A Class of Robust Adaptive Controllers for Innite. Dimensional Dynamical Systems. M. A. Demetriou K. Ito. Center for Research in Scientic Computation

Transcription

1 A Class of Robust Aaptive Controllers for Innite Dimensional Dynamical Systems M. A. Demetriou K. Ito Center for Research in Scientic Computation Department of Mathematics North Carolina State University Raleigh, NC August 5, 995 Abstract An aaptive controller for a perturbe innite imensional plant is evelope to force the state of the plant track the state of a reference moel. The reference moel is base on the nominal plant that has a physical similarity with the plant. Using a Lyapunov stability argument, which is base on the H -Riccati equation of the nominal plant, an aaptive law is evelope for the ajustment of the feeback gain. It is prove that the close-loop system is stable with the tracking error remaining boune, an converging to zero provie that the norm of the structure perturbation is less than a specie attenuation boun. Results of numerical stuies regaring a heat equation an a beam equation are presente to emonstrate the applicability of the propose control algorithm. Introuction We consier an innite imensional plant whose ynamics are governe by a linear evolution equation of the form t x p(t) = A p x p (t) + Bu(t); x p () = x X: (.) The linear operator A p is an unknown innitesimal generator of a C?semigroup on a Hilbert space X with inner prouct an norm h; i an j j X, respectively. The input operator B is known an it is assume that B L(RI m ; X) has nite rank. The objective is to n a robust ynamic feeback control law of the form u(t) =?K(t)x p (t) + r(t) (.) such that the state of the plant x p (t) tracks the state of a reference moel x m (t). This reference moel is base on the nominal close-loop ynamics an is given by t x m(t) = (A? BK m )x m (t) + Br(t) (.3) Research supporte in part by the Air Force Oce of Scientic Research uner grant AFOSR F , an in part by NASA uner grant NAG--6.

2 where r() L loc (; ; RI m ). The open loop operator A is the nominal plant generator in the sense that the unknown plant generator A p has the following perturbation form A p = A + A; A L(X); an K m L(X; RI m ) is an appropriately chosen stabilizing feeback gain operator for the nominal plant that will be ene below. The reference signal r(t) may be etermine by esigning an optimal control for the reference ynamics (.3) with certain performance inex. For example, it might be chosen to minimize the performance inex Z T jx m (t)? xj + jr(t)j t + jx m (T )? xj associate with the problem of regulating the state x m (t) to a esire state x X. Following the moel reference aaptive feeback law [3, 9], the ynamic feeback gain operator K(t) is given by t K(t) = B (x p (t)? x m (t)) x p (t) (.4) where > is the acceleration parameter. The erivation of the aaptation rule (.4) is base on the Lyapunov synthesis approach, [3, 9]. In our approach we choose the nonnegative enite self-ajoint operator to be a solution to the H Riccati equation (.8), iscusse below, an K m = B : (.5) It is shown in Section 3 that, uner certain assumptions on the norm of A an the reference trajectory x m (t), the close-loop system (.) - (.5) has a unique global solution (x p (t); K(t)) an that the asymptotic tracking property Z jx p (t)? x m (t)j X t < ; an e(t) = x p (t)? x m (t)! as t! (.6) is satise (see Theorem 3.). The control law (.) provies a sub-optimal control law for the uncertain plant (.) in the sense of ning an optimal control law for the reference moel (.3) an an asymptotic regulator via the aaptive feeback law (.4). Thus, the problem of constructing a feeback law for the uncertain plant (.) can be ecompose into (i) the optimal control problem of r(t) for the reference moel (.3) an (ii) the asymptotic tracking problem in the sense of (.6). The following matching conition is normally assume in the aaptive control literature [9]. There exists a feeback operator, K : RI m! X such that A p + BK = A m = A? BK m ; (.7) which is equivalent to assume that A R(B): The range conition (.7) is very restrictive for the innite imensional plant (.). Our objective here is to remove this assumption an allow general uncertainty in the plant. Our choice for in

