Genericity of Algebraically Observable Polynomial Systems

E. Sontag has introduced the concept of algebraic observability for n-dimensional polynomial systems. It is a stronger notion than the usual concept of observability and implies the existence of a polynomial expression of the state variables in terms of a finite number of derivatives of the output function. We prove that algebraic observability is a generic property for polynomial systems of bounded degrees. Explicit geometric characterizations of algebraic observability via polynomial embeddings are derived and it is shown that the state variables of an algebraically observable system can be expressed as a polynomial in the first 2n + 1 derivatives of the output.