A multivariable self-tuning controller

0005-1098/80/0701-0351 50100r0

Anima m, V01 . 16, pp. 351-366 Pergamon Press Ltd . 1980. Printed in Great Britain 0 international Federation of Automatic Contra[

A Multivariable Self-Tuning Controller* H . N . KOIVO* A multivariable self-tuning controller, cost function, can stabilize unstable time-varying reference signals.

derived with penalized control terms in the and nonminimum-phase systems and track

Key Words-Adaptive control ; computer control ; digital control ; discrete systems ; identification ; optimal control ; parameter estimation ; process control ; self-adjusting systems ; stochastic control.

et al ., (1978) ; Alam and Burhardt (1979) and Morris, Fenton and Nazer (1977) . In the multivariable case tuning of controllers has not been extensively considered . Recently, results have been established for tuning multivariable 1-controllers (Davison, 1976) and multivariable PI-controllers (Moore, 1979 ; Penttinen and Koivo, 1980). The basic self-tuning controller structure has been extended for the multivariable case by Borisson (1975 ; 1979) . Some further improvements have been suggested by Keviczky and Hetthessy (1977) and this controller has been applied in the control of a cement plant (Keviczky et al., 1979) . In this paper, an extension of the method of Clarke and Gawthrop (1975) for the multivariable case is carried out . The controller structure is obtained by penalizing the control terms in the cost function and then deriving the corresponding control law . In the derivation, the k-stepahead predictor is utilized and is slightly different compared with the one derived by Borisson (1975) . A recursive least-squares algorithm in square-root form (Peterka, 1975) is used for identifying the controller parameters . Set-point changes are considered by augmenting the multivariable selftuning controller of Borisson (1975) in the manner suggested by Wittenmark (1973) and also by utilizing the new resulting structure . Simulated examples illustrate advantages of the derived controller compared with that of Borisson (1975) . Preliminary results of this paper were presented at Optimization Days 1979 (Koivo, 1979) .

Abstract-A multivariable self-tuning controller is derived extending the scalar version of Clarke and Gawthrop (1975) . Because the control terms are penalized in the cost function, fluctuations and peaking in control signals are reduced compared with the multivariablc minimum variance self-tuning controller (Borisson, 1979) . Time-varying reference signals as well as certain nonminimum-phase systems can be handled without difficulty with the proposed controller . Several examples illustrate the power of the derived self-tuning controller . 1 . INTRODUCTION

CONTROL of unknown systems has been a chal-

lenging and difficult problem for a long time . Once the PID-controller structure was invented and its tuning considered, major advances have been hard to come by . Only recently have significant, practicable contributions been made . Selftuning controllers form an important class of controllers which are simple to implement and have proved to be useful in a number of practical applications (.$strum et al., 1977). The basic scalar self-tuning controller is a minimumviarance controller, the parameters of which are identified by a recursive least-squares algorithm . The controller structure works satisfactorily if the system is of minimum phase and if limiters are initially used on the control signal . Improvements in the basic self-tuning controller have been proposed in Clarke and Gawthrop (1975), where the control signal is also penalized in the cost-function . This reduces excessive peaking of the control . In addition, reference signals are taken into account in the costfunction . Self-tuning controllers based on poleplacement have also been proposed (Lstrom, Westerberg and Wittenmark, 1978 ; Wellstead et al., 1978) . Practical problems such as `bursting' of parameters, have been discussed, e .g ., by Astrom

2 . DERIVATION OF CONTROLLER STRUCTURE

Consider the system described by the linear vector difference equation :

"Received 13 August 1979 ; revised 11 February 1980 . The original version of this paper was not presented at any IFAC Meeting . This paper was recommended for publication in revised form by associate editor A . Cauwenberghe . tTampere University of Technology, P .O . Box 527, SF33101 Tampere 10, Finland .

A(q-t)y(t)=B(q-t)u(t-k)+d+C(q-t)~(t)

( 1)

where y e r is the output, u e r is the control, deSPm is the constant, steady-state output re351



H . N . Koivo

352

sponse for a zero input signal, i;e .?m and {~(t)} is a sequence of independent equally distributed random vectors with zero mean value and covariance E[ (t)i'(t)}=rs, k is the time delay, and q - ' is the backward shift operator, q - 'y(t) :y(t- 1) . The polynomial m x m matrices A, Band C are given by A(q

admissible strategies . A control strategy is admissible, if the value of the control signal at time t, u(t), is a function of all the observed outputs up to time t, y(t), y(t-I), . . ., and all previously applied control signals u(1-1), u(t-2)_ . . An optimal strategy is an admissible strategy that minimizes the criterion (2) . Equation (2) can be written as

') :=I+Aiq-'+ . . .+A"q-" I=E{11P(q

')y(i+k)-R(q

')w(t)d 2

B(q ') :=Bo+B,q-'+ . . .+B"q-",

+11Q'(q ')u(t)11 2 '

( 3)

