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A Behavioral Perspective on Models of Linear Dynamical Networks with Manifest Variables

Shengling Shi, Zhiyong Sun, Bart De Schutter

TL;DR

This paper addresses the lack of a unified framework for linear dynamical networks in which input and output channels are not fixed in advance. It adopts Willems’ behavioral theory to define network interconnections based on external variables, and introduces two dual hypergraph-based graphical representations—system graphs (components as vertices) and signal graphs (signals as vertices)—to visualize interconnections without preset inputs/outputs. A key contribution is establishing explicit connections between behavioral networks and structural VAR (SVAR) models, showing that regular interconnections underpin SVAR structure and graph realizations. The work demonstrates how incidence and kernel representations encode interconnections, and it discusses both regular and non-regular interconnections, illustrating how to recover regularity by grouping components. Collectively, the framework provides a principled, model-agnostic way to compare, analyze, and relate diverse linear network models in a way that is amenable to data-driven SVAR interpretation and future algorithmic development.

Abstract

Networks of dynamical systems play an important role in various domains and have motivated many studies on the control and analysis of linear dynamical networks. For linear network models considered in these studies, it is typically pre-determined what signal channels are inputs and what are outputs. These models do not capture the practical need to incorporate different experimental situations, where different selections of input and output channels are applied to the same network. Moreover, a unified view of different network models is lacking. This work makes an initial step towards addressing the above issues by taking a behavioral perspective, where input and output channels are not pre-determined. The focus of this work is on behavioral network models with only external variables. By exploiting the concept of hypergraphs, novel dual graphical representations, called system graphs and signal graphs, are introduced for behavioral networks. Moreover, connections between behavioral network models and structural vector autoregressive models are established. In addition to their connections in graphical representations, it is shown that the regularity of interconnections is an essential assumption when choosing a structural vector autoregressive model.

A Behavioral Perspective on Models of Linear Dynamical Networks with Manifest Variables

TL;DR

This paper addresses the lack of a unified framework for linear dynamical networks in which input and output channels are not fixed in advance. It adopts Willems’ behavioral theory to define network interconnections based on external variables, and introduces two dual hypergraph-based graphical representations—system graphs (components as vertices) and signal graphs (signals as vertices)—to visualize interconnections without preset inputs/outputs. A key contribution is establishing explicit connections between behavioral networks and structural VAR (SVAR) models, showing that regular interconnections underpin SVAR structure and graph realizations. The work demonstrates how incidence and kernel representations encode interconnections, and it discusses both regular and non-regular interconnections, illustrating how to recover regularity by grouping components. Collectively, the framework provides a principled, model-agnostic way to compare, analyze, and relate diverse linear network models in a way that is amenable to data-driven SVAR interpretation and future algorithmic development.

Abstract

Networks of dynamical systems play an important role in various domains and have motivated many studies on the control and analysis of linear dynamical networks. For linear network models considered in these studies, it is typically pre-determined what signal channels are inputs and what are outputs. These models do not capture the practical need to incorporate different experimental situations, where different selections of input and output channels are applied to the same network. Moreover, a unified view of different network models is lacking. This work makes an initial step towards addressing the above issues by taking a behavioral perspective, where input and output channels are not pre-determined. The focus of this work is on behavioral network models with only external variables. By exploiting the concept of hypergraphs, novel dual graphical representations, called system graphs and signal graphs, are introduced for behavioral networks. Moreover, connections between behavioral network models and structural vector autoregressive models are established. In addition to their connections in graphical representations, it is shown that the regularity of interconnections is an essential assumption when choosing a structural vector autoregressive model.
Paper Structure (14 sections, 7 theorems, 12 equations, 3 figures)

This paper contains 14 sections, 7 theorems, 12 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Notation
  4. Systems theory from a behavioral perspective
  5. Manifest networks in a behavioral setting
  6. Algebraic representation
  7. Graphical representation
  8. Regularity of interconnection
  9. Lack of regularity
  10. Structural VAR model
  11. From SVAR to behavioral networks
  12. From behavioral networks to SVAR
  13. CONCLUSIONS
  14. Acknowledgements

Key Result

Lemma III.1

Given a dynamical system $\Sigma = (\mathbb{Z}_+, \prod_{ j \in \mathbb{I}_L } \mathbb{R}^{q_j}, \mathcal{B} ) \in \mathcal{L}^q$ , then $w_j$ is unconstrained if and only if $R_j =0$ for any kernel representation $[R_1\text{ }R_2 \text{ }\cdots R_L ]$ of $\Sigma$, where $R_j \in \mathbb{R}^{\bulle

Figures (3)

  • Figure 1: Two graphical representations of \ref{['eq:exam1']} are shown. The signal graph in (a) represents the $4$ signal vectors as $4$ vertices and the $4$ components as $4$ nets/edges. The dual graph in (b) represents the $4$ components as vertices and the $4$ signals as nets, visualized as edges by convention.
  • Figure 2: The interconnection $\land_{i=1}^3 \Sigma_i$ is not a regular feedback interconnection; however, if merging $\bar{\Sigma}_2 =\Sigma_2\land \Sigma_3$ into a single component, $\Sigma_1 \land \bar{\Sigma}_2$ becomes a regular feedback interconnection of two components.
  • Figure 3: A directed graph of an SVAR model is shown in (a). The model also defines an interconnected system in the behavioral setting with a signal graph (b) and a system graph (c).

Theorems & Definitions (24)

  • Definition III.1: willems1991paradigms
  • Definition III.2
  • Lemma III.1
  • proof
  • Definition III.3
  • Remark III.1
  • Corollary III.1
  • Remark III.2
  • Definition III.4: schrijver2003combinatorial
  • Definition III.5
  • ...and 14 more