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Identifying Network Structure of Nonlinear Dynamical Systems: Contraction and Kuramoto Oscillators

Jaidev Gill, Jing Shuang Li

Abstract

In this work, we study the identifiability of network structures (i.e., topologies) for networked nonlinear systems when partial measurements of the nodal dynamics are taken. We explore scenarios where different candidate structures can yield similar measurements, thus limiting identifiability. To do so, we apply the contraction theory framework to facilitate comparisons between different networks. We show that semicontraction in the observable space is a sufficient condition for two systems to become indistinguishable from one another based on partial measurements. We apply this framework to study networks of Kuramoto oscillators, and discuss scenarios in which different network structures (both connected and disconnected) become indistinguishable.

Identifying Network Structure of Nonlinear Dynamical Systems: Contraction and Kuramoto Oscillators

Abstract

In this work, we study the identifiability of network structures (i.e., topologies) for networked nonlinear systems when partial measurements of the nodal dynamics are taken. We explore scenarios where different candidate structures can yield similar measurements, thus limiting identifiability. To do so, we apply the contraction theory framework to facilitate comparisons between different networks. We show that semicontraction in the observable space is a sufficient condition for two systems to become indistinguishable from one another based on partial measurements. We apply this framework to study networks of Kuramoto oscillators, and discuss scenarios in which different network structures (both connected and disconnected) become indistinguishable.

Paper Structure

This paper contains 7 sections, 4 theorems, 49 equations, 4 figures.

Table of Contents

  1. Introduction and Motivation
  2. Contraction Theory for System Comparisons
  3. Application I: Nonlinear Networked Systems
  4. Application II: Linear Systems
  5. Kuramoto Oscillator Networks
  6. Simulations
  7. Conclusions and Future Work

Key Result

Lemma 1

Assume the auxiliary system sys:aux is contracting with respect to $M_C$ with rate $\mu\in \mathbb{R}_{>0}$, i.e., it satisfies the linear matrix inequality (LMI) then the time-varying length $L(t)$ satisfies the differential inequality

Figures (4)

  • Figure 1: Various network structures can have observed trajectories that contract to one another making them indistinguishable from each other. We derive sufficient conditions on the dynamics for this phenomenon to occur, and in doing so, we are able to generate other candidate structures that behave similarly (i.e., have the same observed trajectories).
  • Figure 2: We display the topology encoded in \ref{['eq:B_matrix']} and \ref{['eq:A_matrix']} (Net 1). The three other topologies are equivalent based on the analysis when an average of nodes 1 and 2 and an average of nodes 3 and 4 are measured.
  • Figure 3: Comparison of different Kuramoto networks displayed in Fig. \ref{['fig:kuramoto_partial']}. When beginning with the same initial condition (Left column), all four network structures' measurements are identical to each other over the entire measurement period and have no phase shifts between them. When the networks start with different initial conditions (Right column) the states differ by a constant phase shift.
  • Figure 4: Comparison of different Kuramoto networks displayed in Fig. \ref{['fig:kuramoto_partial']}. In this case, $x_0$ and $\omega$ are entirely random and thus violate \ref{['eq:condition_nullspace']}. Networks are simulated all with the same initial condition (Left column) and with different initial conditions (Right column).

Theorems & Definitions (15)

  • Definition 1
  • Remark 1
  • Lemma 1
  • proof
  • Corollary 1
  • Remark 2
  • Corollary 2
  • proof
  • Definition 2
  • Theorem 1
  • ...and 5 more