Materials Phase Change PDE Control & Estimation: From Additive Manufacturing to Polar Ice [1st ed.] 9783030584894, 9783030584900

Table of contents :
Front Matter ....Pages i-xiii
Phase Change Model: Stefan Problem (Shumon Koga, Miroslav Krstic)....Pages 1-13
Front Matter ....Pages 15-15
State Feedback Control Design for Stefan System (Shumon Koga, Miroslav Krstic)....Pages 17-58
State Estimator Design for Stefan System (Shumon Koga, Miroslav Krstic)....Pages 59-92
Extended Models and Design (Shumon Koga, Miroslav Krstic)....Pages 93-138
Two-Phase Stefan Problem (Shumon Koga, Miroslav Krstic)....Pages 139-157
Open Problems (Shumon Koga, Miroslav Krstic)....Pages 159-175
Front Matter ....Pages 177-177
Sea Ice (Shumon Koga, Miroslav Krstic)....Pages 179-198
Lithium-Ion Batteries (Shumon Koga, Miroslav Krstic)....Pages 199-219
Polymer 3D-Printing via Screw Extrusion (Shumon Koga, Miroslav Krstic)....Pages 221-245
Metal 3D-Printing via Selective Laser Sintering (Shumon Koga, Miroslav Krstic)....Pages 247-270
Experimental Study with Paraffin Melting (Shumon Koga, Miroslav Krstic)....Pages 271-297
Open Problems (Shumon Koga, Miroslav Krstic)....Pages 299-311
Back Matter ....Pages 313-352

Citation preview

Systems & Control: Foundations & Applications

Shumon Koga Miroslav Krstic

Materials Phase Change PDE Control & Estimation From Additive Manufacturing to Polar Ice

Systems & Control: Foundations & Applications Series Editor Tamer Ba¸sar, University of Illinois at Urbana-Champaign, Urbana, IL, USA

Editorial Board Karl Johan Åström, Lund Institute of Technology, Lund, Sweden Han-Fu Chen, Academia Sinica, Beijing, China Bill Helton, University of California, San Diego, CA, USA Alberto Isidori, Sapienza University of Rome, Rome, Italy Miroslav Krstic, University of California, San Diego, La Jolla, CA, USA H. Vincent Poor, Princeton University, Princeton, NJ, USA Mete Soner, ETH Zürich, Zürich, Switzerland; Swiss Finance Institute, Zürich, Switzerland Former Editorial Board Member Roberto Tempo, (1956–2017), CNR-IEIIT, Politecnico di Torino, Italy

More information about this series at http://www.springer.com/series/4895

Shumon Koga • Miroslav Krstic

Materials Phase Change PDE Control & Estimation From Additive Manufacturing to Polar Ice

Shumon Koga Dept. of Electrical & Computer Engineering University of California, San Diego La Jolla, CA, USA

Miroslav Krstic Mechanical & Aerospace Engineering University of California, San Diego La Jolla, CA, USA

ISSN 2324-9749 ISSN 2324-9757 (electronic) Systems & Control: Foundations & Applications ISBN 978-3-030-58489-4 ISBN 978-3-030-58490-0 (eBook) https://doi.org/10.1007/978-3-030-58490-0 Mathematics Subject Classification: 93-XX © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Ice melts into water in a hot environment. Conversely, water freezes into ice in a cold environment. These phenomena are generally called “phase changes” between the liquid phase and the solid phase. Such a phase change appears in sea ice in the polar region, solidification of molten metal in casting, extrusion in polymer 3D-printing, laser sintering for metal 3D-printing, cryosurgery for cancer treatment, thermal energy storage in buildings, etc. The phase change phenomena can be regarded as a growth or decline of the domain sizes of the liquid/solid phases. Apart from the thermal phase change, a growth of the moving domain can be seen in some chemical, biological, and social dynamics such as tumor growth in a patient’s body, axonal elongation for neurons’ signal transmission, lithiated region in electrodes of lithium-ion batteries, domain walls in ferroelectric nanomaterials, spreading of invasive species to a new environment, etc. A representative mathematical dynamic model that contains moving boundaries is the “Stefan problem,” which is a physical model of thermal phase changes. Since the phase changes are caused by the temperature dynamics, the physical model involves the temperature of each phase which is distributed in space and changes in time. Hence, the mathematical formulation incorporates partial differential equations (PDEs) defined on a time-varying spatial domain, whereas the dynamics of the position of the moving boundary is governed by an ordinary differential equation (ODE) whose input depends on the PDE state. This configuration gives rise to nonlinear coupling of the PDE and ODE dynamics. As a result, though seemingly consisting of just a linear PDE and a linear scalar ODE, the Stefan problem is mathematically peculiar and not amendable to conventional analysis for PDEs and ODEs. The Stefan problem and similar formulations described by moving boundary PDEs have been suggested as computational models for some science and engineering processes, including those listed above. For the purpose of regulating the boundary to a setpoint position, a control algorithm is needed. Likewise, for the purpose of estimating the unmeasured PDE state using other measurable variables, a state estimation algorithm needs to be designed. In earlier work on control of the Stefan model, some researchers have pursued control and state estimation design for v

vi

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the Stefan problem by using an approximation of the derivatives by means of spatial discretization or by imposing a priori assumption on physical states, but without rigorous analysis in closed-loop with the Stefan PDE-ODE system.

What Does the Book Cover? The book introduces a novel approach for control and state estimation of the Stefan system in terms of the design, analysis, applications, and experiments. The design method and analysis are covered in Part I of the book. In the design part, the boundary control and the estimator algorithms are developed via the method of “backstepping.” The backstepping method was originally developed for nonlinear and adaptive systems of ODEs and extended for time-delay systems and systems described by PDEs on fixed boundary domains. This book shows an extension of the backstepping method to moving boundary PDEs by giving a new type of a desired stable closed-loop system (the so-called “target system”) through a spatially causal (second-type Volterra) integral transformation of the PDE state. The key challenge lies in the analysis of the control problem, that is, guaranteeing that there is no appearance of a new phase or disappearance of the existing phases in the domain supposed to be in a given phase. More specifically, once we describe the material which is completely occupied by the liquid phase on one subdomain and occupied by the solid phase on the other complementary domain (e.g., water–ice), and the heat control being located at the outer boundary of the liquid phase, what if the boundary input injects cooling (negative) heat and the freezing process is caused from the outer boundary of the liquid phase? We can imagine that a new solid phase appears from the outer boundary of the liquid phase, and the material turns into a solid–liquid–solid phase configuration (e.g., ice–water–ice). This process violates the modeling postulation. How to avoid such a situation? A mathematical condition is that the temperature in the liquid phase needs to be above the melting temperature which is a threshold level for the material separating the liquid and solid phases. A sufficient condition to guarantee such a temperature constraint is the positivity of the heat input. Physically, we can imagine that as long as the input injects (positive) heat, the domain from the controlled boundary to the liquid–solid interface boundary remains in the liquid phase. Therefore, we impose the positivity of the designed heat input as a “physical constraint.” The positivity is proven after the boundary control is designed via the method of backstepping. The application studies and experimental verifications are covered in Part II. The application problem we introduce first is “sea ice,” which has been studied by researchers in the field of climate and earth science. Several observation data through satellites revealed the recent rapid decline of the amount of the Arctic sea ice, which has motivated the study of estimation for the sea ice and its coupled ocean states. The second application problem we provide is “lithium-ion batteries,” which has become ubiquitous in people’s lives through the development of electronic

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devices such as laptops, smartphones, and electric vehicles. For safe use, it is imperative to estimate how much energy remains in the battery during use. The third problem is polymer 3D-printing, which has emerged in a variety of manufacturing processes and even in consumers’ life. Among some extrusion methods, a screw extrusion is expected to achieve a fast manufacturing process. The screw extrusion causes the phase change from the solid polymer granules to the melt polymer for the ink production, and a control problem is to maintain a desired ratio between the solid polymer granules and the melt liquid polymer in the extruder. In order to verify the designed control and estimation algorithm for the Stefan problem, the experimental investigation is conducted using the cylindrical paraffin as a phase change material which has several attractive features for an energy storage, especially in buildings. The heat input located at the cross-sectional surface is designed as an observer-based output feedback control algorithm through the measured value of the surface temperature of the liquid paraffin and the liquid–solid phase interface position. Moreover, the observer and the control laws are prescribed based on sampled-data measurements and control, of which the design and analysis for a fundamental Stefan problem are developed in Part I.

Who Is the Book For? The book should be of interest to researchers working on control and dynamical systems, including engineers and mathematicians, especially specialists studying partial differential equations, moving boundaries (a.k.a. free boundaries), and their control-related problems. Mathematicians focusing on partial differential equations, especially free boundary problems would find the book stimulating because it tackles a quite challenging problem of global stabilization of a parabolic PDE coupled with a nonlinear ODE of the moving boundary. The book introduces a novel approach to deal with the mathematical problem, and the method could be useful to extend the class of systems we have focused on through further investigation. Thermal engineers and chemical engineers who have been involved with phase change phenomena such as in manufacturing will find a useful methodology in this book since we devote Chap. 9 to ink production stabilization in polymer 3Dprinting. Battery algorithm engineers in the electronics and automotive industry will also recognize exciting opportunities in this book since Chap. 8 presents battery management systems for some commercialized materials that undergo phase transitions. Earth scientists focusing on sea ice dynamics, especially in terms of algorithm development to assimilate the physical modeling with observed data, will learn a new perspective from Chap. 7 in this book. The state estimation for the ocean and climate models has been studied intensively using Kalman filter and adjoint method in other literatures, but our approach using “backstepping” is a new methodology with which the performance is ensured in a systematic and theoretical way.

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The reader is assumed to have a basic graduate-level background on differential equations and calculous to follow the mathematical proof. However, all the notions such as Lyapunov stability and some inequalities such as Young’s, Cauchy–Schwarz, Poincare’, and Agmon’s are given in appendices for the reader’s convenience.

Who Was Stefan? Joseph (Jožef) Stefan was a Slovenian-Austrian physicist of the second half of the nineteenth century. His name is most well known in relation to a physical law relating the total radiation from a black body to the fourth power of its temperature (the Stefan law). The extension to gray bodies is known as the Stefan– Boltzmann law (the latter being a student of the former). Stefan’s other contributions were in thermal conductivity of gases and evaporation, electromagnetics following Maxwell, and, of course, in the mathematical study of the equations of phase change. The largest research institute in Slovenia is named after Jožef Stefan.

Acknowledgements We thank Mamadou Diagne for his contributions to Chaps. 1, 2, 3, 9, and elsewhere. Through the collaboration with Mamadou, we have initiated our effort on control of the Stefan problem and advanced to applications in additive manufacturing. We are also grateful to Rafael Vazquez and Delphine Bresch-Pietri for their contributions to several extensions presented in Chap. 4. We would also like to thank Iasson Karaffylis for his contribution to an analysis of ISS and sampled-data design described in Chap. 4. For the application to sea ice in Chap. 7, we would like to thank Ian Eisenman for giving his suggestion on the physical model and several discussions. We are also grateful to Ian Fenty for providing his knowledge on the state estimation for sea ice thermodynamic model developed in NASA/JPL, which became beneficial for our understanding on the existing results. We would like to thank Leobardo Camacho-Solorio for his contribution to the battery management systems presented in Chap. 8. We would also like to thank Scott Moura for his feedback and comments on our results and motivations of battery managements. We are also grateful to Shuichi Adachi for giving his expertise on battery managements through his book. We would like to thank David Straub for his contribution in Chap. 9. We are also grateful to Joseph Beaman for his contribution to metal additive manufacturing in Chap. 10. Regarding the experimental study presented in Chap. 11, we would like to extend our gratitude to Mitutoshi Makihata, Renkun Chen, and Albert Pisano for their

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contributions to establishing the experimental setup for the thermal control system with phase change materials. A significant inspiration for our work came from papers by Professor Joseph Bentsman and his collaborators. We thank the National Science Foundation for partial support of this research. Shumon Koga would like to give a special thanks to his family Naoki, Yuko, and Shintaro. La Jolla, CA, USA La Jolla, CA, USA

Shumon Koga Miroslav Krstic

Contents

1

Phase Change Model: Stefan Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction and Brief History. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Physical Modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Explicit Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Mathematical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Macroscopic Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 5 7 8 9

Part I Design and Analysis 2

State Feedback Control Design for Stefan System . . . . . . . . . . . . . . . . . . . . . . 2.1 Control Objective for Stefan System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Basic Idea of PDE Control on Fixed Boundary. . . . . . . . . . . . . . . . . . . . . 2.3 Backstepping Control of Stefan System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Gain Tuning to Avoid Input Saturation and Evaporation . . . . . . . . . . . 2.5 Robustness to Diffusivity and Latent Heat Mismatch . . . . . . . . . . . . . . 2.6 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Boundary Temperature Actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Stefan-Like Problem with Dirichlet Interconnection . . . . . . . . . . . . . . . 2.9 Comments and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 20 29 38 39 44 46 53 57

3

State Estimator Design for Stefan System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Basic Idea of PDE Estimation on Fixed Boundary . . . . . . . . . . . . . . . . . 3.2 Temperature Profile Estimation for the Stefan System . . . . . . . . . . . . . 3.3 Observer-Based Output Feedback Control Design . . . . . . . . . . . . . . . . . 3.4 State Estimation Under More Practical Sensors . . . . . . . . . . . . . . . . . . . . 3.5 Estimation of Both Temperature Profile and Moving Interface by Measuring Only a Boundary Temperature . . . . . . . . . . . . 3.6 State Estimation by Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . 3.7 Estimation Under Boundary Temperature Actuation . . . . . . . . . . . . . . . 3.8 Comments and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 59 62 66 74 84 85 89 92

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Contents

4

Extended Models and Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.1 Melting with Advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.2 Actuator Delay Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.3 n-Dimensional Ball Geometry with Symmetry . . . . . . . . . . . . . . . . . . . . . 111 4.4 What Can We Guarantee If the Solid Phase Remains?—ISS . . . . . . 118 4.5 Sampled-Data Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5

Two-Phase Stefan Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Description of the Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Existence, Uniqueness, and Non-monotonicity of Interface Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 State Feedback Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Analysis of the Closed-Loop System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Robustness to Uncertainties of Physical Parameters . . . . . . . . . . . . . . . 5.6 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Comments and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

140 142 146 150 154 156

Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Accelerated Convergence with Added Damping in Target System 6.2 How to Track to a Desired Motion? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Two-Dimensional Disk Geometry with Nonuniformity . . . . . . . . . . . . 6.4 Comments and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159 159 166 171 175

6

139 139

Part II Applications and Experiment 7

Sea Ice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Importance of the Arctic Sea Ice for Global Climate Modeling . . . 7.2 Thermodynamic Model of Arctic Sea Ice . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Annual Cycle Simulation of Sea Ice Thickness. . . . . . . . . . . . . . . . . . . . . 7.4 Temperature Profile Estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Numerical Tests of the Sea Ice Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Comments and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179 179 180 182 183 192 195

8

Lithium-Ion Batteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Battery Management Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Electrochemical Model with Phase Change Electrode . . . . . . . . . . . . . 8.3 State-of-Charge Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Comments and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

199 199 201 204 211 218

9

Polymer 3D-Printing via Screw Extrusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Emergence of 3D-Printing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Screw Extrusion Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Thermodynamic Modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Ink Production Control Based on Screw Speed . . . . . . . . . . . . . . . . . . . . . 9.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Comments and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

221 221 221 222 224 239 244

Contents

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10

Metal 3D-Printing via Selective Laser Sintering . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Selective Laser Sintering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 State Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Observer and Output Feedback Control Design . . . . . . . . . . . . . . . . . . . . 10.5 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Comments and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

247 247 248 250 257 263 270

11

Experimental Study with Paraffin Melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Modeling of PCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Nominal Feedback Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Implementable Control Algorithm Using Sensors and Software . . 11.4 Experimental Setup and Calibration of Unknown Parameters . . . . . 11.5 Experiment with Closed-Loop Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Proof of Theoretical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Comments and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

271 272 274 275 282 285 291 297

12

Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Tumor Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Axonal Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Comments and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

299 299 305 310

A

Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

B

Some Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Cauchy-Schwarz Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Poincare’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Agmon’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

315 315 315 316

C

Stable Systems and Their Proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1 One-Phase Stefan Problem with Monotonic Interface. . . . . . . . . . . . . . C.2 One-Phase Stefan Problem with Convection and Heat Loss . . . . . . . C.3 One-Phase Stefan Problem with Delay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.4 One-Phase Stefan Problem with Non-monotonic Interface and Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

319 319 323 329 335

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

Chapter 1

Phase Change Model: Stefan Problem

1.1 Introduction and Brief History The Stefan system is a well-known moving-boundary PDE system modelling the thermodynamic liquid-solid phase change phenomena. The associated problem of analyzing and finding the solutions to the Stefan model is referred to as the “Stefan problem.” It is named after the Austrian physicist Josef Stefan, who was one of the most distinguished and influential physicists of the nineteenth century, celebrated for his numerous contributions to thermodynamics and heat transfer from the experimental perspective. Perhaps Stefan’s name is more recognized for the Stefan-Bolzman law, which revealed that a material with temperature T in absolute unit emits a radiative heat transfer which is proportional to T 4 , through Stefan’s experimental work and his student Ludwig Boltzman’s work on the theoretical foundation. After the publication of the thermal radiations law, Stefan started to focus of the thickness evolution of polar ice caps motivated by observed data of ice growth and air temperature acquired by British and German explorers during their expeditions. A long time before that, the phase change model by moving boundaries had been studied by Joseph Black in 1762. Franz Neumann developed the solution in his lectures around 1860. However, Neumann’s result had not been published until Weber’s paper in 1901. Stefan developed his analysis of the solution of ice growth and studied the correspondence with the empirical data, which was published in 1891 [194]. Since then, the model has been known as the “Stefan problem” and has been studied widely by researchers from the middle of 1900s [48]. While there are several extended models on the phase change by incorporating additional factors, throughout this chapter we introduce the one-dimensional onephase Stefan problem by assuming • the temperature profile is uniformly distributed along a cross-sectional area • the solid phase temperature is uniformly distributed at the melting temperature © Springer Nature Switzerland AG 2020 S. Koga, M. Krstic, Materials Phase Change PDE Control & Estimation, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-58490-0_1

1

2

1 Phase Change Model: Stefan Problem

• there is no convection in or of the material • the pressure field around the material is static and uniform • the material is pure. In the later chapters, we start to relax the first three assumptions one by one.

1.2 Physical Modeling Consider a pure one-component material of length L in one dimension as depicted in Fig. 1.1. The dynamics of the process depends strongly on the evolution in time of the moving interface (here reduced to a point) at which phase transition from liquid to solid (or equivalently, in the reverse direction) occurs. Therefore, the melting or solidification mechanism that takes place in the physical domain [0, L] induces the existence of two complementary time-varying sub-domains, namely, [0, s(t)] occupied by the liquid phase, and [s(t), L] by the solid phase. Assuming a temperature profile uniformly equivalent to the melting temperature in the solid phase, a dynamical model associated with the melting phenomenon (see Fig. 1.1) involves only the thermal behavior of the liquid phase. Considering a melting material with a density ρ and heat capacity Cp , the local energy conservation law is given by ρCp Tt (x, t) = −qx (x, t),

x ∈ (0, s(t)),

(1.1)

Fig. 1.1 Schematic of one-dimensional one-phase Stefan problem. The temperature profile in the solid phase is assumed to be a uniform melting temperature

1.2 Physical Modeling

3

where q(x, t) is a heat flux profile and T (x, t) is a temperature profile. Moreover, the local energy balance at the position of the liquid–solid interface x = s(t) involved with the latent heat leads to the dynamics of the moving boundary ρΔH ∗ s˙ (t) = q(s(t), t).

(1.2)

At a fundamental level, the thermal conduction for a melting component obeys the well-known Fourier’s Law x ∈ [0, s(t)],

q(x, t) = −kTx (x, t),

(1.3)

where k is the thermal conductivity. Therefore, the time evolution of the temperature profile in the material’s domain can be obtained by combining the energy conservation (1.1) and the thermal condition (1.3), which leads to the following heat equation of the liquid phase Tt (x, t) =αTxx (x, t),

x ∈ (0, s(t)),

(1.4)

where α :=

k . ρCp

(1.5)

At the boundary x = 0, there are two cases of how to impose the boundary condition. One is that a heat flux enters as an external source which can be manipulated as a controlled variable, denoted as qc (t) and the boundary condition is derived using the thermal conduction law (1.3). The other case is that the boundary temperature can be directly controlled as Tc (t). Hence, the boundary condition at x = 0 is either −kTx (0, t) =qc (t),

(1.6)

T (0, t) = Tc (t).

(1.7)

or

The boundary condition prescribed at x = s(t) is involved with the melting temperature which is the constant threshold level to cause the phase change from the solid to liquid under a static pressure, i.e., T (s(t), t) =Tm .

(1.8)

Other than the spacial boundary conditions, the time initial conditions need to be defined as an arbitral spatial function for the temperature profile and a positive valued interface position as T (x, 0) = T0 (x),

s(0) = s0 .

(1.9)

4

1 Phase Change Model: Stefan Problem

If we do not care about the dynamics of the moving boundary s(t), PDE models (1.4)–(1.9) are somewhat simple linear system. However, the tricky property of the Stefan problem lies in the dynamics of the moving boundary s(t). By combining the latent heat energy balance (1.2) and the thermal conduction (1.3), one can derive the so-called “Stefan condition” defined as the following nonlinear ODE: s˙ (t) = −βTx (s(t), t),

(1.10)

where β :=

k , ρΔH ∗

(1.11)

and ΔH ∗ denotes the latent heat of fusion. Equation (1.10) expresses the velocity of the liquid–solid moving interface. As we have presented, there are two problems to describe the one-phase Stefan problem as a nonlinearly coupled PDE-ODE system depending on how to impose the boundary condition at x = 0. Hereafter, we name each problem as PI (Problem I) and PII (Problem II) as follows.

PI : Neumann Boundary Actuation Tt (x, t) =αTxx (x, t),

x ∈ (0, s(t)),

(1.12)

−kTx (0, t) =qc (t),

(1.13)

T (s(t), t) =Tm ,

(1.14)

s˙ (t) = − βTx (s(t), t),

(1.15)

PII : Dirichlet Boundary Actuation Tt =αTxx (x, t),

x ∈ (0, s(t)),

T (0, t) =Tc (t), T (s(t), t) =Tm , s˙ (t) = − βTx (s(t), t).

(1.16) (1.17) (1.18) (1.19)

In addition, for each problem setup, there are two types of mathematical problem of how to obtain the pair of solution as follows: Direct Stefan Problem : Given qc (t) in PI (or Tc (t) in PII) as a prescribed function in time, solve (T (x, t), s(t)).

1.3 Explicit Solutions

5

Inverse Stefan Problem : Given s(t) as a prescribed function in time, solve T (x, t) and qc (t) in PI (or Tc (t) in PII). Since we have established the models (1.12)–(1.15) in PI (or (1.16)–(1.19) in PII) based on the situation that the domain x ∈ [0, s(t)] is occupied by the liquid phase, to maintain a physical validity of the homogeneous melting material, the Stefan problem exhibits an important property that is discussed in the following remark. Remark 1.1 The formulation of the Stefan problem is a reasonable model only if the following condition holds: T (x, t) ≥Tm , 0 < s(t) 0 and the Lipschitz continuity of T0 (x) holds, i.e., 0 ≤ T0 (x) − Tm ≤ H (s0 − x).

(1.23)

Assumption 1.1 is physically reasonable and consistent with Remark 1.1. Hereafter, we always impose Assumption 1.1 without explicitly stating.

1.3 Explicit Solutions Neumann Solution by a Constant Boundary Temperature A well-known analytical solution of the Stefan problem is the so-called “Neumann solution,” named after the discovery of the solution by F. Neumann around 1860 [48]. The name might be a kind of misleading because the solution is equivalent to the one under a constant Dirichlet boundary condition (not Neumann boundary

6

1 Phase Change Model: Stefan Problem

condition) at x = 0 as a direct Stefan problem in PII. Thus, the condition is prescribed as Tc (t) = Tb ,

(1.24)

in PII. The Neumann solution to the Eqs. (1.16)–(1.19) with the above condition is given by   x Tb − Tm , (1.25) erf √ T (x, t) =Tb − erf (λ) 2 αt √ s(t) =2λ αt, (1.26) where erf(·) is the error function defined by  x 2 2 erf(x) = √ e−t dt. π 0

(1.27)

We can see that the pair of solutions (1.25) and (1.26) satisfies the model equations (1.16)–(1.19) with the implicit parameter λ which is a solution to the following nonlinear algebraic equation: √ 2 π λerf(λ)eλ = Ste,

(1.28)

where Ste is the so-called “Stefan number” defined by Ste =

Cp (Tb − Tm ). ΔH ∗

(1.29)

Since the boundary temperature condition (1.24) is reasonable and simple, this pair of solution (1.25) and (1.26) has been very popular among thermal and chemical engineers. Once we consider the inverse Stefan problem of PI by prescribing the interface solution as (1.26), the boundary heat flux is obtained as qc (t) =

k(Tb − Tm ) 1 √ √ . erf (λ) π α t

(1.30)

In other words, for the boundary temperature T (0, t) to maintain a constant value, the boundary heat flux qc (t) must be a decaying function in time which is proportional to √1t .

Case 2: Linear Growth of the Interface Another known analytical solution can be obtained by the inverse Stefan problem of both PI and PII. There, the interface dynamics is set as growing linearly in time, which can be described as

1.4 Mathematical Analysis

7

s(t) =At,

(1.31)

where A > 0 is a positive parameter. Then, one can see that the following solution of the temperature profile  α  A (At−x) eα T (x, t) = − 1 + Tm (1.32) β satisfies the governing equations (1.12)–(1.15) (or (1.16)–(1.19)). Thus, the associated boundary condition in PI is given by qc (t) =

kA A2 t eα , β

(1.33)

in PI, or Tc (t) =

α β

 2  A e α t − 1 + Tm

(1.34)

in PII.

1.4 Mathematical Analysis This section is devoted to a rigorous analysis which is especially of interest to mathematicians, that is, the existence and uniqueness of the classical solution. To begin, referring to [75], the definition of the classical solution is defined as follows: Definition 1.1 Under Assumption 1.1, a pair (T (x, t), s(t)) is the classical solution of the one-phase Stefan problem (1.12)–(1.15) with qc (t) ≥ 0 in PI (or (1.16)–(1.19) with Tc (t) ≥ Tm in PII) for all t < σ , where 0 < σ ≤ ∞ if 1. Txx and Tt are continuous for 0 < x < s(t), 0 < t < σ ; 2. T and Tx are continuous for 0 ≤ x ≤ s(t), 0 < t < σ ; 3. T is also continuous for t = 0, 0 < x ≤ s0 and 0 ≤ lim inf T (x, t) ≤ lim sup T (x, t) < ∞ as t → 0, x → 0; 4. s(t) is continuously differentiable for 0 ≤ t < σ ; 5. the Eqs. (1.12)–(1.15) are satisfied. Thus, the explicit solutions introduced in Sect. 1.3 are the classical solution. Again by referring to [75], the existence and uniqueness of the classical solution can be guaranteed by the following lemma. Lemma 1.2 Assume that qc (t) in PI (or Tc (t) in PII) and T0 (x) are continuously differentiable functions for ∀t > 0 and ∀x ∈ [0, s0 ]. Then there exists a unique classical solution (T (x, t), s(t)) of the system (1.12)–(1.15) provided that qc (t) ≥ 0 in PI (or the system (1.16)–(1.19) provided that Tc (t) ≥ Tm in PII) and Assumption 1.1 for all t > 0.

8

1 Phase Change Model: Stefan Problem

Once the existence and uniqueness of the solution is established, the validity of the model is ensured by the following lemma. Lemma 1.3 If there is a unique classical solution of (1.12)–(1.15), then for any qc (t) ≥ 0 in PI (or Tc (t) ≥ Tm in PII) for all t < σ where 0 < σ ≤ ∞, the condition (1.20) holds, and (1.22) is also satisfied by Lemma 1.1. In addition, if qc (t) > 0 in PI (or Tc (t) > Tm in PII) holds, then the strong inequality of (1.20) and (1.22) holds. The proof of Lemma 1.3 is based on Maximum principle as shown in [82]. Furthermore, both Definition 1.1 and Lemma 1.2 can be extended to the generalized parabolic PDE Tt = α(x, t)Txx + b(x, t)Tx + h(x, t)T ,

(1.35)

provided that h(x, t) ≤ 0 and the functions αx , αxx , αt , b, bx , and h are Hölder continuous for 0 ≤ x < ∞, t ≥ 0.

1.5 Macroscopic Energy Balance This section provides a physical perspective of the system. The conventional thermodynamics gives the heat balance in macroscopic scale, known as the first law of thermodynamics. For the Stefan problem, the first law of thermodynamics is obtained by taking the integration of the local energy balance on the whole domain as 

s(t)

ρCp

Tt (x, t)dx = −(q(s(t), t) − q(0, t)).

(1.36)

0

Combining the specific heat with the latent heat, the internal energy is defined as 

s(t)

E(t) = ρCp

(T (x, t) − Tm )dx + ρΔH ∗ s(t).

