On parameter and structural identifiability: Nonunique observability/reconstructibility for identifiable systems, other ambiguities, and new definitions

Abstract
A number of inconsistencies, misunderstandings and ambiguities have arisen in the recent literature regarding identifiability concepts, evidenced in part by the number of "point-counterpoint" notes and comments published on this subject. It appears that the problem is due fundamentally to a lack of a unified set of definitions. A partially new, simple and hopefully complete set of definitions is proposed here for deterministic models. They are based on an extended model, termed the "constrained structure"-appropriate for complete analysis of identifiability properties of a system. They rely minimally on special jargon and are consistent with reported definitions for notions of stochastic identifiability. Relationships with other definitions are discussed and it is also shown that an important class of identifiable models are not necessarily uniquely observable/reconstructible. The latter problem has important implications in applications.