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Faster Consensus via a Sparser Controller

Luca Ballotta, Vijay Gupta

TL;DR

This work addresses accelerating consensus in networks of $N$ agents with architecture-dependent communication delays. By modeling delays as a function of hop count and formulating the delayed system via a delay-augmented state, the authors develop a tractable method to jointly optimize controller architecture and feedback gains. They derive stability conditions under delays, reduce the gain-design to a two-step SDP-based procedure, and demonstrate through numerical experiments that sparse controller architectures can yield faster convergence than dense ones when delays grow with link density. The findings highlight the importance of considering latency effects in topology design for scalable, fast consensus in large networks.

Abstract

In this paper, we investigate the architecture of an optimal controller that maximizes the convergence speed of a consensus protocol with single-integrator dynamics. Under the assumption that communication delays increase with the number of hops from which information is allowed to reach each agent, we address the optimal control design under delayed feedback and show that the optimal controller features, in general, a sparsely connected architecture.

Faster Consensus via a Sparser Controller

TL;DR

This work addresses accelerating consensus in networks of agents with architecture-dependent communication delays. By modeling delays as a function of hop count and formulating the delayed system via a delay-augmented state, the authors develop a tractable method to jointly optimize controller architecture and feedback gains. They derive stability conditions under delays, reduce the gain-design to a two-step SDP-based procedure, and demonstrate through numerical experiments that sparse controller architectures can yield faster convergence than dense ones when delays grow with link density. The findings highlight the importance of considering latency effects in topology design for scalable, fast consensus in large networks.

Abstract

In this paper, we investigate the architecture of an optimal controller that maximizes the convergence speed of a consensus protocol with single-integrator dynamics. Under the assumption that communication delays increase with the number of hops from which information is allowed to reach each agent, we address the optimal control design under delayed feedback and show that the optimal controller features, in general, a sparsely connected architecture.
Paper Structure (11 sections, 6 theorems, 24 equations, 7 figures)

This paper contains 11 sections, 6 theorems, 24 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Setup
  3. System Model
  4. Problem Formulation
  5. Control Design
  6. Stability Analysis
  7. Optimization of Feedback Gains
  8. Numerical Experiments
  9. Conclusion
  10. Proof of \ref{['prop:monotonicity-real-roots']}
  11. Proof of \ref{['lem:monotonicity-complex-roots']}

Key Result

Lemma 1

Let eq:controlled-dynamics be equivalently written as the following delay-free system, Then, the spectrum of $A_n$ is given by where $\lambda_{j}^{n} \in\sigma(K_n)$ is the $j$th eigenvalue of $K_n$ in non-decreasing order, such that $0 = \lambda_{1}^{n} < \lambda_{2}^{n} \le \dots \le \lambda_{N}^{n}$, and the characteristic polynomial associated with $\lambda_{}^{}$ is

Figures (7)

  • Figure 1: Root locus of $h_{}(z;\lambda)$ with $\lambda \in [0,\bar{\lambda}_{\tau}]$ for $\tau = 5$ (left) and $\tau = 6$ (right). All branches expand from $z = 0$ except for the one starting at $z = 1$.
  • Figure 2: Graphic of $\rho(\lambda)$ with $\lambda \! \in \!(0,\bar{\lambda}_{\tau})$ for $\tau = 5$.
  • Figure 3: Convergence rate with $3$-regular graph $\mathcal{G}_{1}$, $N = 100$, $\tau_n = n$.
  • Figure 4: Convergence rate with $3$-regular graph $\mathcal{G}_{1}$, $N = 100$, $\tau_n = n^2$.
  • Figure 5: Convergence rate with $4$-regular graph $\mathcal{G}_{1}$, $N = 100$, $\tau_n = n$.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Lemma 1: Eigenvalues of delay system ballotta2023tcns
  • Remark 1: Impact of delays
  • Proposition 1: ballotta2023tcns
  • Corollary 1: Stabilizing uniform gains
  • Lemma 2: Monotonicity of real eigenvalues
  • proof
  • Lemma 3: Monotonicity of complex eigenvalues
  • proof
  • Proposition 2: Convergence rate
  • Remark 2: Optimal uniform gains