Product of nonnegative operators and infinite-dimensional H∞ Riccati equations

Systems & Control Letters 41 (2000) 183–188

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Product of nonnegative operators and inÿnite-dimensional H∞ Riccati equations Akira Ichikawa ∗ Department of Electrical and Electronic Engineering, Shizuoka University, Hamamatsu 432-8561, Japan Received 20 April 1999; received in revised form 22 May 2000; accepted 20 June 2000

Abstract In this paper we collect some useful properties of the product of nonnegative operators in a Hilbert space. We then apply them to the standard H∞ -control problem for inÿnite-dimensional time-varying systems and give necessary and sucient conditions for the existence of a suboptimal controller by three conditions involving two independent Riccati equations with c 2000 Elsevier Science B.V. All rights reserved. a coupling inequality. Keywords: Spectrum; Nonnegative operators; H∞ -control; Time-varying; Riccati equations

1. Introduction The standard H∞ control problem for a continuoustime system is to ÿnd necessary and sucient conditions for the existence of a so-called -suboptimal controller and the characterization of all such controllers [3]. As is well-known, necessary and sucient conditions are given either by solutions of two coupled Riccati equations (X∞ ; Ytmp in [3]) or solutions of two Riccati equations (X∞ ; Y∞ in [3]) with a coupling condition r(X∞ Y∞ ) ¡ 2 , where r(M ) denotes the spectral radius of M . In application, the conditions of the second type are usually convenient to use as we can solve the two Riccati equations independently. The H∞ theory has been extended to time-varying systems in [11] and necessary and sucient conditions of the ÿrst type are given. The H∞ problem on the ÿnite horizon has also been considered in [9] and conditions of the second type are given. Necessary and sucient ∗ Tel/Fax: +81-53-478-1088. E-mail address: [email protected] (A. Ichikawa).

conditions of the second type on the inÿnite horizon have been established recently in [6]. The H∞ theory has been extended to inÿnite-dimensional systems and results analogous to [3] are obtained in [7]. The time-varying case with initial uncertainty is also considered in [5] and the conditions of the ÿrst type are given. In this paper we shall collect some properties of the product of operators in Hilbert space. We shall show that three nonnegative operators X; Y and Z satisfying the equality Z − Y − (1= 2 )ZXY = 0 have useful properties. We then establish necessary and sucient conditions of the second type for the inÿnite-dimensional time-varying H∞ problem.

2. Some properties of the product of operators Let P be a bounded linear operator on a Hilbert space H . Let (P) and r(P) be the spectrum and the spectral radius of P,p respectively. Recall that r(P) = sup|(P)| = limn→∞ n |P n |. Inner products and norms

c 2000 Elsevier Science B.V. All rights reserved. 0167-6911/00/$ - see front matter PII: S 0 1 6 7 - 6 9 1 1 ( 0 0 ) 0 0 0 5 5 - 4

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A. Ichikawa / Systems & Control Letters 41 (2000) 183–188

in Hilbert spaces will be denoted by h·; ·i and | · |. The space of bounded linear operators mapping H into another Hilbert space K is denoted by L(H ; K ) and L(H ) = L(H ; H ). Lemma 1. (i) Let M; N ∈ L(H ). Then the resolvent sets of MN and NM coincide. If I −MN has a bounded inverse; then so does I −NM and N (I −MN )−1 =(I − NM )−1 N . (ii) Let M and N be nonnegative operators in L(H ). Then (MN ) ⊂ [0; |MN |]. Proof. (i) The ÿrst part is found in [8, p. 221]. (ii) It suces to show that (MN ) is nonnegative since  ¿ |MN | lies in the resolvent set of MN . But this follows from (MN )=(M 1=2 NM 1=2 ) and the fact that M 1=2 NM 1=2 is nonnegative and has a nonnegative spectrum [2, p. 615]. A direct proof using the deÿnition of the spectrum is also possible. Lemma 2. Let P ∈ L(H ) and assume (P) ⊂ [0; |P|]. If I + (1=a)P has a bounded inverse for some a ¿ 0; then r(P(I + 1=aP)−1 ) ¡ a. Proof. Since (P) ⊂ [0; |P|], f(x) = x(1 + x=a)−1 is analytic in a neighbourhood of (P) and (f(P)) = f((P)) by the spectral mapping theorem [4, p. 569]. Hence, −1 !   1 ¡ a: = sup r P I+ P a 1 + (=a)  ∈ (P) Lemma 3. Let X; Y and Z in L(H ) satisfy the equation Z − Y − (1=a)ZXY = 0 for some a ¿ 0. (i) Then I +(1=a)XZ and I −(1=a)XY have bounded inverses and −1 −1   1 1 ; Y = Z I + XZ : Z = Y I − XY a a (ii) If X and Z are nonnegative; then (XZ) and (XY ) are bounded and nonnegative and r(XY ) =

