Adaptive Control Based on Exact Model Matching

Copyright © IFAC Adaptive Systems in Control and Signal Processing. Glasgow . UK. 1989

ADAPTIVE CONTROL BASED ON EXACT MODEL MATCHING K. Ichikawa Department of Mechanical Engineering. Sophia University. Tokyo. Japan

Absract. A model reference adaptive control is an adaptive version of exact model matchIng. For the simplicit y of argument. a continuous-time scalar plant is cosidered . Control parameters. the total nu mber of which is 2n. are replaced by their respective estimates to construct an adaptive control s ystem. In this paper. t he estimates are updated by an adaptive law. which is based on projection method. Proof of global stability of the adaptive control s ystem is established by comparing the order of varios time functions within the system. Keywords. Adaptive control: model reference: continuous time: scalar s ystem: exact model matching: stabilit y proof.

INTRODUCT ION author presented a fundam entally revised proof (1987). which reached an almost perfect proof. but it sti 11 has a sI ight defect. The perfect proof is given in this oaoer. Moreover, the significance of augmented error signal whch is not used explicitly in this paper but was emphasized in Monopoli (1974) is expilicitly demonstrated.

Global stability of continuous time model reference adaptive control systems was solved in several ways (Monopoli . 1974 : Narendra. Lin and Valavani.1980: Feuer and Morse. 1978: Kreisselmeier. 1980). The structure of control systems as well as the proof of stability in the above literatures is somewhat different to each other. A model reference adaptive control system is developed in this paper based on exact model matcing techniques. Exact model matching is a design method such that given. explicitly . the plant to be controlled. the closed loop system transfer function exactly coincides wi th a prescdbed reference model transfer function (Ichikawa. 1985 : Wolovich. 1974) . In exact model matching . the plant output y(t) tracks the model output Ym(t) ex-

STRUCTURE OF PROPSED ADAPTIVE CONTROL SYSTEM Let u(t)

the plant transfer function and output y(t) be t(s) = r(s)p-I(s)

with

input (1)

For both e xact model maching and adaptive control. we need the following assumptions : (A. 1) p(s) is an n degree manic polynomial. (A. 2) r(s) is an m degree polynomial. with

actly as t tends to infinity. The term model following is used to emphasize this tracking propert y. Adaptive control is basicall y an adaptive version of exact model matching . where the plant transfer function is not completely known. The essential objective of 'ldaptive control is. therefore. to J'ealize exact model matching adaptively. and hence the author holds a standpoint that adaptive control is nothing but an extension of exact model matching. contrary to the thought kept by many researchers that adaptive control is adaptive tracking of Ym(t) by y(t). Control

O

~

m

~

n-l.

(A. 3) r(s) and p(s) are relatively prime . (A. 4) r(s) is stable. (A. 5) The sign of high frequency gain of the plant is known. which is denoted by kT (the inverse of

kT

is denoted by kI' o

Let the reference model transfer function with input v(t) and output y(t) be

law for adaptive control is constructed by re placing controller paramet e rs in exact model matching system by their estimates. which are continually updated by an appropriate adaptive law. Among many related matters. the proof of global stability is the most important matter. The clear-cut structure of this t ype of adaptive control system seems to allow us a straightforward proof of stability . The author ever presented a proof of stability (1985). but it was incorrect as was pointed out by Ortega and Lazano (1988). Later. the

td(s)

=

rd(s)Pd

-I

(s)

(2)

Signals v(t) and Ym(t) are called outer refer ence input and output . respectively . Signal v(t) is assumed to be uniformly bounded and piecewise continuous . Since td(s) prescribes the resulting closed loop transfer function. its choice is an important matter (Melsa and Schultz. 1969). but discussion about the choice of td(s) is omitted here. Of course.

89

90

K. Ichikawa

must be

stable.

For both exact model

matching and adaptive control. the assumption is required in order differentiating operation.

following to avoid (3)

It is to be noticed that signal v(t) can vary with time quite arbitrarily. but Ym(t) cannot do so. The signal Ym(t) has a spectrum limited to low

frequency region by the filt e r

td(s).

Thus. the objective of adaptive control is by no ~eans to track Ym(t) which is supposed to fluctuate arbitrarily.

u(t) = eT(t)L(t) instead of (11).

