Immersion of non-linear systems into linear systems up to output injection: characteristic equation approach

There exists a class of non-linear systems which cannot be transformed into non-linear observer form via diffeomorphism, but can be immersed into higher dimensional non-linear observer form. This paper investigates when n dimensional non-linear systems can be immersed into (n + m)-dimensional non-linear systems in non-linear observer form, and provides a necessary and sufficient condition. We also present an algorithm to check the immersibility. In general, at the first step of the algorithm, one should solve a differential equation with m + I unknowns rather than solving one with n + m unknowns simultaneously. When a solution to the differential equation can be obtained, the immersibility can be checked by a recursive algorithm. Moreover, when 2m less than or equal to n, a straightforward algorithm is provided to check the immersibility.