Description 
88 p. 
Note 
Advisor: G. George Yin. 
Thesis 
Thesis (Ph.D.)  Wayne State University, 2012 
Summary 
This work is concerned with asymptotic properties of consensustype algorithms for networked systems whose topologies switch randomly. The regimeswitching process is modeled as a discretetime Markov chain with a nite state space. The consensus control is achieved by designing stochastic approximation algorithms. In the setup, the regimeswitching 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 u that denes 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(u), a continuoustime interpolation of the iterates converges weakly to a system of randomly switching ordinary dierential equations modulated by a continuoustime Markov chain. In this case, a scaled sequence of tracking errors converges to a system of switching diusion. When 0 < E << u, the network topology is almost nonswitching during consensus control transient intervals, and hence the limit dynamic system is simply an autonomous dierential equation. When u << E, the Markov chain acts as a fast varying noise, and only its average is relevant, resulting in a limit dierential equation that is an average with respect to the stationary measure of the Markov chain. Simulation results are presented to demonstrate these findings. By introducing a postiteration averaging algorithm, this dissertation demonstrates that asymptotic optimality can be achieved in convergence rates of stochastic approximation algorithms for consensus control with structural constraints. The algorithm involves two stages. The first stage is a coarse approximation obtained using a sequence of large stepsizes. Then the second stage provides a renement by averaging the iterates from the rst stage. We show that the new algorithm is asymptotically efficient and gives the optimal convergence rates in the sense of the best scaling factor and "smallest" possible asymptotic variance. Numerical results are presented to illustrate the performance of the algorithm. 
Subject 
Mathematics

Added Title 
Wayne State University thesis (Ph.D.) : Mathematics

