Learning a Contracting KKL-observer with Local Optimal Guarantees

TL;DR

This paper introduces a deep learning-based method to learn KKL observers with stability guarantees and local optimality, improving nonlinear state estimation performance.

Contribution

It develops a neural network approach to optimize latent dynamics in KKL observers, ensuring contraction and local optimality guarantees.

Findings

Neural networks can effectively approximate KKL transformations.

The method achieves robust state estimation under noise.

Numerical simulations validate improved performance.

Abstract

The Kazantzis-Kravaris-Luenberger (KKL) observer provides a general framework for nonlinear state estimation by immersing the system dynamics into a stable linear or nonlinear latent dynamics. However, the performance of KKL observers relies heavily on the specific choice of these latent dynamics, which is often heuristic. This paper proposes a methodology to learn a KKL observer that combines global stability guarantees with local optimality. We derive a condition on the latent dynamics such that the observer locally mimics the behavior of a Minimum Energy Estimator (Mortensen observer). We then employ Deep Learning to approximate the KKL transformation and the latent dynamics, using neural network architectures that structurally enforce the contraction property. The proposed strategy is validated through numerical simulations on nonlinear benchmarks, demonstrating a good performance in the presence of state and measurement noise.