Solid Boundary Output Feedback Control of the Stefan Problem: The Enthalpy Approach

Abstract

By taking enthalpy-an internal energy of a diffusion-type system-as the system state and expressing it in terms of the temperature profile and the phase-change interface position, the output feedback boundary control laws for a fundamentally nonlinear single-phase one-dimensional (1-D) PDE process model with moving boundaries, referred to as the Stefan problem, are developed. The control objective is tracking of the spatiotemporal temperature and temporal interface (solidification front) trajectory generated by the reference model. The external boundaries through which temperature sensing and heat flux actuation are performed are assumed to be solid. First, a full-state single-sided tracking feedback controller is presented. Then, an observer is proposed and proven to provide a stable full-state reconstruction. Finally, by combining a full-state controller with an observer, the output feedback trajectory tracking control laws are presented and the closed-loop convergence of the temperature and the interface errors proven for the single-sided and the two-sided Stefan problems. Simulation shows the exponential-like trajectory convergence attained by the implementable smooth bounded control signals.

Keywords: Control; Stefan problem; enthalpy; nonlinear partial differential equations; solidification.

Fig. 1.

Schematic of one-dimensional (1-D) Stefan problem.

Fig. 2.

Relationship between enthalpy $h$ and temperature $T$.

Fig. 3.

Block diagram for the control law with the full-state enthalpy feedback and its calculations in terms of temperature and solidification front position.

Fig. 4.

Initial condition for simulations.

Fig. 5.

Reference temperature $\bar{T}(x,t)$ and solidification front position $\bar{s}\left(t\right)$ used in the trajectory tracking simulations.

Fig. 6.

Simulation of the open-loop system (1)–(4) with initial condition mismatch shown in Fig. 4 and $u\left(t\right)=\bar{u}\left(t\right)$. (a) Reference temperature error $\bar{T}(x,t)$. (b) Solidification front $s\left(t\right)$.

Fig. 7.

Simulation of system (1)–(4) with initial condition mismatch in Fig. 4 under feedback control law (29). (a) Boundary control $u\left(t\right)$ from (29). (b) Solidification front $s\left(t\right)$. (c) Reference temperature error T(x, t).

Fig. 8.

Illustration of the contradiction for case ii) in Theorem 3.1. Since it is assumed that $\parallel T-\hat{T}{\parallel }_{\mathrm{\infty }}=0,\hat{T}$ cannot assume the form of ${\hat{T}}_{\text{nat}}$, and is forced to assume the shape of ${\hat{T}}_{\text{forced}}$, which implies the existence of a phase change interface at $s$ and $\hat{s}$. This contradicts the uniform convergence argument, and if it were to exist, it would imply $\dot{\hat{s}}<0$.

Fig. 9.

Simulation of system (1)–(4) with initial condition mismatch and output feedback control law (56) with estimator (40)–(43). (a) Boundary control $u\left(t\right)$ from (56). (b) Temperature estimation error $T-\bar{T}(x,t)$. (c) Solidification front $s\left(t\right)$. (d) Temperature reference error $\bar{T}(x,t)$.

Fig. 10.

Schematic of the two-sided Stefan problem.

Fig. 11.

Initial condition for simulation with symmetric enthalpy error.

Fig. 12.

Temperature error $\tilde{T}$ for two-sided Stefan problem (62)–(65), with initial conditions from Fig. 11 and different boundary control. (a) Using Neumann boundary control (80). (b) Using Neumann boundary control (75).

Fig. 13.

Block diagram for the proposed two-sided Stefan problem control law (90).