Automatica, Vol.47, No.7, 1366-1378, 2011
Asymptotic properties of consensus-type algorithms for networked systems with regime-switching topologies
This paper is concerned with asymptotic properties of consensus-type algorithms for networked systems whose topologies switch randomly. The regime-switching process is modeled as a discrete-time Markov chain with a finite state space. The consensus control is achieved by using stochastic approximation methods. In the setup, the regime-switching process (the Markov chain) contains a rate parameter epsilon > 0 in the transition probability matrix that characterizes how frequently the topology switches. On the other hand, the consensus control algorithm uses a stepsize mu that defines how fast the network states are updated. Depending on their relative values, three distinct scenarios emerge. Under suitable conditions, we show that when 0 < epsilon = O(mu), a continuous-time interpolation of the iterates converges weakly to a system of randomly switching ordinary differential equations modulated by a continuous-time Markov chain. In this case a scaled sequence of tracking errors converges to a system of switching diffusion. When 0 < epsilon << mu p, the network topology is almost non-switching during consensus control transient intervals, and hence the limit dynamic system is simply an autonomous differential equation. When mu << epsilon, the Markov chain acts as a fast varying noise, and only its averaged network matrices are relevant, resulting in a limit differential equation that is an average with respect to the stationary measure of the Markov chain. Simulation results are presented to demonstrate these findings. (C) 2011 Elsevier Ltd. All rights reserved.