Sampled-Data Control of the Stefan System
This paper presents results for the sampled-data boundary feedback control to the Stefan problem. The Stefan problem represents a liquid-solid 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  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 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 . Numerical simulation illustrates the desired performance of the control law implemented to vary at each sampling time and keep constant during the period.
UCSD]Shumon Koga, iasson]Iasson Karafyllis, UCSD]Miroslav Krstic
Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411 USA
Department of Mathematics, National Technical University of Athens, 15780 Athens, Greece
Key words: Sampled-data system, Stefan problem, moving boundaries, distributed parameter systems, nonlinear stabilization
Liquid-solid phase transitions are physical phenomena which appear in various kinds of science and engineering processes. Representative applications include sea-ice melting and freezing , continuous casting of steel , cancer treatment by cryosurgeries , additive manufacturing for materials of both polymer  and metal , crystal growth , lithium-ion batteries , and thermal energy storage systems . Physically, these processes are described by a temperature profile along a liquid-solid material, where the dynamics of the liquid-solid interface is influenced by the heat flux induced by melting or solidification. A mathematical model of such a physical process is called the Stefan problem, which is formulated by a diffusion PDE defined on a time-varying spatial domain. The domain’s length dynamics is described by an ODE dependent on the gradient of the PDE state. Apart from the thermodynamical model, the Stefan problem has been employed to model several chemical, electrical, social, and financial dynamics such as tumor growth process , domain walls in ferroelectric thin films , spreading of invasive species in ecology , information diffusion on social networks , and optimal exercise boundary of the American put option on a zero dividend asset .
While the numerical analysis of the one-phase Stefan problem is broadly covered in the literature, their control related problems have been rarely addressed. In addition to it, most of the proposed control approaches are based on finite dimensional approximations with the assumption of an explicitly given moving boundary dynamics [8, 2]. For control objectives, infinite-dimensional approaches have been used for stabilization of the temperature profile and the moving interface of a 1D Stefan problem, such as enthalpy-based feedback  and geometric control . These works designed control laws ensuring the asymptotical stability of the closed-loop system in the norm. However, the results in  are established based on the assumptions on the liquid temperature being greater than the melting temperature, which must be ensured by showing the positivity of the boundary heat input.
Recently, boundary feedback controllers for the Stefan problem have been designed via a “backstepping transformation” [31, 41] which has been used for many other classes of infinite-dimensional systems. For instance,  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 norm without imposing any a priori assumption. Based on the technique,  designed an observer-based output feedback control law for the Stefan problem,  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,  designed a state feedback control for the Stefan problem under the material’s convection,  developed a control design with time-delay in the actuator and proved a delay-robustness,  investigated an input-to-state stability of the control of Stefan problem with respect to an unknown heat loss at the interface, and  developed a control design for the two-phase Stefan problem.
The aforementioned results assumed the control input to be varying continuously in time; however, in practical implementation of the control systems it is impossible to dynamically change the control input continuously in time due to limitations of the sensors, actuators, and software. Instead, the control input can be adjusted at each sampling time at which the measured states are obtained or the actuator is manipulated. One of the most fundamental and well known method to design such a “sampled-data” control is the so-called “emulation design” that applies “Zero-Order-Hold” (ZOH) to the nominal “continuous-time” control law. A general result for nonlinear ODEs to guarantee the global stability of the closed-loop system under such a ZOH-based sampled-data control was studied in , and the sampled-data observer design under discrete-time measurement is developed in  by introducing inter-sampled output predictor. As further extensions, the stability of the sampled-data control for general nonlinear ODEs under actuator delay is shown in [18, 19] by applying predictor-based feedback developed in , and results for a linear parabolic PDE are given in  by employing Sturm-Liouville operator theory. The sampled-data control for parabolic PDEs has been intensively developed by Fridman and coworkers by utilizing linear matrix inequalities [1, 12, 13, 39]. However, none of the existing work on the sampled-data control has studied the class of the Stefan problem described by a parabolic PDE with state-dependent moving boundaries “(a nonlinear system)”.
