

If the parameters of the model are known, the controller can easily be tuned. The proportional gain and integral time constant of the controller are selected based on model static gain and effective time constant as

1 K K , = - and 7 ; = r (3)

In process control, the parameters of the system and thus the model change and therefore the controller needs con- stant retuning. Hagglund's P I P is designed for a known system and the first-order model is derived offline and as such cannot cope with system parameter changes. More- over, this controller is essentially a Smith predictor and it is well known that a Smith predictor is sensitive to mis- match between the process and the model. This problem can be solved if we somehow estimate online the model parameters namely K, s and T from the input u(t) and the output y(f) of the original high-order system, we will be able to update and hence retune the controller from eqns. 2 and 3 as the system change.

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3 Adaptive PIP controller

To design an adaptive controller around the predictive PI controller, the parameters of the system including the unknown time delay should be identified recursively. In this Section, we first present the identification algorithm and then outline the design procedure for the adaptive PIP controller.

3.1 Parameter estimation The, possibly, high-order dominant delay process is mod- elled by eqn. 1. However, the adaptive controller requires simultaneous estimation of time delay ( T ) , gain ( K ) and time constant (5). Standard least squares cannot be directly applied since the presence of unknown delay imposes a nonlinear in the parameter configuration. Let us define the parameter vector O ( t ) = [ T , aO, a l ]T where a,, = l/K and a, = T/K. Now, we can use Polynomial identification method [ S I which extends the continuous- least squares criterion to polynomial in the parameter system; i.e. the parameters of a general nth order system with unknown time delay can be identified. This is achieved by minimising the following cost function:

~ [ e ( t ) , t ] = +e-@'[d( t ) - d o l T ~ o [ d ( t ) - do]

+ l e - 8 ( r - r ) t 2 ( t , T) dr (4)

where e(t) is the parameter vector, and e(t, T) is the estimation error defined as

qt, T) = d Y ( T ) , U(T), &t)l ( 5 )

In eqn. 4, P is the nonnegative scalar forgetting factor: and So is the positive definite matrix initial cost weight- ing.

Theorem: If the optimal estimate of d of 0 exists and is unique for every t , then it satisfies the following system of differential equations:

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

where

?J[d ( t ) , t ] iie Ji[d(t), t ] =

p ( t ) = J , [ d ( t ) , t ] - (94

40) = eo J,(o) = so (10)

The only nonzero initial conditions are

Proof: The proof is obtained by repeated differentiation of the cost function eqn. 4. The general algorithm con- siders a transfer function with both numerator and denominator as well as time delay which makes it not

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feasible for single processor implementation. However, since eqn. 1 can be represented as a system with no numerator dynamics, the identification is very much simplified and is implemented by first partitioning the parameter vector O ( t ) as

0 = [ z ] where A = [ao, U , ] ~ (11)

Let us define hi(t), gi(t) and Y ( t ) as

l%(t) d'u(t - T ) q i ( t ) = 7 7 = ( - 1 y + 1 80' ~ dt' for i 2 1

and

Y(t) = - = [":':I, - y( t ) ] T

?A

In the above equations, u(t) is the control signal, i is the truncation order and is selected by the user and y(t) is the system output. Similarly, H,( t ) is partitioned as

Differentiating once again with respect to B( t ) gives

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

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

3.2 Design of the adaptive controller A step by step procedure for the design of the adaptive PIP controller is as follows:

Step I : The high-order system is modelled by a first- order with time delay (eqn. I), the initial parameter vector [6(0)], initial cost [J2(0)], the forgetting factor (P), the truncation order of estimation ( i ) are chosen. Based on the a priori knowledge of the system bandwidth, the state variable filter 161 is selected.

Step 2: The parameters of this model are identified from the input and output of the original high-order system (eqns. 6 and 12-17).

Step 3 : The parameters of the PIP are updated (eqn. 2 and 3).

Step 4 : Go back to step 2.

4 Experimental studies

In this Section we demonstrate the performance of the adaptive algorithm proposed in this paper on two real systems. The first system is a flow control problem and the second one is a temperature control problem. Since the algorithm reported in this paper is applicable to systems with dominant time delay, we added artificial

