Adaptive compensators for perturbed positive real infinite ... - CiteSeerX

Int. J. Appl. Math. Comput. Sci., 2003, Vol. 13, No. 4, 441–452

ADAPTIVE COMPENSATORS FOR PERTURBED POSITIVE REAL INFINITE-DIMENSIONAL SYSTEMS RUTH F. CURTAIN∗ , M ICHAEL A. DEMETRIOU∗∗ K AZUFUMI ITO∗∗∗ ∗

Department of Mathematics, University of Groningen P.O. Box 800, 9700 AV Groningen, The Netherlands e-mail: [email protected]

∗∗ Department of Mechanical Engineering Worcester Polytechnic Institute, Worcester, MA 01609, USA e-mail: [email protected] ∗∗∗ Center for Research in Scientific Computation, Box 8205 North Carolina State University, Raleigh, NC 27695-8205, USA e-mail: [email protected]

The aim of this investigation is to construct an adaptive observer and an adaptive compensator for a class of infinitedimensional plants having a known exogenous input and a structured perturbation with an unknown constant parameter, such as the case of static output feedback with an unknown gain. The adaptive observer uses the nominal dynamics of the unperturbed plant and an adaptation law based on the Lyapunov redesign method. We obtain conditions on the system to ensure uniform boundedness of the estimator dynamics and the parameter estimates, and the convergence of the estimator error. For the case of a known periodic exogenous input we design an adaptive compensator which forces the system to converge to a unique periodic solution. We illustrate our approach with a delay example and a diffusion example for which we obtain convincing numerical results. Keywords: infinite dimensional systems, positive real systems, adaptive controllers

1. Introduction

The proposed observer is of the form

In this paper we construct adaptive observers for the infinite-dimensional linear system with structured perturbations on a complex Hilbert space X d x(t) = (A0 + BΓC) x(t) + Bu(t) + f (t), dt x(0) = x0 ∈ X,

(1)

y(t) = Cx(t), where A0 is an infinitesimal generator of an exponentially stable C0 semigroup T (t), t ≥ 0 on X (Pazy, 1983). The signals u(t) and y(t) are the vector-valued inputs and outputs, respectively, and f (t) is an X-valued known exogenous input. The operators B ∈ L(Rm , X) and C ∈ L(X, Rm ) are known, but the gain matrix Γ ∈ Rm×m is unknown. The structured perturbation term BΓC may represent a passive feedback loop. The gain Γ may depend on other factors such as temperature and age, and consequently it needs to be estimated in real time.

d b x b(t) = A0 x b(t) + B Γ(t)y(t) + Bu(t) + f (t), dt x b(0) = x b0 ,
db Γ(t) = G y(t) − C x b(t) y T (t), dt b b0 , Γ(0) = Γ

(2)

where G = GT > 0 is a pre-selected adaptation matrix gain. The objective of the paper is to analyze the stability and convergence properties of the proposed adaptive observer. Simplified versions of this adaptive observer scheme for special classes of systems were studied in earlier papers (Curtain et al., 1997; Demetriou and Ito, 1996; Demetriou et al., 1998). Our main result (Theorem 2) uses a Lyapunov equation of the form A∗ Qx + QAx = −L∗ Lx, B ∗ Q = C,

x ∈ D(A0 ), (3)

R.F. Curtain et al.

442 with A0 + µI replacing A, where µ is a certain positive constant, Q ∈ L(X) and L is a bounded operator on D(A0 ). That is, if there exist a pair of operators b (Q, L) satisfying (3), then the gain estimate Γ(t) and the observer error e(t) = x(t) − x b(t) are bounded and 1 eµt kQ 2 e(t)kX → 0 as t → ∞ provided that u, y ∈ ∞ L`oc (0, ∞; Rm ). Moreover, under a persistence of excib → Γ as tation condition the parameter convergence Γ(t) t → ∞ is proved. The earlier results on the existence of solutions to Lur’e equations in the literature are too restrictive for our application. Balakrishnan (1995) assumes that A is a Riesz spectral operator and the scalar inputs and outputs are very smooth; both Curtain (1996a; 1996b) and Pandolfi (1997) require the exact controllability, which is never satisfied by our class of systems. In the recent results by Curtain (2001) and Pandolfi (1998), the latter assumption is removed. The results in (Curtain, 2001) provide the type of the positive-real lemma suited to our applications. A key assumption to ensure the existence of µ, L, Q satisfying (3) is that there exists a positive number µ so that (A0 + µI, B, C) satisfies a positive-real condition (Curtain, 2001) as follows with A = A0 + µI. Definition 1. Let A be an infinitesimal generator of an exponentially stable semigroup on X. If the transfer function G(s) = C(s I − A)−1 B : C0+ → L(C m ), where C0+ = {s ∈ C : Re s > 0} satisfies (i)

G(s) = G(¯ s),

(4)

(ii)

G(s) is holomorphic on C0+ ,

(5)

(iii)

G(s) + G(s)∗ ≥ 0 for all s = ı ω, ω ∈ R, (6)

In Section 4, we propose an adaptive compensator design using a separation scheme with an LQR design on the resulting adaptive observer. To illustrate the above results we present some numerical results on our three examples in Section 5.