1.2 Contributions and results
This paper presents the first theoretical result for the sampled-data boundary feedback control for the Stefan problem. The approach employed in this paper is distinct from the methodology developed in literature. Namely, we solve the growth of the system’s energy analytically in time under the proposed sampled-data feedback control that is in the form of an energy-shaping design. Then, a perturbation that is incorporated in the closed-loop system due to the error between the continuous-time design and the sampled-data design can be represented analytically, and the closed-loop stability is proven by using Lyapunov method.
First, we consider the one-phase Stefan problem by assuming that the solid phase temperature is maintained at the melting temperature and focusing on the single melting process. We employ ZOH to the nominal continuous-time feedback controller for the one-phase Stefan problem developed in , and prove the required conditions for the model validity and the global exponential stability of the closed-loop system under explicit conditions for the setpoint position and the control gain with respect to the sampling scheduling. Next, we consider the two-phase Stefan problem by incorporating the dynamics of the solid phase temperature and prove the analogous results for the sampled-data control for the two-phase Stefan problem. The results established in this paper hold for arbitrary sampling schedules, and not necessarily uniform sampling schedules.
The mathematical model the one-phase Stefan problem for a single phase change is presented in Section 2 with stating some important properties. The sampled-data control law and the stability proof of the closed-loop system is given in Section 3. The extension of the presented procedure to the two-phase Stefan problem is described in Section 4. The numerical simulation of the proposed control law is provided in Section 5. The paper ends with the concluding remarks in Section 6.
2 Description of the One-Phase Stefan Problem
Consider a physical model which describes the melting or solidification mechanism in a pure one-component material of length in one dimension. In order to mathematically describe the position at which phase transition occurs, we divide the domain into two time-varying sub-domains, namely, the interval which contains the liquid phase, and the interval that contains the solid phase. A heat flux enters the material through the boundary at (the fixed boundary of the liquid phase) which affects the liquid-solid interface dynamics through heat propagation in liquid phase. As a consequence, the heat equation alone does not provide a complete description of the phase transition and must be coupled with the dynamics that describes the moving boundary. This configuration is shown in Fig. 1.
Assuming that the temperature in the liquid phase is not lower than the melting temperature of the material , the energy conservation and heat conduction laws yield the heat equation of the liquid phase as follows
with the boundary conditions
and the initial values
where , , , , and are the distributed temperature of the liquid phase, the manipulated heat flux, the liquid density, the liquid heat capacity, and the liquid heat conductivity, respectively. Moreover, the local energy balance at the liquid-solid interface yields
where represents the latent heat of fusion.
There are two underlying assumptions to validate the model (2)-(2). First, the liquid phase is not frozen to the solid phase from the boundary . This condition is ensured if the liquid temperature is greater than the melting temperature. Second, the material is not completely melt or frozen to single phase through the disappearance of the other phase. This condition is guaranteed if the interface position remains inside the material’s domain. In addition, these conditions are also required for the well-posedness (existence and uniqueness) of the solution in this model. Taking into account of these model validity conditions, we emphasize the following remark.
Based on the above conditions, we impose the following assumption on the initial data.
, for all , and is continuously differentiable in .
3 Sampled-Data Control for the One-Phase Stefan Problem
3.1 Problem statement and main result
The steady-state solution of the system (2)–(2) with zero manipulating heat flux yields a uniform melting temperature and a constant interface position given by the initial data. In , the authors developed the exponential stabilization of the interface position at a desired reference setpoint through the design of as
where is the controller gain. However, in practical implementation, the actuation value cannot be changed continuously in time. Instead, by obtaining the measured value as signals discretely in time, the control value needs to be implemented at each sampling time. One of the most typical design for such a sampled-data control is the application of ”Zero-Order-Hold”(ZOH) to the nominal continuous time control law. Through ZOH, during the time intervals between each sampling, the control maintains the value at the previous sampling time. Let be the -th sampling time for , and be defined by
The application of ZOH to the nominal control law (3.1) leads to the following design for the sampled-data control
of which the right hand side is constant during the time interval . Let us denote for . Hereafter, all the variables with subscript denote the variables at . First, we introduce the following assumptions on the setpoint and the sampling scheduling.
