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Compositional design for time-varying and nonlinear coordination

Jonas Hansson, Emma Tegling

TL;DR

The paper addresses high-order multi-agent coordination by proposing a compositional framework that constructs $n$-th order consensus by serially connecting stable first-order consensus operators. By enforcing relative feedback and mild regularity conditions, the authors show that the resulting high-order system inherits stability properties from its components, enabling modular and scalable design. The framework is demonstrated across saturated inputs, time-varying linear dynamics, and time-delayed communication, with theoretical guarantees and simulations indicating improved stability and performance over conventional or naive serial designs. The approach offers practical benefits for vehicular formations and other large-scale networks, and lays groundwork for extending scalable, robust coordination to nonlinear and delay-prone settings. Overall, the work provides a principled method to achieve high-order consensus with localized implementation and broad applicability in dynamic, constrained multi-agent systems.

Abstract

This work addresses the design of multi-agent coordination through high-order consensus protocols. While first-order consensus strategies are well-studied -- with known robustness to uncertainties such as time delays, time-varying weights, and nonlinearities like saturations -- the theoretical guarantees for high-order consensus are comparatively limited. We propose a compositional control framework that generates high-order consensus protocols by serially connecting stable first-order consensus operators. Under mild assumptions, we establish that the resulting high-order system inherits stability properties from its components. The proposed design is versatile and supports a wide range of real-world constraints. This is demonstrated through applications inspired by vehicular formation control, including protocols with time-varying weights, bounded time-varying delays, and saturated inputs. We derive theoretical guarantees for these settings using the proposed compositional approach and demonstrate the advantages gained compared to conventional protocols in simulations.

Compositional design for time-varying and nonlinear coordination

TL;DR

The paper addresses high-order multi-agent coordination by proposing a compositional framework that constructs -th order consensus by serially connecting stable first-order consensus operators. By enforcing relative feedback and mild regularity conditions, the authors show that the resulting high-order system inherits stability properties from its components, enabling modular and scalable design. The framework is demonstrated across saturated inputs, time-varying linear dynamics, and time-delayed communication, with theoretical guarantees and simulations indicating improved stability and performance over conventional or naive serial designs. The approach offers practical benefits for vehicular formations and other large-scale networks, and lays groundwork for extending scalable, robust coordination to nonlinear and delay-prone settings. Overall, the work provides a principled method to achieve high-order consensus with localized implementation and broad applicability in dynamic, constrained multi-agent systems.

Abstract

This work addresses the design of multi-agent coordination through high-order consensus protocols. While first-order consensus strategies are well-studied -- with known robustness to uncertainties such as time delays, time-varying weights, and nonlinearities like saturations -- the theoretical guarantees for high-order consensus are comparatively limited. We propose a compositional control framework that generates high-order consensus protocols by serially connecting stable first-order consensus operators. Under mild assumptions, we establish that the resulting high-order system inherits stability properties from its components. The proposed design is versatile and supports a wide range of real-world constraints. This is demonstrated through applications inspired by vehicular formation control, including protocols with time-varying weights, bounded time-varying delays, and saturated inputs. We derive theoretical guarantees for these settings using the proposed compositional approach and demonstrate the advantages gained compared to conventional protocols in simulations.

Paper Structure

This paper contains 21 sections, 7 theorems, 95 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Consensus
  4. Problem Setup
  5. Assumptions
  6. Main Results
  7. Example: serial consensus
  8. Implementation of compositional consensus
  9. Applications of Compositional Consensus
  10. Saturated consensus
  11. Time-varying linear dynamics
  12. Time-delayed consensus
  13. Case Studies
  14. Time-varying graph Laplacians
  15. Saturated coordination
  16. ...and 6 more sections

Key Result

Theorem 1

Let each subsystem $\mathcal{L}_k$, implement relative feedback according to Assumption ass:kisinvariant, and be chosen such that each unperturbed system admits a unique solution for every initial condition $z_k(0)$ that satisfies for some function $a_k$. Assume additionally that each $\mathcal{L}_k$ satisfies Assumptions ass:kislipschitz--ass:kiscont for $k = 1, \dots, n-1$. Then, the compositi

Figures (3)

  • Figure 1: Simulation results using time-varying graph Laplacians whose connection strengths vary sinusoidally. The compositional controller achieves stable second-order consensus. The naïve serial consensus exhibits a significant transient but eventually converges, while the conventional consensus seems to be truly unstable.
  • Figure 2: Simulation of compositional, conventional, and naïve serial consensus under saturated control inputs. The compositional and naïve serial consensus protocols achieve smooth convergence, while conventional consensus exhibits an undesired transient indicative of string instability.
  • Figure 3: Simulation of second-order consensus under absolute velocity feedback with and without delay. The compositional consensus controller is robust to feedback delays. This is unlike the conventional controller, which shows oscillatory behavior under the same conditions.

Theorems & Definitions (22)

  • Definition 1: Consensus
  • Definition 2: $n\textsuperscript{th}$-Order consensus
  • Example 1
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Remark 1
  • Proposition 4
  • Example 1: continued
  • Proposition 5
  • ...and 12 more