  The design of Nash equilibrium seeking strategies for games in which the
involved players are of second-order integrator-type dynamics is investigated
in this paper. Noticing that velocity signals are usually noisy or not
available for feedback control in practical engineering systems, this paper
supposes that the velocity signals are not accessible for the players. To deal
with the absence of velocity measurements, two estimators are designed, based
on which Nash equilibrium seeking strategies are constructed. The first
strategy 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 strategy is designed based on a high-pass
filter and is motivated by the incentive to reduce the order of the closed-loop
system which in turn reduces the computation costs of the seeking algorithm.
Extensions to Nash equilibrium seeking for networked games are provided. Taking
the advantages of leader-following consensus protocols, it turns out that both
the observer-based method and the filter-based method can be adapted to deal
with games in distributed systems, which shows the extensibility of the
developed strategies. Through Lyapunov stability analysis, 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 verifications of the proposed algorithms.
