Robust Stability of Switched Linear Systems With Polytopic Uncertainty

In this paper, we study robust stability of a class of switched linear systems (SLSs) that is composed of a finite number of linear continuous variable systems (CVSs) and the switchings of CVSs are predefined and represented by a finite state machine. Each CVS consists of a cluster of unstable matrices, which are simultaneously block-triangularizable. This type of SLSs cannot be addressed by existing results. A decomposition method is used to decompose the original system into several lower dimensional SLSs at the diagonal block positions. Our results show that the original higher dimensional SLS is asymptotically stable if all these lower dimensional SLSs are asymptotically stable. The synergy between the continuous of each lower dimensional SLSs and the discrete dynamics is fully utilized by the cycle analysis method to derive sufficient conditions on the robust stability of each lower dimensional SLS. With our method, all CVSs in a cycle instead of each individual CVS is taken as an analysis unit. The condition on the stability of SLSs can then be relaxed in the sense that Lyapunov functions are only required to be non-increasing along each type of cycle instead of each CVS.