Observability at an initial state for polynomial systems

We consider observability at an initial state for polynomial systems. When testing for local observability for nonlinear systems, the observability rank condition based on the inverse function theorem is commonly used. However, the rank condition is a sufficient condition, and we cannot test for global observability using the rank condition. In this paper, first we derive necessary and sufficient conditions for global observability at an initial state for continuous-time polynomial systems. Then, necessary and sufficient conditions for local observability are derived from one of the global observability conditions using the localization of a polynomial ring. Using the same procedure, we derive both global and local observability conditions for discrete-time polynomial systems. Each condition is characterized by a finite set of equations, since polynomial rings are Noetherian. Finally, examples demonstrate the proposed criteria for testing for observability.