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1 CDC-REG467 Optimal Hankel norm approximation for the Pritchard-Salamon class of non-exponentially stable innite-dimensional systems A.J. Sasane and R.F. Curtain Department of Mathematics, University of Groningen, P.O.Box 8, 97 AV Groningen, The Netherlands. f A.J.Sasane, R.F.Curtaing@math.rug.nl Abstract. The optimal Hankel norm approximation problem is solved for a class of innite-dimensional systems without assuming exponential stability. Introduction. The optimal Hankel norm approximation problem has received a lot of attention, both in the mathematical and engineering literature (see Adamjan et al. [], Ball and Helton [], Ball and Ran [3], Curtain and Ran [4], Glover [7] and Ran [3]). Its importance in control theory is due to its connections with the model reduction problem (see, for instance, Glover [7], Glover et al. [8] and Young [9]). Given a transfer function G(s) L (C pm ), we suppose that G(s) has a compact Hankel operator? : L ([; ); C m )! L ([; ); C p ) which is dened by (?u)(t) h(t + s)u(s)ds 8u L (; ; C m ); where h() denotes the impulse response of the system.? then has countably many singular values ::: and these are also called the Hankel singular values of G. The suboptimal Hankel norm approximation problem associated with G is now dened as follows: Find all K(?s) H ;l (C pm ) such that jjg + Kjj for l > > l+ ; in the L? norm; where H ;l (C pm ) denotes the set of complex p m matrix valued functions X() of a complex variable with a decomposition X ^G +F, where ^G is the matrix transfer function of a system of MacMillan degree at most equal to l, with all its poles in the open right half-plane, and F H (C pm ). It is well-known that (see Adamjan et al. []) inf jjg + Kjj l+ : K(?s)H;l(C pm ) In the same paper the optimal Hankel norm approximation problem is solved for the scalar case, and solutions for other classes of functions G have been obtained in Ball and Helton [] and Nikol'skii []. In this paper we consider the state-space solution to the optimal Hankel norm approximation problem in terms of the system parameters A, B, C. This problem has also been studied in literature assuming that A is the innitesimal generator of an exponentially stable C?semigroup and B and C are linear operators. Glover et al. [8] was the rst paper to do this. In Sasane and Curtain [6], B and C are bounded operators. Extensions to the case in which B and C are unbounded can be found in Sasane and Curtain [7] or Curtain and Ran [4]. In all these papers, it is assumed that A generates an exponentially stable C?semigroup. However, there exists an important class of systems with a transfer function G H (C pm ), for which A does not generate an exponentially stable C?semigroup. In [8], approximating solutions to the optimal Hankel norm approximation problem were obtained without assuming exponential stability, but only for the case that the Hankel operator is nuclear; this is a strong assumption. It is the aim of this paper to nd solutions to the suboptimal Hankel norm approximation problem in terms of the A, B, C operators for the Pritchard-Salamon class of non-exponentially stable systems. The specic class of systems we consider in this paper is dened below: Denition. Let V and W be separable Hilbert spaces with continuous, dense injections and which satisfy W,!,! V: Suppose that A is the innitesimal generator of strongly continuous semigroups T W (t), T (t) and T V (t) on W, and V, respectively, such that T V (t)j T (t) and T (t)j W T W (t). Since these p.

