Feedback control of the two-phase Stefan problem, with an application to the continuous casting of steel

Authors

Bryan Petrus, Joseph Bentsman, Brian G. Thomas

Type: Article

Publication Date: 2010-12-01

Citations: 33

DOI: https://doi.org/10.1109/cdc.2010.5717456

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Abstract

A full-state feedback control law is derived that stabilizes the two-phase Stefan problem with respect to a reference solution using control of the Neumann boundary condition. Stability and convergence are shown via a Lyapunov functional on the error system with moving boundaries. A second control law is also derived, for which stability is proved and convergence is conjectured due to the clearly convergent simulation results. A simple Dirichlet controller is also considered, and is used to design a boundary-output-based estimator that, in combination with full-state feedback controllers, yields a plausible output feedback control law with boundary sensing and actuation. The performance of the control laws is demonstrated using numerical simulation.

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2021 60th IEEE Conference on Decision and Control (CDC)
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Solid Boundary Output Feedback Control of the Stefan Problem: The Enthalpy Approach

2022-08-09

Bryan Petrus , Zhelin Chen , Hamza El-Kebir +2 more

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 … 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.

Control and State Estimation of the One-Phase Stefan Problem via Backstepping Design

2017-01-01

Shumon Koga , Mamadou Diagne , Miroslav Krstić

This paper develops a control and estimation design for the one-phase Stefan problem. The Stefan problem represents a liquid-solid phase transition as time evolution of a temperature profile in a … This paper develops a control and estimation design for the one-phase Stefan problem. The Stefan problem represents a liquid-solid phase transition as time evolution of a temperature profile in a liquid-solid material and its moving interface. This physical process is mathematically formulated as a diffusion partial differential equation (PDE) evolving on a time-varying spatial domain described by an ordinary differential equation (ODE). The state-dependency of the moving interface makes the coupled PDE-ODE system a nonlinear and challenging problem. We propose a full-state feedback control law, an observer design, and the associated output-feedback control law via the backstepping method. The designed observer allows estimation of the temperature profile based on the available measurement of solid phase length. The associated output-feedback controller ensures the global exponential stability of the estimation errors, the H1- norm of the distributed temperature, and the moving interface to the desired setpoint under some explicitly given restrictions on the setpoint and observer gain. The exponential stability results are established considering Neumann and Dirichlet boundary actuations.

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Single-boundary control of the two-phase Stefan system

2019-11-15

Shumon Koga , Miroslav Krstić

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Single-Boundary Control of the Two-Phase Stefan System

2019-01-01

Shumon Koga , Miroslav Krstić

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 … 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.

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Control and State Estimation of the One-Phase Stefan Problem via Backstepping Design

2018-01-01

Shumon Koga , Mamadou Diagne , Miroslav Krstić

This paper develops a control and estimation design for the one-phase Stefan problem. The Stefan problem represents a liquid-solid phase transition as time evolution of a temperature profile in a … This paper develops a control and estimation design for the one-phase Stefan problem. The Stefan problem represents a liquid-solid phase transition as time evolution of a temperature profile in a liquid-solid material and its moving interface. This physical process is mathematically formulated as a diffusion partial differential equation (PDE) evolving on a time-varying spatial domain described by an ordinary differential equation (ODE). The state-dependency of the moving interface makes the coupled PDE-ODE system a nonlinear and challenging problem. We propose a fullstate feedback control law, an observer design, and the associated output feedback control law of both Neumann and Dirichlet boundary actuations via the backstepping method. Also, the state-feedback control law is provided when a Robin boundary input is considered. The designed observer allows the estimation of the temperature profile based on the available measurements of liquid-phase length and the heat flux at the interface. The associated output feedback controller ensures the global exponential stability of the estimation errors, the H 1 -norm of the distributed temperature, and the moving interface at the desired setpoint under some explicitly given restrictions on the setpoint and observer gain.

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Control of Two-Phase Stefan Problem via Single Boundary Heat Input

2018-12-01

Shumon Koga , Miroslav Krstić

This paper presents the control design of the two-phase Stefan problem via a single boundary heat input. The two-phase Stefan problem is a representative model of liquid-solid phase transition by … This paper presents the control design of the two-phase Stefan problem via a single boundary heat input. 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. As an extension from our previous study on the one-phase Stefan problem, we design a state feedback control law to stabilize the interface position to a desired setpoint by employing the backstepping method. 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 L2 norm. Numerical simulation is provided to illustrate the good performance of the proposed control law.

