Calculating the \(H_{\infty}\)-norm using the implicit determinant method
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.
scientific article; zbMATH DE number 6356189