Robust Stability of a Modified Clarke-Gawthrop Self-Tuning Controller

Copyright © IFAC 12th Triennial World Congress. Sydney. Australia. 1993

ROBUST STABILITY OF A MODIFIED CLARKEGAWTHROP SELF-TUNING CONTROLLER Xing Yuan Gu and Cheng Shao Department of Automatic Control. Northeast University of Technology. Shenyang. Liaoning. 110006. PRC

Abstract. Tn the paper, the Clarke-Gawthrop type self-tuning controller is modified and shown to be robust stable with respect to unmodeled plant uncertainties and bounded disturbances. The self-tuning controller is proposed here by the two steps: the first, an optimal control law i~ derived by means of minimizing a quadratic cost function of the Clarke-Gawthrop type; the second, the optimal control law is modified by introducing an estimate of the modeling error as a feedback. The robustnes s results are derived by neither requiring too much a priori knowledge of the plant parameters, nor using any assumptions on the adaptive signals. Key Words. Self-tuning, robustness, stability, unmodeled dynamics.

1. INTRODUCTION

crucial assumptions that the time delay is minimal (k=l) and the quantity aCt) related to the plant inputs and outputs is small, were required by the proposed method. Recently the robust stability of the adaptive pole placement algorithms for the unmode led dynamics and bounded disturbance case was investigated in Middleton, et al. (1988, 1989), where, however,a convex region including the model parameters and an upper bound function of the ununmodeled dynamics needed to be known.

Adaptive control issues for linear systems with constant but unknown parameters have been widely discussed in the literature. An important class of discrete-time adaptive algorithms for the linear systems are the self-tuning regulators with minimum variance strategy (Astrom and Wittenmark,1973) Since then Clarke and Gawthrop (1975; 1979) developed the self-tuning controller which penalizes not only output error as does the self-tuning regulator, but also the excessive control fluctuation, and it allows the system to be non-minimum phase and the set point to change. The controller is based on a cost function with a priori chosen constant weights. This gives an advantage over the adaptive pole placement controllers that the control law is explicit and does not require to solve the Bezoutian equations on-line, thus avoiding the associated ill-conditioning prohlems, which might arise in adaptive pole placement algorithms (Kumar and Moore, 1983; Goodwin and Sin, 1984; etc.).

This paper is to exam the robust stability problem of the Clarke-Gawthrop type self-tuning controller with respect to unmodeled plant uncertainties and bounded disturban ces . The plant considered here is a class of high-order linear systems with unknown and perturbed parameters. By means of minimizing a quadratic cost function of the Clarke-Gawthrop type, and employing a modified least squares estimator with a relative deadzone, a new self-tuning controller is propo sed in the paper. It is shown that the self-tuning controller provides robust stability with respect to the high-order unmodeled dynamics and bounded disturbances under rather relaxed condit~on. The robustness results illustrate that the Clarke-Gawthrop type self-tuning controller is both suitable to the nonminimum phase systems and insensitive to small plant changes.

A first stability analysis for self-tuning controllers of the Clarke-Gawthrop type was performed in Gawthrop (1980) and later in Goodwin et al.(1981), where the stochastic disturbances of a moving average process were considered and the assumptions of persistent excitation or strictly pos itive realness were employed. The deterministic convergence of a Clarke-Gawthrop type self-tuning controller was given in Tsiligiannis and Svoronos (1986) for the disturbance-free case . However, for the practical applicability of adaptive algorithms the stability established in the ideal ca ses is not sufficient . It is known that a stable adaptive control algorithm is unneces sarily robu s t s table (Rohrs et al.,1982). Therefore, lots of att ention has been recently attracted to the robust stability issues of adaptive control systems. A basic attempt was made in Shao (1991) where the robust stability of the Clarke-Gawthrop type self-tuning controller was established for bounded disturbance case. Although Gawthrop and Lim (1982) gave the robust stability of the self-tuning controller in the presence of plant non-linearities, unmodeled disturbances and plant-model order mismatCh, some

2. THE PLANT DESCRIPTION Tn most applications the design model is of lower order than the plant because of some unmodeled dynamics present in the plant. In particular, the plant, including mod e ling uncertainties, is timeinvariant,finite dimensional and can be denoted by - 1 _

G(q

)-

B(l +/lB' ) A(l+/lA , )

(la)

y( t)=G(q - l) 1I( t-d)+v( t)

(lb)

where yand 11 ar e the scalar output and input,respec tively. j) is the bounded disturbances, d ~ 1 is the plant delay;A, A' ,B, B'are polynomials of unit

253

delay operator

q-l

of orders

respectively.