Sampled-Data Control of the Stefan System

2019-01-01

Shumon Koga , Iasson Karafyllis , Miroslav Krstić

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 … 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 [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.

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Observer-based event-triggered boundary control of the one-phase Stefan problem

2024-02-07

Bhathiya Rathnayake , Mamadou Diagne

This paper presents an observer-based event-triggered boundary control strategy for the one-phase Stefan problem, utilising the position and velocity measurements of the moving interface. The design of the observer and … This paper presents an observer-based event-triggered boundary control strategy for the one-phase Stefan problem, utilising the position and velocity measurements of the moving interface. The design of the observer and controller is founded on the infinite-dimensional backstepping approach. To implement the continuous-time observer-based controller in an event-triggered framework, we propose a dynamic event triggering condition. This condition specifies the instances when the control input must be updated. Between events, the control input is maintained constant in a Zero-Order-Hold manner. We demonstrate that the dwell-time between successive triggering moments is uniformly bounded from below, thereby precluding Zeno behaviour. The proposed event-triggered boundary control strategy ensures the well-posedness of the closed-loop system and the satisfaction of certain model validity conditions. Additionally, the global exponential convergence of the closed-loop system to the setpoint is established using Lyapunov approach. A simulation example is provided to validate the theoretical findings.

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Observer-based Event-triggered Boundary Control of the One-phase Stefan Problem

2022-01-01

Bhathiya Rathnayake , Mamadou Diagne

This paper provides an observer-based event-triggered boundary control strategy for the one-phase Stefan problem using the position and velocity measurements of the moving interface. The infinite-dimensional backstepping approach is used … This paper provides an observer-based event-triggered boundary control strategy for the one-phase Stefan problem using the position and velocity measurements of the moving interface. The infinite-dimensional backstepping approach is used to design the underlying observer and controller. For the event-triggered implementation of the continuous-time observer-based controller, a dynamic event triggering condition is proposed. The triggering condition determines the times at which the control input needs to be updated. In between events, the control input is applied in a \textit{Zero-Order-Hold} fashion. It is shown that the dwell-time between two triggering instances is uniformly bounded below excluding \textit{Zeno behavior}. Under the proposed event-triggered boundary control approach, the well-posedness of the closed-loop system along with certain model validity conditions is provided. Further, using Lyapunov approach, the global exponential convergence of the closed-loop system to the setpoint is proved. A simulation example is provided to illustrate the theoretical results.

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Output Feedback Control of the One-Phase Stefan Problem

2016-01-01

Shumon Koga , Mamadou Diagne , Miroslav Krstić

In this paper, a backstepping observer and an output feedback control law are designed for the stabilization of the one-phase Stefan problem. The present result is an improvement of the … In this paper, a backstepping observer and an output feedback control law are designed for the stabilization of the one-phase Stefan problem. The present result is an improvement of the recent full state feedback backstepping controller proposed in our previous contribution. The one-phase Stefan problem describes the time-evolution of a temperature profile in a liquid-solid material and its liquid-solid moving interface. This phase transition problem is mathematically formulated as a 1-D diffusion Partial Differential Equation (PDE) of the melting zone defined on a time-varying spatial domain described by an Ordinary Differential Equation (ODE). We propose a backstepping observer allowing to estimate the temperature profile along the melting zone based on the available measurement, namely, the solid phase length. The designed observer and the output feedback controller ensure the exponential stability of the estimation errors, the moving interface, and the ${\cal H}_1$-norm of the distributed temperature while keeping physical constraints, which is shown with the restriction on the gain parameter of the observer and the setpoint.

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Riccati-feedback Control of a Two-dimensional Two-phase Stefan Problem

2022-01-01

Björn Baran , Peter Benner , Jens Saak

We discuss the feedback control problem for a two-dimensional two-phase Stefan problem. In our approach, we use a sharp interface representation in combination with mesh-movement to track the interface position. … We discuss the feedback control problem for a two-dimensional two-phase Stefan problem. In our approach, we use a sharp interface representation in combination with mesh-movement to track the interface position. To attain a feedback control, we apply the linear-quadratic regulator approach to a suitable linearization of the problem. We address details regarding the discretization and the interface representation therein. Further, we document the matrix assembly to generate a non-autonomous generalized differential Riccati equation. To numerically solve the Riccati equation, we use low-rank factored and matrix-valued versions of the non-autonomous backward differentiation formulas, which incorporate implicit index reduction techniques. For the numerical simulation of the feedback controlled Stefan problem, we use a time-adaptive fractional-step-theta scheme. We provide the implementations for the developed methods and test these in several numerical experiments. With these experiments we show that our feedback control approach is applicable to the Stefan control problem and makes this large-scale problem computable. Also, we discuss the influence of several controller design parameters, such as the choice of inputs and outputs.

