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A Multiplex Approach Against Disturbance Propagation in Nonlinear Networks with Delays

Shihao Xie, Giovanni Russo

TL;DR

The paper addresses disturbance propagation in nonlinear networks with time-varying delays by introducing a multiplex control architecture grounded in contraction theory. It defines $\mathcal{L}_\infty^p$-Input-to-State Scalability and $\mathcal{L}_\infty^p$-Input-Output Scalability, and derives sufficient C1–C3 conditions that guarantee convergence to a desired trajectory while rejecting polynomial disturbances and preventing amplification of residual disturbances, uniformly in network size. The design problem is cast as a convex optimization with LMIs to compute controller gains, enabling distributed synthesis for both leaderless and leader-follower topologies under bounded delays. The approach is validated through in-silico simulations and hardware experiments (Robotarium) demonstrating scalable formation tracking and robust disturbance rejection. Collectively, the results extend prior work on string stability and delay-free contraction to nonlinear networks with delays, providing a practically implementable, scalable framework for distributed control in large networks.

Abstract

We consider both leaderless and leader-follower, possibly nonlinear, networks affected by time-varying communication delays. For such systems, we give a set of sufficient conditions that guarantee the convergence of the network towards some desired behaviour while simultaneously ensuring the rejection of polynomial disturbances and the non-amplification of other classes of disturbances across the network. To fulfill these desired properties, and prove our main results, we propose the use of a control protocol that implements a multiplex architecture. The use of our results for control protocol design is then illustrated in the context of formation control. The protocols are validated both in-silico and via an experimental set-up with real robots. All experiments confirm the effectiveness of our approach.

A Multiplex Approach Against Disturbance Propagation in Nonlinear Networks with Delays

TL;DR

The paper addresses disturbance propagation in nonlinear networks with time-varying delays by introducing a multiplex control architecture grounded in contraction theory. It defines -Input-to-State Scalability and -Input-Output Scalability, and derives sufficient C1–C3 conditions that guarantee convergence to a desired trajectory while rejecting polynomial disturbances and preventing amplification of residual disturbances, uniformly in network size. The design problem is cast as a convex optimization with LMIs to compute controller gains, enabling distributed synthesis for both leaderless and leader-follower topologies under bounded delays. The approach is validated through in-silico simulations and hardware experiments (Robotarium) demonstrating scalable formation tracking and robust disturbance rejection. Collectively, the results extend prior work on string stability and delay-free contraction to nonlinear networks with delays, providing a practically implementable, scalable framework for distributed control in large networks.

Abstract

We consider both leaderless and leader-follower, possibly nonlinear, networks affected by time-varying communication delays. For such systems, we give a set of sufficient conditions that guarantee the convergence of the network towards some desired behaviour while simultaneously ensuring the rejection of polynomial disturbances and the non-amplification of other classes of disturbances across the network. To fulfill these desired properties, and prove our main results, we propose the use of a control protocol that implements a multiplex architecture. The use of our results for control protocol design is then illustrated in the context of formation control. The protocols are validated both in-silico and via an experimental set-up with real robots. All experiments confirm the effectiveness of our approach.
Paper Structure (16 sections, 4 theorems, 48 equations, 7 figures)

This paper contains 16 sections, 4 theorems, 48 equations, 7 figures.

Table of Contents

  1. INTRODUCTION
  2. RELATED WORKS
  3. STATEMENT OF CONTRIBUTIONS
  4. MATHEMATICAL PRELIMINARIES
  5. THE SET-UP
  6. MULTIPLEX ARCHITECTURE
  7. NETWORK SCALABILITY AND CONTROL GOAL
  8. RUNNING EXAMPLE: MULTI-ROBOT FORMATION CONTROL
  9. MAIN METHODOLOGICAL RESULTS
  10. SUFFICIENT CONDITIONS FOR SCALABILITY
  11. RUNNING EXAMPLE: MULTI-ROBOT FORMATION CONTROL (CONTINUE)
  12. VALIDATION
  13. CONCLUSIONS AND FUTURE WORK
  14. PROOF OF PROPOSITION \ref{['prop: weighted']}
  15. FULFILLING C2 AND C3 VIA OPTIMISATION
  16. ...and 1 more sections

Key Result

Lemma 1

For any structured vector norm $\vert \cdot \vert_G$ on $\mathbb{R}^{nN\times nN}$ and any $p$-vector norm $\vert \cdot \vert_S$ on $\mathbb{R}^N$, we have: (i) $\mu_G(A) \le \mu_S(\hat{A})$; (ii)$\Vert A \Vert_G \le \Vert {\bar{A}} \Vert_S$.

Figures (7)

  • Figure 1: The multiplex architecture. One disturbance is highlighted and the reference signal is omitted. Layers can have different topologies, which can be both directed and undirected.
  • Figure 2: Reference trajectory of the hand position provided by the virtual leader together with an example of desired formation.
  • Figure 3: Maximum hand position deviation (in meters) of each circle across as number of circles increases from $1$ to $30$.
  • Figure 4: Top panel: hand position deviations of all the robots (in meters). Bottom panel: hand position deviations of the unperturbed robots only (in meters). Robots on the same circle have the same color. Disturbances are the ones in \ref{['eqn:simulation_disturbances']}.
  • Figure 5: Hand position deviation when the control protocol is designed according to 9353260 (top), SILVA2021109542 (middle) and when the protocol in \ref{['equ: fbl_control']} is used (bottom).
  • ...and 2 more figures

Theorems & Definitions (16)

  • Lemma 1
  • Lemma 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Definition 1
  • Proposition 1
  • Remark 6
  • ...and 6 more