Gershgorin–Brualdi perturbations and Riccati equations

For uncertain linear systems with complex parameter perturbations of static output feedback type a quadratic Liapunov function of maximal robustness was constructed in [D. Hinrichsen, A.J. Pritchard, Stability radius for structured perturbations and the algebraic Riccati equation, Syst. Control Lett. 8 (1986) 105–113]. Such Liapunov functions can be used to ensure the stability of uncertain systems under arbitrary nonlinear and time-varying perturbations which are smaller than the stability radius. In this paper we establish analogous results for structured Gershgorin–Brualdi type perturbations of diagonal matrices where all the matrix entries at an arbitrarily prescribed set of positions are independently perturbed. We also derive explicit and computable formulae for the associated μ-values, stability radii and spectral value sets.