Jj ~

0 is a small singular perturba-

n , n " A A

n

B

and nB"

prerequisite for adaptive control. Assumption A2 provides necessary parameter structure of the reduced-order model.

tion scalar. It is seen below that the existence of ~ leads to the high-order unmodeled dynamics . From (1) one obtains

-f u( t-d)+TJ

y( t)=

B'-A ' B

A

~ 1+~A'

TJ p ( tl=

p

(

t)

u( t-d)+IJ( t)

(2)

Regarding the unmodeled dynamics ing knowledge is available.

only the follow-

(3)

Assumpti on A3 . An upper bound ~~ of ~ is known.

Using (2) gives 3. A NEW SELF-TUNING CONTROLLER

(4)

Then a singular perturbation from to the reduced-ord er mode l

!

y( t)=

~>O

to

~O

The objective here i s to propose a self-tuning controller on the basis of the reduced-order model,or the knowl edge of A and 8, but to apply it to the plant (1) such that the closed loop system tracks the desired output and the robust stability is esta blished with respect to unmodeled dynamics and bounded disturbances.

leads (5)

u( t-d)+IJ( t)

Assume the designer to be given only the reducedorder model (5), and thi s without knowledge of the coefficients of A and B.Thu s the modeling error TJ

Let P be arbitrary monic polynomial in q-1 0f order np' Introduce

p

includes the high-order unmodeled dynamics related to y. The following results establish the fact that ~p is overbounded r elat ively to the output y.

(9)

where the orders nr and Lemma 1 . Let C(q-1) be a polynomial in q-1 with finite order nc' Then for arbitrary o~(O,I) there exists ~ >0 such that C (z - 1)=I+~C(z-1)_0 for all ~E(O,

F and G sat isfy n = F

d-l; nG=nA-l or np which is the greater, and

F is

moni c also. Then from (2) one obtains

o ~ ~E (O, ~] and Izl>o, i.e. C (q-1) is strictly Huro ~

witz uniformly in

nGof

Py( t+d)=Gy( t)+8FIl(

~o] .

t)+AFll p (

t+d)

(10)

whi ch can be written in a regressive form

nc Proof. Let C(Z-1)= I c . z - i , then for all Izl>o

<1>( t+d)=eTX( t)+q( t+d)

(11 )

1

j:O

where q,( t+d)=Py( t+d); e is parameter vlfctor compor.ed of the coefficient~; of G and BF; X (t)=(y( t) ... ,y( t-n ) ,Il(t), ... ,Il(t-n -d+1); TI( t+d) =AFl"lp ( t). G B

nc OtI-~II Ci l o

If (q( t) 1 i s a whit e noi se

~equence,an control law con be obtained by meanr. of mlnlmzing the following quadrilti c cost function of the Clarke-Gawthrop type with respect to 11(t)

- i

i=O

nc

Take

~o= 1/(1+ I IC 1 a-i) i j.Q

the r esu lt follows.

o

J=E{ (P(y( t+d) - y* ( t+d» ]7.+ (0' (u( t) -lI( t-l))]

The relative boundedne ss of TJpi s then established. Lemma 2. There exir,t consta nts independent of Ot 0

~,

k~,

21

(12)

where P and 0' are conr.tant weighting polynomials in q -1, y* (t) i~; the bounded de~i red output. In fHct, from (11) one obtains

k;>O, which are

such that f or suf ficiently small

T (P(1'( t+ (/- y* ( t+ d) )] 7.= (X ( t)

~

e- Py* ( t+ d)] 2+

T

(6)

+2(X (t)O-Py*(t+d)]TJ(t+d)+r,(t+(/)

2

and 50 Proof.

Define B'-A'

g( t)= 1+~B'

J=E{ (XT (t) O-Py* (t+d) ] 2+ (0' (u( t)-u(

Using lemma lone obtains

Minimi z ing J as in Gawthrop (1980) giver. that there exi5ts

~

o >0

xT(oe+o(u(t)-u(t-l»=Py*(t+d)

such that for all ~c(O,~o]' R~=I+~8' is strictly Hurwitz. Then referring to Shao (1991) it follows that. for all

~c(O,

ma x 1y("t) 1+ k

(8)

O~l.~t

where k denotes generically a positive real constant which is independent of ~. Using (4) and (8) leads to (6).

estimate OU) and then app lying (14) to (11) give s T

P( y( t+ d- y* ( t+ (/) ) = X

[J

Assumption A2 .

nA , n and B

11 and

(

t) (e-O ( t) )-

-O( u(

In summary the plant to be controlled is a system with the high-order unmod e l ed dynamics, and only a reduced-order model i s available to the designer. The following assumptions are made on A and B. Ass11mption Al . A is a monic polynomial, are relatively prime.

