Optimal Hankel norm approximation for infinite-dimensional systems

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1 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. Keywords: Infinite-dimensional systems, optimal Hankel norm approximation. Abstract The optimal Hankel norm approximation problem is solved under the assumptions that the system Σ(A, B, C) is an exponentially stable infinite-dimensional system with bounded input and output operators. An explicit parameterization of all solutions is obtained in terms of the system parameters A, B, C. 1 Introduction An important question in the design of controllers for large scale systems is whether a model can be simplified without undue loss of accuracy. One approach is to use a low-order optimal Hankel norm approximation; for finite-dimensional systems this was solved in Glover 9 and extended to a class of infinite-dimensional systems in Glover et al. 10. To approximate stable transfer functions G(s) H (C p m ) in the L norm we suppose that G(s) has a compact Hankel operator Γ : L 2 (0, ; C m ) L 2 (0, ; C p ) which is defined by (Γu)(t) 0 h(t + s)u(s)ds u L 2 (0, ; C m ), (1) where h( ) L 1 (0, ; C p m ) denotes the impulse response of the system. Γ then has countably many singular values σ 1 σ 2... and these are also called the Hankel singular values of G. In Glover 9 and Glover et al. 10, it was shown that if h L 1 L 2 and Γ is nuclear (i.e., i1 σ i < ), then there exist finite-dimensional L approximations to G; error estimates were also provided. These approximations were in terms of truncated balanced realizations. These also yielded approximate solutions to the related optimal Hankel norm approximation problem: Find all K( s) H,l (C p m ) such that G + K σ for σ l > σ > σ l+1, in the L norm, where H,l (C p m ) denotes the set of complex p m matrix valued functions X( ) of a complex variable with a decomposition X Ĝ +F, where Ĝ is the matrix transfer function (2) 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 p m ). It is well-known that (see Adamjan et al. 1) inf G + K σ l+1. (3) K( s) H,l (C p m ) 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 2 and Nikol skii 12. A different approach to solving the optimal Hankel norm approximation problem was taken in Ball and Ran 3, Ran 13 and Curtain and Ran 6, which led to explicit formulas for all solutions K( s) in terms of an arbitrary realization of G(s). The starting point was to quote a result from Ball and Helton 2, which states that the optimal Hankel norm approximation problem is equivalent to solving a certain J-spectral factorization problem. Then a solution is constructed from a given realization of G(s). However, if one looks for the result quoted from 2 in these three papers, one realizes that this is not an obvious corollary of the very abstract and general theory in 2. There remains a gap between 2 and the results obtained in 3, 13 and 6. This motivated the self-contained proofs of the suboptimal Nehari extension problem in Curtain and Zwart 7, Curtain and Ichikawa 4 and Oostveen and Curtain 5 which did not rely on the abstract results in 2. The suboptimal Nehari extension problem is a special case of the optimal Hankel norm approximation problem; one seeks all K( s) H (C p m ) satisfying G + K σ for a given σ > Γ. In this paper we provide a self-contained solution to the optimal Hankel norm approximation problem for the class of exponentially stable infinite-dimensional systems with bounded input and output operators and finite-dimensional input and output spaces. A key step in our derivation is to appeal to the recent inertia results for Lyapunov equations from Sasane and Curtain 15. If we specialize our results to the Nehari problem case, we obtain a streamlined version of the proof in Curtain and Zwart 8. We outline the contents of the following sections. In section 2 we introduce some function spaces and some of their properties that will be used in the sequel. In section 3, using the results from Sasane and Curtain 15, we prove the existence of a solution to the key J-spectral factorization problem and in section 4 we prove the existence of a solution to

