  Structural identifiability concerns the question of which unknown parameters
of a model can be recovered from (perfect) input-output data. If all of the
parameters of a model can be recovered from data, the model is said to be
identifiable. However, in many models, there are parameters that can take on an
infinite number of values but yield the same input-output data. In this case,
those parameters and the model are called unidentifiable. The question is then
what to do with an unidentifiable model. One can either adjust the model, if
experimentally feasible, or try to find a reparametrization to make the model
identifiable. In this paper, we take the latter approach. While existing
approaches to find identifiable reparametrizations were limited to scaling
reparametrizations or were not guaranteed to find a globally identifiable
reparametrization even if it exists, we significantly broaden the class of
models for which we can find a globally identifiable model with the same
input-output behavior as the original one. We also prove that, for linear
models, a globally identifiable reparametrization always exists and show that,
for a certain class of linear compartmental models, an explicit
reparametrization formula exists. We illustrate our method on several examples
and provide detailed analysis in supplementary material on github.
