Identifying necessary and sufficient conditions for the observability of models of biochemical processes

The notions of observability and controllability of non-linear systems are a cornerstone of mathematical control theory and cover a wide scope of applications including process design, characterization, monitoring and control. Synthetic biology - which aims to (re)-program living functionalities - and bio-based process engineering - which aims to develop biotechnological manufacturing processes based on industrial and natural living agents - remarkably benefit of methodological improvements inspired to control theory for countless reasons including the huge variety of control mechanisms in living organisms, experimental limitations in terms of measurement feasibility, design of controllers - at single cell or population level - of synthetic production processes and process optimization purposes. Many fundamental problems of control theory such as stabilisability of unstable systems and optimal control may be solved under the assumption that the system is observable/controllable. Observability and controllability are mathematical duals, that means that the observability property can be determined analysing the controllability of the dual system and vice versa. Given this duality, we focus on observability. In this work, we revisit a generalization of the Fujisawa and Kuh theorem as a tool to explore the possibility that a system is observable. We show that the theorem, when applicable, is a sufficient but not necessary condition for observability. We revisit the theorem to propose a necessary and sufficient condition for observability for non-linear systems. Finally, we show how it is possible to identify regions of the phase space of the model in which the model is observable.