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Backstepping Control of the One-Phase Stefan Problem

2016-01-01

Shumon Koga , Mamadou Diagne , Shu‐Xia Tang +1 more

In this paper, a backstepping control of the one-phase Stefan Problem, which is a 1-D diffusion Partial Differential Equation (PDE) defined on a time varying spatial domain described by an … In this paper, a backstepping control of the one-phase Stefan Problem, which is a 1-D diffusion Partial Differential Equation (PDE) defined on a time varying spatial domain described by an ordinary differential equation (ODE), is studied. A new nonlinear backstepping transformation for moving boundary problem is utilized to transform the original coupled PDE-ODE system into a target system whose exponential stability is proved. The full-state boundary feedback controller ensures the exponential stability of the moving interface to a reference setpoint and the ${\cal H}_1$-norm of the distributed temperature by a choice of the setpint satisfying given explicit inequality between initial states that guarantees the physical constraints imposed by the melting process.

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Toward Model-Based Control of the Vertical Gradient Freeze Crystal Growth Process

2021-02-26

Stefan Ecklebe , Frank Woittennek , Christiane Frank‐Rotsch +2 more

In this contribution tracking control designs using output feedback are presented for a two-phase Stefan problem arising in the modeling of the Vertical Gradient Freeze process. The two-phase Stefan problem, … In this contribution tracking control designs using output feedback are presented for a two-phase Stefan problem arising in the modeling of the Vertical Gradient Freeze process. The two-phase Stefan problem, consisting of two coupled free boundary problems, is a vital part of many crystal growth processes due to the temporally varying extent of the solid and liquid domains during growth. After discussing the special needs of the process, collocated as well as flatness-based state feedback designs are carried out. To render the setup complete, an observer design is performed, using a flatness-based approximation of the original distributed parameter system. The quality of the provided approximations as well as the performance of the open and closed loop control setups is analysed in several simulations.

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Towards model based control of the Vertical Gradient Freeze crystal growth process

2019-01-01

Stefan Ecklebe , Frank Woittennek , Jan Winkler +2 more

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Enthalpy-based Output Feedback Control of Two-sided Stefan Problem with Input Saturation

2022-12-06

Zhelin Chen , Hamza El-Kebir , Bryan Petrus +2 more

In the steel casting process, the deficient surface temperature history of the solidifying steel slab can induce the transverse surface cracks, while the improper relative location of the solidification front … In the steel casting process, the deficient surface temperature history of the solidifying steel slab can induce the transverse surface cracks, while the improper relative location of the solidification front inside the slab down the caster can create the internal cracks. Furthermore, if the slab is not fully solidified when it leaves the support rolls of the caster, the pressure from the molten steel will cause the steel shell to bulge out drastically, creating a defect called "whale", which damages the casting machine and causes a long work stoppage. Therefore, regulation of both the steel temperature and the liquid-solid interface location history is the key to maintaining the high steel quality and operational safety. This regulation can be achieved by the water spray cooling. The latter, however, is characterized by saturation when the water flow rate reaches its available maximum. The previous work of the authors started addressing this problem by presenting the full state feedback enthalpy-based control law for the two-sided Stefan problem with actuator saturation, describing the steel slab solidification under spray rate constraint. However, in the actual system, the full state feedback is not possible, since only the solid boundary temperature sensing is available. The present work closes this fundamental gap by combining a full state controller with an observer based on the temperature of the solid boundary. This combination produces the output feedback control law capable of tracking the desired temperature and solidification front trajectory under input saturation in the two-sided Stefan problem. The closed-loop convergence of the temperature and the interface errors for the output feedback system obtained are proven. Simulation shows the exponential-like trajectory convergence attained by the implementable smooth bounded control signals.

