Piecewise-affine Lyapunov functions for continuous-time linear systems with saturating controls

Abstract

This work is concerned with piecewise-affine functions as Lyapunov function candidates for stability analysis of time-invariant continuous-time linear systems with saturating closed-loop control inputs. Using a piecewise-affine model of the closed-loop system, new necessary and sufficient conditions for a piecewise-affine function be a Lyapunov function are presented. Based on linear programming formulation of these conditions, an effective backward set simulation procedure is proposed for construction of such Lyapunov functions for estimation of the region of local asymptotic stability. Compared to quadratic functions, piecewise-affine functions showed to be more adequate to capture the dynamical effects of saturation nonlinearities, giving strictly less conservative results.