Analytical Solution to Optimal Distributed Bipartite Consensus for Heterogeneous Multi-Agent Systems on Coopetition Networks: A Fast Convergent Algorithm

Abstract

This note makes the first attempt to investigate the optimal bipartite consensus problem in the general case for heterogeneous multi-agent systems with cooperative-competitive interactions, specifically those containing a spanning tree. Unlike traditional control protocols based on the gradient descent method, our proposed optimal bipartite algorithm demonstrates a fast superlinear convergence speed. The key innovation of this method lies in the design of a fully distributed optimal controller, which is based on a distributed observer utilizing local neighbor information. The analytical solution for the optimal controller is derived with the help of the Riccati equation, aligning with classical optimal control theory. This approach unifies the design method for both the bipartite consensus problem and the well-studied consensus problem. Additionally, the proposed optimal algorithm can be directly extended to homogeneous systems. A numerical example is provided to demonstrate the effectiveness and rapid convergence of our control algorithm.