I E E Proc.-Control Theory Appl., Vol . 142, N o . 5, September 1995

time delay to outputs of both systems. For flow control, a fixed delay and for temperature control, a variable delay was added to their respective outputs. Furthermore, to assess 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 control Flow control is a typical process control situation in food, drink and petrochemical industries. A laboratory- scale process control unit (PCU) from Bytronic [SI was used to demonstrate the performance of the algorithm on a real system. The system rig consisted of a sump, a pump, a manual/computer control diverter valve, a cooler and a drain valve. The sump water was pumped through the pipeline, the cooler and the manual flow control valve to the process tank. The flow rate was measured by an impeller type flow meter located near the process tank. The water was fed back to the sump via the drain valve, thus completing the flow cycle. The objective was to control the flow rate by manipulating the pump voltage. Fig. 1 shows the PCU layout for the flow control

strrrer

process

I

s u m p pump

Flg 1 menf

BItronir process control lajour for Jon rate control erperr-

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

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

The flow rate was measured every 0.05 s and was delayed for 7.8 s before being used to calculate the updated control signal (i.e., the pump drive voltage). Three param- eters were estimated. The estimator initial parameters and 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 truncation order i = 4, respectively. The design was implemented in continuous-time and the state variable filter of the form

I E E Proc.-Conrrol Theory Appl., Vol. / 4 2 , No. 5, September 1995

1/(0.5s + 1)4 was applied to the system input and output for generating filtered high-order derivatives of input and output which are needed for parameter estimation (eqns. 12 and 13). The set-point was a square-wave with an amplitude 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 0 z\ 0 50 100 150 2 00

t1me.s Li

01 0 50 100 150 200

t1me.s b

-

0 50 100 150 200 t1me.s

C Fig. 2 (I Set point and output flow rale IF) against time h Pump drive voltage (VI against time c Estimated parameters ( T d . K . r ) against time

Adaptiae.Pow control oJByfronic process control unit

overall experimental results. In particular, Fig. 2a shows the set-point, output pair, Fig. 26 shows the control signal (pump driving voltage) and the estimated param- eters are shown in Fig. 2c. As shown in this figure, the adaptive PIP controller shows a very satisfactory per- formance.

Next, to compare the proposed controller with an industrial controller, an experiment for Foxboro EXACT self-tuning controller was also carried out. The Foxboro EXACT self-tuning controller uses a pattern recognition technique 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 the user-specified damping and overshoot constraints when a series of set point changes or load disturbance changes is applied to the process.

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The initial estimates of the controller parameters were determined using the prerune function. The MAIN NB (noise band) MAIN cornax (maximum wait time) and MAIN DFCT (derivative factor) were determined auto- matically through this pretune function. The MAIN DMP (damping factor) and the MAIN OVR (overshoot) were both selected at the default value of 0.25. The process was then subjected to a series of set-point changes (three or more set point changes), hence the EXACT self-tuning algorithm obtained the opimal con- troller parameters. This industrial PID controller imple- ments a series PID control algorithm which has the following form:

where U ( s ) and E(s ) are the Laplace transform of the control and the error signals, respectively. After a series of set-point changes, the EXACT tuning algorithm con- verged and the final settings were obtained. The EXACT controller performance after tuning period (about 300 s) is shown in Fig. 3. In particular, Fig. 3a shows the set

: 4 ,

' 2.

U- ' 3 4 !

2 2j i

2 r ~ . ? L

3 3;

1 2-

&' , I I-

-0 B O 123 :60 . rne.5 5

Fig. 3 o O ~ i p ~ i and sei poini flou raie iFI against time h Pump dri\e boltage I I I against time

Conrrol 0: Brr ronrc process control unir h i F o r h o r o EX.4CT

point and the output and Fig. 3b shows the control action. The PID parameters obtained from Foxboro EXACT controller were K , = 0.3125 (proportional band = 320%), 7; = 2.4 and & = 0.6 (MAIN P = 320, MAIN I =0 .4min and MAIN D =0.01 min). The closed-loop response after 300 s for the proposed control- ler PIP and the EXACT controller are shown in Fig. 4. I t is apparent that the proposed controller performs better than the EXACT controller for this dominant time delay process.

4.2 Temperarure control A popular piece of control engineering laboratory equipment, the feedback process trainer PT326 [SI, was used to demonstrate the performance of the adaptive PIP

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controller on a variable time delay system. The system consisted of an adjustable air blower, a length of tube and a heater. The air was drawn through the throttle

:;i , , , , , , , ,

0 40 80 120 160 time,s

0

a

'.O > O 8 8 04 6 -

0 40 80 120 160 time. s

b

0 2

0

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

Compnrison of the proposed algorithm with E X A C T

PIPI Foxboro

~~

opening by the centrifugal blower. It was then warmed up by the heater as it passed through the tube length and sent back to the atmosphere. The system input was the heater drive voltage and with its output the air tem- perature which was measured by a thermistor fitted to the end of the tube. The schematic diagram of the heater is shown in Fig. 5.

adjustable a i r f low

controller measurement I t

I 1 ,

t external

ac tuot ion slgna I

Fig. 5 Block dingram for the feedhnck process trainer (PT326)

