Constructing a Minimal Linear Realization to Analyze Observability of Nonlinear Dynamical Systems over Finite Fields

This paper develops a linear approach for analyzing the observability of nonlinear dynamical systems over finite fields using the Koopman operator framework. It constructs a minimal linear realization (LOR) that can reproduce all the output sequences of the original nonlinear system, and provides necessary and sufficient conditions for the observability of the nonlinear system through the LOR. The paper also establishes an upper bound on the number of outputs required for the unique reconstruction of the initial condition.