(14)

where O=q~O'lbo; q~ nnd b o ore the leading coefficients of 0' and 8, respectively. Since TJ(t) includes unmodeled dynamic s , the control law (14) will not be suitable for our purpose. Let us further investigate what are met if (14) is used to the self-tuning case. Replacing e in (14) by it s

~()]

1 g( t) 1~ k

t-l»] G1+lJTj (13)

(7)

y( t)

t) -lI( t-l)

+rl(

t+(/)

(15)

Thus it is observed that because of TJ the tracking error e( t)=y( t)-y*( t) will hardly be ensured to tend to zero even though the parameter estimates

8

e( t) converu~ to their true value s. To remove the e ff ect of unmodel ed dynamics the Clarke-Gawthrop type control law (11) is modified by introducing an estimate of TJ

d are known.

Remark 1. Assumption Al implie s that the reducedorder model is controllable, this is an obvious

~(t)=4>( tl-e(

254

t) T X( t-d)

(16)

and Soderstram, 1983) it tollows that eqn.(18d) is equivalent to

a new selt-tuning control law IS then gIven by tl e( tl +Q( u( t) -lIt t-ll) =py* ( t+d) -~( t)

XT (

For parameter estimation the following least squares s cheme i s employed e( t)=e(

(laa)

t-l) T X(

(l8b)

t-d)

L( t)=K( t-2)X( t-d) 11 a+ XT ( t-d)K( t-2)X( t-d»)

t-ll=K(

t-?)-).( t)L(

t) [a+XT(

K(

t-ll-I=K(

t-2)

T

-1+

A( t)X( t-d)X (t-d) a+ (l-A( tll XT ( t-d)K( t-2) X(

modified

t-l )+).( tl L( t) e ( tl

e ( t)=4>( t)-e(

K(

(l7)

t - d)K( t-2)X( t-d»)L(

V( t)-V(

t) T

/4

+ (1-A(

0<0'0<1/7 ,

a and

parameten;, (K(tll is a arbitrary initial K( - I»O.

matrix

sequence

with

are independent of JJ such that for all JJE:[O, JJ*] ITJ(t) Is JJk' max Iy(s) I+k' lossst 0

2. The parameter estimation is of the r e lative deadzone approach which is widely used for the solution of the robust adaptive control issues (Kreisselmeier and Anderson,1986; Middleton et al. 1988; 1989). Howe ver, using the relative deadzone approach the parameter estimates will not converge to their true values in general, which probably results in the potential existence of steady state errors of tracking the desired output. No results are given to solve this problem. In this paper, it is shown theoretically that the modified ClarkeGawthrop type self-tuning controller ensures the average tracking error to decay to zero, which implies that the proposed algorithm is capable of removing various steady state e rrors. P and

Q

~

If

(24) ~ max(k~,

in (18e) is chosen such that

k~},

then from (18e) when A(t)#Q it follows that V(t) is bounded and non-increasing sequence, and thus converges. In view of (24) and (18e) the result is easily derived. (ii) Using (18a)-(18d),(23) and (19) leads to (20) readily. (iii) This is straightforward from the boundedness of V( t) and (22) . o Lemma 4. For sufficiently small error and input dynamics satisfy

ar e chosen such that

f(q-I)=P(q-I)B(q-I)+O(q-I)A(q-I),

(23)

Using lemma 4 one obtains that t hre exist sufficiently small JJ*>O and constants k~, k~>O which

Remark

Assumption M.

t) z/

t» XT ( t-d)K( t-2) X( t-d)j

xXT (t-d)K( t-2)X( t-d»)

( 18e)

are positive adjustable

~

t-l)

-A(t)(c(t)z/4 -TJ(t)z)/[a+(l-)'(t»x

oSs~;;t

'I, otherwise, '1 10 [0'0' 3(1-0'0)/4),

where

follows

T t)/3)X ( t-d)K( t-2)X( t-d)jc( T [a+X (t-d)K( t-2) X( t-d)] [a+

(l8d)

C

it

t) [a+(l-4>.(

-)A(

' i f lL(t) I <2/i(JJ* max ly(s)l+ll A( t) =

(22)

e(

Def ine V( t) =e( t) T K( t-l) -I t) , then from (21), (18a), (lac) and (22) that

(18c)

t-d)

(~(l _ q-I)Q)

(P&l-OA) e(

t)=B[lld£(

t)+(e( t-1)

~O,

the tracking

~(t-d» TX( t-d)]+(\( t) (25)

is stable, i.e.

