1. Hardy spaces<\/em>: the space \\( L^2 \\) of square integrable functions on the unit disk splits into two orthogonal subspaces: the space \\(H^2\\) of \\(L^2\\) functions having an analytic continuation in the unit disk (negative Fourier coefficients are zero) and the space \\(H_-^2\\) of \\(L^2\\) functions having an analytic continuation outside the unit disk and vanishing at infinity (non-negative Fourier coefficients are zero)<\/p>\n 2. stable<\/em>: poles inside the unit disk<\/p>\n 3. McMillan degree<\/em>: size of the A-matrix in a minimal realization\/ number of poles counting multiplicity<\/p>\n We use a special matrix fraction description known has Douglas-Shapiro-Shields factorization, and write a rational matrix of degree n as<\/p>\n <\/p>\n By the Hilbert projection theorem <\/strong>, at a local minimum \\(H(z)\\)<\/span> of the criterion, \\(C(z)\\)<\/span> has to be the projection \\(P^+\\) of \\(G^{-1}F\\) onto \\(H^2\\). The approximation problem reduces to the minimization of a concentrated criterion [4,5]<\/p>\n Advantage: minimization over a bounded set<\/strong><\/p>\n 4. lossless matrix: rationnal matrix analytic outside the unit disk which takes unitary values on the circle.<\/p>\n In practice, the concentrated criterion is computed using balanced realizations5<\/sup> to represent lossless matrix functions. We have the nice following property Advantage: computation with unitary matrices <\/strong><\/p>\n 5. A balanced realization<\/em> is such that both grammians are equal and diagonal<\/p>\n The optimization space has a manifold structure [6]. We use an atlas of charts, like for the earth, where each chart provides a local flat (Euclidien) representation which allows for the use of differential tools. In the actual version of RARL2, we use the atlas presented in [7], called lossless mutual encoding <\/em><\/p>\n Advantage: use of differential tools safe <\/strong><\/p>\n In RARL2, the minimization process makes use of the Matlab solver fmincon<\/strong>. [1] L. Baratchart, Existence and generic properties for L2 approximants of linear systems I.M.A. Journal of Math. Control and Identification<\/i> 3, 89-101 (1986)<\/p>\n [2]L. Baratchart and M. Olivi Critical points and error rank in matrix H2 rational approximation and the strong differential consistency of output error identification from white noise inputs<\/a>Constructive Approximation <\/i>14, pp. 273-300 (1998)<\/p>\n [3] L. Baratchart, M. Cardelli, M. Olivi, Identification and rational L2 approximation: a gradient algorithm, Automatica<\/i> 27(2) 413-418 (1991)<\/p>\n [4] P. Fulcheri and M. Olivi Matrix rational H2 approximation: a gradient algorithm based on Schur analysis <\/a> SIAM Journal on Control and Optimization<\/i> Vol. 36, No. 6, 2103-2127(1998)<\/p>\n [5] M. Olivi, F. Seyfert, J.P. Marmorat, Identication of microwave filters by analytic and rational H2<\/sup>approximation, <\/a>Automatica, <\/i> 49, 317-325(2013)<\/p>\n [6] D. Alpay, L. Baratchart, A. Gombani, On the differential structure of matrix-valued rational inner functions, Operator Theory: Advances and Applications, <\/i> 73, 30–66(1994)<\/p>\nReduction of the optimization space<\/h3>\n
\n
\nover the (quotient) set of lossless matrices of McMillan degree n <\/strong> up to a right unitary matrix.<\/p>\n State-space formulas <\/h3>\n
\n
\nD+C(zI-A)^{-1}B &\\hbox{ lossless }\\\\
\n(A,B,C,D) &\\hbox{ balanced }\\end{array}\\right. \\Leftrightarrow \\begin{bmatrix} D & C \\\\
\nB & A \\end{bmatrix} \\hbox{ unitary }\\) <\/center><\/p>\n\n
\n
\nwhere the pair \\((C,A)\\)<\/span> is assumed output normal <\/strong>: \\(A^*A+C^*C=I\\)<\/span><\/li>\n
\n
\n
\ndefined over the set of equivalence classes under similarity of balanced output pairs (BOP) <\/strong>.\n<\/li>\n<\/ul>\nParametrization of the optimization space<\/h3>\n
\n
\n
\nis positive definite. <\/li>\n
\nOptimization over a manifold<\/h3>\n
\nThe optimization is carried over the manifold structure as illustrated below:
\n![\"manifold\"](\"https:\/\/project.inria.fr\/rarl2\/files\/2016\/09\/manifold.png\")
<\/a><\/center><\/p>\n\n
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\n References <\/h3>\n