3 the aaptive law (.4) is base on the fact that the H feeback control is evelope to construct a feeback law that stabilizes all perturbe plants with a specie uncertainty boun for kak. That is, if for given >, there exists a nonnegative, self-ajoint operator on X satisfying the game-theoretic Riccati equation A + A? BR? B? DD + C C x = ; (.8) for all x D(A ), then for K = R? B, the close-loop plant operator A p? BK = A + D?C? BK generates an exponentially stable semigroup on X provie that Here we assume the structure perturbation k?k < : A = D?C;? L(X) with known operators D L(X) an C L(X). It is shown (e.g. [4,, 3, 4, 6]) that the least >, such that (.8) has a nonnegative solution, gives the best stability boun in the robust linear feeback control, i.e. 8 >< = inf > >: 9 a linear feeback K L(RI m ; X) such that A? BK + D?C generates an exponentially stable semigroup, provie that k?k < Moreover, it can be seen that this robust stabilization problem is equivalent to the stanar H - attenuation problem for the state feeback case (see [6, 7]). Throughout this paper we consier the case when D; C are chosen to be the ientity operators in orer to account for unstructure perturbation, an the control weight is taken, for simplicity, to be R = I. All iscussions can be simply extene to the structure perturbation case. For the sake of completeness of our iscussions, the asymptotic tracking property of the constant feeback synthesis 9 >= : >; u(t) =?K m x p (t) + r(t) with K m = B (.9) is given in Section. Comparison of the performance between the constant feeback synthesis (.9) an the aaptive control law (.), (.4) is of our interest an will be analyze in Section 3 in terms of the Lyapunov stability argument, an in Section 5 through numerical testings. An outline of the paper is as follows. In Section we summarize the results for the robust moel reference control of innite imensional systems. In Section 3 we present our main results on the moel reference aaptive control base on the Lyapunov stability argument. The well-poseness of the propose control law (.) with the aaptation rule (.4) is given in Section 4. Numerical 3

4 stuies of a heat equation with spatially varying parameter an of an Euler-Bernoulli beam with piezoceramic actuators are shown in Section 5 an conclusions an iscussion of possible extensions of this work are given in Section 6. Robust Moel Reference Control When the control law (.9) is applie to the plant equation (.), it yiels the close loop system t x p(t) = (A p? BK m )x p (t) + Br(t): (.) The close loop system (.) combine with the reference moel (.3) give, for e(t) = x p (t)?x m (t), the following state error equation t e(t) = (A p? BB )x p (t) + Br(t)? (A? BB )x m (t)? Br(t) = (A p? BB )e(t) + (A p? A )x m (t) = A p e(t) + Ax m (t) = (A? BB )e(t) + (A p? A )x p (t) = A e(t) + Ax p (t); (.) where A p = A p? BB, A = A? BB an A = A p? A = A p? A. Here, solutions to (.3), (.), (.) are unerstoo in the sense of the mil solution, see []. In aition, if x p (); x m () D(A ) an r(t) is continuously ierentiable, then equation (.3) as well as (.) have a classical solution x(t) C (; ; X) \ C(; ; D(A )). Hence, we have that (.) hols in the strong sense. The corresponing Lyapunov function is given by V (t) = h e(t); e(t)i: (.3) We assume here that is the solution to the Riccati equation A + A? BB? I + Q x = (.4) for all x D(A ) an Q I. Or, equivalently A + A x =? We assume that an thus we have that BB + I kak < + Q x: (.5) (.6) kak =? ; < : (.7) The following calculations are unerstoo in the sense of Theorem 3. presente in x 3. Taking the erivative of (.3) along the solutions of (.) an using (.5) - (.7), we have that t V (t) =?h( BB + + Q)e(t); e(t)i + h(a p? A )e(t); e(t)i 4