2. Positive-Real Lemmas The adaptive observer scheme is only applicable to positive-real systems and the key is a positive real lemma. As is well known, it is possible to have different versions corresponding to spectral factors of different dimensions. We have found three useful versions. The first version is particularly useful for dissipative systems with collocated actuators and sensors, and it was utilized in (Demetriou and Ito, 1996). These systems are always positive-real, and the following lemma is trivial. Lemma 1. Suppose that A is the infinitesimal generator of a contraction semigroup on X and B ∈ L(Rm , X). Then Q = I is a solution to the constrained Lyapunov equation for x ∈ D(A) hAx, Qxi + hQx, Axi ≤ 0, B∗Q = B∗. In the adaptive observer application, one also needs to suppose that A generates an exponentially stable C0 semigroup. An example satisfying Lemma 1 is the following. Example 1. Consider the diffusion equation ∂z ∂2z = 2 + b(ξ)u(t), ∂t ∂ξ

z(ξ, 0) = z0 (ξ), Z 1 b(ξ)z(ξ, t) dξ, y(t) =

then G is positive real. Although there are many results on the positivereal lemma in the infinite-dimensional literature (e.g., see (Staffans, 1995; 1997; 1998; 1999; Weiss, 1994; 1997; Weiss and Weiss, 1997), most are in terms of a certain Riccati equation. For our main result we need the singular equation (Lur’e) (3), for which no corresponding Riccati equation exists. In Section 2, we state and discuss three distinct versions of a positive-real lemma that are in essence existence theorems for a Lur’e equation like (3). Moreover, we collect various sets of verifiable sufficient conditions. We discuss three examples which satisfy at least one version of the positive-real lemma. In Section 3 we prove the main theorem based on a Lyapunov method and a solution to the Lur’e equation, and state the persistence of the excitation condition we use for the gain estimate convergence.

z(0, t) = 0 = z(1, t),

0

where b ∈ L2 (0, 1) = X. We let   dh 2     h ∈ L (0, 1) : h, are absolutely continuous,   dξ D(A)= 2   d h    ∈ L2 (0, 1) and h(0) = 0 = h(1)  2 dξ and define Ah =

d2 h for h ∈ D(A). dξ 2

Then A has compact resolvent, eigenvalues λn = √ −n2 π 2 , n ∈ N and eigenvectors φn = 2 sin(nπξ),

Adaptive compensators for perturbed positive real infinite-dimensional systems n ∈ N, which form an orthonormal basis for L2 (0, 1). A is exponentially stable, self-adjoint and for x ∈ D(A)

443

where ( b(ξ) =

0 elsewhere.

hx, Axi ≤ −kxk2 , and it generates a contraction semigroup. Finally, note that y(t) = hb, z(·, t)i = Cz(·, t) and B = C ∗.  To treat systems for which the actuators and sensors were not collocated, the following version was proven in (Curtain et al., 1997). Lemma 2. Suppose that A is self-adjoint, has compact resolvent, it generates an exponentially stable semigroup, its eigenvalues {λn , n ∈ N} are simple and its eigenvectors {φn , n ∈ N} form an orthonormal basis for X. Suppose that b, c ∈ X satisfy

Take X = L2 (0, 1) to be the Hilbert space with the weighted inner product Z hf, gi =

h(A + µI)x, Qyi + hQx, (A + µI)yi = −hLx, Lyi, (7) hx, ci = hx, Qbi. Proof. Show by direct substitution that the following operators satisfy the constrained Lyapunov equation: ∞ X hc, φn i hx, φn iφn , hb, φn i n=1

(8)

1

Lx =

 ∞  X −2(µ + λn )hc, φn i 2 n=1

hb, φn i

hx, φn iφn .

This lemma applies to Example 1, where we can take µ = π 2 −  for any  > 0. The following example from (Curtain et al., 1997) does not satisfy the conditions of Lemma 1, but Lemma 2 does apply. Example 2. Consider the diffusion equation ∂z ∂2z ∂z = 2 −α + b(ξ)u(t) + f, ∂t ∂ξ ∂ξ z(0, t) = 0,

z(1, t) = 0,

with the output given by Z y(t) = 0

α > 0,

z(ξ, 0) = z0 (ξ),

1

e−αξ z(ξ, t) dξ,

e−αξ f (ξ)g(ξ) dξ.

Then, defining  dh    h : h, dξ are absolutely continuous D(A0 ) =  d2 h   and ∈ X, h(0) = 0 = h(1) dξ 2

A0 h =

Then there exist operators 0 ≤ Q = Q∗ ∈ L(X), L ∈ L(D(A), X) and µ > 0 such that for x, y ∈ D(A)

1

0

and hc, φn ihb, φn i > 0, n ∈ N, hc, φn i < ∞. sup n∈N hb, φn i

Qx =

1 on [0, 1/2),

      

d2 h dh −α for h ∈ D(A0 ), 2 dξ dξ

it is straightforward to show that A0 is self-adjoint with 2 eigenvalues λn = − α4 − n2 π 2 and normalized eigen√ αξ/2 vectors φn (ξ) = 2e sin(nπξ), n ∈ N. The set {φn , n ∈ N} forms an orthonormal basis for X. Let √
α 4nπ 2 1 − e− 2 (−1)n cn := hc, φn i = , n ∈ N, 4n2 π 2 + α2 bn := hb, φn i √
α α −α nπ 4 sin( 4nπ 2 1 + e− 4 cos( nπ 2 ) 2 ) − 2nπ e . = 4n2 π 2 + α2 So bn cn > 0 for all n and for certain constants m and M c n m ≤ sup ≤ M. n≥1 bn So the assumptions of Lemma 2 are satisfied, Q given by (8) satisfies the constrained Lyapunov equation (7) and it is boundedly invertible; L is unbounded.  Note that in both Lemmas 1 and 2 the L term will be unbounded in general, even though B and C are bounded, and that L maps into the state-space X. This is in contrast to the usual finite-dimensional version for which L maps into the output space Rm . The latter version is much harder to prove for infinite-dimensional systems, and earlier versions in (Balakrishnan, 1995; Curtain, 1996a; 1996b; Pandolfi, 1997) assumed very strong conditions on the system, such as exact controllability. Here we extract some useful results from (Curtain, 2001), where only mild conditions are assumed on the system operators (A, B, C).

