Performance Bounds and Optimal Design of Randomly Switching Linear Consensus Networks

We analyze the steady-state performance of randomly switching linear consensus networks subject to an additive noise. Our focus is on a class of discrete-time linear consensus networks whose dynamics evolve in time by switching between symmetric doubly-stochastic matrices that are drawn from a given finite set of realizations, which are dictated by a discrete random variable with a known probability mass function (pmf). The mean square deviation of the output from zero is utilized as a performance measure to quantify the noise propagation across the network. The direct computation of the performance measure, however, suffers from the curse of dimensionality. The first objective of this paper is to report a rather tight lower-bound for the performance measure that requires comparably less computations. Our second goal is to design a randomly switching network by minimizing the overall performance measure with respect to the vector of switching probabilities between consecutive realizations. Indeed, we prove that the performance measure is a convex function of the vector of switching probabilities, and furthermore, the optimal network design problem can be cast as a Semi-Definite Program (SDP). Moreover, we show that the performance measure is a convex function of the link weights of all underlying graphs, which implies that one can perform a reweighting procedure to minimize the performance measure using convenient convex programming tools. Different aspects of our theoretical results are illustrated via numerous examples and simulations.