Targeted agreement of multiple Lagrangian systems

In this paper, we study the targeted agreement problem for a group of Lagrangian systems. Each system observes a convex set as its local target and the objective of the group is to reach a generalized coordinate agreement towards these target sets. Typically, the generalized coordinate represents position or angle. We first consider the case when the communication graphs are fixed. A control law is proposed based on each system's own target sensing and information exchange with neighbors. With necessary connectivity, the generalized coordinates of multiple Lagrangian systems are shown to achieve agreement in the intersection of all the local target sets while generalized coordinate derivatives are driven to zero. We also discuss the case when the intersection of the local target sets is empty. Exact targeted agreement cannot be achieved in this case. Instead, we show that approximate targeted agreement can be guaranteed if the control gains are properly chosen. In addition, when communication graphs are allowed to be switching, we propose a model-dependent control algorithm and show that global targeted agreement is achieved when joint connectivity is guaranteed and the intersection of local target sets is nonempty. Simulations are given to validate the theoretical results.