Distributed Nash equilibrium seeking for second-order systems with finite/fixed-time convergence in the absence of velocity measurement

In this paper, the distributed Nash equilibrium seeking problem is addressed for the second-order systems without velocity measurement. The Nash equilibrium seeking methods with finite/fixed-time convergence are designed to obtain fast convergence speed and excellent steady-state performance, respectively. Firstly, a finite-time velocity observer is proposed to observe the players' own velocity in a finite time for second-order systems in the absence of velocity measurement. A distributed finite-time estimator is designed to obtain the position of other players using topology information. Based on the designed observer and estimator, a Nash equilibrium seeking strategy is proposed with finite time convergence. What's more, a strategy to find a Nash equilibrium with a fixed time convergence is suggested. This is based on a fixed time velocity observer and a distributed position estimator. The idea is to avoid being dependent on the settling time of the initial states of the system. Then, based on Lyapunov function as well as finite- and fixed-time stability, the convergence conditions of the studied closed-loop system are deduced. Finally, numerical examples are given to verify the effectiveness of the proposed algorithm.