Immersion of nonlinear systems into linear systems modulo output injection

This paper deals with the problem of the immersion of a SISO system into a linear up to an output injection one. For this class of systems Luenberger-like observers can be designed: the dynamics of the error between the states of the observer and the immersed controlled system is linear. Necessary and sufficient conditions are first stated within a very-general framework. Effective computations and examples are then provided for uncontrolled and control-affine systems. In particular the control-affine case is completely (with a very slight restriction) solved.