Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nash Equilibrium Seeking for General Linear Systems with Disturbance Rejection

Xin Cai, Feng Xiao, Bo Wei, Mei Yu, Fang Fang

Abstract

This paper explores aggregative games in a network of general linear systems subject to external disturbances. To deal with external disturbances, distributed strategy-updating rules based on internal model are proposed for the case with perfect and imperfect information, respectively. Different from existing algorithms based on gradient dynamics, by introducing the integral of gradient of cost functions on the basis of passive theory, the rules are proposed to force the strategies of all players to evolve to Nash equilibrium regardless the effect of disturbances. The convergence of the two strategy-updating rules is analyzed via Lyapunov stability theory, passive theory and singular perturbation theory. Simulations are presented to verify the obtained results.

Nash Equilibrium Seeking for General Linear Systems with Disturbance Rejection

Abstract

This paper explores aggregative games in a network of general linear systems subject to external disturbances. To deal with external disturbances, distributed strategy-updating rules based on internal model are proposed for the case with perfect and imperfect information, respectively. Different from existing algorithms based on gradient dynamics, by introducing the integral of gradient of cost functions on the basis of passive theory, the rules are proposed to force the strategies of all players to evolve to Nash equilibrium regardless the effect of disturbances. The convergence of the two strategy-updating rules is analyzed via Lyapunov stability theory, passive theory and singular perturbation theory. Simulations are presented to verify the obtained results.

Paper Structure

This paper contains 14 sections, 7 theorems, 40 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph Theory
  4. Input-to-state Stability
  5. Passive linear system
  6. Aggregative Games
  7. Problem Formulation
  8. Main Results
  9. Strategy-updating Rule with Perfect Information
  10. Strategy-updating Rule with Imperfect Information
  11. Simulations
  12. Network of Double-integrator Agents
  13. Generation Systems in Electricity Markets
  14. Conclusion

Key Result

Theorem 1

(Khalil.2002) Let $V:\it{R}^{n_1}\rightarrow \it{R}$ be a continuously differentiable function such that $\forall x_1\in \it{R}^{n_1}, x_2\in \it{R}^{n_2}$, where $\alpha_1$, $\alpha_2$ are class $\mathcal{K}_\infty$ functions, $\rho$ is a class $\mathcal{K}$ function, and $W(x_1)$ is a continuous positive definite function on $\it {R}^{n_1}$. Then, the system (11) is input-to-state stable with $

Figures (7)

  • Figure 1: Block diagram of the designed system with perfect information
  • Figure 2: Communication graphs.
  • Figure 3: Outputs of double-integrator agents by \ref{['pi']}.
  • Figure 4: Outputs of double-integrator agents by \ref{['sur2']}.
  • Figure 5: Output power of generator systems by strategy-updating rule \ref{['pi']}.
  • ...and 2 more figures

Theorems & Definitions (13)

  • Theorem 1
  • Lemma 1
  • Definition 1
  • Lemma 2
  • Lemma 3
  • Definition 2
  • Definition 3
  • Remark 1
  • Lemma 4
  • Theorem 2
  • ...and 3 more