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

TL;DR

This paper introduces a method for identifying nonlinear acyclic networks in continuous time using measurements of sinks, derivatives, and initial conditions, enabling the reconstruction of complex network structures.

Contribution

It provides necessary and sufficient conditions for network identification from sink measurements and develops algorithms for trees and general DAGs with nonlinear dynamics.

Findings

Successful identification of network structures from sink measurements.

Recovery of nonlinear edge functions using derivatives and initial conditions.

Method for distinguishing parallel paths of equal length in DAGs.

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.