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Linear Regulator-Based Synchronization of Positive Multi-Agent Systems

Alba Gurpegui, Mark Jeeninga, Emma Tegling, Anders Rantzer

TL;DR

This paper addresses the problem of synchronizing homogeneous positive multi-agent systems on undirected graphs using relative-state measurements. It proposes a Linear Regulator (LR)–based static protocol, computable via a linear program, to guarantee synchronization under graph families with known Laplacian bounds, while enforcing positivity of trajectories. It provides necessary and sufficient conditions for positivity and establishes when the LR protocol achieves synchronization for all graphs in a family, supported by LP-based construction of the gain. Simulations on large regular graphs validate positive behavior and show faster convergence with increased connectivity, highlighting the method's scalability and applicability to positive networked systems.

Abstract

This paper addresses the positive synchronization of interconnected systems on undirected graphs. For homogeneous positive systems, a static feedback protocol design is proposed, based on the Linear Regulator problem. The solution to the algebraic equation associated to the stabilizing policy can be found using a linear program. Necessary and sufficient conditions on the positivity of each agent's trajectory for all nonnegative initial conditions are also provided. Simulations on large regular graphs with different nodal degree illustrate the proposed results.

Linear Regulator-Based Synchronization of Positive Multi-Agent Systems

TL;DR

This paper addresses the problem of synchronizing homogeneous positive multi-agent systems on undirected graphs using relative-state measurements. It proposes a Linear Regulator (LR)–based static protocol, computable via a linear program, to guarantee synchronization under graph families with known Laplacian bounds, while enforcing positivity of trajectories. It provides necessary and sufficient conditions for positivity and establishes when the LR protocol achieves synchronization for all graphs in a family, supported by LP-based construction of the gain. Simulations on large regular graphs validate positive behavior and show faster convergence with increased connectivity, highlighting the method's scalability and applicability to positive networked systems.

Abstract

This paper addresses the positive synchronization of interconnected systems on undirected graphs. For homogeneous positive systems, a static feedback protocol design is proposed, based on the Linear Regulator problem. The solution to the algebraic equation associated to the stabilizing policy can be found using a linear program. Necessary and sufficient conditions on the positivity of each agent's trajectory for all nonnegative initial conditions are also provided. Simulations on large regular graphs with different nodal degree illustrate the proposed results.

Paper Structure

This paper contains 14 sections, 7 theorems, 39 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Model Description
  4. Notation
  5. Graph Description
  6. Multi-agent Systems
  7. Positive Systems
  8. Preliminaries
  9. ARE-based Protocol
  10. Linear Regulator
  11. Problem Formulation and Main Results
  12. Graph Families with Eigenvalue Bounds
  13. Simulations
  14. Conclusions

Key Result

Lemma 1

A graph $\mathcal{G}$ is connected if and only if the associated Laplacian matrix $\mathcal{L}$ has a simple eigenvalue at the origin. Furthermore, in this case, the eigenvector associated with the eigenvalue at the origin is $\mathds{1}$ and all other eigenvalues lie in the open right half-plane.

Figures (2)

  • Figure 1: Evolution over time of the first (left panels) and the second (right panels) state of each agent $i=1, \dots, 150$ synchronizing over $5$-regular graphs (upper panels) and $7$-regular graphs.
  • Figure 2: Trajectories of agents synchronizing over a $5$-regular graph (left panel) and $7$-regular graph.

Theorems & Definitions (21)

  • Lemma 1
  • Definition 1
  • Definition 2: Metzler
  • Definition 3: Positive System
  • Theorem 2: Luenberger
  • Definition 4: State synchronization
  • Remark 1
  • Lemma 3: Theorem 2.5 book_ref
  • proof
  • Remark 2
  • ...and 11 more