Consensus formation in a switched Markovian dynamical system

We address the problem of distributively obtaining average-consensus among a connected network of sensors that each respectively track, by linear stochastic approximation, the stationary distribution of an ergodic Markov chain with slowly switching regimes. A hyper-parameter modeled as a Markov process on a slow time-scale modulates the regime of each observed Markov chain, thus at any given time the hyper-parameter determines what stationary distribution will be estimated by each sensor. If the Markov chains share a common stationary distribution conditional on the regime, it is shown the sequence of sensor state-values weakly-converge to an average-consensus under the distributed linear consensus-filter for all network communication graphs. Conversely, if the Markov chains have unique stationary distributions in each regime, then the average-consensus can be achieved only when sensors communicate state-values at a frequency that is on the same time-scale as the frequency at which they observe the fast Markov chain. In this scenario, unlike a static consensus filter, the state-value communication graph need not be connected for an average-consensus to be reached, however this is true only when the communication graph of observation data satisfies a specific connectivity condition. Simulations illustrate our conclusions and observation model.