Schur parametrizations of stable all-pass discrete-time systems and balanced realizations. Application to rational L 2 approximation.
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1 Schur parametrizations of stable all-pass discrete-time systems and balanced realizations. Application to rational L 2 approximation. Martine Olivi INRIA, Sophia Antipolis 12 Mai
2 Outline. Introduction : lossless functions and rational L 2 approximation Schur parametrizations and balanced realizations The software RARL2 New atlases 2
3 Lossless functions: transfer functions of stable allpass systems. Lossless function (discrete-time): G(z) square rational stable such that with equality on the unit circle. G(z) G(z) I, z > 1 A lossless function G(z) has a minimal realization s.t. (D, C, B, A) balanced R = D C unitary. B A lossless transfer functions in discrete-time can be represented by unitary realization matrices. Möbius transform: continuous-time dicrete-time. 3
4 Lossless functions in linear systems theory. Douglas Shapiro Shields factorization: S rational stable S = P G G lossless, P analytic in the unit disk (unstable). Many problems in linear system theory can be stated in terms of lossless functions : stochastic identification (Separable Least Squares) model reduction problems via rational approximation control design? 4
5 Rational L 2 approximation. H 2 Hardy space of matrix valued functions analytic in D, H 2 Hardy space of matrix valued functions analytic outside D, vanishing at F 2 = 1 { 2π 2π Tr 0 } F (e it )F (e it ) dt. Given F H 2, minimize F H 2 H rational, stable of McMillan degree n. Applications developped in the APICS team: identification from partial frequency data (hyperfrequency filters) sources and cracks detection (MEG, EEG) 5
6 The criterion. H = P G a best approximant of F : P proj. of F G 1 onto H 2 For G lossless (up to a left unitary factor) J(G) = F P (G) G 2. F (z) = C(zI N A) 1 B + D, H(z) = γ(zi n A) 1 B + D. For (A, B) input-normal pair (AA + BB = I) J(A, B) = F 2 Tr (γγ ), where γ = CW, W solution to AW A + BB = W. 6
7 Parametrization issue in the MIMO case In view of optimization, it is better use an atlas of charts attached to a differential manifold M a collection of local coordinate maps (charts) (W i, ϕ i ) W i open in M, ϕ i : W i R d diffeomorphism their domains cover the manifold: W i = M the changes of coordinates ϕ ij are smooth 7
8 Advantages of such a parametrization... in the representation of stable allpass systems: ensures identifiability a small perturbation of the parameters preserves the stability and the order of the system allows for the use of differential tools: a search algorithm can be run over the manifold changing from one local coordinate map to another when necessary. 8
9 Two approaches. L p n: p p (complex) lossless functions of McMillan degree n is a real differentiable manifold of dimension 2np + p 2 Subclass of real functions G( z) = G(z) dim np + p(p 1)/2 charts from realizations theory (state-space) M. Hazewinkel, R.E. Kalman (1974) for proper linear systems using nice selections B. Hanzon, R.J. Ober (1998) charts from Schur type algorithms (functional) D. Alpay, L. Baratchart, A. Gombani (1994) interpolation theory, reproducing kernel Hilbert spaces Ball, Gohberg, Rodman, Dym, Alpay,... An atlas which combines these two approaches B. Hanzon, M.O., R. Peeters (1999) 9