Input-to-State Stability for the Control of Stefan Problem with Respect to Heat Loss at the Interface

2018-06-01

Shumon Koga , Iasson Karafyllis , Miroslav Krstić

This paper develops an input-to-state stability (ISS) analysis of the one-phase Stefan problem with a prescribed heat loss at the liquid-solid interface. We focus on the closed-loop system under the … This paper develops an input-to-state stability (ISS) analysis of the one-phase Stefan problem with a prescribed heat loss at the liquid-solid interface. We focus on the closed-loop system under the control law proposed in [7] which is designed to stabilize the interface position at a desired position for the one-phase Stefan problem without the heat loss. The problem is modeled by a 1-D diffusion Partial Differential Equation (PDE) defined on a time-varying spatial domain described by an ordinary differential equation (ODE) with a time-varying disturbance. The well-posedness and some positivity conditions of the closed-loop system are proved based on an open-loop analysis. The closed-loop system with the designed control law satisfies an estimate of L2 norm in a sense of ISS with respect to the unknown heat loss.

Observer-based Periodic Event-triggered Boundary Control of the One-phase Stefan Problem

2023-01-01

Bhathiya Rathnayake , Mamadou Diagne

The Stefan problem models the phenomenon of phase transition between a liquid and a solid as the time evolution of the temperature profile of a liquid-solid material and its moving … The Stefan problem models the phenomenon of phase transition between a liquid and a solid as the time evolution of the temperature profile of a liquid-solid material and its moving interface. This paper provides a novel observer-based periodic event-triggered boundary control (PETBC) strategy for the one-phase Stefan problem using the position and velocity measurements of the moving interface. We propose a method to convert a specific class of continuous-time dynamic event-triggers that require continuous monitoring to periodic event-triggers that only require periodic evaluation. We achieve this result by finding an upper bound on the underlying continuous-time event-trigger between two successive periodic evaluations. We provide an explicit criterion for choosing a sampling period for periodically evaluating the event-trigger. The control input is updated only at events indicated by the periodic event-trigger and is applied in a zero-order hold fashion between two events. We establish the closed-loop system well-posedness along with certain model validity conditions under the proposed PETBC. Further, we prove that the exponential convergence to the setpoint under continuous-time event-triggered boundary control (CETBC) is preserved under the proposed PETBC. We provide simulation results to validate the theoretical developments.

Event-based Boundary Control of One-phase Stefan Problem: A Static Triggering Approach

2022-06-08

Bhathiya Rathnayake , Mamadou Diagne

In liquid-solid phase change phenomena, the Stefan problem describes the time evolution of the material's temperature profile and the interface position. This paper presents an event-based boundary control strategy for … In liquid-solid phase change phenomena, the Stefan problem describes the time evolution of the material's temperature profile and the interface position. This paper presents an event-based boundary control strategy for the one-phase Stefan problem. The proposed control approach consists of a full-state feedback backstepping boundary control law developed to drive the liquid-solid interface position to a desired setpoint and a static event-trigger mechanism which determines the time instants at which the control input needs to be updated. It is shown that the dwell-time between two triggering instances is uniformly bounded from below. Due to the existence of a minimal dwell-time, the closed-loop system is free from the so-called Zeno behavior. The control input is updated at event times and applied in a Zero-Order-Hold (ZOH) fashion. The well-posedness of the closed-loop system along with certain model validity conditions is proved. Furthermore, using the Lyapunov approach, it is shown that the proposed control approach globally exponentially stabilizes the temperature profile to the melting temperature of the material in L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> -norm and the moving interface to the desired setpoint. A simulation example is provided to validate the theoretical developments.

Input-to-State Stability for the Control of Stefan Problem with Respect to Heat Loss