5 +h e(t); (A p? A )e(t)i + h(a? A p )x m (t); e(t)i +h e(t); (A? A p )x m (t)i? j e(t)j X? je(t)j X + j e(t)j X (kakje(t)j X + kakjx m (t)j X )? (? ) je(t)j X? j e(t)j X? je(t)j X? j e(t)j X + j e(t)j X jx m (t)j X kak?je(t)j X?? min ;? min ; j e(t)j X + jx kak m (t)j X je(t)j X + j e(t)j X + jx kak m (t)j X je(t)j X + j e(t)j X + (? ) jx m (t)j X (??c je(t)j ) X + h e(t); e(t)i + jx m (t)j X; =?c je(t)j X + V (t) + c jx m (t)j X; (.8) for some positive constants c ; c, where we use the completion of squares an the fact that V (t) = h e(t); e(t)i j e(t)j X je(t)j X j e(t)j X + je(t)j X : (.9) This implies that for all t, the tracking error e(t) satises V (t) e?ct V () + c e?c (t?s) jx m (s)j X s (.) an je(t)j X t V () + c jx m (s)j X s : (.) c By imposing aitional conitions on the state of the reference moel, we can show that the function je(t)j X (; ) with h L t e(t); e(t)i X <. If je(t)j X is uniformly continuous an integrable, then it follows from Barbalat's lemma ([3, 9, ]) that je(t)j X! as t!. If the operator is coercive, i.e. h e(t); e(t)i je(t)j X, >, then we have that je(t)j X is inee uniformly continuous an integrable an thus we can apply Barbalat's lemma to conclue convergence of e(t) to zero. While this coercivity conition is true in nite imensional systems, it is not necessarilly satise by innite imensional systems. Since no such information of coercivity of can be extracte from the game-theoretic Riccati equation (.4), we must n an alternate route to show convergence of the state error to zero. Thus, assuming Z jx m (t)j X t < ; (.) i.e. x m () L (; ; X), we can show that V (t)! as t!. Moreover, we can prove that p h x; xi enes a norm in X. Thus, we conclue that e(t)! as t!. 5

6 We present this in the form of the theorem below, but rst we prove that p h x; xi inee enes a norm on X. Lemma. The seminorm p h x; xi is a norm on X. Proof. We rst note that h x; xi Z js(t)xj X t (.3) for x X, provie that Q I, where S(t) is the semigroup generate by A on X. Suppose there exists an element x X such that h x; xi =. Then, from (.3) we have that Z js(t)xj t = which implies that S(t)x =, t since t! js(t)xj is continuous in [; ]. Hence, S()x = an therefore x =. Theorem. Given >, assume that the Riccati equation (.4) has a nonnegative solution. Then, if the control (.9) is applie to the plant (.), the tracking error e(t) satises Z je(t)j Xt Z V () + c jx m (s)j c Xs an lim t! V (t) =, provie that x m () L (; ; X). Moreover, lim e(t) = t! in the sense that the norm p h e(t); e(t)i! as t!. Proof. We rst show that the norm V (t)! as t! using the estimates (.8) - (.). Then, an application of Lemma. woul give the esire results. In aition to the estimate (.), using (.8), we get that j e(t)j X t V () + c jx m (s)j X s : (.4) c which implies that R V (t) t <. Using the above (an implicitly (.)), we get that Z Z Z t V (t) t M je(t)j X + j e(t)j X t + jx m (t)j X t ; (.5) for some positive constant M >. It thus follows from (.9), (.), (.), (.4) an (.5) that Z t V (t) t < ; (.6) an Z jv (t)j t < : (.7) Equation (.7) implies that lim inf V (t) =. But, (.6) implies that lim V (t) exists since V (t) = V () + R t V (s) s. Hence, lim V (t) = lim inf V (t) =. t Using Lemma. we have that lim e(t) = ; t! which conclues our proof. 6

7 3 Moel Reference Aaptive Control We now procee, as in the nite imensional case, see [3, 9], to introuce the aaptive control law for the plant given by equation (.). Let us ientify the space L(X; RI m ) with X m. We consier the aaptive control law of the form u(t) =?K(t)x p (t) + r(t); (3.) where we aim to ajust K(t) X m to the optimal feeback operator of the uncertain plant (.) an not to the optimal feeback operator for the nominal plant, (i.e. K m = B ). The rationale behin such a choice is the esire to improve the performance of the nominal controller K m. The upate law for K(t) is given by t K(t) = B e(t) x p (t) (3.) where > ; in this case can be viewe as the weighting on the X m -inner prouct, or as the reciprocal of the aaptive gain, see [3, 9]. In the aaptive law (3.), (3.), the parameter is a esign parameter that is not necessarily the optimal one as in the non-aaptive case, i.e. (.4), because it is esire to stuy the combine eect of robust feeback an aaptation. The Lyapunov stability argument use below requires regularity of solutions. This is gurantee uner the following theorem. Theorem 3. (Ito-Powers, [5]) Consier the abstract evolution equation in X with x x(t) = (A? BK(t)) x(t) + f(t) (3.3) t D(A), f C (; T ; X), where we assume that A is the innitesimal generator of C?semigroup S(t) on X an t! BK(t)x; x X is continuously ierentiable for each x X. Then, the mil solution ene by x(t) = S(t)x + S(t? s)(?bk(s)x(s) + f(s)) s (3.4) is a strong (classical) solution [], in the sense that x(t) C(; T ; D(A)), x(t) AC (; T ; X), tx(t) C(; T ; X). From Theorem 3. all functions that appear in the following stability arguments are strongly ifferentiable, assuming x D(A), r C (; T ; RI m ) an thus the calculations make sense. Then, we use the continuity of solutions x(t) C(; T ; X) to (.) with respect to (x ; f()) in X L (; T ; X) an the Lyapunov functions to generalize the stability results to the general initial conition x X an the reference signal r() L (; T ; RI m ). It follows from (.), (.) an (3.) that t e(t) = (A p? BK(t))x p (t) + Br(t)? (A? BK m )x m (t)? Br(t) 7

