Neural Luenberger state observer for nonautonomous nonlinear systems

TL;DR

This paper introduces a data-driven, model-free neural Luenberger observer for nonlinear, nonautonomous systems, trained offline with neural networks, with proven error bounds and validated on industrial case studies.

Contribution

It extends the Kazantzis-Kravaris/Luenberger observer to nonautonomous systems using neural networks, providing theoretical guarantees and practical validation.

Findings

Neural networks can learn the nonlinear injective mapping of states.

The observer guarantees a bounded error when applied to new data.

Validation on bioreactor and Williams-Otto reactor demonstrates effectiveness.

Abstract

This work proposes a method for model-free synthesis of a state observer for nonlinear systems with manipulated inputs, where the observer is trained offline using a historical or simulation dataset of state measurements. We use the structure of the Kazantzis-Kravaris/Luenberger (KKL) observer, extended to nonautonomous systems by adding an additional input-affine term to the linear time-invariant (LTI) observer-state dynamics, which determines a nonlinear injective mapping of the true states. Both this input-affine term and the nonlinear mapping from the observer states to the system states are learned from data using fully connected feedforward multi-layer perceptron neural networks. Furthermore, we theoretically prove that trained neural networks, when given new input-output data, can be used to observe the states with a guaranteed error bound. To validate the proposed observer synthesis method, case studies are performed on a bioreactor and a Williams-Otto reactor.