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Lyapunov-Based Graph Neural Networks for Adaptive Control of Multi-Agent Systems

Brandon C. Fallin, Cristian F. Nino, Omkar Sudhir Patil, Zachary I. Bell, Warren E. Dixon

TL;DR

This paper provides the first result on GNNs with stability-driven online weight updates to address the multi-agent target tracking problem and develops new Lyapunov-based distributed GNN and graph attention network (GAT)-based controllers to adaptively estimate unknown target dynamics and address the second-order target tracking problem.

Abstract

Graph neural networks (GNNs) have a message-passing framework in which vector messages are exchanged between graph nodes and updated using feedforward layers. The inclusion of distributed message-passing in the GNN architecture makes them ideally suited for distributed control and coordination tasks. Existing results develop GNN-based controllers to address a variety of multi-agent control problems while compensating for modeling uncertainties in the systems. However, these results use GNNs that are pre-trained offline. This paper provides the first result on GNNs with stability-driven online weight updates to address the multi-agent target tracking problem. Specifically, new Lyapunov-based distributed GNN and graph attention network (GAT)-based controllers are developed to adaptively estimate unknown target dynamics and address the second-order target tracking problem. A Lyapunov-based stability analysis is provided to guarantee exponential convergence of the target state estimates and agent states to a neighborhood of the target state. Numerical simulations show a 20.8% and 48.1% position tracking error performance improvement by the GNN and GAT architectures over a baseline DNN architecture, respectively.

Lyapunov-Based Graph Neural Networks for Adaptive Control of Multi-Agent Systems

TL;DR

This paper provides the first result on GNNs with stability-driven online weight updates to address the multi-agent target tracking problem and develops new Lyapunov-based distributed GNN and graph attention network (GAT)-based controllers to adaptively estimate unknown target dynamics and address the second-order target tracking problem.

Abstract

Graph neural networks (GNNs) have a message-passing framework in which vector messages are exchanged between graph nodes and updated using feedforward layers. The inclusion of distributed message-passing in the GNN architecture makes them ideally suited for distributed control and coordination tasks. Existing results develop GNN-based controllers to address a variety of multi-agent control problems while compensating for modeling uncertainties in the systems. However, these results use GNNs that are pre-trained offline. This paper provides the first result on GNNs with stability-driven online weight updates to address the multi-agent target tracking problem. Specifically, new Lyapunov-based distributed GNN and graph attention network (GAT)-based controllers are developed to adaptively estimate unknown target dynamics and address the second-order target tracking problem. A Lyapunov-based stability analysis is provided to guarantee exponential convergence of the target state estimates and agent states to a neighborhood of the target state. Numerical simulations show a 20.8% and 48.1% position tracking error performance improvement by the GNN and GAT architectures over a baseline DNN architecture, respectively.

Paper Structure

This paper contains 18 sections, 7 theorems, 180 equations, 6 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Graphs
  5. Graph Neural Network Architectures
  6. Graph Neural Network
  7. Graph Attention Network
  8. Approximation Capabilities of Graph Neural Networks
  9. Problem Formulation
  10. System Dynamics and Network Topology
  11. Objective
  12. Control Synthesis
  13. Observer Design
  14. Controller Design
  15. Adaptive Update Law Design
  16. ...and 3 more sections

Key Result

Lemma 1

For each node $i\in V$, the first partial derivative of the GNN architecture at node $i$ in (eq:deep-gnn-architecture) with respect to (eq:deep-gnn-weights) is where $\nabla_{\theta_{i}}\phi_{i}\in\mathbb{R}^{d^{(out)}\times p_{{\rm GNN}}}$ and the partial derivatives of $\phi_{i}$ with respect to $\text{vec}(W_{i}^{(\ell)})$ for all $\ell=0,\ldots,k$ is defined in Table tab:deep-jacobians.

Figures (6)

  • Figure 1: Overview of the message-passing framework employed at each GNN layer in which nodes aggregate messages from their local neighborhoods. This figure shows a two-layer implementation of the message-passing model Hamilton2020.
  • Figure 2: Model of a single node's forward pass for the $j^{\text{th}}$ layer of the GNN architecture. The output of the previous layer is aggregated over the neighborhood of the target node. The aggregated output is passed into a NN. The output of the NN is the input to the $j^{\text{th}}$ layer activation function. The activation function also appends a bias. The activated output serves as an input for the next GNN layer Zeng.Tang2021.
  • Figure 3: Visualization of communication topologies examined in Table \ref{['tab:compare-results']}. Shaded agents are connected to the target, where $b_{i}=1$.
  • Figure 4: Trajectory visualization for the $\text{GNN}+\text{GNN}$ configuration and acyclic graph communication topology.
  • Figure 5: Mean position tracking error response for each NN architecture with the acyclic graph communication topology.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Remark 1
  • Remark 2
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Theorem 1