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Frequency-weighted optimal Hankel-norm approximation of stable transfer functions. (English) Zbl 0559.93036
This paper shows how to modify the optimal Hankel-norm approximation problem for scalar, finite-dimensional, linear, time-invariant systems to allow for frequency weighting G. Here G(z) is assumed to be a strictly minimum phase, strictly stable, real rational scalar transfer function. Let r be the smallest integer such that $$\lim_{z\to \infty}z^ rG(z)$$ is nonzero. Then, define the tilde operation by $$\tilde G(z)=z^{- r}G(z^{-1}).$$ We first restate the optimal Hankel-norm approximation method. Let $$\Gamma$$ (F) denote the Hankel matrix of a proper stable rational transfer function F(z) which has McMillan degree n. Then, for any integer $$k<n$$ there exists a unique bounded Hankel matrix $${\bar \Gamma}$$ of rank k such that $$\| \Gamma (F)-{\bar \Gamma}\| =\sigma_{k+1}(F),$$ where $$\sigma_{k+1}(F)$$ is the $$(k+1)$$-st singular value of $$\Gamma$$ (F) (the singular values being ordered in descending magnitude). Further the matrix $${\bar \Gamma}$$ is the Hankel matrix of some rational $$\chi$$ (z) with $$\chi (z)=F(z)-\sigma_{k+1}(F)\phi (z),$$ where $$\phi$$ is an all-pass function with exactly k strictly stable poles. $$\chi$$ (z) is a unique best $$L^{\infty}$$-approximation to F(z). When setting $$F=H\tilde G$$, we identify $$\chi$$ $$\tilde G^{-1}$$ as an $$L^{\infty}$$-approximation to H(z), and $$\bar W=[W]_+=[\chi \tilde G^{-1}]_+$$ as a stable reduced order approximation to H(z) where $$[W]_+$$ denotes the strictly stable part of W. $$\bar W$$ can be expressed as $$\bar W=H-\sigma_{k+1}(H\tilde G)[\phi \tilde G^{- 1}]_+.$$
We can take $$\bar W$$ as a stable k-th order frequency-weighted approximation to H(z). A lower bound of $$L^{\infty}$$-errors of the approximation is given. For the continuous-time case, the tilde operation is defined by $$\tilde G(s)=G(-s)$$ where G(s) is proper, strictly stable, and minimum phase and $$\lim_{s\to \infty}G(s)$$ is finite and nonzero. An optimal Hankel-norm reduction can be performed on $$F(s)=[H(s)\tilde G(s)]_+,$$ where H(s) is strictly stable and proper of order n. We can take $$\tilde W=[\chi \tilde G^{-1}]_+$$ as the k-th order frequency- weighted approximation to H(s). A numerical example is also given.
Reviewer: M.Kono

##### MSC:
 93B50 Synthesis problems 41A50 Best approximation, Chebyshev systems 93C05 Linear systems in control theory 93C99 Model systems in control theory 41A20 Approximation by rational functions 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
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