2 the Hankel norm approximation problem. Finally, we give a complete characterization of all solutions to the optimal Hankel norm approximation problem in section 5. 2 An algebra of transfer functions Before discussing the problem, let us introduce some function spaces which will be used in the sequel. 1. H (C c p m ) denotes the set of complex p m matrix valued functions defined in the closed right half-plane, which are bounded and holomorphic in C + 0, and continuous in C + 0. Hc, with point-wise addition and multiplication, is a commutative ring with identity. 2. H,l c (Cp m ) denotes the set of complex p m matrix valued functions X( ) of a complex variable with a decomposition X Ĝ+F, where Ĝ 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 c p m ). 3. H,l c (Cp m ) denotes the set of complex p m matrix valued functions X( ) of a complex variable with a decomposition X Ĝ+F, where Ĝ is the matrix transfer function of a system of MacMillan degree equal to l, with all its l poles in the open right half-plane, and F H (C c p m ). 4. S denotes the set of complex valued functions g H c that have a nonzero limit at infinity in C 0 +, finitely many zeros in C 0 +, and they are all contained in the open right half-plane. 5. MH c denotes the set of matrices (of any size) with elements in H c. We now list a few elementary facts concerning elements from this class of transfer functions. Lemma 2.1 Let f( ), g( ) H c. If g S has at most l zeros in C + 0, then fg 1 H c,l (C). Lemma 2.2 If M, N MH c have the same number of columns, M is a square matrix with det(m) S and det(m) has l zeros in the open right half-plane, then NM 1 H,l c (Cp m ). Definition 2.3 Suppose M, N MH c, and have the same number of columns. Then the pair (M, N) is right coprime over MH c if there exist X, Y MH c such that the following Bezout identity holds: XM Y N I s C + 0. (4) Suppose that G H c,l (Cp m ), that the pair (M, N) is right coprime over MH c. If M is such that det M S and G NM 1, we call this a right coprime factorization of G over MH c. Lemma 2.4 If K H,l c (Cp m ), then there exists a right coprime factorization K NM 1, where M is rational, det(m) R (0) has exactly l zeros in C + 0 and they are all contained in C + 0. Lemma 2.5 If K H c,l (Cp m ) and K 1 H c (C p p ), K 2 H c (C m m ), then K 1 KK 2 H c,l (Cp m ). Lemma 2.6 If (N, M) is a right coprime factorization of K H c,l (Cp m ) and V H c (C m m ) is invertible as an element of MH c, then (NV, MV ) is also a right coprime factorization of K. Moreover, any two right coprime factorizations of K H c,l (Cp m ) are unique up to a common right multiplication by an invertible element in MH c. Lemma 2.7 If K NM 1 H,l c (Cp m ), N H (C c p m ), M H c (C m m ), with N, M right coprime, then det(m) has exactly l zeros in C + 0, and they are all contained in the open right half-plane. The following technical lemma will be used in the characterization of solutions. Lemma 2.8 If K( ) MH c,, then given any ɛ > 0, there exists a δ > 0 such that whenever 0 ζ δ, we have K(ζ + j ) K(j ) + ɛ, where denotes the L norm. Proof Let K(s) G(s) + F (s) where G(s) is the matrix rational transfer operator of a system of MacMillan degree, say l, with all its poles in the open right half-plane and F MH. c Let the poles of G be contained in the half-plane C + r for some r > 0. Consider the function θ(s) K(s) (where denotes the standard Euclidean 2-norm, namely P i,j p ij 2 for P C ) defined for s belonging to the infinite strip Ω : {s C 0 Re(s) r}. Clearly, θ( ) is continuous in Ω and holomorphic in the interior of Ω. Using the triangle inequality, it is easy to see that θ( ) is bounded in Ω: s G(s) is bounded in Ω (since all its poles are in C + r ) and s F (s) is bounded in Ω (in fact, in C + 0 ). For any ζ > 0, define M(ζ) sup ω R {θ(ζ +jω)}. Using Theorem 12.8 (page 257, W. Rudin 14), we obtain M(ζ) ( K(j ) ) 1 ζ ζ r M(r) r ( M(r) ( K(j ) ) K(j ) ) ζ r. ( ) ζ Since lim M(r) r ζ 0 K(j ) 1, there exists a δ such that 0 < δ < r, and for any ζ satisfying 0 ζ δ, we have K(ζ + j ) K(j ) + ɛ.

