Vector incremental L2-gain and incremental stability for switched nonlinear systems

This paper studies vector incremental L₂-gain and incremental stability for switched nonlinear systems using individual incremental gains and multiple storage functions. Firstly, a vector incremental L₂-gain concept for switched nonlinear systems is proposed. Each subsystem is not required to have incremental L₂-gain, but it has its own incremental L₂-gain and the related storage function, when it is active. The transformation of “energy” from the active subsystem to each inactive subsystem is characterized by cross-supply rates. Then, we show that a switched nonlinear system who has vector incremental L₂-gain can be incrementally stabilized under some constraints on the energy change of inactive subsystems. Secondly, a state-dependent switching law is designed to achieve vector incremental L₂-gain, even if each subsystem does not have incremental L₂-gain in the classic sense. Thirdly, each switched system is not required to have vector incremental L₂-gain, but the feedback interconnection of switched nonlinear systems is incrementally stabilized by the design of a composite switching law. The switching law allows the two switched systems switch asynchronously. Two examples are provided to verify the effectiveness of the proposed method.