2 semigroups are consistent, we shall simply use the notation T (t). Assume further that U and Y are separable Hilbert spaces (the input and output spaces), respectively.. B L(U; V ) is an admissible control operator for T (t) if there exists a constant > such that t T (t? s)bu(s)ds W kuk L([;t);U) for all nite t >.. C L(W; Y ) is an admissible observation operator for T (t) if there exists a constant > and a t > such that for all z W. kct ()zk L([;t);Y ) kzk V Under the above assumptions, the state linear system (A; B; C; D) is called a Pritchard-Salamon system for any D L(U; Y ). If, in addition, D(A V ),! W, (A; B; C; D) is called a regular Pritchard-Salamon system. Moreover, we assume that the following assumptions are satised: A. (A) \ C + is empty and A satises the spectrum determined growth assumption where C + : fs C j Re(s)> g. A. U C m, Y C p. A3. (A; B; C) is input stable, i.e., the controllability map B from L ([; ); U) to dened by Bu T (t)bu(t)dt; is bounded. u L ([; ); U); A4. (A; B; C) is output stable, i.e., the observability map C from to L ([; ); Y ) dened by is bounded. (Cz)(t) CT (t)z; z ; A5. G H (C pm ) is such that! 7! G(j!) : R! C pm is continuous and has a (unique) limit G C pm at. This means that the boundary function G(j) : R! C pm L is equal almost everywhere to a continuous function, which we denote by the same symbol G(j). This convention is used throughout this paper. We outline the contents of the following sections. In section we rst develop the mathematical tools which we need for the proof of our main results. In section 3 we prove a few properties of the class of systems that we consider which will be used in the proofs in the subsequent sections. Finally, in section 4, we prove our main result. Mathematical Preliminaries The key to the proof of our new result is Corollary. which is an easy consequence of the following lemma. Lemma. If G H (C pm ) and! 7! G(j!): R! C pm is uniformly continuous, then given any " >, 9 > such that sup!r kg(j!)?g( + j!)k < " whenever. It follows from Theorem 5.8 (page 96, M. Rosenblum and J. Rovnyak [4]) that G( + j!)? Since for >, R? G(jt) dt; > : (t?!) + (t?!) + dt, we have kg(j!)? G( + j!)k G(jt)? (t?!) + dt? G(j!) G(jt)? G(j!) (t?!) + dt :? Choose a > such that kg(jt)? G(j!)k < " for every t and! satisfying jt?!j <. Now choose a > such that for any satisfying, we have Rn[!?;!+] (t?!) + dt < " : 4 kg(j)k Thus kg(j!)? G( + j!)k G(jt)? G(j!)? (t?!) + dt!+ kg(jt)? G(j!)kdt!? (t?!) + + kg(jt)? G(j!)kdt Rn[!?;!+] (t?!) + "!+!!? (t?!) + dt! + kg(j)k (t?!) + dt Rn[!?;!+] " " : + kg(j)k : 4 kg(j)k ": p.

3 Since the choice of! is arbitrary, this completes the proof.. L B : BB L() is a self-adjoint, nonnegative solution of the operator Lyapunov equation AL B z + L B A z?bb z 8z D(A ); (4) Corollary. If G H (C pm ) and! 7! G(j!): R! C pm is continuous and has limits G(j) at, then given any " >, 9 > such that sup!r kg(j!)?g( + j!)k < " whenever. that Given any " >, M > and M > such sup kg(j!)? G(j)k < " ; and ()![M ;) sup kg(j!)? G(?j)k < "!(?;M ] : () Moreover, since! 7! G(j!): R! C pm is continuous, it is uniformly continuous in [M? ; M + ], and so given any " >, 9 such that > > and whenever!,! [M? ; M + ] and j!?! j <, kg(j! )? G(j! )k < ": (3) Thus it follows from (), () and (3) that whenever!,! R and j!?! j <, then kg(j! )?G(j! )k < ". Hence! 7! G(j!): R! C pm is uniformly continuous, and so the result follows from Lemma.. 3 Properties of the System. Next we prove a few properties of our class of systems which will be used in the sequel. Lemma 3. If (A) \ C + is empty, A satises the spectrum determined growth assumption and C, and Re() >, then A?I is the innitesimal generator of the exponentially stable semigroup fe?t T (t)g t on. From Exercise.4 (Curtain and wart [5]), A? I is the innitesimal generator of the C?semigroup fe?t T (t)g t on with a growth bound equal to the sum of the growth bound of ft (t)g t and?re(). Since (A) \ C + is empty and A satises the spectrum determined growth assumption, it follows that the growth bound of ft (t)g t is non-positive and so A? I generates the exponentially stable semigroup fe?t T (t)g t on. Lemma 3. If the regular Pritchard-Salamon system (A, B, C) satises the assumptions A-4, then and L C : C C L() is a self-adjoint, nonnegative solution of the operator Lyapunov equation.? CB. A L C z + L C Az?C Cz 8z D(A): (5) 3. In addition, if the assumption A5 is satised, then? and L C L B are compact, and the nonzero Hankel singular values of (A; B; C) are equal to the square roots of the nonzero eigenvalues of L C L B. If l+ < < l, then I?? L B L C is invertible.. This follows from Theorem 3., page, Hansen and Weiss [] and its dual statement.. Since C(Bu)(t) CT (t) T (s)bu(s)ds CT (t + s)bu(s)ds (?u)(t); it follows that? CB. 3. It follows from Corollary 4. (Hartman's theorem for the half-plane, page 46, Partington []), that? is compact. Now the proof is analogous to the proof of Lemma 8..9 (page 4, Curtain and wart [5]). Notation. It is easy to see that (A? I; B; C) is also a regular Pritchard-Salamon system for any C. If A satises the assumption A and >, then A?I generates the exponentially stable semigroup fe?t T (t)g t on, and (A? I; B; C) is an exponentially stable, regular Pritchard-Salamon system. Denote the controllability map of this system by B [], the observability map by C [], the Hankel operator by? [], and the l th Hankel singular value by [] l. If [] l+ < < [] l, let N [] : (I?? L [] B L[] C )?, where L [] B : B[] B [], L [] C : C[] C []. Lemma 3.3 If the regular Pritchard-Salamon system (A; B; C) satises the assumptions A-5, then p. 3

