Observability and forward-backward observability of discrete-time nonlinear systems

In this paper we study the observability properties of nonlinear discrete-time systems. Two types of contributions are given. First we present observability criteria in terms of appropriate codistributions. For particular, but significant, classes of systems we provide criteria that require only a finite number of computations. Then we consider invertible systems (which includes discrete-time models obtained by sampling continuous-time systems) and prove that the weaker notion of forward-backward observability is equivalent to the stronger notion of (forward) observability