Polynomial Lyapunov Functions for Consensus of Multi-agent Systems via PD control

This paper investigates the consensus problem of multi-agent systems under the proportional-derivative (PD) control protocol. It devises an automatic search algorithm for the polynomial Lyapunov functions beyond the conventional quadratic form. With a relaxation of Lipschitz-like condition, a criterion is established for consensus of multi-agent systems via the PD control. Then, the relaxed nonnegativity conditions can be transformed into the sum-of-squares conditions for polynomial systems. And the polynomial Lyapunov functions are then obtained by solving the sum-of-squares optimization problem within the convex programming framework, which can be addressed efficiently in polynomial time. Finally, a numerical example is presented to verify the effectiveness of the proposed algorithm.