Robust, diagonal and D-stability via QLF´s: The discrete-time case

The robust stability problem of discrete-time linear systems subjected to a class of sector-bounded nonlinear time-varying perturbations is considered. The idea of a simultaneous quadratic Lyapunov function-a single Lyapunov function that guarantees the asymptotic stability of the whole class of perturbed system-is used to analyze the problem. It is shown that certain classes of nominal system matrices play an important role in this simultaneous robust stability problem. In particular, simultaneous stability is implied by diagonal stability and it is shown that, for several important classes of matrices, the converse also holds. D-stability is defined for the discrete-time case and is shown to be equivalent to the stability of a finite set of matrices which are the vertices of a certain polytope. This gives an easily tested necessary condition for diagonal stability