(1.37)

0

Taking the time derivative of (1.37) along the solution of (1.12)–(1.15), we can see that dE (t) = qc (t). dt

(1.38)

Taking the time integration of (1.38) yields the following description of the macroscopic energy conservation law 

t

E(t) − E(0) =

qc (τ )dτ. 0

(1.39)

1.6 Numerical Methods

9

The left-hand side of (1.39) denotes the growth of internal energy, and its right-hand side denotes the external work provided by the injected heat flux. This form directly captures the classical first law of thermodynamics without heat dissipation.

1.6 Numerical Methods This section presents the numerical method of 1D one-phase Stefan problem by referring to [132]. While there are several methods, in this book we introduce a boundary immobilization method (BIM) technique which shows an enough accurate data for 1D Stefan problem in cartesian coordinate. The idea of BIM is to scale the original coordinate on time-varying domain to the new coordinate on fixed domain. The resulting system through the scaling leads to a nonlinear coupled PDE-ODE system. Then, we utilize some approximations for the spatial and time derivatives by finite difference and Euler method, that yields a set of nonlinear difference equations.

Algorithm Development by BIM Let us introduce the following scaling of the spatial coordinate and the associated state variable as ξ :=

x , s(t)

v(ξ, t) := T (x, t).

(1.40)

Then, the relations of the spatial and time derivatives are given by ∂ξ 1 vξ (ξ, t) = vξ (ξ, t), ∂x s(t)  2 ∂ξ 1 vξ ξ (ξ, t) = vξ ξ (ξ, t), Txx (x, t) = ∂x s(t)2 Tx (x, t) =

Tt (x, t) =

(1.41) (1.42)

∂ξ x s˙ (t) vξ (ξ, t) + vt (ξ, t) = − vξ (ξ, t) + vt (ξ, t) ∂t s(t)2

=−

ξ s˙ (t) vξ (ξ, t) + vt (ξ, t). s(t)

(1.43)

First, we derive the numerical algorithm for Neumann boundary actuation setup in PI. Substituting the above derivatives in (1.12)–(1.15), we obtain vt (ξ, t) =

α ξ s˙ (t) vξ (ξ, t), vξ ξ (ξ, t) + s(t) s(t)2

0 0, the internal energy for a given setpoint must be greater than the initial internal energy. Thus, the following assumption is imposed. Assumption 2.1 The setpoint sr is chosen to satisfy β s0 + α



s0

(T0 (x) − Tm )dx < sr < L.

(2.11)

0

Therefore, Assumption 2.1 stands as the least restrictive condition for the choice of setpoint and can be consequently viewed as a setpoint restriction.

Open-Loop Control by Energy Shaping For any given open-loop control signal qc (t) satisfying (2.10), the asymptotic convergence of the system (2.1)–(2.4) to sr can be established and the following lemma holds. Lemma 2.1 Consider an open-loop setpoint control law qc (t) which satisfies (2.10). Then, the interface converges asymptotically to the prescribed setpoint sr and consequently, conditions (2.7) and (2.8) hold. The proof of Lemma 2.1 can be derived straightforwardly from (2.10). To illustrate the introduced concept of open-loop “energy shaping control” action, we define ΔE as the left-hand side of (2.10), i.e., ΔE =

k k (sr − s0 ) − β α



s0

(T0 (x) − Tm )dx.

(2.12)

0

For instance, the rectangular pulse control law given by  qc (t)

=

q¯ for t ∈ [0, ΔE/q] ¯ 0 for t > ΔE/q¯

 (2.13)

20

2 State Feedback Control Design for Stefan System

satisfies (2.12) for any choice of the boundary heat flux q¯ and thereby ensures the asymptotic convergence of (2.1)–(2.4) to the setpoint (Tm , sr ).

Towards Closed-Loop Feedback Control It is remarkable that adopting an open-loop control strategy, such as the rectangular pulse (2.13), does not allow to improve the convergence speed. Moreover, the physical parameters of the model need to be known accurately. In engineering processes, the practical implementation of an open-loop control is limited by performance and robustness issues, thus closed-loop control laws have to be designed to deal with such limitations. In the following sections, we aim the design of closed-loop backstepping control law for the one-phase Stefan model toward achieving faster exponential convergence to the desired setpoint (Tm , sr ) while ensuring the robustness of the closed-loop system to the uncertainty of the physical parameters. With such setpoint regulation in mind, for a given reference setpoint (Tm , sr ), we define the reference error states (u, X) as u(x, t) = T (x, t) − Tm ,

X(t) = s(t) − sr ,

(2.14)

respectively. Then, the reference error system associated with the coupled system (2.1)–(2.4) is written as ut (x, t) =αuxx (x, t),

0 ≤ x ≤ s(t),

ux (0, t) = − k −1 qc (t), u(s(t), t) =0, ˙ X(t) = − βux (s(t), t).

(2.15) (2.16) (2.17) (2.18)

Hence, from the perspective of the reference error system (u, X), the objective is to design qc (t) to stabilize the state variables (u, X) at (0, 0), which is a standard objective in control theory.

2.2 Basic Idea of PDE Control on Fixed Boundary Overview In this section, we introduce a well-known method of the control design for systems described by PDEs. The method is “backstepping” which was first developed for nonlinear and adaptive systems [129], and successfully extended to PDE of

2.2 Basic Idea of PDE Control on Fixed Boundary

21

parabolic type [22, 180, 182], hyperbolic type [7, 127], delays [16, 124], and adaptive systems [126, 183, 184]. Further advances on the backstepping control of diffusion equations defined on a multidimensional space or involving in-domain coupled systems can be found in [10, 53, 206, 207]. PDE backstepping design has been utilized for the applications to oil drilling [85, 171, 211, 217], multi-agent system [79, 165], turbulent flows [203], traffic control [223], battery management [150, 151], mining cables [213–215], etc. The fundamental idea of the backstepping is to introduce a “state transformation” to convert the original system to an ideal stable system, and then derive the control design to be consistent with the transformation. Results devoted to the backstepping stabilization of coupled systems described by a diffusion PDE in cascade with a linear ODE have been primarily presented in [122] with Dirichlet type of boundary interconnection and extended to Neumann boundary interconnection in [195, 198]. For systems related to the Stefan problem, [91] designed a backstepping output feedback controller that ensures the exponential stability of an unstable parabolic PDE on a priori known dynamics of moving interface which is assumed to be an analytic function in time. Moreover, for PDEODE cascaded systems under a state-dependent moving boundary, [32] derived a local stability result for nonlinear ODEs with actuator dynamics governed by a wave PDE defined on a time- and state-dependent moving domain. Such a technique is based on the input delay and wave compensation for nonlinear ODEs designed in [15, 125] and its extension to state-dependent input delay compensation for nonlinear ODEs is provided in [14]. While the results in [32] and [14], which cover state-dependent problems, do not ensure global stabilization due to a so-called feasibility condition that needs to be satisfied a priori, such a restriction was recently removed in [58], which provides a global stability result. However, the result in [58] is limited to the case of a hyperbolic PDE in cascade with a nonlinear ODE.

Unstable Reaction-Diffusion PDE As an introductory example of diffusion type systems whose solution diverges in time, let us consider the following diffusion-reaction PDE: ut =uxx (x, t) + λu(x, t),

0 < x < 1,

(2.19)

u(0, t) =0,

(2.20)

u(1, t) =0,

(2.21)

u(x, 0) =u0 (x).

(2.22)

The solution to (2.19)–(2.21) is uniquely given by (see Section 3.1 in [128] for details)

22

2 State Feedback Control Design for Stefan System

u(x, t) =

∞ 

Cn e(λ−π

2 n2 )t

sin (π nx) ,

(2.23)

n=1

where 

1

Cn = 2

u0 (y) sin (π ny) dy.

(2.24)

0

The important characteristic of the solution (2.23) is the time dependency, namely, 2 2 the exponential term e(λ−π n )t . For the solution not to diverge, the coefficient in the exponent λ − π 2 n2 needs not to be positive for all n = 1, 2, · · · . Clearly, this condition holds if λ ≤ π 2 by considering the case n = 1. Moreover, if λ < π 2 , then the solution converges to zero as t → ∞. In order for the solution to converge to zero under λ ≥ π 2 , some actuation needs to be applied. There are two distinct types of control problems for PDEs. One is “in-domain control,” which renders actuators to be located inside the domain of the PDE, resulting in the control term to appear in PDE as a mathematical structure. The other is “boundary control,” in which the actuator is located only on the boundary of the PDE. Then, the control term only appears in the boundary condition of the PDE. In general, the boundary control is more challenging to design compared to the in-domain control and is also more practically relevant. Referring to [128], we consider the boundary control at x = 1, with U (t) as a control input. Thus, the resulting problem we consider is ut =uxx (x, t) + λu(x, t),

0 < x < 1,

(2.25)

u(0, t) =0,

(2.26)

u(1, t) =U (t).

(2.27)

The “backstepping design” is arguably the most systematic method for boundary control of PDEs, studied widely since the first work in [180]. The method introduces a state transformation (called “backstepping transformation”) from the original state u(x, t) to newly defined state w(x, t) in the form 

x

w(x, t) = u(x, t) −

k(x, y)u(y, t)dy,

(2.28)

0

where k(x, y) is the so-called “gain kernel” function which is introduced and solved later. The idea of backstepping is to design a controller to force the system behave like a stable “target system.” Mostly, the target system is chosen to have a similar structure as the original system with canceling some undesired (or adding some desired) terms for the stabilization. For instance, in the reaction-diffusion PDE (2.25)–(2.27), the undesired term is λu(x, t) in (2.25) as we have seen in the analytical solution (2.23). Hence, a natural choice of the target system is

2.2 Basic Idea of PDE Control on Fixed Boundary

23

wt =wxx (x, t),

0 < x < 1,

(2.29)

w(0, t) =0,

(2.30)

w(1, t) =0.

(2.31)

The solution of the target system (2.29)–(2.29) converges to zero as we can see by substituting λ = 0 in the analytic solution (2.23). Next task is to find the gain kernel k(x, y) in the transformation (2.28) to achieve consistency between the original usystem (2.25)–(2.27) and the target w-system (2.29)–(2.31). Taking the first and second spatial derivatives of (2.28), we obtain  wx (x, t) =ux (x, t) − k(x, x)u(x, t) −

x

kx (x, y)u(y, t)dy,

(2.32)

0

wxx (x, t) =uxx (x, t) − k(x, x)ux (x, t)    x d k(x, x) u(x, t) − kxx (x, y)u(y, t)dy. − kx (x, x) + dx 0 (2.33) Taking the time derivative of (2.28) along the solution of (2.25)–(2.27) leads to 

x

wt (x, t) = uxx (x, t) + λu(x, t) −

k(x, y)(uyy (y, t) + λu(y, t))dy.

(2.34)

0

Using the integration by parts twice and the boundary condition (2.26), 

x

 k(x, y)uyy (y, t)dy =k(x, x)ux (x, t)−k(x, 0)ux (0, t)−

0

x

ky (x, y)uy (y, t)dy

0

=k(x, x)ux (x, t) − k(x, 0)ux (0, t)  x − ky (x, x)u(x, t) + kyy (x, y)u(y, t)dy.

(2.35)

0

Substituting (2.35) into (2.34), we have   wt (x, t) =uxx (x, t) − k(x, x)ux (x, t) + ky (x, x) + λ u(x, t)  x   + k(x, 0)ux (0, t) − kyy (x, y) + λk(x, y) u(y, t)dy. 0

Subtracting (2.33) from (2.36), we have   d wt (x, t) − wxx (x, t) = 2 k(x, x) + λ u(x, t) + k(x, 0)ux (0, t) dx

(2.36)

24

2 State Feedback Control Design for Stefan System



x

+



 kxx (x, y) − kyy (x, y) − λk(x, y) u(y, t)dy.

0

(2.37) To satisfy target PDE (2.29), the right-hand side of (2.37) must be zero for any u(x, t), and thus the following conditions of the gain kernel function must be satisfied: kxx (x, y) − kyy (x, y) =λk(x, y),

(2.38)

k(x, 0) =0,

(2.39)

d λ k(x, x) = − . dx 2

(2.40)

The solution to the PDE (2.38)–(2.40) is given by I1 (z) , k(x, y) = −λy z

z :=



λ(x 2 − y 2 ),

(2.41)

where I1 (z) is a modified Bessel function of the first kind defined by I1 (z) =

∞  m=0

 z 2m+1 1 . m!(m + 1)! 2

(2.42)

Evaluating (2.28) at x = 1 together with the boundary conditions of the original system (2.27) and the target system (2.31), the control law is designed by  U (t) =

1

k(1, y)u(y, t)dy 0



=−λ 0

1

 I1 ( λ(1 − y 2 )) y  u(y, t)dy. λ(1 − y 2 )

(2.43)

The conclusion here is that the designed backstepping feedback controller (2.43) stabilizes the unstable reaction diffusion PDE (2.25)–(2.27). This is how the design problem of PDE control is solved via backstepping. For the mathematical analysis to conclude with the guarantee of closed-loop stability, we need more analysis by guaranteeing the invertibility of the transformation and equivalence of the norm, but we omit it here. We refer the readers to [128] for more detailed procedure.

2.2 Basic Idea of PDE Control on Fixed Boundary

25

Unstable ODE Cascaded with Diffusion PDE For an example of an unstable ODE cascaded with a diffusion PDE as in the Stefan problem, we consider the following system: ut = uxx (x, t),

0 < x < 1,

ux (0, t) = 0,

(2.44) (2.45)

u(1, t) = U (t),

(2.46)

˙ X(t) = AX(t) + Bu(0, t),

(2.47)

where X ∈ Rn is an ODE state, and A ∈ Rn×n and B ∈ Rn×1 are time invariant matrices of a controllable pair. If there exists an eigenvalue of the matrix A which has a strictly positive value on its real part, then there exists at least one element in the state vector X which diverges in time. In such a case, the system becomes unstable, and some actuation is needed to stabilize it. If the control input can directly affect the ODE state, u(0, t) in the right-hand side of (2.47) is replaced by the control input U (t). In such a case, a simple choice of a control input is U (t) = KX(t), where K ∈ R1×n is a control gain to be chosen. Then, the closed-loop system ˙ becomes X(t) = (A + BK)X(t). Hence, by choosing K such that the closed-loop matrix A + BK is “Hurwitz” matrix (i.e., all the eigenvalues have a strictly negative real part), all of the elements in the state X converge to zero, and hence the system is stabilized. However, if the effect of the input is propagated through the heat equation, the cascaded PDE-ODE system (2.44)–(2.47) needs to be considered. The backstepping method can be applied for the control design of such systems as well. Let us introduce the following backstepping transformation:  w(x, t) = u(x, t) −

x

k(x, y)u(y, t)dy − φ(x)T X(t),

(2.48)

0

which maps to wt = wxx (x, t),

0 < x < 1,

(2.49)

wx (0, t) = 0,

(2.50)

w(1, t) = 0,

(2.51)

˙ X(t) = (A + BK)X(t) + Bw(0, t),

(2.52)

where K ∈ R1×n is a control gain. The difference between u-system and w-system is on the system matrix of ODE, namely, A and A + BK. Since A + BK is chosen to be Hurwitz matrix, ODE (2.52) is a stable system under the setting w(0, t) ≡ 0. In addition, the heat equation (2.49) with the boundary conditions (2.50) and (2.51)

26

2 State Feedback Control Design for Stefan System

is stable system as we have discussed in (2.29)–(2.31). Hence, we can see that the target system (2.49)–(2.52) is stable. (See [122] for a rigorous proof using Lyapunov analysis.) Taking the derivatives of (2.48) along the solution of (2.44)–(2.47), we obtain wx (x, t) = ux (x, t) − k(x, x)u(x, t) −

 x 0

kx (x, y)u(y, t)dy − φ  (x)T X(t),



wxx (x, t) = uxx (x, t) − k(x, x)ux (x, t) − kx (x, x) + −

 x 0

(2.53)



d k(x, x) u(x, t) dx

kxx (x, y)u(y, t)dy − φ  (x)T X(t),

(2.54)

wt (x, t) = uxx (x, t) − k(x, x)ux (x, t) + ky (x, x)u(x, t) − (ky (x, 0) + φ(x)T B)u(0, t)  x kyy (x, y)u(y, t)dy − φ(x)T AX(t). (2.55) − 0

Therefore, wt (x, t) − wxx (x, t)   d = 2 k(x, x) u(x, t) − (ky (x, 0) + φ(x)T B)u(0, t) dx  x + (kxx (x, y) − kyy (x, y))u(y, t)dy + (φ  (x)T − φ(x)T A)X(t).

(2.56)

0

Substituting x = 0 in (2.48) and (2.53), we have w(0, t) = u(0, t) − φ(0)T X(t), wx (0, t) = −k(0, 0)u(0, t) − φ  (0)T X(t).

(2.57) (2.58)

On the other hand, by the boundary condition (2.50), we need wx (0, t) = 0. Also, by comparing ODEs of (2.47) and (2.52), we require u(0, t) = KX(t) + w(0, t). Therefore, (2.56)–(2.58) yield kxx (x, y) = kyy (x, y), k(x, x) = 0,

ky (x, 0) + φ(x)T B = 0,

φ  (x)T = φ(x)T A, φ(0)T = K,

φ  (0)T = 0.

The solution to (2.59)–(2.62) is given by

(2.59) (2.60) (2.61) (2.62)

2.2 Basic Idea of PDE Control on Fixed Boundary

27





φ(x) = K 01,n e T



x−y

k(x, y) =

Ax



I 0n,n

 (2.63)

,

φ(z)T Bdz,

(2.64)

0

where 0i,j ∈ Ri×j is a zero matrix, I ∈ Rn×n is an identity matrix, and A ∈ R2n×2n is a matrix defined by  A =

 0n,n A . I 0n,n

(2.65)

Evaluating (2.48) at x = 1, the controller design is derived as  U (t) =

1

k(1, y)u(y, t)dy + φ(1)T X(t)

0



= K 01,n





1  1−y

e 0

Az

0



I 0n,n



 Bdz u(y, t)dy + e

A



I 0n,n



 X(t) . (2.66)

Hence, the designed controller (2.66) stabilizes PDE-ODE cascades given in (2.44)– (2.47).

Stefan-Like Cascaded Diffusion PDE-ODE on Fixed Domain Finally, we introduce PDE-ODE cascades which has a similar structure to the Stefan problem given in (2.15)–(2.18) but on fixed domain. Let us replace the moving boundary terms s(t) in (2.15)–(2.18) with a constant domain length D, and consider the following system: ut = αuxx (x, t),

0 < x < D,

(2.67)

ux (0, t) = −k −1 qc (t),

(2.68)

u(D, t) = 0,

(2.69)

˙ X(t) = −βux (D, t).

(2.70)

Next, we introduce the following backstepping transformation: 

D

w(x, t) = u(x, t) − x

k(x, y)u(y, t)dy − φ(x − D)T X(t),

(2.71)

28

2 State Feedback Control Design for Stefan System

which transforms onto wt = αwxx (x, t),

0 < x < D,

(2.72)

wx (0, t) = 0,

(2.73)

w(D, t) = 0,

(2.74)

˙ X(t) = −cX(t) − βwx (D, t).

(2.75)

Since ODE state X ∈ R is a scalar variable in this system, Hurwitz matrix discussed in the previous PDE-ODE system can be described by the coefficient −c in (2.75) with a control gain c > 0. Taking the derivatives of (2.71) along the solution of (2.67)–(2.70), we obtain 

D

wx (x, t) = ux (x, t) + k(x, x)u(x, t) −

kx (x, y)u(y, t)dy − φ  (x − D)T X(t),

x

(2.76)   d k(x, x) u(x, t) wxx (x, t) = uxx (x, t) + k(x, x)ux (x, t) + kx (x, x) + dx  D − kxx (x, y)u(y, t)dy − φ  (x − D)T X(t), (2.77) x

wt (x, t) = αuxx (x, t) + αk(x, x)ux (x, t) − αky (x, x)u(x, t)  x − (αk(x, D) − φ(x − D)β)ux (D, t) − α kyy (x, y)u(y, t)dy. 0

(2.78) By (2.77) and (2.78), wt (x, t) − αwxx (x, t)   d = − α 2 k(x, x) u(x, t) − (αk(x, D) − φ(x − D)T β)ux (D, t) dx  x +α (kxx (x, y) − kyy (x, y))u(y, t)dy + αφ  (x − D)X(t).

(2.79)

0

Substituting x = D in (2.71) and (2.76), we have w(D, t) = −φ(0)X(t), wx (D, t) = ux (D, t) − φ  (0)X(t).

(2.80) (2.81)

2.3 Backstepping Control of Stefan System

29

Applying the boundary condition (2.74) to (2.80), and comparing ODE of (2.70) with (2.75) by using (2.81), the following conditions are obtained: φ  (x) = 0,

φ(0) = 0,

φ  (0) =

c . β

(2.82)

The solutions are given by φ(x) = k(x, y) =

c x, β

(2.83)

β c φ(x − y) = (x − y). α α

(2.84)

Substituting (2.83) and (2.84) into (2.71), the backstepping transformation is derived as c w(x, t) = u(x, t) − α



D

(x − y)u(y, t)dy −

x

c (x − D)X(t). β

(2.85)

Evaluating the spatial derivative of (2.85) at x = 0 and using (2.68) and (2.73), the control law is obtained by  qc (t) = −c

k α

 0

D

u(y, t)dy +

 k X(t) . β

(2.86)

Motivated by the transformation (2.85) and the control law (2.86), in the following sections we develop the control design and its closed-loop analysis of the Stefan problem.

2.3 Backstepping Control of Stefan System This section presents the main theorem of this book. We provide how the backstepping method for PDEs can be extended from fixed boundary PDE to the moving boundary PDE (Fig. 2.2). Let us state the theorem first. Theorem 2.1 Consider the closed-loop system consisting of the plant (2.1)–(2.4) and the control law    k s(t) k qc (t) = −c (T (x, t) − Tm )dx + (s(t) − sr ) , (2.87) α 0 β where c > 0 is an arbitrary controller gain. Let Assumption 1.1 and Assumption 2.1 hold, and assume that the initial conditions (T0 (x), s0 ) are compatible with the

30

2 State Feedback Control Design for Stefan System

Fig. 2.2 Block diagram of the state feedback closed-loop control. Both the temperature profile and the interface position are assumed to be available for the control input

control law. Then, the closed-loop system has a unique classical solution which satisfies the model validity conditions T (x, t) >Tm ,

s˙ (t) > 0,

∀x ∈ (0, s(t)),

∀t > 0,

∀t > 0,

s0 < s(t) 0. Motivated by the energy conservation (2.9), we take the time derivative of (2.87) along the solution of (2.1)–(2.4), which yields   k s(t) k (T (x, t) − Tm )dx + (s(t) − sr ) , −c α 0 β    1 1 s(t) 1 s˙ (t)(T (s(t), t) − Tm ) + = − ck Tt (x, t)dx + s˙ (t) , α α 0 β  

d q˙c (t) = dt





s(t)

= − ck

Txx (x, t)dx − Tx (s(t), t) ,

0

=ckTx (0, t), = − cqc (t).

(2.112)

Solving (2.112) leads to qc (t) = qc (0)e−ct .

(2.113)

Since the setpoint restriction (2.11) implies qc (0) > 0, we have qc (t) > 0,

∀t ≥ 0.

(2.114)

Hence, one can deduce that the closed-loop system has a unique classical solution. Then, using Lemma 1.3, the conditions in (2.88) are satisfied. By the control law (2.87), we have k k qc (t) (s(t) − sr ) = − − β c α



s(t)

(T (x, t) − Tm )dx.

(2.115)

0

Applying (2.114) and (2.88) to (2.115), we obtain s(t) < sr for all t > 0. In addition, the second condition in (2.88) implies that s0 < s(t). Combining these two later inequalities leads to (2.89). In the next section, the inequalities (2.88) and (2.89) are used to establish the Lyapunov stability of the target system (2.97)–(2.100).

34

2 State Feedback Control Design for Stefan System

Convergence of Liquid Length to a Desired Value Next, we prove the exponential stability of the closed-loop system based on the analysis of the target system (2.97)–(2.100). We employ the Lyapunov method, which has been applied to general nonlinear ODEs [107] and extended to PDEs [128]. We consider a functional V1 defined by 

1 V1 = 2

s(t)

w(x, t)2 dx.

(2.116)

0

Taking the time derivative of (2.116), we have 

s(t)

1 w(x, t)wt (x, t)dx + s˙ (t)w(s(t), t)2 2 0  s(t)  s(t) c =α w(x, t)wxx (x, t)dx + s˙ (t)X(t) w(x, t)dx β 0 0  s(t)  s(t) c y=s(t) 2 = αw(x, t)wx (x, t)|y=0 − α wx (x, t) dx + s˙ (t)X(t) w(x, t)dx β 0 0  s(t)  s(t) c 2 wx (x, t) dx + s˙ (t)X(t) w(x, t)dx. (2.117) = −α β 0 0

V˙1 =

Next, we consider V2 defined by 1 V2 = 2



s(t)

wx (x, t)2 dx.

(2.118)

0

Taking the time derivative of (2.118), we get 

s(t)

1 wx (x, t)wxt (x, t)dx + s˙ (t)wx (s(t), t)2 2 0  s(t) 1 x=s(t) =wx (x, t)wt (x, t)|x=0 − wxx (x, t)wt (x, t)dx + s˙ (t)wx (s(t), t)2 2 0  s(t) =wx (s(t), t)wt (s(t), t) − α wxx (x, t)2 dx

V˙2 =

−

c s˙ (t)X(t) β

 0

0 s(t)

1 wxx (x, t)dx + s˙ (t)wx (s(t), t)2 . 2

(2.119)

Recall the boundary condition (2.99), i.e., w(s(t), t) = 0. Taking the total time derivative on both sides, we obtain

2.3 Backstepping Control of Stefan System

35

d w(s(t), t) = wt (s(t), t) + s˙ (t)wx (s(t), t) = 0, dt

(2.120)

which yields wt (s(t), t) = −˙s (t)wx (s(t), t).

(2.121)

Moreover, the integration in first term in the last line in (2.119) is given by 

s(t)

wxx (x, t)dx = wx (s(t), t).

(2.122)

0

Therefore, plugging (2.121) and (2.122) into (2.119), we arrive at V˙2 = −α



s(t)

wxx (x, t)2 dx −

0

c 1 s˙ (t)X(t)wx (s(t), t) − s˙ (t)wx (s(t), t)2 . β 2 (2.123)

Next, we consider V3 defined by V3 =

1 X(t)2 . 2

(2.124)

Using (2.100), the time derivative of (2.124) is given by ˙ V˙3 =X(t)X(t) = − cX(t)2 − βX(t)wx (s(t), t).

(2.125)

Let V be the functional defined by V = V1 + V2 + pV3 .

(2.126)

By (2.117), (2.123), and (2.125), the time derivative of (2.126) is given by V˙ = − α



s(t)

 wxx (x, t) dx − α

0

c + s˙ (t)X(t) β

2

 0

s(t)

wx (x, t)2 dx − pcX(t)2 − pβX(t)wx (s(t), t)

0 s(t)

c s˙ (t) wx (s(t), t)2 . w(x, t)dx − s˙ (t) X(t)wx (s(t), t) − β 2 (2.127)

Using the fact that s˙ (t) > 0 and applying Young’s inequality yield   p β2 2 2 wx (s(t), t) , cX(t) + −pβX(t)wx (s(t), t) ≤ 2 c

(2.128)

36

2 State Feedback Control Design for Stefan System

c s˙ (t)X(t) β



s(t)

0

⎛ 2 ⎞  2  s(t) s˙ (t) ⎝ c X(t) + w(x, t)dx ≤ w(x, t)dx ⎠ , 2 β 0

c s˙ (t) −˙s (t) X(t)wx (s(t), t) ≤ β 2





2 c X(t) + wx (s(t), t)2 . β

(2.129) (2.130)

Also, by Cauchy-Schwarz inequality, we have 

2

s(t)



s(t)

≤ sr

w(x, t)dx 0

w(x, t)2 dx.

(2.131)

0

Applying (2.128)–(2.131) to (2.127), the following inequality on V is derived: V˙ ≤ − α



s(t) 0





s(t)

wxx (x, t) dx − α

sr + s˙ (t) 2

2



wx (x, t)2 dx −

0

s(t)

0

 c2 2 w(x, t) dx + 2 X(t) . β

pc pβ 2 X(t)2 + wx (s(t), t)2 2 2c

2

(2.132)

Applying Pointcare’s and Agmon’s inequality which give  s(t)  s(t)  s(t) w(x, t)2 dx ≤ 4sr2 0 wx (x, t)2 dx and wx (s(t), t)2 ≤ 4sr 0 wxx (x, t)2 dx, 0 the inequality (2.132) becomes   s(t)   s(t) 2pβ 2 sr pc X(t)2 V˙ ≤ − α − wxx (x, t)2 dx − α wx (x, t)2 dx − c 2 0 0    s(t) 2 sr c + s˙ (t) w(x, t)2 dx + 2 X(t)2 . (2.133) 2 0 β Therefore, by choosing p=

cα , 4β 2 sr

(2.134)

we arrive at α V˙ ≤ − 2 8sr



s(t)

wx (x, t)2 dx −

0



sr + s˙ (t) 2

 0

s(t)

α 4sr2



s(t)

w(x, t)2 dx −

0

c2 w(x, t) dx + 2 X(t)2 β 2



pc X(t)2 2

2.3 Backstepping Control of Stefan System

≤ − bV + a s˙ (t)V ,

37

(2.135)

where   8sr c , a = max 1, α   α b = min ,c . 4sr2

(2.136) (2.137)

However, the second term on the right-hand side of (2.135) does not let us to directly conclude exponential stability. To deal with it, we introduce a new Lyapunov function W defined by W = V e−as(t) .