2019-01-01

Shumon Koga , Iasson Karafyllis , Miroslav Krstić

This paper develops an input-to-state stability (ISS) analysis of the Stefan problem with respect to an unknown heat loss. The Stefan problem represents a liquid-solid phase change phenomenon which describes … This paper develops an input-to-state stability (ISS) analysis of the Stefan problem with respect to an unknown heat loss. The Stefan problem represents a liquid-solid phase change phenomenon which describes the time evolution of a material's temperature profile and the liquid-solid interface position. First, we introduce the one-phase Stefan problem with a heat loss at the interface by modeling the dynamics of the liquid temperature and the interface position. We focus on the closed-loop system under the control law proposed in [16] that is designed to stabilize the interface position at a desired position for the one-phase Stefan problem without the heat loss. The problem is modeled by a 1-D diffusion Partial Differential Equation (PDE) defined on a time-varying spatial domain described by an ordinary differential equation (ODE) with a time-varying disturbance. The well-posedness and some positivity conditions of the closed-loop system are proved based on an open-loop analysis. The closed-loop system with the designed control law satisfies an estimate of {L}_2 norm in a sense of ISS with respect to the unknown heat loss. The similar manner is employed to the two-phase Stefan problem with the heat loss at the boundary of the solid phase under the control law proposed in [25], from which we deduce an analogous result for ISS analysis.

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Boundary Geometric Control of a Nonlinear Diffusion System with Time‐Dependent Spatial Domain

2015-11-17

Ahmed Maidi , Jean‐Pierre Corriou

Abstract A Stefan problem represents a distributed parameter system with a time‐dependent spatial domain. This paper addresses the boundary control of the position of the moving liquid–solid interface in the … Abstract A Stefan problem represents a distributed parameter system with a time‐dependent spatial domain. This paper addresses the boundary control of the position of the moving liquid–solid interface in the case of nonlinear Stefan problem with Neumann actuation. The main idea consists in deriving an equivalent linear model by means of Cole‐Hopf tangent transformation, i.e. under a certain physical assumption, the original nonlinear Stefan problem is converted to a linear one. Then, the geometric control law is deduced directly from that developed, by the authors of the present paper, for the linear Stefan problem. Based on the fact that the Cole‐Hopf transformation is bijective, it is shown that the developed control law yields a stable closed‐loop system. The performance of the controller is evaluated through numerical simulation in the case of stainless steel melting characterized by a temperature‐dependent thermal conductivity, which is nonlinear. The objective is to control the position of the liquid–solid interface by manipulating a heat flux at the boundary.

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

2022-08-09

Bryan Petrus , Zhelin Chen , Hamza El-Kebir +2 more

Single-Boundary Control of the Two-Phase Stefan System

2019-01-01

Shumon Koga , Miroslav Krstić

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Control and State Estimation of the One-Phase Stefan Problem via Backstepping Design

2018-01-01

Shumon Koga , Mamadou Diagne , Miroslav Krstić

This paper develops a control and estimation design for the one-phase Stefan problem. The Stefan problem represents a liquid-solid phase transition as time evolution of a temperature profile in a … This paper develops a control and estimation design for the one-phase Stefan problem. The Stefan problem represents a liquid-solid phase transition as time evolution of a temperature profile in a liquid-solid material and its moving interface. This physical process is mathematically formulated as a diffusion partial differential equation (PDE) evolving on a time-varying spatial domain described by an ordinary differential equation (ODE). The state-dependency of the moving interface makes the coupled PDE-ODE system a nonlinear and challenging problem. We propose a fullstate feedback control law, an observer design, and the associated output feedback control law of both Neumann and Dirichlet boundary actuations via the backstepping method. Also, the state-feedback control law is provided when a Robin boundary input is considered. The designed observer allows the estimation of the temperature profile based on the available measurements of liquid-phase length and the heat flux at the interface. The associated output feedback controller ensures the global exponential stability of the estimation errors, the H 1 -norm of the distributed temperature, and the moving interface at the desired setpoint under some explicitly given restrictions on the setpoint and observer gain.

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Online recalibration of the state estimators for a system with moving boundaries using sparse discrete-in-time temperature measurements

2016-07-01

Bryan Petrus , Zhelin Chen , Joseph Bentsman +1 more

In this paper, the problem of estimation is considered for a class of processes involving solidifying materials. These processes have natural nonlinear infinite-dimensional representations, and measurements are only available at … In this paper, the problem of estimation is considered for a class of processes involving solidifying materials. These processes have natural nonlinear infinite-dimensional representations, and measurements are only available at particular points in the caster, each corresponding to a single discrete-in-time boundary measurement in the Stefan problem partial differential equation (PDE) mathematical model. The results for two previous estimators are summarized. The first estimator is based on the Stefan problem, using continuous instead of discrete-in-time boundary measurements. The second estimator employs a process model that is more detailed than the Stefan Problem, but with no output injection to reduce estimation error, other than model calibration. Both of these estimation frameworks are extended in the current paper to a more realistic sensing setting. First, an estimator is considered that uses the Stefan Problem under some simplifying but practically justified assumptions on the unknowns in the process. The maximum principle for parabolic PDEs is employed to prove that online calibration using a single discrete-in-time temperature measurement can provide removal of the estimation error arising due to mismatch of a single unknown parameter in the model. Although unproven, this result is then shown in simulation to apply to the more detailed process model.

