Nonlinear Network Identifiability with Full Excitations

TL;DR

This work tackles identifiability in nonlinear networks with edge dynamics under full excitation. It shows that for DAGs with analytic edge functions satisfying $f(0)=0$, measuring all sinks is necessary and sufficient, while introducing constant terms destroys identifiability when a node has multiple in-neighbors. For graphs with cycles, the authors assume additively separable nonlinearities and finite memory, proving that measuring one node from each sink of the condensation digraph suffices to identify all edge functions; this is established via unfolded digraph constructions and NFIR representations. The results reveal fundamental differences from linear network identifiability and provide a principled sensor placement rule for nonlinear network identification with full excitation, along with open avenues for extending to broader nonlinear classes.

Abstract

We derive conditions for the identifiability of nonlinear networks characterized by additive dynamics at the level of the edges when all the nodes are excited. In contrast to linear systems, we show that the measurement of all sinks is necessary and sufficient for the identifiability of directed acyclic graphs, under the assumption that dynamics are described by analytic functions without constant terms (i.e., $f(0)=0$). But if constant terms are present, then the identifiability is impossible as soon as one node has more than one in-neighbor. In the case of general digraphs that may contain cycles, we consider additively separable functions for the analysis of the identifiability, and we show that the measurement of one node of all the sinks of the condensation digraph is necessary and sufficient. Several examples are added to illustrate the results.

Figures (6)

• Figure 1: DAG where the function associated with the measurement of the node $3$ only depends on a finite number of inputs.
• Figure 2: If a DAG has at least a node $i$ with two or more in-neighbors, then it is not identifiable in the class $\mathcal{F}_{ALL}$.
• Figure 3: Network with functions in $\mathcal{F}_Z$ that cannot be identified by only measuring the sink when $f_{2,1}$ and $f_{3,1}$ are linear due to the superposition principle.
• Figure 4: If the node 2 is measured and $f_{5,2}$ is identifiable, the measurement of the node 5 provides the identifiability of the edges $f_{5,1}$, $f_{5,3}$ and $f_{5,4}$ and functions $F_1$, $F_3$ and $F_4$ if and only if they are identifiable in the induced subgraph $G_{V\setminus \{ 2\}}$.
• Figure 5: Cycle graph with 2 nodes, where the function associated with the measurement of node 2 depends on an infinite number of past inputs due to the cycle.

Theorems & Definitions (41)

• Example 1: Finite inputs
• Definition 1: Set of measured functions
• Definition 2: Identifiability
• Proposition 1: Sinks and sources
• proof
• Definition 3: Class of functions $\mathcal{F}_{ALL}$
• Proposition 2: Unidentifiability
• proof
• Proposition 3: Arborescence
• proof