Frequency domain solution to delay-type Nehari problem
type: Item
instance of: scholarly article
title: Frequency domain solution to delay-type Nehari problem
zbMATH Open document ID: 1013.93010
DOI: 10.1016/S0005-1098(02)00246-7
author: Qing-Chang Zhong
published in: Automatica
publication date: 2003-04-27T00:00:00Z
review text: 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 \(W^{-1}(s)\).
Mathematics Subject Classification ID: 93B36
Mathematics Subject Classification ID: 93B20
Mathematics Subject Classification ID: 93C80
Mathematics Subject Classification ID: 93C23
zbMATH DE Number: 1900990
zbMATH Keywords: \(H_\infty\)-control
zbMATH Keywords: Nehari problem
zbMATH Keywords: time delay systems
zbMATH Keywords: Riccati equation
zbMATH Keywords: Smith predictor
zbMATH Keywords: frequency-domain results
zbMATH Keywords: minimal state-space realization
zbMATH Keywords: chain-scattering transform
reviewed by: Dmitry Kaliuzhnyi-Verbovetskyi
MaRDI profile type: Publication
cites work: \(H^ \infty\)-optimal control and related minimax design problems. A dynamic game approach.
cites work: The Nevanlinna–Pick Problem for Matrix-Valued Functions
cites work: Explicit formulas for optimally robust controllers for delay systems
cites work: \({\mathcal H}^\infty\) sensitivity minimization for delay systems
cites work: Robust control of infinite dimensional systems. Frequency domain methods
cites work: A course in \(H_{\infty}\) control theory
cites work: Classes of linear operators. Vol. II
cites work: AJ-Spectral Factorization Approach to $\mathcal{H}_\infty $
cites work: On ℋ/sub ∞/ control for dead-time systems
cites work: Control of systems with I/O delay via reduction to a one-block problem
cites work: ${\cal H}_\infty$ Control and Estimation Problems with Delayed Measurements: State-Space Solutions
cites work: The Nehari problem in systems with distributed input delays is inherently finite dimensional
cites work: Robust control in the gap: a state-space solution in the presence of a single input delay
cites work: Weighted Sensitivity Minimization in Systems with a Single Input Delay: A State Space Solution
cites work: The standard H/sub ∞/ problem in systems with a single input delay
cites work: On standard H∞ control of processes with a single delay
cites work: Q4368716
cites work: On the weighted sensitivity minimization problem for delay systems
description: scientific article; zbMATH DE number 1900990