Sum-of-squares polynomials and the stability of discrete-time systems

The purpose of this paper is to gather several recent applications of sum-of-squares polynomials in analyzing the stability of discrete-time systems, as well as to present a new one. We review a stability test for multidimensional systems. Also, a method to describe a convex stability domain is presented in the general case. The new application regards a robust stability test for polynomials whose coefficients depend polynomially on some bounded parameters. The test uses a Positivstellensatz relaxation in the form of a semidefinite programming problem. We give some examples and point out possible applications.