Enthalpy-based Output Feedback Control of Two-sided Stefan Problem with Input Saturation

2022-12-06

Zhelin Chen , Hamza El-Kebir , Bryan Petrus +2 more

Predictor-feedback stabilization of two-input nonlinear systems with distinct and state-dependent input delays

2022-07-16

Yan Zhao , Huijun Gao , Shengyuan Xu +1 more

PDE-Based Modeling and Non-collocated Feedback Control of Electrosurgical-Probe/Tissue Interaction

2021-05-25

Hamza El-Kebir , Joseph Bentsman

The first control-oriented model of the interaction of an electrosurgical probe with organic tissue, based on a 1-D PDE with a moving boundary, is introduced. To attain the desired electrosurgically-induced … The first control-oriented model of the interaction of an electrosurgical probe with organic tissue, based on a 1-D PDE with a moving boundary, is introduced. To attain the desired electrosurgically-induced tissue changes using this model, a non-collocated output feedback moving boundary control law is proposed. The latter is realized through a novel non-collocated pointwise temperature-based state observer for the two-phase Stefan problem. Simulation demonstrates that the controller proposed meets the performance objective. The controller implementation is also discussed.

Backstepping Control of the One-Phase Stefan Problem

2016-01-01

Shumon Koga , Mamadou Diagne , Shu‐Xia Tang +1 more

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Sampled-Data Control of the Stefan System

2019-01-01

Shumon Koga , Iasson Karafyllis , Miroslav Krstić

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Input-to-State Stability for the Control of Stefan Problem with Respect to Heat Loss

2019-01-01

Shumon Koga , Iasson Karafyllis , Miroslav Krstić

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Backstepping control of the one-phase stefan problem

2016-07-01

Shumon Koga , Mamadou Diagne , Shu‐Xia Tang +1 more

In this paper, a backstepping control of the one-phase Stefan Problem, which is a 1-D diffusion Partial Differential Equation (PDE) defined on a time varying spatial domain described by an … In this paper, a backstepping control of the one-phase Stefan Problem, which is a 1-D diffusion Partial Differential Equation (PDE) defined on a time varying spatial domain described by an ordinary differential equation (ODE), is studied. A new nonlinear backstepping transformation for moving boundary problem is utilized to transform the original coupled PDE-ODE system into a target system whose exponential stability is proved. The full-state boundary feedback controller ensures the exponential stability of the moving interface to a reference setpoint and the ℋ 1 -norm of the distributed temperature by a choice of the setpint satisfying given explicit inequality between initial states that guarantees the physical constraints imposed by the melting process.

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Online Recalibration of the State Estimators for a System With Moving Boundaries Using Sparse Discrete-in-Time Temperature Measurements

2017-08-07

Bryan Petrus , Zhelin Chen , Joseph Bentsman +1 more

In this paper, the problem of estimation is considered for a class of processes involving solidifying materials. These processes have natural nonlinear infinite-dimensional representations, and measurements are only available at … In this paper, the problem of estimation is considered for a class of processes involving solidifying materials. These processes have natural nonlinear infinite-dimensional representations, and measurements are only available at particular points in the caster, each corresponding to a single discrete-in-time boundary measurement in the Stefan problem partial differential equation (PDE) mathematical model. The results for two previous estimators are summarized. The first estimator is based on the Stefan problem, using continuous instead of discrete-in-time boundary measurements. The second estimator employs a process model that is more detailed than the Stefan problem, but with no output injection to reduce estimation error, other than model calibration. Both of these estimation frameworks are extended in the current paper to a more realistic sensing setting. First, an estimator is considered that uses the Stefan problem under some simplifying but practically justified assumptions on the unknowns in the process. The maximum principle for parabolic PDEs is employed to prove that online calibration using a single discrete-in-time temperature measurement can provide removal of the estimation error arising due to mismatch of a single unknown parameter in the model. Although unproven, this result is then shown in simulation to apply to the more detailed process model.

