LQR-based Optimal Linear Consensus Algorithms

Laplacian matrices play an important role in linear consensus algorithms. This paper studies linear-quadratic regulator (LQR) based optimal linear consensus algorithms for multi-vehicle systems with single-integrator kinematics in a continuous-time setting. We propose two global cost functions, namely, interaction-free and interaction-related cost functions. With the interaction-free cost function, we derive the optimal (nonsymmetric) Laplacian matrix. It is shown that the optimal (nonsymmetric) Laplacian matrix corresponds 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 pre-specified symmetric Laplacian matrix associated with the interaction graph. Illustrative examples are given as a proof of concept.