(2.138)

The time derivative of (2.138) is written as   W˙ = V˙ − a s˙ (t)V e−as(t) ,

(2.139)

and using (2.135) the following estimate can be deduced: W˙ ≤ −bW.

(2.140)

Hence, W (t) ≤ W (0)e−bt , and using (2.89) and (2.138), we obtain V (t) ≤ easr V (0)e−bt .

(2.141)

From the definition of V in (2.126) the following holds:   ||w||2H1 + pX(t)2 ≤ easr ||w0 ||2H1 + pX(0)2 e−bt .

(2.142)

Finally, with the help of (2.89), the direct transformation (2.95) and its associated inverse transformation (2.110)–(2.111) combined with Young’s and CauchySchwarz inequalities enable us to state the existence of a positive constant D > 0 such that   ||u||2H1 + X(t)2 ≤ D ||u0 ||2H1 + X(0)2 e−bt , (2.143) which completes the proof of Theorem 2.1.

38

2 State Feedback Control Design for Stefan System

2.4 Gain Tuning to Avoid Input Saturation and Evaporation In practical control systems, the capability of the actuator is often limited to a certain range. Thus is related to as “input saturation” [89]. The Stefan problem, as a melting process, is particular in this regard. The heat input should not go beyond a given upper bound, while it is feasible to assume that the lower bound is zero, i.e., the actuator does not work as a cooler. Furthermore, the liquid temperature must be lower than the boiling temperature to avoid an evaporation which is another phase transition, from liquid to gas. Such an overall input and state constrained problem, from control algorithm perspective, can be treated by restricting the control gain. First, we state the following well-known lemma for the Stefan problem. Lemma 2.2 If Tm ≤ T0 (x) ≤ T¯0 (1 − x/s0 ) + Tm and 0 ≤ qc (t) ≤ q¯ for all t ≥ 0, then Tm ≤ T (x, t) ≤ T¯ (x, t) := K(s(t) − x) + Tm ,

(2.144)

∀x ∈ (0, s(t)), ∀t ≥ 0, where K = max{q/k, ¯ T¯0 /s0 }. Proof Let v(x, t) := T¯ (x, t) − T (x, t). Taking the time and second spatial derivatives yields vt = K s˙ (t) − Tt (x, t),

vxx = −Txx (x, t).

(2.145)

Since 0 ≤ qc (t), we have s˙ (t) ≥ 0. Thus, we obtain vt ≥αvxx , vx (0, t) ≤0,

v(s(t), t) = 0.

(2.146) (2.147)

Applying the maximum principle to (2.146)–(2.147), we can state that if v(x, 0) ≥ 0 for all x ∈ (0, s0 ), then v(x, t) ≥ 0 for all x ∈ (0, s(t)) and all t ≥ 0, which concludes Lemma 2.2. Next, we state the following theorem on the input and state constraint on the closed-loop analysis of the designed control law. Theorem 2.1 Assume T¯0 ≤ ss0r (Tb − Tm ), where Tb is the boiling temperature. By choosing the control gain c > 0 as 0 −1 and ε2 > −1. Theorem 2.2 Consider the closed-loop system (2.151)–(2.154) and the control law (2.87) under Assumptions 1.1 and 2.1. Then, for any pair of perturbations (ε1 , ε2 ) such that ε1 ≥ ε2 , and for any control gain c satisfying 0 < c ≤ c∗ , where ∗

c =



3 10

1/4

α 1 + ε1 , 8sr2 ε1 − ε2

the closed-loop system is exponentially stable in the sense of the H1 norm (2.90). Proof The perturbed reference error system (u, X) is given by ut (x, t) =α(1 + ε1 )uxx (x, t),

0 ≤ x ≤ s(t),

(2.155)

40

2 State Feedback Control Design for Stefan System

−kux (0, t) =qc (t),

(2.156)

u(s(t), t) =0,

(2.157)

s˙ (t) = − β(1 + ε2 )ux (s(t), t).

(2.158)

Taking the spatial and time derivatives of the backstepping transformation (2.95) along the solution to the “perturbed system” (2.155)–(2.158), we obtain  c s(t) c wx (x, t) =ux (x, t) − u(y, t)dy − X(t) α x β c wxx (x, t) =uxx (x, t) + u(x, t), α

(2.159) (2.160)

wt (x, t) =α(1 + ε1 )uxx (x, t) + c(1 + ε1 )u(x, t) +

c(ε1 − ε2 ) c s˙ (t)(x − s(t)) + s˙ (t)X(t). β(1 + ε2 ) β

(2.161)

Hence, the associated target system is derived as wt (x, t) =α(1 + ε1 )wxx (x, t) +

c ε1 − ε2 c s˙ (t)X(t) + s˙ (t)(x − s(t)), β β 1 + ε2 (2.162)

wx (0, t) =0,

(2.163)

w(s(t), t) =0,

(2.164)

˙ X(t) = − c(1 + ε2 )X(t) − β(1 + ε2 )wx (s(t), t).

(2.165)

Next, we prove that the control law (2.87) applied to the perturbed system (2.151)–(2.154) satisfies (2.114) and (2.89). Taking the time derivative of (2.87) along with (2.151)–(2.154), we arrive at q˙c (t) = −c(1 + ε1 )qc (t) − ck (ε1 − ε2 ) ux (s(t), t).

(2.166)

The positivity of the control law (2.87) applied to the perturbed system (2.151)– (2.154) can be shown using a contradiction argument. Assume that there exists t1 > 0 such that qc (t) > 0, ∀t ∈ (0, t1 ), and qc (t1 ) = 0. Then, Lemma 1.3 leads to ux (s(t), t) < 0, ∀t ∈ (0, t1 ). Since ε1 ≥ ε2 , (2.166) implies that q˙c (t) ≥ −c(1 + ε1 )qc (t),

∀t ∈ (0, t1 ).

(2.167)

Using the comparison principle, (2.167) and Assumption 2.1 lead to qc (t1 ) ≥ qc (0)e−c(1+ε1 )t1 > 0.

(2.168)

2.5 Robustness to Diffusivity and Latent Heat Mismatch

41

Thus qc (t1 ) = 0 which is in contradiction with the assumption qc (t1 ) = 0. Consequently, (2.114) holds by this contradiction argument. Accordingly, (2.89) is established using (2.114) and the control law (2.87). Now, consider V1 defined by V1 =

1 ||w||2L2 . 2

(2.169)

The time derivative is obtained by V˙1 = − α(1 + ε1 )||wx ||2L2 − ce¯ c + s˙ (t)X(t) β





c X(t) + wx (s(t), t) β



s(t)

(x − s(t))w(x, t)dx

0

s(t)

(2.170)

w(x, t)dx, 0

where e¯ = ε1 − ε2 .

(2.171)

By Young’s and Cauchy inequalities, we have 

  s(t) c X(t) + wx (s(t), t) (x − s(t))w(x, t)dx β 0   2  c c2 e¯2 γ1 s(t) 2 X(t) + wx (s(t), t) + ≤ (x − s(t))2 dx||w||2 γ1 β 2 0   2 c c2 e¯2 γ1 sr3 2 X(t) + wx (s(t), t) + ||w||2 ≤ γ1 β 6   2 c 2γ1 sr5 c2 e¯2 2 2 ||wx ||L2 + X(t) + wx (s(t), t) . ≤ (2.172) 3 γ1 β − ce¯

Choosing γ1 =

3α(1 + ε1 ) 4sr5

(2.173)

and applying to (2.170), we get α(1 + ε1 ) 4sr5 c2 e¯2 V˙1 ≤ − ||wx ||2L2 + 2 3α(1 + ε1 )



 2 c 2 X(t) + 4sr ||wxx || β

42

2 State Feedback Control Design for Stefan System

+

c s˙ (t)X(t) β



s(t)

(2.174)

w(x, t)dx. 0

Consider V2 defined by V2 =

1 2



s(t)

wx (x, t)2 dx.

(2.175)

0

The time derivative is obtained by c ¯ t) + cew ¯ x (s(t), t)w(0, t) V˙2 = −α(1 + ε1 )||wxx ||2L2 + ecX(t)w(0, β   c 1 + s˙ (t) − X(t)wx (s(t), t) − wx (s(t), t)2 . (2.176) β 2 Applying Young’s and Agmon’s inequality leads to

+ 2γ2 sr ||wxx ||2L2



2 c ecw(0, ¯ t) β  2 1 s˙ (t) c 2 + ¯ t)) + X(t) . (cew(0, 2γ2 2 β

γ1 1 V˙2 ≤ − α(1 + e1 )||wxx ||2L2 + X(t)2 + 2 2γ1

(2.177)

Thus, choosing γ2 =

α(1 + e1 ) , 4sr

(2.178)

it follows that α(1 + e1 ) γ1 1 V˙2 ≤ − ||wxx ||2L2 + X(t)2 + 2 2 2γ1 +



c ecw(0, ¯ t) β

2sr c2 e¯2 c2 w(0, t)2 + s˙ (t) 2 X(t)2 . α(1 + e1 ) 2β

2

(2.179)

Consider Y defined by Y =

1 X(t)2 . 2

The time derivative is obtained and bounded by Y˙ = − (1 + ε2 )cX(t)2 − (1 + ε2 )βX(t)wx (s(t), t)

(2.180)

2.5 Robustness to Diffusivity and Latent Heat Mismatch

≤−

(1 + ε2 )c 2sr (1 + ε2 )β 2 X(t)2 + ||wxx ||2 , 2 c

43

(2.181)

where Young’s inequality and Agmon’s inequality are used from the first line to the second line. Thus, choosing V4 = V2 + pY,

(2.182)

with setting p as p=

cα(1 + ε1 ) , 8sr (1 + ε2 )β 2

(2.183)

we have α(1 + ε1 ) (1 + ε2 )c γ1 ||wxx ||2L2 − p X(t)2 + X(t)2 V˙4 ≤ − 4 2 2  2 2 2 c 1 2sr c e¯ c2 + ecw(0, ¯ t) + w(0, t)2 + s˙ (t) 2 X(t)2 . 2γ1 β α(1 + ε1 ) 2β

(2.184)

Choosing γ1 = p

(1 + e2 )c , 2

(2.185)

we get α(1 + e1 ) (1 + e2 )c ||wxx ||2L2 − p X(t)2 V˙4 ≤ − 4 4 +

10c2 e¯2 sr c2 w(0, t)2 + s˙ (t) 2 X(t)2 . α(1 + e1 ) 2β

(2.186)

Let V be the overall Lyapunov functional defined by V = dV1 + V4 .

(2.187)

The time derivative satisfies the following inequality:    s(t) 2 e¯ 2 s 2  c ) α(1 + e 40c 1 r 2 ||wx ||L2 + d V˙ ≤ − d w(x, t)dx − s˙ (t)X(t) 2 dα(1 + e1 ) β 0   16dsr f (e) ¯ α(1 + e1 ) 1− ||wxx ||2L2 − 4 α(1 + e1 ) 

44

2 State Feedback Control Design for Stefan System

  32dsr f (e) ¯ c2 α(1 + e1 ) c2 2 1 − X(t) − + s ˙ (t) X(t)2 , α(1 + e1 ) 32β 2 sr 2β 2

(2.188)

where f (e) ¯ =

4sr5 c2 e¯2 . 3α(1 + e1 )

(2.189)

Setting d=

160c2 e¯2 sr2 α 2 (1 + e1 )2

(2.190)

leads to    c 4  α(1 + ε1 ) α(1 + ε1 ) ||wxx ||2L2 ||wx ||2L2 − 4− ∗ 4 12 c     c 4  c2 α(1 + ε1 ) c2 2− ∗ X(t)2 + s˙ (t) d 2 sr2 ||w||2L2 + 2 X(t)2 . − 2 64sr c β β (2.191)

V˙ ≤ − d



From (2.191) we deduce that for all 0 < c < c∗ , there exists positive parameters a and b such that V˙ ≤ − bV + a s˙ (t)V .

(2.192)

The exponential stability of the target system (2.162)–(2.165) can be straightforwardly established following the proof procedure used in (2.138)–(2.142), which completes the proof of Theorem 2.2.

2.6 Numerical Simulation Following the numerical method presented in Sect. 1.6, the simulation of the closedloop system is performed by applying the feedback control law to the boundary heat input time sequence. The initial values are set to s0 = 1 cm, T0 (x) = T¯ (1−x/s0 )+Tm with T¯ = 100 ◦ C, and the setpoint is chosen as sr = 35 cm which satisfies the setpoint restriction (2.11).

2.6 Numerical Simulation

45

40

30

20

10

0 0

0.5

1

1.5

2

2.5

3

2

2.5

3

(a) 40

30

20

10

0 0

0.5

1

1.5

(b) Fig. 2.3 The moving interface responses of the plant (2.151)–(2.154) with the open-loop pulse input (2.13) (dashed line) and the backstepping control law (2.87) (solid line) in Neumann boundary actuation. (a) The plots with accurate parameters (1 , 2 ) = (0, 0). Our backstepping control achieves the faster convergence of the interface position. (b) The plots under parameters perturbation (1 , 2 ) = (0.3, −0.2). Our backstepping control is robust to uncertainties of the system parameters

Comparison of the Pulse Input and the Backstepping Control Law Figure 2.3 shows the responses of the plant (2.151)–(2.154) with the open-loop pulse input (2.13) (dashed line) and the backstepping control law (2.87) (solid line). The time window of the open-loop pulse input is set to 50 min. The gain of the backstepping control law is chosen sufficiently small, c = 0.001, to avoid numerical instabilities. Figure 2.3a shows the response of s(t) without the

46

2 State Feedback Control Design for Stefan System

parameters perturbations, i.e., (ε1 , ε2 ) = (0, 0) and clearly demonstrates that s(t) converges to sr applying both rectangular pulse input and backstepping control law. However, the convergence speed is faster with the backstepping control. Moreover, from the dynamics of s(t) under parameters’ perturbations (ε1 , ε2 ) = (0.3, −0.2) shown in Fig. 2.3b, it can be seen that the convergence of s(t) to sr is only achieved with the backstepping control law. On both Fig. 2.3a, b, the responses with the backstepping control law show that the interface position converges faster without the overshoot beyond the setpoint, i.e., s˙ (t) > 0 and s0 < s(t) < sr , ∀t > 0.

Closed-Loop System’s Validity with Respect to the Physical Constraints The dynamics of the controller qc (t) and the temperature at the initial interface T (s0 , t) with the backstepping control law (2.87) are described in Fig. 2.4a b, respectively, for the system without parameter’s uncertainties, i.e., (ε1 , ε2 ) = (0, 0) (red) and the system with parameters’ mismatch (ε1 , ε2 ) = (0.3, −0.2) (blue). As presented in Fig. 2.4a, the boundary heat controller qc (t) remains positive, i.e., qc (t) > 0 in both cases. Moreover, Fig. 2.4b shows that T (s0 , t) converges to Tm with T (s0 , t) > Tm for the system with accurate parameters and the system with uncertainties on the parameters. Physically, Fig. 2.4b means that the temperature at the initial interface location increases above the melting temperature Tm , which enables the melting of the solid phase to the setpoint sr . After this significant transient dynamics, T (s0 , t) settles back to Tm . An identical behavior is observed when the system is subject to parameters’ uncertainty. Therefore, the numerical results are consistent with our theoretical result.

2.7 Boundary Temperature Actuation Some actuators such as a thermo-electric cooler require the direct controlling of the temperature at the boundary, which corresponds to a Dirichlet boundary control problem [21], noted as PII in Sect. 1.2. In this section, backstepping feedback control for PII is developed. We define the control problem consisting of the following system: Tt (x, t) =αTxx (x, t),

0 ≤ x ≤ s(t),

T (0, t) =Tc (t) + Tm , T (s(t), t) =Tm , s˙ (t) = − βTx (s(t), t),

(2.193) (2.194) (2.195) (2.196)

2.7 Boundary Temperature Actuation

10

8

47

5

6

4

2

0

0

0.5

1

1.5

2

2.5

3

2

2.5

3

(a) 750 700 650 600 550 500 450 400 0

0.5

1

1.5

(b) Fig. 2.4 The closed-loop responses with accurate parameters (red) and parameters perturbation (blue) under the backstepping control in Neumann boundary actuation. (a) Positivity of the controller remains, i.e., qc (t) > 0. (b) T (s0 , t) warms up from Tm and returns to it

where Tc (t) is a controlled temperature relative to the melting temperature. As shown in Lemma 1.3, the designed temperature controller needs to ensure the positivity, i.e., the following conditions are required to hold as physical constraints: Tc (t) > 0,

(2.197)

s0 < s(t) < sr .

(2.198)

48

2 State Feedback Control Design for Stefan System

Setpoint Restriction For boundary temperature control, the conservation law obeys the following: d dt



1 α



s(t) 0

1 s(t)2 x(T (x, t) − Tm )dx + 2β

 = Tc (t).

(2.199)

Considering the same control objective T (x, t) → Tm ,

s(t) → sr ,

(2.200)

taking the limit of (2.199) from 0 to ∞ yields 

∞

ΔE =

Tc (t)dt,

(2.201)

0

s 1 (sr2 − s02 ) − α1 0 0 x(T0 (x) − Tm )dx. Hence, by imposing the where ΔE := 2β physical constraint (2.197), the least restrictive condition for the choice of setpoint is derived, and the open-loop stabilization is presented in the following text. Lemma 2.3 Consider an open-loop setpoint control law Tc (t) which satisfies (2.201). Then, for any setpoint sr satisfying sr >

s02 +

2β α



s0

x(T0 (x) − Tm )dx,

(2.202)

0

the control objective (2.200) is satisfied. As in Sect. 2.1, a simple rectangular pulse input achieves (2.200). Such a control action given by  Tc (t) =

 T¯ for t ∈ [0, ΔE/T¯ ] , 0 for t > ΔE/T¯

(2.203)

stands as an open-loop “energy shaping” approach.

State Feedback Controller Design First, we suppose that the physical parameters are accurately known and state the following theorem. Theorem 2.3 Consider a closed-loop system consisting of the plant (2.193)– (2.196) and the control law

2.7 Boundary Temperature Actuation



1 Tc (t) = −c α



s(t)

0

49

 1 x (T (x, t) − Tm ) dx + s(t) (s(t) − sr ) , β

(2.204)

where c > 0 is the controller gain under Assumption 1.1. Then, for any reference setpoint sr and control gain c which satisfy  β s0 x sr >s0 + (T0 (x) − Tm )dx, α 0 s0 α c≤ √ , 2 2sr

(2.205) (2.206)

respectively, the closed-loop system is exponentially stable in the sense of the norm (2.90). Proof We use the same backstepping transformation as in (2.95), which leads to the following target system: wt (x, t) =αwxx (x, t) +

c s˙ (t)X(t), β

(2.207)

w(0, t) =0,

(2.208)

w(s(t), t) =0,

(2.209)

˙ X(t) = − cX(t) − βwx (s(t), t)

(2.210)

and the control law (2.204). Next, we show that the physical constraints (2.197) and (2.198) are insured if (2.205) holds. Taking the time derivative of (2.204), we have c T˙c (t) = −cTc (t) − s˙ (t)X(t). β

(2.211)

Assume that ∃t2 such that Tc (t) > 0, ∀t ∈ (0, t2 ) and Tc (t2 ) = 0. Then, by Lemma 1.3, we get u(x, t) > 0 and s˙ (t) > 0 for ∀t ∈ (0, t2 ). Hence, s(t) > s0 > 0. Applying these inequalities to (2.204), we deduce X(t) < 0, ∀t ∈ (0, t2 ). Hence, (2.211) verifies the differential inequality T˙c (t) > −cTc (t), ∀t ∈ (0, t2 ). Comparison principle and (2.205) yield Tc (t2 ) > Tc (0)e−ct2 > 0 in contradiction to Tc (t2 ) = 0. Therefore, t2 such that Tc (t) > 0 for ∀t ∈ (0, t2 ) and Tc (t2 ) = 0, which implies Tc (t) > 0, ∀t > 0 assuming (2.205). Finally, we consider a functional V =

d 1 p ||w||2L2 + ||wx ||2L2 + X(t)2 . 2 2 2

(2.212)

50

2 State Feedback Control Design for Stefan System

With an appropriate choice of the positive parameters d and p, the time derivative of (2.212) yields  α √ dα ||w||2H1 V˙ ≤ − − 2csr ||wxx ||2 − 2 2(4sr2 + 1)   2 c αc2 d 2 sr2 2 2 . − 2 X(t)2 + s˙ (t) X(t) + ||w|| 2 4β β2

(2.213)

Thus, choosing the controller gain to satisfy (2.206), it can be verified that there exist positive constants b and a such that V˙ ≤ − bV + a s˙ (t)V .

(2.214)

Similarly in the Neumann boundary actuation case, under the physical constraint (2.197), the exponential stability of the target system (2.207)–(2.210) can be established from the inequality (2.214), which completes the proof of Theorem 2.3.

Robustness to Parameter Uncertainty Next, we investigate the controller (2.204) to perturbations on the plant’s physical parameters α and β, considering the following perturbed system: Tt (x, t) =α(1 + ε1 )Txx (x, t),

0 ≤ x ≤ s(t),

T (0, t) =Tc (t) + Tm ,

(2.215) (2.216)

T (s(t), t) =Tm ,

(2.217)

s˙ (t) = − β(1 + ε2 )Tx (s(t), t),

(2.218)

where ε1 and ε2 are parameter perturbations such that ε1 > −1 and ε2 > −1. Theorem 2.4 Consider the closed-loop system consisting of the plant (2.215)– (2.218) and the control law (2.204) under the assumption on (2.205) to hold. Then, for any perturbations (ε1 , ε2 ) that satisfy ε1 ≥ ε2 , there exists c¯∗ > 0 such that for all controller gain values c satisfying 0 < c ≤ c¯∗ , the closed-loop system is exponentially stable in the sense of the norm (2.90). Proof By the same transformation (2.95), the target w-system is given by wt (x, t) =α(1 + ε1 )wxx (x, t) + w(0, t) =0,

c ε1 − ε2 c s˙ (t)X(t) + s˙ (t)(x − s(t)), β β 1 + ε2 (2.219) (2.220)

2.7 Boundary Temperature Actuation

51

w(s(t), t) =0,

(2.221)

˙ X(t) = − c(1 + ε2 )X(t) − β(1 + ε2 )wx (s(t), t).

(2.222)

To prove the physical constraints (2.197) and (2.198), taking the time derivative of (2.204) along the system (2.215)–(2.218), we obtain c T˙c (t) = − c(1 + ε1 )Tc (t) − s˙ (t)X(t) β − c(ε1 − ε2 )ux (s(t), t).

(2.223)

Thus, the inequality ε1 ≥ ε2 enables to state the positivity of the controller Tc (t) > 0 and the physical constraints (2.197) and (2.198) are verified. Finally, we consider the functional defined in (2.212). After a lengthy calculation and applying inequalities in a similar way as in Sect. 2.5, with an appropriate choice of d and p and imposing c < c1 where c1 :=

α(1 + ε2 ) , √ 2 2sr

(2.224)

we have  dα(1 + ε1 ) α(1 + ε1 )  2 − Ac3 ||wxx ||2L2 ||wx ||2L2 − V˙ ≤ − 4 8   2 c α(1 + ε1 ) 3 2 − Ac − − Bc X(t)2 32β 2 sr   c2 2 2 2 2 + s˙ (t) d sr ||w||L2 + 2 X(t) , β

(2.225)

where √ 29 2sr6 (1 + sr )(ε1 − ε2 )2 A= , 3α 3 (1 + ε1 )2 (1 + ε2 ) √ 16 2sr2 B= . α(1 + ε2 )

(2.226) (2.227)

Let c2 be a positive root of Ac23 + Bc2 = 1.

(2.228)

Then, for 0 < ∀c < c¯∗ := min{c1 , c2 }, there exist positive constants a¯ and b¯ which ¯ + a¯ s˙ (t)V , which concludes the proof of Theorem 2.4. verifies V˙ ≤ −bV

52

2 State Feedback Control Design for Stefan System 40

30

20

10

0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

40

30

20

10

0

Fig. 2.5 The moving interface response of the plant (2.193)–(2.196) with the open-loop pulse input (2.203) (dashed line) and the backstepping control law (2.204) (solid line), under the accurate parameters (1 , 2 ) = (0, 0) (top) and under the parameter perturbations (1 , 2 ) = (0.3, −0.2). We observe the faster convergence and the parametric robustness of our backstepping control law

Numerical Simulation Numerical simulation is studied for the designed Dirichlet boundary actuation. Analogous plots to the ones given in Sect. 2.6 are depicted in Figs. 2.5 and 2.6, respectively. We observe similar good performance and properties to the ones for Neumann boundary actuation in terms of the convergence to the setpoint and the validity of the physical constraints.

2.8 Stefan-Like Problem with Dirichlet Interconnection

53

1800 1600 1400 1200 1000 800 600 400 0

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

1600 1400 1200 1000 800 600 400 0

Fig. 2.6 The closed-loop responses with accurate parameters (red) and parameters perturbation (blue) under the backstepping control in Dirichlet boundary actuation. Both positivity of the control input and the constraints of temperature are validated

2.8 Stefan-Like Problem with Dirichlet Interconnection In this section, we study whether the procedure for control design and analysis of the Stefan problem is applicable to an analogous moving boundary problem. Specifically, one might be interested in the following diffusion PDE under Dirichlet interconnection on the moving boundary dynamics: ut (x, t) =αuxx (x, t), u(0, t) =U (t), ux (s(t), t) =0,

0 < x < s(t)

(2.229) (2.230) (2.231)

54

2 State Feedback Control Design for Stefan System

s˙ (t) = − βu(s(t), t).

(2.232)

The dynamics (2.232) leads to the shrinking moving interface under the positivity of PDE state, contrary to the Stefan problem. Applying maximum principle, the following lemma is stated. Lemma 2.4 If u0 (x) > 0 and U (t) > 0 for all t > 0, then u(x, t) > 0, ∀x ∈ (0, s(t)) and s˙ (t) < 0, ∀t > 0. Therefore, we consider the stabilization of the interface position s(t) driven “back” to a setpoint sr . The following theorem is presented. Theorem 2.5 Assume s0 > 0, u0 (x) > 0 for ∀x ∈ (0, s0 ), and that the setpoint is chosen to satisfy 0 < sr < s0 −

β α



s0

xu0 (x)dx.

(2.233)

0

Then, the closed-loop system under the control law  U (t) = −c

1 α



 1 xu(x, t)dx − (s(t) − sr ) , β

s(t)

0

(2.234)

satisfies the following properties U (t) >0,

u(x, t) > 0,

s˙ (t) 0. Assume there exists t ∗ > 0 such that s(t) > sr for ∀t ∈ [0, t ∗ ) and s(t ∗ ) = sr . Then, (2.239) yields U˙ (t) > −cU (t) for ∀t ∈ [0, t ∗ ], which leads to U (t) > U (0)e−ct for ∀t ∈ [0, t ∗ ]. By Lemma 2.4, it holds that u(x, t) > 0, ∀x ∈ (0, s(t)), ∀t ∈ [0, t ∗ ). Thus, by (2.234), we have U (0)e−ct
0 due to the setpoint condition (2.233), the inequality (2.240) contradicts with the imposed assumption s(t ∗ ) = sr . Thus, there does not exist such a finite time t ∗ , which yields s(t) > sr for ∀t > 0. Applying this to (2.239) and using comparison principle, the inequalities (2.235) are satisfied. By Lemma 2.4, the inequality (2.236) is derived, and finally (2.237) is proved by applying these inequalities to (2.234). Next, we consider the following transformation: w(x, t) = u(x, t) +

c α



s(t)

(y − x)u(y, t)dy −

x

c X(t). β

(2.241)

Taking the time and spatial derivatives, we obtain the following target system:   c c X(t) + w(s(t), t) , wt (x, t) =αwxx (x, t) + s˙ (t)(s(t) − x) α β

(2.242)

w(0, t) =0,

(2.243)

wx (s(t), t) =0,

(2.244)

˙ X(t) = − cX(t) − βw(s(t), t).

(2.245)

Finally we prove the stability of the target system (2.242)–(2.245) by utilizing the proven inequalities (2.236) and (2.237). Consider 1 1 ||w||2 = V1 = 2α 2α



s(t)

w(x, t)2 dx.

(2.246)

0

The time derivative is given by 

s˙ (t) w(s(t), t)2 2α 0   s(t)  c c X(t) + w(s(t), t) + 2 s˙ (t) (s(t) − x)w(x, t)dx. β α 0

V˙1 = −

s(t)

wx (x, t)2 dx +

(2.247)

By Young’s inequality with the help of s˙ (t) < 0, we have   s(t) c X(t) + w(s(t), t) (s(t) − x)w(x, t)dx s˙ (t) β 0 ⎛ 2 ⎞  2  s(t) s˙ (t) ⎝ 1 c ≤− X(t) + w(s(t), t) + (s(t) − x)w(x, t)dx ⎠ γ 2 β γ 0 

56

2 State Feedback Control Design for Stefan System

 c2 γ s(t)3 2 2 2 ||w|| . ≤ − s˙ (t) X(t) + γ w(s(t), t) + 2γ β2 

(2.248)

Applying (2.248) to (2.247), the following bound is obtained: V˙1 ≤ −



s(t)

0

c − s˙ (t) 2 α Setting γ =

α 2c

s˙ (t) w(s(t), t)2 2α

wx (x, t)2 dx + 

 c2 γ s(t)3 2 2 2 ||w|| . X(t) + γ w(s(t), t) + 2γ β2

(2.249)

leads to

c V˙1 ≤ − ||wx ||2 − s˙ (t) 2 α



 cα cs(t)3 2 2 . ||w|| X(t) + α 2β 2

(2.250)

Consider Y =

1 X(t)2 . 2

(2.251)

Taking the time derivative and applying Young’s and Agmon’s inequalities yield c 2β 2 s0 ||wx ||2 . Y˙ = −cX(t)2 + βX(t)w(s(t), t) ≤ − X(t)2 + 2 c

(2.252)

Therefore, by defining V = V1 + pY

(2.253)

c , 4β 2 s0

(2.254)

with p= we have   2 c2 s03 1 c pc 2 2 2 2 X(t) − s˙ (t) V˙ ≤ − ||wx || − X(t) + 3 ||w|| 2 2 2αβ 2 α ≤ − aV − b˙s (t)V .

