Neural Network-Based KKL observer for nonlinear discrete-time systems

For non-autonomous multivariable discrete-time nonlinear systems, we address the state estimation problem using a Kazantzis-Kravaris-Luenberger (KKL) observer. We aim to build a mapping that transforms a nonlinear dynamics into a stable linear system modulo an output injection and to design an asymptotic observer. However, this mapping is difficult to compute and its numerical approximation may be badly conditioned during the transient phase. We propose an algorithm based on ensemble learning techniques to improve the numerical approximation of the mapping and its extension in the transient phase. This ensures a good asymptotic convergence of the observer and avoids peaking phenomena. The algorithm demonstrates good performance in high-dimensional and multi-input-multi-output examples.