Sampled-Data Control of the Stefan System

Abstract

This paper presents results for the sampled-data boundary feedback control to the Stefan problem. The Stefan problem represents a liquidsolid phase change phenomenon which describes the time evolution of a material's temperature profile and the interface position. First, we consider the sampled-data control for the one-phase Stefan problem by assuming that the solid phase temperature is maintained at the equilibrium melting temperature. We apply Zero-Order-Hold (ZOH) to the nominal continuous-time control law developed in [23] which is designed to drive the liquid-solid interface position to a desired setpoint. 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.

Partial differential equation modelling of the solid–liquid phase process in iron sintering and interface control with backstepping method

“…25 Stefan problem describes a liquid-solid phase change phenomenon which describes the time evolution of the temperature distribution and the location of the interface of a material. 26 This physical process is mathematically represented as a diffusive PDE, which evolves over a timevarying spatial domain described by an ordinary differential equation (ODE). 27 Many studies of Stefan-type problems have been carried out for the one-and two-phase cases.…”

Control of the Stefan System and Applications: A Tutorial

“…This review has summarized recent results on backstepping control of the Stefan system and provided a tutorial introduction to the basic design ideas and representative applications. In addition to the topics discussed in this article, we have also developed several other problems, such as the state observer of the Stefan system (48) and its applications to polar ice (54) and lithium-ion batteries (55), control design in the presence of actuator delay (49), robustness analysis by input-to-state stability (121), stability analysis with sampled-data control (51), an output feedback control for the Stefan problem with advection with applications to screw extruder-based 3D printing (52), and experimental validation by melting paraffin (56).…”

“…The initial article (48) introduced the designs of a state feedback control law, an observer, and an output feedback law for the one-phase Stefan system by proposing a nonlinear backstepping transformation for the moving-boundary Stefan PDE; it proved the exponential stability of the closed-loop system without imposing the a priori assumption that the temperature state respects the phase constraints, instead showing that such constraints are actually maintained under proposed feedback. Extensions have been provided in articles that developed a control design with time delay in the actuator and proved delay robustness (49), developed a control design for the twophase Stefan problem (50), and showed the stability of the closed-loop system under sampled-data control (51). The backstepping controller and estimator for the Stefan system have been successfully applied to screw extruder-based polymer 3D printing (52), laser sintering-based metal 3D printing (53), polar ice in the Arctic (54), lithium-ion batteries (55), and energy storage by paraffin with experimental validation (56).…”