Relaxed ISS Small-Gain Theorems for Discrete-Time Systems

TL;DR

This paper introduces relaxed ISS small-gain theorems for discrete-time systems that do not require each subsystem to be ISS, reducing conservatism and establishing conditions for exponential ISS systems.

Contribution

It develops a new class of dissipative finite-step ISS Lyapunov functions and proves their necessity and sufficiency for system ISS without requiring subsystem stability.

Findings

Relaxed small-gain theorems are non-conservative for exponential ISS systems.

Dissipative finite-step ISS Lyapunov functions characterize system ISS.

The approach broadens applicability of ISS small-gain theory.

Abstract

In this paper ISS small-gain theorems for discrete-time systems are stated, which do not require input-to-state stability (ISS) of each subsystem. This approach weakens conservatism in ISS small-gain theory, and for the class of exponentially ISS systems we are able to prove that the proposed relaxed small-gain theorems are non-conservative in a sense to be made precise. The proofs of the small-gain theorems rely on the construction of a dissipative finite-step ISS Lyapunov function which is introduced in this work. Furthermore, dissipative finite-step ISS Lyapunov functions, as relaxations of ISS Lyapunov functions, are shown to be sufficient and necessary to conclude ISS of the overall system.