On the admissible equilibrium points of nonlinear dynamical systems affected by parametric uncertainty: Characterization via LMIs

This paper investigates the set of admissible equilibrium points of nonlinear dynamical systems affected by parametric uncertainty. As it is well-known, determining this set is a difficult problem since one should compute the solutions of a system of nonlinear equations for all the admissible values of the uncertainty, which typically amounts to an infinite number of times. In order to address this problem, this paper proposes a characterization of this set via convex optimization for the case of polynomial nonlinearities and uncertainty constrained in a polytope. Specifically, it is shown that an upper bound of the smallest outer estimate with a freely selectable fixed shape can be obtained by solving a linear matrix inequality (LMI) problem built through the square matrix representation (SMR). Then, a necessary and sufficient condition is provided for establishing the tightness of the found upper bound. The proposed methodology and its benefits are illustrated through several numerical examples.