The H∞ norm of a transfer matrix function for a continuous-time control system is the maximum of the norm of the transfer matrix on the imaginary axis, or equivalently, the reciprocal of the largest value of ε such that the associated ε-spectral value set is contained in the left half-plane. We start by defining spectral value sets and discussing some of their fundamental properties, including the intricate relationship between the singular vectors of the transfer matrix and the eigenvectors of the corresponding perturbed system matrix. We then introduce an iterative method for approximating the ε-spectral value set abscissa (the maximum of the real part of the points in the set), characterizing the fixed points of the iteration, and explain how the procedure can be combined with a Newton-bisection outer iteration to approximate the H∞ norm. We then explain why this idealized algorithm sometimes breaks down and introduce a method called hybrid expansion-contraction to address this deficiency. Under reasonable assumptions, the new algorithm finds locally maximal values of the norm of the transfer matrix on the imaginary axis and although these are only lower bounds on the H∞ norm, it typically finds good approximations in cases where we can test this. It is much faster than the standard Boyd-Balakrishnan-Bruinsma-Steinbuch algorithm to compute the H∞ norm when the system matrices are large and sparse. The main work required by the algorithm is the computation of the rightmost eigenvalues of a sequence of matrices that are rank-one perturbations of a sparse matrix.
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