Switching Between Linear Consensus~Protocols: A Variational Approach

Abstract

We consider a linear consensus system with n agents that can switch between r different connectivity patterns. A natural question is which switching law yields the best (or worst) possible speed of convergence to consensus? We formulate this question in a rigorous manner by relaxing the switched system into a bilinear consensus control system, with the control playing the role of the switching law. A best (or worst) possible switching law then corresponds to an optimal control. We derive a necessary condition for optimality, stated in the form of a maximum principle (MP). Our approach, combined with suitable algorithms for numerically solving optimal control problems, may be used to obtain explicit lower and upper bounds on the achievable rate of convergence to consensus. We also show that the system will converge to consensus for any switching law if and only if a certain (n-1) dimensional linear switched system converges to the origin for any switching law. For the case n=3 and r=2, this yields a necessary and sufficient condition for convergence to consensus that admits a simple graph-theoretic interpretation.