B . nonsingular, C(q-') :=I+C,q-'+ . . .+C"q-", where the roots of detC(a) lie outside the unit disc . The assumptions here are essentially the same as in Borisson (1975) except for B(a) . In Borisson (1975) only minimum-phase systems were allowed, when here on the other hand, also certain non-minimum phase systems can be handled . Summarizing the basic assumptions about system (1) : the number of outputs is equal to the number of inputs ; B„ is nonsingular ; roots of det C(a) lie outside unit disc . Remark 1 . The assumption on C(a) is a weak one . A stricter requirement of having the same number of outputs as inputs and having a nonsingular B o can be relaxed by using a different minimization procedure as was done by Peterka and Astrdm (1973) . The penalty that is paid is that of solving the corresponding Riccati equation on-line . As indicated in the sequel an optimal control law can be determined with the method of the paper without this assumption, when the system parameters are known, but this form has not been useful in self-tuning . The cost function to be considered is of the form k-1 Y

p ;y(l+k-i)-

-O

7,

R,w(t-i)

1_0

(2)

+ E, Q ;u(t-

a-n

}

where w(t)e .r is the known reference signal, pis are real numbers and p 0 = 1, R ; and Q ; m x m matrices . Notation 11x11 2 =x T x has been used . The upper bounds N and M in the sums of w and u terms are not crucial . Most common ones are N=O and M=O or M=1 . The criterion (2) is minimized over all the

where P, R and Q' are polynomial matrices obtained from (2) in an obvious way . The cost function (3) includes the useful cases : 1,

=

Eilly(t+k) - w(t)I1 2 +a T (t) diag {2, 2m ;u(t)?

I 2 =E{lly (t+k)-w(t )11 2 +(u(t) -u(t-I))'diag{2 ", X (u(r)-u(t-1))1 .

Remark 2 . The first criterion 1, penalizes deviations of the output from the reference signal and variations in control . It is well-known (e .g . Clarke and Gawthrop (1975)) that the cost function I, does not guarantee in general that the mean value of y(t), y(t), would equal the mean The level of w(t), w(t), unless 2,=0, second criterion I 2 introduces additional integrators into the loop and assures the equality of y(t) and w(t), but the dynamic response becomes more sluggish . Remark 3 . By extending P and Q' in (3) to include rational transfer functions, Gawthrop (1977) has made some interesting interpretations of the self-tuning controller in the scalar case . These comprise a model-reference adaptive control and a self-tuning least-squares predictor in conjunction with conventional compensation . Such an extension can also be considered in the multivariable case, although it is not done in this paper . The optimal control law is deduced by using the optimal (least-squares) predictor y*(t+klt), which at time t predicts the output k steps ahead using the previous data {Y(tby0 - I ), . . . ; u(t),u(t-I),

The optimal k-step-ahead predictor for (1) is derived in appendix and is given by (A .5) as y*(t+k[t)=C '(q-')[F'(q-')Y(r)

+E'(q ')B(q-')u(t)+Y] .



A multivariable self-tuning controller From (A .4), y(t+k) can be expressed as

from which an admissible optimal control law can be computed even in the case that B is not square . For self-tuning purposes the form has not proved successful and the development is now limited to the case where B is square .

y (t+ k) = C - ' (q - ' )[E' (4 - ' )Y (t) +E'

353

(q - ' )B (q -' )u (t) + y] +i;(t+k)+ . . .+E'_,~(t+1).

By defining a matrix polynomial The prediction error is given by Q(q-') :=(BO) '(Q'(0))'Q'(q-') e(t+k)=y(t+k)-y*(t+kit) =g(t+k)+ . . .+Ek ,4(t+1)

(4)

and is therefore uncorrelated with y(t), y(t 1), . . ., and u(t), u(t-1), . . . and hence with y*(t +kit) . Substitution of (4) into the cost function (3) results in I=E{11P(q ')(y*(t+klt)+e(t+k))

-R(q-')w(t)Ih+IIQ'(q-')n(t)112}

(5)

and because P(q-')e(t+k) is uncorrelated with u(t-t), w(t-i), and y(t-i), and with P(q - ')y*(t + i I t), 0 < i < t, by (A .6), (5) can be written as I = I1P(q -' )y * (t+klt) - R(q - ')w(t)112 +IIQ'(q - ')u(t)11 2 +Ei11P(q - ' )e (t+k)112} .

and a vector function ¢*(t+kit), q5*

(t + k i t) : =P(q - ' )y* (t + k i t) - R(q - ') w(t) +Q(q ')u(t),

the optimal control is found by setting 0*(t+kl t) =0. In the sequel, a closely related vector function 0, defined by 4 (t+ k) :=P(q-')y(t+k)

(7)

- R(q - ' )w(t)+Q(q-')u(t),

is needed . It is now shown, as in Clarke and Gawthrop (1975), that an equivalent minimization problem to (3) can be formulated . Subtracting (6) from (7) and using (4) results in ¢(t+k)=O*(t+kit)+e(t+ k),

The problem has now been reduced into a deterministic optimization problem . The minimum of I is found by forming the gradient of I :

(6)