3 3 The existence of a J-spectral factorization A key step in the solution of the Nehari problem was the construction of a solution X(s) to a certain J-spectral factorization provided that σ > Γ. For the optimal Hankel norm problem we have σ l+1 < σ < σ l, but nonetheless, the X(s) factor is precisely the same. The proof of the following theorem is analogous to that of Lemma in Curtain and Zwart 8: Lemma 3.1 Suppose that Σ(A, B, C) is an exponentially stable infinite-dimensional system with B L(C m, Z), C L(Z, C p ) and σ l+1 < σ < σ l. Let X( ) be given by X(s) + σ 2 CLB 0 σ σb Nσ (si + A ) 1 C L C B (5) where N σ (I σ 2 L B L C ) 1. It follows that 1. X( s) MH c is invertible in MH c, and X(s) 1 0 σ 1 σ 2 CLB B (si + A ) 1 Nσ C σ 1 L C B (6) 2. If G(s) C(sI A) 1 B, then I W (s) : p 0 G (s) 0 σ 2 Ip G(s), 0 where we use the notation F (s) : F ( s), has a J- spectral factorization W (s) X (s) X(s), (7) 0 for s jω, ω R. If σ > Γ, it was shown in Curtain and Zwart 8 that X 11 ( s) belongs to MH, c is invertible in MH c and X11 1 ( s) is the transfer operator of an exponentially stable infinite-dimensional system. In the case that σ l+1 < σ < σ l, X11 1 ( s) still exists, but it is not stable. We can show under the extra assumption that there exists no k N such that σ k 0, i.e., the given system system is truly infinite-dimensional (this will be a standing assumption in the rest of the paper), X11 1 ( s) is the sum of a stable part in H (C c p p ) and an antistable rational part with at most l unstable poles. In order to do so we will use Theorem 1.4 from Sasane and Curtain 15, which we quote below: Theorem 3.2 If P L(Z) is a self-adjoint solution of A P z + P Az C Cz z D(A), (8) C has finite rank, and Σ(A,, C) is exponentially detectable, then P has a pure point spectrum in the open left half-plane, and ν(p ) π(a), where ν( ) denotes the number of eigenvalues in the open left half-plane, and π( ) denotes the number of eigenvalues in the open right half-plane. For σ l+1 < σ < σ l, let N σ : (I σ 2 L B L C ) 1 L(Z). We now show that N σ L B is self-adjoint and it has at most l negative eigenvalues. Lemma 3.3 Suppose that Σ(A, B, C) is an exponentially stable infinite-dimensional system with B L(C m, Z), C L(Z, C p ) and σ l+1 < σ < σ l. Then N σ L B L(Z) is a self-adjoint operator, and ν(n σ L B ) l. Proof Sublemma 3.4 J : I σ 2 Γ Γ L(L 2 (0, ; U)) has a spectrum σ(j) contained in (, δ) (δ, ) for some δ > 0, and σ(j) (, δ) consists of exactly l negative eigenvalues. Proof Γ Γ is compact and has a pure point spectrum with 0 as the accumulation point. Considering the resolvent ((I σ 2 Γ Γ) λi) 1, it is easy to see that J has a spectrum which is a shifted version of the spectrum of σ 2 Γ Γ. Finally, since Γ Γ has a pure point spectrum {σ 2 1, σ 2 2,...} with 0 as the accumulation point, and since σ l+1 < σ < σ l, J has exactly l negative eigenvalues. 0 σ(j). Consider the spectral decomposition of L 2 (0, ; U) into L and L + induced by the self-adjoint operator J. L 2 (0, ; U) L L +, and if v + L +, Jv +, v + 0. Let Z Z Z 0+ be the spectral decomposition induced by the self-adjoint bounded operator N σ L B. We have N σ L B z, z (I σ 2 L B L C ) 1 L B z, z L B z, (I σ 2 L C L B ) 1 z L B (I σ 2 L C L B )z 0, z 0, where z 0 (I σ 2 L C L B ) 1 z. Thus N σ L B z, z B z 0, B z 0 σ 2 B L C BB z 0, B z 0 u, u σ 2 B L C Bu, u, where u B z 0, and so N σ L B z, z (I σ 2 Γ Γ)u, u. (9) Now define Ψ : Z L as follows: If z Z, Ψz : Π L B Nσz, where Π L denotes the canonical projection from L 2 (0, ; U) onto L. Ψ is clearly linear. Moreover, Ψ is injective. For if Π L B Nσz 0, v : B Nσz L + and 0 N σ L B z, z N σ L B z, z (I σ 2 Γ Γ)v, v 0.

