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

Abstract

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.