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Controlling identical linear multi-agent systems over directed graphs

Nicola Zaupa, Luca Zaccarian, Isabelle Queinnec, Sophie Tarbouriech, Giulia Giordano

TL;DR

This work tackles synchronization of N identical LTI agents connected by a directed, time-invariant graph using a common static state-feedback $K$. It develops μ-synchronization conditions and relaxes the resulting BMIs into an iterative LMI-based procedure that yields a feasible $K$ while allowing linear performance constraints, notably a bound on $\|K\|$. The proposed two-step synthesis/analysis algorithm, solvable as generalized eigenvalue problems and accelerated by bisection on the rate $\mu$, demonstrates superior convergence rates across multiple dynamics and network topologies compared to Riccati and related LMI baselines, at the cost of higher computation. The results enable scalable, performance-aware synchronization design for directed MAS and point to extensions toward convex-concave approaches and static output feedback.

Abstract

We consider the problem of synchronizing a multi-agent system (MAS) composed of several identical linear systems connected through a directed graph.To design a suitable controller, we construct conditions based on Bilinear Matrix Inequalities (BMIs) that ensure state synchronization.Since these conditions are non-convex, we propose an iterative algorithm based on a suitable relaxation that allows us to formulate Linear Matrix Inequality (LMI) conditions.As a result, the algorithm yields a common static state-feedback matrix for the controller that satisfies general linear performance constraints.Our results are achieved under the mild assumption that the graph is time-invariant and connected.

Controlling identical linear multi-agent systems over directed graphs

TL;DR

This work tackles synchronization of N identical LTI agents connected by a directed, time-invariant graph using a common static state-feedback . It develops μ-synchronization conditions and relaxes the resulting BMIs into an iterative LMI-based procedure that yields a feasible while allowing linear performance constraints, notably a bound on . The proposed two-step synthesis/analysis algorithm, solvable as generalized eigenvalue problems and accelerated by bisection on the rate , demonstrates superior convergence rates across multiple dynamics and network topologies compared to Riccati and related LMI baselines, at the cost of higher computation. The results enable scalable, performance-aware synchronization design for directed MAS and point to extensions toward convex-concave approaches and static output feedback.

Abstract

We consider the problem of synchronizing a multi-agent system (MAS) composed of several identical linear systems connected through a directed graph.To design a suitable controller, we construct conditions based on Bilinear Matrix Inequalities (BMIs) that ensure state synchronization.Since these conditions are non-convex, we propose an iterative algorithm based on a suitable relaxation that allows us to formulate Linear Matrix Inequality (LMI) conditions.As a result, the algorithm yields a common static state-feedback matrix for the controller that satisfies general linear performance constraints.Our results are achieved under the mild assumption that the graph is time-invariant and connected.
Paper Structure (10 sections, 2 theorems, 18 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 10 sections, 2 theorems, 18 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Feedback Design
  4. Revisited synchronization conditions
  5. Iterative algorithm
  6. Comparison and Simulations
  7. Dynamical system and network
  8. Compared approaches
  9. Results
  10. Conclusions

Key Result

Proposition 1

Consider the system in eq:interconnection and the attractor $\mathcal{A}$ in (eq:setA). The synchronization set $\mathcal{A}$ is $\mu$--UGES if and only if any of the following conditions holds:

Figures (3)

  • Figure 1: Left: topologies of the considered graphs. Right: eigenvalues of the Laplacian matrix; the black cross denotes $\lambda_0=0$, green full dots denote all the other eigenvalues. The values considered in method "b" are visualized as squares and the relative set is delimited by a dashed line.
  • Figure 2: Time evolution of the distance of the states from the synchronization set $\mathcal{A}$ for agents with dynamics $A_{\text{X-29}}$. The methods "a", "b" and "e" are compared. The inset figures zoom into the second-half time.
  • Figure 3: Time evolution of the distance of the states from the synchronization set $\mathcal{A}$ for agents with dynamics $A_{\text{osc}}$. The methods "a", "b" and "e" are compared. The inset figures zoom into the second-half time.

Theorems & Definitions (6)

  • Definition 1: $\mu$--Synchronization
  • Proposition 1
  • Remark 1
  • Proposition 2
  • proof
  • Remark 2