Non-linear estimators for the observation and stabilisation of falling liquid films

TL;DR

This paper develops a non-linear estimator coupled with feedback control to stabilize falling liquid films using limited measurements, achieving near full-information control performance at moderate Reynolds numbers.

Contribution

It introduces a non-linear estimator framework for stabilizing liquid films with limited observations, enhancing control performance compared to linear estimators.

Findings

Estimator restores control performance close to full-information scenarios.

Effective stabilization achieved at moderate Reynolds numbers.

Actuator and observer placement influence system dynamics.

Abstract

Falling liquid films are complex physical processes modelled by infinite-dimensional dynamical systems. The interfacial behaviour is characterised by the Reynolds number, a nondimensional parameter, and above a critical value the uniform film becomes unstable and travelling waves occur. By injecting and removing fluid from the base at discrete locations, we aim to stabilise otherwise unstable flat interfaces, with the additional limitation that observations of the state are restricted to finitely many measurements of the film height. Successful feedback control has recently been achieved in the case of full observations using linear-quadratic regulator controls coupled to asymptotic approximations of the Navier-Stokes equations, but restricted observations severely curtailed their performance. In this study we couple the well understood full-information feedback control strategy to a non-linear estimator. The dynamics of the estimator are designed to approximate those of the film, and we apply a forcing term chosen to ensure that measurements of the estimator match the available measurements of the film. Using this method, we restore the performance of the controls to a level approaching their full information counterparts, even at moderately large Reynolds numbers. We also briefly investigate the effects of the relative positioning of actuators and observers on the resulting dynamics.