L2 Gain Performance Analysis of Linear Switched Systems: Fast Switching Behavior

We investigate the L₂ gain of switched linear systems under fast switching. For systems that possess a suitable notion of a time-average system, we characterize the relationship between the L₂ gain of the switched system and the L₂ gain of its induced time-average system when the switching rate is sufficiently fast. We show that the switched system L₂ gain is in general different from the average system L₂ gain if the input or output matrix switches. If only the state coefficient matrix switches, the input-output energy gain for a fixed L₂ input signal is bounded by the L₂ gain of the average system as the switching rate grows large. Additionally for a fixed L₂ input, the maximum pointwise in time difference between the switched and average system outputs approaches zero as the switching rate grows.