16 Remarks on the observability of nonlinear discrete time systems Francesca Albertini Dipartimento di M atematica Via Belzoni 7, 35131 Padova, Italy. Tel: +39(49)8275966. e-mail: albertini@pdmatl.math.unipd.it Domenico D'Alessandro Universita' di Padova, Dipartimento di Elettronica ed Informatica Via Gradenigo 6a, 35100 Padova, Italy. Tel: +39(49)8287791. e-mail: daless@maya.dei.unipd.it Abstract The observability of discrete time nonlinear systems is studied. Criteria of observability are given in terms of codistributions. This leads naturally to decompositions similar to the ones known in the continuous time case. Some observability properties of invertible systems are also investigated. In particular, it is shown that, under regularity hypotheses, the weaker notion of forward-backward observability is equivalent to the one of (forward) observability, for these systems. Keywords Discrete-time, nonlinear systems, observability. 1 INTRODUCTION We deal with observability questions for nonlinear discrete-time systems of the form x(t + 1) y(t) f(x(t), u(t)), t == 0, 1,2, ... h(x(t)). We consider single input single output systems, since the general case involves only no- tational changes. In we assume that x(t) E M,y(t) E Y and u(t) E U, with M and Y connected, second countable, Hausdorff, differentiable manifolds, of dimensions nand 1, respectively. We also assume that the control space U is an open interval of 1R, such that o E U. Such a system is said to be of class Ck, if the manifolds M and Yare of class Ck, and the functions f: M x U ---> M and h: M ---> Y, are of class Ck. We shall often use the abbreviated notation fu(x):= f(x,u). J. Doležal et al. (eds.), System Modelling and Optimization © Springer Science+Business Media Dordrecht 1996