A Necessary and Sufficient Condition for Robust Stability of LTI Discrete-Time Systems using Sum-of-Squares Matrix Polynomials

This paper deals with the robust stability of discrete-time systems with convex polytopic uncertainties. First, it is proved that the parameter-dependent Lyapunov function can be assumed to be a polynomial with a specific bound on its degree. Then, it is shown that the robust stability of any system is equivalent to the existence of two matrix polynomials with some bounds on their degrees, where these two polynomials and also the corresponding Lyapunov matrix polynomial satisfy a specific relation. Furthermore, a method is presented to convert the problem of existence of such polynomials to a set of linear matrix inequalities and equalities, which is referred to as semidefinite programming (SDP), and can be solved by using a number of available softwares. One of the capabilities of the proposed method is that the bounds obtained for the degrees of the related polynomials can be replaced by any smaller numbers in order to simplify the computations, at the cost of a potentially conservative result. Moreover, in the case when it is desired to accurately solve the robust stability problem while the degrees of the related polynomials are large, a computationally efficient method is proposed to convert the problem to the SDP with a reduced number of variables. The efficacy of this work is demonstrated in two numerical examples