Discovering multiple Lyapunov functions for switched hybrid systems with global exponential stability

We in this paper analyze the global exponential stability of switched hybrid systems, whose subsystems have polynomial vector fields, by discovering multiple Lyapunov functions in quadratic forms. We start with an algebraizable sufficient condition for the existence of quadratic multiple Lyapunov functions. Then, since different discrete modes are considered, we apply real root classification together with a projection operator to under-approximate this sufficient condition step by step, arriving at a set of semi-algebraic sets which only involve the coefficients of the pre-assumed multiple Lyapunov function. Afterwards, we compute a sample point in the corresponding semi-algebraic set for the coefficients, resulting in a multiple Lyapunov function. Finally, we test our approach on some examples using a prototypical implementation. These computation results show the applicability and promise of our approach. Especially, our present approach can further be extended for discovering multiple homogeneous Lyapunov functions of even degree.