A necessary and sufficient condition for stability of LMS-based consensus adaptive filters

This paper investigates the stability and performance of the standard least mean squares (LMS)-based consensus adaptive filters under a changing network topology. We first analyze the stability for possibly unbounded, non-independent and non-stationary signals, by introducing an information condition that can be shown to be not only sufficient but also necessary for the global stability. We also demonstrate that the distributed adaptive filters can estimate a dynamic process of interest from noisy measurements by a set of sensors working in a collaborative manner, in the natural scenario where none of the sensors can fulfill the estimation task individually. Furthermore, we give an analysis of the filtering error under various assumptions without stationarity and independency constraints on the system signals, and thus do not exclude applications to stochastic systems with feedback. In contrast to the analyses of the normalized LMS-based distributed adaptive filters, we need to use stochastic averaging theorems in the stability analysis due to possible unboundedness of the system signals.