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Dynamic Nash Equilibrium Seeking for a Class of Nonlinear Uncertain Multi-agent Systems

Weijian Li, Yutao Tang

TL;DR

This work tackles distributed NE seeking for a class of monotone $N$-player games where each agent exhibits nonlinear uncertain dynamics with disturbances. It introduces a three-part framework: a fully distributed reference signal generator to compute NE, internal models to reject exogenous disturbances, and an augmented system that combines the agents, reference generator, and internal models. By applying a backstepping-based, distributed state-feedback controller, the paper proves semi-global stabilization of the augmented system, ensuring that each agent’s output converges to the NE despite uncertainties and disturbances. The approach generalizes prior NE-seeking methods to broader nonlinear dynamics and highlights a practical pathway for robust distributed coordination in networked systems.

Abstract

We consider seeking a Nash equilibrium (NE) of a monotone game, played by dynamic agents which are modeled as a class of lower-triangular nonlinear uncertain dynamics with external disturbances. We establish a general framework that converts the problem into a distributed robust stabilization problem of an appropriately augmented system. To be specific, we construct a virtual single-integrator multi-agent system, as a reference signal generator, to compute an NE in a fully distributed manner. By introducing internal models to tackle the disturbances, as well as embedding the virtual system, we derive an augmented system. Following that, we show that the outputs of all agents reach an NE of the game if the augmented system can be stabilized by a control law. Finally, resorting to a backstepping procedure, we design a distributed state-feedback controller to stabilize the augmented system semi-globally.

Dynamic Nash Equilibrium Seeking for a Class of Nonlinear Uncertain Multi-agent Systems

TL;DR

This work tackles distributed NE seeking for a class of monotone -player games where each agent exhibits nonlinear uncertain dynamics with disturbances. It introduces a three-part framework: a fully distributed reference signal generator to compute NE, internal models to reject exogenous disturbances, and an augmented system that combines the agents, reference generator, and internal models. By applying a backstepping-based, distributed state-feedback controller, the paper proves semi-global stabilization of the augmented system, ensuring that each agent’s output converges to the NE despite uncertainties and disturbances. The approach generalizes prior NE-seeking methods to broader nonlinear dynamics and highlights a practical pathway for robust distributed coordination in networked systems.

Abstract

We consider seeking a Nash equilibrium (NE) of a monotone game, played by dynamic agents which are modeled as a class of lower-triangular nonlinear uncertain dynamics with external disturbances. We establish a general framework that converts the problem into a distributed robust stabilization problem of an appropriately augmented system. To be specific, we construct a virtual single-integrator multi-agent system, as a reference signal generator, to compute an NE in a fully distributed manner. By introducing internal models to tackle the disturbances, as well as embedding the virtual system, we derive an augmented system. Following that, we show that the outputs of all agents reach an NE of the game if the augmented system can be stabilized by a control law. Finally, resorting to a backstepping procedure, we design a distributed state-feedback controller to stabilize the augmented system semi-globally.

Paper Structure

This paper contains 11 sections, 4 theorems, 51 equations, 1 figure.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARY AND FORMULATION
  3. Mathematical Preliminary
  4. Problem Statement
  5. GENERAL FRAMEWORK
  6. Reference Signal Generator
  7. Internal Model
  8. Augmented System
  9. MAIN RESULTS
  10. EXAMPLE
  11. CONCLUSION

Key Result

Lemma 1

Let Assumptions ass:convex and ass:graph hold. If $\gamma_2 \ge (\overline l_F^2 / \underline l_F + \overline l_F)/\lambda_{\min}(\mathcal{L})$, then $\mathbf{p}(t)$ approaches $\mathbf{p}^*$ with an exponential rate, where $\mathbf{p}^* = 1_N \otimes p^*$, $p^*$ is the NE of $\mathbf{G}$, and $\lam

Figures (1)

  • Figure 1: (a) The trajectory of $\log(\Vert \mathbf{p}(t) - \mathbf{p}^*\Vert)$. (b) The trajectories of $e_i(t)$.

Theorems & Definitions (11)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Remark 3
  • Remark 4
  • Lemma 2
  • Remark 5
  • Theorem 1
  • Remark 6
  • Theorem 2
  • ...and 1 more