4 Thus, Jv, v 0. Using Jv +, v + 2 Jv +, v + Jv +, v +, Jv, Jv 0, which implies that Jv 0. But J is injective, and so v 0. Now B N σz 0 BB N σz 0 L B N σz 0 L C L B N σz 0, and so Nσz z, which means that 1 belongs to the spectrum of Nσ. So 1 { σ2 σ 2 σ } n 2 n N {0}, i.e., σ k 0 for some k N, contradicting the infinite-dimensionality of the system. Thus, dim Z dim L l. Hence ν(n σ L B ) l (From Kato 11, pages , it follows that since the dim Z <, the essential spectrum in the open left half-plane is empty and so dim Z ν(n σ L B )). We now relate the number of negative eigenvalues of N σ L B to the unstable part of X 1 11 ( s). Lemma 3.5 Suppose that Σ(A, B, C) is an exponentially stable infinite-dimensional system with B L(C m, Z), C L(Z, C p ) and σ l+1 < σ < σ l. Let X( ) be given by (5). Then X 11 ( s) 1 H c,l (Cp p ). Proof We have X 11 ( s) I p + σ 2 CL B N σ(si A ) 1 C. (10) It follows from Corollary (Curtain and Zwart 8) that I p σ 2 CL B N σ (si A + σ 2 C CL B N σ) 1 C is the inverse of X 11 ( s). Consider the Riccati equation P A z +AP z +N σ BB N σ z σ 2 P C CP z 0 (11) for all z D(A ). As in the proof of Lemma b (Curtain and Zwart 8), it is readily verified that N σ L B L(Z) is a self-adjoint solution of this Riccati equation. Thus if  : A σ 2 C CL B N σ, then N σ L B L(Z) is a selfadjoint solution of the Riccati equation P Âz +  P z + N σ BB N σz + σ 2 P C CP z 0, (12) for all z D(Â). Consequently, N σl B L(Z) is a selfadjoint solution of the Lyapunov equation B P Âz+ P z Nσ B N σ 1 σ CP σ 1 z z CP D(Â). Moreover, ( Â, B N σ σ 1 CL B N σ ) (13) (14) is exponentially detectable, since 0 σ 1 C L(U Y, Z) and  + 0 σ 1 C B Nσ σ 1 CL B Nσ A σ 2 C CL B Nσ + σ 2 C CL B Nσ A generates an exponentially stable semigroup on Z. Thus applying Theorem 3.2, we obtain that π(â) ν(n σl B ) l. Hence, Z Z + Z ; Â+ 0  0  where Â+ : Z + Z +, dim Z + r l, and Â+ has all its r eigenvalues in the open right half-plane.  : D(Â) Z Z is the infinitesimal generator of an exponentially ( s) can be written as a sum of the transfer function of a system with MacMillan degree at most l with all poles in the open right half-plane and a function in MH. c stable semigroup on Z. Thus, X Existence of a solution to the optimal Hankel norm problem In this section we show the existence of a solution to the optimal Hankel norm approximation problem for the class of exponentially stable infinite-dimensional systems Σ(A, B, C) with bounded input and output operators and finite-dimensional input and output spaces. Although it is known (see Adamjan et al. 1) that ; inf G + K σ l+1, (15) K( s) H,l (C p m ) we only need the inequality stated in Theorem 4.2 below which for completeness is proven in the appendix, together with the following technical lemma. Lemma 4.1 Let S : Z 1 Z 2 be a bounded operator from the Hilbert space Z 1 to the Hilbert space Z 2, with Schmidt vector pairs (v i, w i ): Sv i σ i w i, S w i σ i v i, (16) where σ i σ i+1, w i v i 1, i N. If Ŝ : Z 1 Z 2 is an arbitrary bounded linear map of rank l, then S Ŝ σ l+1. (17) Theorem 4.2 Let G L (jr; C p m ) and K( s) H,l (C p m ). Then inf G + K σ l+1. (18) K( s) H,l (C p m ) As in Curtain and Zwart 8, we construct a solution to the optimal Hankel norm problem using K o ( s) V 12 ( s)v 1 22 ( s), where V ( s) X( s) 1. Lemma V 22 ( s) is invertible as an element of MH,, c and V22 1 ( s) Hc,l (Cm m ). 2. V 12 ( s)v 1 22 ( s) Hc,l (Cp m ).

