Piecewise switching-dependent Lyapunov functional for analyzing systems with hysteresis

This paper presents a new class of Lyapunov functional to effectively analyze systems with hysteresis, which is a class of linear hybrid systems where the switching logic can be expressed as affine equalities of the continuous state. This class extends an existing piecewise quadratic Lyapunov functional in a way that the proposed Lyapunov function is an explicit function of the switching logic as well as the continuous state. Conservatism is thereby reduced in the new analysis approach using these Lyapunov functions because the analysis uses a more general class of Lyapunov functions than the previously published Popov/Yakubovich and piecewise quadratic Lyapunov functions. Furthermore, the derived analysis conditions can be cast as a semi-definite program (SDP), which can be efficiently solved using widely available software. Included are examples to show the effectiveness of this class of Lyapunov functional in analyzing relay control systems.