On integrability of observable space for discrete-time analytic systems

The goal of the paper is to extend the results on the observability decomposition of continuous-time nonlinear systems into the discrete-time nonlinear systems. Unlike the continuous-time case, the observable space related to the concept of single-experiment observability of the discrete-time nonlinear systems cannot always be locally spanned by exact one-forms whose integrals would define the observable state coordinates. For the special subclass of reversible polynomial systems, it has been shown that the observable space is integrable. In this paper, we extend this fact to the reversible analytic systems, which implies that the observability decomposition can always be achieved for the reversible analytic system. Moreover, by combining the observability decomposition and the conventional accessibility decomposition, we extend the Kalman decomposition in the linear case.