Stability of networked nonlinear systems: Generalization of small-gain theorem and distributed testing

This paper studies the stability problem for networked control systems. A general result is presented to determine either global uniform boundedness (GUB), global asymptotic stability (GAS) or input-to-state stability (ISS), for interconnected nonlinear systems. This result checks stability in terms of a scalar called the network gain, hence we call the result the network gain theorem. The result generalizes the previously known matrix small-gain theorem and cyclic small-gain theorem for ISS. As in these results, our theorem does not assess the stability of a given networked system, but of a whole family of networked systems satisfying certain common assumption. An advantage of our stability condition is that it is not only sufficient, but also necessary, in the sense that, if not met, there exists an unstable networked system within that family. To complement our theoretical result, we propose a fully distributed algorithm to compute the network gain. We present simulation results to illustrate the proposed algorithm.(c) 2023 Elsevier Ltd. All rights reserved.

Automated summary

This paper studies the stability of networked control systems and presents a general result that determines global uniform boundedness, global asymptotic stability, or input-to-state stability for interconnected nonlinear systems. The result is checked using a scalar called the network gain, leading to the name "network gain theorem". The result extends the previously known matrix small-gain theorem and cyclic small-gain theorem for ISS. Additionally, the stability condition proposed is both sufficient and necessary, meaning that if it is not met, there exists an unstable networked system within the considered family. To complement the theoretical result, a fully distributed algorithm for computing the network gain is proposed, and simulation results are presented to illustrate its effectiveness.