  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.
