A necessary condition, a sufficient condition for structural identifiability

In this paper we give a necessary condition for structural identifiability of uncontrolled autonomous systems. This condition only turns on the identifiability of the right hand side of the differential system: x˙(t)=f(x(t),θ) x(t₀)=x₀(θ). It is applied to a well-known unidentifiable nonlinear model of microbial growth. We then prove that this necessary condition becomes sufficient when the state is one-dimensional. This is obtained by a classical series expansion of the input-output map. This condition is easy to check, as it is shown by the study of some examples, in which we do much less computation than the involved literature to prove identifiability properties