# Control and State Estimation of the One-Phase Stefan Problem via Backstepping Design

## TL;DR

This paper introduces a backstepping-based control and observer design for the one-phase Stefan problem, enabling stable temperature regulation and interface estimation in phase transition processes modeled by coupled PDE-ODE systems.

## Contribution

It presents a novel backstepping approach for simultaneous control and state estimation of the nonlinear coupled PDE-ODE Stefan problem.

## Key findings

- Achieved global exponential stability of temperature and interface estimates.
- Designed an observer based on temperature measurements of the solid phase.
- Validated stability under Neumann and Dirichlet boundary conditions.

## Abstract

This paper develops a control and estimation design for the one-phase Stefan problem. The Stefan problem represents a liquid-solid phase transition as time evolution of a temperature profile in a liquid-solid material and its moving interface. This physical process is mathematically formulated as a diffusion partial differential equation (PDE) evolving on a time-varying spatial domain described by an ordinary differential equation (ODE). The state-dependency of the moving interface makes the coupled PDE-ODE system a nonlinear and challenging problem. We propose a full-state feedback control law, an observer design, and the associated output-feedback control law via the backstepping method. The designed observer allows estimation of the temperature profile based on the available measurement of solid phase length. The associated output-feedback controller ensures the global exponential stability of the estimation errors, the H1- norm of the distributed temperature, and the moving interface to the desired setpoint under some explicitly given restrictions on the setpoint and observer gain. The exponential stability results are established considering Neumann and Dirichlet boundary actuations.