A networked small-gain theorem based on discrete-time diagonal stability

TL;DR

This paper introduces a new networked small-gain theorem based on discrete-time diagonal stability, providing a generalized and more applicable criterion for finite-gain $L_2$ stability in interconnected systems.

Contribution

It develops a novel sufficient condition for networked system stability using DTDS, extending the classical small gain theorem and including new criteria for rank-one perturbations.

Findings

The new theorem generalizes the standard small gain theorem.

It allows application of known DTDS conditions to network stability analysis.

Examples demonstrate the effectiveness of the proposed conditions.

Abstract

We present a new sufficient condition for finite-gain $L_2$ input-to-output stability of a networked system. The condition requires a matrix, that combines information on the $L_2$ gains of the sub-systems and their interconnections, to be discrete-time diagonally stable (DTDS). We show that the new result generalizes the standard small gain theorem for the negative feedback connection of two sub-systems. An important advantage of the new result is that known sufficient conditions for DTDS can be applied to derive sufficient conditions for networked input-to-output stability. We demonstrate this using several examples. We also derive a new necessary and sufficient condition for a matrix that is a rank one perturbation of a Schur diagonal matrix to be DTDS.