An LMI-based technique for robust stability analysis of linear systems with polynomial parametric uncertainties

Robust stability analysis of state space models with respect to real parametric uncertainty is a widely studied challenging problem. In this paper, a quite general uncertainty model is considered, which allows one to consider polynomial nonlinearities in the uncertain parameters. A class of parameter-dependent Lyapunov functions is used to establish stability of a matrix depending polynomially on a vector of parameters constrained in a polytope. Such class, denoted as Homogeneous Polynomially Parameter-Dependent Quadratic Lyapunov Functions (HPD-QLFs), contains quadratic Lyapunov functions whose dependence on the parameters is expressed as a polynomial homogeneous form. Its use is motivated by the property that the considered matricial uncertainty set is stable if and only there exists a HPD-QLF. The paper shows that a sufficient condition for the existence of a HPD-QLF can be derived in terms of Linear Matrix Inequalities (LMIs).