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Construction of optimal Hankel approximations in the guise of stochastic processes. (English) Zbl 0774.41016

Probability theory and applications, Essays to the Mem. of J. Mogyoródi, Math. Appl. 80, 77-98 (1992).
[For the entire collection see Zbl 0755.00022.]
The author considers the problem of approximating a given rational function \(K(z)=C(zI-A)^{-1} B\), where \(A\) is a stable (in discrete time sense) matrix, with another rational matrix valued function of the same type \(G(z)=H(zI-F)^{-1} G\), where \(F\) is again a stable matrix, but under the restriction that \(G(z)\) has fixed number of poles inside the unit circle. By a result due to K. Glover [Int. J. Control 39, 1115-1193 (1984; Zbl 0543.93036)], the above problem reduces to the examination of the approximation of type \(K(z)-\sigma\phi(z)\), where \(\Phi(z)\) is unitary on the unit circle. In the extensive literature on this problem (see, e.g., the review of the book “Interpolation of rational matrix functions” by J. A. Ball, I. Gohberg and L. Rodman, due to M. A. Kaashoek in Bull. Am. Math. Soc., New Ser. 28, No. 2, 426-434 (1993)], the author found his own way approaching this question “using an essentially elementary method”. “Our aim is only to prove that using the so-called one-step extension of the transfer function \(\theta\)” (from the Ball-Ran’s paper [J. A. Ball and A. C. M. Ran, SIAM J. Control Optimization 25, 362-382 (1987; Zbl 0623.93016)], “in the canonical case, and the equation given by Glover” (loc. cit.) “in the singular case. We show that the construction can be reduced to the solution of a system of linear equations. We also give a short singular value analysis of the Hankel operator corresponding to the error transfer function \(K-G\)”.

MSC:

41A20 Approximation by rational functions

Keywords:

Hankel operator

Biographic References:

Mogyoródi, József
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