An Adaptive I Regulator for Pure Delay Processes

Copyright © IFAC Evaluation of Adapti'"e Control Strategies, Tbilisi, USSR, 1989

DESIG"I A"ID EVALUATION OF A.IlAPTIVE ALGORITHMS

AN ADAPTIVE I REGULATOR FOR PURE DELAY PROCESSES Cs. Banyasz and L. Keviczky Compuler and Aulomalion In slilulf Hungarian Acadnn\" of Sciencl'S H -1502, Budapesl, Kende u 13-17, Hungary

Abstract. This paper introduces a direct recursive estimation scheme for ~he pure delay of a discrete time, linear model. An adaptive I regulator 1S also presented for controlling this special class of processes. Keywords. Time delay estimation, approximation, adaptive I regulator.

INTRODUCTION Discrete PlO regulators can be implemented in many different ways. Different structures correspond to different continuous PlO regulators. The most common sampled data PlO regulator is given by the discrete transfer function -1 -2 PO+P1 z + P z 2 1- z-l

where z-l is the backward shift operator. The gain, the integral and derivative time constants of the corresponding continuous regulator can easily be obtained by applying different approximations for the integral and derivative actions. This form includes a PI regulator, too using P (z-l) with P2"O. We can obtain an I regulator using p(z-l)=Po=Kr, i.e.:

G (z

R

-1

K

I

)=---

1- z

(2 )

-1

(Note that there are many other form a proper integral action.)

ways

to

Many rules exist for tuning continuous and discrete time PlO regulators, which procedures are, however, in many cases non-trivial. Therefore, the automatic tuning of PlO regulators has a great practical importance. The REFERENCES give a good collection of papers dealing with automatically tuned and/or adaptive PlO regulators. The same stands for PI and I regulators. In our investigation the I regulator is applied in a closed-loop system given in Fig. 1. This figure shows the continuous process given by the transfer function GpC(s)=H(s) e-s't/G(s), the zero order holding (Z. O. H.) element and the discrete transfer function p(z-l)/(l-z-l) of the regulator. Here 't is the time delay of the process.

recursive

estimation,

stochastic

The final form of the discrete-time closed loop system is given in Fig. 2 for I regulator (KI/ (1-z- 1 )) and a pure time delay process (z-k). To obtain an adaptive I regulator we introduce a recursive estimation algorithm first, then a very simple method to tune the gain of the regulator using the estimated delay. Besides the many unclarified convergence properties of the recursive estimation schemes one of the most important drawback of these algorithms is that they do not estimate the process time delay. Most of these methods use apriori known delay to the parameter identification. The few approaches published for delay estimation use certain more or less intelligent, mostly recursive probing methods for the selection of the time delay from some neighbouring class of models. This paper introduces a new recursive parameter estimation scheme which directly estimates the process time delay. The applied model is nonlinear in the unknown delay. The suggested new method is a direct recursive estimation algorithm based on the canonic form of the stochastic approximation schemes for nonlinear problems. CANONIC FORM OF RECURSIVE SCHEMES I t is well known (see e.g. Tsypkin (1968)), that most of the recursive parameter identification methods can be led back to the canonic form

pt =p t-1 +R t g t

-1

(3)

where dQ gt-1=~

(4 )

t-1

is the gradient of a realization Q of the stochastic loss function V and (3) solves the stochastic approximation type problem

64

Cs, Banyasz an d L. Ke\'icz k}'

minV=minE{Q}

p

Here E{ ... ) stands for expected values.

Consider the discrete time mode l

The different recursive schemes generally correspond to different selections of the convergence (or weighting) matrix Rt . An optimal select i on of Rt , based on a second order approximation of Q, is given by

R t

=[

t

L H

RECURSIVE ESTIMATION OF A PURE TIME DELAY

(5)

p

]-1

(6)

j =O t

Pt-l

y(t) =z

-k

u (t) + e(t) = u(t - k) +e (t)

(11)

of a linear pure time de l ay p r ocess . Here u(t) , y(t) and e(t) are the input , the output and the indepe n de n t , zero mean additive output noise of the process, respectively. Furthermore k denotes thE} discrete dead ti me and the argument t' stands for the discrete (integer) time of the sampled process. The approximate model is given by

where Ht is the Hessian matrix of the second order derivatives of Q. Thus the first and second order derivatives of Q play an important role in completing the recursive scheme . Most of the widely used parameter estimation methods use a quadratic loss function (Least Squares (LS) methods)

y(t)=Z - kU(t)