Output feedback control of the one-phase Stefan problem

2016-12-01

Shumon Koga , Mamadou Diagne , Miroslav Krstić

In this paper, a backstepping observer and an output feedback control law are designed for the stabilization of the one-phase Stefan problem. The present result is an improvement of the … In this paper, a backstepping observer and an output feedback control law are designed for the stabilization of the one-phase Stefan problem. The present result is an improvement of the recent full state feedback backstepping controller proposed in our previous contribution. The one-phase Stefan problem describes the time-evolution of a temperature profile in a liquid-solid material and its liquid-solid moving interface. This phase transition problem is mathematically formulated as a 1-D diffusion Partial Differential Equation (PDE) of the melting zone defined on a time-varying spatial domain described by an Ordinary Differential Equation (ODE). We propose a backstepping observer allowing to estimate the temperature profile along the melting zone based on the available measurement, namely, the solid phase length. The designed observer and the output feedback controller ensure the exponential stability of the estimation errors, the moving interface, and the ℋ 1 -norm of the distributed temperature while keeping physical constraints, which is shown with the restriction on the gain parameter of the observer and the setpoint.

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Observer for heat equation with state-dependent switched parameters

2016-06-01

Jan Dimon Bendtsen , John Leth

This paper considers estimation of an unknown state function in the heat equation with state-dependent parameter values. The work is motivated by phase transitions in physical media, e.g., thawing of … This paper considers estimation of an unknown state function in the heat equation with state-dependent parameter values. The work is motivated by phase transitions in physical media, e.g., thawing of water or foodstuff, welding and casting processes. We point out that known solution to standard Stefan problem solutions can be recovered with this formalism, and then propose a simple phase transition estimator that relies only on boundary measurements. Simulations indicate that the estimates converge for noise-free measurements.

Control of Two-Phase Stefan Problem via Single Boundary Heat Input

2018-12-01

Shumon Koga , Miroslav Krstić

Input-to-State Stability for the Control of Stefan Problem with Respect to Heat Loss at the Interface

2018-06-01

Shumon Koga , Iasson Karafyllis , Miroslav Krstić

Control of the Vertical Gradient Freeze crystal growth process via backstepping

2020-01-01

Stefan Ecklebe , Frank Woittennek , Jan Winkler

This contribution presents a backstepping-based state feedback design for the tracking control of a two-phase Stefan problem which is encountered in the Vertical Gradient Freeze crystal growth process. A two-phase … This contribution presents a backstepping-based state feedback design for the tracking control of a two-phase Stefan problem which is encountered in the Vertical Gradient Freeze crystal growth process. A two-phase Stefan problem consists of two coupled free boundary problems and is a vital part of many crystal growth processes due to the time-varying extent of crystal and melt during growth. In addition, a different approach for the numerical approximation of the backstepping transformations kernel is presented.

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Towards model based control of the Vertical Gradient Freeze crystal growth process

2019-01-01

Stefan Ecklebe , Frank Woittennek , Jan Winkler +2 more

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State estimation of the Stefan PDE: A tutorial on design and applications to polar ice and batteries

2022-01-01

Shumon Koga , Miroslav Krstić

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Toward Model-Based Control of the Vertical Gradient Freeze Crystal Growth Process

2021-02-26

Stefan Ecklebe , Frank Woittennek , Christiane Frank‐Rotsch +2 more

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Boundary Control of PDEs: A Course on Backstepping Designs

2008-01-01

Miroslav Krstić

This concise and highly usable textbook presents an introduction to backstepping, an elegant new approach to boundary control of partial differential equations (PDEs). Backstepping provides mathematical tools for converting complex … This concise and highly usable textbook presents an introduction to backstepping, an elegant new approach to boundary control of partial differential equations (PDEs). Backstepping provides mathematical tools for converting complex and unstable PDE into elementary, stable, and physically intuitive target PDE systems that are familiar to engineers and physicists. The text s broad coverage includes parabolic PDEs; hyperbolic PDEs of first and second order; fluid, thermal, and structural systems; delay systems; PDEs with third and fourth derivatives in space; real-valued as well as complex-valued PDEs; stabilization as well as motion planning and trajectory tracking for PDEs; and elements of adaptive control for PDEs and control of nonlinear PDEs. It is appropriate for courses in control theory and includes homework exercises and a solutions manual that is available from the authors upon request. Audience: This book is intended for both beginning and advanced graduate students in a one-quarter or one-semester course on backstepping techniques for boundary control of PDEs. It is also accessible to engineers with no prior training in PDEs. Contents: List of Figures; List of Tables; Preface; Introduction; Lyapunov Stability; Exact Solutions to PDEs; Parabolic PDEs: Reaction-Advection-Diffusion and Other Equations; Observer Design; Complex-Valued PDEs: Schrodinger and Ginzburg Landau Equations; Hyperbolic PDEs: Wave Equations; Beam Equations; First-Order Hyperbolic PDEs and Delay Equations; Kuramoto Sivashinsky, Korteweg de Vries, and Other Exotic Equations; Navier Stokes Equations; Motion Planning for PDEs; Adaptive Control for PDEs; Towards Nonlinear PDEs; Appendix: Bessel Functions; Bibliography; Index