 ¡ a: 1 + (=a)  ∈ (XZ) sup

1 =I + X a

1 Z − Y − ZXY a

and I − (1=a)XY and hence by Lemma 1 and I − (1=a)YX have bounded inverses

−1 1 = I + ZX Z; a −1  −1  1 1 = I − YX Y: Z = Y I − XY a a 1 XZ a

−1



(ii) Since XY = XZ(I + (1=a)XZ)−1 , the assertion readily follows from Lemma 2. 3. H∞ Riccati equations Consider x˙ = A(t)x + B1 (t)w + B2 (t)u;

x(t0 ) = Eh;

z = C1 (t)x + D12 (t)u; zT = Fx(T );

T ¿ 0;

(1)

y = C2 (t)x + D21 (t)w; where x ∈ H is the state, h ∈ H1 , w ∈ W the disturbance, u ∈ U the control input, z ∈ Z and zT ∈ Z1 are the controlled outputs, y ∈ Y the output to be used for control, H ; H1 ; W ; U ; Y ; Z and Z1 are separable Hilbert spaces, A(t) generates an evolution operator S(t; r) ∈ L(H ) on H [1,10], all other operators are linear and bounded in appropriate spaces and operators depending on t are strongly continuous and bounded on [t0 ; ∞). For a given ¿ 0, the H∞ control problem with initial uncertainty on [t0 ; T ] is to ÿnd necessary and sucient conditions for the existence of a controller with |zT |2 + kzk2L2 (t0 ;T ; Z) 6d2 (|h|2 + kwk2L2 (t0 ;∞;W ) ) for any h ∈ H1 and w ∈ L2 (t0 ; ∞; W ) for some 0 ¡ d ¡ , where k · kL2 denotes the norm in L2 spaces. Similarly, the H∞ control problem on [t0 ; ∞) is to ÿnd necessary and sucient conditions for the existence of an internally stabilizing controller with kzk2L2 (t0 ;T ; Z) 6d2 (|h|2 + kwk2L2 (t0 ;∞; W ) )

Proof. (i) Since    1 1 I − XY I + XZ a a 

I + (1=a)XZ I + (1=a)ZX and  Y =Z I +

for any h ∈ H1 and w ∈ L2 (t0 ; ∞; W ) for some 0 ¡ d ¡ : We assume the following conditions:  = I;

A1: (i) For each t A(t) generates a C0 -semigroup on H .

A. Ichikawa / Systems & Control Letters 41 (2000) 183–188

(ii) There exists a strongly continuous mapping S(t; s) ∈ L(H ); t¿s¿t0 such that S ∗ (t; s) is strongly continuous. For each x ∈ D(A(s)), S(t; s)x ∈ D(A(t)) and @S(t; s)x = A(t)S(t; s)x; @t

S(s; s) = I:

