A gram-SOS approach for robust stability analysis of discrete-time systems with time-varying uncertainty

This paper addresses the problem of establishing robust asymptotical stability of discrete-time systems affected by time-varying parametric uncertainty. Specifically, it is supposed that the coefficients of the system depend linearly on the uncertainty, and that the uncertainty is confined into a polytope. In the continuous-time case, the problem can be addressed by imposing that the system admits a common homogeneous polynomial Lyapunov function (HPLF) at the vertices of the polytope. Unfortunately, such a strategy cannot be used in the discrete-time case since the derivative of the HPLF is nonlinear in the uncertainty. The problem is addressed in this paper through linear matrix inequalities (LMIs) by proposing a novel method for establishing decrease of the HPLF. This method consists, firstly, of introducing a Gram matrix built with respect to the state and parametrized by an arbitrary vector function of the uncertainty, and secondly, of requiring that a transformation of the introduced Gram matrix is a sum of squares (SOS) of matrix polynomials. The proposed method provides a condition for robust asymptotical stability that is sufficient for any degree of the HPLF candidate and that includes quadratic robust stability as special case.