Optimal Cooperative Control For Multi-Agent Systems Based On LQR Theory

In recent years, cooperative optimal control for multi-agent systems is gradually becoming a very active research topic of different fields at home and abroad. Coordinated control for multi-agent systems has a broad practical application in the fields of industry, society, management and control science, such as formation control for unmanned aerial vehicles, attitude control for multiple satellites, distributed sensor network, etc. Multi-agent systems usually contain a large number of agents. Seeking optimal control strategies for cost performance index has important theoretical significance and practical value.From the perspective of energy optimization, this paper studies distributed coordinated optimization problems on the basis of previous research which enriches the research achievement of distributed optimal control. The specific research works and main contributions of this paper include the following aspects:1. Considering continuous-time multi-agent systems which satisfy certain conditions, distributed optimal consensus control and optimal tracking control without communication topology constraints are studied. The LQR control theory is used to discuss the global optimal control protocol which can minimize quadratic performance indexes. For the optimal consensus control, the condition for the global optimal control protocol to satisfy the distributed control forms is received by using the matrix decomposition methods. It is proved that the LQR-based optimal control protocol can guarantee consensus. For the optimal tracking control, considering the leader-follower system which has integrator dynamics or positive definite dynamics, the decomposability of M-matrix is further studied and the optimal control protocol is decomposed to a optimal laplacian matrix and a optimal pinning gain matrix. It is also proved that the optimal control protocol can ensure that followers track the leader's status.2. For discrete-time multi-agent systems with integrator dynamics, the distributed cooperative optimal algorithms are considered. Based on discrete-time LQR, we discuss an optimal consensus protocol for the leaderless multi-agent system and an optimal tracking protocol for the leader-follower system. In the process of designing the optimal tracking control protocol, the decomposability of laplacian matrix and pinning gain matrix corresponding to the communication topology is further discussed to prove that the optimal control protocol based on the discrete-time Riccati equation meets the requirements of distributed control.3. Cooperative tracking suboptimal control strategies under fixed graph topologies for leader-follower systems are studied. For both cases of continuous-time and discrete-time, the maximum principle is used to design the distributed suboptimal tracking control protocol when the weight matrix of performance index is composed of a laplacian matrix and a pinning gain matrix. The optimal scaling factor is choosed to realize suboptimization of the performance index and guarantee that the followers can progressively track the leader's states at the same time. Finally, the necessary condition for the existence of distributed suboptimal control protocol is discussed.