Tuning of Multivariable PI-Controllers for Unknown Systems with Input Delay

ADAPTIVE CONTROL III

Copyright © IFAC Control Science and Technology (8th Triennial World Congress) Kyoto. Japan. 1981

TUNING OF MUL TIV ARIABLE PI-CONTROLLERS FOR UNKNOWN SYSTEMS WITH INPUT DELAY H . N. Koivo and S. Pohjolainen Department of Electrical Engineering, Tampere University of Technology, Tampere, Finland

Abstract . A design method for on- plant tuning of multi variable PI-controllers for unknown systems is presented. The plant is assumed to be linear, open - loop stable with input delays and subject to step disturbances. Only step - response experiments are utilized in the tuning. An example illustrates the procedure. Keywords . Linear systems; multivariable control systems; process control; PID- control; robust controllers; tracking systems; feedback; state-space methods; step response; time-delay systems INSTRODUCTION

PROBLEl·f STATEMENT

It is well-known that control of time-delay systems is much harder than those with no delay. In practice, a PI-controller is often tried and although i t does not function perfectly, it often works satisfactorily and lS used.

Consider the following line ar time-invariant stable time - delay system

In this paper, the system is assumed to be unknown, but stable and linear with input delays. The problem is to tune a multivariable PI- controller for the system using only step- response experiments. This is an analogous problem with the scalar case (Ziegler and Nichols, 1942). The problem has previously been discussed by Penttinen and Koivo (1980) , when there are no delays in the system . A reasonably complete theory is presented here , when delays occur in the input. The method has also direct links with the characteristic locus technique (MacFarlane, Kouvaritakis , and Edmunds, 1978), which has been applied to the delay case without proper tehoretical justification .

x(t)

Ax(t) + Bu(t-·r) + Ez

y(t)

Cx(t) + Fz(t ) ,

,

e(t) m where xE if', input u E R , error e E ~ is the difference between constant reference signal q Yr and the output y, and z ER is a constan t disturbance and the time-delay T > O. The matrices A, B, C, E , and F are constant but unknown and have appropriate dimensions. A m~v~able of the form

PI-con~olle~

for (1 )-( 3) is

u(t) = Pe(t) + Kv(t) , ~(t) = e(t), (4) where v E ~ and the constant matrices P and K are the gains of the proportional and the integral part of the controller, respectively.

After defining the problem, the P-controller is investigated. It is shown that the closed-loop system remains stable, if P is small enough . Further, a reasonable choice for the P- controller gain is indicated and how the gain can be determined experimentally. These extend the results of Penttinen and Koivo (1980). It is also shown that the application of the I-controller eliminates the steady- state error . If the gain of the 1controller is small enough, the closed-loop system is proved to be stable. The results of Davison (1976) are thus extended to the input-delay case. The paper concludes with an illustrative example .

The assumptions made are the same as in the undelayed case (Penttinen and Koivo, ( 1980) : 1.

Matri x A is (asymptotically ) stable, i.e., the real parts of the eigenvalues are negative.

2.

The control input u can be excited and the output y, which is to be regulated, can be measured. This implies that m and k are known.

3.

The order n and the plant parameters A, B, C, E , and F are unknown.

Note also that the disturbance z might not be measurable.

951

H. N. Koivo and S. Pohjolainen

952

It is desired to find a multi variable PI-controller (4) for the system (1) - (3) so that asymptotic regulation occurs for all constant reference signals Yr and for all constant disturbances z.

1 -ST - ST Y(s)=-C(I+(A-BPCe )/s + ... )BPye Is. s r

PlWblem.

MAIN RESU1TS

A P-controller for systems with input delay has been suggested in Penttinen and Koivo (1980). It will be shown here that the idea can be given a sounder basis and that it can be extended to a practically important case of different delays in the input. The latter case is very common , because in process control it is customary to fit a transfe r function of the form G(s) = Ke- ST /(sT+l) to the step response .

