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Distributed Differential Graphical Game for Control of Double-Integrator Multi-Agent Systems with Input Delay

Hossein B. Jond

TL;DR

This work addresses distributed Nash equilibrium computation for cooperative control of noncooperative double-integrator MASs with input delay on directed graphs. By converting the N-player DGG into an edge-based set of optimal control problems and transforming the delayed dynamics into a delay-free TPBVP, the authors derive explicit distributed Nash actions and associated state trajectories through a careful eigenstructure (including Jordan form) analysis of a key edge-dynamics matrix $M$. A critical result is that the existence of a unique distributed Nash equilibrium hinges on the invertibility of $H(T-\tau)$, which is established via the defectiveness of $M$ and its spectral decomposition, yielding closed-form expressions for $y(t)$ in terms of initial data and delay. The illustrative example demonstrates fully distributed consensus trajectories with input delay, highlighting practical applicability to networked MAS control where only local information is available.

Abstract

This paper studies cooperative control of noncooperative double-integrator multi-agent systems (MASs) with input delay on connected directed graphs in the context of a differential graphical game (DGG). In the distributed DGG, each agent seeks a distributed information control policy by optimizing an individual local performance index (PI) of distributed information from its graph neighbors. The local PI, which quadratically penalizes the agent's deviations from cooperative behavior (e.g., the consensus here), is constructed through the use of the graph Laplacian matrix. For DGGs for double-integrator MASs, the existing body of literature lacks the explicit characterization of Nash equilibrium actions and their associated state trajectories with distributed information. To address this issue, we first convert the N-player DGG with m communication links into m coupled optimal control problems (OCPs), which, in turn, convert to the two-point boundary-value problem (TPBVP). We derive the explicit solutions for the TPBV that constitute the explicit distributed information expressions for Nash equilibrium actions and the state trajectories associated with them for the DGG. An illustrative example verifies the explicit solutions of local information to achieve fully distributed consensus.

Distributed Differential Graphical Game for Control of Double-Integrator Multi-Agent Systems with Input Delay

TL;DR

This work addresses distributed Nash equilibrium computation for cooperative control of noncooperative double-integrator MASs with input delay on directed graphs. By converting the N-player DGG into an edge-based set of optimal control problems and transforming the delayed dynamics into a delay-free TPBVP, the authors derive explicit distributed Nash actions and associated state trajectories through a careful eigenstructure (including Jordan form) analysis of a key edge-dynamics matrix . A critical result is that the existence of a unique distributed Nash equilibrium hinges on the invertibility of , which is established via the defectiveness of and its spectral decomposition, yielding closed-form expressions for in terms of initial data and delay. The illustrative example demonstrates fully distributed consensus trajectories with input delay, highlighting practical applicability to networked MAS control where only local information is available.

Abstract

This paper studies cooperative control of noncooperative double-integrator multi-agent systems (MASs) with input delay on connected directed graphs in the context of a differential graphical game (DGG). In the distributed DGG, each agent seeks a distributed information control policy by optimizing an individual local performance index (PI) of distributed information from its graph neighbors. The local PI, which quadratically penalizes the agent's deviations from cooperative behavior (e.g., the consensus here), is constructed through the use of the graph Laplacian matrix. For DGGs for double-integrator MASs, the existing body of literature lacks the explicit characterization of Nash equilibrium actions and their associated state trajectories with distributed information. To address this issue, we first convert the N-player DGG with m communication links into m coupled optimal control problems (OCPs), which, in turn, convert to the two-point boundary-value problem (TPBVP). We derive the explicit solutions for the TPBV that constitute the explicit distributed information expressions for Nash equilibrium actions and the state trajectories associated with them for the DGG. An illustrative example verifies the explicit solutions of local information to achieve fully distributed consensus.
Paper Structure (14 sections, 1 theorem, 86 equations, 4 figures)

This paper contains 14 sections, 1 theorem, 86 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kronecker Product
  4. Graph Theory
  5. Differential Graphical Game
  6. Main Results
  7. Illustrative Example
  8. Conclusion
  9. Decomposition of M
  10. Eigenvalues of M
  11. Eigenvectors of M
  12. M Is Defective
  13. Jordan Blocks of M
  14. Decomposition of H

Key Result

Theorem 1

$H(T-\tau)$ is nonsingular and where

Figures (4)

  • Figure 1: Block diagram representation of the proposed systematic approach to solving the DGG.
  • Figure 2: Agents' trajectories under the non-distributed (solid gray lines) and distributed solutions (dashed color lines).
  • Figure 3: Delay-free time histories under the non-distributed and distributed solutions.
  • Figure 4: Delayed ($\tau=0.5$) time histories under the non-distributed and distributed solutions.

Theorems & Definitions (2)

  • Theorem 1
  • proof