5 Proof 1. X 1 11 ( s) Hc,l (Cp p ). Using XV 1 V 1 X I, it can be checked that V 22 ( s) 1 X 22 ( s) X 21 ( s)x 1 11 ( s)x 12( s). Thus, V 22 ( s) is invertible as an element of MH,. c Moreover, it follows from Lemma 2.5 that V 22 ( s) 1 X 22 ( s) X 21 ( s)x11 1 ( s)x 12( s) H,l c (Cm m ). 2. V 12 ( s)v22 1 ( s) X 1 11 ( s)x 12( s), and so the result follows from Lemma 2.5. Theorem 4.4 There exists K o ( s) H c,l (Cp m ) such that G + K < σ < σ l. Proof Define K o ( s) : V 12 ( s)v22 1 ( s) H,l c (Cp m ). We know from Lemma that V22 1 ( s) G u +G s, where G s MH, c and G u is a strictly proper rational transfer matrix of a system with all its poles in the open right half-plane. Thus, V22 1 (jω) is defined ω R. Hence, G + Ko Ip G Ko 0 Ip G 0 V 0 V22 1, with s jω, ω R, and so we have (G + K o ) (G + K o ) σ 2 G + Ko G + Ko 0 σ 2 0 V22 1 V Ip G 0 0 σ 2 Ip G 0 V 0 V V V 1 22, where we have used equation (7), the definition of V and Lemma Thus it follows that (G + K o )(jω)u 2 σ 2 u 2 V 1 22 (jω)u 2 (19) for u C m and ω R. Since V 22 (jω)v22 1 (jω) I, u V 22 (jω) V22 1 (jω)u. (20) Since V 22 ( s) MH c, there exists a constant M such that V 22 ( s) M. Since V 22 (jω) is invertible on the imaginary axis, M > 0, and we have u C m and ω R. Hence it follows that G + K o < σ < σ l. (22) In the above theorem, we have constructed a solution K o ( s) H c,l (Cp m ) with at most l unstable poles. In fact, any solution to the optimal Hankel norm approximation problem has an unstable rational part of MacMillan degree exactly l. Corollary 4.5 If K( s) H,l (C p m ) is such that G + K σ < σ l, then K( s) Ĝ +F, where Ĝ has MacMillan degree exactly l, with all l poles in the open right half-plane and F H (C p m ), i.e., K( s) H,l (C p m ). Proof Suppose that K( s) Ĝ1 +F 1, where Ĝ1 has MacMillan degree r and all r poles in the open right halfplane, and F 1 H (C p m ). Since K( s) H,l (C p m ), r l. From Theorem 4.2, it follows that σ l > σ G + K inf K( s) H,r (C p m ) G + K σ r+1. Thus σ l > σ r+1, which implies that l < r + 1, and so l r. Hence l r. Finally we collect more precise information concerning our constructed solution K o ( s) V 12 ( s)v22 1 ( s) to the optimal Hankel norm problem from Theorem 4.4. Corollary 4.6 K 0 (s) V 12 ( s)v22 1 ( s) has the following properties 1. K o ( s) H c,l (Cp m ). 2. (V 12 ( s), V 22 ( s)) is a right coprime factorization over MH c of K o ( s) H c,l (Cp m ). 3. det(v 22 ( s)) has no zeros on the imaginary axis, and exactly l zeros in C + 0. Proof 1. From Lemma 4.3.2, we know that K o ( s) : V 12 ( s)v22 1 ( s) Hc,l (Cp m ), and from the proof of Theorem 4.4, we know that it satisfies G + K < σ < σ l. Thus from Corollary 4.5 above, V 12 ( s)v22 1 ( s) Hc,l (Cp m ). 2. V 12 ( s)v 1 22 ( s) Hc,l (Cp m ), V 12 ( s), V 22 ( s) MH c. Moreover, X 22 V 22 ( X 21 )V 12 I, (23) where X 22 ( s), X 21 ( s) MH c. Hence it follows that (V 12 ( s), V 22 ( s)) is a right coprime factorization of V 12 ( s)v 1 22 ( s). 3. This follows from the above and Lemma 2.7. u M V22 1 (jω)u, (21)

6 5 Characterization of solutions In this section we obtain a nice parameterization of all solutions to the optimal Hankel norm approximation problem. First we prove a few properties that will be used in the proof of the characterization theorem. Lemma 5.1 Suppose that Σ(A, B, C) is an exponentially stable infinite-dimensional system with B L(C m, Z), C L(Z, C p ) and σ l+1 < σ < σ l. Let X( ) be given by (5) and V ( s) : X 1 ( s). Then 1. det(v 22 ( s)) has a nonzero limit at infinity in C V 21 ( s) is strictly proper. 3. X 21 ( s) is strictly proper. 4. det(x 22 ( s)) has a nonzero limit at infinity in C + 0. Proof Since V 22 ( s) σ 1 + σ 3 B (si A ) 1 NσL C B V 21 ( s) σ 2 B (si A ) 1 NσC X 21 ( s) σ 1 B N σ(si A ) 1 C X 22 ( s) σ σ 1 B N σ(si A ) 1 L C B, and A is the infinitesimal generator of an exponential semigroup, the results follow. We now parameterize a family of solutions to the optimal Hankel norm approximation problem. Theorem 5.2 Suppose that Σ(A, B, C) is an exponentially stable infinite-dimensional system with B L(C m, Z), C L(Z, C p ) and σ l+1 < σ < σ l. Let X( ) be given by (5). If Q( s) H (C c p m ) satisfies Q 1, and K( s) : R 1 ( s)r 2 ( s) 1, where R1 ( s) R 2 ( s) : X 1 ( s) Q( s), (24) then K( s) H c,l (Cp m ) and G + K σ. Proof Step 1: We show that V22 1 V 21 < 1, where denotes the L norm. X(s) satisfies equation (7); and so taking inverses, we obtain V V 0 1 Ip G 0 0 σ 2 1 G