Consensus Of Second-order Multi-agent Systems

In this paper,we consider the consensus problem of a second-order multi-agent system with switching topologies with a certain consensus protocol. We first design a consensus protocol basing on the neighbor rule, then we perform an invertible transformation to turn the original system into an equivalent system, whose state matrices are Laplacian. This paper shows that consensus under switching topologies can be achieved asymptotically if the union of the directed interaction graphs is frequently jointly connected. In the proof process we mainly use a Lyapunov function from the contraction property of the first order multi-agent system equation. Frequently jointly connected, which allows the interaction topology arbitrarily vary in some sufficient large time intervals, thus is a milder condition than connected and uniformly jointly connected, and so it is more suitable for practical applications.A necessary and sufficient condition is also given for the fixed topology case. In addi-tion, we introduced some definitions, notations and basic information about the research history of multi-agent systems and some necessary tools like graph and matrix.