(z)#O,

Izlsl.

(P&l-QA) u( t-d)=A[lldc( t)+(e( t-1)

Remark 3. This condition is basic in the ClarkeGawthrop type self-tuning controller (e.g . Clarke and Gawthrop, 1979; Gawthrop and Lim, 1982; Tsiligiannis and Svoronos, 1986; Shao, 1991). However, it i s made here on the reduced-order model, and the robust adaptive control issues with respect to high-order unmodeled dyn ami cs are investigated .

~(t-d» T x(

t-d)]+

+(\( t)

(26)

10. (t) Is C +C (JJ+(,l( tll max IIX("t-d) 11, (i=I,2)

(27)

1

1

oS'tst

2

where lld=l-q -d ,(,l( tl ->0 as t-J0 are constant, and Cz is independent of JJ. Proof .

Using (2) one obtains

4. ROBUST STABILITY ANALYSIS Ae(

To establish robust stability of the closed system the following lemmas are necessary.

lim t->

A( t) 1/2 e ( t)

---":"':"';'-'--~"-'---

a+XT(

where h( t) -> 0 as t Euclidean norm. e( t)

Pet t)

-><1>,

h(t)IIX(t-d)1I

11· 11 denotes the

=ll

c( t)+(e( t-l)

~(t-d»

+(e(t-d)

t-d) -Oll( t-d)+

~(t-d-lll T X(

t-2d)

(20)

o 1 (t)=O(ATJ p (t)

-Ay'(

tll+B(e(

t-d)

~(t-d-l)

T X(

•

(i) Let e(t)=e(t)-e. Substituting (Ill into (18b) one obtains

On the basis of the Matrix Inversion Lemma

•

oz( t)=P(Ay*( t)-ATJ p ( t»+A(e(

Proof.

t-l)

t-2d) (30)

vectorMultiplying (28) by P and (29) by -A, adding together results in (26) with

d)e(

(29)

Multiplying (28) by Q and (29) by B, and then adding together results in (25) with

is bounded.

c( t)=~( tl_XT (t -

T X(

(l9)

t-d)K( t - 2)X( t-d)

(ii) I (e(t-I)-e(t-dllTX(t-d) I s

(iii)

= 0

(28)

t-d)+ATJ p ( t) -Ay'( t)

Substituting (16) into (17) gives

Lemma 3. If ~ in (18e) is taken sufficiently large and JJ* is sufficiently small, then for all JJE:[O, JJ*) the parameter estimation scheme (18) satisfies

(i)

tl =Bu(

loop

and then T

t-d) ~(t-d-lll X( t-2d) (31)

From (3) one obtains

(2ll ATJp(t)= JJ

(Ljung

255

(B'-A' )B

I+JJA'

u(t-d)+Av(t)

(32)

Applying lemma 1 to

A~~l+~'

and referring to the

O'IZ

~

proof of lemma 2 it is easy to prove that 6 (t), i (i~l,2) defined in (30) and (31) satisfy (27). o

Lemma 5. Let I z( t) I. I v( t) " I a( t)' and 16( t) , be non-negative real scalar sequences and lim a(t)~O. tIf for arbitrary t ' ~O and all t~t ' there exist constants k" kz, k3>0 such that

(ii) For arbitrarily large N, using (29) gives

t·d

- Qu(N-d)+Qu(O)+

k +k max v(~)+k max 6(~) 4 St'~~~t 6t ' ~~~t

In this paper, it is shown that the useful ClarkeGawthrop self-tuning controller can be modified to give robust stability properties in the presence of the high-order unmodeled dynamics and bounded external disturbances . In addition, such a modification removes all steady state errors appearing in the adaptive control system.