8 = (A p? BK(t) + BK m? BK m )x p (t)? (A? BK m )x m (t) = (A p? BK m )x p (t)? (A? BK m )x m (t)? B(K(t)? K m )x p (t) = A p x p (t)? A p x m (t)? B(K(t)? K m )x p (t) + Ax m (t) = A p e(t)? B(t)x p (t) + Ax m (t) = A e(t)? B(t)x p (t) + Ae(t) + Ax m (t); (3.5) where (t) = K(t)? K m X m an K m = B. Let us ene the Lyapunov function Then, we have V (t) = h e(t); e(t)i + j(t)j Xm: (3.6) t V (t) =?h( BB + + Q)e(t); e(t)i? h e(t); Ae(t)i +h e(t); Ax m (t)i? h e(t); B(t)x p (t)i + h t (t); (t)i Xm? min ; je(t)j X + j (? e(t)j ) X + jx m (t)j X =? je(t)j X + j e(t)j X + c jx m (t)j X; (3.7) for some positive constants, c, where we use the fact that This implies that V (t) + h t (t); (t)i Xm = h e(t); B(t)x p (t)i: (3.8) je(s)j X + j e(s)j X s V () + c jx m (t)j X s (3.9) for all t >. If we assume Z jx m (t)j X s < (3.) then j(t)j X m const. for t. Moreover, we can show state error convergence to zero using the same arguments as in the proof of Theorem.. Theorem 3. Assume that for given > an Q I, the Riccati equation (.4) has a nonnegative self-ajoint solution, that the uncertainty boun kak < is satise, an that the state of the reference moel has nite energy (i.e. equation (3.)), then the close loop system for the aaptive law (3.) - (3.) satises lim e(t) = ; t! in the sense that the norm p h e(t); e(t)i! as t!, an u() L (; ; RI m ). Proof. Dene W (t) = h e(t); e(t)i: 8

9 Using (3.7), (3.9) an (3.) we have that Z Z t W (t) t M je(t)j X + j e(t)j X t + Z jx m (t)j X t ; for some positive constant M, where we use the fact that j(t)j X m const. It thus follows from (3.9), (3.) that Z t W (t) t < ; (3.) an that Z jw (t)j t < : (3.) Using arguments similar to those for Theorem., we conclue that It thus follows from Lemma. that lim W (t) = lim h e(t); e(t)i = : (3.3) t! t! lim e(t) = : (3.4) t! Finally, it follows from (3.9) an (3.) that x p () L (; ; X) an j(t)j X m < an thus u() L (; ; RI m ). It is not easy to analyze the performance of the aaptive feeback synthesis against the one of the constant feeback law K m = B in the quantitative manner. However, we can argue the qualitative improvement of the aaptive law (3.) over the constant feeback (.9) as follows. Assume that there exists a feeback law K p X m such that where K = K p? K m. Then, as above we have ka? BKk kak; < : (3.5) t e(t) = (A p? BK p )e(t)? B b (t)x p (t) + (A? B(K p? K m ))x m (t) = (A? BK m )e(t)? B b (t)x p (t) + (A? BK)(e(t) + x m (t)); (3.6) where b (t) = K(t)? K p. Let Using the same calculation as above, we obtain bv (t) = h e(t); e(t)i + j b (t)j Xm (3.7) V t b (t) =?h( BB + + Q)e(t); e(t)i?h e(t); (A? BK)e(t)i + h e(t); (A? BK)x m (t)i? j e(t)j X? je(t)j X + j e(t)j X (ka? BKkje(t)j X + ka? BKkjx m (t)j X )? j e(t)j X? je(t)j X + j e(t)j X (kakje(t)j X + kakjx m (t)j X ) 9