The setpoint is chosen to verify
The sampling schedule has a finite upper diameter and a positive lower diameter, i.e., there exist constants such that
Our main theorem is given next.
Consider the closed-loop system (2)–(2), (2), (3.1) under Assumptions 1, 2. Then for every , there exists a constant for which the following property holds: for every sequence with for which Assumption 3 holds, the initial-boundary value problem (2)–(2) with (3.1) has a unique solution satisfying (2), (2) as well as the following estimate:
where , for all , in the norm .
3.2 Some key properties of the closed-loop system
We first provide the following lemma.
PROOF. We introduce the following reference error states:
Define the internal energy of the reference error system as follows:
Noting that is constant for as under ZOH-based sampled-data control, taking the integration of (25) from to yields
which leads to the explicit solution as follows:
Substituting (29) into (27) yields as (17). Therefore, the closed-loop system under the sampled-data feedback control (3.1) is equivalent to the open-loop solution with the control input (17). Moreover, under Assumptions 2, 3, and the fact that , the input (17) is shown to be a bounded piecewise continuous function and for all . Thus, the existence and uniqueness of the solution is ensured by Lemma 1, from which we conclude Lemma 2.
The closed-loop system satisfies the following properties:
which leads to for all .
3.3 Stability analysis
To conclude Theorem 1, this section is devoted to the stability proof of the closed-loop system under the designed sampled-data control law. First, we introduce the backstepping transformation developed in  for the continuous-time design, and apply the transformation to the closed-loop system under the sampled-data control in this paper.
3.3.1 State transformation
Introduce the following backstepping transformation
which maps into
The objective of the transformation (3.3.1) is to add a stabilizing term in (41) of the target -system which is easier to prove the stability than -system. By taking the derivative of (3.3.1) with respect to and respectively, to satisfy (38), (40), (41), we derive the conditions on the gain kernel solution, and they leads to the following solution:
By taking the derivative of the transformation (3.3.1) in and substituting , we have
where is an explicit function in time defined by
The closed form representation of (45) using variables is given after the inverse transformation is obtained in the next section.
3.3.2 Inverse transformation
Consider the following inverse transformation
where , , , and is to be chosen later. Finally, using the inverse transformation, the boundary condition (39) is rewritten as
3.3.3 Lyapunov method
and we firstly apply Lyapunov method for the time interval for all , and next for the interval from to . For both cases, we consider the following functional
Applying Young’s inequality to the second line of (3.3.3) twice, we get
where . Since and for all , there exists such that for and . Thus, setting , the inequality (3.3.3) leads to
Consider the following functional
(i) For , for all
Applying comparison principle to (61) for leads to
Setting and recalling , we get
where , and is defined by
where is defined by
Applying (66) from to inductively, we get
Since , by using given in Assumption 3, the following inequality holds
Thus, the inequality (69) leads to
In the similar way, we get
(ii) For ,
Applying comparison principle to (61) from to , we get
We consider the -norm of -system defined by
Due to the invertibility of the transformation from to together with the boundedness of the domain , there exist positive constants and such that the following inequalities hold:
Moreover, due to the definition of the reference energy given in (24), using Young’s and Cauchy Schwarz inequalities one can show that
which completes the proof of Theorem 1.
4 Sampled-Data Design for Two-Phase Stefan Problem
In this section, we extend the results we have established in the previous section to the ”two-phase” Stefan problem, where the temperature dynamics in the solid phase is governed by the heat equation with different physical parameters from the liquid phase, following the work in . This configuration is depicted in Fig. 2.
4.1 Problem statement
The governing equations are descried by the following coupled PDE-ODE-PDE system:
where , and all the variables denote the same physical value with the subscript ”l” for the liquid phase and ”s” for the solid phase, respectively. The solid phase temperature must be lower than the melting temperature, which serves as one of the conditions for the model validity, as stated in the following remark.
To keep the physical state of each phase meaningful, the following conditions must be maintained: