Optimality Aspect

Abstract

This chapter considers the optimality aspect in distributed multi-agent coordination. We study optimal linear coordination algorithms for multi-agent systems with single-integrator dynamics in both continuous-time and discrete-time settings from a linear quadratic regulator perspective. We propose two global cost functions, namely, interaction-free and interaction-related cost functions. With the interaction-free cost function, we derive the optimal state feedback gain matrix in both continuous-time and discrete-time settings. It is shown that the optimal gain matrix is a nonsymmetric Laplacian matrix corresponding to a complete directed graph. In addition, we show that any symmetric Laplacian matrix is inverse optimal with respect to a properly chosen cost function. With the interaction-related cost function, we derive the optimal scaling factor for a prespecified symmetric Laplacian matrix associated with an undirected interaction graph in both continuous-time and discrete-time settings. Illustrative examples are given as a proof of concept.