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In this paper, we design an optimal energy cost controller for linear systemsasymptotic consensus given the topology of the graph. The controller dependsonly on relative information of the agents. Since finding the control gain forsuch controller is hard, we focus on finding an optimal controller among aclassical family of controllers which is based on Algebraic Riccati Equation(ARE) and guarantees asymptotic consensus. Through analysis, we find that theenergy cost is bounded by an interval and hence we minimize the upper bound. Inorder to do that, there are two classes of variables that need to be optimized:the control gain and the edge weights of the graph and are hence designed fromtwo perspectives. A suboptimal control gain is obtained by choosing $Q=0$ inthe ARE. Negative edge weights are allowed, and the problem is formulated as aSemi-definite Programming (SDP) problem. Having negative edge weights meansthat "competitions" between the agents are allowed. The motivation behind thissetting is to have a better system performance. We provide a different prooffrom the angle of optimization and show that the lowest control energy cost isreached when the graph is complete and with equal edge weights. Furthermore,two sufficient conditions for the existence of negative optimal edge weightsrealization are given.

Consensus Control for Linear Systems with Optimal Energy Cost