Fast Consensus with Chebyshev Polynomials

“No matter how fast your computer system runs, you will eventually think of it as slow.” When the number of robots in the network is large, distributed averaging methods usually have a slow convergence rate. In this chapter, we analyze the use of Chebyshev polynomials in the distributed consensus problem to reduce the number of iterations required to achieve a good consensus. We present a distributed linear iteration using these polynomials that compared to other approaches, is able to achieve the average of the initial conditions in a small number of iterations. In this chapter we characterize the main properties of the algorithm for both, fixed and switching communication topologies. Additionally, we provide a second algorithm for the adaptive selection of the parameters to optimize the convergence rate. We validate the studied method with extensive simulations.