Stability region analysis using simulations and sum-of-squares programming

The problem of computing bounds on the region-of-attraction for systems with polynomial vector fields is considered. Invariant sets of the region-of-attraction are characterized as sublevel sets of Lyapunov functions. Finite dimensional polynomial parameterizations for the Lyapunov functions are used. A methodology utilizing information from simulations to generate Lyapunov function candidates satisfying necessary conditions for bilinear constraints is proposed. The suitability of the Lyapunov function candidates are assessed solving linear sum-of-squares optimization problems. Qualified candidates are used to compute provably invariant subsets of the region-of-attraction and to initialize various bilinear search strategies for further optimization. We illustrate the method on several small examples drawn from the literature.