The adaptive law is derived from the alternative representation of plant dynamics . which is represented in terms of 9 instead of coefficients of r(s) and pis). Two kinds of alternative representation can be considered: output signal representation and input signal one. Input signal representation is especially preferable in our case. From (4). we obtain 1 T I TTPTu(t) 8 TTPT"(t) + kly(t) 9TQlt)

k(s)p(s)+h(s)r(s) = q(s)!p(s) -k lr(s)r(s) ] (4) has unique polynomial solutions k(s) and his) with degree n-2 and n- l at most . re spectively (lchikawa.1985: Wolovich. 1974). The control law defined by k(s) his) u(t) = -u(s)+--y(s)+k r (s)t (s)v(s) (5) q(s) q(s) I d achieves exact mod e l matching: i. e .. yts) = td(s)v(s) and all signals within the control

(16) Q(t) = [1 .T(t) y(t)]T TTPT We denote an available signal [ l/r(p» ) u(t) by wit).

By using the prese nt value of e(t). we

can compute eT(t)Q(t). which is an estimate of wit).

We denote this signal by w(t). That is.

wit) = eT(t)Q(t) Then . we define ,It) by ,It)

system are uniformly bounded. (proof) See Ichikawa(1985) and Wolovich(1974). The meaning of q(s) is such that it is a characteristic polynomial of low order state observer. Let us represent k(s) and his) as k(s) k sn-2 + (6) n-2 h sn-I + (7) his) n-l respectively. We define 2n-l parameter vector I and signal vecor • by = [k - ... kO h - ... h ] T (8) n l O n 2 n-2 n-l • (t) = [p_-u(t) ... _I_u(t) P_-y(t) q(p) q(p) q(p) I

(11 )

(18)

~ ~T(t)Q(t)

(19)

which evaluates the identification error of controller parameters . Several kinds of adaptive law for e(t) are available. each of which can solely aim at diminishing ,It). We e~ploy the following adaptive law based on projection algorithm. Q (t), (t) -r --~--9(t) = Ht) c + QT(t)Q(t) (20)

Use of control law (14) together with adaptive law (20) definitely specifies the total adaptive control scheme. It is to be noticed that L(t) is quite different from Q(t) and ,It) is not the same as e (t) =y (t) -y m(t). the adapti ve error . PRELIMINARIES FOR ADAPTIVE CONTROL

(9)

respectively. Then. the control law (5) can be written as 8T(t) + kl r(p)Y m(t) (10) u( t)

wit) - wit)

(17)

T re(t) - 91 Q(0

Lemlla 1.

~ 9\(t)

(15)

where

Let us consider. for the moment. solving an exact model matching problem. Let q(s) and r(s) be any stable monic polynomials with degree n-I and n- m. respectively. The polynomial equation. which is often called Diophantine equation.

1 T q(pYy (t)]

(14 )

The stability proof much relies on the concept of time function order and exponential function of time. Consider any two vector functions of time x(t) and y(t). each with any dimension.

where [8 T

k ]T I

(12)

and L(t) = [. T(t)

r (p)y (t) ] T (13) m It is to be noticed that although rip) is a differentiating operator. r(p)Ym( t )=r(p)td(s) xv(t) is available because of (A. 6).

Definition I. Let sup 1x (01 O ~r~ t = a lim I y (t) I sup t~oo

(21)

O ~,H

Now. let us proceed to adaptive control. Since rls) and pis) and . of course. kl are unknown.

If a=O. then x(t) is said small order of y(t). and is denoted by x(t)=.(y(t». If a is nonzero finite. x(t) and y(t) are said same order to each other. and is denoted by x(t)= O(y(t» or y(t)=O(x(t». If a is infinite. it is clear that y(t)=.(x(t».

vector 9 is unknown. Then. we replac e 9 by its

Consider any dynamical system described by

estimate e(t) which is conducted and updated by an adaptive law which will be introduced later. Thus . the control law for adaptive control is

~(t) = A(p(t»x(t) + b(q(t»u(t)

(22)

where both pit) and q(t) are uniformly bounded. The initial state x(t ) is assumed to be O

91

Adaptive Control Based on Exact Model Matching finil e. Al s o , u(l ) is assumed lo be uniformly bounded and pi ecewise continuous . LeMMa 2. Le l x(l ) be a soluli on of (22 ). Then, Ix(l)1 is bound ed by exponenlial func lion (i. ,e. , x (l) is of exponential orde r). (proof ) x(l) is represe nled by x(l) ;

~(l,

l )x(l )+ S

o

wh e re ~(t, I) is frOM A(p(l» as ~(l , I)