(12)

where the unknown delay k is to be identified. The one dimensional , scalar recursive estimator based on (3) - (9) is given by k"

t

=k

t- 1

e

- r [de(t)] (t) - ,-

t

(I)

Le.

where e (t) is a prediction error, residual or equation error depending on the applied model and estimation scheme. In this case

k

t- l

and t

de(t) q = - ,- - e(t) = [f e (t)~ t -1 dp t , t-1 P

(13)

dk

=

r (8)

t

L q,

]-1

[ j =1 J

k

t-l

r

_ .,---=-t--=1_ 1+ qt

(14)

\-1

t- l

where

and

Ht

de(t)

de(t)

=~ - T-

Pt-1 d' P t -1

d~~" 1

2

+

d e(t) T

e(t)

(9)

d' d' P t- 1 Pt- 1

(15)

and Because e(t) is generally a nonlinear function of p , therefore the computation of the first order f t =de (t) Id p and second order d 2 e (t ) Id p d p T error sensitivity functions, their complexity is the most important factor in the algorithms of these recursive schemes . In many cases a proper approximation is used instead of the exact computation of Rt . These strategies with positive definite weighting matrices are usual l y called quasi - gradient schemes . One of the simplest approximation is to use the approximate Hessian -

de(t)

de(t)

H =----t dp ,T

t-1 dp

(10)

t- 1

instead of the Ht. The most important advantage of the form (10) is that assuming the vicinity of the convergence point Rt is the inverse of a dyadic sum , therefore it can easily be computed by the well - known recursive matrix inversion scheme , which is the base of the Recursive Least Squares (RLS) techniques. In this case forgetting strategies can also be taken into consideration easily.

(16)

The computation of the first order error sensitivity function can be completed as

de(t) dy(t) -k -, - = - - -,- =- ( - lnz)z u(t) = dk dk (17)

where the first order approximation of ln z was applied . Because of the well - known exponential transformation lnz=sh =1-z

-1

(sh

~

d

dt

~

-1

1- z

)

(l8)

the above approximation is a very good one for small sampling interval h and for h ~ O. Using (13) and (17) the recursive LS delay estimation equation is

An Adapt ive I Regul ator (19)

The computation of qt needs the order error sensitivity function

second

65

integral gain KI estimated delay k.

on

the

Let us consider a time delay channe l discrete transfer function of

with

-3

(21)

The recursive computation of rt can be completed by using (21) and (14), or t- 1

is used if a forgetting factor be applied.

O
should

THE APPLIED ADAPTIVE REGULATOR DESIGN METHOD

The solution of the absolute value and phase equations ensuring the prescribed $ az1 . 07=60o results in a very simple explicit formula (Banyasz and Keviczky (1982)) for the integral gain =_1_ 2k - 1

(27 )

when k relates to the argument of a variable (e.g. u(t-k)) otherwise the nonintege r ~ in the recursive equation. Open loop experiments

The discrete - time closed- loop system in Fig. 2 is easy to be handled from designer ' s point of view. This loop consists of a serially-connected integrator and a pure delay z-k. In this simple closed-loop system there is a strong relationship between the overshoot of the unit step response and the prescribed phase - advance $ a ' as our earlier investigations showed (see Banyasz and coworkers (1982 , 1985)).

I

(26)

In the examples below the recursive time delay estimation is performed according to (19) , (21) and (22). Note that (19) gives an estimated delay, denoted by ~, which is not an integer number yet . In the practical application the best integer approximation of (19) should be used, i.e.:

(22)

t

K

u(t) + e(t)

where the input excitation u (t) is a square - wave with the half time period of TPH and the true time delay is k=3 .

Thus finally qt is given by

r

based

SIMULATION EXAMPLE

y(t)= z

r

( 23)

(23)

Here the discrete time delay of the process is

Fig. 3 shows the recursive estimation of the pure time delay in open loop for an outer excitation (at u (t)) slower (TPH=5) than the true delay (k=3). The two sections are for different initial estimates ko and for different ro and w2 . Fig. 4 shows the operation of the recursive estimation in case of slowly varying time delay. The simulation studies showed that the recursive time delay estimation is very sensitive for the proper selection of the forgetting factor (w) and sometimes for the initial (co) variances (r o ) and very special consideration is neccessary to the optimal selection (design if possible) of the applied input excitation. (From numerical point of view the approximate computati?n of the numeric differences , e . g. 6y (t) or 62~ (t) is crucial! ! ! )

(24 )

k=entier ('t / h)

Closed loop experiments with the applied sampling time h.

In the closed loop experiments the square wave excitation was applied at the reference signal Yr(t) with TPH=40.