(2.255)

As in the previous way, consider W = V ebs(t) and taking the time derivative with applying (2.255), it holds that W˙ ≤ −aW.

(2.256)

2.9 Comments and Remarks

57

Thus, by (2.236) and (2.237), the following estimate of the norm is derived: V ≤ V0 eb(s0 −sr ) e−at ,

(2.257)

from which we conclude the exponential stability stated in Theorem 2.5. Numerical simulation is studied by using the same value of α and β as of zinc. Figure 2.7 shows that under the closed-loop system the interface position is driven from s0 = 10 cm to the setpoint sr = 2 cm without overshoot. Figure 2.8 illustrates that the designed controller maintains the positivity as proven in the theorem.

2.9 Comments and Remarks While the numerical analysis of the one-phase Stefan problem is broadly covered in the literature, there have been only a few control results. In addition, most of the proposed control approaches are based on finite-dimensional approximations with the assumption of an explicitly given moving boundary dynamics [9, 44, 51, 160]. Diffusion-reaction processes with an explicitly known moving boundary dynamics are investigated in [9] based on the concept of inertial manifold [42] and the partitioning of the infinite-dimensional dynamics into slow and fast finitedimensional modes. Motion planning boundary control has been pursued in [160] to ensure asymptotic stability of a one-dimensional one-phase nonlinear Stefan problem assuming a prior known moving boundary and deriving the manipulated input from the solutions of the inverse problem. However, the series representation introduced in [160] leads to highly complex solutions that reduce controller design possibilities. For control objectives, infinite-dimensional frameworks that lead to significant challenges in the process characterization have been developed for the stabilization

10

8

6

4

2 0

2

4

6

8

10

12

Fig. 2.7 The interface response of the Stefan-like problem with Dirichlet interconnection. s(t) is driven back to the setpoint sr = 2 cm

58

2 State Feedback Control Design for Stefan System 600 500 400 300 200 100 0 0

2

4

6

8

10

12

Fig. 2.8 The closed-loop response of the control input of the Stefan-like problem with Dirichlet interconnection. Positivity of U (t) is maintained

of the temperature profile and the moving interface of the Stefan problem. An enthalpy-based boundary feedback control law that ensures asymptotic stability of the temperature profile and the moving boundary at the desired reference has been employed in [161, 163]. Lyapunov analysis is performed in [141] based on a geometric control approach which enables to adjust the position of a liquid–solid interface to the desired setpoint while exponentially stabilizing the L2 -norm of the distributed temperature. However, the results in [141] are stated based on physical assumptions on the liquid temperature being greater than the melting point, which needs to be guaranteed by proving strictly positive boundary input. The significant contribution of the proposed method using backstepping is to prove all the physical properties which need to be addressed and global exponential stability (for all initial temperature profiles above freezing) of the closed-loop system by focusing on the stability analysis of the target system. This chapter presented control designs for the one-phase Stefan problem via backstepping method. The novelties of our results are summarized below: 1. A new approach to globally stabilizing a class of nonlinear parabolic PDEs with moving boundary via a nonlinear backstepping transformation is proposed. 2. The closed-loop responses satisfy the physical constraints needed for the validity of the model. 3. A novel formulation of the Lyapunov function for moving boundary PDEs was applied and it showed the exponential stability of the closed loop system. Even though our state feedback controller for the Neumann boundary actuation is the same as the one proposed in [162], we ensure the exponential stability of the interface and temperature in H1 norm, which is stronger than the asymptotic stability presented in [162]. The application of extremum seeking control with static maps to the Stefan problem following the recent results of [154] could be an interesting design that can be applied to the optimization of phase-change phenomena in building use [133].

Chapter 3

State Estimator Design for Stefan System

The state feedback control presented in Chap. 2 requires the entire profile of the temperature in the liquid phase as a given information. Some imaging-based sensors such as thermographic camera (a.k.a. infrared camera or IR camera) enables to capture the temperature profile; however, they include relatively high noise as compared to single point thermal sensors, such as thermocouples. Thus, estimating the entire temperature profile given a boundary measurement is a significant task for the implementation of the control algorithm. The problem of estimating variables of interest given some measured value is widely known as “state estimation.” One of the most popular state estimation methods is the “Kalman filter” [95] which is an optimal filter for linear dynamical systems with white Gaussian noise in the model and measurements. Another wellknown concept is the “Luenberger observer” [140], which stabilizes the estimation error at zero in linear deterministic systems. In finite-dimensional systems, the observer gain is designed by means of pole placement or a linear matrix inequality [23]. In this book, we focus on the state estimation of the Stefan system by a Luenberger-type observer approach with designing the observer gain via the backstepping method [181]. As a comparison, we also introduce the Extended Kalman Filter (EKF) for a reduced-order model of the Stefan system.

3.1 Basic Idea of PDE Estimation on Fixed Boundary Consider the unstable reaction-diffusion PDE presented in Sect. 2.2 ut =uxx (x, t) + λu(x, t),

0 < x < 1,

(3.1)

u(0, t) =0,

(3.2)

u(1, t) =U (t),

(3.3)

© Springer Nature Switzerland AG 2020 S. Koga, M. Krstic, Materials Phase Change PDE Control & Estimation, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-58490-0_3

59

60

3 State Estimator Design for Stefan System

where U (t) is a time-varying input which can be either an open-loop forcing or a feedback control. Suppose that a boundary flux ux (0, t) is available for measurement y(t): y(t) = ux (0, t)

(3.4)

The observer is constructed as a copy of the plant (3.1)–(3.3) plus the product of the observer gain and the measurement error, given by ˆ t) + p1 (x)(y(t) − uˆ x (0, t)), uˆ t =uˆ xx (x, t) + λu(x,

0 < x < 1,

(3.5)

u(0, ˆ t) =0,

(3.6)

u(1, ˆ t) =U (t).

(3.7)

The objective is to find the gain function p1 (x) such that uˆ converges to u. First, we introduce the estimation error variable defined by u˜ = u − u. ˆ

(3.8)

Subtraction of (3.5)–(3.7) from (3.1)–(3.3) yields the estimation error dynamics as ˜ t) − p1 (x)u˜ x (0, t), u˜ t =u˜ xx (x, t) + λu(x,

0 < x < 1,

(3.9)

u(0, ˜ t) =0,

(3.10)

u(1, ˜ t) =0.

(3.11)

Consider the backstepping transformation from a newly defined variable w˜ to the estimation error u, ˜ (which is, in fact, an inverse transformation) given by 

x

u˜ = w˜ −

p(x, y)w(y)dy, ˜

(3.12)

0

where w˜ obeys the following stable diffusion PDE: w˜ t =w˜ xx ,

0 < x < 1,

(3.13)

w(0, ˜ t) =0,

(3.14)

w(1, ˜ t) =0.

(3.15)

Taking the spatial and time derivatives of (3.12), we obtain  u˜ xx =w˜ xx − p(x, x)w˜ x (x) −  − 0

x

pxx (x, y)w(y)dy ˜

 d ˜ p(x, x) + px (x, x) w(x) dx (3.16)

3.1 Basic Idea of PDE Estimation on Fixed Boundary

u˜ t =w˜ xx − p(x, x)w˜ x (x) + p(x, 0)w˜ x (0) + py (x, x)w(x) ˜  x − pyy (x, y)w(y)dy. ˜

61

(3.17)

0

Thus,   d ˜ u˜ t − u˜ xx − λu˜ + p1 (x)u˜ x (0, t) =(p1 (x) + p(x, 0))w˜ x (0) + 2 p(x, x) − λ w(x) dx  x (pxx (x, y) − pyy (x, y) + λp(x, y))w(y)dy. ˜ + 0

(3.18)

By boundary condition, we get p(1, y) = 0.

(3.19)

pxx − pyy = − λp,

(3.20)

λ d p(x, x) = , dx 2

(3.21)

Therefore,

p(1, y) =0.

(3.22)

Introduce a change of coordinates x¯ = 1 − y,

y¯ = 1 − x,

p( ¯ x, ¯ y) ¯ = p(x, y),

(3.23)

we have p¯ x¯ x¯ − p¯ y¯ y¯ =λp, ¯

(3.24)

λ d p( ¯ x, ¯ x) ¯ =− , d x¯ 2

(3.25)

p(x, ¯ 0) =0.

(3.26)

This PDE (3.24) and the boundary conditions (3.25), (3.26) are equivalent to the one introduced in (2.38)–(2.40). Hence, taking back to the original coordinate x and y, the solution is given by p(x, y) = −λ(1 − x)

I1 (z) , z

The observer gain is given by

z :=



λ((1 − y)2 − (1 − x)2 ).

(3.27)

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3 State Estimator Design for Stefan System

p1 (x) = −p(x, 0) = λ(1 − x)

I1

  λ(1 − (1 − x)2 )  . λ(1 − (1 − x)2 )

(3.28)

3.2 Temperature Profile Estimation for the Stefan System In this section, we develop the observer design of temperature profile for the Stefan problem given available measurements of the interface position and the temperature gradient at the interface. While measuring the temperature gradient at the interface lacks on the practical feasibility in sensing technique, the estimator under the setting is relatively easy and enables the analysis of the output feedback control. Recall the Stefan problem modeling the liquid temperature dynamics under the melting, described by Tt (x, t) =αTxx (x, t),

x ∈ (0, s(t)),

(3.29)

−kTx (0, t) =qc (t),

(3.30)

T (s(t), t) =Tm ,

(3.31)

s˙ (t) = − βTx (s(t), t).

(3.32)

Denoting the estimates of the temperature Tˆ (x, t), the following theorem holds: Theorem 3.1 Consider the plant (3.29)–(3.32) with the measurements Y1 (t) = s(t),

Y2 (t) = Tx (s(t), t),

(3.33)

and the following observer   Tˆt (x, t) =α Tˆxx (x, t) + p1 (x, Y1 (t)) Y2 (t) − Tˆx (Y1 (t), t) ,

(3.34)

−k Tˆx (0, t) =qc (t),

(3.35)

Tˆ (Y1 (t), t) =Tm ,

(3.36)

where x ∈ [0, Y1 (t)], and the observer gain p1 (x, Y1 (t)) is     λ 2 − x2 I1 Y (t) 1 α p1 (x, Y1 (t)) = −λY1 (t)    , λ 2 2 α Y1 (t) − x

(3.37)

with a gain parameter λ > 0. Assume that the model validity condition T (x, t) ≥ Tm is satisfied. Then, for all λ > 0, the observer error system has a unique classical solution and is exponentially stable in the sense of the norm

3.2 Temperature Profile Estimation for the Stefan System

||T − Tˆ ||2H1 .

63

(3.38)

Since the observer PDE (3.34)–(3.35) is a cascaded system of the plant PDEODE (3.29)–(3.32), the observer state Tˆ (x, t) admits a classical solution only if the plant states (T (x, t), s(t)) admits a classical solution. Note that the observer gain (3.37) is dependent on the measured value Y1 (t) = s(t), which renders the derivation of the observer gain require online computation.

Observer Gain Derivation by Backstepping Transformation Let us define the estimation error state u˜ as u(x, ˜ t) = T (x, t) − Tˆ (x, t).

(3.39)

Subtraction of (3.34)–(3.36) from (3.29)–(3.31) leads to the estimation error system given by u˜ t (x, t) =α u˜ xx (x, t) − p1 (x, s(t))u˜ x (s(t), t),

(3.40)

u˜ x (0, t) =0,

(3.41)

u(s(t), ˜ t) =0.

(3.42)

As for the full-state feedback case, the backstepping transformation for moving boundary PDEs,  u(x, ˜ t) =w(x, ˜ t) +

s(t)

P (x, y)w(y, ˜ t)dy,

(3.43)

x

is constructed to convert the following exponentially stable target system: ˜ t), w˜ t (x, t) =α w˜ xx (x, t) − λw(x,

(3.44)

w˜ x (0, t) =0,

(3.45)

w(s(t), ˜ t) =0,

(3.46)

into the u-system ˜ (3.40)–(3.42). Taking the derivative of (3.43) with respect to t and x along the solution of (3.44)–(3.46), respectively, for any continuous function w(x, ˜ t), the gain kernel P (x, y) and the observer gain p1 (x, s(t)) must satisfy Pxx (x, y)−Pyy (x, y) +

λ P (x, y) = 0, α

(3.47)

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3 State Estimator Design for Stefan System

P (x, x) =

λ x, 2α

(3.48)

Px (0, y) =0,

(3.49)

p1 (x, s(t)) = − αP (x, s(t)),

(3.50)

in order to map (3.40)–(3.42) into (3.44)–(3.46). Introduce the change of coordinates and the state as x¯ = y,

y¯ = x,

P¯ (x, ¯ y) ¯ = −P (x, y),

λ¯ =

λ . α

(3.51)

Then, the system (3.47)–(3.49) is rewritten using the new coordinates as P¯x¯ x¯ − P¯y¯ y¯ =λ¯ P¯ ,

(3.52)

λ¯ ¯ P¯ (x, ¯ x) ¯ = − x, 2 P¯y¯ (x, ¯ 0) =0.

(3.53) (3.54)

The solution to (3.52)–(3.54) is obtained as in Eqs. (4.64)–(4.66) in [128]. Taking back to the original coordinates and variables, the gain kernel function is solved as  λ P (x, y) = y α

I1 

 λ 2 α (y

− x2) .

λ 2 α (y

(3.55)

− x2)

Finally, using (3.50), the observer gain (3.37) is derived.

Inverse Transformation The inverse transformation is formulated as  w(x, ˜ t) =u(x, ˜ t) −

s(t)

Q(x, y)u(y, ˜ t)dy,

(3.56)

x

where the gain kernel Q(x, y) satisfies λ Qxx (x, y) − Qyy (x, y) = Q(x, y), α λ Q(x, x) = x, 2α

(3.57) (3.58)

3.2 Temperature Profile Estimation for the Stefan System

65

Qx (0, y) =0.

(3.59)

The solution to (3.57)–(3.59) is written as  λ Q(x, y) = y α

J1 

 λ 2 α (y

− x2) ,

λ 2 α (y

(3.60)

− x2)

where J1 (x) is a Bessel function of the first kind.

Temperature Profile Estimate Converges to the Real Temperature To show the stability of the target w-system ˜ (3.44)–(3.46), we consider a functional 1 ˜ 2H1 . V˜ = ||w|| 2

(3.61)

Taking the time derivative of (3.61) along the solution of (3.44)–(3.46) leads to s˙ (t) w˜ x (s(t), t)2 . ˜ 2H1 − V˙˜ = −α||w˜ x ||2H1 − λ||w|| 2

(3.62)

As stated in Lemma 1.1 in Sect. 1.2, the model validity condition T (x, t) ≥ Tm leads to s˙ (t) ≥ 0. Hence, the following differential inequality in V˜ is derived from (3.62): V˙˜ ≤ −λV˜ .

(3.63)

Hence, w-system ˜ (3.44)–(3.46) is exponentially stable, which induces the exponential stability of the original u-system ˜ (3.40)–(3.42), which completes the proof of Theorem 3.1.

Algorithm Development of the Designed Observer The designed observer (3.34)–(3.36) obeys the PDE on moving boundary; however, the numerical model needs to be described by a finite-dimensional state in a discrete time. Following the procedure in Sect. 1.6, the scaled PDE on the fixed domain is first derived, which requires the information on s(t) and s˙ (t). Through the available measurements, these variables are obtained as Y1 (t) and −βY2 (t). Since

66

3 State Estimator Design for Stefan System

the observer utilizes the measured values as inputs, the fixed domain PDE observer can be expressed by vˆt (ξ, t) =

α ξβY2 (t) vˆξ (ξ, t) vˆξ ξ (ξ, t) − 2 Y1 (t) Y1 (t)   vˆξ (1, t) , + p1 (ξ, Y1 (t)) Y2 (t) − Y1 (t)

0 H , the closedloop system is exponentially stable in the sense of the norm ||T − Tˆ ||2H1 + ||T − Tm ||2H1 + (s(t) − sr )2 .

(3.71)

Backstepping Transformation For the output feedback analysis, we introduce the estimator state of a reference error uˆ by defining

68

3 State Estimator Design for Stefan System

uˆ = Tˆ − Tm .

(3.72)

Then, u-system ˆ is described by   uˆ t (x, t) =α uˆ xx (x, t) + p1 (x, Y1 (t)) Y2 (t) − uˆ x (Y1 (t), t) ,

(3.73)

−k uˆ x (0, t) =qc (t),

(3.74)

u(Y ˆ 1 (t), t) =0,

(3.75)

with the observer gain p1 in (3.37). We can see that the estimation error state defined by (3.39) is equivalent to the estimation error in a reference error state, namely, u˜ = T − Tˆ = u − u. ˆ Since u-system ˜ in (3.40)–(3.42) is independent on the control input U (t), by separation principle, the output feedback controller is designed by utilizing the estimator state uˆ instead of the plant state u in the full-state feedback control. The transformation of the variables (u, ˆ X) into (w, ˆ X) is performed using the gain kernel functions of backstepping transformation in state feedback control. Thus, we consider w(x, ˆ t) = u(x, ˆ t) −

c α



s(t)

(x − y)u(y, ˆ t)dy −

x

c (x − s(t))X(t). β

(3.76)

Note that wˆ = w − w˜ due to the different transformation between wˆ and w. ˜ Taking the derivatives of (3.76) along with the solution of (3.73)–(3.75) with the help of the transformation (3.43), the associated target system is obtained by wˆ t (x, t) =α wˆ xx (x, t) +

c s˙ (t)X(t) + f (x, s(t))w˜ x (s(t), t), β

(3.77)

wˆ x (0, t) =0,

(3.78)

w(s(t), ˆ t) =0,

(3.79)

˙ X(t) = − cX(t) − β wˆ x (s(t), t) − β w˜ x (s(t), t),

(3.80)

where c f (x, s(t)) = P (x, s(t)) − α



s(t)

(x − y)P (y, s(t))dy − c(s(t) − x).

(3.81)

x

Evaluating the spatial derivative of (3.76) at x = 0, we derive the output feedback controller as    k s(t) k (3.82) qc (t) = −c u(x, ˆ t)dx + X(t) . α 0 β

3.3 Observer-Based Output Feedback Control Design

69

After a lengthy calculation, one can see that the inverse transformation is also equivalent to the one of the state feedback control, namely β u(x, ˆ t) =w(x, ˆ t) + α



s(t)

ψ(x − y)w(y, ˆ t)dy + ψ(x − s(t))X(t),

(3.83)

x

where the gain kernel is (2.111).

Observer Gain Restriction for Positivity of Heat Input As presented in Chap. 2, the closed loop system must verify the two physical constraints qc (t) >0,

∀t > 0

s0 0.

(3.85)

We derive sufficient conditions to guarantee that the physical constraints (3.84) and (3.85) are not violated when the output feedback control law (3.82) is applied to the plant. First, we state the following lemma. Lemma 3.1 Suppose that w(0, ˜ t) < 0. Then, the solution to (3.44)–(3.46) satisfies w(x, ˜ t) < 0, ∀x ∈ (0, s(t)), ∀t > 0. The proof of Lemma 3.1 is constructed using the maximum principle [156]. Next, we state the following lemma. Lemma 3.2 For any initial temperature estimate Tˆ0 (x) and any observer gain parameter λ satisfying (3.68) and (3.69), respectively, the following properties hold: u(x, ˜ t) < 0, u˜ x (s(t), t) > 0, ∀x ∈ (0, s(t)),

∀t > 0.

(3.86)

Proof Lemma 3.1 states that if w(x, ˜ 0) < 0, then w(x, ˜ t) < 0. In addition, from (3.43), w(x, ˜ t) < 0 leads to u(x, ˜ t) < 0 due to the positivity of the solution to the gain kernel (3.55). Therefore, with the help of (3.56), we deduce that u(x, ˜ t) < 0 if the following holds: 

s0

u(x, ˜ 0)
0. In addition, from the boundary condition (3.42) and Hopf’s lemma, it follows that u˜ x (s(t), t) > 0.

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3 State Estimator Design for Stefan System

The final step is to prove that the output feedback closed-loop system satisfies the physical constraint (3.84). Proposition 3.1 Suppose the initial values Tˆ0 (x) and s0 satisfy (3.68) and the setpoint sr is chosen to satisfy (3.70). Then, the physical constraints (3.84) and (3.85) are satisfied by the closed-loop system consisting of the plant (3.29)–(3.32), the observer (3.34)–(3.36), and the output feedback control law (3.67). Proof Taking the time derivative of (3.82) along with the solution (3.73)–(3.75), with the help of the observer gain (3.50), we obtain 



s(t)

q˙c (t) = −cqc (t) + 1 +

 P (x, s(t))dx u˜ x (s(t), t).

(3.88)

0

From the positivity of the gain kernel solution (3.55) and the Neumann boundary value (3.86), the following differential inequality holds: q˙c (t) ≥ −cqc (t).

(3.89)

Hence, if the initial values satisfy qc (0) > 0, (3.70) is then satisfied from (3.82) and (3.68), and we get qc (t) > 0,

∀t > 0.

(3.90)

Then, using (3.86) given in Lemma 3.2 and the positivity of u(x, t) (see Lemma 1.3), the following inequality is established: u(x, ˆ t) > 0,

∀x ∈ (0, s(t)),

∀t > 0.

(3.91)

Finally, substituting the inequalities (3.90) and (3.91) into (3.82), we arrive at X(t) < 0, ∀t > 0, which guarantees that the second physical constraint (3.85) is satisfied.

Convergence of Estimation Error and the Liquid Length We prove the stability of the overall closed-loop system under the output feedback. We consider 1 Vˆ1 = 2 Its time derivative is

 0

s(t)

w(x, ˆ t)2 dx.

(3.92)

3.3 Observer-Based Output Feedback Control Design

V˙ˆ1 = − α



s(t)

wˆ x (x, t)2 dx +

0



s(t)

+

c s˙ (t)X(t) β

71



s(t)

w(x, ˆ t)dx

0

f (x, s(t))w(x, ˆ t)dx w˜ x (s(t), t),

0

(3.93)

  c2 sr s(t) 2 ≤−α wˆ x (x, t) dx + s˙ (t) X(t) + w(x, ˆ t) dx 2 0 2β 2 0     s(t) s(t) γ0 1 + f (x, s(t))2 dx w(x, ˆ t)2 dx + w˜ x (s(t), t)2 , 2 2γ 0 0 0 (3.94) 



s(t)

2

where we used s˙ (t) > 0 and Young’s inequalities. Let  A1 = sups0 ≤s(t)≤sr

s(t)

f (x, s(t))2 dx.

(3.95)

0

By Poincare’s inequality, we obtain 

α V˙ˆ1 ≤ − 2 +

s(t)

0



c2 sr wˆ x (x, t) dx + s˙ (t) X(t) + 2 2 2β



s(t)

2

2sr2 A1 w˜ x (s(t), t)2 , α

 w(x, ˆ t) dx 2

0

(3.96)

where we chose γ0 =

α . 4sr2 A1

(3.97)

Next, we consider 1 Vˆ2 = 2



s(t)

wˆ x (x, t)2 dx.

(3.98)

0

The time derivative leads to V˙ˆ2 = − α



s(t)

0

wˆ xx (x, t)2 dx −

c s˙ (t) s˙ (t)X(t)wˆ x (s(t), t) − wˆ x (s(t), t)2 β 2

! − f (s(t), s(t))wˆ x (s(t), t) + fx (0, s(t))w(0, ˆ t)   s(t)

+ 0

fxx (x, s(t))w(x, ˆ t)dx w˜ x (s(t), t).

(3.99)

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3 State Estimator Design for Stefan System

Applying Young’s and Poincare’s inequalities with the help of s˙ (t) > 0, we have   2  s(t)  c α s(t) ˙ 2 2 ˆ V2 ≤− wˆ xx (x, t) dx+˙s (t) X(t) +2sr (γ2 +γ3 sr A2 ) wˆ x (x, t)2 dx 2 0 2β 2 0   1 2sr f (s(t), s(t))2 fx (0, s(t))2 w˜ x (s(t), t)2 , + (3.100) + + α 2γ2 2γ3 where  A2 = maxs0 ≤s(t)≤sr

s(t)

fxx (x, s(t))2 dx,

(3.101)

0

and γ2 and γ3 are positive parameters to be determined. Let Y be a Lyapunov function such that Y =

1 X(t)2 . 2

(3.102)

Taking time derivative and applying Young’s inequality, c β2 β2 wˆ x (s(t), t)2 + w˜ x (s(t), t)2 . Y˙ ≤ − X(t)2 + 2 c c

(3.103)

We define the Lyapunov functional Vˆ = Vˆ1 + Vˆ2 + pY,

(3.104)

where p > 0 is to be determined. Then, combining the inequalities, we get  s(t)  α  s(t) α 4pβ 2 sr ˙ 2 ˆ V ≤− − wˆ xx (x, t) dx− −2sr (γ2 +γ3 sr A2) wˆ x (x, t)2 dx 2 c 2 0 0    c2 pc sr s(t) 2 2 X(t) + s˙ (t) − X(t) + w(x, ˆ t) dx 2 2 0 β2   2sr f (s(t), s(t))2 fx (0, s(t))2 pβ 2 1 2s 2 A1 + + + w˜ x (s(t), t)2 . + + r α 2γ2 2γ3 α c (3.105) Choosing p= we obtain

cα , 16β 2 sr

γ2 =

α , 16sr

γ3 =

α , 16sr2 A2

(3.106)

3.3 Observer-Based Output Feedback Control Design

α V˙ˆ ≤ − 4



s(t)

0



wˆ xx (x, t)2 dx −

α 4



s(t)

73

wˆ x (x, t)2 dx −

0

pc X(t)2 2

 c2 sr s(t) 2 + s˙ (t) X(t) + w(x, ˆ t) dx 2 0 β2   2sr f (s(t), s(t))2 8sr fx (0, s(t))2 8sr2 A2 2sr2 A1 α + w˜ x (s(t), t)2 . + + + + α α α α 16sr (3.107) 

Thus, defining Vall = Vˆ + d V˜ ,

(3.108)

we have  s(t)  s(t) α α pc 2 ˙ wˆ x (x, t) dx − w(x, ˆ t)2 dx − X(t)2 Vall ≤ − 2 16sr2 0 16sr2 0    c2 sr s(t) 2 + s˙ (t) X(t) + w(x, ˆ t) dx 2 0 β2    dα α 2sr f (s(t), s(t))2 8sr fx (0, s(t))2 8sr2 A2 2sr2 A1 − + + + + w˜ x (s(t), t)2 − 4sr α α α α 16sr  s(t)  s(t) − d(λ + α) w˜ x (x, t)2 dx − dλ w(x, ˜ t)2 dx. (3.109) 0

0

Choosing d sufficiently large, we arrive at V˙all ≤ −bVall + a s˙ (t)Vall ,

(3.110)

where   c2 a = max sr2 , 2 , pβ

 b = min

 α , c, 2λ . 8sr2

(3.111)

As we have studied stability analysis of state feedback, the inequality (3.110) with the help of s˙ (t) > 0 and s0 < s(t) < sr leads to the exponential stability of (w, ˜ w, ˆ X)-system. By the invertibility and boundedness of the transformations, we deduce the proof of Theorem 3.2.

74

3 State Estimator Design for Stefan System

Numerical Simulation We use parameters of zinc given in Table 1.1 in Sect. 1.6. The initial interface is set to s0 = 1 cm, and the setpoint is chosen as sr = 35 cm. The initial estimation of the temperature profile is set to Tˆ0 (x) = T¯ˆ (1 − x/s0 ) + Tm with T¯ˆ = 30 ◦ C while the initial temperature is set to T0 (x) = T¯ (1 − x/s0 ) + Tm with T¯ = 10 ◦ C, and the observer gain is chosen as λ = 0.01. Then, the restriction on Tˆ0 (x), λ, and sr described in (3.68)–(3.70) is satisfied. The control gain is chosen as c = 0.001 . The dynamics of the moving interface s(t), the output feedback controller qc (t), and the temperature at the initial interface T (s0 , t) are depicted in Fig. 3.2. The first plot shows that the interface s(t) converges to the setpoint sr without overshoot which is guaranteed in Proposition 3.1. The second figure shows that the output feedback controller remains positive as stated in Proposition 3.1. The model validity can be seen in the third figure, which illustrates T (s0 , t) increases from the melting temperature Tm to enable melting of material and settles back to its equilibrium. Figure 3.3 shows a short-time dynamics of the temperature profile (solid) and the estimated temperature (dash). Clearly, the estimated temperature converges to the true temperature quickly.