Taking only the first term into account gives y(t) ~ (t-T)CBPyr1(t-T). The system can be de coupled for t slightly larger than T by choosing P so that CBP = 1 1 , where 1, = diag{Pl""'Pk} and Pi's are called "t ming parameters". I frank (CB) = k, the generalized inverse of CB, (CB)7 , exists (Penrose, 1955). Thus t P=P 1 =(CB) 1 , 1 1 1 where (CB)t = BTCT(CBB TCT )-1

The experimental determination of the matrix CB is based on the following observation. Assume again that x(O) = 0, z(t) = 0 for all t, and u(t) is a step-function. Since A lS stable, (1) and (2) imply

A state space representation, which includes G(s), is Ax(t) +

J L

j=l

B. u(t-T.) J

y(t)=Cx(t)=CA- 1 (e A(t-T)-I)Bu,t ~ T, and

J

where TJ ~ ? T2 ~ Tl ~ O. delay case is treated first.

The single

The idea of the multi variable P-controller lS to try to decouple the system for small t, t ~ 0 , or correspondingly in the frequency domain, at high frequencies. This is exactly the same idea as used in characterist ic locus design technique (MacFarlane and Kouvaritakis, 1977), where the model is known and there is no input delay. In fact, the Cambridge design package (Edmunus, 1978) allows delays in the transfer-function matrix without theoretical justification. It must be emphasized that in this paper the model ~ ~ot k.~ow~ and theoltetic-al jU6ti6..i.wtio~ wd.t be g~ve~ 60lt the ~put delay c-~e. The results bear great significance in solving practical tuning problems of PI-controllers for multivariable systems with input delays. Consider now the system (1) with one input delay. Assume x(O) = 0 , z(t) = 0 for all t, and u(t) " 0 for tE [-T, O]. These are no restrictions, since x(O) = 0 implies on ly that we just wait until steady-state is reached. Setting z = 0 means that the experiment is performed when no (deterministic ) disturbances are present. Substitute the P-controller (4) (K 0) into (1). Then , in the case that Yr lS a constant A x (t) + BP e (t-T)

Matrix CB can be evaluated from m step-responses by computing the derivative of the output at t = T+, say graphically. In fact, the algorithm of Penttinen and Koivo (1980) lS applicable with slight modifications:

Step. 1.

Let u(t) " 0. When the steadystate (y(oo) 0) is achieved, choose a constant input u(t) = u1(* 0) and apply it to the plant. Then Y(T+) = Y1 = CBu1'

Step 2.

Repeat step with a constant control . +which. is linearly independent of u1' Then y T ) = Y2 = CBu2'

Ut'

Step m.

Repeat step 1 with a constant control Urn, which is linearly independent of u1' u2""'Urn' Then Y(T+) = Ym = CBu m·

As a result

It might happen that rank (CB) < k, where (CB)t does not exist. Equation (9) suggests that the rank-deficiency could be avoided by choosing At (Ce 0 B)t 1 , for small 1 t

- C x( t-T) ) , t

(8)

~

T ,

(6 )

where l(t) is a unit step-function. Taking 1aplace-transform of (6) and (2) yields after some manipulation for large s

o

> 0 .

(1 1 )

This just means that the derivative y has to be evaluated at time T + to instead of T. The P in (11) also confirms with the first phase of a PI-controller design using characteristic locus technique (MacFarlane, Kouvaritakis and Edmunds, 1978). Edmmds (1980)

953

Tuning of Mult i va ri a bl e PI- Co ntroller s suggests that since no physical system can respond at infinite freque ncies , it i s r easonable to choose a cut- off freque ncy at which decoup l ing 0;' alignment occurs. The out- off frequency corresponds then in some sens e to ti me t o ·

Treating the single de lay ca se first , system ( 1) , (2) is extende d by introducing a new state var iab l e v appearin g in (16) . The augmented system be co mes

[~ ( t )J

that the proportional gai n is small enough. The followi ng theorem guar antees that the closed-loop system r emains stab l e after add i ng a P- contr olle r, if IP I is s mal l enough. T re.oJtem 1.