where s jω, ω R. Considering the (2, 2) block of the above gives V 21 V 21 V 22V 22 σ 2 (25) where s jω, ω R. Lemma shows that V 22 is invertible as an element of L, and thus V 1 22 V 21u 2 u 2 σ 2 V 1 22 u 2. (26) Choosing a M > 1 such that V 22 M, we obtain u 2 V 22 2 V 1 22 u 2 M 2 V 1 22 u 2. (27) Hence V ( ) 1 1 M < 2 1, and so we have V < 1. Step 2: We show that det(v 21 ( s)q( s) + V 22 ( s)) has exactly l zeros in C 0 +, and they are contained in the open right half-plane. By Corollary 4.6.3, we know that s det(v 22 ( s)) has no zeros on the imaginary axis, and exactly l zeros in C + 0. So there exists an ɛ > 0 such that all its zeros are contained in the half-plane C ɛ +. From Corollary 4.6.1, we know that K 0 ( s) : V 22 ( s) 1 V 21 ( s) H,l c (Cp m ), and so it follows from Lemma 2.8 applied to K 0 (s), that a δ > 0 such that ɛ > δ and whenever 0 ζ δ, V 1 22 ( ζ j )V 21( ζ j ) < 1. (28) Fix such a ζ > 0. Consider φ(α, s) : det(αv 21 ( ζ s)q( ζ s) + V 22 ( ζ s)), where α 0, 1. φ(0, ) and φ(1, ) are meromorphic (actually holomorphic in C + 0 ζ/2 ) functions on an open set containing C + 0 at infinity in C + 0. We have: with nonzero limits 1. φ(α, s) : 0, 1 jr C is a continuous function. 2. φ(0, jω) det(v 22 ( ζ jω)), and φ(1, jω) det(v 21 ( ζ jω)q( ζ jω) + V 22 ( ζ jω)) αv 22 ( ζ jω) 1 V 21 ( ζ jω)q( ζ jω) is invertible, since αv 1 22 ( ζ jω)v 21( ζ jω)q( ζ jω) < 1. (29) Moreover, since s det(v 22 ( ζ s)) has no zeros on the imaginary axis, it follows that φ(α, jω) is nonzero for all α 0, 1 and ω R. 4. φ(α, ) is nonzero for all α 0, 1, since det(αv 21 ( s ζ)q( s ζ) + V 22 ( s ζ)) has a nonzero limit at infinity in C + 0. So the assumptions of Lemma A.1.18 (Curtain and Zwart 8, page 570) are satisfied by φ and so the Nyquist indices of φ(0, s) and φ(1, s) are the same. Consequently, the number of zeros are the same (the number of poles in each case is zero, since φ(0, s), φ(1, s) are holomorphic in C + 0 ζ/2 ). But when α 0, det(v 22 ( s ζ)) has l zeros in the closed right half-plane C + 0. Thus the number of zeros of φ(1, s) the number of zeros of φ(0, s) l. But since ζ can be chosen

7 arbitrarily small, det(v 21 ( s)q( s) +V 22 ( s)) has exactly l zeros in C 0 +. Moreover, det(v 21 ( s)q( s) +V 22 ( s)) has no zeros on the imaginary axis, since det(i +V22 1 ( jω) V 21( jω) Q( jω)) 0 for all ω R. Step 3: We show that K( s) H,l c (Cp m ). det(v 21 ( s)q( s) +V 22 ( s)) has a nonzero limit at infinity in C + 0, since Q( s) MHc is proper, V 21 ( s) is strictly proper, V 22 ( s) is proper in C + 0 and det(v 22( s)) has a nonzero limit at infinity in C + 0 (see Lemma 5.1). So by Lemma 2.2 and using Step 2 above, it follows that K defined by K( s) (V 11 ( s)q( s) + V 12 ( s)) (V 21 ( s)q( s) + V 22 ( s)) 1 is a well-defined element of H c,l (Cp m ). Step 4: We show that G + K σ. R 2 (jω) V 21 (jω)q(jω) + V 22 (jω), and since det(v 22 ( s)) has no zeros on the imaginary axis and V 1 22 (jω)v 21(jω)Q(jω) < 1 for all ω R, it follows that R 2 (jω) is invertible for every ω R. Thus with s jω, for all ω R, (G + K) (G + K) σ 2 (R2 1 ) Q V Ip G 0 Ip G Q 0 σ 2 V R 0 I 2 1 m (R 1 2 ) (Q Q )R 1 2. Thus G + K σ. Step 5: We show that K( s) H c,l (Cp m ). Finally, from Corollary 4.5, and Steps 3 and 4 above, it follows that K( s) H c,l (Cp m ). Next we show that any continuous solution K( s) H c,l (Cp m ) to the optimal Hankel norm approximation problem has the form (24). Theorem 5.3 Suppose that Σ(A, B, C) is an exponentially stable infinite-dimensional system with B L(C m, Z), C L(Z, C p ) and σ l+1 < σ < σ l. Let X( ) be given by (5). If K( s) H,l c (Cp m ) and G + K σ, then K( s) R 1 ( s)r 2 ( s) 1, where R1 ( s) X 1 Q( s) ( s) (30) R 2 ( s) for some Q( s) H c (C p m ) satisfying Q 1. Proof Let K( s) H,l c (Cp m ) satisfy G + K σ and suppose it has the coprime factorization K( s) N(s)M 1 (s) over MH, c where N, M MH, c M is rational, and det M R (0) has exactly l zeros in C + 0 and none on the