Frequency domain solution to delay-type Nehari problem
The paper under review generalizes some known frequency-domain results on the delay-type Nehari problem in the stable case to the unstable case. The delay-type Nehari problem is described as follows: given a minimal state-space realization of a rational matrix \(G_\beta(s)=\left[\begin{smallmatrix} A & B\cr -C & 0\end{smallmatrix}\right]\), which is not necessarily stable, and \(h\geq 0\), characterize the optimal value \(\gamma_{\text{opt}}=\inf\{\|G_\beta(s)+e^{-sh}K(s)\|_{L_\infty}:\;K(s)\in H_\infty\}\), and for a given \(\gamma>\gamma_{\text{opt}}\) parameterize the suboptimal set of proper and causal \(K(s)\in H_\infty\) (compensators) such that \(\|G_\beta(s)+e^{-sh}K(s)\|_{L_\infty}<\gamma\). The solvability condition for this problem is formulated in terms of the non-singularity of three matrices. The optimal value \(\gamma_{\text{opt}}\) is the maximal \(\gamma\in (0,\infty)\) such that one of the three matrices becomes singular. A parameterization of all suboptimal compensators is given in the form of a chain-scattering transform of a parameter function \(Q(s)\) subject to \(\|Q(s)\|_{H_\infty}<\gamma\), where the coefficient matrix of this transform is given in a transparent form which incorporates an infinite-dimensional block (a modified Smith predictor) \(\Delta(s)\) and a finite-dimensional block \(W^{-1}(s)\).
scientific article; zbMATH DE number 1900990