On the convergence of time-varying fusion algorithms: Application to localization in dynamic networks

In this paper, we study the convergence of dynamic fusion algorithms that can be modeled as Linear Time-Varying (LTV) systems with (sub-) stochastic system matrices. Instead of computing the joint spectral radius, we partition the entire set of system matrices into slices, whose lengths characterize the stability (convergence) of the underlying LTV system. We relate the lengths of the slices to the rate of the information flow within the network, and show that fusion is achieved if the unbounded slice lengths grow slower than an explicit exponential rate. As a motivating application, we provide a distributed algorithm to track the positions of an arbitrary number of agents moving in a bounded region of interest. At each iteration, agents update their position estimates as a convex combination of the states of the neighbors, if they are able to find enough neighbors; and do not update otherwise. We abstract the corresponding position tracking algorithm as an LTV system, and introduce the notion of slices to provide sufficient conditions under which the system asymptotically converges to the true positions of the agents. We demonstrate the effectiveness of our approach through simulations.