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For a system, a priori identifiability is a theoretical property depending only on the model and guarantees that its parameters can be uniquely determined from observations. This paper provides a survey of the various and numerous definitions of a priori identifiability given in the literature, for both deterministic continuous and discrete-time models. A classification is done by distinguishing analytical and algebraic definitions as well as local and global ones. Moreover, this paper provides an overview on the distinct methods to test the parameter identifiability. They are classified into the so-called output equality approaches, local state isomorphism approaches and differential algebra approaches. A few examples are detailed to illustrate the methods and complete this survey.

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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 nonlinear differential system. We prove that this necessary condition becomes sufficient when the state is one-dimensional. To the best of our knowledge, the theoretical results obtained in this paper are new although the proofs are trivial. But they give an easy to check condition as it is shown by the study of some typical examples, in which we do much less computation than the involved litterature to prove identifiability properties.Index Terms-Nonlinear autonomous system, power series expansion, structural identifiability.