The Construction of Nonlinear Discrete-time Observers with Local Lipschitz Conditions

In this paper, we present a new methodology for the construction of discrete-time observers for nonlinear discrete-time systems which satisfy Lipschitz condition on bounded sets. The problem under consideration is mathematically addressed through the existence of a continuous mapping in the state space that maps the linear system with an output injection to the observing system provided that the observing orbit is bounded. The existence of such a continuous mapping is proved by Banach's fixed point theorem by making use of a cut-off function, which is used to construct the observer. The proposed method can be directly carried over to the continuous-time cases. An example for discrete-time Hénon system with non-smooth output is provided to illustrate the theoretical outcomes and to show the effectiveness of the proposed observer.