Calculating the \(H_{\infty}\)-norm using the implicit determinant method
type: Item
instance of: scholarly article
zbMATH Open document ID: 1305.65161
DOI: 10.1137/130933228
author: Melina A. Freitag
author: Alastair Spence
author: Paul van Dooren
published in: SIAM Journal on Matrix Analysis and Applications
publication date: 2014-10-15T00:00:00Z
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 93C80
Mathematics Subject Classification ID: 93D09
Mathematics Subject Classification ID: 93C05
Mathematics Subject Classification ID: 65F40
zbMATH DE Number: 6356189
zbMATH Keywords: \(H_{\infty}\)-norm
zbMATH Keywords: Hamiltonian matrix
zbMATH Keywords: transfer matrix
zbMATH Keywords: Newton method
zbMATH Keywords: robust stability
zbMATH Keywords: linear dynamic system
zbMATH Keywords: maximum singular value
zbMATH Keywords: implicit determinant method
zbMATH Keywords: numerical example
describes a project that uses: COMPleib
describes a project that uses: Eigtool
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1137/130933228
OpenAlex ID: W1964334854
title: Calculating the \(H_{\infty}\)-norm using the implicit determinant method
review text: It is well known that the \(H_{\infty}\)-norm of a transfer function matrix is an important property for measuring robust stability in classical control theory. The linear dynamic system NEWLINE\[NEWLINE\begin{aligned} &\dot{x}(t)=Ax(t)+Bu(t), \\ &y(t)=Cx(t)+Du(t), \end{aligned}\tag{1}NEWLINE\]NEWLINE where \(A \in \mathbb{C}^{n\times n}\), \(B\in \mathbb{C}^{n\times p}\), \(C \in \mathbb{C}^{m\times n}\), and \(D \in \mathbb{C}^{m\times p}\) is considered. Let \(G(s)=C(sI-A)^{-1}B+D\) be the transfer matrix of the system (1). The \(H_{\infty}\)-norm of the transfer matrix \(G(s)\) is defined as NEWLINE\[NEWLINE ||G||_{\infty}:=\sup_{\omega \in \mathbb{R}} \sigma_{\max} (G(i \omega)), \tag{2}NEWLINE\]NEWLINE where \(\omega_{\max}\) denotes the maximum singular value of the matrix. The optimization problem (2) is reformulated to one of finding zeros of the determinant of a parameter-dependent Hermitian matrix. The implicit determinant method is described and applied to the problem. The implementation of the implicit determinant method is discussed and numerical examples that illustrate the performance of the method are given.
reviewed by: Mykhaylo Pahirya
description: scientific article; zbMATH DE number 6356189