Global observability of polynomial systems

We considers the global observability of general polynomial systems. To test the observability of nonlinear systems, distinguishability is often used. It is not guaranteed that a finite set of Lie derivatives is sufficient to check the criterion. On the basis of the results of commutative algebra, we show that computing only a finite set of Lie derivatives is sufficient to check the criterion for polynomial systems. Then, we derive the global observability conditions, which give necessity and sufficiency. Finally, examples demonstrate the proposed criterion for testing the observability.