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Combinatorial Admissibility in Control-Affine Networks

Daniel Zelazo, Louis Theran

Abstract

We study synchronization of heterogeneous control-affine nonlinear agents interconnected through diffusive relative-output measurements. Motivated by recent geometric edge-space formulations for formation control, we separate the design into an \emph{edge-space} step, which specifies a stabilizing model evolution for relative outputs, and a \emph{lift} step, which realizes the prescribed edge motion using the agents' allowable input directions under the control-affine geometry. We introduce an admissibility notion that characterizes when an edge-driven diffusive design is feasible. We relate generic admissibility to structured rank and maximum matchings in an associated bipartite graph, yielding checkable combinatorial certificates that show how graph topology and actuation structure jointly limit achievable edge dynamics. The results are illustrated on synchronization of nonlinear oscillators.

Combinatorial Admissibility in Control-Affine Networks

Abstract

We study synchronization of heterogeneous control-affine nonlinear agents interconnected through diffusive relative-output measurements. Motivated by recent geometric edge-space formulations for formation control, we separate the design into an \emph{edge-space} step, which specifies a stabilizing model evolution for relative outputs, and a \emph{lift} step, which realizes the prescribed edge motion using the agents' allowable input directions under the control-affine geometry. We introduce an admissibility notion that characterizes when an edge-driven diffusive design is feasible. We relate generic admissibility to structured rank and maximum matchings in an associated bipartite graph, yielding checkable combinatorial certificates that show how graph topology and actuation structure jointly limit achievable edge dynamics. The results are illustrated on synchronization of nonlinear oscillators.
Paper Structure (12 sections, 3 theorems, 37 equations, 2 figures)

This paper contains 12 sections, 3 theorems, 37 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Edge-Space Geometry and a Model System
  4. Edge-Driven Controller Families and Distributed Implementations
  5. A family of lifts for a prescribed edge flow
  6. Distributed implementation for relative degree-one dynamics
  7. Admissibility of edge-driven designs
  8. Generic admissibility and combinatorial certificates
  9. Simulation: Admissible vs. Non-Admissible Edge Actuation on a Limit Cycle
  10. Case A (admissible)
  11. Case B (non-admissible)
  12. Concluding Remarks

Key Result

Theorem 1

Suppose each agent admits relative-degree-one output dynamics eq:ydot_again on a forward-invariant set $\mathcal{X}_i$, and $H_i(x_i)$ is right invertible for all $x_i\in\mathcal{X}_i$. Apply the local right-inversion controller eq:local_inversion_again and let the virtual input be chosen by the dif with weights $w_{ij}\ge 0$ of a (directed) graph $\mathcal{G}$. If $\mathcal{G}$ contains a (direct

Figures (2)

  • Figure 1: Bipartite graphs associated with the sparsity pattern of $A_\tau(x)$. Gray edges show structural nonzeros; colored edges show a maximum matching.
  • Figure 2: Phase-plane trajectories for the oscillator network under the two actuation patterns.

Theorems & Definitions (13)

  • Remark 1: Edge-driven family in "gain" form
  • Theorem 1
  • Remark 2: Relation to the edge-driven family
  • Definition 1: Admissibility
  • Remark 3: Relationship between admissibility concepts
  • Remark 4: Exact vs. least-squares lifting
  • Lemma 1: Exact realization under admissibility
  • proof
  • Remark 5: Why Theorem \ref{['thm:distributed_outer_loop_consensus']} is always admissible
  • Definition 2: Matching
  • ...and 3 more