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Calculating the \(H_{\infty}\)-norm using the implicit determinant method. (English) Zbl 1305.65161
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 \[ \begin{aligned} &\dot{x}(t)=Ax(t)+Bu(t), \\ &y(t)=Cx(t)+Du(t), \end{aligned}\tag{1} \] 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 \[ ||G||_{\infty}:=\sup_{\omega \in \mathbb{R}} \sigma_{\max} (G(i \omega)), \tag{2} \] 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.

65K10 Numerical optimization and variational techniques
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
93C80 Frequency-response methods in control theory
93D09 Robust stability
93C05 Linear systems in control theory
65F40 Numerical computation of determinants
Eigtool; COMPleib
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