;

exp[

l

~(l ,

I

)b(q( , »U( , ) dl

lO

0

a lransilion malrix

(23) deriv ed

(proof) ( i ) Consider a scalar function V(Hl» Yz .T(l)r - lt(t) (32) Clearl y, V(l) can never be negative , and beCOMes infinile if and only if It(t)1 is infinile . By differentialing , we oblain ,2(t) (33) which is clearly nonpositive . Therefore, V(t) decreases monotonically . Since t(t O) is finile, V(lO) is also finile. Therefore . V(t) is

l S

A(p( , ) d,)

(24 )

Since p(l ) i s uniforml y bound ed, lhere ex isls a finil e T>O s uch lhal IA(p(l » I
IS A(p(, ) d ol

T (l -

Also , we obla in l k I [S A(p(.) d ,) I

uniforMly bounded. (ii ) From (9) and (l0) , we oblain

I)

k T

(l - r)

~ (t) ;

k

; ... ;

s '" I I

I

n

1 0

f " ;':; ': r ···· ·:·· t: ··:; ·; r ..... . "(t) 2n-l 1 n- I : n : 1

Then, from (24), we obla in 1~(l ,T )1 ~ e T(l -, )

(25) Since q(l) i s uniforml y bounded , lhe r e exisls a finil e cl >O s uch lha l Ib(Q(l»I ~ cl for all t.

2n - l

o

Also , lhere ex i s l s

a fini l e c 2>0 such lhat for all l . Then, we oblain

l u(t)l ~ c2

IX(t)I ~e

T (l-l

o

)

+ ............. rlP)YM(t) rn _lkI(t)

0 Ix(tO)1 + [c lc/ rJ x (e

T(t - t

O - 1)

(26)

Lemma 3. Le t x (t) be any time funclion of exponential orde r. Le t zip) be any m degree moni c slabl e pol ynomi a l , whe re p; d/ dl . The n, (i) [l / z(p» )x(t) ; O( x(l » (27) (ii) [pi / z(p» )X(l ) = ",x(l » or O(x(l» , i=1.2 ,"', m- l

T

(l-l )

0 +8

wilh both 83 and 82 be ing 2 posilive and finite . In gene ral , [pi / z(p» )X (l ) is same order as x (l ) , bul in l he s pecial case when x(l ) is a pol ynomial funcli on of l , lhey are small order of x(l ) for i =I,2 ,' " , m-I. Howeve r, [ lIz (p»)x (t ) is always same order as x(t) .

3

(i ) ~(t ) , ed. (ii ) Q(l ) (iii ) ~ (0 (iv ) r( t) ( v )

(vi )

If

1~(lO ) 1

and hence 8(0, is uniformly bound -

lim tit) = 0 l~ oo

Si (t)

8i (l) - Qn-I - i ' i =1. .. , , n-l

li(l)

rn_Is i (t)+r n- l - i , i=1.' " ,n-I

l i (t)

rn - l'i (l) - P2n-l-i i =n , ' , , , 2n-1

(34 . b)

Il is seen lhal (34) belongs lo the class of (22) . Therefore , by LemMa I . "(l) is of exponenlial order . Also , since S(l) is uniforMIy bounded and T(P)Ym(t) is piecewise continuous, • (t) is continuous. Froll lhe relation [pi/Q(p» )y(t) ="2 n- I -I. (l),i=O,"' , n- 1 we oblain y(l) = "nil ) +Qn - 2"n+l(l) + '" +QO"2n- 1 (l) (35) which means lhal y (l) is continuous . Therefor e, Q(t) is continous . (iii) The adaptive l a w (20) can be rewritten as = -

r

Q(t)Q T(Ot(O

(36)

c+QT(l)Q(l)

is finit e, lhen

is con li nuous . is continuous . = "Q(l » Q(t ) r(l ) 0 lim c+QT(t)Q(l) t~ oo

whe r e

~(l)

STABILITY PROOF FOR PROPOSED ADAPTIVE CONTROL Le III Ma 4.

o

(28)

are (proof) Since pi / zip) , i =O, I , "' , m- 1. slricll y prope r slabl e filler and x(l) is T bounded by some lime funclion 8 e (t - l O) +8 I 0 wilh bolh 8 and Po be ing posilive and finile, 1 [pi / z(p»)x(l) are bounded by some lime func lion P e

(34. a)

(29) (30 ) (31 )

