$ {\cal L}_{2}$ Gain of Periodic Linear Switched Systems: Fast Switching Behavior

We investigate the
L

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

₂
gain of the switched system and the
L

₂
gain of its induced time-average system when the switching rate is sufficiently fast. We show that the switched system
L

₂
gain is in general different from the average system
L

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

₂
input signal is bounded by the
L

₂
gain of the average system as the switching rate grows large. Additionally, for a fixed
L

₂
input, the maximum pointwise in time difference between the switched and average system outputs approaches zero as the switching rate grows.