Nonlinear identifiability of directed acyclic graphs with partial excitation and measurement

Abstract

We analyze the identifiability of directed acyclic graphs in the case of partial excitation and measurement. We consider an additive model where the nonlinear functions located in the edges depend only on a past input, and we analyze the identifiability problem in the class of pure nonlinear functions satisfying $f(0)=0$. We show that any identification pattern (set of measured nodes and set of excited nodes) requires the excitation of sources, measurement of sinks and the excitation or measurement of the other nodes. Then, we show that a directed acyclic graph (DAG) is identifiable with a given identification pattern if and only if it is identifiable with the measurement of all the nodes. Next, we analyze the case of trees where we prove that any identification pattern guarantees the identifiability of the network. Finally, by introducing the notion of a generic nonlinear network matrix, we provide sufficient conditions for the identifiability of DAGs based on the notion of vertex-disjoint paths.

Figures (6)

• Figure 1: Model of a network considered for the identification where some nodes are excited (white) and some nodes are measured (gray). Other nodes are excited and measured at the same time (white and gray).
• Figure 2: DAG with a topological ordering where between two consecutive measured nodes $p$ and $q$, all the excited nodes have a path to $q$.
• Figure 3: If the delays of all the nonlinear functions in a multipartite digraph are the same, all the excitation signals associated with a node in the function \ref{['eq:function_Fi']} have the same delay.
• Figure 4: A DAG where a particular choice of functions makes the network unidentifiable. If $f_{3,1}=\gamma f_{4,1}$ and $f_{3,2}=\gamma f_{4,2}$ with $\gamma\neq 0$, the functions $f_{6,4}$ and $f_{6,5}$ cannot be identified.
• Figure 5: Symmetry full excitation/full measurement does not hold for this DAG.

Theorems & Definitions (23)

• Definition 1: Identification pattern
• Definition 2
• Definition 3: Set of measured functions
• Definition 4: Identifiability
• Definition 5: Class of functions $\mathcal{F}_{Z,NL}$
• Lemma 1: Sinks and sources
• proof
• Proposition 1
• proof
• Proposition 2