Stability and Stabilization of Two Time Scale Switched Systems in Discrete Time

In this technical note, stability and stabilization of two time scale switched linear systems in the singular perturbation form are addressed in discrete time. We show that, under an arbitrary switching rule, stability of the slow and fast switched subsystems is not sufficient to assess stability of the original two time scale switched system, even if the singular perturbation parameter tends to zero. Therefore, we propose LMI based conditions that guarantee the asymptotic stability of the two time scale switched system using switched quadratic Lyapunov functions. These conditions express the fact that the coupling between fast and slow subsystems has to be taken into account in addition to stability properties of the two subsystems, when the switching rule is arbitrary. The presented conditions are extended to state feedback control design. A numerical example illustrates the features of the proposed approach.