R.F. Curtain et al.

444 First we need some extra notation:    f : C0+ → Z and f is holomorphic and  H∞ (Z) =  kf k∞ = sup kf (ı ω)kX < ∞ 

Then there exists a Q = Q∗ ∈ L(X) which satisfies, for all x ∈ D(A), the following Lur’e equations: A∗ Qx + QAx = −CΞ∗ CΞ x

(13)

B ∗ Qx = Cx.

(14)

ω∈R

2

H (Z)  +    f : C0 → Z and f is holomorphic and Z ∞ =  2  kf (x + ı ω)k2X dω < ∞  kf k2 = sup x>0

−∞

      


L2 (−ı ∞, ı ∞); Z   f : (−ı ∞, ı ∞) → Z and f is measurable and       Z ∞  21 . =       kf k2 = 0 Z ∞

CΞ T (t)z dt ≤ γkzk2 (11)

(iv) CΞ satisfies (11), where CΞ is given by

0

and Ξ(s) = CΞ (sI − A)−1 B,

(12)

for all s in some right half-plane. 1

Ξ ∈ H ∞ (L(U )) is outer if its range as a multiplication operator on H 2 (U ) is dense in H 2 (U ).

(16)

CΞ φn = Ξ(−λn )∗


−1

Cφn .

(17)

Then, there exists a Q = Q∗ ∈ L(X) satisfying (13) and (14). We note that some easily verifiable conditions for CΞ to satisfy (11) are given in (Hansen and Weiss, 1997) and cited in Lemma 5.6 in (Curtain, 2001).

Adaptive compensators for perturbed positive real infinite-dimensional systems Lemma 4. (Curtain, 2001, Lem. 5.6) Suppose that A, the generator of T (t), is a diagonal matrix on X = `2 with eigenvalue eigenvector pairs {λn , φn | n ∈ N} satisfying the following conditions:

with the transfer function g(s) =

(ii) either T (t) is analytic or there exist numbers α ≥ 0 and 0 < a ≤ b such that a| Im λn |α ≤ − Re λn ≤ b| Im λn |α .

Π(ı ω) = g(ı ω) + g(ı ω)∗ =

(18)

Then T (t) is exponentially stable and C ∈ L(D(A), `2 ) satisfies (11) if and only if there exists m ≥ 0 such that

X

(Cφn )(Cφn )∗ 2 ≤ M h,

L(` ) where R(h, ω) = {z ∈ C | 0 < Re z ≤ h, ω − h ≤ Im z < ω + h}. Finally, in Lemma 5.7 of (Curtain, 2001), various bounds on the spectral factor Ξ(s) are derived which help to verify that Ξ(−λn ) is invertible for the case of singleinput single-output systems. Several examples of SISO parabolic systems are analyzed in Section 6 of that paper, including some with boundary control and point sensing. Applying the approach of Section 6 to our Examples 1 and 2 we see that the transfer functions both have the form X hc, φn ihb, φn i g(s) = , s + λn n where bn = hb, φn i and cn = hc, φn i are both of order 1/n for large n and λn ∼ −n2 in both examples. Consequently, the analysis and the conclusions are the same as in Example 6.1 of (Curtain, 2001): Π(ı ω) ∼ m/|ω|3/2 for some m > 0 and sufficiently large ω, and Π has an outer spectral factor Ξ as in (10) which satisfies Ξ(n2 ) ≥ γ1 n3/2 for some γ1 > 0 and sufficiently large n. So CΞ is well defined by (17) and it satisfies (11). It√is interesting to note that CΞ is unbounded, |CΞ φn | ∼ n. Here we have satisfied all the conditions of Lemma 3 and there exist Q and CΞ ∈ L(D(A), C) satisfying the constrained Lyapunov equations (13) and (14). As has already been noted, Lemmas 1 and 2 only apply to special classes of SISO systems. Lemma 3 applies to a much wider class of partial differential equations, but it is not applicable to delay systems. In the following example we show how Theorem 1 can be applied to a delay system. Example 3. Consider the delay system a, b > 0,

(19) (20)

2(a + b cos(ω)) (a + b cos(ω))2 + (ω − b sin(ω))2 }

≥ 0 if a ≥ |b| . In this case, it is easy to find the spectral factor Ξ ∈ H ∞ given by

−λn ∈R(h,ω)

y(t) = x(t)

1 ∈ H ∞. s + a + be−s

Now

(i) Re λn < 0 for n ∈ N,

x(t) ˙ = −ax(t) − bx(t − 1) + u(t),

445

Ξ(s) =

α + βe−s , α2 + β 2 = 2a, αβ = b. (21) s + a + be−s

The delay system (19), (20) can be formulated on the state-space X = C ⊕ L2 (−1, 0) with generating operators defined by ! ! u r Bu = , C = r, (22) 0 f (·)   !       r     ∈ X | f is absolutely continuous,     f (·) , D(A) =       df   2      dθ (·) ∈ L (−1, 0) and f (0) = r 

 −ar − bf (−1)  A = (23) df f (·) dθ (see Curtain and Zwart, 1995, Ch. 2.4). Clearly, B and C are bounded operators and we recall from (Infante and Walker, 1977) that A generates an exponentially stable semigroup if a − |b| ≥ µ > 0 for some positive constant µ. (A, B) is approximately controllable (see Curtain and Zwart, 1995, Thm. 4.2.10). The candidate for CΞ is r

!

(CΞ x) (t) = αx(t) + βx(t − 1).

(24)

CΞ is not bounded, but it does satisfy (11) (see (Salamon, 1984), and note that T (t) is exponentially stable). We verify that (12) holds: Ξ(s) = CΞ (sI − A)−1 B. The resolvent is now given by (Curtain and Zwart, 1995, Lem. 2.4.5) ! ! r q(0) −1 (sI − A) = , (25) f (·) q(·)

R.F. Curtain et al.

