A Small-Gain Theorem for Discrete-Time Convergent Systems and Its Applications

TL;DR

This paper introduces a small-gain theorem for discrete-time systems ensuring convergence to a reference output, with applications in observer design, nonlinear systems, and reservoir computing.

Contribution

It presents a novel small-gain theorem for discrete-time output-feedback systems that guarantees uniform input-to-output convergence.

Findings

The theorem ensures convergence to a bounded reference output.

Applications include observer-based control and quantum reservoir systems.

The approach extends to time-varying interconnected systems.

Abstract

Convergent, contractive or incremental stability properties of nonlinear systems have attracted interest for control tasks such as observer design, output regulation and synchronization. The convergence property plays a central role in the neuromorphic (brain-inspired) computing of reservoir computing, which seeks to harness the information processing capability of nonlinear systems. This paper presents a small-gain theorem for discrete-time output-feedback interconnected systems to be uniformly input-to-output convergent (UIOC) with outputs converging to a bounded reference output uniquely determined by the input. A small-gain theorem for interconnected time-varying discrete-time uniform input-to-output stable systems that could be of separate interest is also presented as an intermediate result. Applications of the UIOC small-gain theorem are illustrated in the design of observer-based controllers and interconnected nonlinear classical and quantum dynamical systems (as reservoir computers) for black-box system identification.