Since Q(l) is conlinuous and c+QT(l)Q(t) is alwa ys posilive, ~(l) is continuous. (iv) Since bolh t(l) and Q(t) are continuous, , (l) is conlinuous frOM (19). Therefore, , 2(l) / [c+QT(l)Q(l» )

is

turn

iMpli es

V(t) is

e ve r ,

S

00 '

thal

conlinuous,

which in

conlinuous. How-

V(t) dl = V( oo ) - V(O)
92

K. lchikawa

Lemma 5. The followings hold. ( i ) u(t) ,(y(t» or O(y(t» (37) (ii ) »( t) O(y(t» (38) (iii) Q(t) O(y(t» (39) (proof) ( i ) As was proven in Lemma 4 . • (t) is of exponential order. Therefore . u(t) is also exponential order. Now. y(t)=r(p)( [I!p(p»)u(t) I . where r(s) is a stable polynomial by (A. 4). Therefore. y(t)=O([I!p(p»)u(t». or . of course. [l!p(p»)u(t)=O(y(t» . That is . u(t)= p(p)Y(t). where such that

y(t)

is

some

Y(t)=O(y(t».

u(t)=O(Y(t». but

If

time function

p(s) is

otherwise

stable.

u(t) can also be

,(y(t» . Thus. u(t)=,(y(t» or O(y(t». (ii) Since »2n-1 (t) = [l!q(p»)y(t). we obtain (38) by Lemma 3. (iii) Since both [l!r(p»).(t) and y(t) are O(y(t». (39) is evident. Lemma 6. (Key Lemma) The followings hold.

lation

k,(t)Ym(t)-~{k,(t)r(p)Ym(t)1 (44)

= o(r(p)Ym(t» Since

l(p)Ym(t) is uniformly bounded. '(I(p)X

Ym(t»

means

a time function

which tends to

zero as t tends to infinity. Theorem 1. results ( i)

Ii m

t~oo

'f k (t) does not tend to zero . it

(45)

[y (t) - Ym ( t ») = 0

( ii ) All signals wi thin the control system are uniformly bounded . (46) (iii) lim ,(t) = 0 t~oo

That is . the proposed adaptive control system is globally stable. (proof) ( i ) We obtain I ,(t) = -BT(t)TTPT.(t) + k,(t)y(t)

(i ),T(t)-:h»(t)--:h{ST(t).(t)l=o(y(t» (40) I IPI IIPI

I -T TTPT{B (t).(t)+k,(t)r(p)Ym(t)}

(ii) lim [k,(t)y (t)--:h{kI(t)r(p)y (t)I)=O t ..w" m IIPI m (41) (proof) ( i ) We note the follow ing identi ties .

[ST(t)-:h.(t) - -:h{iT(t).(t)l) r IPI r IPI + k,(t){y(t) - Ym(t)1

I

7TPT •

(,T(t) I

TlPT

••••••

i

•

•

•

•

••

»(t)1 = I

7TPT

•••••

•

,

••

•

I

,

•

(ST(t) I

TlPT

•

,

••

••••

•

•

.(t)1 (42 . 1) •••

••••

••

i-I . T

P

7TPT

{,T(t)-:h.(t)1 = ~{S (t)-:h.(t)1 IIPI IIPI IIPI ·T i-I IIPI i

+

~{iT(t)~.(t)1 IIPI

(42 . i)

IIPI

...... .... ............ . , ........ . , .. ... ..... . n-m

~{p){iT(t)TTPT.(t)1 ·T

+ .. . + +

By

n-m-I ·T

=

~{S (t)~.(t)1 n-m-I

~{S (t)~"(t)} r IPI

r IPI n-m

~{iT(t)~.(t)1

(42.n-m)

r IPI IIPI multiplying (42.i) by r i.i

O. I ... ·.n - m.

where rn-m=l. and summing them up. we obtain -T

I

6 ( t ) TTPT» (t) =

1-

k,(t){y(t) - Ym(t)1 = .(y(t»

+ .. , + ~{S (t)~.(t)}

IIPI

-

[k,(t)Ym(t)-TTPT{k,(t)r(p)Ym(t)}) (47) From Lemma 4 (iv) and Lemma 5 (ii ) . ,(t) is .(y(t». The first term of the right hand side of (47) is o(y(t» by Lemma 6 ( i ) . The last term tends to zero by Lemma 6 (ii ) . Therefore. we obtain +

