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Extending identifiability results from isolated networks to embedded networks

Eduardo Mapurunga, Michel Gevers, Alexandre S. Bazanella

TL;DR

This work introduces conditions under which an Excitation and Measurement Pattern ($EMP$) that identifiably captures a subnetwork in isolation remains valid when the subnetwork is embedded in a larger network with partial excitation and measurement. Building on existing graph-based identifiability theory, it extends results to Parallel Paths Networks and proves two sufficient embedding conditions that preserve identifiability, aided by a matrix partitioning framework and the relation $T_{AA}=(I_A-G_A-G_{AB}T_BG_{BA})^{-1}$. The paper also presents constructive examples showing how to synthesize and combine subnet EMPs to achieve identification of the full network with potentially minimal cardinality. The practical impact lies in enabling scalable, sparse EMP design for targeted subnetworks by decomposing complex networks into analyzable components. It suggests a path toward algorithmic network decomposition to facilitate efficient, accurate identification in large dynamical networks.

Abstract

This paper deals with the design of Excitation and Measurement Patterns (EMPs) for the identification of dynamical networks, when the objective is to identify only a subnetwork embedded in a larger network. Recent results have shown how to construct EMPs that guarantee identifiability for a range of networks with specific graph topologies, such as trees, loops, or Directed Acyclic Graphs (DAGs). However, an EMP that is valid for the identification of a subnetwork taken in isolation may no longer be valid when that subnetwork is embedded in a larger network. Our main contribution is to exhibit conditions under which it does remain valid, and to propose ways to enhance such EMP when these conditions are not satisfied.

Extending identifiability results from isolated networks to embedded networks

TL;DR

This work introduces conditions under which an Excitation and Measurement Pattern () that identifiably captures a subnetwork in isolation remains valid when the subnetwork is embedded in a larger network with partial excitation and measurement. Building on existing graph-based identifiability theory, it extends results to Parallel Paths Networks and proves two sufficient embedding conditions that preserve identifiability, aided by a matrix partitioning framework and the relation . The paper also presents constructive examples showing how to synthesize and combine subnet EMPs to achieve identification of the full network with potentially minimal cardinality. The practical impact lies in enabling scalable, sparse EMP design for targeted subnetworks by decomposing complex networks into analyzable components. It suggests a path toward algorithmic network decomposition to facilitate efficient, accurate identification in large dynamical networks.

Abstract

This paper deals with the design of Excitation and Measurement Patterns (EMPs) for the identification of dynamical networks, when the objective is to identify only a subnetwork embedded in a larger network. Recent results have shown how to construct EMPs that guarantee identifiability for a range of networks with specific graph topologies, such as trees, loops, or Directed Acyclic Graphs (DAGs). However, an EMP that is valid for the identification of a subnetwork taken in isolation may no longer be valid when that subnetwork is embedded in a larger network. Our main contribution is to exhibit conditions under which it does remain valid, and to propose ways to enhance such EMP when these conditions are not satisfied.
Paper Structure (9 sections, 4 equations, 3 figures)

This paper contains 9 sections, 4 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions, Notations and Preliminaries
  3. Identifiabillity results for specific network structures
  4. Trees
  5. Loops
  6. Parallel Paths Networks
  7. Isolated versus embedded
  8. Main result
  9. Conclusion

Figures (3)

  • Figure 1: An example of a parallel paths network.
  • Figure 2: A network formed by two loops: the one on the left formed by the solid blue edges, and the other on the right formed by the dotted red edges and the edge $3\to4$.
  • Figure 3: A network with multiple strucures. The nodes $1, 2, 3, 4$ form a PPN. The nodes $3, 4, 5, 6$ form a loop. The dashed blue edges linking nodes $1, 2, 6, 7, 8, 9, 10$ form a tree.