Root-mean-square gains of switched linear systems: A variational approach

We consider the problem of computing the root- mean-square (RMS) gain of switched linear systems. We develop a new approach which is based on an attempt to characterize the "worst-case" switching law (WCSL), that is, the switching law that yields the maximal possible gain. Our main result provides a sufficient condition guaranteeing that the WCSL can be characterized explicitly using the differential Riccati equations corresponding to the linear subsystems. This condition automatically holds for first-order systems, so we obtain a complete solution to the RMS gain problem in this case. In particular, we show that in the first-order case there always exists a WCSL with no more than two switches.