imaginary axis. Define U1 ( s) K( s) X( s) X11 ( s)k( s) + X 12 ( s) X 21 ( s)k( s) + X 22 ( s) (31), and U 1 ( s) U1 ( s) X11 ( s)n(s) + X 12 ( s)m(s) X 21 ( s)n(s) + X 22 ( s)m(s) M(s) (32) Step 1: We show that U 2 (jω) is invertible with U 1 U 1 2 L (jr; C p m ) and U 1 U From (31) we deduce U1 ( s) X( s) For s jω, ω R, we have U 1 U 1 U 2 U 2 U 1 U 2 0 Ip G(s) 1 G(s) + K( s) I p 0 0 U1 and appealing to (7) and (7), we see that for s jω G X 0 0 σ 2. Thus for s jω, U 2 Ip G X 0.,. (33) 1 U 1 U 1 U 2 U 2 (34) G + K G + K 0 σ 2 0. Hence for all u C m and all ω R, we have from equation (34) that U 1 (jω)u 2 U 2 (jω)u 2 (G+K)u 2 σ 2 u 2 0, (35) and so U 1 (jω)u U 2 (jω)u. (36) Since V ( s) X 1 ( s), we deduce that U1 ( s) K( s) V ( s) and so, (37) V 21 ( s)u 1 ( s) + V 22 ( s) (38) for s C + 0. We claim that ker(u 2( jω)) {0} for all ω R. Suppose on the contrary that there exists x 0 such that U 2 ( jω 0 )x 0. Then from (36), we obtain U 1 ( jω 0 )x 0, which violates (38). Concluding, we have that det(u 2 (jω)) 0 for all ω R, and so U 2 (jω) 1 exists for all ω R. From (34), we deduce that U 1 (jω)u 2 (jω) 1 y 2 y 2 ω R, (39)

8 and so U 1 ( s)u2 1 L (jr; C p m ) satisfies U 1 U Step 2: We now construct a Q( s) H, (C c p m ) such that Q(j ) 1 and K( s) (V 11 ( s)q( s) +V 12 ( s))(v 21 ( s)q( s) + V 22 ( s)) 1. Consider MH. We know that X 21 ( s) is strictly proper and both X 22 ( s) and M(s) are proper with a nonzero limit at infinity in C + 0. So there exists a R > 0 such that for every s C 0 +, with s > R, det(u 2( s)) 0. Since det() is holomorphic in C + 0, it follows that if its zeros have an accumulation point in the compact set {s C + 0 s R}, it must lie on the imaginary axis. But det() 0 on the imaginary axis, since det() det() det(m(s)), (40) and neither det(u 2 ) nor det(m) have any zeros on the imaginary axis. Thus det() has only finitely many zeros in C 0 +, and they are all contained in the open right halfplane. So det() S and it follows from Lemma 2.2 that U 1 2 ( s) is an element of MH, c and Q( s) : U 1 ( s)u 1 2 ( s) is a well-defined element of MH,. c We also note that ( M 1 (s)) is invertible as an element of MH, c and Q( s) : U 1 ( s)u2 1 ( s). From Step 1 we see that Q(j ) 1. Now from (31) we obtain and so K( s) X 1 U1 ( s) ( s) U1 ( s) V ( s) K( s) V 11 ( s)u 1 ( s) + V 12 ( s) (41) V 21 ( s)u 1 ( s) + V 22 ( s). (42) Thus K( s) (V 11 ( s)q( s) + V 12 ( s)) (V 11 ( s)q( s)+v 12 ( s)) (V 21 ( s)q( s)+v 22 ( s)) 1, as claimed. Step 3: We show that U 1 ( s), are right coprime over MH c. Since M, N are right coprime, there exist R, S MH c such that RM SN I. Now it is readily verified using V X I that Consider now V 11 U 1 + V 12 U 2 KM N (43) V 21 U 1 + V 22 U 2 M. (44) ( SV 12 + RV 22 )U 2 (SV 11 RV 21 )U 1 R(V 21 U 1 + V 22 U 2 ) S(V 11 U 1 + V 12 U 2 ) RM SN I. Thus U 1 ( s), are right coprime over MH. c Step 4: Q( s) H (C c p m ). The zeros of det(v 22 ( s)), det(m(s)) and det() are contained in some half-plane C + ɛ, where ɛ > 0. Since V22 1 V 21 < 1, there exists a r > 0 such that V22 1 V 21 1 r. It follows from Lemma 2.8 that δ 1 > 0 such that δ 1 < ɛ and for any ζ satisfying 0 < ζ < δ 1, V 1 22 ( ζ )V 21( ζ ) 1 r 2. (45) Similarly it follows Lemma 2.8 that δ 2 > 0 such that δ 2 < ɛ and for any ζ satisfying 0 < ζ < δ 2, Q( ζ j ) 1 + r 4 1 r r. (46) 4 Let δ :min(δ 1, δ 2 ), and fix a ζ satisfying 0 < ζ < δ. Let φ(α, s) det(αv 21 ( s ζ)u 1 ( s ζ) +V 22 ( s ζ)u 2 ( s ζ)), where α 0, φ(0, ) det(v 22 ( ζ)u 1 ( ζ)) and φ(1, ) det(v 21 ( ζ)u 1 ( ζ) + V 22 ( ζ)u 2 ( ζ)) are meromorphic in C + 0 ζ/2. 2. φ(0, ) has a nonzero limit at infinity in C 0 + : det(v 22 ( s)) has a nonzero limit at infinity in C + 0 and det(u 2 ) has a nonzero limit at infinity in C + 0, since det(u 2 ) S. φ(1, ) has a nonzero limit at infinity in C + 0, since V 21 ( s) is strictly proper, U 1 ( s) is proper in C + 0, and the above. 3. φ(α, s) : 0, 1 jr C is a continuous function; and φ(0, jω) det(v 22 ( ζ jω)u 1 ( ζ jω)) det(v 22 ( ζ jω)).det(u 2 ( ζ jω)), φ(1, jω) det(v 21 ( ζ jω)u 1 ( ζ jω) +V 22 ( ζ jω)u 2 ( ζ jω)). 4. We have φ(α, jω) 0 since αv22 1 ( ζ j ) V 21 ( ζ j ) U 1 ( ζ j ) U 1 2 ( ζ j )) < 1 and det(u 2 ( ζ jω)) φ(α, ) 0, since V 21 ( s) is strictly proper, U 1 ( s) is proper in C + 0, and det(v 22( s)). det() has a nonzero limit at infinity in C + 0. Thus the assumptions in Lemma A.1.18 (Curtain and Zwart 8, page 570) are satisfied by φ, and hence it follows that the Nyquist indices of φ(0, s) and φ(1, s) are the same. Consequently, the number of zeros are the same (the