10 ? j e(t)j X? je(t)j X +j e(t)j X kakjx m (t)j X?(? )? ) + ( j e(t)j X je(t)j X je(t)j X? j e(t)j X? (? (? )) je(t)j X? (? (? )) j e(t)j X + j e(t)j X kakjx m (t)j X? (? (? )) je(t)j X? (? (? )) j e(t)j X + kak jx (? (? )) m (t)j X? (? (? )) min ; je(t)j X + j e(t)j X (? ) + jx m (t)j X (? (? )) =? (? (? )) min ; je(t)j X + j e(t)j X (? ) + jx (? (? )) m (t)j X =?~ je(t)j X + j e(t)j X + ~c jx m (t)j X; (3.8) where we use (3.5) an (.6), (.7). The positive constants ~ an ~c are given by ~ = (? (? )) min ; ; an ~c = (? ) (? (? )) : The equivalent constants from equation (.8) are given (as they were explicitly foun in equation (3.7), or simply using the above with ) by (? ) = min ; ; an c = : Using the fact that < (from (.7)) an that < (from (3.5)) along with the inequality? (? ), we can show that ~ an ~c < c. Hence, it woul mean better tracking an asymptotic behavior of the aaptive law (3.) comparing to the constant feeback synthesis control law (.9). Remark 3. In regar to Theorem 3., the existence of the nonnegative solution for the Riccati equation (.4) implies that the pair (A ; B) is stabilizable. Remark 3. Concerning possible limit of K(t), a more precise conition than (3.5) can be assume. Accoring to our energy estimate (3.8), the best K p can be chosen such that (A? BK) + (A? BK) I is satise for the least. In the event that the matching conition (.7) is satise, we then have =.

11 Remark 3.3 For the nite imensional case with the matching conition (.7) satise an with persistence of excitation, [3, 9, 5], we have K(t)! K. But, Theorem 3. only shows that K(t) is boune, which implies that K(t) converges to some K subsequentially, i.e. there exists a sequence ft n g such that K(t n )! K as n! an t n!. So, a natural question to ask is what coul be a set of subsequential limit of K(t) as t!? Accoring to the energy estimate (3.8), t V (t) is more negative if K(t) = K p provies the least upper boun for the following inequality (A? BK) + (A? BK) I; (3.9) for. These heuristic observations make sense in the following argument. Suppose the matching conition (.7) is satise an (3.9) is satise with = an K = K where A p? BK = A m in (.7). This matches with the fact that the asymptotic limit of K(t) is K if it converges. 4 Well-Poseness In this section we establish the well-poseness of the aaptive law (3.), i.e., the existence of solutions to the close loop (.), (3.), (3.) an (3.5). Theorem 4. For arbitrary initial conition (x ; K()) X X m, there exists a > such that the system of equations (.), (3.) (3.) an (3.5) has a solution (x(t); K(t)) in C(; ; X) AC(; ; X m ) with t! K(t)x being continuous for each x X. Moreover, if (.4) is satise then there exists a unique global solution. Proof: The proof is base on the Banach xe-point theorem. Let Y = C(; ; X m ) an ene the mapping on Y by c K = (K) where an x(t) = S(t)x? c K(t) = K() + with e(t) = x(t)? x m (t). Let S be the close subset in Y ene by ( ) Then, we prove S = S(t? s) (BK(s)x(s)? Br(s)) s (4.) (B e(s) x(s)) s; (4.) max jk(t)j X m t[; ] (i) that for jk()j X m there exists a > such that : S! S (ii) an that is a contraction, j (K )? (K )j Y jk? K j Y ; K ; K S; < < : (4.3)

