On noise-to-state stability of stochastic discrete-time systems via finite-step Lyapunov functions

In this paper, we introduce a notion of so-called finite-step simulation functions for discrete-time control systems. In contrast to the existing notions of simulation functions, a finite-step simulation function does not need decay at each time step but after some finite numbers of steps. We show that the existence of such a function guarantees that the mismatch between output trajectories of the concrete and abstract systems lies within an appropriate bound. Using this relaxation, we develop a new type of small-gain conditions which are less conservative than those previously used for compositional construction of approximate abstractions of interconnected control systems. In particular, using finite-step simulation functions, it is possible to construct approximate abstractions, where stabilizability of each subsystem is not necessarily required. The effectiveness of our results is verified by an illustrative example.