Helly-theorem-based time-optimal consensus for multi-agent systems

The final convergence state of multi-agent under ordinary consensus control is restricted by communication topology structure and edge weight. Different convergence states further influence the convergence speed of multi-agent. To attain identical convergence state under different communication topologies, and achieve time-optimal consensus, we designed a time-optimal distributed consensus control strategy for linear multi-agent system with input constraint. Firstly, we proved that the time-optimal consensus state and convergence time uniquely existed based on Helly theorem. More specifically, for the multi-agent system with n agents with input constraint in the d(n >d) dimension state space, the time-optimal state can be determined by d+1 agents at most. When the d+1 crucial agents were obtained, so was the consensus state. According to this theorem, we designed a new distributed coordination algorithm for multi-agent to achieve common knowledge on those critical agents together with the time-optimal consensus state and convergence time, and after that, each of the agents designed its own local optimal control law with terminal-time and terminal-state constraints, which guaranteed the time-optimal consensus of multi-agent. To demonstrate the correctness and efficiency of the algorithm, we applied our algorithm to the second-order dynamic multi-agent systems. Simulation result verifies the feasibility of the distributed algorithm. When the coordinating state dimension is much smaller than the number of agents, the algorithm significantly reduces the amount of computations and increases computation speed.