9 number of poles is zero, as φ(0, s), φ(1, s) are holomorphic in C + 0 δ/2 ) and so the sum of the number of zeros of s det(v 22 ( ζ s)) in C + 0 and the number of zeros of s det(u 2 ( ζ s)) in C 0 + equals the number of zeros of s det(v 21 ( ζ s)u 1 ( ζ s) +V 22 ( ζ s)u 2 ( ζ s)) (det(m(ζ + s), using 44) in C + 0, i.e., Thus S Ŝ σ l+1. Proof (of Theorem 4.2) Let K( s) Ĝ(s) + F (s), where Ĝ has MacMillan degree l, and F H (C p m ). Thus, G + K G + Ĝ H Γ G + ΓĜ σ l+1 (G). (51) l + d(q) l, (47) where d(q) denotes the number of zeros of s det(u 2 ( ζ s)) in C + 0. Thus Q has no poles in C+ ζ. But ζ > 0 can be chosen arbitrarily small, which implies that Q H c (C p m ). Our main result is the following: Theorem 5.4 Suppose that Σ(A, B, C) is an exponentially stable infinite-dimensional system with B L(C m, Z), C L(Z, C p ) and σ l+1 < σ < σ l. Let X( ) be given by (5). K( s) H,l c (Cp m ) and G + K σ iff K( s) R 1 ( s)r 2 ( s) 1, where R1 ( s) R 2 ( s) X 1 ( s) Q( s) for some Q( s) H c (C p m ) satisfying Q 1. (48) Finally we remark that the results in this paper can be extended to the Pritchard-Salamon class of systems which allows for unbounded input and output operators. (The paper is available at the following website: amols/index.html.) 6 Appendix Proof (of Lemma 4.1) Let ˆΠ be the projection from Z 2 onto span{w 1, w 2,...,w l+1 }; then ˆΠ(S Ŝ) S Ŝ. (49) Consider the following restriction of ˆΠŜ: ˆΠŜ : span{v 1, v 2,..., v l+1 } span{w 1, w 2,..., w l+1 } (50) which has rank l, and hence there exists a z ker(ˆπŝ), z 1, say z l+1 i1 a iv i with l+1 i1 a i 2 1. Then ˆΠSz l+1 a i σ i w i, i1 S Ŝ 2 ˆΠSz ˆΠŜz 2 ˆΠSz 2 l+1 σi 2 a 2 i i1 l+1 σl+1 2 a 2 i σl+1. 2 i1 References 1 Adamjan V.M., Arov D.Z. and Krein M.G. Infinite Hankel block matrices and related extension problems. American Mathematical Society Translations, Vol. 111, pp , Ball J.A. and Helton J.W. A Beurling-Lax theorem for the Lie group U(m, n) which contains most classical interpolation theory. Journal of Operator Theory, Vol. 9, pp , Ball J.A. and Ran A.C.M. Optimal Hankel norm model reductions and Wiener-Hopf factorization I: The canonical case. SIAM Journal on Control and Optimization, Vol. 25, No.2, pp , Curtain R.F. and Ichikawa A. The Nehari problem for infinite- dimensional systems of parabolic type. Integral Equations and Operator Theory, Vol. 26, pp.29-45, Curtain R.F. and Oostveen J.C. The Nehari problem for nonexponentially stable systems. Integral Equations and Operator Theory, Vol. 31, pp , Curtain R.F. and Ran A.C.M. Explicit formulas for Hankel norm approximations of infinite-dimensional systems. Integral Equations and Operator Theory, Vol. 13, pp , Curtain R.F. and Zwart H.J. The Nehari problem for the Pritchard-Salomon class of infinite-dimensional linear systems: a direct approach. Integral Equations and Operator Theory, Vol. 18, , Curtain R.F. and Zwart H.J. An Introduction to Infinite- Dimensional Systems Theory. Springer-Verlag, New York, Glover K. All optimal Hankel-norm approximations of linear multivariable systems and their L error bounds. International Journal of Control, Vol. 39, pp , Glover K., Curtain R.F. and Partington J.R. Realization and approximation of linear infinite-dimensional systems with error bounds. SIAM Journal on Control and Optimization, Vol. 26, pp , 1988.

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