Minimal polynomial realizations

Structural identifiability of polynomial and rational systems

“…To introduce polynomial and rational systems we follow Bartosiewicz's definitions in [4,5]. His approach generalizes the common approach of nonlinear control theory from the geometric viewpoint since the state spaces he considers are not necessarily smooth affine varieties.…”

“…The output space is considered to be R r . The following definition of polynomial and rational systems is a slight modification of the definition in [5,Section 2] for polynomial systems, and of [4, Definition 2] for rational systems.…”

“…it is the map x(·; x 0 , u) : [0, T u ) → X such that d dt (ϕ • x)(t; x 0 , u) = (f α ϕ)(x(t; x 0 , u)) and x(0) = x 0 for t ∈ [0, T u ) and for ϕ ∈ A. According to [5,Theorem 1] for any polynomial vector field f on a variety X and any x 0 ∈ X there exists an unique trajectory of f from x 0 defined on maximal interval [0, T ). The corresponding result for any rational vector field f on a variety X and any x 0 ∈ X at which f is defined can be found in [4].…”

“…On the other hand, there exist rationally observable polynomial systems which are not algebraically observable, see Example 2.5. For more details on algebraic and rational observability and algebraic controllability of polynomial and rational systems see [2][3][4][5]24] and others.…”

“…Section 2 introduces the concepts of polynomial and rational systems as they are developed in [4,5,26] and the relevant references therein. The definitions of algebraic controllability, algebraic and rational observability are recalled.…”