(8)

where k_1

s(t+k) :=

I

p ;e(t+k-i)

i=o 0017

2(c'P(q-')y*(t+kl t) 1T

(eu(t)

du(t)

x [P (q - ' )y * (t+ kit)-R (q - ' )w (t )] +2(Q'(0))TQ'(q-' )u(t)=0

or Bu [P (q - ' )y * (t+ kit) - R(q - ' )w(t)] +(Q'(0))TQ'(q-')u(t)=0 . Remark 4 . The requirement of having the same number of inputs as outputs is utilized from this point on . When the system parameters are known, the control law can be determined without this assumption by substituting y*(t+kit) into the above equation . This yields

is uncorrelated with ¢*(t+klt) . It is then apparent that 0* (t+kit) is the optimal predictor of ¢(t+k) at time t and that an equivalent minimization problem to (3), producing the same control law, is to minimize J=E(02(t+k)}=0*(t+klt)2+E{e2(t+k)} . Now substituting the predictor from (A .5) in Appendix into (6) implies that k_1

4~*(t+kit)=Z p . C -' (q -' )

i=o

x [Pk-7(q ' )y(t)+E 1(q ')u(t-J)+yk-J] -R(q-')w(t)+Q(q-' )u(t)=0 .

B T (q - ')C - '(q ')F(q ')y(t) - BoR(q ')w(t)+D(q-')u(t)=0

where D (q - ' ) : =BoP(q - ' )(-' (q-')E(q-')B(q-' )

+ (Q'(O))T Q (q - ' )

Defining matrix polynomials F, G and H as P(q ') : - EPA ; O (q ') :=Ep,q-'Ok- ;+C(q-')Q(q-'), H(q-') :=-C(q-')R(q-'),

(9)



H . N . Koivo

354 and a constant vector 6 : = F' pi7k

The matrix T(q - ') has also been used by Borisson (1975) in a sensitivity study of the optimal system . The stability is determined by the system matrix 7 : In fact, the unstable modes correspond to the zeros with H < 1 of the equation

-, i

yields by premultiplying by C C(q - ' )O*(r+kl t)=F(q - ' )y(t) +c(q-')u(t)+H(q-')w(t)

f

6-0.

(1 det

Solving for u(t) in (10) results in an optimal control law . Note that the premultiplication requires that P and C commute . This is the reason for limiting P to be a polynomial and not a matrix polynomial . This minor limitation is made to serve self-tuning purposes . Summarizing the above, the following proposition is obtained . Consider the system

Proposition l .

l(q-')y(t)=B(q-')u(t-k)+C(q-')~(t)+d, where [ (t)} is a sequence of independent, equally distributed, random vectors with zero mean and covariance ry , and independent of the initial values . If B 0 is nonsingular and detC(x) has all its zeros outside unit disc, the optimal control minimizing (3) is given by equation F(q-' )y(i)+(3(q_ ')u(t) (II)

+H(q-')w(1)+5=0

where the polynomial matrices F, e,, and fl are determined by (9) and (A .6). By setting P=l, and Q=0 in (1I) results in minimum-variance control . Note that the optimal controller (11) is independent of the noise covariance r, . Since G(0)=E,(0)B(0)=I, (11) defines an admissible strategy and since it minimizes (2), it is an optimal strategy . It might not be unique . since E' and F' are not unique . Remark 5 .

The stability of the closed loop system is now analyzed when P=p o =l and w(t)=d=0 . The closed-loop system is obtained from (1), (11), (9) and (A .7) and is described by y(t+k) -u(r)

A(x) F(a)

L-zk

B(a)

=0,

E(a)B(a)+C(a)Q(x)

This determinant can be developed by Schur's formula (Gantmaeher, 1959) . When (A .7) is used and x is dropped as argument, the result is det(EB+CQ)det[A+B(EB+CQ)

'

x (C-EA)]=0 . In the scalar case this reduces to BtQA=0 . which is given as a stability condition in Clarke and Gawthrop (1975) . If Q=0, then the minimum-variance regulator is obtained and the stability is determined by detB=0, since det(EB) det [A+B(EB) - ' ( C-EA )]=det B det C . Thus the overall system is unstable, if B is nonminimum phase . If Q±O, it is possible to stabilize the overall system in certain cases by a proper choice of Q . Some qualitative observations can be made which are analogues of the scalar ease (Clarke and Gawthrop, 1975) . Say Q=AI . If A is openloop unstable but B is minimum phase, then according to the above analysis stable closed-loop system results, if 1 is small enough . If A is openloop stable and B is nonminimum phase, then for a stable closed-loop system, 2 should be large enough . For a large i., det (EB+Ci.) c det det[A+B(EBrC2) '(C-EA)] detA and stability is ensured . An exact stability analysis is fairly tedious . This can be accomplished e .g . by using Rosenbrock's (1975) graphical stability criteria for multivariable systems described by difference equations . These could further be used for determining the value of Q to obtain a good response and a stable overall plant .

