Asymptotic properties of consensus-type algorithms for networked systems with regime-switching topologies |
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Authors: | G Yin Yu Sun Le Yi Wang[Author vitae] |
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Affiliation: | aDepartment of Mathematics, Wayne State University, Detroit, MI 48202, United States;bDepartment of Electrical and Computer Engineering, Wayne State University, Detroit, MI 48202, United States |
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Abstract: | 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 ε>0 in the transition probability matrix that characterizes how frequently the topology switches. On the other hand, the consensus control algorithm uses a stepsize μ 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<ε=O(μ), 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<ε?μ, 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 μ?ε, 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. |
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Keywords: | Regime-switching model Switching diffusion Stochastic approximation Limit ODE Consensus algorithm Convergence |
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