State Estimation of the Stefan PDE: A Tutorial on Design and Applications to Polar Ice and Batteries

TL;DR

This paper introduces a PDE backstepping observer for the Stefan PDE system, enabling accurate state estimation of phase change phenomena like ice melting and battery charge levels using boundary measurements.

Contribution

It develops a novel PDE backstepping observer design for nonlinear PDE-ODE coupled systems with moving boundaries, demonstrated on climate and battery models.

Findings

Robust state estimation for polar ice models despite salinity effects.

Over 15% reduction in State-of-Charge error within 5 minutes in battery models.

Effective boundary measurement-based estimation for phase change systems.

Abstract

The Stefan PDE system is a representative model for thermal phase change phenomena, such as melting and solidification, arising in numerous science and engineering processes. The mathematical description is given by a Partial Differential Equation (PDE) of the temperature distribution defined on a spatial interval with a moving boundary, where the boundary represents the liquid-solid interface and its dynamics are governed by an Ordinary Differential Equation (ODE). The PDE-ODE coupling at the boundary is nonlinear and creates a significant challenge for state estimation with provable convergence and robustness. This tutorial article presents a state estimation method based on PDE backstepping for the Stefan system, using measurements only at the moving boundary. PDE backstepping observer design generates an observer gain by employing a Volterra transformation of the observer error state into a desirable target system, solving a Goursat-form PDE for the transformation's kernel, and performing a Lyapunov analysis of the target observer error system. The observer is applied to models of problems motivated by climate change and the need for renewable energy storage: a model of polar ice dynamics and a model of charging and discharging in lithium-ion batteries. The numerical results for polar ice demonstrate a robust performance of the designed estimator with respect to the unmodeled salinity effect in sea ice. The results for an electrochemical PDE model of a lithium-ion battery with a phase transition material show the elimination of more than 15 \% error in State-of-Charge estimate within 5 minutes even in the presence of sensor noise.