[L.sub.2] gain of periodic linear switched systems: fast switching behavior

We investigate the [L.sub.2] gain of periodic linear switched systems under fast switching. For systems that possess a suitable notion of a time-average system, we characterize the relationship between the [L.sub.2] gain of the switched system and the [L.sub.2] gain of its induced time-average system when the switching rate is sufficiently fast. We show that the switched system [L.sub.2] gain is in general different from the average system [L.sub.2] gain if the input or output coefficient matrix switches. If only the state coefficient matrix switches, the input-output energy gain for a fixed [L.sub.2] input signal is bounded by the [L.sub.2] gain of the average system as the switching rate grows large. Additionally, for a fixed [L.sub.2] input, the maximum pointwise in time difference between the switched and average system outputs approaches zero as the switching rate grows. Index Terms--Fast switching, [L.sub.2] gain analysis, switched systems.