3.4 State Estimation Under More Practical Sensors We also develop the temperature profile estimation design under the available measurement on the liquid temperature at the fixed boundary instead of the temperature gradient at the interface proposed in the last sections. This setup is much more practical, while the drawback is that the analysis for the output feedback control has not been established yet due to a challenge in proving the satisfaction of the physical constraints. Nevertheless, we can show the analysis of the convergent observer by proving the stability of the estimation error state (Fig. 3.4). We consider the same system in last sections, (3.29)–(3.32). The following observer is designed with the statement on the theorem. Theorem 3.3 Consider the plant (3.29)–(3.32) with the measurements Y1 (t) = s(t),

Y2 (t) = T (0, t),

(3.112)

and the following observer   Tˆt (x, t) =α Tˆxx (x, t) + p1 (x, Y1 (t)) Y2 (t) − Tˆ (0, t) ,   qc (t) Tˆx (0, t) = − + p2 (Y1 (t)) Y2 (t) − Tˆ (0, t) , k Tˆ (Y1 (t), t) =Tm ,

(3.113) (3.114) (3.115)

3.4 State Estimation Under More Practical Sensors

75

35 30 25 20 15 10 5 0 0

2.5

10

50

100

150

200

50

100

150

200

50

100

150

200

5

2 1.5 1 0.5 0 0

600

550

500

450

400 0

Fig. 3.2 Simulation of the closed-loop system (3.29)–(3.32) and the estimator (3.34)–(3.37) with the output feedback control law (3.67)

76

3 State Estimator Design for Stefan System 450 445 440 435 430 425 420 0

2

4

6

8

10

Fig. 3.3 Time evolution of true temperature (solid) and the estimate (dash) at x = 0 (black), x = s(t)/4 (red), x = s(t)/2 (blue), and x = 3s(t)/4 (green), respectively

Measurements s(t), T (0, t)

Estimator

Tˆ(x, t)

Fig. 3.4 The estimation problem measuring a boundary temperature and the interface position

where x ∈ [0, Y1 (t)], and the observer gains are

p1 (x, Y1 (t)) =λY1 (t)(Y1 (t) − x) p2 (Y1 (t)) = −

  ! " λ 2 2 I2 α Y1 (t) − (x − Y1 (t)) Y1 (t)2 − (x − Y1 (t))2

λ Y1 (t) 2α

,

(3.116) (3.117)

with a gain parameter λ > 0. Assume that the model validity condition T (x, t) ≥ Tm is satisfied. Then, for all λ > 0, the observer error system is exponentially stable in the sense of the norm ||T − Tˆ ||2H1 . Let u(x, ˜ t) = T (x, t) − Tˆ (x, t) be an estimation error variable. Then, we have a system for error variable as ˜ t), u˜ t (x, t) =α u˜ xx (x, t) − p1 (x, s(t))u(0, u˜ x (0, t) = − p2 (x, s(t))u(0, ˜ t), u(s(t), ˜ t) =0.

0 < x < s(t)

(3.118) (3.119) (3.120)

Unlike the procedure in Sect. 3.2, we cannot establish a good target system using the same form of the transformation. Instead, as developed in the state feedback design in Chap. 2, we first consider the observer design for the analogous system

3.4 State Estimation Under More Practical Sensors

77

on the fixed domain and develop the observer gain with the associated backstepping transformation. After that, we apply the analogous gain and transformation on the moving boundary to the error system (3.118)–(3.120), and prove the stability of the associated target system on the moving boundary.

Fixed Domain Design Consider the analogous estimation error system on the fixed domain x ∈ (0, D) given by ˜ t), u˜ t (x, t) =α u˜ xx (x, t) − p1 (x, D)u(0,

0 max 0, − 2α and c > 0. Suppose that the setpoint is chosen to satisfy f (x) =

sr > s0 +

β α



s0 0

f (x) u(x, 0)dx. f (s0 )

(4.8)

Then, the closed-loop system satisfies the model validity conditions (1.20) and (1.22) and is exponentially stable in the sense of the norm ||T (x, t) − Tm ||2H1 (0,s(t)) + (s(t) − sr )2 .

(4.9)

The proof of Theorem 4.1 is established through the remainder of this section and Sect. C.2.

Analogous PDE-ODE Cascade with Constant Domain We consider an analogous PDE-ODE cascade with constant domain to the Stefan system (4.1)–(4.4), similar to the procedure in Chap. 2, that is ut (x, t) =αuxx (x, t) + bux (x, t) − hu(x, t), ux (0, t) =U (t),

0 < x < D,

(4.10) (4.11)

4.1 Melting with Advection

95

u(D, t) =0,

(4.12)

˙ X(t) = − βux (D, t).

(4.13)

Let us introduce the transformation w(x, t) = u(x, t) −

β α



D

φ(x − y)u(y, t)dy − φ(x − D)X(t),

(4.14)

x

which maps to wt (x, t) =αwxx (x, t) + bwx (x, t) − hw(x, t),

0 0. Finally, by (4.14) and (4.19), to satisfy (4.16), we require wx (x, t) − γ w(0, t)  β D  φ (−y)u(y, t)dy − φ  (−D)X(t), =U (t) − α 0    β D − γ u(0, t) − φ(−y)u(y, t)dy − φ(−D)X(t) = 0, α 0

(4.29)

which leads to the control design β U (t) = γ u(0, t) + α



D

f (x)u(x, t)dy + f (D)X(t),

(4.30)

0

where f (x) =φ  (−x) − γ φ(−x),   c (d1 − γ )e−d1 x − (d2 + γ )e−d2 x . = β(d1 − d2 )

(4.31)

4.1 Melting with Advection

97

Control Design for Stefan System with Advection and Heat Transfer We design the controller for the Stefan system with advection and heat loss governed by (4.1)–(4.4). As presented in Chap. 2, we introduce the transformation which replaces the domain D in (4.14) by the moving boundary s(t), given by w(x, t) = u(x, t) −

β α



s(t)

φ(x − y)u(y, t)dy − φ(x − s(t))X(t),

(4.32)

x

where φ is given by (4.25). Then, applying (4.32) to (4.1)–(4.4), the target system is easily shown as wt (x, t) =αwxx (x, t) + bwx (x, t) − hw(x, t) + s˙ (t)φ  (x − s(t))X(t),

0 < x < s(t)

(4.33)

wx (0, t) =γ w(0, t),

(4.34)

w(s(t), t) =0,

(4.35)

˙ X(t) = − cX(t) − βwx (s(t), t).

(4.36)

Moreover, replacing U (t) → −qc (t)/k and D → s(t) in (4.30), the control design for the Stefan system is given by 

β qc (t) = −k γ u(0, t) + α



s(t)

 f (x)u(x, t)dx + f (s(t))X(t) .

(4.37)

0

Physical Constraint To prove the physical constraint under the closed-loop system, we introduce a variable Z(t) defined by qc + γ u(0, t) k  β s(t) =− f (x)u(x, t)dx − f (s(t))X(t). α 0

Z(t) =

Taking the time derivative of Z(t), we get  β s(t) ˙ ˙ f (x)ut (x, t)dx − f (s(t))X(t) − s˙ (t)f  (s(t))X(t), Z(t) =− α 0

(4.38) (4.39)

98

4 Extended Models and Design



 s(t)

=−β f (s(t))ux (s(t), t)−f (0)ux (0, t)+f  (0)u(0, t)+ −

β α

 −bf (0)u(0) −

 s(t) 0

 f  (x)u(x, t)dy

0

 (bf  (x) + hf (x))u(x, t)dx + βf (s(t))ux (s(t), t)

− s˙ (t)f  (s(t))X(t), =βf (0)ux (0, t)−

 β  αf (0)−bf (0) u(0, t) α

 β s(t) − (αf  (x) − bf  (x) − hf (x))u(x, t)dx − s˙ (t)f  (s(t))X(t). α 0

(4.40)

Here, by f (x) = φ  (−x) − γ φ(−x), we can see that f (0) =φ  (0) − γ φ(0) =

c , β

(4.41) 

b +γ α



h φ(−x), α   h b     − γ φ  (−x), f (x) =φ (−x) − γ φ (−x) = − φ (−x) + α α f  (x) = − φ  (−x) + γ φ  (−x) =

φ  (−x) −

(4.42) (4.43)

and thus αf  (x) − bf (x) = αγ φ  (−x) − (h − bγ ) φ(−x),

(4.44)

which leads to β (αf  (0) − f (0)) = γ c. α

(4.45)

Moreover, αf  (x) − bf  (x) − hf (x)     = αφ  (−x) + bφ  (−x) − hφ  (−x) − γ αφ  (−x) + bφ  (−x) − hφ(−x) =0.

(4.46)

Thus, (4.40) is led to ˙ Z(t) =c(ux (0, t) − γ u(0, t)) − s˙ (t)f  (s(t))X(t) = − cZ(t) − s˙ (t)f  (s(t))X(t).

(4.47)

We prove Z(t) > 0 by contradiction approach. Assume that there exists t ∗ > 0 such that

4.2 Actuator Delay Compensation

99

∀t ∈ (0, t ∗ ),

Z(t) > 0,

Z(t ∗ ) = 0.

(4.48)

By maximum principle and Hopf’s lemma, we get u(x, t) > 0 and s˙ (t) > 0 for all x ∈ (0, s(t)) and t ∈ (0, t ∗ ). Thus, we have s(t) > s0 > 0, ∀t ∈ (0, t ∗ ). In addition, using (4.39) and knowing that f (x) > 0, it leads to X(t) < 0, ∀t ∈ (0, t ∗ ). Therefore, (4.47) leads to ˙ Z(t) > −cZ(t),

∀t ∈ (0, t ∗ ).

(4.49)

Gronwall’s inequality leads to the inequality regarding the solution of the differential equation, written as Z(t) ≥ Z(0)e−ct , ∀t ∈ (0, t ∗ ]. Thus, we have ∗ Z(t ∗ ) ≥ Z(0)e−ct > 0 , which contradicts with the assumption (4.48). Hence, Z(t) > 0,

∀t ≥ 0

(4.50)

is proved. Then, by maximum principle, it holds u(x, t) >0, s˙ (t) >0,

∀x ∈ (0, s(t),

∀t ≥ 0,

ux (s(t), t) < 0,

∀t ≥ 0.

∀t ≥ 0,

(4.51) (4.52)

Imposing (4.50) and (4.51) on (4.39), we obtain X(t) < 0,

∀t ≥ 0.

(4.53)

Thus, the following condition holds: 0 < s(t) < sr ,

∀t ≥ 0.

(4.54)

Finally, with the help of the conditions (4.52) and (4.54), as proven in Sect. C.2, the exponential stability of the closed-loop system is ensured, which completes the proof of Theorem 4.1.

4.2 Actuator Delay Compensation In the presence of actuator delay, a delay compensation technique has been developed for many classes of systems using a backstepping transformation [124]: see [127] for linear ODE systems and [125] for nonlinear ODE systems. Using the Lyapunov method, [121] presented the several analyses of the predictor-based feedback control for ODEs such as robustness with respect to the delay mismatch and disturbance attenuation. To deal with systems under unknown and arbitrarily large actuator delay, a Lyapunov-based delay-adaptive control design was developed in [26, 27] for both linear and nonlinear ODEs with certain systems, and [25]

100

4 Extended Models and Design

extended the design for trajectory tracking of uncertain linear ODEs. For control of unstable parabolic PDE under a long input delay, [123] designed the stabilizing controller by introducing two backstepping transformations for the stabilization of the unstable PDE and the compensation of the delay. By the similar technique, in [199] the coupled diffusion PDE-ODE system in the presence of the actuator delay is stabilized, and in [40] a 2×2 unstable reaction-diffusion PDE system with distinct input delays is stabilized. Implementation issues on the predictor-based feedback are covered in [101] by studying the closed-loop analysis under the sampled-data control. A constrained stabilization of linear ODEs with distinct input delays has been developed in [1] by combining the predictor-based feedback with a control barrier function. In [11], the predictor-based feedback is applied to controlling a robot manipulator in experiments.

Problem Setup and Main Result Here we impose an actuator delay which is caused by several reasons such as computational time or communication delay. Specifically, the communication delay takes place during the time in which the signals are transmitted from sensors to the controller and from the controller to the actuator, and the computational delay is caused during the time when the controller completes the computation after receiving the signals from the sensors. This configuration is shown in Fig. 4.2. Thus, the Stefan problem with the actuator delay is formulated as follows: Tt (x, t) = αTxx (x, t),

0 ≤ x ≤ s(t),

(4.55)

−kTx (0, t) = qc (t − D),

(4.56)

T (s(t), t) = Tm ,

(4.57)

s˙ (t) = −βTx (s(t), t),

(4.58)

and the initial values s(0) = s0 ,

T (x, 0) = T0 (x), ∀x ∈ [0, s0 ],

Fig. 4.2 Schematic of the one-phase Stefan problem with actuator delay

qc (t) = qc,0 (t),

liquid qc (t)

∀t ∈ [−D, 0), (4.59)

solid

T (x, t)

delay 0

s(t)

L

x

4.2 Actuator Delay Compensation

101

where D is the time delay of the input. Since the boundary input (4.56) is now described by qc (t − D), all the statements in Lemmas 1.2 and 1.3 are replaced by qc (t − D). Thus, the boundary input needs to be a bounded piecewise continuous function which is generating nonnegative heat, i.e., qc (t − D) ≥ 0,

∀t > 0.

(4.60)

Hence, we impose the following assumption. Assumption 4.1 The past input qc,0 (t) for t ∈ [−D, 0) is a bounded piecewise continuous nonnegative function, i.e. qc,0 (t) ≥ 0,

∀t ∈ [−D, 0).

(4.61)

With Assumption 4.1, the model validity conditions (1.20) and (1.22) remain if qc (t) ≥ 0 for ∀t > 0. The control objective is the same as in Chap. 2. As addressed in Sect. 2.1, we study the energy growth. The plant (4.55)–(4.58) obeys the following energy conservation law:     t d k s(t) k (T (x, t) − Tm )dx + s(t) + qc (θ )dθ = qc (t). (4.62) dt α 0 β t−D The control objective is achieved if and only if the following limit is satisfied:  lim

t→∞

k α



s(t) 0

k (T (x, t) − Tm )dx + s(t) + β





t

qc (θ )dθ

=

t−D

k sr , β

(4.63)

which can be derived by substituting T (x, t) → Tm , s(t) → sr , and qc (t) → 0 into the left-hand side of (4.63). Taking integration of (4.62) from t = 0 to t = ∞ with the help of qc (t) > 0 for t > 0 and (4.63), the following assumption on the setpoint is made. Assumption 4.2 The setpoint is chosen to satisfy  sr > s0 + β

0

−D

1 qc (t) dt + k α



s0

 (T0 (x) − Tm )dx .

(4.64)

0

Next, we state our main result. Theorem 4.1 Under Assumptions 4.1–4.2, the closed-loop system consisting of the plant (4.55)–(4.58) and the control law

102

4 Extended Models and Design

 qc (t) = − c

t

k qc (θ )dθ + α t−D



s(t)

0

 k (T (x, t) − Tm )dx + (s(t) − sr ) , β (4.65)

where c > 0 is an arbitrary control gain, maintains the model validity conditions (1.20) and (1.22), and is exponentially stable in the sense of the norm  ||T (x, t) − Tm ||2H1 (0,s(t)) + (s(t) − sr )2 +

t

 qc (θ )2 dθ +

t−D

t

q˙c (θ )2 dθ.

t−D

(4.66) The proof of Theorem 4.1 is established through the remainder of this section and the stability proof given in Appendix C.

Backstepping Transformation Change of Variables Introduce reference error variables defined by u(x, t) :=T (x, t) − Tm ,

X(t) := s(t) − sr .

(4.67)

Next, we introduce a variable v(x, t) =

qc (t − x − D) , k

∀x ∈ [−D, 0].

(4.68)

Here, the variable x ∈ [−D, 0] in (4.68) is not the spatial coordinate x ∈ (0, s(t)) of the system (4.55)–(4.58), but a newly introduced variable for an alternative representation of the delayed input, as introduced in [123]. Hence, the variable qc (t − x − D) in (4.68) still represents the boundary heat input (not an input acting on the space x ∈ (0, s(t))), during the time period from t − D to t. Then, (4.68) gives the boundary values of current input v(−D, t) = qc (t)/k and delayed input v(0, t) = qc (t − D)/k, and v(x, t) satisfies a transport PDE. Hence, the coupled (v, u, X)-system is described as vt (x, t) = − vx (x, t),

−D < x < 0

v(−D, t) =qc (t)/k,

(4.70)

ux (0, t) = − v(0, t), ut (x, t) =αuxx (x, t), u(s(t), t) =0,

(4.69)

(4.71) 0 < x < s(t)

(4.72) (4.73)

4.2 Actuator Delay Compensation

103

˙ X(t) = − βux (s(t), t).

(4.74)

Now, the control objective is to design qc (t) to stabilize the coupled (v, u, X)system at the origin.

Direct Transformation We consider backstepping transformations for the coupled PDEs-ODE system as  c c s(t) (x − y)u(y, t)dy − (x − s(t))X(t), α x β  s(t)  0 c c v(y, t)dy + u(y, t)dy + X(t). z(x, t) =v(x, t) + c α 0 β x

w(x, t) =u(x, t) −

(4.75) (4.76)

The transformation (4.75) is the same nonlinear transformation as the one proposed in Sect. 2.3 for delay-free Stefan problem. The formulation of (4.76) is motivated by a design in fixed domain introduced in [123]. Taking derivatives of (4.75) and (4.76) in x and t along with the solution of the system (4.69)–(4.74), we have c wx (x, t) =ux (x, t) − α



s(t)

u(y, t)dy −

x

c X(t), β

zx (x, t) =vx (x, t) − cv(x, t),  0  zt (x, t) = − vx (x, t) − c vy (y, t)dy + c

(4.77) (4.78)

s(t)

uyy (y, t)dy − cux (s(t), t),

0

x

= − vx (x, t) + cv(x, t).

(4.79)

By (4.78) and (4.79), we get zt (x, t) = −zx (x, t). In addition, by substituting x = 0 in (4.76) and (4.77), wx (0, t) = −z(0, t) holds. On the other hand, because w transformation does not depend on v, w system is not changed from the delay-free target system given in Sect. 2.3. Thus, the target (z, w, X)-system is obtained by zt (x, t) = − zx (x, t),

−D < x < 0

(4.80)

z(−D, t) =0,

(4.81)

wx (0, t) = − z(0, t), c wt (x, t) =αwxx (x, t) + s˙ (t)X(t), β w(s(t), t) =0, ˙ X(t) = − cX(t) − βwx (s(t), t).

(4.82) 0 < x < s(t)

(4.83) (4.84) (4.85)

104

4 Extended Models and Design

The control design is achieved through evaluating (4.76) at x = −D together with the boundary conditions (4.70) and (4.81), which yields    0 1 s(t) 1 v(y, t)dy + u(y, t)dy + X(t) . qc (t) = − ck (4.86) α 0 β −D Finally, substituting the definitions (4.67) and (4.68) in (4.86), the control law (4.65) is obtained. In a similar manner, the inverse transformations are obtained by  β s(t) ψ(x − y)w(y, t)dy + ψ(x − s(t))X(t), (4.87) u(x, t) =w(x, t) + α x  0 v(x, t) =z(x, t) − μ(x − y)z(y, t)dy x

−

β μ(x) α



s(t)

ζ (y)w(y, t)dy − ζ (s(t))μ(x)X(t),

(4.88)

0

where

√

cα ψ(x) = sin β μ(x) =ce , cx



 c x , α

1 ζ (x) = cos β

(4.89) 

 c x . α

(4.90)

Physical Constraints Next, we prove that the closed-loop system with the control law (4.65) guarantees some important properties. Lemma 4.1 With Assumption 4.2, the control law (4.65) for the system (4.55)– (4.58) generates a positive input signal, i.e., ∀t > 0.

qc (t) >0,

(4.91)

Proof Taking the time derivative of (4.65) together with the solution of (4.55)– (4.58), we obtain 



q˙c (t) = − c qc (t) − qc (t − D) + k

s(t)

 Txx (x, t)dx − kTx (s(t), t) ,

0

= − c (qc (t) − qc (t − D) − kTx (0, t)) , = − cqc (t).

(4.92)

4.2 Actuator Delay Compensation

105

The differential equation (4.92) yields qc (t) = qc (0)e−ct .

(4.93)

Additionally, substituting t = 0 into the control law (4.65) leads to  qc (0) = −c

0

k qc (θ )dθ + α −D



s0 0

 k (T0 (x) − Tm )dx + (s0 − sr ) . β

(4.94)

Hence, Assumption 4.2 leads to qc (0) > 0.

(4.95)

Applying (4.95) to (4.93), the positivity of the controller (4.91) is satisfied. Hence, the model validity conditions (1.20) and (1.22) hold, i.e., T (x, t) ≥Tm s˙ (t) ≥0,

for all

x ∈ [0, s(t)].

∀t > 0.

(4.96) (4.97)

By the control law (4.65), the following relation holds under the closed-loop system: k 1 (s(t) − sr ) = − qc (t) − β c



t

qc (θ )dθ −

t−D

k α



s(t)

(T (x, t) − Tm )dx.

(4.98)

0

Applying (4.91) and (4.96) to the control law (4.65), it holds s0 0.

(4.99)

Relation Between the Designed Control Law and a State Prediction As developed in some literature for ODE systems, the delay compensated control via the method of backstepping is known to be equivalent to the predictor-based feedback where the control law is derived to stabilize the future state called “predictor state,” see Section 2 in [124] for instance. Hence, one might ask whether our delay compensated control is also equivalent to the predictor-based feedback. This is not a trivial question in the case of Stefan problem due to the complicated structure of ODE dynamics whose state is the domain of the PDE. The nominal control design for delay-free Stefan problem developed in [116] is given by

106

4 Extended Models and Design



k q¯c (t) = − c α



s(t) 0

 k (T (x, t) − Tm )dx + (s(t) − sr ) , β

(4.100)

where we defined the notation q¯c (t) to distinguish with the delay compensated control law (4.65). Thus, our interest lies in proving qc (t) ≡ q¯c (t + D) because q¯c (t + D) is the prediction of the nominal control. We start from the expression of q¯c (t + D) which can be described as 

k q¯c (t + D) = − c α



s(t+D) 0

 k (T (x, t + D) − Tm )dx + (s(t + D) − sr ) . β (4.101)

Integrating ODE dynamics s˙ (t) = −βTx (s(t), t) given in (4.58) from t to t + D yields 

t+D

s(t + D) = s(t) − β

(4.102)

Tx (s(τ ), τ )dτ. t

Next, integrating PDE dynamics Tt = αTxx given in (4.55) in time from t to t + D  t+D leads to T (x, t + D) = T (x, t) + α t Txx (x, τ )dτ . Furthermore, integrating both sides in space from 0 to s(t + D), we obtain  

s(t+D)

s(t+D)

= 



s(t+D)

=

Txx (x, τ )dτ dx, 

0



(Tx (s(t + D), τ ) − Tx (0, τ )dτ,

t t+D

(T (x, t) − Tm )dx + α

0

t t+D

(T (x, t) − Tm )dx + α

0 s(t+D)

s(t+D)  t+D

(T (x, t) − Tm )dx + α

0

= 

(T (x, t + D) − Tm )dx

0

t

α Tx (s(t + D), τ )dτ + k



t

qc (ξ )dξ. t−D

(4.103) Therefore, substituting (4.102) and (4.103) into (4.101), we get  q¯c (t + D) = − c  t

t+D

k α



s(t+D)

(T (x, t) − Tm )dx + k

0

(Tx (s(t + D), τ ) − Tx (s(τ ), τ ))dτ

 k qc (ξ )dξ + (s(t) − sr ) . + β t−D 

t

(4.104)

4.2 Actuator Delay Compensation

107

Consequently, it remains to consider the following term: 

t+D

(Tx (s(t + D), τ ) − Tx (s(τ ), τ ))dτ

t

 =

t+D

=

s(t+D)

Txx (x, τ )dxdτ, t

=



1 α 1 α

s(τ )



s(t+D)  s −1 (x)

Tτ (x, τ )dτ dx, s(t)



t

s(t+D) 

 T (x, s −1 (x)) − T (x, t) dx,

(4.105)

s(t)

where we switched the order of the integrations in time and space from the first line to the second line with defining the inverse function s −1 (x). The existence and uniqueness of s −1 (x) are guaranteed due to the continuous and monotonically increasing property of s(t) provided qc (t) > 0. Thus, boundary condition T (s(t), t) = Tm , ∀t ≥ 0 given in (4.57) implies T (x, s −1 (x)) = Tm from which (4.105) is given by 

t+D t

1 (Tx (s(t + D), τ ) − Tx (s(τ ), τ ))dτ = − α



s(t+D)

(T (x, t) − Tm ) dx.

s(t)

(4.106) Substituting (4.106) into (4.104), we arrive at     t k s(t) k q¯c (t + D) = − c (T (x, t) − Tm )dx + qc (ξ )dξ + (s(t) − sr ) ≡ qc (t). α 0 β t−D (4.107)

Therefore, we conclude that the delay compensated control (4.65) is indeed the prediction of the nominal control law (4.100).

Robustness to Delay Mismatch The results established up to the last section are based on the control design with utilizing the exact value of the actuator delay. However, in practice, there is an error between the exact time delays and the identified delays. Hence, guaranteeing the performance of the controller under the small delay mismatch is important. In this section, D > 0 denotes the identified time delay and ΔD denotes the delay mismatch (can be either positive or negative), which yields D+ΔD as the exact time delay from the controller to the plant. Thus, the system we focus on is described by

108

4 Extended Models and Design

Tt (x, t) =αTxx (x, t),

x ∈ (0, s(t)),

−kTx (0, t) =qc (t − (D + ΔD)), T (s(t), t) =Tm , s˙ (t) = − βTx (s(t), t),

(4.108) (4.109) (4.110) (4.111)

with the control law given in (4.65) which utilizes the identified delay D. Since the control law is not changed, the same backstepping transformation in (4.75) and (4.76) can be applied, but the target (z, w, X)-system needs to be newly derived due to the modification of (4.109). The theorem for the robustness to delay mismatch is provided under the restriction on the control gain, as stated in the following. Theorem 4.2 Under Assumptions 4.1–4.2, there exists a positive constant c¯ > 0 such that ∀c ∈ (0, c) ¯ the closed-loop system consisting of the plant (4.108)–(4.111) and the control law (4.65) maintains the model validity conditions (1.20) and (1.22) and is exponentially stable in the sense of the norm (4.66). An important characteristic to note in Theorem 4.2 is that the existence of c¯ is ensured for any given ΔD as long as D + ΔD > 0. An analogous description with respect to the small delay mismatch is given in the following corollary. Corollary 4.1 Under Assumptions 4.1–4.2, for any given c > 0 there exist positive constants ε > 0 and ε¯ > 0 such that ∀ΔD ∈ (−ε, ε¯ ) the closed-loop system (4.108)–(4.111), (4.65) satisfies the same model validity and stability property as Theorem 4.2. The proof of Theorem 4.2 can be seen in [118] and is omitted here for the sake of brevity.

Numerical Simulation We study the simulation of the proposed delay compensated controller under the accurate value on the delay and the delay mismatch.

Exact Compensation The performance of the proposed delay compensated controller is investigated by comparing to the performance of the nominal controller (4.100). The time delay, the past heat input, and the initial values are set as D = 2 min, qc (t) = 500 W/m for ∀t ∈ [−D, 0), s0 = 0.1 m, and T0 (x) = T¯ (1 − x/s0 ) + Tm with T¯ = 50 K. The setpoint and the controller gain are chosen as sr = 0.15 m and c =0.01/s, which satisfies the setpoint restriction (4.64).

4.2 Actuator Delay Compensation

109

Fig. 4.3 The closed-loop response of (4.55)–(4.58) with the delay compensated control law (4.65) (red) and the uncompensated control law (4.100) (blue). (a) Delay compensated control achieves the monotonic convergence of s(t) to the setpoint sr without overshooting, i.e., s˙ (t) > 0, s0 < s(t) < sr . (b) Delay compensated control keeps injecting positive heat, i.e., qc (t) > 0. (c) T (0, t) converges to the melting temperature Tm with maintaining T (0, t) > Tm

Figure 4.3 shows the simulation results of the closed-loop system of the plant (4.55)–(4.58) with the proposed delay compensated control (4.65) (red) and the uncompensated control law (4.100) (blue). The closed-loop responses of the moving interface s(t), the boundary heat control qc (t), and the boundary temperature T (0, t) are depicted in Fig. 4.3a–c, respectively. As stated in their captions, the proposed delay compensated controller ensures all the derived conditions with the convergence of the interface position to the setpoint, while the uncompensated

110

4 Extended Models and Design

control does not provide such a behavior. Hence, the numerical result is consistent with the theoretical result, and the proposed controller achieves better performances than the uncompensated controller under the actuator delay.

