An optimal cooperative tracking algorithm for discrete-time multi-agent systems

This paper designs the distributed cooperative control algorithm that assure synchronous tracking and are globally optimal with respect to given cost functions. We consider multi-agent systems with identical discrete multi-integrator dynamics and undirected topology graph. The distributed cooperative control protocol and the interaction-free cost function are proposed firstly. With the given cost function, we derive the global optimal cooperative tracking control protocol with the optimal communication graph Laplacian matrix and the optimal pinning gain matrix though discrete LQR method. To achieve this result, we also propose the decomposability of nonsingular M-matrix and the connections between M-matrix and Laplacian matrix. The optimal control protocol requirements the communication graph is a complete graph. Otherwise, considering any symmetric graph Laplacian matrix and any pinning gain matrix, we can derive the corresponding cost function though inverse optimality. Finally, the validity of the result is illustrated through numerical examples.