Optimal energy consensus control for linear multi-agent systems

In this paper, an optimal energy cost controller for linear multi-agent systems' consensus is proposed. It is assumed that the topology among the agents is fixed and the agents are connected through an edge-weighted graph. The controller only uses relative information between agents. Due to the difficulty of finding the controller gain, we focus on finding the optimal controller among a sub-family whose design is based on Algebraic Riccati Equation (ARE) and guarantee consensus. It is found that the energy cost for such controllers is bounded by an interval and hence we minimize the upper bound. To do that, the control gain and the edge weights are optimized separately. The control gain is optimized by choosing Q = 0 in the ARE; the edge weights are optimized under the assumption that there is limited communication resources in the network. Negative edge weights are allowed, and the problem is formulated as a Semi-definite Programming (SDP) problem. The controller coincides with the optimal control in [8] when the graph is complete. Furthermore, two sufficient conditions for the existence of negative optimal edge weights realization are given.