Robustness to Delay Mismatch To evaluate the delay robustness, the performance of the proposed controller is investigated under the delay mismatch. First, the simulation is conducted with the underestimated delay mismatch where the time delay from the actuator to the plant is 60 s while the compensating time delay in the controller is D = 30 s, i.e., the delay mismatch is ΔD = 30 s. The closed-loop responses are depicted in Fig. 4.4 with the choices of the control gain c = 0.01/s (red) and c = 0.1/s (blue). Figure 4.4a illustrates the convergence of the interface position to the setpoint; however, the monotonicity of the interface dynamics is violated with larger gain (red). From Fig. 4.4b, c we can observe that the positivity of the control input and the temperature condition for the liquid phase are satisfied only with the lower gain (blue) for all time, while the simulation with the larger gain (red) violates these conditions too. Hence, with the underestimated delay mismatch, the robustness is well illustrated for sufficiently small gain c > 0, which is consistent with Theorem 4.2. Next, we have studied the simulation with the overestimated delay mismatch with the same value of the time delay from the actuator to the plant 60 s but the compensating time delay in the controller is D = 90 s, i.e., the delay mismatch is ΔD = −30 s. The closed-loop responses are depicted in Fig. 4.5 with the same choices of the control gain as in simulation of underestimated delay mismatch. While the magnitude of the delay mismatch is same as the one conducted in the underestimated delay mismatch, we can observe from Fig. 4.5b, c that the positivity of the control input and the temperature condition for the model validity are satisfied for all time with both smaller control gain (red) and larger control gain (blue). Although Theorem 4.2 guarantees these properties only for sufficiently small control gain c > 0, the numerical results illustrate that the restriction on the control gain to satisfy these properties is not equivalent between the underestimated and overestimated delay mismatch. Indeed, as far as we have investigated the numerical results with the overestimated delay mismatch using other values of the control gain c and the delay perturbation ΔD, the positivity of the control input is satisfied for every cases and the convergence of the interface position to the setpoint is depicted without overshooting. These observations from the numerical simulation lead us to conjecture that the delay-compensated controller might exhibit greater sensitivity to delay mismatch when it is underestimated rather than overestimated in terms of the model validation. Hence, once the user is faced with some range of the uncertainty in the actuator delay, it is better to choose small control gain c > 0, and additionally, it might be better to choose larger value of the compensating delay in the controller to be conservative.

4.3 n-Dimensional Ball Geometry with Symmetry Fig. 4.4 The closed-loop response under the “underestimated” delay mismatch with D = 30 s and ΔD = 30 s. The simulations are conducted with the control gain c = 0.01/s (red) and c = 0.1/s (blue). The delay-robustness is observed only with smaller gain in terms of the model validity. (a) Monotonicity of the interface dynamics is satisfied with smaller gain, but is violated with larger gain. (b) Positivity of the heat input is satisfied with smaller gain, but is violated with larger gain. (c) The boundary temperature keeps above the melting temperature with smaller gain, while it reaches below the melting temperature with larger gain, which violates the temperature condition for the liquid phase

111

0.16 0.15 0.14 0.13 0.12 0.11 0.1 0

10

20

30

40

15

20

(a) 6

10

4 3 2 1 0 -1 -2

0

5

10

(b) 600

400

200

0

-200 0

5

10

15

20

25

30

35

40

(c)

4.3 n-Dimensional Ball Geometry with Symmetry We consider the one-phase Stefan problem with an n-dimensional ball geometry. For instance, n = 2 corresponds to a disk, and n = 3 corresponds to a sphere. The outer annulus of the ball is occupied by the liquid phase and the inner ball is in the solid phase. A heat flux input is located all along the outer surface of the liquid phase. We consider spatial nonuniformity only in the radial coordinate r ≥ 0 and

112 Fig. 4.5 The closed-loop response under the “overestimated” delay mismatch with D = 90 s and ΔD = −30 s. The simulations are conducted with the control gain c = 0.01/s (red) and c = 0.1/s (blue). In this case, all the constraints for the model validity are satisfied with both smaller gain and larger gain. (a) The interface position converges to the setpoint without overshooting. (b) The heat input remains positive. (c) The boundary temperature is greater than melting temperature, which satisfies the temperature condition for the liquid phase

4 Extended Models and Design 0.15 0.14 0.13 0.12 0.11 0.1 0

10

20

30

40

15

20

(a) 6

10

4

3

2

1

0

0

5

10

(b) 400

300

200

100

0

0

5

10

15

20

25

30

35

40

(c) assume uniformity of the temperature in other spherical coordinates. Likewise, we assume that the liquid–solid interface is spherical (Fig. 4.6). Let T (r, t) be the temperature profile in the radial coordinate, and l(t) be the interface radius position. Let qc (t) be the input power (in Watt unit) of the heat flux along the outer surface. Then, referring to [33] for the Stefan model on the disk and sphere, and [205] giving parabolic PDE of n-dimensional ball, the following

4.3 n-Dimensional Ball Geometry with Symmetry

113

Fig. 4.6 The Stefan problem of n-dimensional ball geometry. (a) n = 2, disk. (b) n = 3, sphere

governing equation is provided: ∂t T (r, t) =

  n−1 r ∂ ∂ T (r, t) , r r n−1 α

r

l(t) < r < R

R n−1 k∂r T (R, t) =qc (t),

(4.112) (4.113)

T (l(t), t) =Tm

(4.114)

˙ = − β∂r T (l(t), t). l(t)

(4.115)

Energy Conservation Law We define the internal energy of the system as k E(t) = α



R

r n−1 (T (r, t) − Tm )dr +

l(t)

k (R n − l(t)n ), nβ

(4.116)

where the first term is the specific energy relating the temperature and the last term is a latent heat involved with the volume of the liquid phase. Then, taking the time derivative of the energy yields n−1 ˙ E(t) = kr n−1 ∂r T (r, t)|R ∂r T (l(t), t) l(t) + kl(t)

= kR n−1 ∂r T (R, t) = qc (t),

(4.117)

which serves as the energy conservation law. The energy corresponding to the setpoint is given by

114

4 Extended Models and Design

k (R n − lrn ). nβ

Er =

(4.118)

Due to the constraint qc (t) ≥ 0, to achieve the control objective, the setpoint energy Er must be greater than the initial internal energy E(0), which leads to the requirement of the following assumption on the choice of the setpoint position. Assumption 4.3 The setpoint lr is chosen to satisfy 0
0 is a control gain. Owing to the energy conservation law, the energyshaping design (4.122) satisfies the input constraint qc (t) > 0,

(4.123)

by which we can also guarantee T (r, t) > Tm ,

˙ < 0. l(t)

(4.124)

Moreover, using these properties and the control law (4.122), we additionally have the following property: l(t) > lr .

(4.125)

While the energy-shaping guarantees the desired properties given above, the closedloop stability has not been ensured yet. Next, we show that the energy-shaping control (4.122) is actually equivalent to the backstepping control we derive for the spherical coordinate, and prove the closed-loop stability.

4.3 n-Dimensional Ball Geometry with Symmetry

115

Backstepping Design for Stefan System in n-Dimensional Ball Motivated by the energy conservation law, we define the reference error states as u(r, t) :=T (r, t) − Tm , X(t) := −

1 (l(t)n − lrn ). n

(4.126) (4.127)

Then, we obtain the following reference error system: ∂t u(r, t) =

  n−1 r ∂ ∂ u(r, t) , r r n−1 α

r qc (t) ∂r u(R, t) = n−1 , R k

l(t) < r < R

(4.128) (4.129)

u(l(t), t) =0,

(4.130)

˙ X(t) =βl(t)n−1 ∂r u(l(t), t).

(4.131)

We introduce the backstepping transformation defined by  w(r, t) = u(r, t) −

r

k(r, p)u(p, t)dp − φ(r, l(t))X(t),

(4.132)

l(t)

which transforms into ∂t w(r, t) =

  n−1 ˙ r ∂ ∂ w(r, t) − l(t)∂ r r l(t) φ(r, l(t))X(t), n−1 α

r

l(t) < r < R (4.133)

∂r w(R, t) =0,

(4.134)

w(l(t), t) =0,

(4.135)

˙ X(t) = − cX(t) + βl(t)n−1 ∂r w(l(t), t).

(4.136)

Taking the spatial derivatives of (4.132), we get ∂r w(r, t) =∂r u(r, t) − k(r, r)u(r, t)  r − ∂r k(r, p)u(p, t)dp − ∂r φ(r, l(t))X(t), l(t)

(4.137)

  d k(r, r) u(r, t) ∂rr w(r, t) =∂rr u(r, t) − k(r, r)∂r u(r, t) − ∂r k(r, r) + dr  r − ∂rr k(r, p)u(p, t)dp − ∂rr φ(r, l(t))X(t). (4.138) l(t)

116

4 Extended Models and Design

Taking the time derivative of (4.132), we have ∂t w(r, t) + s˙ (t)∂l(t) φ(r, l(t))X(t)   α = n−1 ∂r r n−1 ∂r u(r, t) r  r   α ˙ k(r, p) n−1 ∂p pn−1 ∂p u(p, t) dp − φ(r, l(t))X(t). − p l(t)

(4.139)

Doing integration by parts, we get 

r

k(r, p) l(t)

  n−1 p ∂ ∂ u(p, t) dp p n−1 p 1

p

p=r



=k(r, p)∂p u(p, t)|p=l(t) −

  k(r, p) ∂p k(r, p) − (n − 1) ∂p u(p, t)dp p l(t) r

=k(r, r)∂r u(r, t) − k(r, l(t))∂r u(l(t), t)   k(r, r) u(r, t) − ∂p k(r, r) − (n − 1) r   r  ∂p k(r, p) k(r, p) ∂pp k(r, p) + (n − 1) u(p, t)dp. + − (n − 1) p p2 l(t) (4.140) Therefore, by (4.139) and (4.140), we obtain ˙ l(t) φ(r, l(t))X(t) ∂t w(r, t) + l(t)∂ =α∂rr u(r, t) +

(n − 1)α ∂r u(r, t) r

− αk(r, r)∂r u(r, t) + (αk(r, l(t)) − βl(t)n−1 φ(r, l(t)))∂r u(l(t), t)   k(r, r) u(r, t) + α ∂p k(r, r) − (n − 1) r   r  ∂p k(r, p) k(r, p) ∂pp k(r, p) + (n − 1) u(p, t)dp. −α − (n − 1) p p2 l(t) (4.141) Combining (4.141) with (4.137) and (4.138) leads to ∂t w(r, t) − α∂rr w(r, t) −

(n − 1)α ˙ l(t) φ(r, l(t))X(t) ∂r w(r, t) + l(t)∂ r

=(αk(r, l(t)) − βl(t)n−1 φ(r, l(t)))∂r u(l(t), t)

4.3 n-Dimensional Ball Geometry with Symmetry

117

   r  ∂r k(r, p) k(r, p) ∂p k(r, p) u(p, t)dp − ∂rr k(r, p)−∂pp k(r, p)+(n − 1) + r p p2 l(t)   d + 2α k(r, r) u(r, t) dr   (n − 1) (4.142) ∂r φ(r, l(t)) X(t). + α ∂rr φ(r, l(t)) + r +α

Thus, for the target PDE (4.133) to hold, the following conditions for the gain kernels must be satisfied: (4.143) αk(r, l(t)) =βl(t)n−1 φ(r, l(t)),     1 k(r, p) 1 ∂r r n−1 ∂r k(r, p) =r n−1 ∂r ∂r k(r, p) + (n − 1) , n−1 n−1 r r p2 (4.144)

1

∂r (r r n−1

n−1

d k(r, r) =0, dr

(4.145)

∂r φ(r, l(t))) =0.

(4.146)

Moreover, for the boundary condition (4.135) and ODE (4.136) to hold, we require φ(l(t), l(t)) =0,

(4.147)

c ∂r φ(l(t), l(t)) = − . βl(t)n−1

(4.148)

By the conditions (4.146)–(4.148), we get ∂r φ(r, l(t)) = −

c βr n−1

.

(4.149)

Finally, taking the integration of (4.149) from r = l(t) to r, the gain kernel solution is obtained by   ⎧ c 1 1 ⎪ if n ≥ 3, − ⎪ n−2 n−2 l(t) ⎨ β(n−2) r c φ(r, l(t)) = − β (ln(r) − ln(l(t))) if n = 2, ⎪ ⎪ ⎩ c − β (r − l(t)) if n = 1.

(4.150)

The condition (4.143) leads to k(r, p) =

β n−1 p φ(r, p). α

(4.151)

118

4 Extended Models and Design

Then, one can derive that the solution (4.151) satisfies the conditions (4.144) and (4.145) for all n ∈ N. Thus, the gain kernel solutions (4.150) and (4.151) lead to the target PDE (4.133)–(4.136). ∂r k(r, p) = −

c pn−1 . α r n−1

(4.152)

Substituting r = R into (4.137), we get qc (t) = R n−1 k



R l(t)

=−

c α

∂r k(R, p)u(p, t)dp + ∂r φ(R, l(t))X(t) 

R

l(t)

pn−1 c u(p, t)dp − X(t). n−1 R βR n−1

(4.153)

Rewriting the equation above by the original state variables, one can derive the backstepping control law  qc (t) = −c

k α



R l(t)

r n−1 (T (r, t) − Tm )dr +

 k n (lr − l(t)n ) , nβ

(4.154)

which corresponds to the energy-shaping control law in (4.122). Thus, we can see that the closed-loop system under the control law (4.154) satisfies the input constraint and model validity shown in (4.123)–(4.125).

4.4 What Can We Guarantee If the Solid Phase Remains?—ISS While all the aforementioned results are based on the one-phase Stefan problem which neglects the cooling heat caused by the solid phase, an analysis on the system incorporating the cooling heat at the liquid–solid interface has not been established. The one-phase Stefan problem with a prescribed heat flux at the interface was studied in [178]. The author proved the existence and uniqueness of the solution with a prescribed heat input at the fixed boundary by verifying positivity conditions on the interface position and temperature profile using a similar technique as in [75]. Regarding the added heat flux at the interface as the heat loss induced by the remaining other phase dynamics, it is reasonable to treat the prescribed heat flux as the disturbance of the system. The norm estimate of systems with a disturbance is often analyzed in terms of Input-to-State Stability (ISS) [186], which serves as a criterion for the robustness of the controller or observer design [8, 74]. The characterizations of ISS have been investigated in [187, 188], which have been utilized for the derivation of small gain theorems [93, 94]. Recently, the ISS for infinite-dimensional systems with respect to the boundary disturbance was

4.4 What Can We Guarantee If the Solid Phase Remains?—ISS

119

developed in [99, 100, 104] using the spectral decomposition of the solution of linear parabolic PDEs in one-dimensional spatial coordinate with Strum-Liouville operators. An analogous result for the diffusion equations with a radial coordinate in n-dimensional balls is shown in [34] with proposing an application to robust observer design for battery management systems [151]. We incorporate the heat loss at the interface in the one-phase Stefan problem, as a cooling effect from the solid phase. Here, we assume that the heat loss is a time-varying function, unlike the two-phase Stefan problem which models the heat loss by the solid phase temperature that is state-dependent. Hence, we consider the following model: Tt (x, t) = αTxx (x, t),

0 ≤ x ≤ s(t),

(4.155)

−kTx (0, t) = qc (t),

(4.156)

T (s(t), t) = Tm ,

(4.157)

s˙ (t) = −βTx (s(t), t) −

β qf (t), k

(4.158)

where qf (t) ≥ 0 is a magnitude of the heat loss. The coefficient β/k is added from the physical modeling, which yields the consistency in the physical unit. We impose the following assumption on the heat loss. Assumption 4.4 The heat loss remains nonnegative, bounded, and continuous for all t ≥ 0, and the total energy is also bounded, i.e., qf (t) ≥0, ∃M >0,

∀t > 0,  ∞ qf (t)dt < M. s.t.

(4.159) (4.160)

0

One critical difference of the system (4.155)–(4.158) from the systems we have studied in previous sections is that the monotonicity of the interface dynamics does not hold, i.e., s˙ (t)  0,

(4.161)

which can cause the following scenario: s(t)  0

(4.162)

even under a physically reasonable situation. Namely, even if we keep injecting a positive heat into the liquid phase, the material can be completely frozen to the solid phase due to the heat loss. Such a situation can be explained in the following lemma proven in [178]:

120

4 Extended Models and Design

Lemma 4.1 Provided that qc (t) ≥ 0 for all t ≥ 0, there exists σ > 0 such that for any t¯ ≤ σ where 0 < σ ≤ ∞, there is a unique classical solution of the system (4.155)–(4.158). If σ = ∞, then s(σ ) = 0. However, to validate the physical model under the feedback control, we need to show ∀t ≥ 0.

s(t) > 0,

(4.163)

Hence, the condition (4.163) stands as an additional constraint to hold under the closed-loop system. Here, the heat loss qf (t) is an unknown variable, and we study how the norm estimate is described under the feedback control law designed in Sect. 2.3, namely, the control law is  qc (t) = −c

k α



s(t) 0

 k (T (x, t) − Tm )dx + (s(t) − sr ) . β

(4.164)

We impose the same assumption on the setpoint position as follows. Assumption 4.5 The setpoint is chosen to satisfy sr > s0 +

β α



s0

(T0 (x) − Tm )dx.

(4.165)

0

Finally, we impose the following condition on the control gain. Assumption 4.6 The control gain c is chosen sufficiently large to satisfy c>

β q¯f , ksr

(4.166)

where q¯f := sup {qf (t)} .

(4.167)

0≤t≤∞

The controller is feedback design of liquid temperature profile and the interface position (T (x, t), s(t)). The heat loss qf (t) is regarded as a disturbance, and the norm estimate of the reference error is derived in the sense of input-to-state stability, as stated in the following theorem. Theorem 4.2 Under Assumptions 4.4–4.6, the closed-loop system consisting of (4.155)–(4.158) with the control law (4.164) satisfies the model validity conditions (1.20) and (4.163), and is ISS with respect to the heat loss qf (t) at the interface, i.e., there exist a class-K L function ζ and a class-K function η such that the following estimate holds:

4.4 What Can We Guarantee If the Solid Phase Remains?—ISS

 Ψ (t) ≤ ζ (Ψ (0), t) + η

121

 sup |qf (τ )| ,

(4.168)

τ ∈[0,t]

for all t ≥ 0, in the L2 norm 

s(t)

Ψ (t) =

1 2

(T (x, t) − Tm )2 dx + (s(t) − sr )2

.

(4.169)

0

Moreover, there exist positive constants M1 > 0 and M2 > 0 such that the explicit functions of ζ and η are given by ζ (Ψ (0), t) =M1 Ψ (0)e−λt ,

(4.170)

η( sup |qf (τ )|) =M2 sup |qf (τ )|,

(4.171)

  1 α λ= min 2 , c , 32 sr

(4.172)

τ ∈[0,t]

τ ∈[0,t]

where

which ensures exponential ISS.

Backstepping Transformation Let u(x, t) and X(t) be reference error variables defined by u(x, t) :=T (x, t) − Tm , X(t) :=s(t) − sr .

(4.173) (4.174)

Then, the system (4.155)–(4.158) is rewritten as ut (x, t) =αuxx (x, t), ux (0, t) = −

qc (t) , k

u(s(t), t) =0, ˙ X(t) = − βux (s(t), t) − d(t), where

(4.175) (4.176) (4.177) (4.178)

122

4 Extended Models and Design

d(t) =

β qf (t). k

(4.179)

We apply the following backstepping transformation: w(x, t) =u(x, t) −

β α



s(t)

φ(x − y)u(y, t)dy − φ(x − s(t))X(t),

(4.180)

x

where the gain kernel φ is given by c φ(x) = x − ε. β

(4.181)

Then, one can derive the following target system: wt (x, t) =αwxx (x, t) + s˙ (t)φ  (x − s(t))X(t) + φ(x − s(t))d(t),

(4.182)

β wx (0, t) = φ(0)u(0), α

(4.183)

w(s(t), t) =εX(t),

(4.184)

˙ X(t) = − cX(t) − βwx (s(t), t) − d(t).

(4.185)

Inverse Transformation Consider the following inverse transformation: u(x, t) =w(x, t) −

β α



s(t)

ψ(x − y)w(y, t)dy − ψ(x − s(t))X(t).

(4.186)

x

Taking the derivatives of (4.186) in x and t along (4.182)–(4.185), to match with (4.175)–(4.178), we obtain the gain kernel solution as ψ(x) =erx (p1 sin (ωx) + ε cos (ωx)) ,

(4.187)

where βε , 2α

ω=

p1 = −

1 2αβω

r=

4αc − (εβ)2 , 4α 2   2αc − (εβ)2 ,

(4.188) (4.189)

4.4 What Can We Guarantee If the Solid Phase Remains?—ISS

123

√

and 0 < ε < 2 βαc is to be chosen later. Finally, using the inverse transformation, the boundary condition (4.190) is rewritten as

 β s(t) β wx (0, t) = − ε w(0, t) − ψ(−y)w(y, t)dy − ψ(−s(t))X(t) . α α 0 (4.190) In other words, the target (w, X)-system is described by (4.182), (4.184), (4.185), and (4.190). Note that the boundary condition (4.184) and the kernel function (4.181) are modified from the one in Sect. 2.3, while the control design is equivalent. The target system derived in Sect. 2.3 requires H1 -norm analysis for stability proof. However, with the prescribed heat loss at the interface, H1 -norm analysis fails to show the stability due to the non-monotonic moving boundary dynamics. The modification of the boundary condition (4.184) enables proving stability in L2 norm, as shown later.

Analysis of Closed-Loop System Here, we prove the well-posedness of the closed-loop solution and the positivity conditions of the state variables. Taking the time derivative of the control law (4.164) along with the energy conservation leads to the following differential equation: q˙c (t) = −cqc (t) + cqf (t),

(4.191)

which has the explicit solution as the following open-loop control: 

t

e−c(t−τ ) qf (τ )dτ,

(4.192)

 k (T0 (x) − Tm )dx + (s0 − sr ) . β

(4.193)

qc (t) = q0 e

−ct

+c 0

where 

k q0 = −c α

 0

s0

Hence, the closed-loop solution is equivalent to the open-loop solution with (4.192). Since Assumption 4.5 leads to q0 > 0, the open-loop controller (4.192) remains positive and continuous for all t > 0 by Assumption 4.4. Hence, applying Lemma 4.1, we can show that there exists σ > 0 such that for any t¯ ≤ σ where 0 < σ ≤ ∞, there is a unique classical solution of the system (4.155)–(4.158). Next, we show σ = ∞ by contradiction. Suppose there exists 0 < σ < ∞ such that s(σ ) = 0. Let E(t) be an internal energy of the physical system defined by

124

4 Extended Models and Design

E(t) =

k α



s(t)

(T (x, t) − Tm ) dx +

0

k s(t). β

(4.194)

Note that E(σ ) = 0 holds by the imposed assumption. Taking the time derivative of (4.194) yields the energy conservation ˙ E(t) = qc (t) − qf (t).

(4.195)

In addition, the time derivative of (4.192) yields q˙c (t) = −c (qc (t) − qf (t)) .

(4.196)

Combining these two and taking integration on both sides give 1 E(t) = E(0) − (qc (t) − qc (0)). c

(4.197)

By (4.193) and (4.194), we get   k q0 = −c E(0) − sr . β

(4.198)

Substituting this and (4.192) into (4.197), we have    t ksr ct E(t) = e−ct E(0) + (e − 1) − ecτ qf (τ )dτ . β 0

(4.199)

Let f (t) be a function in time defined by ksr ct (e − 1) − β

f (t) = E(0) +



t

ecτ qf (τ )dτ.

(4.200)

0

Since E(t) = e−ct f (t), we can see that E(t) > 0 for all t > 0 if and only if f (t) > 0 for all t > 0. By (4.200), we have f (0) = E(0) > 0. Taking the time derivative of (4.200) yields 

f (t) = e

 ct

 ksr c − qf (t) . β

(4.201)

By Assumption 4.6, (4.201) leads to f  (t) > 0 for all t > 0. Therefore, f (t) > 0 for all t > 0, and we conclude E(t) > 0 for all t > 0 which contradicts with the imposed assumption s(σ ) = 0 where σ = ∞. Hence, the existence and uniqueness of the solution hold globally ∀t ≥ 0. Since the open-loop system has a unique solution, the closed-loop solution has a unique solution as well. Thus, the following properties hold:

4.4 What Can We Guarantee If the Solid Phase Remains?—ISS

(4.202)

qc (t) >0, u(x, t) >0,

125

ux (s(t), t) < 0,

(4.203) (4.204)

s(t) >0.

Moreover, applying (4.202) and (4.203) to (4.164), the following is established: 0 < s(t) < sr .

(4.205)

ISS Proof To conclude the ISS of the original system, first we show the ISS of the target system (4.182), (4.184), (4.185), and (4.190) with respect to the disturbance d(t). We consider V (t) =

1 ε ||w||2 + X(t)2 . 2α 2β

(4.206)

Then, as proven in Sect. C.4, for a sufficiently small ε > 0, the following inequality is derived: V˙ (t) ≤ − bV (t) + Γ d(t)2 + a|˙s (t)|V (t),

(4.207)

where   αc2 sr 2βε max 1, 3 3 , α 2β ε   1 α b = min 2 , c , 8 sr  2 2s 3 csr ε + 2r +ε . Γ = βc β α a=

(4.208) (4.209) (4.210)

Let z(t) be defined by 

t

z(t) := s(t) + 2

d(τ )dτ.

(4.211)

2βM . k

(4.212)

0

By (4.160) and (4.205), we have 0 < z(t) < z¯ := sr +

126

4 Extended Models and Design

The time derivative of (4.211) is given by z˙ (t) = −βux (s(t), t) + d(t).

(4.213)

Since s˙ (t) = −βux (s(t), t) − d(t) and recalling ux (s(t), t) < 0 and d(t) > 0, the following inequality holds: |˙s (t)| ≤ −βux (s(t), t) + d(t) = z˙ (t).

(4.214)

Applying (4.214) to (4.207) leads to V˙ (t) ≤ − bV (t) + Γ d(t)2 + a z˙ (t)V (t).

(4.215)

Consider the following functional: W (t) = V (t)e−az(t) .

(4.216)

Taking the time derivative of (4.216) with the help of (4.215) and applying (4.212), we deduce W˙ (t) ≤ −bW (t) + Γ d(t)2 .

(4.217)

Since (4.217) leads to the statement that either W˙ (t) ≤ − b2 W (t) or W (t) ≤ 2 2 b Γ d(t) is true, following the procedure in [186, proof of Theorem 5 in Section 3.3], one can derive b 2 W (t) ≤ W (0)e− 2 t + Γ sup d(τ )2 . b τ ∈[0.t]

(4.218)

By (4.216), we have V (t) = W (t)eaz(t) . Applying (4.218), we get b 2 V (t) ≤ eaz(t) W (0)e− 2 t + Γ sup d(τ )2 . b τ ∈[0.t]

(4.219)

Again by (4.216), we have W (0) = V (0)e−az(0) . Combining these two with the help of (4.212), finally we obtain the following estimate on the L2 norm of the target system: b 2 V (t) ≤V (0)ea z¯ e− 2 t + Γ ea z¯ sup d(τ )2 . b τ ∈[0.t]

(4.220)

Due to the invertibility of the transformation from (u, X) to (w, X) together with the boundedness of the domain 0 < s(t) < sr , there exist positive constants M > 0 and M > 0 such that the following inequalities hold:

4.4 What Can We Guarantee If the Solid Phase Remains?—ISS

127

MΨ (t)2 ≤ V (t) ≤ MΨ (t)2 ,

(4.221)

where Ψ (t) is the L2 norm of the original system defined in (4.169). Finally, applying (4.221) to (4.220), one can derive the norm estimate on the original (T , s)system as Ψ (t) ≤

b Mea z¯ Ψ (0)e− 4 t + M

2Γ ea z¯ sup d(τ ), bM τ ∈[0.t]

(4.222)

which completes the proof of Theorem 4.2.

Numerical Simulation Simulation results are performed for the one-phase Stefan problem by considering a cylinder of paraffin whose physical properties are given in Table 4.1. The setpoint and the initial values are chosen as sr = 2 cm, s0 = 0.1 cm, and T0 (x) − Tm = T¯0 (1 − x/s0 ) with T¯0 = 1 ◦ C. Then, the setpoint restriction is satisfied. The control gain is set as c = 5.0 × 10−3 /s, and the heat loss at the interface is set as qf (t) = q¯f e−Kt ,

(4.223)

where K = 5.0 × 10−6 /s. The closed-loop responses for q¯f = 1.0 × 102 W/m2 (red), 2.0 × 102 W/m2 (blue), and 3.0 × 103 W/m2 (green) are implemented as depicted in Fig. 4.7. Figure 4.7a shows the dynamics of the interface, which illustrates the convergence to the setpoint with an error due to the unknown heat loss at the interface. This error becomes larger as q¯f gets larger, which is consistent with the ISS result. In addition, the property 0 < s(t) < sr is observed. Figure 4.7b shows the dynamics of the proposed closed-loop control law, and Fig. 4.7c shows the dynamics of the boundary temperature T (0, t). Hence, we can observe that the simulation results are consistent with the theoretical result we prove for the model validity conditions and ISS. Table 4.1 Physical properties of the liquid paraffin

Description Density Latent heat of fusion Heat capacity Melting temperature Thermal conductivity

Symbol ρ ΔH ∗ Cp Tm k

Value 790 kg · m−3 210 J · g−1 2.38 J · g−1 ·◦ C−1 37.0 ◦ C 0.220 W · m−1

128 Fig. 4.7 The responses of the system (4.155)–(4.158) with the heat loss qf (t) = q¯f e−Kt under the feedback control law (4.164). (a) Convergence of the interface is observed with offsets from the setpoint depending on the magnitude of the heat loss. (b) Positivity of the closed-loop controller maintains. (c) The model validity of the boundary liquid temperature holds, i.e., T (0, t) > Tm

4 Extended Models and Design

2

1

0 0

1

2

3

4

5

4

5

4

5

(a)

1500 1000 500 0 0

1

2

3

(b)

80

60

40 0

1

2

3 (c)

4.5 Sampled-Data Design The results in the previous sections assumed the control input to be varying continuously in time. However, in digital implementation of control systems it is impossible to dynamically change the control input continuously in time. 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 wellknown 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

4.5 Sampled-Data Design

129

the closed-loop system under such a ZOH-based sampled-data control was studied in [96], and the sampled-data observer design under discrete-time measurement is developed in [97] 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 [98, 101] by applying predictor-based feedback developed in [124], and results for a linear parabolic PDE are given in [103] by employing Sturm-Liouville operator theory. The sampled-data control for parabolic PDEs has been developed by Fridman and coworkers by utilizing linear matrix inequalities [6, 77, 78, 174]. However, none of the existing work on the sampleddata control has studied the Stefan system described by a parabolic PDE with a state-dependent moving boundary governed by a nonlinear ODE. We return to the one-phase Stefan model from Sect. 2.3, Tt (x, t) = αTxx (x, t),

0 ≤ x ≤ s(t),

(4.224)

−kTx (0, t) = qc (t),

(4.225)

T (s(t), t) = Tm ,

(4.226)

s˙ (t) = −βTx (s(t), t).