4 .? []? []!?? uniformly as!,. [] l! l as! for every l N.. Let : L ([; ); C m )! L ([; ); C m ) be the multiplication operator by e?t : ( u)(t) e?t u(t), and : L ([; ); C p )! L ([; ); C p ) be the multiplication operator by e?t : ( y)(t) e?t y(t). Then? []?S[]. Sublemma 3.4! I m strongly as!, and! I p strongly as!. R Let u L ([; ); C m ). Given any " >, choose a M > such that M ku(t)k dt < ". Now choose a > such that < implies that Thus sup je?t? j < t[;m] p?r k u? uk k(e?t? )u(t)k dt " " : " ku(t)k dt : sup je?t? j! M ku(t)k dt t[;m] + je?t? j ku(t)k dt M + M ku(t)k dt " + " Thus! I m strongly as!. Similarly,! I p strongly as! strongly. Since? []? []? S[]?, we have? []? [] h? S[]???i? +?? : Dening K (??), we have that K is compact, since K is compact and K is self-adjoint (using Exercise 8, page 7, Gohberg and Goldberg [9]). We will use the following (Exercise 6.6', page 36, Weidmann [8]). Sublemma 3.5 Suppose that T n, T L(V; W) and that T n! T strongly. Let S L(U; V) be compact. Then T n S! T S uniformly. K! K uniformly, using the Sublemmas 3.4 and 3.5. But K ( K), and ( K)! K ( K) uniformly, and so K! K uni-!?? uni- formly. formly. Consequently,?? Using the Sublemmas 3.4 and 3.5, we obtain that?!? uniformly. Thus? (?)!? uniformly. As a result,? S[]?!?? uniformly, and since k k, we have h? S[]???i?! uniformly. Hence it follows that? []? []!?? uniformly as!.. Since? []? []!?? uniformly as!, l (? []? [] )! l (??) (using Corollary 4.(a), page 9, Dunford and Schwartz [6]). 4 Optimal Hankel Norm Approximation. In this section, we will prove our main result about the existence of solutions to the optimal Hankel norm approximation problem. We will use H c (C pm ) to denote the set of p m matrix valued functions dened in the closed right halfplane, which are bounded and holomorphic in C +, and continuous in C +. Hc ;[l] (C pm ) denotes the set of p m matrix valued functions X() of a complex variable with a decomposition X ^G +F where ^G is the matrix transfer function of a system of MacMillan degree equal to l, with all its poles in the open right half-plane, and F H c (C pm ). We quote the following theorem from Sasane and Curtain [7]. Theorem 4. Suppose that (A; B; C) is an exponentially stable, regular Pritchard-Salamon unnitedimensional system with nite-dimensional input space C m, output space C p and l+ < < l. Let X() be given by X(s) Ip??CLB + I m B N (si + A )? C where N (I?? L B L C )?. L C B p. 4

