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A bisection method for computing the \(H_{\infty}\) norm of a transfer matrix and related problems. (English) Zbl 0674.93020
Summary: We establish a correspondence between the singular values of a transfer matrix evaluated along the imaginary axis and the imaginary eigenvalues of a related Hamiltonian matrix. We give a simple linear algebraic proof, and also a more intuitive explanation based on a certain indefinite quadratic optimal control problem. This result yields a simple bisection algorithm to compute the \(H_{\infty}\) norm of a transfer matrix. The bisection method is far more efficient than algorithms which involve a search over frequencies, and the usual problems associated with such methods (such as determining how fine the search should be) do not arise. The method is readily extended to compute other quantities of system- theoretic interest, for instance, the minimum dissipation of a transfer matrix. A variation of the method can be used to solve the \(H_{\infty}\) Armijo line-search problem with no more computation than is required to compute a single \(H_{\infty}\) norm.

MSC:
93B40 Computational methods in systems theory (MSC2010)
65F35 Numerical computation of matrix norms, conditioning, scaling
93C05 Linear systems in control theory
30D55 \(H^p\)-classes (MSC2000)
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
15A18 Eigenvalues, singular values, and eigenvectors
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