High-order consensus: Optimal controller design

This paper considers the optimal design of a high-order consensus protocol applied to agents modeled with multiple integrators. The problem addressed amounts to determining the optimal values for the coupling gains and for the edge weights of the directed graph that models the network. With that goal in mind, an LQR-like cost function is proposed, which is obtained from the typical LQR cost function by taking the expectation over the initial state of the system. To enable the implementation of optimization algorithms, and ultimately solve the considered problem, it is first necessary to tackle the evaluation of the cost and its derivatives, taking into account the particularities of the consensus problem. Then, to determine the optimal parameters of the high-order consensus protocol, a truncated Newton method is considered. Furthermore, a suboptimal design approach is proposed, which consists in splitting the problem into two simpler ones that are solved sequentially, in a reduced amount of time. It is also shown that if one considers symmetric Laplacian matrices, these simpler problems are convex for second- and third-order consensus under a mild assumption on the initial conditions. Finally, the proposed algorithms are illustrated with examples that demonstrate their efficacy as well as the benefits to the controller synthesis.