+kl J 3 . SELF-TUNING CONTROLLER

A(i ') F(q

C

q

B(q ') ')

E(q -' )B(q ')+C(q

` )Q(q

0

-k-1

0

y(I+k)

p C(q-') I 0

uft) (t+k)

=

0 -'(t+ k)

Consider now controlling the system (1), when the system parameters are unknown . The basic idea in self-tuning is to fix the controller structure to be the one that would be obtained if the system parameters were known . Then, instead of trying to identify the system parameters, the control parameters are recursively identified .



A . multivariable self-tuning controller Here, in place of the system (1) and the cost function (3), the equivalent system of (10) and (8) is used : C(q-')¢*(t+kl t)=P(q - ' )y(t) +O(q-')u(t)+A(q-')w(t)+5=0

(12)

¢(t+k)=4*(t+klt)+e(t+k)

(13)

Consider first the case C(q - )=I . It follows easily that C(q - ')=I (Borisson, 1975) . Equations (12) and (13) can then be written as 4i*(t+klt)=F(q-')y(t)+G(q-')u(t) +l(q-')w(t)+6=0,

Here $ is the exponential forgetting factor, the values of which should be between 0 .9 < Jf <<-1 (cf. Section 5) . Computationally it is more advantageous to use the square-root algorithm (e .g . Peterka, 1975) for updating the parameters . Rounding errors occur, especially in computers (or microprocessors) with short word length, making it possible for the positive definite P-matrix to lose this property . If the dimension of x(t) is denoted by p, the recursive least-squares algorithm (19)-(21) in the square-root form is given as : B, (t+ 1)=A1(t)+K(t)[4,(t)

(14)

-x(t-k)U1(t)], i=l, . . .,m,

i(t+k)=P(q-')y(t)+G(q_')u(t) +H(q-')w(t)+d+t(t+k).

355

(15)

(19)

K(t)=g t " ap,

(22)

P(t+l)=S(t+l)S'(t+l),

(23)

Defining the data vector x(t) by X(t) :=[yT(t),yT(t-t), . . ., UT(t),UT(t-1). . . . ; wT(t),WT(t-1) . . . . ; 1] (16)

where

and the parameter matrix O by (dropping -)

S(t+ 1) ;) =aj _, (S(t),1 _i)/a)i,j=1, . . .,p

-fs,,

where the column vector 0, is 0

8

{0 n

o

o,=(a; ,+f;) 1!2 , i

1

.f;=~ S(t)i)x(t-k) ;, j=1, . . .,p, i=,

1

hi°,, . . .,h°„h ;,, . . .,hm, . . . ;Yi]T,

m,

S(t)ik,fk,, k, =i 0, i>l

(15) can be written componentwise as O ;(t+k)=x(t)9;+e ;(t+ k),

(17)

where the components of x(t) are uncorrelated with a i( t+k). The control u(t) is computed from (14) to be

Go u(t)=-

Z Fiy(t-i)+ L Giu(t-i)

l no

iz1 + I H iw(t-0+71(18) 1O

The control parameters 0 are estimated by a standard recursive least-squares algorithm in the manner indicated in Borisson (1975) : 6 i ( t+

1)=B; (t)

j=l, . . .,p.

1=1, . . .,m,

(19)

x[l+x(t-k)P(t)xT(t-k)]-'

(20)

K(t)=P(t)xT(t-k)

Algorithm (P=1, Q=AI, A is a constant tun-

ing parameter), 1. 2. 3. 4.

Read new output y(t), setpoint w(t) . Compute ¢(t)=y(t)-Rw(t-k)+Au(t-k) . Form data vector x(t-k) as in (16) . Update B ; by recursive least-squares algorithm in square-root form : (19), (22), and (23) . 5 . Generate new control from (18) :

+

P(t+l)={P(t)-K(t) x[1+x(t-k)P(t)xT(t-k)]KT(t)}/(3 .

and ( );j denotes the ijth element of the matrix .

Go ut)=-

+K (t)[¢ i (t)-x (t - k )B i (t )],

t=1,2, . . ..1,

I Fiy(t - i) 1 i?0 G i u(t-i)+ Y H;w(t-i)+9

6 . Set t=t+1 and go to 1 . (21)

It was previously assumed that C=I . If this is



356

H . N . Koivo

not the case, then using (12) in (13) yields (p ;(t+k)=x(t)B ;+a ;(t+k) + (I - c11 (q - ' ))¢* (t + k I t) -Ec . .(q-')(b*(t+klt)

(24)

indicating, as in Clarke and Gawthrop (1975), that ¢* is correlated with x . The least-squares estimates are then biased, but since the control law sets O* equal to zero, the last two terms in (24) disappear . 4 . CONSTANT REFERENCE VALUES

Constant reference values occur frequently in process control, when e .g . the system is changed from one set point to another . The derived controller structure (18) easily handles this special situation . If the steady-state error is to be removed, an integrator can be introduced by using cost function I z . In that case Q=2(1 -q - 'V, where only one tuning parameter i. is chosen for simplicity . If P=1, then the only point of departure in the derived algorithm is in step 2. This would now read : 2 . Compute O(t)=y(t)-Rw(t-k)+i.(u(t--k) -u(t-k-1)) .

