Consensus Acceleration in Multiagent Systems with the Chebyshev Semi-Iterative Method

Abstract

We consider the fundamental problem of reaching consensus in multiagent systems; an operation required in many applications such as, among others, vehicle formation and coordination, shape formation in modular robotics, distributed target tracking, and environmental modeling. To date, the consensus problem (the problem where agents have to agree on their reported values) has been typically solved with iterative decentralized algorithms based on graph Laplacians. However, the convergence of these existing consensus algorithms is often too slow for many important multiagent applications, and thus they are increasingly being combined with acceleration methods. Unfortunately, state-of-the-art acceleration techniques require parameters that can be optimally selected only if complete information about the network topology is available, which is rarely the case in practice. We address this limitation by deriving two novel acceleration methods that can deliver good performance even if little information about the network is available. The first proposed algorithm is based on the Chebyshev semi-iterative method and is optimal in a well defined sense; it maximizes the worst-case convergence speed (in the mean sense) given that only rough bounds on the extremal eigenvalues of the network matrix are available. It can be applied to systems where agents use unreliable communication links, and its computational complexity is similar to those of simple Laplacian-based methods. This algorithm requires synchronization among agents, so we also propose an asynchronous version that approximates the output of the synchronous algorithm. Mathematical analysis and numerical simulations show that the convergence speed of the proposed acceleration methods decrease gracefully in scenarios where the sole use of Laplacian-based methods is known to be impractical.

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