446

Theorem 2. Consider the structurally perturbed system (1), where A0 is the generator of an exponentially stable C0 semigroup on X, B ∈ L(Rm , X), C ∈ L(X, Rm ), f (t) is a known exogenous signal which is locally Bochner integrable, and Γ is an unknown matrix feedback gain. If there exist a positive constant µ, Q ∈ L(X) and L ∈ L(D(A), X) or L(D(A), Rm ) satisfying the constrained Lyapunov equation for x ∈ D(A)

where q(θ) = esθ q(0) −

θ

Z

es(θ−µ) f (µ) dµ

0

1 q(0) = ∆(s)



Z

0 −s(µ+1)

r−b

e

 f (µ) dµ ,

−1

∆(s) = s + a + be−s .

∗

(A0 + µI) Qx + Q (A0 + µI) x = −L∗ Lx,

So CΞ (sI − A)−1

r f (·)

! = CΞ

q(0) q(·)

!

B ∗ Qx = Cx,

 α + βe−s r = αq(0) + βq(−1) = ∆(s)   Z b(α + βe−s ) −s 0 −sθ + 1− e e f (θ) dθ, ∆(s) −1 

(27) (28)

then the state estimator defined by (26) and the adaptation rule with adaptation matrix gain G = GT > 0 given by b˙ Γ(t) = GCe(t)y T (t),

(29)

b b0 Γ(0) = Γ have the following properties:

and CΞ (sI − A)−1 B =

α + βe−s = Ξ(s), s + a + be−s

as required. In fact, it is readily verified that the solution to the Lur’e equation (13), (14) is ! I 0 Q= . 0 β2I For the general retarded system with vector-valued inputs and outputs, see Section 7 in (Curtain, 2001).

3. An Adaptive Observer: Main Results The proposed state estimator is

(26)

b where x b(t) is the state estimate at time t and Γ(t) is the adaptive estimate of the unknown gain. In order to exb tract the adaptation rule for Γ(t), we use the Lyapunov redesign method (Khalil, 1992; Narendra and Annaswamy, 1989), which has proved successful for finite-dimensional systems. In this section, we show that the same adaptive observer that was proposed in (Curtain et al., 1997) for scalar (SISO) systems can be extended to the larger class of multivariable (MIMO) systems considered in this paper. Let X−1 be the completion of X under the norm kφk−1 = kA−1 0 φkX . Then X−1 is a Hilbert space and D(A∗0 ) ⊂ X ⊂ X−1 .

(ii) Moreover, if y ∈ L2 (0, ∞; Rm ), then ke(t)kX → 0 as t → ∞. (iii) If we assume that y ∈ L∞ (0, ∞, Rm ), then the estimation error e(t) = x(t) − x b(t) is bounded in norm for t ≥ 0. Moreover, if Q is coercive, and the plant is persistently exciting, i.e., there exist T0 , δ0 and 0 such that for each admissible gain q ∈ Rm×m with (Euclidean) norm equal to 1 and each sufficiently large t > 0, there exits t¯ ∈ [t, t + T0 ] such that

Z t¯+δ0

B q y(τ ) dτ ≥ 0 ,

−1

t¯

b x b˙ (t) = A0 x b(t) + Bu(t) + B Γ(t)y(t) + f (t), x b(0) = x b0 ,

m b (i) If u, y ∈ L∞ `oc (0, ∞; R ), then the quantities Γ(t) 1 and Q 2 e(t) are bounded in norm for t ≥ 0 and µ 1 e 2 t kQ 2 e(t)kX → 0 as t → ∞.

then the parameter convergence b → Γ as t → ∞ Γ(t) holds. Proof. (i) Consider the dynamics of the state error b e(t) ˙ = A0 e(t) + BΓy(t) − B Γ(t)y(t) e = A0 e(t) + B Γ(t)y(t), e(0) = x0 − x b0 = e0 .

(30) (31)

e b The dynamics for the parameter error Γ(t) = Γ − Γ(t) become e˙ Γ(t) = −GCe(t)y T (t), (32) e b0 = Γ e0 . Γ(0) = Γ−Γ

Adaptive compensators for perturbed positive real infinite-dimensional systems First we need to examine the well-posedness of the coupled system (30), (32), which is, in fact, a linear timedependent system " # " #" # A0 B[ · ]y(t) e(t) d e(t) = . e e dt Γ(t) −GC[ · ]y T (t) 0 Γ(t) (33) The perturbation term D(t) : X ⊕Rm×m → X ⊕Rm×m is given by " # 0 B[ · ]y(t) D(t) = . (34) −GC[ · ]y T (t) 0 m So, if y ∈ L∞ `oc (0, t1 ; R ), (33) has a unique solution given by " # " # e(t) e0 = U (t, 0) , (35) e e0 Γ(t) Γ

where U (t, s) is a mild evolution operator (Curtain and Pritchard, 1978) defined for 0 ≤ s ≤ t ≤ t1 . In fact, y(t) defined by (1) will always be in C(0, t1 ; R m ) for u, f ∈ Lp (0, t1 ; Rm ), p = 1, 2 or ∞. In general, e(t) will not be in D(A0 ), even if e0 ∈ D(A0 ). Sufficient conditions for e(t) ∈ D(A0 ) are that y(·) ∈ C 1 (0, t1 ; Rm ) and e0 ∈ D(A0 ), which are very strong. However, we assume this initially to facilitate the Lyapunov argument. We examine the asymptotic properties of (35) using the followe ing Lyapunov functional for ( Γe ) o n e T G−1 Γ e , e = he, Qei + Tr Γ (36) V (e, Γ) where Q is the solution to (27). Since e(t) ∈ D(A0 ), we may differentiate V along solutions of (33) for 0 ≤ t ≤ t1 to obtain e = hA0 e + B Γy, e Qei + hQe, A0 e + B Γyi e V˙ (e, Γ)   T ˙ −1 e e + 2Tr Γ G Γ