Another way of introducing an integrator in the loop is given by Wittenmark (1973) . The multivariable case does not change the situation essentially . Everywhere where y(r) occurs, Wittenmark uses y(t)-w(t), and instead of u(t), he uses u(t) -u(t-1), if the system does not have integrators . 5 . SIMULATION RESULTS

Example 1 . A model for a head-box of a paper machine was used in Borisson (1979) to illustrate the performance of his self-tuning regulator . Here the same example is treated, when the system is subject to step-changes . The system model is y(t)+A,y(t-1)=B0u(t-1) i- (t).

In Fig . 1 system outputs, control signals, and computed parameter estimates are shown, when step-changes take place in the reference signal and the modified self-tuning minimum-variance regulator is used . The only change in the basic self-tuning regulator is that y is replaced with Ay=y-w . In Fig . 2 the results of the same simulation are presented using the proposed self tuning controller . Comparing Figs . 1 and 2 one can readily see that the overall tracking properties of both controllers are quite similar . The effect of estimating H-parameters is, however, significant . when both inputs are subject to simultaneous stepchanges at T=1000s . The output Y2 of the minimum variance controller is not able to follow the reference signal, while the suggested controller works quite satisfactorily . In addition, fluctuation in control signals is smaller in Fig . 2 . This is due to the penalty on the control . The value A=0.01 was used in Fig . 2 . Figure 3 shows the effect of i., when the Hparameters are not estimated, that is, in step 5 of the algorithm H,=-1 . The value of d was 0 .1 . There is a small, but noticeable decrease in the variance of control signals . In Fig . 4, the proposed controller is used with 1=0.1 . Fluctuations in control signals are smaller when compared with those in Fig . 3 . but the outputs are not as good as in Fig . 2 where d=0 .01 was used . Finally, in Fig . 5 the input reference signal W2 is a square-wave with a period of 400s . The proposed controller with A=0 .01 was used . Throughout simulation the limits of both control signals were +10 and Gaussian random noise with zero mean value and covariance matrix rt =diag{Q1,0.1 was used . Recursive parameter estimation was done with the square-root algorithm and the convergence of parameters is very fast in all cases . Because the process model was time-invariant, the forgetting factor p had the value 1 . A nonminimum-phase system is

Example 2 .

studied next : y(t)=A,y(t-1)+B0u(t-1)+B,u(t-2)

4 ~(t)-

where

where 0. 9

.8 80512x1000 .99101 .80610 -0 .77089 0 .89889 B0- [19 .390

A,=

1-0

.

0 .2]'

B0= [0 .25 0 .2]

59329x10 . 4 -

B,=[0

0 .88052

01, E{C(t)~T(i))=diag{0 .1,0 .1 ;

or Other details of the system are given in Borisson (1975) .

A(q -r )y (t)=B(q-' )u (t-1)+~(1),

40-0o

0-00

10 .00 TIME

80 .00 TIME

120 .00

•1 0 1

120 .00

•1 0 1

160 .00

160 .00

200 .00

200 .00

N 3

m

0

0

40 .00

40 .00

80-00 TIME

80 00 TIME

•1 0 1

120 .00

120-00 -10 1

160 .00

160 .00

200 .00

200 00

^

0

O 0

4

O

9- ' d

w

w

U

6 -

w £ <2

w

0 N a-

0

40 .00

40 .00

40-00

80 .00 TIME

80 .00 TIME

80 .00 TIME

120 .00 -10 1

120 .00 -10 1

120 00 -10 1

160 .00

160 .00

160 00

and 11-parameters .

FIG . I . Example t with the modified minimum-variance controller l . =0) . (a) Output y„ reference signal w„ and control u, . (b) Output y 2 , reference signal w e , and control u 2 . (c) F-, G-,

10 .00

0 .00

O O 0



200 .00

200 .00

200 00

I

120 .00 -10 1

120-00 •1 0 1

160 . D0

160 .00

200 .00

200 .00

80 .00 TIME

80 00 TIME

40,00

40 00

00

120 00 •1 0 1

120 00 •1 0 1

160-00

160 00

200 .00

200 .00

∎rIIVIw"iwuWlYuw

0 00

0

ID

w

o

d '

C

0 No w n w = oo -

0 .00

a1 00 0

0 No W. Iw r0

w^

N ~-

40 .00

40 00

40 .00

80 .00 TIME

80 .00 TIME

80 .00 TIME

120 00 •l 0'

•1 01

120 00

120 .00 01

•1

160-00

160 00

160 .00

tlc .' . Example I with the proposed control tz-0011- 1 aI Output r . reference ,i 1a] a I , and control u . . 161 Output r, .reference signal n„ and control n, . tcl F- . G-. and H-parameter.

