Enthalpy-based Full-State Feedback Control of the Stefan Problem with Hysteresis* Zhelin Chen, Student Member, IEEE, Joseph Bentsman, Senior Member, IEEE,, and Brian G. Thomas Abstract—This paper presents a full–state controller design with respect to a reference solution for the one-phase Stefan problem under input hysteresis. The setting models an industrial casting processes with water cooling hysteresis under Neumann boundary actuation. The control law proposed ensures expo- nential stability of average enthalpy and is proven to provide asymptotic convergence of temperature error and solidification front error. A simulation supports the result. I. INTRODUCTION Consider a continuous steel casting solidification process. The latter is typically modeled by the Stefan problem [1] with moving boundary between liquid and solid phases. Transverse surface cracks may be created when the curved strand of steel undergoes unbending, which causes tensile stress on the inside radius surface, if the steel surface is too brittle. Since ductility of steel is strongly temperature-dependent, these cracks can be avoided by regulating the surface temperature outside of the embrittlement temperature range. The creation of internal cracks, usually found below the strand surface, depends on the relative location of the solidification front down the caster. Another defect, strand bulging past support rolls, called a ”whale”, damages the casting machine and causes a work stoppage. Hence, regulating the entire distributed temperature profile and solidification front of the steel [2] is required. The current industry-standard control method is open-loop control that changes the cooling water spray flow rates ac- cording to the casting speed and given spray patterns. The latter define the flowrates in each spray zone for each casting speed in the table, which depends on the steel grade, product dimensions, and machine design. The spray pattern can be determined by experience, or through offline optimization techniques [3]–[6]. In the control literature on the Stefan problem, there have been many techniques suggested. The approach used in [7] is to solve the inverse Stefan problem, i.e. impose a desired trajectory of the boundary and determine computationally a temperature profile. This would satisfy the whale constraints but could still result in temperature related cracks. In [8], a full-state feedback control law for a single- phase Stefan problem is designed by introducing a nonlinear backstepping transformation. Based on this technique, [9] designed an observer-based output feedback control law that achieved exponentially stability of the moving interface to a fixed reference setpoint. However, the convergence of a *This work was supported by the Continuous Casting Center at Colorado School of Mines, and NSF Award CMMI 1300907. 1 Zhelin Chen and Joseph Bentsman (corresponding author; e-mail: jbentsma@illinois.edu) are with the University of Illinois Urbana-Champaign, IL, USA. 2 Brian G. Thomas is with Colorado School of Mines, Golden, CO USA. moving boundary to a fixed solid/liquid interface is not enough for solidification process like the continuous casting of steel, since neither whale constraints nor cracks constraints are satisfied. These constraints require convergence to the moving interface profile. In [10], the authors control the position of the solidification front, neglecting surface cracks using thermostat- style boundary control inputs. In [11], the authors found a control law that ensured convergence of both temperature and solidification front position to the desired reference profile for a one-phase Stefan problem with practical assumptions on the actual casting process. No previous work has considered the phenomenon of hysteresis. In this paper, a free boundary problem with hysteresis type boundary conditions is considered. Hysteresis effects, which can be characterized as a special type of memory-based relation between an input signal v(t ) and an output signal w(t ), arises in different areas of science, such as physics, engineering, economics, and biology. There is an extensive body of research concerning the modeling of hysteresis [12], [13]. A generic approach to controlling hysteretic systems is to combine inverse compensation with feedback [14]– [16]. There are limited studies on control of Stefan problem with hysteresis. [10] considered a two-phase Stefan problem with hysteresis for a simple situation of thermostat control. Friedman [17] considered optimal control of the free boundary of a two-phase Stefan problem with hysteresis-type boundary conditions. Periodic control of a two-phase Stefan problem with Dirichlet boundary control with hysteresis was considered in [18]. However, all these previous works propose hysteresis effects on the free boundary, rather than at the surface where heat flux is applied. The existence of hysteresis [19] in a certain temperature in- terval, points to significance of considering the thermal history of actual cooling processes when designing controllers, which is the subject of the present work. This paper is organized as follows. The process model is presented in Section II, and the hysteresis model is given in Section III. The control objective and our main result, a control law that guarantees simultaneous asymptotic convergence of the temperature and solidification front location to the reference profile, are stated in Section IV. Supporting numerical simulations are provided in Section V. In Section VI, some extensions of this framework to observer and output feedback control are briefly discussed. II. PROCESS MODELS A. Single-phase Stefan Problem The continuous steel casting process can be modeled accu- rately using a one-dimensional spatial domain of a moving 1- 4035