Dynamic Nash Equilibrium Seeking for Higher-Order Integrators in Networks

A continuous-time distributed generalized Nash equilibrium seeking algorithm over networks for double-integrator agents

“…We have not guaranteed this robustness, i.e., exponential convergence, for the primal-dual dynamics in (8). However, the controller in [18] is designed for games without any constraints (local or shared). On the contrary, the controller in Algorithm 2 drives the system in (10) to a v-GNE of a generalized game, and ensures for the coupling constraints to be satisfied asymptotically.…”

“…Moreover, at steady state, the velocities v i of all the agents must be zero. This scenario has been considered recently in [18], for games without coupling constraints. In ( 10), we consider the input u i = 1 hi (ũ i − v i ), where h i > 0 is a positive scalar andũ i has to be chosen appropriately, for all i ∈ I; moreover, as in [18], let us define the coordinates transformation…”

“…The proof of Theorem 2 is not based on the specific structure of Algorithm 1, but only on its convergence properties, hence the result still holds if another controller with similar features is selected in place of Algorithm 1. In [18], the authors addressed NE problems and chose the inputsũ i according to the algorithm presented in [6,Eq. 47].…”

“…The controller in [6] achieves exponential convergence to a NE, hence ISS with respect to possible additive disturbances [22,Lemma 4.6]. Therefore, in [18], the authors were able to tackle the presence of deterministic disturbances, via an asymptotic observer and by leveraging ISS arguments. We have not guaranteed this robustness, i.e., exponential convergence, for the primal-dual dynamics in (8).…”

“…The authors of [3] considered generally coupled costs games played by linear time-invariant agents, via a discrete-time extremum seeking approach. NE problems arising in systems of multi-integrator agents, in the presence of deterministic disturbances, were addressed in [18]. In all the references cited, the assumption is made of unconstrained action sets and absence of coupling constraints.…”