Distributed Computation of Minimum Step Consensus for Discrete Time Multi-Agent Systems

A problem of achieving consensus in state of multiagent system consisting of N identical agents is considered. The agents are modeled as discrete time linear time invariant (LTI) systems with bounded inputs (|u| ≤ 1) and complete communication among all agents is assumed. The goal of these agents is to achieve consensus in minimum number of time steps. An algorithm to compute the minimum number of time steps in which the consensus can be achieved is devised by computing the attainable set. Key idea of the proposed algorithm is to compute attainable sets by incrementally varying the time steps and checking for intersection of attainable sets corresponding to all N agents. Since, the attainable set is a convex polytope, intersection of attainable sets translates to checking feasibility of multiple linear inequalities. It is shown that for systems with eigenvalues inside the closed unit circle in complex plane, the proposed algorithm terminates in finite number of steps. The number of time steps counted by the proposed algorithm till the intersection of attainable sets becomes feasible is the minimum time steps required to achieve consensus. Lastly, it is shown using Helly's theorem that the computation of intersection of attainable sets in the proposed algorithm can be performed in distributed manner. The proposed algorithm is demonstrated with the help of an example.