Cons i der the open -loop stable system (1 ) wi th z (t ) O. The the r e exists an E* > 0 such that , when u(t) = E P e (t) , the equilibr ium positi on of

=

x = Ax ( t ) - E BPCx(t- T) + E BPYr ' where Yr is a constant vecto r and P an m x k constant matri x, i s asympototic al l y stable for all E, 0 < E < E*. Pr oof can be found i n Koi vo and Pohjolainen (1981).

The experi mental determinat i on of P can be accompli shed by obse r ving the fo llowin g . When x(O) = 0 , z(t) = 0 for al l t , and u (t ) = u i s a step- funct i on , (5 ) with J = 2 and 1 (2 ) imply - 1 A(t- T 1 ) y(t ) = CA (e - I )B 1ul 1 (t - T1 ) + - 1 A( t-T 2 ) CA (e - I )B u 1(t-T ) (13 ) 2 2 1 From (13) ( 14)

Algo ri thm I appl ies as before , e xcept tha t the de r ivati ve of y eval uated at T2 ' If T2 is much l ar ge r than T , the suggested me thod 1 probably does not work . In the mo r e general case of (5) , the delays Ti have to be assumed to be of the s ame magnItude . A wor kable choice fo r P is J

A( TJ -T .) t P = PJ L 1 = [ C .L e J BJ. l L . 1 J =l

( 15)

~C

0J l-X ( t )] + IB] U( t-T : + IE z] , 0 v (t )

Lo

lyr

(17 )

[:~:;] [~ ~J [:~ : ;] =

+

[FOz]

.

(1 8)

I t is appar ent from ( 17 ) t hat t he open - loop system ha s a pole in the origin. Choosing the integral gain K in ( 16) appropriately , however , the closed-l oop syste m can be stab i l i zed . I n additi on , the steady- state error can be el i minated . Substituting u from ( 16 ) into ( 17) with K =E Tt , T := - CA- 1B, results :cn

0 [~ (t)] = L-l A °0J [ X(t )] 1Lo ,:,. ( t )

Theorem 1 can easily be extended to the case of sever al delays . The determination of P in case of different delays is not easy . Cons i der the case of two delays , that is , J = 2 i n (5 ) . A choice that often works and is measurabl e i s given by A(T -T 1) t 2 P = P2 L = [ C(e Bl + B )] Ll (12 ) 1 2

= lA

v (t )

As far as stab i lity is concerned , it suffices

C

v(t

)

+E

( 19) or i n short

~(t ) = AO ~(t ) + E:Al~ ( t-T ) + [EYrZ]

( 20)

with obvi ous notation .

Th e.oJte.m 2.

1

I f rank T = rank (- CA- B) = k ,

then there exists an E* > 0 such that the equilibrium pos i tion of ( 20 ) with z(t ) = 0 is asymptoti cally stable for all E, o < E < E*. Proof can be found in Koivo and Pohjol ainen (198 1) . Multi pl e delay c ase (5) can be handled exactly i n the same manner by choosing the i ntegr al gain to be K = E T}, T = J - 1 J - CA L B .. j =l J I t i s strai ghtfor ward to extend the a l gori thm of Davi son ( 1976) to the case of mul t i ple i np ut de l ays (5 ) in order to experi men tally deter mi ne TJ · Let z(t ) i n (5 ) .

=°

A.tg oltdl .m II Apply a constant input u (t ) = u ' 1 ul 0 , to the open- loop pl ant . Measure the steady- st ate solution y (oo ) = Yl ' which exists s i nce the pl ant (5 ) is stable.

Step 1.

*

Step 2.

r - ConVI.OUe.Jt

Repe a t step 1 with a constant input wh i ch is l inearl y indepent of the previ ous input u l . Measure the steady- state y( oo) Y2 '

The multi variable I-controll er i s gi ven in ( 6 ) by

Step m.

Al gorithm I can be modifi ed by evalua ti ng the de r ivati ve at t = T + to . J

u ( t)

Kv ( t ) , ~ ( t )

e(t)

~,

y r - Cx( t).

