# Dynamic NE Seeking for Multi-Integrator Networked Agents with Disturbance Rejection

## TL;DR

This paper develops a disturbance-resilient Nash equilibrium seeking method for multi-integrator agents with partial information, using gradient-play and internal-model based observers, ensuring convergence despite external disturbances.

## Contribution

It introduces a novel dynamic Nash equilibrium seeking approach combining gradient-play with disturbance observers for multi-integrator agents under partial information.

## Key findings

- Convergence to Nash equilibrium is achieved despite disturbances.
- The method applies to single, double, and multi-integrator agents.
- The approach is proven using input-to-state stability and strong monotonicity.

## Abstract

In this paper, we consider game problems played by (multi)-integrator agents, subject to external disturbances. We propose Nash equilibrium seeking dynamics based on gradient-play, augmented with a dynamic internal-model based component, which is a reduced-order observer of the disturbance. We consider single-, double- and extensions to multi-integrator agents, in a partial-information setting, where agents have only partial knowledge on the others' decisions over a network. The lack of global information is offset by each agent maintaining an estimate of the others' states, based on local communication with its neighbours. Each agent has an additional dynamic component that drives its estimates to the consensus subspace. In all cases, we show convergence to the Nash equilibrium irrespective of disturbances. Our proofs leverage input-to-state stability under strong monotonicity of the pseudo-gradient and Lipschitz continuity of the extended pseudo-gradient.