Hankel norm approximation of real symmetric rational matrix functions (Q580265)

For real symmetric stable rational \(p\times q\) matrix functions \(G(s)=C(sI-A)^{-1}B\) of McMillan degree n, the problem of Hankel norm approximation \(\hat G=F+K(F\in H^{\infty}_{p\times q}\), K of McMillan degree \(\leq \ell)\) is considered for all functions \(\hat F=G-\hat G\) with \(\| F\|_{L^{\infty}}\leq \sigma\), developing a suboptimal case \(\sigma_{\ell}(G)>\sigma >\sigma_{\ell +1}(G)\) and an optimal one \(\sigma =\sigma_{\ell +1}(G)\), \(\sigma_ i(G)=1\), denoting the Hankel singular values of G. If \(\sigma_{\ell}(G)>\sigma >\sigma_{\ell +1}(G)\), it is shown that all \(\hat F(s)\), \(\| \hat F\|_{L^{\infty}}\leq \sigma\) are determined by: \[ \hat F(s)=(\theta_{11}(s)H(s)+\theta_{12}(s))(\theta_{21}(s)H(s)+\theta_{22}(s))^{-1} \] with \(\theta_{ij}(s)\) matrix functions whose computation formulas are given and \(\| H(.)\|_{L^{\infty}}\leq 1\). \(\hat F(s)=\hat F^ T(s)\) iff \(H(s)=H^ T(s)\); \(\hat F(s)\) is real for real s iff H(s) is real for real s. For the optimal case, all \(\hat F(s)\), \(\| \hat F\|_{L^{\infty}}=\sigma_{\ell +1}(G)\) can be obtained by means of \[ \hat F(s)=[\theta_{11}(s),\theta_{12}(s)H(s)+\theta_{13}(s)]\cdot [\theta_{21}(s),\theta_{22}(s)H(s)+\theta_{23}(s)]^{-1} \] where the matrix functions \(\theta_{ij}(s)\) are given and \(\| H(.)\|_{L^{\infty}}\leq 1\). The following necessary and sufficient conditions are derived: \(\hat F(s)=\hat F^ T(s)\) iff \(H(s)=H^ T(s)\); \(\hat F(s)\) is real for real s iff H(s) is real for real s.