Poly-quadratic stability and H∞ performance for discrete systems with time varying uncertainties

We consider discrete time systems with polytopic time varying uncertainty. We look for a class of parameter dependent Lyapunov functions which are quadratic on the system state and depend in a polytopic way on the uncertain parameter. We propose a necessary and sufficient condition for the computation of such a Lyapunov function. This allows to check asymptotic stability of the system under study. A robust H infinity performance problem is presented as an application