Deep Learning-based Luenberger observer design for discrete-time nonlinear systems

In this paper we address the problem of observer design for nonlinear discrete-time systems. Combining the theory of so-called Kazantzis–Kravaris-Luenberger (KKL) observers and Deep Learning, we aim to identify the mapping which transforms a nonlinear dynamics to a stable linear system modulo an output injection and design an asymptotic discrete-time observer. The proposed approach leverages the power of Machine Learning to provide an algorithm based on an unsupervised learning of the mapping, which allows to properly explore the state space.The approach is illustrated on two examples of the autonomous case and two of the non-autonomous one. These examples have been taken from the literature and judiciously chosen to compare the proposed approach with existing results.