Single-boundary control of the two-phase Stefan system

Abstract

This paper presents the control design of the two-phase Stefan problem. The two-phase Stefan problem is a representative model of liquid-solid phase transition by describing the time evolutions of the temperature profile which is divided by subdomains of liquid and solid phases as the liquid-solid moving interface position. The mathematical formulation is given by two diffusion partial differential equations (PDEs) defined on a time-varying spatial domain described by an ordinary differential equation (ODE) driven by the Neumann boundary values of both PDE states, resulting in a nonlinear coupled PDE-ODE-PDE system. We design a state feedback control law by means of energy-shaping to stabilize the interface position to a desired setpoint by using single boundary heat input. We prove that the closed-loop system under the control law ensures some conditions for model validity and the global exponential stability estimate is shown in L 2 norm. Furthermore, the robustness of the closed-loop stability with respect to the uncertainties of the physical parameters is shown. Numerical simulation is provided to illustrate the good performance of the proposed control law in comparison to the control design for the one-phase Stefan problem.

Controllability of One-Dimensional Viscous Free Boundary Flows

“…As a result, in terms of the adjoint problem, one obtains non-local boundary conditions (similar to (5.3)), for which observability inequalities are lacking. We also refer to Dunbar et al [10,9] for motion planning and flatness control, and Krstic et al [21,22,23,24] and the references therein for feedback stabilization via backstepping design of the Stefan problem (1.4), see also Phan & Rodrigues [31] for stabilization to trajectories for general parabolic problems.…”

Sampled-Data Control of the Stefan System

“…For instance, [21] designed a state feedback control law by introducing a nonlinear backstepping transformation for moving boundary PDE, which achieved the exponentially stabilization of the closed-loop system in the H 1 norm without imposing any a priori assumption. Based on the technique, [22] designed an observer-based output feedback control law for the Stefan problem, [23] extended the results in [21,22] by studying the robustness with respect to the physical parameters and developed an analogous design with Dirichlet boundary actuation, [24] designed a state feedback control for the Stefan problem under the material's convection, [27] developed a control design with time-delay in the actuator and proved a delay-robustness, [29] investigated an input-to-state stability of the control of Stefan problem with respect to an unknown heat loss at the interface, and [30] developed a control design for the two-phase Stefan problem.…”

“…Provided that the control gain is bounded by the inverse of the upper diameter of the sampling schedule, we prove that the closed-loop system under the sampled-data control law satisfies some conditions required to validate the physical model, and the system's origin is globally exponentially stable in the spatial L 2 norm. Analogous results for the two-phase Stefan problem which incorporates the dynamics of both liquid and solid phases with moving interface position are obtained by applying the proposed procedure to the nominal control law for the two-phase problem developed in [30]. Numerical simulation illustrates the desired performance of the control law implemented to vary at each sampling time and keep constant during the period.…”

Safe PDE Backstepping QP Control with High Relative Degree CBFs: Stefan Model with Actuator Dynamics

“…In this section, we extend the safety design to the "twophase" Stefan problem, where the interface dynamics is affected by a heat loss modeled by the temperature dynamics in the solid phase, following the work in [30]. Additionally, the solid phase temperature is affected by a heat loss caused at the end boundary of the solid phase, which serves as a disturbance in the system.…”