Simulation-guided lyapunov analysis for hybrid dynamical systems

Abstract

Lyapunov functions are used to prove stability and to obtain performance bounds on system behaviors for nonlinear and hybrid dynamical systems, but discovering Lyapunov functions is a difficult task in general. We present a technique for discovering Lyapunov functions and barrier certificates for nonlinear and hybrid dynamical systems using a search-based approach. Our approach uses concrete executions, such as those obtained through simulation, to formulate a series of linear programming (LP) optimization problems; the solution to each LP creates a candidate Lyapunov function. Intermediate candidates are iteratively improved using a global optimizer guided by the Lie derivative of the candidate Lyapunov function. The analysis is refined using counterexamples from a Satisfiability Modulo Theories (SMT) solver. When no counterexamples are found, the soundness of the analysis is verified using an arithmetic solver. The technique can be applied to a broad class of nonlinear dynamical systems, including hybrid systems and systems with polynomial and even transcendental dynamics. We present several examples illustrating the efficacy of the technique, including two automotive powertrain control examples.