10 The SISO case : a balanced canonical form. A lossless function of degree n has a (unique) realization such that (i) (A, b, c, d) balanced [b, Ab, A 2 b,..., A n 1 b] positive upper triangular or equivalently (ii) R = d c is unitary and positive upper-hessenberg. b A B. Hanzon, R. Peeters (2000) 10
11 R = Then d b c A Parametrization. is unitary and positive upper-hessenberg. γ n κ n 0 κ n γ n I n 1 d c A 0 }{{} R = R n 1, γ n < 1, κ n = 1 γ n 2, γ n = d. By induction: (R k ) k=n,...,0 unitary upper-hessenberg order k and (γ k ) k=n,...,0, R 0 = γ 0, γ 0 = 1. 11
12 Hessenberg form. γ n κ n γ n 1 κ n κ n 1 γ n 2... κ n κ n 1... κ 1 γ 0 κ n γ n γ n 1 γ n κ n 1 γ n 2... γ n κ n 1... κ 1 γ 0 0 κ n 1 γ n 1 γ n 2... γ n 1 κ n 2... κ 1 γ κ n 2 γ n 2 κ n 3... κ 1 γ κ 2 γ 2 γ 1 γ 2 κ 1 γ κ 1 γ 1 γ 0 (1) 12
13 Connection with the Schur algorithm. Let g k (z) = γ k + c k (zi k A k ) 1 b k be the lossless function whose realization is R k. Then g k satisfies the interpolation condition Moreover, g k ( ) = γ k, γ k < 1. g k (z) = γ k z + g k 1 (z) z + γ k g k 1 (z) g k 1 (z) = (g k(z) γ k )z 1 γ k g k (z). This is the Schur algorithm: lossless functions of McMillan degree n are parametrized by the interpolation values γ n, γ n 1,..., γ 1 and γ 0. 13
14 An interpolation problem. Nevanlinna-Pick: find all the lossless functions G(z) G(1/ w)u = v, w < 1, u = 1. A solution exists v < 1. All the solutions are of the form G = (Θ 4 K + Θ 3 )(Θ 2 K + Θ 1 ) 1. for some lossless function K, where ( ( ) z w Θ(z) = I 2p + 1 wz 1 Θ = [ Θ1 Θ 2 Θ 3 Θ 4 ], J = [ I 0 0 I ], C = [ u ref. on interpolation pbs: Ball, Gohberg, Rodman, 1990 v ] ) C C J H (2), H cstarbit. H JH = J 14
15 Charts from a Schur algorithm. Schur algorithm: from G L p n, G = G n,..., G k LF T G k 1,..., G 0 G k (1/ w k )u k = v k, v k < 1 G k has degree k and G 0 is a constant unitary matrix. Chart and Schur parameters: w 1, w 2,..., w n points of the unit circle, u 1, u 2,..., u n unit complex p-vectors, v 1, v 2,..., v n complex p-vectors, v i < 1. W = {G L p n; G k (1/ w k )u k < 1}, ϕ : G (v 1,..., v n, G 0 ) 15
16 Balanced realizations from Schur parameters. For a particular choice of H in (2) depending on the interpolation data, a lossless function G in this chart has a unitary realization matrix computed by induction D k B k C k A k = V k 0 0 I k D k 1 C k 1 0 B k 1 A k 1 where A k is k k, D k is p p, and D 0 = G 0. very nice numerical behavior U k 0 0 I k 1, B. Hanzon, M.O., R. Peeters, INRIA report (2004) 16
17 The matrices U k and V k. U k and V k are the (p + 1) (p + 1) unitary matrices: U k = V k = ξ ku k η k w k ξ kv k η k I p (1 + η k w k )u k u k ξ k u k I p (1 η k ) v kv k v k 2 ξ k v k ξ k = 1 wk 2 1 wk 2 v k 2, η k = 1 vk 2 1 wk 2 v k 2 17
18 Adapted chart From a realization in Schur form (A lower triangular) A = w n w 1, B = 1 wn 2 u n by induction a chart in which v 1 = v 2 =... v n = 0 (Potapov factorization). Doesn t work for real functions: G( z) = G(z) Hanzon, Olivi, Peeters, ECC99 18
19 The RARL2 software This software computes a stable rational L 2 -approximation of specified order n to a multivariable transfer function given in one of the following forms: a realization a finite number of Fourier coefficients some pointwise values on the unit circle. It has been implemented using standard MATLAB subroutines. The optimizer of the toolkit OPTIM is used to find a local minimum, given by a realization, of the nonlinear L 2 -criterion. Implementation : J.P. Marmorat 19
