LQR Control for Homogeneous Agents with Multi-graph Topology

We consider a set of multi-agent systems (MASs) with general linear time-invariant (LTI) dynamics and a control problem where the performance index couples the behavior of the system. The interconnection topology between the agents is modeled as an undirected multi-graph with self-loop, where each system is a node and the control action at each node is a function of its state and the states of its neighbors. The linear quadratic regulator (LQR) control problem considered in this paper can be regarded as a structure optimization problem due to the block diagonal restriction on the feedback gain. It is shown that minimizing the LQR performance limit of the multi-agent system under the distributed controller equals minimizing the sum performance of a single agent system. A sufficient condition is presented in terms of a set of linear matrix inequalities (LMIs) to achieve certain suboptimal performance specifications. In addition to make the control design more applicable, the notion of the LQR performance region is introduced and analyzed, which is shown to be convex with respect to the eigenvalues of the Laplacian matrix. The LQR control of the multi-agent system is then converted to the LQR control of a set of subsystems, which incorporates only two inequality constraints with the minimum and maximum eigenvalues as the coefficients. Numerical examples are presented to illustrate the proposed method.