12 Proof of (i): Let ks(t)k Me!t where S(t) is the semigroup generate by A p. Then, from (4.) jx(t)j X Me!t jx j X + ~ M jx j X + jbj Me!(t?s) jbj ( jx(s)j X + jr(s)j R m) s c + c jx(s)j X s: jr(s)j R m s + MjBj ~ where M ~ = Me! an thus Me!t M ~ for t [; ], Z c = M ~ jx j X + jbj jr(s)j R m s ; c = MjBj: ~ By Gronwall's inequality, we have jx(s)j X s jx(t)j X c e c ; t : (4.4) From (4.) j c K(t)j X m jk()j X m + c je(s)j X jx(s)j X s jk()j X m + c (c 3 + c e c ) c e c F(): where c = jb j X m an we assume that jx m (t)j X c 3 on [; ]. Since! F() is continuous an monotonically increasing, it follows that for jk()j X m, there exists a > such that F( ) ; an thus : S! S with =. Proof of (ii): Let K c i = (K i ) for K i S; i = ;. It follows from (4.) that if we ene (t) = x (t)? x (t), then (t) =? Thus, we have as above j(t)j X c e c S(t? s)b [(K (s)? K (s)) x (s) + K (s)(s)] s jk (s)? K (s)j X mjx (s)j X s c c 4 e c jk? K j Y ; where we assume that jx (t)j X c 4 on [; ]. Hence, using (4.) an the above we have j c K (t)? c K (t)j X m c (jx (s)? x (s)j X jx (s)j X + je (s)j X jx (s)? x (s)j X ) s cc c 4 e c [c 4 + (c 4 + c 3 )]jk? K j Y : We may choose < such that = cc c 4 e c [c 4 +(c 4 +c 3 )] < an thus (4.3) is satise. By the Banach xe point theorem, there exits a unique xe point K() in S which enes a local solution to (.), (3.), (3.) an (3.5).

13 Suppose (.6) is satise, then it follows from (3.9) that jk(t)j X m jk()j X m + const jx j X + jk()? K m j X m + jx m (s)j X s ; t [; ] which implies that a locally ene solution K() is a unique global solution. 5 Examples an Numerical Results Here we present two examples to emonstrate the feasibility of our propose aaptive scheme. Remark 5. The examples presente below cannot be treate within our theoretical framework in Sections 3 an 4, because A = L(X; X). However, the plant generator in these two examples generates an analytic semigroup in X. Thus, we can restate Theorem 3. for the case of analytic semigroups as follows. Theorem 5. We assume that A assume that an A p generate an analytic semigroup on X. Furthermore we (A) (A) + A + Q > I, for some > (A) Ax m () L loc (; ; X). Then the close loop system is well pose. Moreover, if Ax m () L (; ; X), then an u() L (; ; RI m ). lim e(t) = ; t! Proof: It follows from (3.5) - (3.9) t V (t) =?hae(t); e(t)i? hqe(t); e(t)i? h( BB + )e(t); e(t)i +h e(t); Ax m (t)i: Thus, from (A) which implies that V (t) + min t V (t)? min ; ; je(t)j X + j e(t)j j X + jax m (t)j X; je()j X + j e()j X V () + jax m ()j X : Using (A) an similar arguments as in the proof of Theorem 3., we can show that an p h e(t); e(t)i! as t!. Z je()j X < 3

14 Remark 5. A etaile iscussion concerning assumption (A) will appear in a forthcoming paper. In nite imensional systems it resembles the enition of quaratic stability, [8]. assumption of Theorem 5. is satise if x m () D(A p ) an r() L loc (; ; RI m ). Example : We consier the heat iusion equation in the interval [; ] given p (t; with bounary an initial x = p (t; a p + p (t; ) b p + c p x p (t; ) x p (t; ) = = x p (t; ); x p (; ) = x L (; ): In this case we have that the operator A p is given by A p '() = a p q() '() '() + b p + c p '(); with D(A p ) = H (; ) \ H (; ), where we choose q() = :? sin [(? :5)] ; : The secon The nominal plant operator A is chosen to be A '() = a '() + c '(): In our calculations we have chosen the following numerical values for the coecients of A p an A to be a p = :5? b p = :? c p = :? The input operator B is taken to be an the reference signal is given by an a = :5? c = :?3 : B() = [:3;:7](); ; r(t) = t 3: + : sin e?:t : 5 All computations are carrie out by a numerical approximation metho using nite element methos with the linear spline elements an the Fehlberg fourth-fth orer Runge-Kutta metho for time integration. The numerical implementation of the propose aaptive law an its nite imensional approximation an convergence proofs will appear in greater etail in a forthcoming paper. In Figure we plot the state error versus time an we see that the error converges to zero. For comparison, two cases are presente which represent the robust nonaaptive case (ashe line) an the aaptive case (soli line). We can observe that the state error using the aaptive law (3.) 4

