Observer design for implicit state affine systems up to output injection

This paper deals with the problem of observers design for implicit state affine systems for which the observer synthesis depends on the input excitation. Two approaches can be used to solve this problem; The first one consists in designing an observer which requires solving at each time an algebraic equation combined with an ODE system (implicit observer). The second approach proposed in this paper consists in designing an observer described by differential equations only (explicit observer). Moreover, it can be initialized outside the algebraic constraints. A general methodology which guaranties the existence of an explicit observer is presented.