Analysis of hybrid systems resulting from relay-type hysteresis and saturation: a Lyapunov approach

Abstract—This paper studies a class of hybrid systems with linear (or linear plus saturated linear) continuous and discrete dynamics, which are determined by a flow map and jump map, and state-triggered jumps. One motivation for considering this class of systems is that they can model control systems with a relay-type hysteresis element. Based on Lyapunov the-orems for hybrid systems, a Lyapunov function is constructed that effectively incorporates the feature of the jumps. Global asymptotic stability analysis is presented for the case when the flow map is linear, and local asymptotic stability analysis is presented for the case when the flow map is linear plus saturated linear. The stability conditions are derived as matrix inequalities. A numerical example is presented to illustrate the hybrid modeling process for a system experiencing hysteresis. Simulations confirm the effectiveness of the proposed analysis tools and demonstrate the potential of the Lyapunov function