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Identification of Nonlinear Acyclic Networks in Continuous Time from Nonzero Initial Conditions and Full Excitations

Ramachandran Anantharaman, Renato Vizuete, Julien M. Hendrickx, Alexandre Mauroy

Abstract

We propose a method to identify nonlinear acyclic networks in continuous time when the dynamics are located on the edges and all the nodes are excited. We show that it is necessary and sufficient to measure all the sinks to identify any tree in continuous time when the functions associated with the dynamics are analytic and satisfy $f(0)=0$, which is analogous to the discrete-time case. For general directed acyclic graphs (DAGs), we show that it is necessary and sufficient to measure all sinks, assuming that the dynamics are not linear (a condition that can be relaxed for trees). Then, based on the measurement of higher order derivatives and nonzero initial conditions, we introduce a method for the identification of trees, which allows us to recover the nonlinear functions located in the edges of the network under the assumption of dictionary functions. Finally, we propose a method to identify multiple parallel paths of the same length between two nodes, which allow us to identify any DAG when combined with the algorithm for the identification of trees. Several examples are added to illustrate the results.

Identification of Nonlinear Acyclic Networks in Continuous Time from Nonzero Initial Conditions and Full Excitations

Abstract

We propose a method to identify nonlinear acyclic networks in continuous time when the dynamics are located on the edges and all the nodes are excited. We show that it is necessary and sufficient to measure all the sinks to identify any tree in continuous time when the functions associated with the dynamics are analytic and satisfy , which is analogous to the discrete-time case. For general directed acyclic graphs (DAGs), we show that it is necessary and sufficient to measure all sinks, assuming that the dynamics are not linear (a condition that can be relaxed for trees). Then, based on the measurement of higher order derivatives and nonzero initial conditions, we introduce a method for the identification of trees, which allows us to recover the nonlinear functions located in the edges of the network under the assumption of dictionary functions. Finally, we propose a method to identify multiple parallel paths of the same length between two nodes, which allow us to identify any DAG when combined with the algorithm for the identification of trees. Several examples are added to illustrate the results.
Paper Structure (13 sections, 5 theorems, 48 equations, 5 figures, 2 tables, 2 algorithms)

This paper contains 13 sections, 5 theorems, 48 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Model class
  4. Identifiability
  5. Identifiability conditions for trees
  6. Identifiability of Directed acyclic graphs
  7. Algorithms for identification
  8. Algorithm for identification of trees
  9. Algorithm for identification of DAGs
  10. Numerical Examples
  11. Numerical example for trees
  12. Numerical example for DAGs
  13. Conclusions

Key Result

Proposition 1

Under Assumptions ass:graph_topology and ass:full_excitation, a path graph is identifiable in the class $\mathcal{F}_Z$, if and only if the sink is measured. Moreover, the identifiability remains true even if we use zero initial conditions and constant excitation signals, or use multiple initial con

Figures (5)

  • Figure 1: Electrical network where some nodes can be excited with an electrical source (red) and some nodes can be measured through voltmeters (green).
  • Figure 2: A four well-mixed fluid tanks whose dynamics can be expressed as the model \ref{['eq:model']} considered for the identification. The physical system can be abstracted as the network in panel (b) where the edges are characterized by nonlinear functions, all the nodes can be excited (red arrows), and some nodes can be measured (green).
  • Figure 3: Path graph with 4 nodes.
  • Figure 4: DAG whose nonlinear functions can be identified with the measurement of the sink 3 by using Algorithm \ref{['alg:path_graph']} since the two paths between nodes 1 and 3 have different lengths.
  • Figure 5: DAG whose nonlinear functions can be identified with the measurement of the sink 4 by using Algorithms \ref{['alg:path_graph']} and \ref{['alg:bridge_graph']}. The identification of the edges $f_{4,1}$, $f_{4,2}$ and $f_{4,3}$ can be done by using Algorithm \ref{['alg:path_graph']} while Algorithm \ref{['alg:bridge_graph']} can be used for the identification of the edges $f_{2,1}$ and $f_{3,1}$.

Theorems & Definitions (13)

  • Definition 2.1: Set of measured functions
  • Definition 2.2: Identifiability
  • Definition 2.3: Class of functions $\mathcal{F}_Z$
  • Definition 3.1: Source and sink
  • Proposition 1: Path graphs
  • Lemma 1: Lemma 4 vizuete2023nonlinear
  • Theorem 1: Trees
  • proof
  • Definition 4.1: Class of functions $\mathcal{F}_{Z,NL}$
  • Lemma 2: Removal of a node
  • ...and 3 more