  The parameter identifiability problem for a dynamical system is to determine
whether the parameters of the system can be found from data for the outputs of
the system. Verifying whether the parameters are identifiable is a necessary
first step before a meaningful parameter estimation can take place.
Non-identifiability occurs in practical models. To reparametrize a model to
achieve identifiability is a challenge. The existing approaches have been shown
to be useful for many important examples. However, these approaches are either
limited to linear models and scaling parametrizations or are not guaranteed to
find a reparametrization even if it exists. In the present paper, we prove that
there always exists a locally identifiable model with the same input-output
behaviour as the original one obtained from a given one by a partial
specialization of the parameters. As an extra feature of our approach, the
resulting (at least) locally identifiable reparameterization has the same
shape: the monomials in the new state variables in the new model are formed in
the same way as in the original model. Furthermore, we give a sufficient
observability condition for the existence of a state space transformation from
the original model to the new one. Our proof is constructive and can be
translated to an algorithm, which we illustrate by several examples.