15 non aaptive e(t) aaptive Time (sec) Figure : State error, e(t) = x p (t)? x m (t): aaptive an non-aaptive case...9 K m.8.7 K p.6.5 K() ξ Figure : Nominal gain K m, plant gain K p an estimate gain K(). 5

16 converges to zero faster that the non-aaptive case where the constant feeback law (.9) was use. This agrees with the theoretical estimates of x an x3 concerning the constants an ~ that are relate to the tracking of the state error. Our calculations were carrie out by choosing the initial values of the plant an reference moel states an the initial guess of the estimate K(t) at t = as x p (; ) = :? sin(), x m (; ) = an K(), respectively. In Figure, we plot the nominal feeback gain K m (soli line), an the nal value of the estimate K() (ashe line). In aition, we plot the feeback gain K p (otte line) which is the feeback gain that correspons to the \optimal" gain base on the actual plant operator A p, i.e. K p = B p where p solves A p p + p A p? p BB? I p + Q x = ; 8x D (A p ): This shows that the aaptive law K(t) is ajuste to some optimal feeback gain for the uncertain plant but not to the nominal gain K m or the optimal gain K p. 6

17 Example : As a secon example, we consier a one imensional Euler-Bernoulli beam with Kelvin-Voigt viscoelastic amping. It is assume that a single piezoceramic patch locate in the center of the beam is use for actuation, see []. The unerlying equation is given x p (t; x p (t; 3 x p (t; ) EI p + c D I p [ ; ]()u(t) + f(t; ) (5.) with bounary an initial conitions x p (t; ) p(t; = x p (t; l) p (; ) x p (; ) = w () an = w (); If we write (5.) in a rst orer form (see []) on the state space X = H (; l) (; l), then the L plant generator is given by A p = " I K p with D(A p ) = f(; ) X : H (; l) an K p + D p L (; l)g, where the plant stiness an amping operators are given by K p '() = EI p '()! D p # D p '() = Similarly, the nominal plant generator is given by " # I A = ; K D with the corresponing stiness an amping operators given by! K '() = '() EI D '() = ; = ;! 3 '() c D I p 3 '() c D I an D(A ) = f(; ) X : H (; l) an K +D L (; l)g. We have chosen the following numerical values for the coecients of A p an A to be EI p = 5 c D I p = :?3 an The (unboune) input operator B L(RI ; H? (; l)) is given by EI = c D I = :?3 : K B [ ; ]()u(t) where [ ; ]() is the characteristic function on the interval [ ; ] for l = :6 with [ ; ] = [:5; :45], an the constant K B is a piezoceramic constant that is given by K B = :33655?3. The reference signal r(t) is given by r(t) = 5 + sin t cos t :

18 In Figure 3 we plot the norm of the isplacement error for the aaptive (soli line) an nonaaptive cases (ashe line). Once again we observe that the state error using the aaptive law (3.) converges to zero faster than the non-aaptive case with the constant feeback law (.9). The isplacement feeback gains K m (soli line), K p (otte line) an the nal value K() (ashe line) are plotte in Figure 4. It is observe, like the previous example, that the isplacement feeback gain K(t) converges to some optimal feeback gain but not to the nominal isplacement gain K m or the optimal isplacement gain for the uncertain plant K p. The 3-D plots of the isplacement errors for the aaptive an non-aaptive cases are plotte in Figures 5 & 6 respectively. Inspection of the last two gures reveals that the state error e(t; ) converges to zero faster in the aaptive case than that of the non-aaptive case. 8

19 .. non aaptive.8 e(t).6.4 aaptive Time (sec) Figure 3: State error, e(t) = x p (t)? x m (t): aaptive an non-aaptive case. 3 x 7 soli: K m.5.5 otte: K p ashe: K() ξ Figure 4: Nominal gain K m, plant gain K p an estimate gain K(). 9