447

Notice that although we have assumed that e0 ∈ D(A0 ) and y ∈ C 1 (0, t1 ; Rm ) to derive (38), all terms make perfectly good sense for e0 ∈ X and y ∈ C(0, t1 ; Rm ). Moreover, (35) and the facts that sup0≤s≤t≤t1 kU (t, s)k < ∞ and that D(A0 ) is dense in X show that (38) can be extended to all e0 ∈ X. We m now extend (38) to all y ∈ L∞ `oc (0, t; R ) by appealing to Lemma A1 in Appendix, which shows that if we approximate y by a sequence yn ∈ C 1 (0, t1 ; Rm ) satisfying Z t1

y(s) − yn (s) 2 ds → 0 as n → ∞, 0

then there holds

sup U (t, s) − Un (t, s) → 0 as n → ∞. (39) 0≤s≤t≤t1

So the respective solutions to (33) satisfy 

 e n (t) → 0

e(t) − en (t) + e sup Γ(t) − Γ X

0≤s≤t≤t1

as n → ∞ (40) and this suffices to show that (38) holds for any y ∈ m e L∞ `oc (0, t1 ; R ) and e0 ∈ X. This implies that Γ ∈ 1 ∞ ∞ m×m L (0, ∞; R ) and Q 2 e ∈ L (0, ∞; X). 1

Next, we define q(t) := kQ 2 e(t)k2 and deduce the following from (38): Z t1 n o e T0 G−1 Γ e 0 = V (0). q(t1 )+2µ q(s) ds ≤ q(0)+Tr Γ 0

(41) Equation (41) and the Bellman-Gronwall Lemma imply that q(t1 ) ≤ e−2µt1 V (0) or, equivalently,

1

Q 2 e(t1 ) 2 ≤ e−2µt1 V (0).

e = −kLek2 − 2µhe, Qei − 2(Ce)T Γy n o e + 2Tr y(Ce)T Γ using (27), (28) and (32) = −kLek2 − 2µhe, Qei using bT a = Tr(abT ). (37) We now integrate (37) from t = 0 to t = t1 to obtain n o e T (t1 )G−1 Γ(t e 1) he(t1 ), Qe(t1 )i + Tr Γ Z t1 Z t1 2 + kLe(t)k dt + 2µ hQe(t), e(t)i dt 0 0 n o e T0 G−1 Γ e 0 . (38) = he0 , Qe0 i + Tr Γ

(42)

Now t1 can be chosen arbitrarily large and so µ 1 ke 2 t Q 2 e(t)k → 0 as t → ∞. (ii) If y ∈ L2 (0, ∞; Rm ), then it follows from (i) that e the forcing term B Γ(t)y(t) in (30) is in L2 (0, ∞; X). Note that for all τ ∈ [0, t] Z τ h i e e(t) = T (t−τ ) T (τ )x(0)+ T (τ − s)B Γ(s)y(s) ds 0

Z

t

e T (t − s)B Γ(s)y(s) ds.

+ τ

Since T (t) is exponentially stable and B has finite rank, this implies that ke(t)kX → 0 as t → ∞. This follows from the asymptotic property of the convolution of two L2 (0, ∞) functions (see Titchmarsh, 1962). e (iii) Since Γ(t) and y(t) are uniformly bounded in norm for t ≥ 0 and A0 generates an exponentially stable

R.F. Curtain et al.

448 semigroup, (30) shows that e(t) is uniformly bounded in norm for t ≥ 0. The parameter convergence is proven by applying the results in Section 3 of (Baumeister et al., 1997). Our persistent excitation condition coincides with the one in Definition 3.3 of (Baumeister et al., 1997). The results in (Baumeister et al., 1997) are based on the fact that

Z t¯+δ0



e )y(τ ) dτ

e(t¯ + δ0 ) ≥ B Γ(τ

− e(t¯) −1 −1 −1

t¯

Z

−

t¯

t¯+δ0

A0 e(τ ) dτ

−1

and that for 0 ≤ t1 ≤ τ e ) e Γ(t1 ) − Γ(τ Z τ = GCe(t)y T (t) dt ≤ |G| kCk |y|∞ |τ − t1 |

1 T →∞ T

Z

T

J(u) = lim

0



 1 kC2 x b(t)k2 + kR− 2 u(t)k2 dt

(45) over all controls satisfying ku(t)k dt < 0 ∞ and for which the corresponding closed loop trajectory is bounded on t ≥ 0. They showed that if f (t) is periodic, the minimizing control law is given by   u2 (t) = −R−1 B ∗ P x b(t) + r(t) , (46) limT →∞ T1

RT

2

where P = P ∗ ∈ L(X) is the solution to the Riccati equation for x ∈ D(A0 ) A∗0 P x + P A0 x − P BR−1 B ∗ P x + C2∗ C2 x = 0 (47)

t1

1 2

We use the LQR control design from (Prato and Ichikawa, 1988). Suppose that (A0 , B, C2 ) is exponentially stabilizable and detectable and 0 < R = RT ∈ L(Cm ). We seek to minimize the average cost

sZ

and r(t) is the solution to

τ

t1

ke(t)k2X dt.

