Stability regions for saturated linear systems via conjugate Lyapunov functions

We use a recently developed duality theory for linear differential inclusions (LDIs) to enhance the stability analysis of systems with saturation. Based on the duality theory, the condition of stability for a LDI in terms of one Lyapunov function can be easily derived from that in terms of a Lyapunov function conjugate to the original one in the sense of convex analysis. This paper uses a particular conjugate pair, the convex hull of quadratics and the maximum of quadratics, along with their dual relationship, for the purpose of estimating the domain of attraction for systems with saturation nonlinearities. To this end, the nonlinear system is locally transformed into a LDI system with an effective approach which enables optimization on the local LDI description. The optimization problems are derived for both the convex hull and the max functions, and the domain of attraction is estimated with both the convex hull of ellipsoids and the intersection of ellipsoids. A numerical example demonstrates that the estimation of the domain of attraction by this paper's methods drastically improve those by the earlier methods.