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

This paper deals with piecewise-affine functions as Lyapunov function candidates for stability analysis of time-invariant discrete-time linear systems with saturating closed-loop control inputs. Considering the nonlinear behavior of the closed-loop system, new necessary and sufficient conditions for a piecewise-affine function as a Lyapunov function are presented. Based on linear programming formulation of these conditions, an effective procedure is proposed for determination of such Lyapunov functions and associated polyhedral regions of local asymptotic stability, with reduced conservativeness. Compared to piecewise-linear functions, like the Minkowski functions, piecewise-affine functions are more adequate to capture the dynamical effects of saturation nonlinearities, giving strictly less conservative results.