80 .00 TIME

40 .00

0 .00

1

80 .00 TIME

4000

I

0 .Do

0 0

O

00



1 200 .00

1 200 00

200 .00

40,00

0

1

TIME

80100

TIME

80100

•1 0 1

120 .00 1

•1 0 1

120 .00 r

160 .00 1

160 .00 I

200 .00

200-00

Flu . 3 . Example L with the proposed control (i,=0 .1) and

40 .00

0 .00

O

O

I moo °



00

400o

4000

TIME

8o .oo

TIME

80 .00

•1 0 1

120 00

120 00 1

200 00

160 .0o ~ 2oo-oo

60 00

0

'0 .00

0

8-I

t ° o

w

In 0

1 0 .00

40 .00

40 .00

40 .00

N I r r

O

0_ I Z -

L

w ow w

1

TIME

80 .00

TIME

80100

TIME

80 .00

120 .00

•1 o'

120 .00

•1 0 1

120 .00

•1 0'

160 00

160 .00

16000

200 00

200 00

200-00

=-I . (a) Output y„ preference signal w„ and control u, . (b) Output r z , reference signal w -,, and control u z . (e) F- and Gparameters .

N 3

0 .00

N

0

u

b

w

00

0

00

0 .00

0

r 00

Ic. . 1 .

1

120 .00 -10 1

120 .00 •I 0'

1

11 ,Ih tile {vo1n e

80 .00 TIME

8o .00 TIME

Es ,nnp ..~I c ':

4D .00

40 .00



160 00

160 .00

.

v- 1) 1 ;.

200 .00

200 .00

I ' ll

00

0

m

t -.

0

Icmncc

,I

40 .00

40 .00

120 00 •1 0'

120 .00 -10 1

-2nul n, . and lontn?I u,

80 .00 TIME

80'00 TIME

0 0 .00

r-i 40 .00

1 40 .00

4000 .

I

1

8000 TIME

80 .00 TIME

80,00 , T I ME

I

I 120 .00 •1 0 1

1

120,00

20 .00

0 1

•1 0

.

1

1

200 .00

200 . DD

1 160 .00 , 200 .00

160 .00

160 .00

rye fence signal ii -. .and eomr0l !,_._ IC) I . G-- and H-parameters .

200 .00

200 .00

!'o! Oulput 1

160 .00

160 .00

0-

0 0

w t ¢ .-

In

m

< o.

L

w

cn 0

0

0

40,00

40 .00

40 .00

TIME

80 .00

TIME

80-t10

TIME

80 00

120 .00

•1 0 1

120 .00

•1 0 1

160 .00

160 .00

200 .00

200 .00

120 00 ~ 160 .00 ~ 200 .00

•1 0 1

N 3

0 0 00 .00 40 .00 80 00

TIME

120-00

•1 0 1

160 .00

200 .00

W

0

s .ca

w

n

0

n

0 N 0 0_

W

0 N o K W

0 0

x -

,o o=

w

w i°

0 N 0-

I

40 .00

40 .00

40 .00

TIME

80 .00

TIME

80-00

TIME

80 .00

120 .00

•1 0 1

120 .00

•1 0 1

120 .00

•1 0 1

160 00

160 .00

160 .00

Flu- 5 . Example 1 with the proposed control (d=0 .01). (a) Output y„ reference signal w„ and control u, . (b) Output p i , reference signal w 2 , and control ui . (c) F-, G-, and Hparameters .

00

0 .00

0

'0 .00

0 0 0



200 00

200 .00

200 .00

w

0

-I

0 0

n

a

S

F

0

0

c



40 .00

TIME

40 .00

TIME

50 .00

•1 0 1

60 .00

•1 0 1

80 .00

.00 60

1 100 .00

100 . CC

w r

I

2 0 .00

I 20 .00

TIME

40-00

TIME

4C . C O

I

•1 0 1

60 .00

•1 0'

1 60 .00

80 .00

I 80 .00

I

C0

100 .00

1 100 .00

Fic . 6 . Control of the unstable and nonminImum-phase system of example 2 with the proposed control ( .-=I ) . (a) Output reference signal n •„ and control u 2 . (c) F-, G-, and H-parameters .

3 .00

20 .00

0 .30

F-

V

G L.

W F

1

.

20 .GC

2 ; .00

_J

2C 30

I

reference signal

0, C.0

'0 03

X 0 .03

00 n

Ic l ,

•1 0 1

60-00

•1 0

60 .00

•1 0 1

I 50 .00

60 .00

80 .00

8000

and control u, . (h) Output ?'_,

TIME

40 .00

TIME

40 .00

TIME

'000

100 .00

130 .00

'100 .00

120 .00

160 .00

200 .00

0 .00

0 0

TIME

80 .00

•1 0'

120 .00

•1 0 1

N

0

160 .00

200 .00

00

0 .00

0 40 .00

80 .00

40 .00

TIME

80 .00

TIME

120 .00

•1 0 1

120 00

•1 0 1

160 .00

200 .00

160 00

200 .00

0

0 .00

0

W ~ WW = O ~ O C

0 N o-

X 0 .00

n 0-

< 0-

C O

0 N O~

'

40 .00

40 .00

40'00

'

80'00

TIME

80 .00

TIME

80 .00

TIME

'

120 .00 '

•I 0 t

120 .00

•1 0'

120 00

•1 0 1

160 .00

160 .00

160 .00 '

200 .00

200 .00

200 .00

and control u, . (c) F-, G-, and H-parameters .