It is interesting to see that in case of K

lirn _ I = lirn __ 1_=...L h~

h

h~

2't- h

2't

(25)

we obtain the well - known classical design rule of thumb back, avai l able for integrators, compensating dead-time processes. The adaptive I regulator can finally be completed by using recurSl.ve time delay estimation scheme according to (19), (21) and (22), furthermore applying the optimal

Fig. 5 shows the recursive estimation k, the output signal y(t) and the reference signal Yr (t) for ko =2. In this case r o =10, w2 =O.95 was used. Similar results can be seen in Fig. k o =6 and using the same ro and w2 .

6 for

The adaptation to the required closed loop transient is fast and this behaviour is very nice for even practical applications.

Cs.

66

B,iny~lsz

and L Keviczky

The operation of the adaptive I regulator can be seen in Fig. 7 and Fig. 8 in case of slow ly varying time delay . Fig. 7 shows the

DISCRETE PtO REouunOA

CONT I HUOUS PROCESS

G

recursive estimation k of the time delay k. Th e reference signal and the output signal are shown in Fig. 8 . Here ro=lO and w2 =O . 9 were applied.

( _ H(.) -s' PC") - G(.)·

Fig . 1 Th e block scheme of the closed loop system (p(z - l ) = Kr;

CONCLUSIONS

t

Our paper prescribed a recursive estimation algorithm for the identification of the pure delay of linear dynamic process . The introduced technique is the direct application of the canonic form of stochastic approximation type algorithms. Then an adaptive from the above explicite design integral gain is

y(t)

e(t)

I regulator is constructed recursive scheme and an formula providing optimal derived.

Simple simulation examples illustrate the transient behaviours and the applicability of the introduced technique.

H(s)/G(s) = l)

Fig 2 The block scheme of the d isc ret.e- t ime closed loop syst em with I regulator

10~-------------------,

10~--------------------,

REFERENCES Agarval, M. and C. Camidas (1986). On-line estimation of time delay and continuous - time process parameters. American Contro l Conference , Seattle, 728-733. Banyasz, Cs. and L. Keviczky (1988). A new recursive time delay estimation method for ARMAX models. 8th IFAC Symp on Ident and Syst Par Est . , Beijing,

4 (\L -_ _ _ _ _--j

O+-~_.--~-.----._

o

10

20

__-+~

30

10

40

20

30

40

(r o =10 ; w2 ",O .9 S; TPH=S) fig . 3 Recursive estimation of a pure ti me

delay in open lo op

Bokor, J. and L. Keviczky (1985). Recursive structure, parameter and delay time estimation using ESS representations. 7th IFAC Symp on Ident and Syst Par Est , York (UK), 867- 872 .

14 12

Gawthrop, P.J. and M. T . Nihtila (1984). Identification of time delays using a polynomial identification method. E..e.pQtl CE/Till School of Engineering & Applied

10

Sciences, University of Sussex.

Habermayer, M. and L. Keviczky (1985) . Investigation of an adaptive Smith controller by simulation . 7th Conference on Digital Computer Applications

to

Process

10

20

30

40

Control ,

Vienna. fig . 4 Recurs i ve estimation in c a se of s lowl y varying time delay (open l oop)

Keviczky, L., J. Bokor and Cs. Banyasz (1979). A new identification method with special parametrizati on for model structure determi nation. 5th IFAC Symp on Ident and Syst Par Est, Darmstadt. Kurz, H. (1979). Digital parameter- adaptive control of processes with unknown or time-varying dead- time. 5th IFAC Symp on Ident and Syst Par Est, Darmstadt. Kurz, H. and W. Goedecke (1981). Digital parameter adaptive control of processes with unknown dead- time. Automatica,17, 245 - 252. Tsypkin, Y.Z.

(1968). Adaptation and Learning in

Automatic Systems , Nauka , Moscow.

-1+---~---r-------'--------r---~--~--~

o

100

200

300

400

Fig.5 Th e oper at ion of th e adapt i ve 1: regu l ator for constant k =3

An Ad aptive I Regulato r

5

-1+---------~--~----r_--~--_,----~--_+----~

200

100

(k o :6;

400

300

ro"'10; w2 =O . 95)

Fig . 6 The operation of the adaptive :I regulator for constant k=3

k

~

I

\

I

\ 200

100

300

400

Fig.7 Recursive estimation of the time delay

in closed loop

2.0,------------------------------------,

y(t)

Yr

1.5

1.0

0.5

0 .0

-0 .5

+---__-----,---____ 100

-~---__.----I___-~

200

300

400

Fig . 8 The operation of the adaptive I regulator in case of slowly varying time delay (see Fig. 7)

67