[r(p)-rO)!p .:T I r( P ) {8 ( t) TTPT· ( t) I

It is clear that y(t) ~ Ym(t) . provided

(48) k,(t)

does not converge to zero . Also. it results that y(t) is uniformly bounded . (ii) Since .(t)=O(y(t» and u(t)=,(y(t» or O(y(t» . we obtain (ii ) . (iii) Since Q (t) is un i form 1y bounded by Lemlla 5 (iii) . r(t)=o(Q(t» (Lemma 4 (iv» implies (iii ). 'n Theorem I. we assumed that k, (t) does not tend to ze ro. Since sgn(k,) is known by we

(A. 5) .

can choose

k, (to)

such

that

[r(p)-rIP-rO)!p ~T P + r{p) {B (t)TTPT"(t)}

sgn(k,(t ) = sgn(k,). The adaptive law (20) O can be slightly modified so that k\(t) never

+ ........ ...... .............. .. ..

goes into the region with opposite sign of

2

·T

+

n-m-I

during the adaptive process .

~{i (t)~.(t)1 r IPI

k,

71PI

+ ~(ST(t).(t)1 rlPI

(43)

Since . by Lemma 4 (vi) . i(t) tends to •T

e

zero as

.

t tends to infinity . (t) [p1/r(p») .(t) = o(.(t». i=O . l. .. · .n-m-I. Moreover . ( [ r (p)10)/p}!r(p) . . . . . l!r(p) are all strictly proper stable filters. Therefore. all terms of the right hand side of (43) become small order of .(t) . except the last term. However . »(t)=O(y(t» by Lemma 5 (ii) . we obtain ( i ). ( ii ) Silli lar argument to ( i ) leads to the re-

AUGMENTED ERROR SIGNAL We rewrite (20) . using (47). That is . Q(t) ~ t) = -k\(t)r T [e(t) cH (t)Q(t) + _1_[eT (t)-:h.(t)_~{ST (t).(t) I) k (t) 71PI r IPI I

+ _1_[kI(t)y (t)-~{kI(t)r(p)y (t)I)) k(t) m IIPI m \

(49)

Adaptive Control Based on Exact Model Matching

Thus . the augmenting err0r sinal introduced by Monopoli(1974) is e (t) = _1_(9T (t)~. (t)

k

a

I

(t)

T ,PI

~(8T(t)L(t)l) T ,PI

(50)

Therefore . dt) . which is used in our adaptive law and is the evaluation of identification error for controller parameters. can be regarded as an augmented e rror signal . The output form of plant dynamics r epresenta tion is obtained from (4) ; that is . y(t)

~(u(t)

"IT

- oT.(t)1

(51)

,PI

Therefore . e(t)

(52) This is an adaptive-control e rror model . The crucial difference be tween our model (19) and the model (52) is that in (52) e(t) i s an output of the transfe r function of i/[kIT(p) ). which is not positive real unless m=n - l. while in (19) ,(t) is an output of transfe r function of 1. which is of course strictly positive real . This is the cause why our adaptive con trol system is of extremely simpl e structure. CONCLUS IONS A model reference adaptive control s ystem was constructed as an extension of exact mode l matching . The proof of golbal stabilit y was carried out . based on comparion of order of various time functions . Controll e r paramete r identification error was employed in adaptive law . instead of adaptive control e rror . This scheme does not necessitate the employment of positive real function or augmented e rror signal. REFERENCES Feuer . A. and S. Morse (1978) . Adaptive con trol of single- input single-output linear systems. IEEE Trans . Autom . Contr ., AC - 23 , 557-570 . System Design Ichikawa, le (1985) . Control Based On Exact Model Matching Techniques . Springer-Verlag , Berlin . Ichikawa . K. (1987). A design method of a model reference adaptive control system and its stability . JSME Int. J. , 30 , 830 835 . Kreisselmeier , G. (1980) . Adaptive control with adaptive observation and asymptotic feedback matrix s ynthesis. IEEE Trans . Autom. Contr ., AC-25 , 712-722 . Melsa. J. L. and D. G. Schultz (1969) . Linear Control System. McGraw-Hill . New York-.- ----

Monopoli. R. v. (1974). Model reference adaptive control with an augmented error signal. IEEE Trans. Autom . Contr . . AC-19. 474-484 . L. S. Valavani Narendra. S.. Y. H. Lin and (1980). Stable adaptive controller design . Part ~ : proof of stability . IEEE Trans . Autom . Contr .. AC- 25 . 440-448 . Ortega. R. and R. Lozano (1988). Global stable adaptive controller for systems with delay . Int. J. Control. 47. 17-23. Wolovich . w. A. (1974) . Linear Multivariable Systems . Springer - Verlag . New York.

93