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Path-Based Conditions for the Identifiability of Non-additive Nonlinear Networks with Full Measurements

Renato Vizuete, Julien M. Hendrickx

TL;DR

Problem: identifiability of nonlinear networks with non-additive node dynamics under full measurement and path-independent delays. Approach: define generic identifiability via a nonlinear matrix $J_G(\mathbf{u}_G^k)$, analyze DAGs, and establish a vertex-disjoint-path condition as a sufficient criterion; for polynomial functions this condition is also necessary. Key results: path-independent delays imply identifiability extends to general delay configurations; in polynomial class the condition is necessary; a counterexample shows the sufficiency is not necessary for the additive model; a conjecture is stated for analytic functions. Significance: links graph-theoretic path structures to identifiability, guiding excitation/measurement design in nonlinear network identification.

Abstract

We analyze the identifiability of nonlinear networks with node dynamics characterized by functions that are non-additive. We consider the full measurement case (all the nodes are measured) in the path-independent delay scenario where all the excitation signals of a specific node have the same delay in the output of a measured node. Based on the notion of a generic nonlinear matrix associated with the network, we introduce the concept of generic identifiability and characterize the space of functions that satisfies this property. For directed acyclic graphs (DAGs) characterized by analytic functions, we derive a sufficient condition for identifiability based on vertex-disjoint paths from excited nodes to the in-neighbors of each node in the network. Furthermore, when we consider the class of polynomial functions, by using well-known results on algebraic varieties, we prove that the vertex-disjoint path condition is also necessary. Finally, we show that this identifiability condition is not necessary for the additive nonlinear model. Some examples are added to illustrate the results.

Path-Based Conditions for the Identifiability of Non-additive Nonlinear Networks with Full Measurements

TL;DR

Problem: identifiability of nonlinear networks with non-additive node dynamics under full measurement and path-independent delays. Approach: define generic identifiability via a nonlinear matrix , analyze DAGs, and establish a vertex-disjoint-path condition as a sufficient criterion; for polynomial functions this condition is also necessary. Key results: path-independent delays imply identifiability extends to general delay configurations; in polynomial class the condition is necessary; a counterexample shows the sufficiency is not necessary for the additive model; a conjecture is stated for analytic functions. Significance: links graph-theoretic path structures to identifiability, guiding excitation/measurement design in nonlinear network identification.

Abstract

We analyze the identifiability of nonlinear networks with node dynamics characterized by functions that are non-additive. We consider the full measurement case (all the nodes are measured) in the path-independent delay scenario where all the excitation signals of a specific node have the same delay in the output of a measured node. Based on the notion of a generic nonlinear matrix associated with the network, we introduce the concept of generic identifiability and characterize the space of functions that satisfies this property. For directed acyclic graphs (DAGs) characterized by analytic functions, we derive a sufficient condition for identifiability based on vertex-disjoint paths from excited nodes to the in-neighbors of each node in the network. Furthermore, when we consider the class of polynomial functions, by using well-known results on algebraic varieties, we prove that the vertex-disjoint path condition is also necessary. Finally, we show that this identifiability condition is not necessary for the additive nonlinear model. Some examples are added to illustrate the results.
Paper Structure (13 sections, 10 theorems, 37 equations, 7 figures)

This paper contains 13 sections, 10 theorems, 37 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Model class
  4. Generic identifiability
  5. Generic nonlinear matrix for $g_i=0$
  6. Path-independent delay
  7. Zero measure set of functions
  8. Directed Acyclic Graphs
  9. Sufficient condition
  10. Necessary condition for polynomial functions
  11. Non-additive and additive model
  12. Conclusions and future work
  13. Proof of Lemma \ref{['lemma:algebraic_variety']}

Key Result

Proposition 1

For any DAG $G$, there exists a particular choice of delays $m_{i,j}$ of the functions $f_{i,j}$ such that all the variables $u_j$ associated to a node $j$ have the same delays in $F_i(u_1^{k-T_{i,1}}\!,\ldots,u_{n_i}^{k-T_{i,n_i}})$.

Figures (7)

  • Figure 1: Model of a network considered for the identification where all the nodes are measured (gray) and some nodes can be excited (white and gray). The dynamics of each node is determined by a nonlinear function $\Phi_i$ of the outputs of the in-neighbors.
  • Figure 2: Each node function $\Phi_i$ can be expressed as the sum of edge functions $f_{i,j}$ and a function $g_i$ composed by products of the outputs of the in-neighbors. Unlike the additive model, the presence of a static component $\gamma$ does not affect the identifiability.
  • Figure 3: DAG where a particular choice of delays $m_{2,1}$, $m_{3,2}$ and $m_{3,1}$ might generate a function $F_3$ with a path-independent delay where the information coming through $f_{3,1}$ and $f_{3,2}$ cannot be distinguished.
  • Figure 4: DAG with a topological ordering where there exists always a particular choice of delays $m_{i,j}=i-j$ such that all the excitation signals $u_j$ in $F_i$ have the same delays.
  • Figure 5: A DAG where a particular choice of functions makes the network unidentifiable. If $f_{3,1}= f_{4,1}$ and $f_{3,2}= f_{4,2}$, the outputs $y_3$ and $y_4$ are the same, which implies that the functions $f_{5,3}$ and $f_{5,4}$ cannot be identified.
  • ...and 2 more figures

Theorems & Definitions (30)

  • Remark 1: Functions $\Phi_i$ and $F_i$
  • Definition 1: Set of measured functions
  • Definition 2: Generic Identifiability
  • Definition 3: Class of functions $\mathcal{F}_A$
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Example 1: Particular choice
  • Definition 4: $K$-generic property
  • ...and 20 more