(4.227)

In practical implementation, the input qc (t) cannot be changed continuously in time. Instead, by taking the measured output signal discretely in time, the control is to be implemented at each sampling time. The typical devise for a sampled-data implementation of the nominal continuous time control law is the “Zero-OrderHold” (ZOH). Through ZOH, during the time intervals between samples, the control maintains the input value from the previous sampling instant. Let tj be the j th sampling time for j = 0, 1, 2, . . ., and let the sampling interval length τj be defined as τj = tj +1 − tj .

(4.228)

The application of ZOH to the nominal control law (4.164) leads to the following design for the sampled-data control: 

k qc (t) = − c α

 0

s(tj )

 k (T (x, tj ) − Tm )dx + (s(tj ) − sr ) , β

∀t ∈ [tj , tj +1 ), (4.229)

where the right-hand side is constant during the time interval t ∈ [tj , tj +1 ). Let us denote qj = qc (t) for t ∈ [tj , tj +1 ). Hereafter, all the variables with subscript j denote the variables at t = tj . We introduce the following assumption on the sampling scheduling. Assumption 4.7 The sampling schedule has a finite upper diameter and a positive lower diameter, i.e., there exist constants 0 < r ≤ R such that

130

4 Extended Models and Design

sup {τj } ≤R,

(4.230)

inf {τj } ≥r.

(4.231)

j ∈Z + j ∈Z +

Our main theorem is given next. Theorem 4.3 Consider the closed-loop system (4.224)–(4.227), (4.229) under Assumptions 4.5, 4.7. Then for every 0 < r ≤ R < 1/c, there exists a constant M := M(r) such that the closed-loop system has a unique solution satisfying the following estimate: Ψ (t) ≤ MΨ (0) exp(−bt), where b =

1 8

0 min

α ,c sr2



1

, for all t ≥ 0, in the L2 norm

s(t)

Ψ (t) =

(4.232)

(T (x, t) − Tm )2 dx + (s(t) − sr )2 .

(4.233)

0

The positive constant M in (4.232) has a dependency on r > 0 as M(r) = M1 +

M2 cr

1 − (1 − cr)2 e 8

,

(4.234)

for some positive constants M1 > 0 and M2 > 0, which are not dependent on r > 0.

Analysis of the Closed-Loop System We introduce the following reference error states: u(x, t) = T (x, t) − Tm ,

X(t) = s(t) − sr .

(4.235)

The governing equations (4.224)–(4.227) are rewritten as the following reference error system: ut (x, t) =αuxx (x, t),

(4.236)

ux (0, t) = − qc (t)/k,

(4.237)

u(s(t), t) =0, ˙ X(t) = − βux (s(t), t). Define the internal energy of the reference error system as follows:

(4.238) (4.239)

4.5 Sampled-Data Design

131

2 = k E(t) α



s(t)

u(x, t)dx +

0

k X(t). β

(4.240)

Taking the time derivative of (4.240) along the solution of (4.236)–(4.239) leads to d 2 E(t) = qc (t). dt

(4.241)

Noting that qc (t) is constant for t ∈ [tj , tj +1 ) as qc (t) = qj under ZOH-based sampled-data control, taking the integration of (4.241) from t = tj to t = tj +1 yields 2j = τj qj , 2j +1 − E E

(4.242)

2j = E(t 2 j ) and τj = tj +1 − tj . The sampled-data control (4.229) and the where E internal energy (4.240) at each sampling time satisfy the following relation: 2j . qj = −cE

(4.243)

Substituting (4.243) into (4.242), we obtain   2j , 2j +1 = 1 − cτj E E

(4.244)

which leads to the explicit solution as follows: 20 2j =E E

j3 −1

(1 − cτi ) .

(4.245)

i=0

Substituting (4.245) into (4.243) yields qc (t) = qj =q0

j3 −1

(1 − cτi ) ,

∀t ∈ [tj , tj +1 ),

∀j ∈ Z + ,

(4.246)

i=0

where  q0 = −c

k α



s0 0

(T0 (x) − Tm )dx +

 k (s0 − sr ) . β

(4.247)

Therefore, the closed-loop system under the sampled-data feedback control (4.229) is equivalent to the open-loop solution with the control input (4.246). Moreover, under Assumptions 4.5 and 4.7, and the fact that c < R1 , the input (4.246) is shown to be a bounded piecewise continuous function and qc (t) ≥ 0 for all t ≥ 0. Thus, the existence and uniqueness of the solution are ensured. One can deduce

132

4 Extended Models and Design

s˙ (t) > 0,

∀t ≥ 0,

(4.248)

and thus s0 < s(t) for all t ≥ 0. Integrating (4.241) from t = tj to t ∈ [tj , tj +1 ) leads to 2 −E 2j = (t − tj )qj , E(t)

∀t ∈ [tj , tj +1 ).

(4.249)

With the help of (4.243) and (4.245), Eq. (4.249) yields   2j , 2 = 1 − c(t − tj ) E E(t) By Assumption 4.7 and since c
0, for all t ∈ [tj , tj +1 ) and for all j ∈ Z + . Applying this to (4.250) and noting that 2j < 0, E

∀j ∈ Z + ,

(4.251)

one can obtain 2 < 0, E(t)

∀t ≥ 0.

(4.252)

Substituting (4.252) into (4.240) and applying u(x, t) > 0 for all x ∈ (0, s(t)) and t ≥ 0, we have X(t) < 0,

∀t ≥ 0,

(4.253)

which leads to s0 < s(t) < sr ,

∀t ≥ 0.

(4.254)

Target System We use the same backstepping transformation as (4.180), (4.181). Thus, we get the target system wt (x, t) =αwxx (x, t) + s˙ (t)φ  (x − s(t))X(t), w(s(t), t) =εX(t), ˙ X(t) = − cX(t) − βwx (s(t), t). The boundary condition at x = 0 is obtained by

(4.255) (4.256) (4.257)

4.5 Sampled-Data Design

wx (0, t) = −

133

c qc (t) β − εu(0, t) − k α α



s(t)

u(y, t)dy −

0

c X(t). β

(4.258)

2j for all Substituting the design of the sampled-data control qc (t) = qj = −cE + 2 t ∈ [tj , tj +1 ) and for all j ∈ Z , and recalling the definition of E(t) in (4.240), the boundary condition (4.258) can be written as wx (0, t) = −

 c 2 2j − β εu(0, t). E(t) − E k α

(4.259)

Moreover, substituting (4.250), we can describe (4.259) as wx (0, t) =f (t) −

β εu(0, t), α

(4.260)

where f (t) is an explicit function in time defined by f (t) =

c2 2 Ej · (t − tj ), k

∀t ∈ [tj , tj +1 ),

j ∈ Z +.

(4.261)

The closed form representation of (4.260) using variables (w, X) is given by using the same inverse transformation as (4.186), (4.187), which yields

 β s(t) β wx (0, t) =f (t) − ε w(0, t) − ψ(−y)w(y, t)dy − ψ(−s(t))X(t) . α α 0 (4.262) Therefore, the closed form of the target (w, X)-system is described by (4.255), (4.256), (4.257), and (4.262).

Stability Proof For a given t ≥ 0, we define the most recent sampling number as n := {n ∈ Z + |tn ≤ t < tn+1 },

(4.263)

and we firstly apply Lyapunov method for the time interval t ∈ [tj , tj +1 ) for all j = 0, 1, · · · , n − 1, and next for the interval from tn to t. For both cases, we consider V =

1 ε ||w||2 + X(t)2 , 2α 2β

(4.264)

134

4 Extended Models and Design

 s(t) where ||w|| denotes L2 norm defined by ||w|| = w(x, t)2 dx. As proven 0 in Sect. C.4, and applying s˙ (t) > 0, for a sufficiently small ε > 0, the following inequality is derived: V˙ ≤ − bV + 2sr f (t)2 + a s˙ (t)V ,

(4.265)

where   α 1 b = min 2 , c , 8 sr

  αc2 sr 2βε max 1, 3 3 . a= α 2β ε

(4.266)

W = V e−as(t) .

(4.267)

Consider the following functional:

Taking the time derivative of (4.267) with the help of (4.265), we deduce W˙ ≤ −bW + 2sr f (t)2 e−as(t) ≤ −bW + 2sr f (t)2 .

(4.268)

(i) For t ∈ [tj , tj +1 ), for all j = 0, 1, · · · , n − 1, Applying comparison principle to (4.268) for t ∈ [tj , tj +1 ) leads to W (t) ≤ W (tj )e−b(t−tj ) + 2sr e−bt



t

ebτ f (τ )2 dτ.

(4.269)

tj c2 2 k Ej (t

Setting t = tj +1 and recalling f (t) =

Wj +1 ≤Wj e−bτj +

− tj ), ∀t ∈ [tj , tj +1 ), we get

2c4 sr −bτj 22 e Ej Ij , k2

(4.270)

where Wj = W (tj ), and Ij is defined by  Ij :=

tj +1

eb(τ −tj ) (τ − tj )2 dτ.

(4.271)

tj

Then, by introducing the variable s = b(τ − tj ) and integration by substitution, with the help of bτj < 18 cτj < 18 for all j ∈ Z + derived by (4.266), Assumption 4.7 and the fact that c < R1 , one can derive the following inequality: Ij =

1 b3

 0

bτj

es s 2 ds ≤

J , b3

(4.272)

4.5 Sampled-Data Design

135

where J is defined by 

1 8

J :=

es s 2 ds.

(4.273)

0

Applying (4.272) to (4.270) yields Wj +1 ≤Wj e−bτj + Bj ,

(4.274)

2J c4 sr −bτj 22 e Ej . k 2 b3

(4.275)

where Bj is defined by Bj =

Applying (4.274) from j = n − 1 to j = 0 inductively, we get Wn ≤ W0 e−b

4n−1 i=0

τi

+ Bn−1 +

n−2 

Bi e−b

4n−1

j =i+1 τj

(4.276)

.

i=0

2j given in (4.245), we have By (4.275) and the solution of E n−2 

Bi e−b

4n−1

j =i+1 τj

i=0

22 e−b 2J c4 sr E 0 ≤ k 2 b3 22 e−b 2J c4 sr E 0 ≤ k 2 b3

4n−1

j =0 τj

4n−1

j =0 τj

 1+

 i−1 n−2 3  i=1

 1+

(1 − cτk )

e

b

k=0

 i−1 n−2 3  i=1

 2

4i−1



j =0 τj

 (1 − cτk ) e

2 bτk

.

(4.277)

k=0

Since b=

  α c 1 min 2 , c < , 8 8 sr

(4.278)

by using r = infj ∈Z + {τj } > 0 given in Assumption 4.7, the following inequality holds: cr

(1 − cτi )2 ebτi ≤ (1 − cr)2 e 8 := δ < 1, Thus, the inequality (4.277) leads to

∀j ∈ Z + .

(4.279)

136

4 Extended Models and Design n−2 

Bi e

−b

4n−1

j =i+1 τj

i=0

22 e−b 2J c4 sr E 0 ≤ k 2 b3 ≤

4n−1

j =0 τj

 1+

n−2 

 δ

i

i=1

22 −b 4n−1 τ 2J c4 sr E 0 j =0 j . e k 2 b3 (1 − δ)

(4.280)

In a similar way, we get 22 −b 4n−1 τ 2J c4 sr E 0 j =0 j . e k 2 b3 (1 − δ)

Bn−1 ≤

Recalling that τj = tj +1 − tj and t0 = 0, we get and (4.281) to (4.276), we arrive at

4n−1

j =0 τj

(4.281) = tn . Applying (4.280)

202 )e−btn , Wn ≤ (W0 + AE

(4.282)

where A=

2J c4 sr . 2 k b3 (1 − δ)

(4.283)

(ii) For t ∈ [tn , tn+1 ), Applying comparison principle to (4.268) from tn to t ∈ [tn , tn+1 ), we get W (t) ≤Wn e−b(t−tn ) + Bn e−b(t−tn+1 ) 202 e−bt . ≤Wn e−b(t−tn ) + AE

(4.284)

Finally, combining (4.282) and (4.284), the following bound is obtained: 202 )e−bt . W (t) ≤ (W0 + 2AE

(4.285)

Recalling the relation W = V e−as(t) defined in (4.267), and applying 0 < s(t) < sr , the norm estimate for W in (4.285) leads to the following estimate for V : 202 )e−bt . V (t) ≤ easr (V0 + 2AE

(4.286)

We consider the L2 -norm of (u, X)-system defined by  Ψ (t) = 0

s(t)

u(x, t)2 dx + X(t)2 .

(4.287)

4.5 Sampled-Data Design

137

Due to the invertibility of the transformation from (u, X) to (w, X) together with the boundedness of the domain 0 < s(t) < sr , there exist positive constants M > 0 and M > 0 such that the following inequalities hold: MΨ (t) ≤ V (t) ≤ MΨ (t).

(4.288)

 2 = k s(t) u(x, t)dx + Moreover, due to the definition of the reference energy E(t) α 0 k β X(t) given in (4.240), using Young’s and Cauchy Schwarz inequalities one can show that 202 ≤ KΨ0 , E

(4.289)

where K = 2k 2 max{ αsr2 , β12 }. Applying (4.288) and (4.289) to (4.286), we deduce that there exists a positive constant M > 0 such that the following inequality holds: Ψ (t) ≤ MΨ0 e−bt ,

(4.290)

which completes the proof of Theorem 4.3.

Numerical Simulation We use the same physical parameters of paraffin, the initial conditions, and the setpoint as those used in the last section for ISS. We consider periodic sampling with period given by τj = R = 10 min, for all j ∈ Z. The control gain is set as c = 5.0 × 10−3 /s, by which the requirement R < 1c is satisfied. The time responses of the interface position, the control input, and the boundary temperature under the closed-loop system are depicted in Fig. 4.8a–c, respectively. Figure 4.8a illustrates that the interface position s(t) converges to the setpoint sr monotonically and smoothly without overshooting, i.e., s˙ (t) > 0 and s0 < s(t) < sr hold for all t ≥ 0. Figure 4.8b shows that the proposed sampled-data control law maintains constant positive value for every sampling period and is monotonically decreasing to zero. Figure 4.8c illustrates that the boundary temperature T (0, t) remains greater than the melting temperature Tm with accompanying “spikes” at every sampling time t = τj up to 2 hours. Such spikes are caused by the large drop of the control input qc (t) at sampling time observed from Fig. 4.8b, which affects the boundary temperature directly as given in the boundary condition. Therefore, the numerical results are consistent with the theoretical results we have established for the required properties and in Theorem 4.3 for closed-loop stability.

138 Fig. 4.8 The responses of the system (4.224)–(4.227) under the ZOH-based sampled-data control (4.229). (a) Convergence of the interface to the setpoint is observed without the overshoot. (b) The sampled-data controller maintains positive value. (c) The boundary temperature features “spikes” at every sampling time

4 Extended Models and Design

2

1

0 0

1

2

3

4

5

4

5

4

5

(a)

1500 1000 500 0 0

1

2

3

(b)

80

60

40 0

1

2

3 (c)

Chapter 5

Two-Phase Stefan Problem

While in Sect. 4.4 we already dealt with an approximation of the influence of the solid phase, where non-monotonic interface dynamics arose due to incorporating a heat loss at the interface in the one-phase Stefan model, such a heat loss should be exactly modeled by a PDE for the solid phase and with the effect of the heat flux from the solid phase on the interface ODE [114]. The control design to asymptotically stabilize the resulting PDE-ODE-PDE system is quite challenging. We present it in this chapter.

5.1 Description of the Physical Model The two-phase Stefan system describes the model of the phase change phenomena of melting and freezing (solidification) processes. As depicted in Fig. 5.1, two complementary time-varying sub-domains x ∈ [0, s(t)] and x ∈ [s(t), L] are occupied by the liquid phase and the solid phase, respectively. Let Tl (x, t) and Ts (x, t) be the temperature profiles of liquid and solid, respectively, and s(t) be the position of the interface between liquid and solid. Then, the energy conservation and heat transfer laws give the following PDE-ODE-PDE model of the temperature profile: ∂Tl ∂ 2 Tl (x, t) =αl 2 (x, t), ∂t ∂x

0 < x < s(t),

∂Ts ∂ 2 Ts (x, t) =αs 2 (x, t), s(t) < x < L, ∂t ∂x qc (t) ∂Tl ∂Ts (0, t) = − (L, t) = 0, , ∂x kl ∂x

© Springer Nature Switzerland AG 2020 S. Koga, M. Krstic, Materials Phase Change PDE Control & Estimation, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-58490-0_5

(5.1) (5.2) (5.3)

139

140

5 Two-Phase Stefan Problem

Fig. 5.1 Schematic of the two-phase Stefan problem. The temperature profiles of both the liquid phase and the solid phase are dynamic

qc (t)

liquid

solid

Tl (x, t)

Ts (x, t )

Tm

0

Tl (s(t), t) =Tm , γ s˙ (t) = − kl

s(t)

Ts (s(t), t) = Tm

L

x

(5.4)

∂Tl ∂Ts (s(t), t) + ks (s(t), t), ∂x ∂x

(5.5)

with the initial data Tl,0 (x) := Tl (x, 0), Ts,0 (x) := Ts (x, 0), s0 := s(0), where qc (t) > 0 is a boundary heat input. Here, αi = ρki ci i , where ρi , ci , ki for i ∈{l, s} are the density, the heat capacity, the thermal conductivity, and the heat transfer coefficient, respectively and the subscripts “l” and “s” are associated with the liquid or solid phase, respectively. Also, γ = ρl ΔH ∗ where ΔH ∗ denotes the latent heat of fusion.

5.2 Existence, Uniqueness, and Non-monotonicity of Interface Dynamics There are underlying assumptions to validate the model (5.1)–(5.5). First, the liquid phase is not frozen near the boundary x = 0. This condition is ensured if the liquid temperature Tl (x, t) is greater than the melting temperature Tm . Second, in a similar manner, the solid phase is not molten near the boundary x = L, which is ensured if the solid temperature Ts (x, t) is less than the melting temperature. Third, the material is not completely molten or frozen to a 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. Remark 5.1 To keep the physical state of each phase meaningful, the following conditions must be maintained: Tl (x, t) ≥Tm ,

∀x ∈ (0, s(t)),

∀t > 0,

(5.6)

5.2 Existence, Uniqueness, and Non-monotonicity of Interface Dynamics

Ts (x, t) ≤Tm ,

∀x ∈ (s(t), L),

∀t > 0,

∀t > 0.

0 0 and Hs > 0 such that Tm ≤Tl,0 (x) ≤ Tm + Hl (s0 − x), Tm + Hs (s0 − x) ≤Ts,0 (x) ≤ Tm ,

∀x ∈ [0, s0 ],

∀x ∈ [s0 , L].

(5.9) (5.10)

The existence and uniqueness of the two-phase Stefan problem were proven in [36, Theorem 1 in p. 4 and Theorem 4 in p. 8] by employing the maximum principle, which is stated in the following lemma. Lemma 5.1 Under Assumption 5.1, and provided that qc (t) is a piecewise continuous function that satisfies ∀t ∈ [0, t ∗ ),

qc (t) ≥ 0,

(5.11)

there exists a finite time t := supt∈(0,t ∗ ) {t|s(t) ∈ (0, L)} > 0 such that a classical solution to (5.1)–(5.5) exists, is unique, and satisfies the model validity condition (5.6)–(5.8) for all t ∈ (0, t). Moreover, if t ∗ = ∞ and it holds  0 < γ s∞ +

t

qc (s)ds < γ L,

(5.12)

0

for all t ≥ 0, where s∞ := s0 +

kl αl γ

 0

s0

(Tl,0 (x) − Tm )dx +

ks αs γ



L

(Ts,0 (x) − Tm )dx,

(5.13)

s0

then t = ∞, namely, the well-posedness and the model validity conditions are satisfied for all t ≥ 0. The variable s∞ defined in (5.13) is s∞ = limt→∞ s(t) under the zero input qc (t) ≡ 0 for all t ≥ 0. For (5.12) to hold for all t ≥ 0, we at least require it to hold at t = 0, which leads to the following assumption. Assumption 5.2 The initial conditions that appear in s∞ in (5.13) satisfy 0 < s∞ < L.

(5.14)

142

5 Two-Phase Stefan Problem

5.3 State Feedback Control Design As in the previous chapters, the control objective is to stabilize the temperature profile and the interface position (Tl , Ts , s) at a reference setpoint (Tm , Tm , sr ). We approach this problem by means of energy shaping control, that is originally developed for underactuated mechanical systems such as robot manipulators [69, 87]. The thermal internal energy of the overall system in (5.1)–(5.5) is given by kl E(t) = αl

 0

s(t)

ks (Tl (x, t) − Tm )dx + αs



L

(Ts (x, t) − Tm )dx + γ s(t),

s(t)

(5.15) which includes the specific heat of both liquid and solid phases and the latent heat. Taking the time derivative of (5.15) along the solution of (5.1)–(5.5), one can obtain the energy conservation law formulated as d E(t) = qc (t). dt

(5.16)

To achieve the control objective, the internal energy (5.15) must converge to the following setpoint energy: lim E(t) = γ sr .

t→∞

(5.17)

Taking the time integration of (5.16) from t = 0 to ∞, and imposing the input constraint (5.11) required for the model validity as stated in Lemma 5.1, in order to achieve (5.17) we deduce that the following restriction on the setpoint is necessary: Assumption 5.3 With s∞ defined in (5.13), the setpoint sr is chosen to satisfy s∞ < sr < L.

(5.18)

Graphic illustration of Assumptions 5.1–5.3 and the control objective is depicted in Fig. 5.2. With Assumption 5.3, due to the energy conservation (5.16), the following control law (5.19) qc (t) = −c(E(t) − Er ),     kl s(t) ks L = −c (Tl (x, t)−Tm )dx+ (Ts (x, t)−Tm )dx+γ (s(t)−sr ) αl 0 αs s(t) (5.20) drives the internal energy E(t) to the reference energy Er . We state the following theorem, which established that (5.20) achieves much more than E(t) → Er .

5.3 State Feedback Control Design

qc (t)

Tl,0 (x)

143

t) ≡ q c( der

Ts,0 (x)

s0

L

s∞

0 sign

0

Tm

Tm

Un

De

Tm

0

qc ( t) >

0

Tm

x 0

L

x

Tm sr

L

x

Fig. 5.2 Illustration of Assumptions 5.1–5.3 and control objective

Theorem 5.1 Under Assumptions 5.1–5.3, the closed-loop system consisting of the plant (5.1)–(5.5) and the control law (5.20), where c > 0 is an arbitrary controller gain, maintains the conditions (5.6)–(5.8), and there exists a positive constant M > 0 such that the following exponential stability estimate holds: Ψ (t) ≤ MΨ (0)e−dt for all t ≥ 0, where d = 

s(t)

Ψ (t) =

1 2

0 min

αl , αs , c 2L2 L2

 (Tl (x, t) − Tm )2 dx +

0

(5.21)

1 , in the L2 -norm

L

(Ts (x, t) − Tm )2 dx + (s(t) − sr )2 .

s(t)

(5.22)

Error Variables Relative to Melting Temperature Let u(x, t) and v(x, t) be reference error temperature profiles of the liquid and the solid phase, respectively, defined as u(x, t) = Tl (x, t) − Tm ,

v(x, t) = Ts (x, t) − Tm .

(5.23)

Then the system (5.1)–(5.5) is rewritten as ut (x, t) =αl uxx (x, t), ux (0, t) = −

qc (t) , kl

0 < x < s(t) u(s(t), t) = 0,

(5.24) (5.25)

144

5 Two-Phase Stefan Problem

vt (x, t) =αs vxx (x, t), vx (L, t) =0,

s(t) < x < L

(5.26)

v(s(t), t) = 0,

(5.27)

s˙ (t) = − βl ux (s(t), t) + βs vx (s(t), t).

(5.28)

The system (5.24)–(5.28) shows the two PDEs coupled with the ODE that governs the moving boundary. The stabilization of states (u, v, s) at (0, 0, sr ) is approached by designing the control law of qc (t) in (5.25); however, the multiple PDEs are difficult to deal with using only one input.

Change of Variable to Absorb the Solid Phase into the Interface To reduce the complexity of the system’s structure in (5.24)–(5.28), we introduce another change of variable. Let X(t) be a state variable defined by βs X(t) = s(t) − sr + αs



L

v(x, t)dx.

(5.29)

s(t)

This state represents the (scaled) thermal energy of the solid phase and the interface. Taking the time derivative of (5.29) and with the help of (5.26)–(5.28), we get ˙ X(t) = −βl ux (s(t), t) which eliminates the v-dependency in ODE dynamics (5.28). Thus, the (u, v, s)-system in (5.24)–(5.28) can be reduced to (u, X)-system as ut (x, t) =αl uxx (x, t), ux (0, t) = −

qc (t) , kl

0 < x < s(t) u(s(t), t) = 0,

˙ X(t) = − βl ux (s(t), t).

(5.30) (5.31) (5.32)

Therefore, the control problem is now formulated as designing the boundary control qc (t) in (5.31) to stabilize the (u, X)-system in (5.30)–(5.32) at the zero states (0, 0), which is equivalent to the problem of the stabilization of the one-phase Stefan model studied in Sect. 2.3. The main difference relative to Sect. 2.3 is that the monotonicity of the velocity of the moving interface, i.e., s˙ (t) > 0, is not guaranteed in the twophase Stefan problem due to the reversible melting and freezing process. Remark 5.2 Considering the output (5.29) of the state-space system (5.24)–(5.28), the Eqs. (5.30)–(5.32), along with (5.26), (5.27), can be considered as the system’s input–output “normal form” in the sense of Byrnes and Isidori [31, 90] (see Chapter 4 in [90]). As indicated above, a control design will be conducted for the input–output dynamics (5.30)–(5.32), i.e., the dynamics of the liquid phase with a modified interface output map X = h(s, v) using the control qc . The stability of

5.3 State Feedback Control Design

145

the inverse dynamics (5.26), (5.27), which happen to be the dynamics of the solid phase, is studied later.

Backstepping Transformation We use the same backstepping transformation as (4.180), (4.181), namely, βl w(x, t) =u(x, t) − αl φ(x) =



s(t)

φ(x − y)u(y, t)dy − φ(x − s(t))X(t),

(5.33)

x

1 (cx − ε), βl

(5.34)

where ε > 0 is a parameter to be determined in the stability analysis. Thus, the associated target system is derived as wt (x, t) =αl wxx (x, t) + w(s(t), t) =

c s˙ (t)X(t), βl

(5.35)

ε X(t), βl

(5.36)

˙ X(t) = − cX(t) − βl wx (s(t), t).

(5.37)

Taking the derivative of (5.33) in x, we obtain ε c wx (x, t) =ux (x, t) − u(x, t) − αl αl



s(t) x

u(y, t)dy −

c X(t). βl

(5.38)

For the standard backstepping procedure, a homogeneous boundary condition at x = 0 of the target system leads to a control law. If we chose wx (0, t) = 0, we obtain a stable target system in the case of fixed domain. However, the Stefan problem necessitates that the heat input maintain positivity in order to guarantee the model validity. The choice of wx (0, t) = 0 leads to a control design that does not ensure positivity. Instead, by the energy conservation law (5.15), (5.16), we can see that the following choice of the state feedback controller ensures positivity: 

1 qc (t) = − c αl



s(t)

 u(y, t)dy + X(t) ,

(5.39)

0

as developed in “energy shaping control.” Hence, we design the control law with respect to X(t) as (5.39) and obtain a boundary condition of the target system. Setting x = 0 in (5.38) and applying (5.39), the boundary condition at x = 0 is obtained as

146

5 Two-Phase Stefan Problem

wx (0, t) = −

ε u(0, t), αl

(5.40)

whose right-hand side should be rewritten with respect to (w, X) by using the same inverse transformation as (4.186) (4.187).

 ε βl s(t) ψ(−y)w(y, t)dy − ψ(−s(t))X(t) . wx (0, t) = − w(0, t) − αl αl 0 (5.41) The target (w, X)-system written as (5.35)–(5.37) and (5.41) has a complicated structure, with coupling between w and X at both boundaries, but is nevertheless proven to satisfy the exponential stability estimate in L2 norm with the help of the properties derived in the next section.

