An Algebraic Approach to Local Observability at an Initial State for Discrete-Time Polynomial Systems

In this paper, we consider local observability at an initial state for discrete-time autonomous polynomial systems. When testing for observability, for discrete-time nonlinear systems, a condition based on the inverse function theorem is commonly used. However, it is a sufficient condition. In this paper, first we derive a necessary and sufficient condition for global observability at an initial state for these polynomial systems. Then, a necessary and sufficient condition for local observability is derived from the global observability condition using the localization of a polynomial ring. Each condition is characterized by a finite set of equations, since polynomial rings are noetherian. Finally, an example demonstrates the proposed criteria for testing the observability.