5 K(?s) H c ;[l] (C pm ) and kg + Kk i K(?s) : R (?s)r (?s)?, where R (?s) : X R (?s) Q(?s) (?s)? ; for some Q(?s) H c (C pm ) satisfying kqk. We now state our main result: Theorem 4. Suppose that the regular Pritchard- Salamon system (A, B, C) satises the assumptions A-5 and let l+ < < l. If Q(?s) H (C pm ), and kqk, then there exists a > such that for every satisfying < <, I m K [] (?s) R [] (?s)r[] (?s)? ; where " # [] R (?s) Q(?s? ) R [] (?s) X []? (?s) I m and X [] (s) Ip I m?h B N [] (si + A? I) C? [] +?CL B L [] C B i ; is such that K [] (?s) H c ;[l] (C pm ) and kg+k [] k.. For any >, consider the system (A? I, B, C). From Lemma 3., A?I is the innitesimal generator of the exponentially stable semigroup fe?t T (t)g t on. Thus (A? I; B; C) is an exponentially stable, regular Pritchard-Salamon system.. Moreover, for any >, Q(??) H c (C pm ), and kq(??)k. 3. Let " :? l+ >. Choose a > small enough so that whenever < <, sup j!r kg(j!)?g( + j!)k < ". This can be done, owing to assumption A5 and Corollary.. 4. Next choose a >, such that whenever < <, we have (see Lemma 3.3.) [] l+ < l+ + < [] l : 5. Let : minf ; g and consider any satisfying < <. Applying Theorem 4., we have that K [] (?s) H c ;[l] (C pm ) and sup!r kg(j! + ) +K [] (j!)k + l. Thus, kg(j!) + K [] (j!)k kg(j!)? G(j! + ) +G(j! + ) + K [] (j!)k kg(j!)? G(j! + )k +kg(j! + ) + K [] (j!)k? l+ + l+ + : This completes the proof. Acknowledgements. I would like to thank Professor Erik Thomas from the University of Groningen and Professor Thomas Azizov from the University of Voronezh for many useful discussions. References [] V.M. Adamjan, D.. Arov, and M.G. Krein. Innite Hankel block matrices and related extension problems. American Mathematical Society Translations, :33{56, 978. [] J.A. Ball and J.W. Helton. A Beurling-Lax theorem for the Lie group U(m; n) which contains most classical interpolation theory. Journal of Operator Theory, 9:7{4, 983. [3] J.A. Ball and A.C.M. Ran. Optimal Hankel norm model reductions and Wiener-Hopf factorization I: The canonical case. SIAM Journal on Control and Optimization, 5():36{38, 987. [4] R.F. Curtain and A. Ran. Explicit formulas for Hankel norm approximations of innite-dimensional systems. Integral Equations and Operator Theory, 3:455{469, 989. [5] R.F. Curtain and H.J. wart. An Introduction to Innite-Dimensional Linear Systems Theory. Springer- Verlag, New York, 995. [6] N. Dunford and J.T. Schwartz. Linear Operators, volume. John Wiley and Sons, 964. [7] K. Glover. All optimal Hankel-norm approximations of linear multivariable systems and their L error bounds. International Journal of Control, 39:5{ 93, 984. [8] K. Glover, R.F. Curtain, and J.R. Partington. Realization and approximation of linear innitedimensional systems with error bounds. SIAM Journal on Control and Optimization, 6:863{898, 988. p. 5

6 [9] I. Gohberg and S. Goldberg. Basic Operator Theory. Birkhauser, 98. [] S. Hansen and G. Weiss. New results on the operator Carleson measure criterion. IMA Journal of Mathematical Control and Information, 4:3{3, 997. [] N.K. Nikol'skii. Ha-plitz operators: a survey of some recent results. In S.C. Power, editor, Operators and Function Theory. Reidel, Boston, 985. [] J.R. Partington. An Introduction to Hankel Operators. London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 988. [3] A. Ran. Hankel norm approximation for innitedimensional systems and Wiener-Hopf factorization. In R.F. Curtain, editor, Modelling Robustness and Sensitivity Reduction in Control Systems, NATO ASI Series, pages 57{7. Springer-Verlag, 986. [4] M. Rosenblum and J. Rovnyak. Topics in Hardy Classes and Univalent Functions. Birkhauser Advanced Texts. Birkhauser Verlag, Basel, 994. [5] A.J. Sasane. Optimal Hankel norm approximation for non-exponentially stable innite-dimensional systems. Systems and Control Letters, Submitted for Publication. [6] A.J. Sasane and R.F. Curtain. Optimal Hankel norm approximation for innite-dimensional systems. In Proceedings of the MTNS, Perpignan, France,. to appear. [7] A.J. Sasane and R.F.Curtain. Optimal Hankel norm approxiamtion for the Pritchard-Salamon class of innite-dimensional systems. Integral Equations and Operator Theory, Submitted for Publication. [8] J. Weidmann. Linear Operators in Hilbert Spaces. Number 68 in Graduate Texts in Mathematics. Springer-Verlag, 98. [9] N.J. Young. The Nehari problem and optimal Hankel norm approximation. In Analysis and optimization of systems: state and frequency domain approaches for innite-dimensional systems, number 85 in Lecture Notes in Control and Information Sciences. Springer, Berlin, 993. p. 6

Optimal Hankel norm approximation for infinite-dimensional systems

Optimal Hankel norm approximation for infinite-dimensional systems A.J. Sasane and R.F. Curtain Department of Mathematics, University of Groningen, P.O.Box 800, 9700 AV Groningen, The Netherlands. {A.J.Sasane,R.F.Curtain}@math.rug.nl