A Relaxed Lyapunov Condition for Input-to-State Stability of Discrete-Time Nonlinear Systems

This paper investigates conditions for input-to state stability (ISS) and input-to-state practical stability (ISpS) of nonlinear discrete-time systems in terms of Lyapunov functions. A well known condition for input-to-state stability is the existence of an ISS-Lyapunov function for which, in every time step, the rate of decrease is bounded in the norm of the state, and the rate of increase due to an input is bounded in the norm of the input. We show that input-to-state stability can be established by means of ISS-Lyapunov functions which satisfy a weaker decrease condition defined over a bounded time interval. Furthermore, we characterize ISS (ISpS) with respect to a set of states (e.g. a consensus subspace) instead of only considering the origin. We illustrated the applicability of the proposed relaxed conditions by studying conditions under which suboptimal model predictive control (MPC) is input-to state stable.