Theorem 1. Subject to assumptions AI-A4 , provided ~*>O is sufficiently small, the self-tuning algorithm applied to plant (1) ensures that (i) y and u are bounded for arbitrary bounded initial conditions and all ~€[O, ~*l , (ii) the tracking error satisfies

6 . REPERENCES

" e( t) ~ 0 (35) lim I1f L "t.o Proof. (i) According to assumption A4 and referring to Shao (1991) , it follows from lemmas 4, 5 and (20) that there exist constants C" C2>0 such that for sufficiently small ~* and all ~[ 0, ~*l ~

C +C max IE(~) 1 ' 2o~~~t

AstrOm , X.J. and B. Wittenmark (1973). On selftuning regulators. Automatica, 9, 195-199. Clarke , D.W . and P.J. Gawthrop (1975) . Self-tuning controller . Proc. lEE, 122, 929-934. Clarke , D.W. and P.J. Gawthrop (1979) . Self-tuning control. Proc . lEE, 126, 633-640. Gawthrop, P.J. (1980) . On the stability and conve rgence of a s elf-tuning controller . Int. J . Control, 31, 973-998 . Gawthrop, P.J . and X. W. Lim (1982) . Robustness of self-tuning controllers . Proc. IEE-D,129,21-29. Goodwin, G. C., C. R. Johnson and X. S. Sin (1981) . Global convergence for adaptive one-step-ahead optimal controllers based on input matching. IEEE Trans., AC-26, 1269-1273. Goodwin, G.C.and X.S. Sin (1984) . Adaptive Filtering Prediction and Control . Pretice-Hall,Englewood Cliffs , NJ. Xreisselmei e r, G. and B.D . O. Anderson (1986). Robust model reference adaptive control. IEEE Trans., AC-31, 127-133 . Kumar, R. and J.B. Moore (1983) . On adaptive minimum variance r egulation for nonmlnlmum phase plants. Automatica , 18, 449-453 . Ljung, L. and T. SoderstrOm (1983). Theory and Practi ce of Recursive Identification .The MIT Press London. Middleton, R.H ., G. C. Goodwin, D. J. Hill and D.O. Mayne (1988). Design issues in adaptive control IEEE Trans., AC-33, 50- 58 . Middleton, R. H. , G. C. Goodwin and Y. Wang (1989) . On the robustness of adaptive controllers using relative deadzones. Automatica, 25, 889-896. Rohrs, C.E . , L. Valavani, M. Athans and G. Stein (1982) . Robustness of adaptive control algorithms in the presence of unmodeled dynamics. Proc . 21st IEEE CDC, Orlando, FL. 3. Shao, C. (1991) . Stable adaptive control system subject to bounded disturbances . Int. J. ACSP, 5, 121-134 . Tsiligiannis, C. A. and S. A. Svoronos (1986) .Deterministic convergence of a Clarke-Gawthrop selftuning controller. Automatica, 22, 193-199.

(36)

Thus the boundedness of X(t) can be ensured by that of E(t). The conclusion is proved by contradiction. Now assume that E(t) is not bounded . Then for arbitrarily large n define the sequence tn~

minI tl IE(t)

I~

n,

t~O

I

Along it one obtains max

IE(~)

o~~~tn

1

~

IE(t ) 1_; t _ as n _

n

n

(37)

Divide Itn' into the following two subsets T , ~ { tn 1 ). ( t n ) ~O "

T2~ { tn 1 ). ( t n ) jI()

t-2d) (39)

5. CONCLUSION

~ is sufficiently large and

11

T X(

Since all signals in the adaptive control system are bounded, using (39),(20) and Assumption A3 the conclusion is completed .

The robust stability results are given as follows .

IIX(~-d)

".(a( t-d)-a(t-2d» . L

t.o

(34)

Proof. The conclusion can be de rived by referring to Tsiligiannis and Svoronos (1986). o

max

d-'

E(t)-L£(t)+ t.o

. t-ll-a(t-d» . L" (a( T X( t-d)

+

then for some constants k 4 , ks' k6>0

o~~~t

L"

L" Pet t)~ t.d

k +k max [v(~)+a(~)z(~»)+k max 6(~) (33) , 2t'~~~t 3t'~~~t

z( t)~

(38)

provided m is sufficiently large . This contradicts (19), which implies that the assumption that E(t) is not bounded , is false. The result thus follows.

The following lemma is an extension of lemma 3.2.4 in Tsiligiannis and Svoronos (1986) .

z(t)~

>0

0

[l+A.ax(K(-ll) O+C ) z) ,IZ z

,

For t n € T" using 08e). (33), (20), (32) and (36) it follows that for sufficiently small ~* and all E(t) is bounded on T,. Since E(t) is assumed to be not bounded on T, then E(t) should be unbounded on Tz.Let Tz be expressed as a sub-sequence {t~Z),:., Along it from (18d), (18e) one obtains

256