( 16)

Repeat step 1 with a constant input is l inear ly independent of the pre VlOUS Inputs Ul ' ~ " . . ' ~-1 ' Measure the steady- state y( oo ) = Ym. ~ , wh~ch

H. N. Koivo and S. Pohjo l ainen

954

As a

where typical values for the parameters at an operating point are a11 = - 0.015, a21 = -0.011, a22 = -0 . 023, b 11 = 0.012, and b 22 0 . 019. The delay values are T1 = 0, T2 = 10, but also the case T1 = 5, T2 = 5 is treated.

result

PI-c.oYl-Vtoilelt The multi variable PI-controller is constructed in the same way as shown in Penttinen and Koivo (1980) . Determine first the gain matrices PJ and TJ using algorithms I and 11, respectively. The step-inputs needed can be the same for both algoritms. Then t 'me the P- controller for the plant by adjusting the tuning parameters in L 1 . Start with small postive values and stop when the closed- loop step-response is as good as possible. Then make the tuning parameters slightly smaller, say 5-1 0 % smaller. Now begin with a small positive value for £, the tuning parameter of I - con troller, and adjust it until the maximum speed of step response for the closed- loop system is achieved. In Penttinen and Koivo (1980) a quikc way of determining the tuning parameters p . is pre sented. It is repeated here for th~ sake of completeness:

06

Method

1.

i

2. 3.

= 1, . .. ,k.

The second case is slightly harder, but again a fairly good response is achieved and the steady- state error lS eliminated. In Fig . 3 the case of T1 = T2 = 5 s is shown when input is [-1.2, 1.2 ]. Comparing the response in Fig. 2 and Fig . 3, it can be seen that the latter case in slightly earier to handle . CONCLUSIONS

6-mcUng tuntYlg paJtametelt-6

Apply the P-controller with Pi

The multi variable PI - controller was tuned for (21) using algorithms I and rI. The step responses are presented in Fig's 1 and 2. In Fig. 1 the reference signal is [0, 1.2 ] and in Fig. 2 [-1.2, 1.2 ], respectively. The response in the first case quite good. The maximum interaction at the beginning is about 20 %. Note the scaling in Fig. 1.a. The sharp edge in Y1 is explained by the fact that the control ~ starts its action after the initial delay . The steady-state erros is zero, as expected.

= p,

Adjust p to a small value and l et each component of Yr be a step- function, scaled to uni ty . Measure the output y(t) and decide if the responses are satisfactory. If not, adjust p until the fastest mode, say Y1' is satisfactory . Let P1 = p. Then in crease Pi's related to the other modes by multiplying them by K1/K i , i = 1,2, ... ,k, where Ki is the steady-state gain of the ith output. If necessary , determine first the p correnponding to the second fastest mode, say P2' and then p corresponding to the third fastest mode, etc.

RemaJtk .

In some cases it is also possibl e to set up a cost function based on rise time, overshoot, settling time etc. Then rrunlrruze the cost with respect to the tuning parameter&

Tuning of multi variable PI - controllers for unknown , but linear and stable systems with input delays have been consireded. Such models appear regularly as approximations for most common processes. Control des i gn for linear systems with input delays is quite difficult. Perhaps, the most successful procedures have been the frequency domain techniques, inverse Nyquist array (Rosenbrock, 1974) and characteristic locus (MacFarlane, Kouvari takis , and Edmunds, 1978). In both of them, controllers are designed for delay systems without rigrous justification and the transfer function matrices are assumed to be known. Here the model is unknown and theoretical basis is given for PI- controller tuning . The resulting controller is roughly the same as the one obtained by the frequency domain techniques for high and low frequencies. REFERENCES

EXAMPLE

Consider a linearized model for a concentration-flow process (Penttinen and Koivo, 1980)

Davison , E. J. (1976). Multivariable tuning regulators: The feedforward and robust con trol of a general servomechanism problem. IEEE Trans. Aut. Control AC-21 , 35 - 47 . Edmunds, J.M. (1979). Cambridge linear analysis and design programs . Report CUED/ F- CAMS/TR 170 Engineering Department, University of Cambridge, England.

(21 )

Edmunds, J . M. (1980). Control system and analysis using closed- loop Nyquist and Bode arrays . Int. J. Control 30, 773 - 802 .

