On Constructing Multiple Lyapunov Functions for Tracking Control of Multiple Agents With Switching Topologies

Distributed consensus tracking for linear multiagent systems (MASs) with directed switching topologies and a dynamic leader is investigated in this paper. By fully considering the special feature of Laplacian matrices for topology candidates, several new classes of multiple Lyapunov functions (MLFs) are constructed in this paper for leader-following MASs with, respectively, an autonomous leader and a nonautonomous leader. Under the condition that each possible topology graph contains a spanning tree rooted at the leader node, some efficient criteria for achieving consensus tracking in the considered MASs are provided. Specifically, it is proven that consensus tracking in the closed-loop MASs can be ensured if the average dwell time for switching among different topologies is larger than a derived positive quantity and the control parameters in tracking protocols are appropriately designed. It is further theoretically shown that the present Lyapunov inequality based criteria for consensus tracking with an autonomous leader are much less conservative than the existing ones derived by the M-matrix theory. The results are then extended to the case where the topology graph only frequently contains a directed spanning tree as the MASs evolve over time. At last, numerical simulations are performed to illustrate the effectiveness of the analytical analysis and the advantages of the proposed MLFs.