Sufficient and necessary conditions for discrete-time nonlinear switched systems with uniform local exponential stability

In this paper, we investigate sufficient and necessary conditions of uniform local exponential stability (ULES) for the discrete-time nonlinear switched system (DTNSS). We start with the definition of T-step common Lyapunov functions (CLFs), which is a relaxation of traditional CLFs. Then, for a time-varying DTNSS, by constructing such a T-step CLF, a necessary and sufficient condition for its ULES is provided. Afterwards, we strengthen it based on a T-step Lipschitz continuous CLF. Especially, when the system is time-invariant, by the smooth approximation theorem, the Lipschitz continuity condition of T-step CLFs can further be replaced by continuous differentiability; and when the system is time-invariant and homogeneous, due to the extension of Weierstrass approximation theorem, T-step continuously differentiable CLFs can even be strengthened to be T-step polynomial CLFs. Furthermore, three illustrative examples are additionally used to explain our main contribution. In the end, an equivalence between time-varying DTNSSs and their corresponding linearisations is discussed.