Asymptotic properties of consensus-type algorithms for networked systems with regime-switching topologies

• DOI: 10.1016/j.automatica.2011.02.028
• Publication Date: 2011-03-16T18:30:18Z
• Authors: G. Yin; Yu Sun; Le Yi Wang
• Keywords: 519; XXXXXX - Unknown; 0202 electrical engineering, electronic engineering, information engineering; 02 engineering and technology

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 @e>0 in the transition probability matrix that characterizes how frequently the topology switches. On the other hand, the consensus control algorithm uses a stepsize @m 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<@e=O(@m), 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<@[email protected][email protected], 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 @[email protected][email protected], 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.