Commutativity of immersion and linearization

As recently shown it is possible to represent a given nonlinear system via immersion as rational or polynomial functions leading to a simplified model structure. An immersion is a mapping of the initial state from the original state space to another state space, while exactly preserving the input- output map. This paper shows that linearization of the system after immersion has the identical input-output map as the linearization of the original system before immersion. In other words, immersion and linearization commute. This is potentially useful for, e.g., sensitivity analysis, linear control design after nonlinear identification, and has important implications for system approximation by linearization.