Partial Differential Equations of Parabolic Type

1964-01-01

Avner Friedman

Automatic Control via Thermostats of a Hyperbolic Stefan Problem with Memory

1999-03-04

Pierluigi Colli , Maurizio Grasselli , Juergen Sprekels

Optimal control of the free boundary in a two-phase Stefan problem

2006-11-29

Michael Hinze , Stefan Ziegenbalg

Boundary control of a nonlinear Stefan problem

2004-07-08

William B. Dunbar , Nicolas Petit , Pierre Rouchon +1 more

The classical Stefan problem is a linear one-dimensional heat equation with a free boundary at one end, modelling a column of liquid (e.g. water) in contact with an infinite strip … The classical Stefan problem is a linear one-dimensional heat equation with a free boundary at one end, modelling a column of liquid (e.g. water) in contact with an infinite strip of solid (ice). Given the fixed boundary conditions, the column temperature and free boundary motion can be uniquely determined. In the inverse problem, one specifies the free boundary motion, say from one steady-state length to another, and seeks to determine the column temperature and fixed boundary conditions, or boundary control. This motion planning problem is a simplified version of a crystal growth problem. In this paper, we consider motion planning of the free boundary (Stefan) problem with a quadratic nonlinear reaction term. The treatment here is a first step towards treating higher order nonlinearities as observed in crystal growth furnaces. Convergence of a series solution is proven and a detailed parametric study on the series radius of convergence given. Moreover, we prove that the parametrization can indeed be used for motion planning purposes; computation of the open loop motion planning is straightforward and we give simulation results.

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Control of the freezing interface motion in two‐dimensional solidification processes using the adjoint method

1995-01-15

Shinill Kang , Nicholas Zabaras

Abstract The aim of this work is to calculate the optimum history of boundary cooling conditions that, in two‐dimensional conduction driven solidification processes, results in a desired history of the … Abstract The aim of this work is to calculate the optimum history of boundary cooling conditions that, in two‐dimensional conduction driven solidification processes, results in a desired history of the freezing interface location/motion. The freezing front velocity and heat flux on the solid side of the front, define the obtained solidification microstructure that can be selected such that desired macroscopic mechanical properties and soundness of the final cast product are achieved. The so‐called two‐dimensional inverse Stefan design problem is formulated as an infinite‐dimensional minimization problem. The adjoint method is developed in conjunction with the conjugate gradient method for the solution of this minimization problem. The sensitivity and adjoint equations are derived in a moving domain. The gradient of the cost functional is obtained by solving the adjoint equations backward in time. The sensitivity equations are solved forward in time to compute the optimal step size for the gradient method. Two‐dimensional numerical examples are analysed to demonstrate the performance of the present method.

Partial differential equations of parabolic type

1965-06-01

John S. Maybee

Boundary Control of PDEs

2008-01-01

Miroslav Krstić , Andrey Smyshlyaev

Partial Differential Equations

1999-03-01

Steve Abbott , Lawrence C. Evans

Partial Differential Equations

1988-01-01

Shiing-Shen Chern

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Optimal control of nonlinear parabolic systems : theory, algorithms, and applications

1994-01-01

Pekka Neittaanmäki , Dan Tiba

Motivating Examples * Preliminaries * Finite Element Approach for Partial Differential Equations * Control Problems * Numerical Analysis of Control Problems * State Constrained Control Problems. Motivating Examples * Preliminaries * Finite Element Approach for Partial Differential Equations * Control Problems * Numerical Analysis of Control Problems * State Constrained Control Problems.