KKL Observer Synthesis for Nonlinear Systems via Physics-Informed Learning

TL;DR

This paper introduces a physics-informed neural network approach to synthesize KKL observers for nonlinear systems, providing robustness guarantees and demonstrating superior generalization in simulations.

Contribution

It presents a novel neural network-based method for designing KKL observers, combining physics-informed learning with inverse mapping, and offers theoretical robustness guarantees.

Findings

Robust state estimation with approximation error bounds

Superior generalization outside training domain

Effective in benchmark nonlinear systems

Abstract

This paper proposes a novel learning approach for designing Kazantzis-Kravaris/Luenberger (KKL) observers for autonomous nonlinear systems. The design of a KKL observer involves finding an injective map that transforms the system state into a higher-dimensional observer state, whose dynamics is linear and stable. The observer's state is then mapped back to the original system coordinates via the inverse map to obtain the state estimate. However, finding this transformation and its inverse is quite challenging. We propose learning the forward mapping using a physics-informed neural network, and then learning its inverse mapping with a conventional feedforward neural network. Theoretical guarantees for the robustness of state estimation against approximation error and system uncertainties are provided, including non-asymptotic learning guarantees that link approximation quality to finite sample sizes. The effectiveness of the proposed approach is demonstrated through numerical simulations on benchmark examples, showing superior generalization capability outside the training domain compared to state-of-the-art methods.