Fast approximation of the H∞ norm via optimization over spectral value sets.

The H∞ norm of a transfer matrix function for a control system is the reciprocal ofthe largest value of ε such that the associated ε-spectral value set is contained in the stability regionfor the dynamical system (the left half-plane in the continuous-time case and the unit disk in thediscrete-time case). After deriving some fundamental properties of spectral value sets, particularlythe intricate relationship between the singular vectors of the transfer matrix and the eigenvectors ofthe corresponding perturbed system matrix, we extend an algorithm recently introduced by Guglielmiand Overton [SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1166–1192] for approximating the maximalreal part or modulus of points in a matrix pseudospectrum to spectral value sets, characterizing itsfixed points. We then introduce a Newton-bisection method to approximate the H∞ norm, forwhich each step requires optimization of the real part or the modulus over an ε-spectral value set.Although the algorithm is guaranteed only to find lower bounds on the H∞ norm, it typically findsgood 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 matricesare large and sparse and the number of inputs and outputs is small. The main work required by thealgorithm is the computation of the spectral abscissa or radius of a sequence of matrices that arerank-one perturbations of a sparse matrix.