Differentially algebraic immersion of nonlinear systems into rational-in-the-state representations

A system immersion is a mapping of an initial state such that two different systems have an identical input-output mapping. This paper considers a particular class of system immersion, called differentially algebraic (DA) immersion, which is suitable to investigate geometric characteristics of a system after immersion. It is shown that a given system is DA immersible into a polynomial-in-the-state representation (PSR) if and only if it is so into a rational-in-the-state representation (RSR). Then, necessary and sufficient conditions are given for DA immersibility into a RSR in terms of differential algebraic structure of the state equation. Some examples are given to highlight differences between related theoretical results.