Distributed Nash Equilibrium Seeking for Games in Second-Order Systems Without Velocity Measurement

The design of distributed Nash equilibrium-seeking strategies for games in which the involved players are of second-order integrator-type dynamics is investigated in this article. Noticing that velocity signals are usually noisy or not available for feedback control in practical engineering systems, this article supposes that the velocity signals are not accessible for the players. To deal with the absence of velocity measurements, two estimators are designed. The first estimator is established by employing an observer, which has the same order as the players’ dynamics, to estimate the unavailable system states (e.g., the players’ velocities). The second estimator is designed based on a high-pass filter and is motivated by the incentive to reduce the order of the estimator, which in turn saves the computation costs of the seeking algorithms. On the basis of the designed observers/filters, distributed Nash equilibrium-seeking strategies are then established through incorporating them with consensus and gradient algorithms. It is analytically proven that the players’ actions can be regulated to the Nash equilibrium point and their velocities can be regulated to zero by utilizing the proposed velocity-free Nash equilibrium-seeking strategies. A numerical example is provided for the verification of the proposed algorithms.