Stability analysis of nonlinear systems via multiple mixed max-min based Lyapunov functions

This paper presents stability analysis of nonlinear systems using a new type of Lyapunov functions. We define so-called multiple mixed max-min based Lyapunov functions that contain existing piecewise Lyapunov functions as a special case. The extension is realized by multiple mixed connection of max and min operations for plural local quadratic functions. Hence, stability analysis discussed in this paper is more relaxed than those based on existing piecewise Lyapunov functions in addition to quadratic Lyapunov functions. Relaxed stability conditions based on multiple mixed max-min based Lyapunov functions are derived by considering switching conditions among the plural local quadratic functions. The derived stability conditions are represented in terms of bilinear matrix inequalities (BMIs). Unfortunately, BMIs cannot be generally solved via LMI solvers. Therefore, to simply solve BMIs, we propose a practical way that combines an LMI solver and particle swarm optimization. In this paper, two analytical examples are provided. The examples illustrate that our approach provides more relaxed stability results than existing piecewise Lyapunov functions approaches.