The algebraic theory of nonlinear observability revisited

The algebraic theory of nonlinear observability is revisited by first enlarging the class of systems previously considered by the author (1991). We also provide new insights in the singularity of the notion of observability. The basic geometric picture of observability is that of a projection map of the system trajectories onto the data trajectories. Observability is then seen as the condition for this map to be finite in the sense that its fibres are generically finite. It is this definition which easily passes to differential algebraic geometric characterizations. Next, most of the picture is kept by defining the notion of singular observation data, i.e. the special data for which the generic observability property is lost. In this paper, we present as many elements of this theory as possible