Lyapunov Analysis of a Tunable Consensus Protocol in Discrete Time: Application to Distributed Estimation

In this paper, we construct a strict Lyapunov function to certify the uniform exponential stability of the consensus subspace for a class of discrete-time networks. This class of networks results from the interconnection of first-order systems using a recently established consensus protocol. In particular, we provide an explicit expression for a quadratic Lyapunov function that certifies the desired property without constraining the interconnection gains. Our assumptions are that the communication graph contains a directed spanning tree and that the agents can perform a sequence of exchanges before updating their states. This approach allows us to propose a new distributed gradient-descent algorithm, in which the considered protocol governs the interactions among the different estimators. The effectiveness of the proposed estimation algorithm is compared, via simulations, to an existing algorithm based on a classical interconnection protocol.