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Gradient-Based Distributed Controller Design Over Directed Networks

Yuto Watanabe, Kazunori Sakurama, Hyo-Sung Ahn

TL;DR

This article addresses the dynamic matching problem of two distinct groups of agents with different sensing ranges, a novel coordination task that involves pairing agents from two distinct groups to achieve a convergence of the paired agents' states to the same value.

Abstract

In this study, we propose a design methodology of distributed controllers for multi-agent systems on a class of directed interaction networks by extending the gradient-flow method. Although the gradient-flow method is a common design tool for distributed controllers, it is inapplicable to directed networks. First, we demonstrate how to construct a distributed controller for systems over a class of time-invariant directed graphs. Subsequently, we establish better convergence properties and performance enhancement than the conventional gradient-flow method. To illustrate its application in time-varying networks, we address the dynamic matching problem of two distinct groups of agents with different sensing ranges. This problem is a novel coordination task that involves pairing agents from two distinct groups to achieve a convergence of the paired agents' states to the same value. Accordingly, we apply the proposed method to this problem and provide sufficient conditions for successful matching. Lastly, numerical examples for systems on both time-invariant and time-varying networks demonstrate the effectiveness of the proposed method.

Gradient-Based Distributed Controller Design Over Directed Networks

TL;DR

This article addresses the dynamic matching problem of two distinct groups of agents with different sensing ranges, a novel coordination task that involves pairing agents from two distinct groups to achieve a convergence of the paired agents' states to the same value.

Abstract

In this study, we propose a design methodology of distributed controllers for multi-agent systems on a class of directed interaction networks by extending the gradient-flow method. Although the gradient-flow method is a common design tool for distributed controllers, it is inapplicable to directed networks. First, we demonstrate how to construct a distributed controller for systems over a class of time-invariant directed graphs. Subsequently, we establish better convergence properties and performance enhancement than the conventional gradient-flow method. To illustrate its application in time-varying networks, we address the dynamic matching problem of two distinct groups of agents with different sensing ranges. This problem is a novel coordination task that involves pairing agents from two distinct groups to achieve a convergence of the paired agents' states to the same value. Accordingly, we apply the proposed method to this problem and provide sufficient conditions for successful matching. Lastly, numerical examples for systems on both time-invariant and time-varying networks demonstrate the effectiveness of the proposed method.
Paper Structure (22 sections, 13 theorems, 54 equations, 9 figures)

This paper contains 22 sections, 13 theorems, 54 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and Definitions
  4. Graph Theory
  5. Problem Setting
  6. Target System
  7. Control Objective
  8. Main Result
  9. Controller Design
  10. Solution to Problem 1
  11. Numerical Experiments
  12. Application to Distributed Dynamic Matching
  13. System Description
  14. Problem Formulation of Dynamic Matching
  15. Preliminaries for Controller Design
  16. ...and 7 more sections

Key Result

Lemma 1

Let time-invariant directed graph $G$ satisfy Assumption assu:g. Assume that $\bar{V}_\mathrm{ud}$ and $V_\mathrm{ud}$ are nonnegative, continuously differentiable, and radially unbounded. Then, for the system dynamics with controller and ctr:digraph, the set $\Omega$ in def:Omega is an equilibrium

Figures (9)

  • Figure 1: Examples of directed graph $G$ satisfying Assumption \ref{['assu:g']} in Example \ref{['ex:assu_G']}, undirected graph $G_\mathrm{ud}$ with the set of bidirectional edges in $G$, and undirected graph $\bar{G}_\mathrm{ud}$ with $\bar{\mathcal{E}}_\mathrm{ud}$ in \ref{['def:ebar']}. This graph $G$ has a leader-follower structure.
  • Figure 2: Sketch of \ref{['def:problem_eql']} in Problem 1. The undesired equilibrium set $\Omega\setminus\mathcal{T}$ of the designed controller is expected to be smaller than that of the gradient-flow method $\nabla V_\mathrm{ud}^{-1}(0)\setminus \mathcal{T}$.
  • Figure 3: Simulation results of distance-based formation with $4$ agents: (a) desired formation and the network topology; (b)-(d) the results of the proposed method in \ref{['ctr:digraph']}, the previous method (A) in \ref{['eq:dist_ud']}, and the previous method (B) in \ref{['eq:dist_di']}. Here, the black lines and the red arrows are undirected and directed edges, and the pink numbered circles and the blue ones are the agents in $V_\mathrm{t}$ and $\mathcal{N}\setminus\mathcal{V}_\mathrm{t}$. Black numbered circles are the initial states, and dotted lines are the trajectories.
  • Figure 4: Simulation results of distance-based formation with $6$ agents.
  • Figure 5: Comparison of the box plots of the formation error and input norm at $t=80$s for the proposed method and the previous methods (A) and (B). The upper and lower plots correspond to the cases of $n=4$ and $n=6$, respectively. Scattered dots represent the distribution map of the formation error and the input norm.
  • ...and 4 more figures

Theorems & Definitions (34)

  • Example 1
  • Example 2
  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 24 more