Our assertion then simply follows from the corresponding results to Lemmas 3.5–3.6 and Theorem 3.4 in (Baumeister et al., 1997). Remark 1. The assumption y ∈ L2 (0, ∞) in (ii) can be verified when A0 + BΓC generates an exponentially stable C0 semigroup on X and u, f ∈ L2 (0, ∞). The assumption y ∈ L∞ (0, ∞) in (iii) can be verified when A0 + BΓC generates an exponentially stable C0 semigroup on X and u, f ∈ L∞ (0, ∞).


r(t) ˙ = A∗0 − P BR−1 B ∗ r(t) − P f (t), r(t) → 0 as t → ∞. (48) Equation (48) has the solution Z ∞ r(t) = TP∗ (s − t)P f (s) ds,

where TP (t) is the exponentially stable C0 -semigroup generated by A − BR−1 B ∗ P . The closed loop trajectory converges exponentially fast to the periodic solution Z

  TP (t − s) f (s) − BR−1 B ∗ r(s) ds, (50)

−∞

In this section, we propose an adaptive compensator for the perturbed plant (1) where f (t) is a known exogenous signal. We obtain results for f (t) a periodic signal and for f ∈ L2 (0, ∞; X). First we apply output injection to obtain a modified control problem: b u(t) = u2 − Γ(t)y(t).

(43)

This has the advantage of producing the new estimator dynamics x b˙ (t) = A0 x b(t) + Bu2 (t) + f (t), x b(0) = x b0 and the same error dynamics (33) for

t

p(t) =

4. Adaptive Compensators

(49)

t

(44)

i.e., lim eνt kb x(t, t0 ) − p(t)kX = 0,

t→∞

(51)

where ν is the decay rate of TP (t). In the case of a constant exogenous signal, i.e., f (t) = f0 , we get  −1   p(t) = − A0 −BR−1 B ∗ P A∗0 A0 −BR−1 B ∗ P f0 . We note that for the case when f ∈ L2 (0, ∞; X), the feedback control law u2 (t) = −R−1 B ∗ P x b(t)

e(t) ( Γ(t) ) b

as before.

So it remains to design a controller u2 (t) for the system (44).

ensures that kx(t)k → 0 as t → ∞ as argued in Theorem 3.1, (ii), see also Lemma 12 in (Oostveen and Curtain, 1998).

Adaptive compensators for perturbed positive real infinite-dimensional systems We propose the following adaptive compensator for the case of a known periodic exogenous input:
x b˙ (t) = A0 − BR−1 B ∗ P x b(t) − BR−1 B ∗ r(t) + f (t), x b(0) = x b0 ,
r(t) ˙ = A∗0 − P BR−1 B ∗ r(t) − P f (t), r(t) → 0 as t → ∞, b u(t) = −R−1 B ∗ (P x b(t) + r(t)) − Γ(t)y(t), b˙ Γ(t) = GCe(t)y T (t),

b b0 Γ(0) =Γ

for our structurally perturbed plant (1). In Section 3 we showed that

µ 1 e 2 t Q 2 (x(t) − x b(t)) X → 0 as t → ∞, independently of the choice of the control. So combining this with the results in this section, we conclude that for the case of a known periodic input f (t)

1

eβt Q 2 (p(t) − x b(t)) X → 0 as t → ∞, where β = min(ν, µ/2) and p(t) is given by (50).

5. Examples and Numerical Results We present some numerical results for the three examples considered in Section 2. For each of these examples, there exists a solution Q ∈ L(X) satisfying (27) for a certain µ > 0, and in all three cases Q is invertible. Consequently, we can conclude that for the adaptive observer and adaptation rule (2)

eµt (x(t) − x b(t)) X → 0 and with the adaptive compensator of Section 4

βt

e (x(t) − p(t)) → 0. X All the computations described below were carried out on a Digital Personal Workstation 433 au-Series in the Mechanical Engineering Department at Worcester Polytechnic Institute. A finite element Galerkin approximation scheme based on spline elements was used for the spatial discretization of the two PDEs similar to the one developed in (Baumeister et al., 1997). The resulting finite dimensional ODE systems were integrated in time using a Fehlberg fourth-fifth Runge-Kutta method. The delay system in Example 3 was discretized using the method presented in the paper by Ito and Kappel (1991). The resulting evolution (finite dimensional) system was similarly integrated using the Runge-Kutta code rkf45.f.

449

Example 4. As was already mentioned in Section 2, we can choose in this case µ = π 2 −  in (7), and define the operators Q and L via (8). Alternatively, when Lemma 1 is used, we have Q = I and L = 0 with the same µ. The input operator b(x) was chosen as ( 1 on [0, 1/2), b(ξ) = 0 elsewhere. The unknown gain was chosen as Γ = 1, and as initial conditions we chose z(ξ, 0) = sin(πξ) and zb(ξ, 0) = cos(2πξ) − 1. The exogenous input was f (ξ, t) = b 50χ[0,1] (ξ) sin(2πt). The initial guess for Γ(0) = 0 with an adaptive gain of G = 20. Figure 1(a) depicts the time R1 evolution of the output state error Ce(t) = 0 (z(ξ, t) − zb(ξ, t)) dξ. The convergence to zero is achieved within b 0.5 seconds. The parameter estimate Γ(t) (dashed) and the actual value of Γ = 1 are depicted in Fig. 1(b). Parameter convergence is achieved in 4 seconds.  Evolution of output error, Ce(t)

1

0.5

0

−0.5 0

0.5

1

1.5

Γ and

1.5

2

2.5

3

4

5

6

^ Γ(t)

1

0.5

0 0

1

2

3 Time (sec)

Fig. 1. Evolution of (a) output error and (b) parameter estimate b Γ(t) (dashed) – actual parameter Γ (solid).