(iii) limn→∞ Sn (t; s)x = S(t; s)x for any x ∈ H uniformly on any bounded sets t0 6s6t6T , where Sn (t; s) is the evolution operator generated by the Yosida approximation An (t) = n2 [nI − A(t)]−1 − nI of A(t) [1,10]. ∗ (t)[C1 (t) D12 (t)] = [0 I ] for each t. A2: (i) D12 ∗ (t)] = [0 I ] for each t. (ii) D21 (t)[B1∗ (t) D21 A3: (i) (A(t); B1 (t); C1 (t)) is stabilizable and detectable. (ii) (A(t); B2 (t); C2 (t)) is stabilizable and detectable. The solutions to these problems are known [5]. To introduce the results in [5] we need a deÿnition. Consider the Riccati equations − X˙ (t) = A∗ (t)X (t) + X (t)A(t) +P(t) + X (t)R(t)X (t);

(2)

Y˙ (t) = A(t)Y (t) + Y (t)A∗ (t) +Q(t) + Y (t)S(t)Y (t)

(3)

on [t0 ; ∞) where P; Q; R and S in L(H ) are bounded strongly continuous self-adjoint operators. Recall that X (t) is a mild solution of (2) if it is strongly continuous and satisÿes Z t S ∗ (r; s)[P(r) X (s)x = S ∗ (t; s)X (t)S(t; s)x +

185

3.1. H∞ Riccati equations on ÿnite horizon First we consider the H∞ -problem on [t0 ; T ]. We assume the conditions A1 and A2. Then the following result is known [5]. Theorem 1. There exists a -suboptimal controller if and only if the two conditions below hold. (i) There exists a bounded nonnegative mild solution to the Riccati equation − X˙ (t) = A∗ (t)X (t) + X (t)A(t)   1 ∗ ∗ +X (t) 2 B1 B1 − B2 B2 (t)X (t)

(4) +C1∗ (t)C1 (t); X (T ) = F ∗ F; E ∗ X (t0 )E6d2 I;

(5) for some 0 ¡ d ¡ :

(6)

(ii) For the X (t) given in (i); there exists a bounded nonnegative mild solution to the Riccati equation   1 ∗ ˙ Z(t) = A + 2 B1 B1 X (t)Z(t)

∗  1 ∗ +Z(t) A + 2 B1 B1 X (t)

  1 +Z(t) 2 XB2 B2∗ X − C2∗ C2 (t)Z(t)

(7) +B1 (t)B1∗ (t);  −1 1 ∗ E∗: Z(t0 ) = E I − 2 E X (t0 )E

(8)

We shall establish the following equivalent necessary and sucient conditions.

s

+X (r)R(r)X (r)]S(r; s)x dr for any s6t and x ∈ H [1,5]. The mild solution of (3) is similarly deÿned. Deÿnition 1. (a) A bounded self-adjoint mild solution X (t) of (2) is called a stabilizing solution if A(t) + R(t)X (t) is exponentially stable, i.e., |SX (t; s)|6M e−(t−s) ;

t0 6 s 6 t ¡ ∞

for some M ¿ 0 and  ¿ 0, where SX (t; s) is the evolution operator generated by A(t) + R(t)X (t). (b) A bounded self-adjoint mild solution Y of (3) is called a stabilizing solution if A(t) + Y (t)S(t) is exponentially stable.

Theorem 2. There exists a -suboptimal controller if and only if the three conditions below hold. (i) There exists a bounded nonnegative mild solution to the Riccati equation (4)–(6). (ii) There exists a bounded nonnegative mild solution to the Riccati equation Y˙ (t) = A(t)Y (t) + Y (t)A∗ (t)   1 +Y (t) 2 C1∗ C1 − C2∗ C2 (t)Y (t)

(9) +B1 (t)B1∗ (t); Y (t0 ) = E ∗ E; FY (T )F ∗ 6d2 I;

(10) for some 0 ¡ d ¡ :

(11)

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A. Ichikawa / Systems & Control Letters 41 (2000) 183–188

