Computation and stability of limit cycles in hybrid systems

In this note, a practical way to compute limit cycles in context of hybrid systems is investigated. As in many hybrid applications the steady state is depicted by a limit cycle, control design and stability analysis of such hybrid systems require the knowledge of this periodic motion. Analytical expression of this cycle is generally an impossible task due to the complexity of the dynamic. A fast algorithm is proposed and used to determine these cycles in the case where the switching sequence is known. The proposed method is based on the rule played by the switching times in the sensitivity functions. The stability of the cycle is also deduced at the end of the run thanks to the computation of the Jacobian matrix of the linearized sampled time systems. This work can be used as a starting point for sensibility analysis, measurement of attraction area and control design. Index Terms limit cycles, hybrid systems, sensitivity functions, stability, power converters.