Example 5. Equations (7) and (8) can be satisfied with 2 µ = α4 π 2 − , where the parameter α = 0.2. Initial conditions were set as z(ξ, 0) = sin(πξ) and zb(ξ, 0) = −0.25 sin(πξ). A constant in space and time exogenous function is implemented as f (ξ, t) = 50χ[0,1] (ξ) and b Γ(0) = 0 with G = 2. With these values of initial conditions, it is observed in Fig. 2(a) that the output state error converges to zero in 3.5 seconds. Furthermore, the pab rameter Γ(t) converges to the actual value Γ = 1 in 4 seconds as shown in Fig. 2(b). Example 6. The plant parameters were chosen as a = 3, b = 1. In this case the solution to the constrained Lya-

R.F. Curtain et al.

450 Evolution of output error, Ce(t)

0.2

2 seconds. Parameter convergence is also achieved in about 5 seconds. For numerical results for a multivariable example the reader is directed to (Demetriou et al., 1998).

0.15 0.1 0.05 0 −0.05 −0.1 0

0.5

1

1.5

2

2.5

Γ and

1.5

3

3.5

4

4.5

5

^ Γ(t)

Acknowledgment The research of the third autor was supported in part by the Airforce Office of Scientific Research under Grant No. AFOSR-49620-95-1-0447.

1

0.5

0 0

0.5

1

1.5

2

2.5 Time (sec)

3

3.5

4

4.5

5

Fig. 2. Evolution of (a) output error and (b) parameter estimate b Γ(t) (dashed) – actual parameter Γ (solid).

1.5 1 0.5

Curtain R.F. (1996a): Corrections to the Kalman-YakubovichPopov Lemma for Pritchard-Salamon systems. — Syst. Contr. Lett., Vol. 28, No. 4, pp. 237–238.

0 −0.5 −1 −1

0

1

2

3

Γ and

20

4

5

6

^ Γ(t)

Curtain R.F. (1996b): The Kalman-Yakubovich-Popov Lemma for Pritchard-Salamon systems. — Syst. Contr. Lett., Vol. 27, No. 1, pp. 67–72. Curtain R.F. (2001): Linear operator inequalities for stable weakly regular linear systems. — Math. Contr. Sign. Syst., Vol. 14, No. 4, pp. 299–337.

10 0 −10 −20 0

Balakrishnan A.V. (1995): On a generalization of the KalmanYakubovic lemma. — Appl. Math. Optim., Vol. 31, No. 2, pp. 177–187. Baumeister J., Scondo W., Demetriou M. and Rosen I. (1997): On-line parameter estimation for infinite dimensional dynamical systems. — SIAM J. Contr. Optim., Vol. 3, No. 2, pp. 678–713.

Evolution of x(t) and ^ x(t)

2

References

1

2

3 Time (sec)

4

5

6

Fig. 3. Evolution of (a) plant state x(t) (solid) and state esb timate x b(t) (dashed); (b) parameter estimate Γ(t) (dashed) – actual parameter Γ (solid).

Curtain R.F., Demetriou M.A. and Ito K. (1997): Adaptive observers for structurally perturbed infinite dimensional systems. — Proc. IEEE Conf. Decision and Control, San Diego, California, pp. 509–514. Curtain R.F. and Pritchard A.J. (1978): Infinite Dimensional Linear Systems Theory. — Berlin: Springer. Curtain R.F. and Zwart H.J. (1995): An Introduction to Infinite Dimensional Linear Systems Theory. — Berlin: Springer.

punov equations (13), (14) is " # I 0 √ Q= , 0 (3 ± 8)I

Demetriou M.A., Curtain R.F. and Ito K. (1998): Adaptive observers for structurally perturbed positive real delay systems. — Proc. 4th Int. Conf. Optimization: Techniques and Applications, Curtin University of Technology, Perth, Australia, pp. 345–351.

which is boundedly invertible. The actual value of the parameter was Γ = 0.4 with b the initial condition for its estimate chosen as Γ(0) = 0.2. The initial state was set at x(t − 1) = sin(4t − 1) − sin(−1) and the state estimate as x b(t − 1) = 0.5 sin(4t − 1); thus x(0) = sin(3) + sin(1) and x b(0) = 0.5 sin(3). Here we had f (t) = 0 for the exogenous signal and chose an adaptive gain of G = 500. It is observed from Fig. 3(a) that the state estimate converges to the plant state in about

Demetriou M.A. and Ito K. (1996): Adaptive observers for a class of infinite dimensional systems. — Proc. 13th World Congress, IFAC, San Francisco, CA, pp. 409–413. Hansen S. and Weiss G. (1997): New results on the operator Carleson measure criterion. — IMA J. Math. Contr. Inf., Vol. 14, No. 1, pp. 3–32. Infante E. and Walker J. (1977): A Lyapunov functional for scalar differential difference equations. — Proc. Royal Soc. Edinburgh, Vol. 79A, Nos. 3–4, pp. 307–316.

Adaptive compensators for perturbed positive real infinite-dimensional systems Ito K. and Kappel F. (1991): A uniformly differentiable approximation scheme for delay systems using splines. — Appl. Math. Optim., Vol. 23, No. 3, pp. 217–262. Khalil H.K. (1992): Nonlinear Systems. — New York: Macmillan. Narendra K.S. and Annaswamy A.M. (1989): Stable Adaptive Systems. — Englewood Cliffs, NJ: Prentice Hall. Oostveen J.C. and Curtain R.F. (1998): Riccati equations for strongly stabilizable bounded linear systems. — Automatica, Vol. 34, No. 8, pp. 953–967.

451

Appendix Lemma A1. Suppose that A generates a C0 -semigroup on the Hilbert space X and consider Pkthe mild evolution operator U (t, s) generated by A + i=1 Di yi (t), Di ∈ L(X) and yi ∈ L∞ (0, t1 ). Let Un (t, s) be the evolution Pk operator generated by A + i Di yin (t), where for each i yin (t) is a sequence of functions in C 1 (0, t1 ) satisfying Z t1

n

yi (t) − yi (t) 2 dt → 0 as n → ∞. 0

Pandolfi L. (1997): The Kalman-Yacubovich theorem: An overview and new results for hyperbolic boundary control systems. — Nonlin. Anal. Theory Meth. Applic., Vol. 30, No. 30, pp. 735–745.

There holds

sup U (t, s) − Un (t, s) L(X) → ∞ as n → ∞.

Pandolfi L. (1998): Dissipativity and Lur’e problem for parabolic boundary control systems. — SIAM J. Contr. Optim., Vol. 36, No. 6, pp. 2061–2081.

Proof. We only give a detailed proof for k = 1, since the arguments extend readily to any finite k. We recall from (Curtain and Zwart, 1995) the defining equations for U (t, s) and Un (t, s):

Pazy A. (1983): Semigroups of Linear Operators and Applications to Partial Differential Equations. — New York: Springer. Prato G.D. and Ichikawa A. (1988): Quadratic control for linear periodic systems. — Appl. Math. Optim., Vol. 18, No. 1, pp. 39–66. Salamon D. (1984): Control and Observation of Neutral Systems. — Research Notes in Mathematics, 91, Boston: Pitman Advanced Publishing Program. Staffans O.J. (1995): Quadratic optimal control of stable abstract linear systems. — Proc. IFIP Conf. Modelling and Optimization of DPS with Applications to Engineering, Warsaw, Poland, pp. 167–174. Staffans O.J. (1997): Quadratic optimal control of stable well-posed linear systems. — Trans. Amer. Math. Soc., Vol. 349, No. 9, pp. 3679–3715. Staffans O.J. (1998): Coprime factorizations and well-posed linear systems. — SIAM J. Contr. Optim., Vol. 36, No. 4, pp. 1268–1292. Staffans O.J. (1999): Quadratic optimal control of well-posed linear systems. — SIAM J. Contr. Optim., Vol. 37, No. 1, pp. 131–164. Titchmarsh E.C. (1962): Introduction to the Theory of Fourier Integrals. — New York: Oxford University Press. Weiss M. (1994): Riccati Equations in Hilbert Spaces: A Popov Function Approach. — Ph.D. Thesis, Rijksuniversiteit Groningen, The Netherlands. Weiss M. (1997): Riccati equation theory for PritchardSalamon systems: a Popov function approach. — IMA J. Math. Contr. Inf., Vol. 14, No. 1, pp. 45–83. Weiss M. and Weiss G. (1997): Optimal control of stable weakly regular linear systems. — Math. Contr. Signals Syst., Vol. 10, No. 4, pp. 287–330.

0≤s≤t≤t1

U (t, s)x = T (t − s) Z t T (t − α)Dy(α)U (α, s)x dα +

(A1)

s

and Un (t, s)x = T (t − s) Z t + T (t − α)Dyn (α)Un (α, s)x dα, (A2) s

and the estimate kU (α, s)k ≤ M e(α−s)(ω+µ) ,

(A3)

where kT (t)k ≤ M eωt ,

t ≥ 0,

(A4)

and µ = M kDkkykL∞ (0,t1 ) > 0. Consider the following estimates obtained using (A1) and (A2):

U (t, s) − Un (t, s) Z t



≤ T (t − α)D |y(α) − yn (α)| U (α, s) dα s

Z t



+ T (t − α)D kyn kL∞ U (α, s) − Un (α, s) dα s

Z t ≤ kDk M eω(t−α) M e(ω+µ)(α−s) |y(α) − yn (α)| dα s

Z t

+kDk kyn kL∞ M eω(t−α) U (α, s)−Un (α, s) dα. s

R.F. Curtain et al.

452

Defining fn (t, s) = e−ω(t−s) U (t, s) − Un (t, s) , we obtain Z t fn (t, s) ≤ M 2 kDk eµα e−µs |y(α) − yn (α)| dα

We integrate from t to s noting that fn (s, s) = 0 to obtain e−C2 t fn (t, s)

s

Z

≤ 2C1 µe−µs

t

+ kDk kyn kL∞

fn (α, s) dα

Z

t

|y(α) − yn (α)|2 dα



1 2

 21

t 2µ(α−s)

×

e

dα Z

+ kDk kyn kL∞

t

fn (α, s) dα s

12 21 Z t1 2µ(t−s) 2 = C1 e − 1 |y(α) − yn (α)| dα s

+ C2

=

2C1 µ −µs  (µ−C2 )t (µ−C2 )s  e e −e ky−yn kL2 (0,t1 ) µ−C2

fn (t, s) ≤

s

Z

e(µ−C2 )β dβ ky − yn kL2 (0,t1 )

and

s

Z

t

s

s

≤ M 2 kDk

Z

 2C1 µ  µ(t−s) e − eC2 (t−s) ky−yn kL2 (0,t1 ) µ − C2

and

U (t, s) − Un (t, s) 2C1 µ h (ω+µ)(t−s) (ω+C2 )(t−s) i ≤ e −e ky−yn kL2 (0,t1 ) , µ − C2 which proves our claim.

t

fn (α, s) dα, s

Received: 5 September 2002 Revised: 30 June 2003

where C1 and C2 only depend on t1 . Thus fn (t, s) ≤ 2C1 eµ(t−s) ky−yn kL2 (0,t1 ) +C2

Z

t

fn (α, s) ds s

and differentiating this inequality with respect to t for fixed s yields dfn (t, s) ≤ 2C1 µeµ(t−s) ky − yn kL2 (0,t1 ) + C2 fn (t, s) dt and
d −C2 t e fn (t, s) ≤ 2C1 µeµ(t−s) e−C2 t ky−yn kL2 (0,t1 ) . dt