Small-gain theorem for stability, cooperative control and distributed observation of infinite networks

TL;DR

This paper introduces a computationally feasible small-gain theorem for analyzing stability, control, and observation in infinite networks, unifying various problems under a common framework.

Contribution

It develops a novel small-gain theorem applicable to infinite networks, enabling stability analysis, consensus, and distributed observer design in a unified manner.

Findings

The small-gain condition can be efficiently computed via spectral radius.

The theorem applies to time-varying infinite networks.

It facilitates stability analysis, consensus, and observer design for infinite systems.

Abstract

Motivated by a paradigm shift towards a hyper-connected world, we develop a computationally tractable small-gain theorem for a network of infinitely many systems, termed as infinite networks. The proposed small-gain theorem addresses exponential input-to-state stability with respect to closed sets, which enables us to analyze diverse stability problems in a unified manner. The small-gain condition, expressed in terms of the spectral radius of a gain operator collecting all the information about the internal Lyapunov gains, can be numerically computed for a large class of systems in an efficient way. To demonstrate broad applicability of our small-gain theorem, we apply it to the stability analysis of infinite time-varying networks, to consensus in infinite-agent systems, as well as to the design of distributed observers for infinite networks.