The adjustable air blower was set a t 40" and was maintained at this angle for the duration of the experi- ment. An open loop test was then carried out on the system and from the process reaction curve, a first-order

I E E Pro?.-Control Theorv A p p l , Vol. 142, N o 5 , September IYYS

model was deduced as

Since the time delay of the system was fixed and very small as compared to its time constant, a variable delay was added to the measured system output. Therefore, the system transfer function was

V ( s ) l.le-Ls V,(s) 0.5s + 1

G,(s) = - = - (22)

L = 2.0

L = 2.5

L = 3.0

for 0 < t < 60

for 0 < t < 120

for 0 < t < 180

A state variable filter of the form 1/(0.6s + was applied to the system input and output for polynomial identification. The estimator initial parameters, the initial cost, the forgetting factor and the estimator truncation order were selected as O(0) = Cl.0, 0.5, O.5lr, Jz(0) =

1 x 10-513 where 1, is a 3 x 3 identity matrix, = 0.05 and i = 4, respectively. The set-point was a square-wave with amplitude of 1.0 and period of 40 s superimposed on DC offset level 4.0. Fig. 6a-c shows the overall results. In

> L 10

," 8-

x olL.._---, * o 30 60 90 120 150 180

t1me.s a

o h - , 30 60 90 120 150 180

h m e , s b

30 60 90 120 150 180 time.5

C Fig. 6 n Set point and output against time h Control action agamrt time c Estimated parameters agamrt time

Adaprire ronrrol uJPT326 tiirh iuriahle delay

particular, Fig. 6u shows the setpoint, output pair, Fig. 6h shows the control signal and Fig. 6c shows the estimated parameters. As shown in this figure, the adaptive PIP controller shows excellent performance under variable time delay condition. The polynomial identification can

I E E Proc.-Control Theory .4ppl.. Vol . 142. No. 5, September 1995

track the variable time delay and PIP can give a very good control for this system. The model of the heater was also used to compare simulation results with the actual experimental results. Fig. 7a-c shows the simulation results and compare very well with those of Fig. 6a-c.

>

times a

> lo r

P 81 c' '

0 30 60 90 120 150 180 time. s

b

I i I-

30 60 90 120 150 180 E o!

0 time, s C

Fig. 7 dela! U Set point and output against time h Control action against time c Estimated parameters against time

Adaprirr control of sirnuluted model of PT326 with variable

5 Conclusions

A new adaptive controller is proposed for the special class of systems with dominant time delay. This class of systems are known to be difficult to control and there are few algorithms which can successfully control this type of processes. However, for the purpose of identification and control, dominant time delay systems can be represented by a first-order with time delay model. The parameters of this model are recursively identified by the polynomial identification and are used to tune Hagglund's PIP con- troller. Experimental studies were carried out to show the performance of algorithm under noisy environment, parameter variations and input disturbances. The con- troller performance deteriorates as the ratio of the time delay to the effective time constant of the system decreases. Therefore, it can be concluded that for the dominant time delay systems, this approach can be very useful 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 proposed controller 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 real systems.

6 References

I CLARKE, D.W.. MOHTADI, C., and TUFFS, P.S.: 'Generalised predictive control. Part I : The basic aleorithm'. Auromarica. 1987. 23. ( 2 ) pp 137-148

2 DUMONT. G A ELNAGGAR, A and ELSHAFEI, A 'Adaptive predictive control of systems with varying time delay', Int. J. Adaptioe Conrrol & Signal Processing, 1993, 7 , pp. 91-101

3 HAGGLUND, T.: 'A dead time compensating three term controller'. IFAC Identification and system parameter estimation, Pergamon press plc, Budapest. Hungary, 1991,pp. 1167-117?

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4 HAGGLUND, T.: 'A predictive PI controller for processes with long dead times'. IEEE Control Systems, 1992, 12, ( I ) , pp. 57-60

5 GAWTHROP, P.J., NIHTILA, M.T., and BESHARATI RAD, A.: 'Recursive parameter estimation of continuous systems with unknown time delay', Control, Theory & Adu. Techno/. , 1989, 5, (3) pp. 227-248

6 GAWTHROP, P.J.: 'Continuous self-tuning control'. Vol. I , 'Design' (Research Studies Press, UK, 1987)

7 SMITH, O.J.M.: 'Closer control of loops with dead time', Chem. Engng, 1957,53, (5 ) . pp. 217-219

8 BYTRONIC (UK): 'Documentation for the Bytronic process control unit' (Bytronic, UK, 1994)

9 FEEDBACK INSTRUMENTS LTD.: 'Manual for PT326' (Feedback Inst. Ltd.. UK, 1982)

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