(iii) r(X (t)Y (t))6d2 ; ∀t¿t0 for some 0 ¡ d ¡ . Under condition (i) above we shall show that (ii) and (iii) imply condition (ii) in Theorem 1. We shall establish this theorem by a series of lemmas which give useful properties of these Riccati equations. For simplicity we omit the argument t in A(t), X (t), etc. in the proofs below. Lemma 4. Suppose there exists a -suboptimal controller. Then conditions (i); (ii) of Theorem 1 and condition (ii) of Theorem 2 hold. Proof. Necessity of Theorem 1 is proved in [5]. Condition (ii) of Theorem 2 is the dual to (i) and follows from the adjoint system of (1). Lemma 5. Suppose there exists a -suboptimal controller and let X (t); Y (t) and Z(t) be the solutions of (4)–(6); (9)–(11); and (7) and (8); respectively. Then Z(t) − Y (t) − 1= 2 Z(t)X (t)Y (t) = 0; ∀t ∈ [t0 ; T ]. Proof. Set Q = Z − Y − (1= 2 )ZXY . Suppose ÿrst A(t) ∈ L(H ). Then by dierentiation we have    1 1 ∗ ∗ ∗ ˙ XB2 B2 X − C2 C2 Q Q = A + 2 B1 B1 X + Z

2 ∗   1 ∗ ∗ C C − C C : +Q A + Y 1 2 2

2 1 Hence Q(t) = SZ (t; t0 )Q(t0 )SY∗ (t; t0 ) where SZ and SY are evolution operators generated by A + (1= 2 )B1 B1∗ X + Z((1= 2 )XB2 B2∗ X − C2∗ C2 ) and A + Y ((1= 2 )C1∗ C1 − C2∗ C2 ), respectively. By direct calculation we obtain Q(t0 ) = 0 and hence Q(t) = 0; ∀t¿t0 . Next, consider system (1) with A(t) replaced by An (t). Then for n large there exists a -suboptimal controller and from the ÿrst part we obtain nonnegative solutions Xn , Yn and Zn to (4) – (6), (9) – (11), (7) and (8), respectively, with A(t) replaced by An (t). Furthermore Zn − Yn − (1= 2 )Zn Xn Yn = 0; ∀t ∈ [t0 ; T ]. As in [1, Vol. II, Part II], we can show that Xn , Yn and Zn converge strongly as n → ∞ to X , Y and Z respectively. Thus, we obtain Z − Y − (1= 2 )ZXY = 0; ∀t ∈ [t0 ; T ]. Lemma 6. Suppose there exists a -suboptimal controller. Then condition (iii) in Theorem 2 holds. Proof. By Lemmas 2, 3 and 5 the spectrum of XY has the form ( 2 =( 2 + )),  ∈ (XZ). Since X and

Z are nonnegative and bounded on [t0 ; T ], (XZ) is bounded and nonnegative. Hence r(XY )6d2 for some 0 ¡ d ¡ . Lemma 7. Suppose X (t) and Y (t) satisfy the conditions of Theorem 2. Then Z(t) = Y (t)(I − (1= 2 )X (t)Y (t))−1 is a bounded nonnegative mild solution of (7) and (8). Proof. By the condition (iii), (I − (1= 2 )XY )−1 is bounded and so is Z. Moreover, by Lemma 1 Z = Y 1=2 (I − (1= 2 )Y 1=2 XY 1=2 )−1 Y 1=2 ¿0. Note that Z − Y − (1= 2 )ZXY = 0. Suppose ÿrst A(t) ∈ L(H ). Then X; Y and Z are dierentiable and     1 1 Z˙ I − 2 XY = A + 2 B1 B1∗ X Y

   1 1 I − 2 XY +Z A + 2 B1 B1∗ X

  1 +B1 B1∗ I − 2 XY

  1 ∗ ∗ XB B X − C C +Z 2 2 2 2 Y:

2 Hence Z satisÿes (7). Moreover, Z(t0 ) equals to the right-hand side of (8). For a general A(t) we replace it by the Yosida approximation An (t), derive the Riccati equation for Zn and then pass to the limit n → ∞. Now the proof of Theorem 2 follows immediately from Lemmas 4, 6 and 7. Lemma 8. (a) Suppose X (t) and Y (t) satisfy the conditions of Theorem 2 and deÿne Z(t) = Y (t) (I − (1= 2 )X (t)Y (t))−1 . Suppose that x(t) satisÿes   ∗ 1 ∗ ∗ C C1 − C2 C2 (t)x(t): (12) − x˙ = A + Y

2 1 Then x(t) ˜ = (I − 1= 2 X (t)Y (t))x(t) satisÿes  1 ˙ − x(t) ˜ = A + 2 B1 B1∗ X

∗  1 ∗ ∗ XB B X − C C (t)x(t) ˜ (13) +Z 2 2 2 2

2 and (12) and (13) are equivalent. Proof. Dierentiating x˜ we obtain   ∗  1 1 ∗ ∗ ∗ ˙ −x˜ = A + 2 B1 B1 X +Y 2 XB2 B2 X − C2 C2 Y x

 ∗ 1 1 ∗ − 2 A + 2 B1 B1 X XYx

A. Ichikawa / Systems & Control Letters 41 (2000) 183–188

 ∗   1 1 I − 2 XY x = A + 2 B1 B1∗ X

  1 + 2 XB2 B2∗ X − C2∗ C2

−1    1 1 I − 2 XY x ×Y I − 2 XY

 1 = A + 2 B1 B1∗ X

∗  1 ∗ ∗ XB2 B2 X − C2 C2 x: ˜ +Z

2

187

Theorem 4. There exists a -suboptimal controller if and only if the three conditions below hold. (i) There exists a bounded nonnegative stabilizing mild solution to the Riccati equation (14) and (15). (ii) There exists a bounded nonnegative stabilizing mild solution to the Riccati equation Y˙ (t) = A(t)Y (t) + Y (t)A∗ (t)   1 +Y (t) 2 C1∗ C1 − C2∗ C2 (t)Y (t)

This procedure can be justiÿed using An (t); Xn (t) and so on.

+B1 (t)B1∗ (t); Y (t0 ) = E ∗ E:

(18) (19)

3.2. H∞ Riccati equations on inÿnite horizon

(iii) r(X (t)Y (t))6d2 ; ∀t¿t0 for some 0 ¡ d ¡ .

Now, we consider the H∞ -problem on [t0 ; ∞). We assume further that the condition A3 holds. Then the following is known [5].

Again from [5] we have the following.

Theorem 3. There exists a -suboptimal controller if and only if the two conditions below hold. (i) There exists a bounded nonnegative stabilizing mild solution to the Riccati equation − X˙ (t) = A∗ (t)X (t) + X (t)A(t)   1 ∗ ∗ +X (t) 2 B1 B1 − B2 B2 (t)X (t)

(14) +C1∗ (t)C1 (t); E ∗ X (t0 )E6d2 I; for some 0 ¡ d ¡ :

(15)

(ii) For the X (t) given in (i); there exists a bounded nonnegative stabilizing mild solution to the Riccati equation   1 ∗ ˙ Z(t) = A + 2 B1 B1 X (t)Z(t)

∗  1 ∗ +Z(t) A + 2 B1 B1 X (t)

  1 +Z(t) 2 XB2 B2∗ X − C2∗ C2 (t)Z(t)

+B1 (t)B1∗ (t);  −1 1 E∗: Z(t0 ) = E I − 2 E ∗ X (t0 )E

(16) (17)

We shall establish the following equivalent necessary and sucient conditions.

Lemma 9. Suppose there exists a -suboptimal controller. Then conditions (i); (ii) of Theorem 3 and condition (ii) of Theorem 4 hold. Lemma 10. Suppose there exists a -suboptimal controller. Let X (t); Y (t) and Z(t) be the bounded nonnegative stabilizing solutions of (14); (15); (18); (19) and (16); (17); respectively. Then Z(t) − Y (t) − (1= 2 )Z(t)X (t)Y (t) = 0; ∀t ∈ [t0 ; ∞). Proof. Note that any -suboptimal controller is also