20 Main features state-space representation parametrized by Schur parameters optimization over a manifold adapted chart obtained from a realization in Schur form recursive search on the degree (optional): minimum of degree k initial point of degree k + 1 (error-norm preserved) 20
21 Automobile gas turbine Hung, MacFarlane (1982) ; Glover (1984) ; Yan, Lam (1999) 21
22 Nyquist diagrams 2 2; order
23 Approximants: order 1 23
24 Approximants: order 2 24
25 Approximants: order 3 25
26 Approximants: order 4 26
27 Approximants: order 5 27
28 Approximants: order 6 28
29 Approximants: order 7 29
30 Approximants: order 8 30
31 Hyperfrequency Filter The problem: find a 8th order model of a MIMO (2 2) hyperfrequency filter, from experimental pointwise values in some range of frequencies provided by the CNES (French space agency). Data pre-processing (interpolation/completion): compute a stable matrix transfer function of high order which approximates these data, given by a great number (800) of Fourier coefficients. PRESTO-HF: software by F. Seyfert; HYPERION: software by J. Grimm. 31
32 Data and approximant at order 8 Nyquist diagrams 32
33 Data and approximant at order 8 Bode diagrams 33
34 1 2iπ Nudelman interpolation. T G(1/z)U (z I W ) 1 dz = V, (U, W ) output normal pair W W + U U = I (normalization) condition for a solution : positive definite solution to Lyapunov eq. P W P W = U U V V, P > 0. Then, G is the LFT of some K lossless G = (Θ 4 K + Θ 3 )(Θ 2 K + Θ 1 ) 1. Θ(z) = [ I 2p + (z 1)C(I zw ) 1 P 1 (I W ) C J ] H. Θ = [ Θ1 Θ 2 Θ 3 Θ 4 ], J = [ I 0 0 I Nudelman, 1977; Ball, Gohberg, Rodman, 1990 ], C = [ U V ] 34
35 Atlases of charts A chart: (W l, U l ),..., (W 1, U 1 ) output normal pairs, W k : n k n k, U k : p n k, k n k =n G L p n is in the chart if we can construct : G = G l,..., G k LF T G k 1,..., G 0. where G k satisfies Nudelman for (W k, U k, V k ) and P k > 0. The coordinate map: G (V 1,..., V l ) W k = w k, w k D, Nudelman G k (1/ w k )u k = v k, P k > 0 : v k < u k For a suitable choice of the LFT s, balanced realizations computed recursively from interpolation data (W k, U k, V k ). 35
36 Charts for real lossless functions W k, U k real, W k : 1 1 or 2 2 with two conjugate eighenvalues. Adapted chart : realization in real Schur form V 1 = V 2 =... V l = 0. W l 0 0 A =.... 0, B = U l. W 1 36
37 Allpass mutual encoding. W : n n 1-step Schur algorithm chart C Ω : (W, U) Ω(z) = Y + U(zI n W ) 1 X L p n P W P W = U U V V G(z) C Ω P > 0 (positive definite) adapted chart : Ω(z) = G(z), then P = I V = 0 Implementation : J.P. Marmorat 37
38 A finite atlas. A sub-atlas of Nevanlinna-Pick atlas (real case): Charts: w 1 = w 2 = = w n = 0; u 1, u 2,..., u n standard basis vectors. Balanced realization matrix= orthogonal positive p-upper Hessenberg matrix, up to a column permutation. Connection with the canonical form of (Hanzon, Ober, 1998) Peeters, Hanzon, M.O. 38
39 Hessenberg form. v n M n v n 1 M n M n 1 v n 2... M n M n 1... M 1 κ n vnv n 1 vnm n 1 v n 2... vnm n 1... M 1 0 κ n 1 vn 1v n 2... vn 1M n 2... M κ n 2 vn 2M n 3... M κ 2 v2v 1 v2m κ 1 v1, κ k = 1 v k 2, M k = I p + (κ k 1) v kv k v k 2. 39
40 Conclusion. parametrizations which present a nice behavior with respect to optimization problems an interesting connection between these parametrizations and realizations with particular structure a rich framework subclasses of functions or realizations with particular structure optimization parameters physical parameters olivi@sophia.inria.fr 40
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