Chebyshev Polynomials in Distributed Consensus Applications

Abstract

4In this paper we analyze the use of Chebyshev polynomials in distributed consensus applications. We 5 study the properties of these polynomials to propose a distributed algorithm that reaches the consensus 6 in a fast way. The algorithm is expressed in the form of a linear iteration and, at each step, the 7 agents only require to transmit their current state to their neighbors. The difference with respect to 8 previous approaches is that the update rule used by the network is based on the second order difference 9 equation that describes the Chebyshev polynomials of first kind. As a consequence, we show that our 10 algorithm achieves the consensus using far less iterations than other approaches. We characterize the 11 main properties of the algorithm for both, fixed and switching communication topologies. The main 12 contribution of the paper is the study of the properties of the Chebyshev polynomials in distributed 13 consensus applications, proposing an algorithm that increases the convergence rate with respect to 14 existing approaches. Theoretical results, as well as experiments with synthetic data, show the benefits 15 using our algorithm. 16

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