-suboptimal for the ÿnite horizon problem with F =0. Let XT ; YT and ZT be the corresponding solutions of three Riccati equations. Then by Lemma 5 we obtain ZT − YT − (1= 2 )ZT XT YT = 0. Since XT (t) and ZT (t) converge strongly to X (t) and Z(t) as T → ∞ [5] and since YT (t) = Y (t) on [t0 ; T ], we conclude Z − Y − (1= 2 )ZXY = 0, ∀t ∈ [t0 ; ∞). Lemma 11. Suppose X (t) and Y (t) satisfy the conditions of Theorem 4. Then Z(t) = Y (t)(I − (1= 2 )X (t)Y (t))−1 is a bounded nonnegative stabilizing solution of (16) and (17). Proof. As in the proof of Lemma 7 we can show that Z satisÿes (16) and (17). Under the assumptions above (12) and (13) are still equivalent on [t0 ; ∞). Hence Z is a stabilizing solution of (16) since Y is a stabilizing solution of (18). Now the proof of Theorem 4 follows immediately from Lemmas 9–11.

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A. Ichikawa / Systems & Control Letters 41 (2000) 183–188

As for stabilizing solutions we have the following properties. Lemma 12. (a) A stabilizing solution of (2) is unique. (b) Let Y (t) and Y (t) be two stabilizing solution of (3). Then Y (t) − Y (t) → 0

as t → ∞:

Proof. (a) Let X and X be two stabilizing solutions of (2). Then −

d (X − X ) = (A + RX )∗ (X − X ) dt +(X − X )(A + RX ):

Hence, X (t) − X (t) = SX∗ (T; t)[X (T ) − X (T )]SX (T; t); where SX is the evolution operator generated by A + RX . Hence |X (t) − X (t)|6M1 e−1 (T −t) cM2 e−2 (T −t) for some positive constant Mi ; i ; i = 1; 2 and c. Letting T → ∞ we obtain X (t) − X (t) = 0; ∀t¿t0 . (b) Since d (Y − Y ) = (A + YS)(Y − Y ) + (Y − Y )(A + Y S)∗ ; dt we have Y (t) − Y (t) = SY (t; t0 )[Y (t0 ) − Y (t0 )]SY∗ (t; t0 ) where SY is the evolution operator generated by A+YS. Hence Y (t) − Y (t) → 0 as t → ∞, since A + YS and A + Y S are exponentially stable.

Acknowledgements The author would like to thank a reviewer for helpful comments. References [1] A. Bensoussan, G. Da Prato, M.C. Delfour, S.K. Mitter, Representation and Control of Inÿnite-Dimensional Systems, Vols. I, II, Birkhauser, Boston, 1992, 1993. [2] R.F. Curtain, H.J. Zwart, An Introduction to Inÿnite-Dimensional Linear Systems Theory, Springer, New York, 1995. [3] J.C. Doyle, K. Glover, P.P. Khargonekar, B.A. Francis, State-space solutions to standard H2 and H∞ control problems, IEEE Trans. Automat. Control 34 (1989) 831–847. [4] N. Dunford, J.T. Schwartz, Linear Operators, Part I: General Theory, Interscience, New York, 1957. [5] A. Ichikawa, H∞ -control and ÿltering with initial uncertainty for inÿnite-dimensional systems, Int. J. Robust Nonlinear Control 6 (1996) 431–452. [6] A. Ichikawa, H. Katayama, Remarks on the time-varying H∞ Riccati equations, Systems Control Letters 37 (1999) 335– 345. [7] B.A.M. van Keulen, The H∞ -problem with measurement feedback for linear-inÿnite dimensional systems, J. Math. Systems Estimation Control 3 (1993) 374–411. [8] B.A.M. van Keulen, H∞ -control for Distributed Parameter Systems: A State Space Approach, Birkhauser, Boston, 1993. [9] D.J.N. Limebeer, B.D.O. Anderson, P.P. Khargonekar, M. Green, A game theoretic approach to H∞ control for time-varying systems, SIAM J. Control Optim. 30 (1992) 262–283. [10] A. Pazy, Semigroups of Linear Operators and Applications to Partial Dierential Equations, Springer, Berlin, 1983. [11] R. Ravi, K.M. Nagpal, P.P. Khargonekar, H∞ control of linear time-varying systems: A state-space approach, SIAM J. Control Optim. 29 (1991) 1394–1413.