Stability test of multidimensional discrete-time systems via sum-of-squares decomposition

A new stability test for d-dimensional discrete-time systems is presented. It consists of maximizing the minimum eigenvalue of a positive definite Gram matrix associated with a polynomial positive on the unit d-circle. This formulation is based on expressing the polynomial as a sum-of-squares and leads to a semidefinite programming (SDP) problem. Several heuristics are introduced for reducing the complexity of the problem in the case of sparse polynomials. Although in its practical form the test is based on a sufficient condition, the experimental results show that correct stability decisions are given. Comparisons with previous methods are favorable.