Root mean square gain of discrete-time switched linear systems under dwell time constraints

This paper deals with discrete-time switched linear systems and considers the problem of computing an upper bound to the dwell time ensuring a pre-specified root mean square (RMS) gain. As a natural consequence of treating general systems of this class in terms of the order and the number of subsystems, only sufficient conditions are worked out. They depend on the complete separation of the stabilizing and anti-stabilizing solutions of the algebraic Riccati equation associated to each subsystem. Moreover, as positive features, it is shown that the dwell time preserving the specification can be calculated through linear matrix inequalities (LMIs) and line search, being thus numerically solvable in polynomial time, and this allows the treatment of stable switched linear systems which do not admit a common Lyapunov function. The case of a guaranteed RMS gain for arbitrary switching signals is also addressed. A simple academic example constituted by three subsystems of third order is included for illustration.