9 55

Tuning o f Mul tivaria bl e PI-Cont r o l le r s Koivo , H.N . and S . Pohjolainen (1981). Tuning of multivariable PI- controllers fo r unknown systems with time - delay . Tampe r e University of Techno l ogy Report, Tarnpere , Finland . 4-----~------~>~,~====,,~.=_---------

MacFarlane , A. G. J ., B. Kouvaritakis and J . M. Edmunds (1977 ) . Complex variabl e me thods for multi vari able feedback systems an alys i s and design. Alternatives for Linear ~Iult i­ var i able Control , National Engi nee r ing Consortium, Chicago , 1977, 189- 228.

Si

cl N

>-

Penrose , R. (1955) . A gene r al ized i nverse for matr ices . Pr oe . Cambridge Phil . Soc . 2..l, 406-413 . Penttinen , J. and H. N. Koivo (1980) . Multivariable tuni~g regulato r s for unknown systems . Autornat i ca 1£, July 1980 . Rosenbrock , H. N. ( 1974) . Compute r -ai ded con trol system design . Academic Pr ess, New Yo rk .

8 2, !~I---rLn--.oa~I--~i--~i--~--~i--~~--I~~ 6o.r~ go.oo IZJ.~ O.,UU

Wilkinson, J . H. (1965). Algeb r aic eigenvalue problem. Oxford University Press, London.

Fi g . 1 b.

Ziegl er , J.G . and J.G. Nichols (1942) . Opti mum settings for automatic controllers . Trans . ASME 64, 75 - 91 .

TIME

[0 , 1. 2], when '1 =0 , ' 2=10 s.

~rl---ri--~i'--~I--~.---'I---'i---'--J:u,~~:ruo-80.00 00.00 00.00

C.txi

~

q 1'---1i--~i~~,----r---Ti--~i--~--~r-----

0.00

20.00

~.OO

00.00

TIME

Fi g . 1 a.

Output r esponses for the input

JMl.O
Fig. 2 a

lIM(

956

H. J. Koivo and S. Pohjolainen

s.

-

i_,...

i

t. ••• CU

i

l

ro.m

TIME

Fig. 2 b. Output responses for the input [-1.2,1.2] when T =O, T =10 s. 1 2 Discussion to Paper 33.1 C.P. Jeff e r son (Australia): There are two comments I would like to make about the exp e rimental system chosen by the authors both for this paper and the previous one. Firstly, the method of control is unrealistic there are many established process control techniques which a process control engineer could use to decouple the flow and concentration loops. The authors make no attempt to control level, which must be an important parameter in a real system. Secondly, a general point about adaptive control - an understanding of the process dynamics and gain can often be exploited to good advantage. For the model head box system chosen, if level is closely controlled, the process gain for concentration control is inversely proportional to flow demand and so are time constants and dead times (if present) . Hence, it is easy to show that the process is selfadapting, the only controller parameter to be changed being reset rate which should be proportional to flow. In general, it is better to understand the process rather than to treat it like a "bl ack box". H. Koivo (Japan): I believe Dr. Jefferson has misunderstood the use of the experimental system which is not a head-box but mimicks the short cycle of the paper machine. The methods presented in both papers are of a general nature. They are not connected to the pilot plant used. The idea was rather to see how the algorithms would work in a real environment, since both the self-timing controller and the multivariable PI-controller have worked well in simulations. I agree with Dr. Jefferson that one must know the process before designing the control system it would be quite unfor~ivable and irresponsible, if you would not. To completely ignore the possibility of using "black box" methods on the other hand is rather shortsighted. A good example of using the transfer function method to design a multi variable controller for a difficult process is given in another paper (110.5).

Fig. 3 a

~~

"~,I I~

1';.,

.: I

i /; \'. rI~-.-=--...-.::----I I

\/

I

i

• ~--_,.-__r - - ..--T-----r .- . -!---.----r---

G~

Z::.G3

8~~.~~

.":!.(:L.) j

11-:::

cJ .. JJ

Fig. 3 b. Output responses for the input [-1.2,1.2] when T = T =S s. 1 2