Cooperative Optimal Control for Multi-Agent Systems on Directed Graph Topologies

This note brings together stability and optimality theory to design distributed cooperative control protocols that guarantee consensus and are globally optimal with respect to a positive semi-definite quadratic performance criterion. A common problem in cooperative optimal control is that global optimization problems generally require global information, which is not available to distributed controllers. Optimal control for multi-agent systems is complicated by the fact that the communication graph topology interplays with the agent system dynamics. In the note we use an inverse optimality approach together with partial stability to consider the cooperative consensus and pinning control. Agents with identical linear time-invariant dynamics are considered. Communication graphs are assumed directed and having fixed topology. Structured quadratic performance indices are derived that capture the topology of the graph, which allows for global optimal control that is implemented using local distributed protocols. A new class of digraphs is defined that admits a distributed solution to the global optimal control problem, namely those with simple graph Laplacian matrices.