FIG . 7 . Control of an unstable system of example 3 with the proposed control with k=2 ( ;.=0 .01) . (a) Output y 1 , reference signal w„ and control u, . (b) Output ye , reference signal w2,

40 .00

TIME

'0 .00

80 .00

0

40 .00

'0 .00

a i 0-

<-

= C 0o

W

'0. W °1Y 0

0

0

7 G n

Y

8



0 .00

0



80 .00 TIMF

120 .00 -10 ,

160,00

FIr'- . 6 . Cm,trol of an unstable s'stern of ex

40 .00

0 .oo

<0 .00

40 .00

80 .00 TIME

8o .00 TIME

,20 .00 -10 1

120 .00 ' -l01

1 ,60 .00

,60 .00

200 .00

200 .00

0

0 .00

0 .o0

a. LU

¢ 0

w

0' 0 0

0

40 .00

40 .110

40 .00

so .or TIME

eo .oo TIME

80 00 TIME

i2c 00 -10'

120 .00 -10 1

120 .00 -ID'

uple T u,th the proposed control in an integrator form l, -(1,011 . (a) Output r reference signal n, . and colulol u, Ihl Output signal u„ and control u, . 1cl F- . G- . and H-parameters .

200 .00

a o

0

K

w

w

0 cn 0 D

r, .

1 200-00

2110 .00

200 .00

reference

160 00

i so .oc

160 .00



365

A . multivariable self-tuning controller where .9q - ' A(q')_ 1-0 I O.Sq - ' B( )

0 .5q - ' 1-0.2y - ` '

1 [0.2+q - ' 0.25 0 .2+q -

After T=200s the input reference signals were : WI square wave with an amplitude 1 .0 and a period of 400s ; W2 square wave with amplitude 0 .5 and the same period as W l . 6 . CONCLUSIONS

The zeros of detA(a) are x,=0.86 and a,-16.58 . Because x, is inside the unit circle, the open-loop system is unstable . The system is also of nonminimum phase, as both zeros, x,=-0 .7 and )t, -03, of del B(a) are inside the unit circle . In this case, the choice of d was crucial, since there is a lower limit for A for which the closedloop system is stable . Values of ).?I turned out to be suitable. In Fig . 6 the simulation results are presented . Example 3 . The effect of the delay k was studied by Borisson (1979) with the example :

This paper has extended the self-tuning controller of Clarke and Gawthrop (1975) into a multivariable case, where the system has the same number of inputs as outputs . In this manner Borisson's (1979) minimum variance selftuning regulator has been broadened to include nonminimum-phase systems. In addition, the developed self-tuning controller is able to track time-varying reference signals . Simulation examples indicate that the derived controller has the desired property of limiting the control signal significantly compared with the minimum variance controller . Acknowledgement-I would like to thank Mr. J . Tanttu for excellent computer programming . This research has been supported in part by the Finnish Academy .

y(t)+A,y(t-I)=u(t-2)+~(t)+C,~(t-1), where

REFERENCES

A,[

0.5

-0.2]'

C`

[

0.2

-0.81

E{~(t)eT(t)}=diag {0.1, 0 .1} . Here det C(a) has its zeros outside the unit circle while det A (a) has zeroes a, ~ -16 .6 and x 2 =0 .86 whence the open loop system is unstable . This time the parameters of the model

y(t)+F,y(t-t)+Gau(t-2) +G,u(t-3)+H,w(I-2)=0 were estimated and control law (18) was used, as usual . In Fig. 7 0(t)=y(t)-w(t-2)+J.u(t-2) with 2=0 .01 was employed, while in Fig . 8 an integrator was introduced to the system by using 4 (t)=y(t)-w(t-2)+ ;,[u(t-2)-u(t-3)]

with A=0 .01 . The difference between these two control strategies is not obvious, but in the beginning, the integrator version tends to oscillate more . However, after some 50 steps, the output and control signals are quite good . This example also displays that the model in the estimation can differ from the actual model of the system .