20 x ξ Time (sec).8 Figure 5: State error, e(t; ) = x p (t; )? x m (t; ): aaptive case. x ξ Time (sec).8 Figure 6: State error, e(t; ) = x p (t; )? x m (t; ): non-aaptive case. 6 Conclusions an Future Research An aaptive control law was propose an shown to actually enhance the robustness properties of the close loop system. Its performance excees the performance of a non-aaptive robust controller. Through numerical nings, the above claim was justie an shown that even with unknown state initial conitions the aaptive controller yiels better tracking. Possible exten-

21 sions of the above aaptive scheme woul involve systems with unboune input operators an uncertainties in input operators as well as a more general class of system operators in orer to inclue a wier class of innite imensional systems. This is the subject of our current research. The propose aaptive control law (.), (.4) assumes full state observation. We will exten our stuy to the construction of aaptive compensator ynamics base on a partial state observation. Moreover, the convergence properties of the approximation of the aaptive law that uses a nite imensional approximation of the Riccati solution an an approximate reference moel x m (t) will be stuie. References [] H. T. Banks an K. Ito, A unie framework for approximation an inverse problems for istribute parameter systems, Control Theory an Avance Technology, 4 (988), pp. 73{9. [], Approximation in LQR problems for innite imensional systems with unboune input operators, CRSC Report 94-, Center for Research in Scientic Computation, North Carolina State University, Raleigh, NC , November, 994; Journal of Mathematical Systems, Estimaton an Control, to appear. [3] P. A. Ioannou an J. Sun, Robust Aaptive Control, Prentice Hall, Englewoo Clis, NJ, 995. [4] K. Ito, Well-poseness an Stability of Perturbe Linear Evolution Equations, in Proceeings of Conference on Sensing, Ientication, an Control of Flexible Structures, K. A. Morris, e., Fiels Institute Communications, The Fiels Institute for Research in Mathematical Sciences,Waterloo, Canaa, 993, AMS, pp. 79{96. [5] K. Ito an R. K. Powers, Chanrasekhar equations for innite imensional systems, SIAM J. Control an Optimization, 5 (987), pp. 596{6. [6] B. V. Keulen, H -Control for Distribute Parameter Systems: A State-Space Approach, Birkhauser, Boston-Basel-Berlin, 993. [7] B. V. Keulen, M. Peters, an R. Curtain, H control with state feeback: The inniteimensional case, Journal of Mathematical Systems, Estimation, an Control, 3 (993), pp. { 39. [8] P. Khargonekar, I. R. Petersen, an K. Zhou, Robust stabililization of uncertain linear systems: quaratic stabilizability an H theory, IEEE Trans. on Automat. Control, 35 (99), pp. 356{36.

22 [9] K. S. Narenra an A. M. Annaswamy, Stable Aaptive Systems, Prentice Hall, Englewoo Clis, NJ, 989. [] A. Pazy, Semigroups of Linear Operators an Applications to Partial Dierential Equations, Springer-Verlag, New York, 983. [] V. M. Popov, Hyperstability of Control Systems, Springer-Verlag, Berlin, 973. [] A. J. Pritchar an S. Townley, Robustness of linear systems, J. of Di. Eqns, 77 (989), pp. 54{86. [3], Robustness optimization for abstract, uncertain control systems: unboune inputs an perturbations, in Proc. of IFAC Symposium on Distribute Parameter Systems, E. Jai, e., Perpignan, France, July 989, Pergamon Press, pp. 7{. [4], Robustness optimization for uncertain innite-imensional systems with unboune inputs, IMA J. of Math. Control an Information, 8 (99), pp. {34. [5] S. Sastry an M. Boson, Aaptive Control: Stability, Convergence an Robustness, Prentice-Hall, Englewoo Clis, NJ, 989. [6] S. Townley, Robust stability raii for istribute parameter systems: A survey, in Analysis an Optimization of Systems: State an Frequency Domain Approaches for Inifnite Dimensional Systems, R. F. Curtain, e., Proceeings of the th International Conference Sophia- Antipolis, France, June 9-99, Springer-Verlag, pp. 3{33.