Observability of Nonlinear Dynamical Systems over Finite Fields
Ramachandran Anantharaman, Virendra Sule
TL;DR
This work develops a Koopman-operator–based framework to study observability of nonlinear DSFF over finite fields. It introduces the Linear Output Realization (LOR), the smallest linear system that can reproduce all outputs of a nonlinear DSFF by operating within a minimal invariant subspace that contains the output functions. The paper proves that the DSFF is observable iff the map $\\hat{\\psi}: \\mathbb{F}^n o \\mathbb{F}^N$ is injective, and shows the maximum number of outputs needed to uniquely recover the initial condition equals the LOR dimension $N$, with Cayley-Hamilton enabling reduction of longer sequences. Importantly, the LOR remains observable even when the original nonlinear system is not, providing a robust linear surrogate for analysis and potential observer design in finite-field settings.
Abstract
This paper discusses the observability of nonlinear Dynamical Systems over Finite Fields (DSFF) through the Koopman operator framework. In this work, given a nonlinear DSFF, we construct a linear system of the smallest dimension, called the Linear Output Realization (LOR), which can generate all the output sequences of the original nonlinear system through proper choices of initial conditions (of the associated LOR). We provide necessary and sufficient conditions for the observability of a nonlinear system and establish that the maximum number of outputs sufficient for computing the initial condition is precisely equal to the dimension of the LOR. Further, when the system is not known to be observable, we provide necessary and sufficient conditions for the unique reconstruction of initial conditions for specific output sequences.