Mean-square stability of discrete-time switched systems under modeled random switching

This paper studies the mean-square stability of discrete-time linear switched systems with random switching, where the switching signal is the output of a logic dynamical switching model driven by an independent and identically distributed (i.i.d.) process. This class of switching is referred to as the modeled random switching. By regarding the switching model as a part of the system, a combined switched system with i.i.d. switching is obtained, which is of a hybrid nature, that is, the augmented state vector has both logic and continuous components. The semi-tensor product of matrices and the vector representation of logic are applied to merge the logic and the continuous components of the state. The equivalence between the mean-square stability of the original switched system and that of the merged switched system under i.i.d. switching is proved. By deriving the dynamics of the column-stacking form of the covariance matrix, necessary and sufficient conditions of mean-square stability for discrete-time linear switched systems with modeled random switching are obtained. An illustrative example is provided to demonstrate the usage of the proposed results.