Sampled-Data Observers for 1-D Parabolic PDEs with Non-Local Outputs

TL;DR

This paper develops systematic sampled-data observer designs for 1-D parabolic PDEs with non-local outputs, providing explicit conditions for exponential convergence and stability estimates even with noise and errors.

Contribution

It introduces two novel sampled-data observer methods for nonlinear parabolic PDEs with non-local outputs, including explicit convergence conditions and robustness analysis.

Findings

Explicit conditions for sampling schedule ensure exponential convergence.

Observer error estimates remain valid under measurement noise and modeling errors.

Method can be extended to boundary point measurements.

Abstract

The present work provides a systematic approach for the design of sampled-data observers to a wide class of 1-D, parabolic PDEs with non-local outputs. The studied class of parabolic PDEs allows the presence of globally Lipschitz nonlinear and non-local terms in the PDE. Two different sampled-data observers are presented: one with an inter-sample predictor for the unavailable continuous measurement signal and one without an inter-sample predictor. Explicit conditions on the upper diameter of the (uncertain) sampling schedule for both designs are derived for exponential convergence of the observer error to zero in the absence of measurement noise and modeling errors. Moreover, explicit estimates of the convergence rate can be deduced based on the knowledge of the upper diameter of the sampling schedule. When measurement noise and/or modeling errors are present, Input-to-Output Stability (IOS) estimates of the observer error hold for both designs with respect to noise and modeling errors. The main results are illustrated by two examples which show how the proposed methodology can be extended to other cases (e.g., boundary point measurements).