Constructing piecewise quadratic Lyapunov functions with linear programming for continuous-time switched and conewise linear systems

In this paper, we present a method for constructing continuous piecewise quadratic (CPQ) Lyapunov functions for continuous-time switched and conewise linear systems using linear programming (LP). Key in our approach is the formulation of effective sufficient conditions for the copositivity of matrices via diagonal dominance. This formulation consists of linear constraints and can be expressed as an LP. It is shown that the sufficient conditions are also necessary conditions for the existence of a Lyapunov function, given a sufficiently refined CPQ function. We provide an in-depth comparison between our new method and other computational methods in the literature, and provide extensive numerical experiments on various switched and conewise linear systems. In particular, we show that the proposed method is the most accurate of the LP-based methods for constructing Lyapunov functions and is a numerically competitive alternative to LMI-based methods.