Alam, M . A . and K . K . Burhardt (1979) . Further work on self-tuning regulator. Preprints of the /8th IEEE Cony. on Decision and Control . Fort Lauderdale, Florida . Astrom, K . J ., U . Borisson, L . Ljung and B . Wittenmark (1977) . Theory and applications of self-tuning regulators . Automari ( a 13, 457 476 . Astrom . K . J ., B . Westerberg and B. Wittenmark 11978) . Selftuning controllers based on pole-placement design . Report TFRT-3148, Dept . of Automatic Control . Lund Inst . of tech., Lund, Sweden . Borisson. U . (1975) . Self-tuning regulators--industrial application and multivariable theory. Report 7513, Dept . of Automatic Control, Lund Inst. of Tech ., Land, Sweden . Borisson, U . (1979) . Self-tuning regulators for a class of multivariable systems . Automarica 15, 209-215 . Clarke, D. W . and P . J . Gawthrop (1975). Self-tuning controller . Proc . IEE 122, 929-934. Davison, E. J . (1976) . Multivariable tuning regulators : The feedforward and robust control of a general servomechanism problem . IEEE Trans . Aut. Control AC-21, 35-47 . Gantmacher, F.R . (1959). The Theory of Matrices. Vol . 1 . Chelsea Pub] . Co ., New York . Gawthrop, P . J . (1977) . Some interpretations of the self-tuning controller . Proe. [EE 124, 889-894. Keviczky, L . and L Hetthessy (1977) . Self-toning minimum variance control of mimo discrete time systems. Automatic Control Theory Appl. 5, 11-17 . Keviczky, L ., J . Hetthessy. M . Hilger and J . Kolostori (1978). Self-tuning adaptive control of cement raw material blending . Automatica 14, 525-532. Koivo, H . N . (1979) . A multivariable self-tuning regulator . Optimiration Days 1979, McGill Univ ., Montreal . Canada . Moore, B . C . (1979) . On-line tuning of multivariable PIcontrollers using principal component analysis : Preliminary results . Systems Control Report No . 7905 . Dept . of Elect . Eng ., University of Toronto, Toronto, Canada . Morris, A . J ., T . P . Fenton and Y . Nazer (1977) . Application of self-tuning regulators to the control of chemical processes . Digit a l Computer Applications to Process Control . (Edited by Van Nauta Lemke and Verbruggen) . IFAC and North-Holland Publishing Co ., Amsterdam, pp . 447-455 . Penttinen . J. and H . N . Koivo (1980). Multivariable tuning regulators for unknown systems . .4utomarica 16, 393-398 .



H . N . Koivo

366

Peterka, V . (1975). A square root filter for real-time multivariable regression . Kybernefika 11, 53-67 . Peterka, V . and K . J . Astrom (1973) . Control of multivariable systems with unknown but constant parameters . Preprints of the 3rd IFAC Symposium on Identification and Systems Parameter Estimation . The Hague, Netherlands, 535-544 . Rosenbrock . H . H . (1970). State-space and Multivariable Theory . Chap . 6, sect . 6 .2 . Nelson, London . Wellstead, P. E., D. Prager, P . Zankcr and J . M . Edmunds (1978). Self-tuning pole zero assignment regulators. Control Systems Centre, Report 404, University of Manchester Inst . of Tech . Manchester, England. Wittenmark, B . (1973) . A self-tuning regulator . Report 7311, Dept . of Automatic Control, Lund Inst. of Tech ., Lund, Sweden .

[he idea is now to separate the disturbance into two parts, one representing future disturbances and the other one representing disturbances that occurred up to and including time i . In order to do that one has to consider the inverse of C(q - ' ) or [C(q

')]

adj C (q -' l = detC(q - '1

Since det C(q - ' I is an nth order polynomial and adj C;(q - ' 1 is of equal or lesser degree than det C(q - ' ), it is clear that re(q-')]-'=.I+q-'(')+q It now follows from IA .3) that

APPENDIX k-step-ahead predictor By Euclidean algorithm applied to polynomial matrices

ylr+k)=C '(q ')[F'(q

~)y(f)+E'(q ')B(q ')uit)

.F" (IW]+E'(q-')C(r-k)

('(q-')= A(q - ' )E (q - ' ) + q - " F' (q - ' ).

=C" '(q ')[E(q

where

+E'(1M]+4(r

I)v(f)+E'(q ')B(9 ')u(t) k)I_-+Ej_,g(t+1)

IAA)

E'(q ')=1+E',q-' + . . .+F _ 1 q' - ', F'(4

'1=Fo+F~q-'+ ._+Fp , q' - ' .

and thus the optimal k-step-ahead predictor is given by

Furthermore, as in Borisson (19791, there exists nonunique matrices f and F such that

y'(t+klt)=C-'(9-')tP(q 'WO 1 .45)

+E'tq-')B(q-')u(t)+y]

E'(q-')F'(q where detE'(q-')=detE'(q-') and E'10)=1 In addition define a polynomial matrix C(q - ' ) : C(q ') :=E'(q 'IA(q-')+q-'E'(q ')

where y : =E'(1)d . Because of the form of the cost function, y- it +1 It) also is needed for j<=k . The only point of difference in the course of derivation appears in (A .2). where the argument of u will be t+j- k instead oft. The j-step-ahead predictor is then given by

for which C

(A .1)

C(q-')E'(q-')=E'(q ') C(q ').

y' It +j] r) =3 System (1) can be written as

'(q '1CEi(q )Y(T) +C;(q-')u(i+j -k)-y,] . 0<)

y(f+,j).

A(q-')v(t+k)=B(q ')u(t)+C(q ')f(t+k)+d .

k,

(A-61

j<= o.

(A .2) where P , Cr;, and y, are obtained at step j from

Multiplying this by E'(q - ') from the right and using (A .1) results in Cl4 ')y(t+k)=F'(q

')y(r)+E'(q ')B(q

'N(t1

'(q - ')E'(q '1 It+k)+E'(q-')d .(A .3)

e'(9')E,(1-')A(q ')+q 'F"(q ;(q ')=Ej(q 'IB(q ~), 43 y,-E; f 1)d .

), (A7J