Improved Estimation of Stability Regions for Uncertain Linear Systems with Saturating Actuators: an LMI-based Approach

This paper deals with the stability analysis of uncertain linear systems whose matrices belong to a polytope and whose actuators are subject to amplitude saturation. Two linear matrix inequality conditions are given: the first one is based on quadratic stability and its solution allows to determine a region of robust asymptotic stability written in terms of the matrix of the Lyapunov function (a state feedback gain can also be determined); the second condition is based on parameter-dependent Lyapunov functions and is written as sets of linear matrix inequalities that are progressively less conservative, thanks to the use of relaxations provided by Polya´s theorem, allowing the improvement of the estimation of the stability domain obtained by the first condition. The proposed conditions are written as convex optimization problems, avoiding the solution of bilinear matrix inequality based problems, and can also deal with the case where the system matrices are not precisely known but belong to a polytope. Numerical results illustrate that the conditions given in the paper allow to improve the estimates of domains of stability for the class of systems under investigation