5.4 Analysis of the Closed-Loop System Guaranteeing the Conditions of Model Validity Analogously to the problems so far, we prove the positivity of input. Taking the time derivative of the control law (5.39) along the solution of the system yields q˙c (t) = −cqc (t).

(5.42)

Hence, the state feedback control law is an exponentially decaying function of time, given by qc (t) = qc (0)e−ct .

(5.43)

Since Assumption 5.3 is equivalent to qc (0) > 0, (5.43) ensures qc (t) >0,

∀t > 0.

(5.44)

Applying Lemma 5.1 directly leads to u(x, t) >0,

ux (s(t), t) < 0,

(5.45)

v(x, t) e−az(0) , of which the existence is ensured by Assumptions 5.1–5.3. Hence, (w, X)-system is shown to be exponentially stable. Consider the Lyapunov function  V1 = ||u||2 =

s(t)

u(x, t)2 dx.

(5.58)

0

Due to the invertibility of the transformations, there exist positive constants M > 0, M¯ > 0 such that the following norm equivalence between (u, X)-system and (w, X)-system holds:     M V1 (t) + X(t)2 ≤ V (t) ≤ M¯ V1 (t) + X(t)2 .

(5.59)

Hence, by (5.57), the following exponential stability estimate of the (u, X)-system is shown: V1 (t) + X(t)2 ≤

M¯ δ (V1 (0) + X(0)) e−bt . M

(5.60)

Stability Analysis for the Solid Phase Let V2 be the L2 -norm of the reference error of the solid phase temperature v defined by 1 1 V2 (t) = ||v||2 = 2 2



L s(t)

v(x, t)2 dx.

(5.61)

5.4 Analysis of the Closed-Loop System

149

Taking the time derivative of (5.61) along the solution of (5.26)–(5.27), and applying Poincare’s inequality with the help of 0 < s(t) < L, we obtain  L s˙ (t) 2 ˙ v(s(t), t) − αs vx (x, t)2 dx, V2 = − 2 s(t) αs αs ≤− V2 < − 2 V2 . 2(L − s(t))2 2L

(5.62)

By comparison principle, the differential inequality (5.62) yields the following exponential decay of the norm: V2 (t) ≤ V2 (0)e

−

αs t 2L2

(5.63)

.

Stability of Overall Liquid-Interface-Solid System Applying Young’s and Cauchy Schwartz inequalities to the definition of X given in (5.29) with the help of 0 < s(t) < L yields the following norm estimate: X(t)2 ≤2Y (t) +

2Lβs2 V2 , αs2

(5.64)

where we defined Y (t) = |s(t) − sr |2 . On the other hand, a bound on Y (t) with respect to X(t)2 and V2 is also obtained in a similar manner to (5.64), which yields Y (t) ≤ 2X(t)2 +

2Lβs2 V2 . αs2

(5.65)

Finally, summing the norms of the liquid temperature, the interface position, and the solid temperature, respectively, and applying (5.65), (5.57), (5.63), and (5.64), we see that there exists a positive constant M such that the following estimate of the norm holds: V1 (t) + Y (t) + V2 (t) ≤M (V1 (0) + Y (0) + V2 (0)) e which completes the proof of Theorem 5.1.

0 − min b,

αs 2L2

1 t

,

(5.66)

150

5 Two-Phase Stefan Problem

5.5 Robustness to Uncertainties of Physical Parameters The control design (5.20) requires the physical parameters of both the liquid and solid phases. However, in practice these parameters are uncertain. Guaranteeing the robustness of the stability of the closed-loop system with respect to such parametric uncertainties is significant. Suppose that the proposed control law is replaced by 

kl qc (t) = −c (1 + εl ) αl



s(t) 0

ks + (1 + εs ) αs



(Tl (x, t) − Tm )dx 

L

(Ts (x, t) − Tm )dx + γ (1 + εf )(s(t) − sr ) ,

s(t)

(5.67) where εl , εs , and εf are the uncertainties of physical parameters satisfying εl > −1, εs ≥ −1, and εf ≥ −1. We state the following theorem. Theorem 5.2 Under Assumptions 5.1, 5.2, and assuming that the setpoint is chosen to satisfy qc (0) > 0 with (5.67) and sr < L, consider the closed-loop system consisting of the plant (5.1)–(5.5) and the control law (5.67). Then, for any perturbations (εl , εs , εf ) satisfying εl ≥ εf ≥ εs ,

(5.68)

there exists (possibly small) R > 0 such that if / / / εf − εl / / / / 1 + ε / < R, l

(5.69)

then the closed-loop system maintains model validity (5.6)–(5.8) and the exponential stability at the origin holds for the norm defined in (5.22). Theorem 5.2 implies that if we know lower and upper bounds of the physical parameters as k l ≤ kl ≤ k l , α l ≤ αl ≤ α l , and γ ≤ γ ≤ γ , then the most conservative choice of the control law to satisfy the condition (5.68) is given by  qc (t) = −c

kl αl



s(t)

 (Tl (x, t) − Tm )dx + γ (s(t) − sr ) ,

(5.70)

0

which does not incorporate the solid phase temperature. This design requires less information than the exact feedback design (5.20); however, the conditions qc (0) > 0 and sr < L, which lead to s0 +

kl αlγ

 0

s0

(Tl,0 (x) − Tm )dx < sr < L,

(5.71)

5.5 Robustness to Uncertainties of Physical Parameters

151

are more restrictive than Assumption 5.3 for the unperturbed design (5.20), which causes a tradeoff between the parameter uncertainty and the restriction of the setpoint. The proof of Theorem 5.2 is established by following similar steps to the proof of Theorem 5.1.

Closed-Loop Analysis First, we derive an analogous result on the properties of the closed-loop system to those derived in Sect. 5.4 by employing contradiction approach twice. Assume that there exists a finite time t ∗ > 0 such that qc (t) > 0,

∀t ∈ [0, t ∗ ),

(5.72)

∗

qc (t ) = 0.

(5.73)

Then, by Lemma 5.1, for t ∈ (0, t¯) where t¯ := supt∈(0,t ∗ ) {t|s(t) ∈ (0, L)}, the solution exists, is unique, and satisfies (5.6)–(5.8). If t¯ < t ∗ , then it follows that either s(t¯) = 0 or s(t¯) = L holds. However, under Assumption 5.2 and qc (t) > 0 for all t ∈ [0, t ∗ ), it holds that s(t) > s∞ > 0,

∀t ∈ (0, t ∗ ),

(5.74)

and hence s(t¯) = 0. Moreover, applying qc (t) > 0 and (5.6) and (5.7) for all t ∈ (0, t¯) to the feedback design (5.67), one can see that s(t¯) = L. Hence, t¯ = t ∗ . Taking the time derivative of the control law (5.67), we get the following differential equation: q˙c (t) = − c(1 + εl )qc (t) − (εl − εf )ckl + (εs − εf )cks

∂Tl (s(t), t) ∂x

∂Ts (s(t), t), ∂x

≥ − c(1 + εl )qc (t),

∀t ∈ (0, t ∗ ),

(5.75) (5.76)

where the inequality from (5.75) to (5.76) follows from (5.68) and Hopf’s lemma with the help of (5.6) and (5.7) for all t ∈ (0, t ∗ ). Therefore, applying comparison principle to (5.76), one can show that qc (t) > qc (0)e−ct ,

∀t ∈ (0, t ∗ ),

(5.77)

which leads to the contradiction with the imposed assumption (5.73). Thus, there does not exist such t ∗ , from which we conclude

152

5 Two-Phase Stefan Problem

qc (t) ≥ 0,

∀t ≥ 0,

(5.78)

and the well-posedness and the conditions (5.6)–(5.8) hold for all t ≥ 0. Next, we prove the stability of the perturbed closed-loop in the similar manner as the proof of Theorem 5.1. Let c¯ = c(1 + εl ),

(5.79)

and redefine the gain kernel function as φ = β1l (cx ¯ − ε) associated with the backstepping transformation (5.33). Then, the target systems are described as wt (x, t) =αl wxx (x, t) + w(s(t), t) =

c¯ s˙ (t)X(t), βl

ε X(t), βl

(5.80) (5.81)

˙ X(t) = − cX(t) ¯ − βl wx (s(t), t),

(5.82)

and the boundary condition at x = 0 is given by wx (0, t) = −

ε u(0, t) + d(t), αl

(5.83)

where d(t) is the perturbation caused by the parametric uncertainties, given by d(t) =

¯ s εs − εf ck 1 + εl kl αs



L

s(t)

v(x, t)dx +

εf − εl c¯ X(t). 1 + εl kl

(5.84)

We consider the Lyapunov function defined by (5.48). The time derivative of V (t) = 1 ε 2 2 2 X(t) along the perturbed target system (5.80)–(5.83) satisfies the 2αl ||w|| + 2βl

following inequality:    32cL ¯ 2 2εL 3+ ||wx ||2 V˙ ≤ − 1 − αl αl    32cL ¯ 2 ε c¯ ε2 3+ X(t)2 − w(0, t)d(t), − − 2 αl αl βl 2   2  ε s˙ (t) c¯ s(t) + X(t) + 2 w(x, t)dxX(t) . 2αl βl βl 0 Applying Young’s and Agmon’s inequalities, the perturbation is bounded by −w(0, t)d(t) ≤

1 w(0, t)2 + 2Ld(t)2 , 8L

(5.85)

5.5 Robustness to Uncertainties of Physical Parameters

153

≤

1 1 w(s(t), t)2 + ||wx ||2 + 2Ld(t)2 , 4L 2

≤

1 ε2 X(t)2 + ||wx ||2 + 2Ld(t)2 . 4L 2

(5.86)

Moreover, applying Young’s and Cauchy Schwarz inequalities to the square of (5.84), we get 

¯ s εs − εf ck d(t) =2 1 + εl kl αs 2

2





εf − εl c¯ L v(x, t) dx + 2 1 + εl kl s(t) L

2

2 X(t)2 .

(5.87)

Applying (5.86) and (5.87) to (5.85), we can see that there exists sufficiently small ε > 0 such that the following inequality holds:  / /  / εf − εl /2 1 ε 4L c ¯ 2 / X(t)2 V˙ ≤ − ||w|| − c¯ − 2 // 16L2 4βl2 kl 1 + εl /    εs − εf ck ¯ s 2 L + 4L2 v(x, t)2 dx 1 + εl kl αs s(t)   2  ε s˙ (t) c s(t) + X(t) + 2 w(x, t)dxX(t) . 2αl βl βl 0

(5.88)

Therefore, if / / 2 / εf − εl /2 / / < εkl , /1+ε / 32βl2 Lc¯ l

(5.89)

then the differential inequality (5.88) becomes    1 ¯ s 2 L εc¯ 2 2 2 εs − εf ck V˙ ≤ − ||w|| − X(t) + 4L v(x, t)2 dx 1 + εl kl αs 16L2 8βl2 s(t)   2  ε s˙ (t) c s(t) + X(t) + 2 w(x, t)dxX(t) . (5.90) 2αl βl βl 0 Since v-system is equivalent to the one in previous sections, the time derivative of V2 = ||v||2 satisfies the inequality (5.62), which is αs V˙2 ≤ − 2 V2 . 2L

(5.91)

Combining (5.62) and (5.90) with applying comparison principle, one can derive that there exist positive constants M1 > 0 and d1 > 0 such that the following decay of the norm holds:

154

5 Two-Phase Stefan Problem

Table 5.1 Physical properties of zinc of both the liquid phase and the solid phase

Description Liquid density Solid density Liquid heat capacity Solid heat capacity Liquid thermal conductivity Solid thermal conductivity Melting temperature Latent heat of fusion

Symbol ρl ρs cl cs kl ks Tm ΔH ∗

Value 6570 kg · m−3 6890 kg · m−3 390 J · kg−1 · K−1 390 J · kg−1 · K−1 130 W · m−1 100 W · m−1 420 ◦ C 120,000 J · kg−1

V (t) + V2 (t) ≤ M1 (V (0) + V2 (0))e−d1 t .

(5.92)

Using the procedure in the last section, we conclude Theorem 5.2.

5.6 Numerical Simulation Simulation results are performed by considering a zinc whose physical parameters are given in Table 5.1. Under the identical choice of the physical parameters in the plant and control, we compare the performance of the proposed design in (5.20) (hereafter “two-phase design”) and the design  qc (t) = −c

kl αl



s(t)

 (Tl (x, t) − Tm )dx + γ (s(t) − sr ) ,

(5.93)

0

(hereafter “one-phase design”). The stability under the “one-phase design” is guaranteed by Theorem 5.2 for the robustness analysis of the closed-loop system under the restriction of the setpoint to satisfy qc (0) ≥ 0 for (5.93). The material’s length, the initial interface position, and the setpoint position are chosen as L = 1.0 m, s0 = 0.4 m, and sr = 0.5 m. The initial temperature profiles are set as Tl,0 (x) = T¯l,0 (1 − x/s0 ) + Tm and Ts,0 (x) = T¯s,0 (1 − (L − x)/(L − s0 )) + Tm with T¯l,0 = 10 ◦ C and T¯s,0 = −200 ◦ C. Then, the setpoint restrictions for both “twophase design” and “one-phase design” are satisfied. The control gain is set as c = 1.0 × 10−2 /s. The closed-loop responses are implemented as depicted in Fig. 5.3a–c for both “two-phase design” (solid) and “one-phase design” (dash). Figure 5.3a shows the dynamics of the interface s(t). We can observe that s(t) decreases at first due to the freezing caused by the initial temperature of the solid phase, and after some time the interface position increases and converges to the setpoint owing to the melting heat input. Moreover, the interface dynamics under the “two-phase design” achieves faster convergence than that under the “one-phase design” with having a little overshoot as seen from Fig. 5.3a. Figure 5.3b shows the dynamics of the closed-loop

5.6 Numerical Simulation

155

0.5 0.45 0.4 0.35 0

100

200

300

400

300

400

300

400

(a) 4

10

5

3 2 1 0 0

100

200

(b) 800 700 600 500 400 0

100

200

(c) Fig. 5.3 The closed-loop responses under the proposed “two-phase” design (pink solid) and the “one-phase” design (pink dash). (a) Convergence of the interface to the setpoint sr is observed for both controls; however, the proposed two-phase design achieves faster convergence as seen in the settling time in Fig. 5.4. (b) Positivity of the heat input is satisfied for both control designs. (c) The boundary temperature maintains above the melting temperature, and hence there is no appearance of a new solid phase from the controlled boundary x = 0 in the liquid phase

156

5 Two-Phase Stefan Problem 400 300 200 100 0 0

2

4

6

8

10

Fig. 5.4 Settling time of the interface convergence in Fig. 5.3a with respect to the error ε

control, and Fig. 5.3c shows the dynamics of the boundary temperature of the liquid phase Tl (0, t). Figure 5.3b illustrates the positivity of the heat input qc (t) > 0, and Fig. 5.3c illustrates the liquid boundary temperature being greater than the melting temperature, both of which are consistent with the derived properties. Hence, we observe that the simulation results are consistent with the theoretical result we prove as model validity conditions and the stability analysis. To compare the performance on the convergence speed between “two-phase design” and “one-phase design,” we investigate the settling time τε with respect to the error ε [%] of the interface position relative to the setpoint, mathematically defined by  / / ε , τε := inf τ //|s(t) − sr | ≤ |s0 − sr | τ ≥0 100

 ∀t ≥ τ .

(5.94)

Figure 5.4 shows the value of τε with ε = 10%, 5%, 2%, 1%. From the figure, it is observed that the convergence speed of “two-phase design” compared to the speed of the “one-phase design” is approximately four times faster for ε = 10%, two times faster for ε = 5%, and one and half times faster for both ε = 2% and 1%, respectively. Hence, Fig. 5.4 validates superior performance of the proposed “two-phase design” compared to the “one-phase design.”

5.7 Comments and Remarks In this chapter, we presented the full state feedback control law of a single heat boundary input for the two-phase Stefan problem to stabilize the moving interface position at a desired setpoint. The main contribution is that we theoretically prove the global exponential stability of the closed-loop system of the two-phase Stefan problem with designing the state feedback control law by employing energy shaping and backstepping. While our present result is only on the stabilization of the moving

5.7 Comments and Remarks

157

interface at the setpoint with restricting the equilibrium temperature to only the uniform melting temperature, the simultaneous stabilization of the interface position and the temperature profile at arbitrary setpoint and temperature profiles following recent results in [225] for traffic congestion control with moving shockwave by bilateral control [41] is an opportunity for future work. The application of extremum seeking control for online optimization of static maps with actuation through the Stefan system, following the recent results of [71, 154], is also a potential direction.

Chapter 6

Open Problems

This chapter introduces several open control-theoretic problems for the Stefan system, which may be of interest to researchers and students who are inspired and tackle a challenging problem in this domain. In each problem, we have provided our insights, idea, intermediate analysis, and some further challenges. The readers are welcome to pursue the topics.

6.1 Accelerated Convergence with Added Damping in Target System While the control design for the Stefan system has been achieved throughout the previous chapters, the convergence speed is limited since the decay rate derived in Lyapunov analysis cannot be chosen arbitrarily large [see Eq. (2.137)]. In this section, we consider the control design for the Stefan system to speed up the convergence. While the theoretical proof has not been derived for this problem, the control design is proposed and the numerical simulation is provided.

The Target (w, X)-System Through the First Transformation We consider the Stefan system given in Chap. 2. For the reference error states u(x, t) := T (x, t) − Tm and X(t) = s(t) − sr , the reference error system is given by ut (x, t) =αuxx (x, t),

(6.1)

ux (0, t) = − qc (t)/k,

(6.2)

© Springer Nature Switzerland AG 2020 S. Koga, M. Krstic, Materials Phase Change PDE Control & Estimation, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-58490-0_6

159

160

6 Open Problems

Fig. 6.1 Schematic of each transformation (u, X) → (w, X) → (z, X)

u(s(t), t) =0,

(6.3)

˙ X(t) = − βux (s(t), t). As shown in Chap. 2, the backstepping transformation  c c s(t) (x − y)u(y, t)dy − (x − s(t))X(t) w(x, t) = u(x, t) − α x β

(6.4)

(6.5)

converts the system (6.1)–(6.4) to the following (w, X)-system: wt (x, t) =αwxx (x, t) +

c s˙ (t)X(t), β

wx (0, t) =W (t),

(6.7)

w(s(t), t) =0,

(6.8)

˙ X(t) = − cX(t) − βwx (s(t), t), where

 W (t) := − k

−1

c qc (t) + α

(6.6)

 0

s(t)

 c u(x, t)dx + X(t) . β

(6.9)

(6.10)

The backstepping control in Chap. 2 is designed to make W (t) = 0. However, for the sake of accelerating the convergence speed by adding the damping term in the target PDE, we leave W (t), and introduce a second transformation as follows (Fig. 6.1).

The Target (z, X)-System Through Second Transformation Adding Damping Consider the second transformation from w(x, t) to a newly introduced variable z(x, t) given by  s(t) z(x, t) = w(x, t) − K(x, y, s(t))w(y, t)dy, (6.11) x

6.1 Accelerated Convergence with Added Damping in Target System

161

which converts (w, X)-system in (6.6)–(6.9) to zt (x, t) =αzxx (x, t) − λz(x, t) + s˙ (t)F (x, s(t), w(x, t), X(t)),

(6.12)

zx (0, t) =0,

(6.13)

z(s(t), t) =0,

(6.14)

˙ X(t) = − cX(t) − βzx (s(t), t),

(6.15)

where λ > 0 is a damping coefficient, and c F (x, s(t), w(x, t), X(t)) := β





1−



s(t)

K(x, y, s(t))dx X(t) x



s(t)

−

Ks(t) (x, y, s(t))w(y, t)dy.

(6.16)

x

A desired property of this (z, X)-system is that, owing to the damping term −λz in PDE (6.12), Lyapunov analysis enables us to obtain an arbitrary choice of the decay rate, at least in the case of the fixed domain. The additional term s˙ (t)F in (6.12) appears due to the moving boundary s(t) included in the transformations (6.5) and (6.11). Taking the spatial derivative of (6.11) twice, we get 

s(t)

zx (x, t) =wx (x, t) + K(x, x, s(t))w(x, t) −

Kx (x, y, s(t))w(y, t)dy, x

(6.17) zxx (x, t) =wxx (x, t) + K(x, x, s(t))wx (x, t)   d K(x, x, s(t)) w(x, t) + Kx (x, x, s(t)) + dx  s(t) − Kxx (x, y, s(t))w(y, t)dy.

(6.18)

x

Taking the time derivative of (6.11) along the solution of (6.6)–(6.9), we get  s(t)  s(t) zt (x, t) =wt (x, t)− K(x, y, s(t))wt (y, t)dy−˙s (t) Ks(t)(x, y, s(t))w(y, t)dy, x

x

 =α wxx (x, t) + K(x, x, s(t))wx (x, t) − Ky (x, x, s(t))w(x, t) −

 s(t) x



Kyy (x, y, s(t))w(y, t)dy − K(x, s(t), s(t))wx (s(t), t)

+ s˙ (t)F (x, s(t), w(x, t), X(t)).

(6.19)

162

6 Open Problems

Therefore, by using (6.11), (6.18), and (6.19), one can derive (6.20) zt − αzxx + λz − s˙ (t)F (x, s(t), w(x, t), X(t))   d = λ − 2α K(x, x, s(t)) w(x, t) dx   s(t)  λ +α Kyy (x, y, s(t)) − Kyy (x, y, s(t)) − K w(y, t)dy α x − αK(x, s(t), s(t))wx (s(t), t).

(6.21)

Thus, to satisfy PDE (6.12), the gain kernel function K(x, y, s(t)) must satisfy the following conditions: λ¯ K(x, x, s(t)) = (x − s(t)), 2 ¯ Kxx (x, y, s(t)) − Kyy (x, y, s(t)) =λK(x, y, s(t)), K(x, s(t), s(t)) =0,

(6.22) (6.23) (6.24)

where λ . α

λ¯ :=

(6.25)

Let us introduce the change of coordinate and variable defined by x¯ =s(t) − x,

y¯ = s(t) − y,

¯ x, K( ¯ y) ¯ =K(x, y, s(t)).

(6.26) (6.27)

Then, the conditions (6.22)–(6.24) are rewritten using these variables as λ ¯ x, ¯ K( ¯ x) ¯ = − x, 2 ¯ x, ¯ y) ¯ − K¯ y¯ y¯ (x, ¯ y) ¯ =λK( ¯ y), ¯ K¯ x¯ x¯ (x, ¯ x, K( ¯ 0) =0.

(6.28) (6.29) (6.30)

As performed in Chap. 3, the solution to the conditions (6.28)–(6.30) is given by ¯ x, K( ¯ y) ¯ = −λy¯

I1

  λ(x¯ 2 − y¯ 2 )  . λ(x¯ 2 − y¯ 2 )

(6.31)

6.1 Accelerated Convergence with Added Damping in Target System

163

Hence, the gain kernel function K(x, y, s(t)) is obtained as I1

K(x, y, s(t)) = − λ(s(t) − y)

 

λ((s(t) − x)2 − (s(t) − y)2 )

 .

(6.32)

Kx (0, y, s(t))w(y, t)dy.

(6.33)

λ((s(t) − x)2 − (s(t) − y)2 )

Control Law with Damping Substituting x = 0 in (6.17), and applying (6.7) and (6.13), we get  0 = W (t) + K(0, 0, s(t))w(0, t) −

s(t)

0

We rewrite the equation above using the original variable (u, X). Applying the transformation (6.5), we get w(0, t) =u(0, t) +  s(t) 0

c α

Kx (0, y, s(t))w(y, t)dy =



s(t)

yu(y, t)dy +

0

c s(t)X(t), β

(6.34)

 s(t) Kx (0, y, s(t))u(y, t)dy

0

   s(t) c s(t) Kx (0, y, s(t)) (y − z)u(z, t)dz dy − α 0 y  c s(t) Kx (0, y, s(t))(s(t) − y)dyX(t). (6.35) + β 0

By switching the order of the double integral in (6.35), we have 

s(t)  s(t)

0



s(t)

Kx (0, y, s(t))(y − z)u(z, t)dzdy =

ψ(y)u(y, t)dy,

(6.36)

0

y

where 

y

ψ(y) = 0

(z − y)Kx (0, z, s(t))dz.

(6.37)

164

6 Open Problems

Substituting (6.34)–(6.36) and (6.10) into (6.33), the control law with damping is given by 

s(t)

qc (t) =k 0

 λ Ψ (y, s(t))u(y, t)dy + φ(s(t))X(t) − s(t)u(0, t) , 2

(6.38)

where c Ψ (y, s(t)) = − α

φ(s(t)) = −

c β





λ 1 + s(t)y − 2

 1+

λ s(t)2 + 2

y

 (z − y)Kx (0, z, s(t))dz − Kx (0, y, s(t)),

0



Kx (x, y, s(t)) =λ(s(t) − y)(s(t) − x)

s(t)

(6.39)

 Kx (0, y, s(t))(s(t) − y)dy ,

(6.40)

0

I2



 λ((s(t) − x)2 − (s(t) − y)2 )

(s(t) − x)2 − (s(t) − y)2

.

(6.41)

To conclude stability of the closed-loop system with accelerated convergence, we apply Lyapunov analysis to the target (z, X)-system given by (6.12)–(6.15) first, which would enable us to obtain an arbitrary choice of the decay rate using free gain parameters (c, λ). However, due to the presence of the nonlinear term in target PDE (6.12), it is not easy to prove stability. Once we assume the monotonicity of the interface dynamics, i.e., s˙ (t) ≥ 0 to hold, which is required for model validity, the stability proof can be performed by Lyapunov analysis by using the condition, as done in Chap. 2. Recall that this condition was ensured by proving the positivity of the control input, which is not guaranteed to hold for the control law with damping (6.38). Hence, we have not arrived at a theoretical guarantee of stability and model validity under the damping control law. We investigate the performance in numerical simulation next.

Simulation Result The simulation result with the damping control law is provided in Fig. 6.2, in which the value of λ are tested by λ = 0 (green), λ = 10 (blue), and λ = 1000 (red). Figure 6.2a shows the evolution of the interface position s(t), which apparently achieves a successful convergence to the setpoint for all choice of the damping ratio. In particular, we see that the plot with the large damping λ = 1000 (red) shows the accelerated convergence. However, as depicted in Fig. 6.2b, the boundary temperature reaches below the melting temperature under the large damping, which physically implies that the material gets frozen from the controlled boundary, and hence violates the model validity. Such a freezing is caused by too much cooling by

6.1 Accelerated Convergence with Added Damping in Target System

165

0.3 0.2 0.1 0

0

20

40

60

80

100

60

80

100

60

80

100

(a)

1000

500

0 0

20

40 (b)

105

6 4 2 0 -2 -4

0

20

40 (c)

Fig. 6.2 Simulation results of the closed-loop system under the control input with damping (6.38). (a) The convergence of the interface position is accelerated by increasing damping. (b) The boundary temperature reaches below melting temperature under large damping. (c) The controller injects negative heat even under a modest damping

166

6 Open Problems

the negative control input as depicted in Fig. 6.2c. On the other hand, a modest value of the damping λ = 10 (blue) also reaches the negative heat input, but the boundary temperature keeps above the melting temperature, which satisfies the model validity. Hence, if the actuator can inject both positive and negative heat, the control with a modest damping becomes a better choice than the control without damping for the sake of accelerating the convergence with keeping the model validity.

Further Challenges As addressed above, one of the further challenges is proving the model validity condition T (x, t) ≥ 0 for all x ∈ (0, s(t)) and for all t ≥ 0 under the closedloop system, by which we can guarantee the stability of the closed-loop system through Lyapunov analysis for the target z-system in (6.12)–(6.15). An observation from the simulation result is that there would be a condition for the free parameters (c, λ) to ensure the model validity, which limits the convergence speed at the end. A potential way to relax such a limitation is to apply the damping control to the trajectory tracking problem introduced in the next section, though the problem is still challenging even for the damping-free control. While we have focused on accelerating the convergence speed of the exponential stability, recently a stronger notion of stability has emerged as “prescribed-time” stability, in which the convergence is achieved within a prescribed finite time regardless of the initial state. The prescribed-time stabilization for PDEs has been developed for reaction-diffusion equation in [192] and Schrödinger equation in [193], by introducing an unbounded time-varying damping coefficient in the target system and the associated time-varying backstepping transformation. The extension of the prescribed-time stabilization for the Stefan system also remains as an open problem which is more challenging than the exponential stabilization with accelerated convergence.

6.2 How to Track to a Desired Motion? In this section, we consider the trajectory tracking of the Stefan system, where the objective is to track the liquid–solid interface position to a desired “time-varying” reference trajectory. In general, for linear dynamical systems, the trajectory tracking is a trivial extension from the constant setpoint regulation of the state variables by suitably defining the reference states. However, for nonlinear systems, the problem is nontrivial. Especially, for the Stefan system, the problem is challenging due to the presence of the moving boundary state.

6.2 How to Track to a Desired Motion?

167

Problem Statement We consider the following Stefan model: Tt (x, t) =αTxx (x, t),

0 < x < s(t)

(6.42)

−kTx (0, t) =qc (t),

(6.43)

T (s(t), t) =Tm ,

(6.44)

s˙ (t) = − βTx (s(t), t).

(6.45) (r)

Our objective is to track the system into the reference states (Tt (x, t), sr (t)) which satisfy (r)

(r) Tt (x, t) =αTxx (x, t),

0 < x < sr (t)

(6.46)

−kTx(r) (0, t) =qc(r) (t),

(6.47)

T (r) (sr (t), t) =Tm ,

(6.48)

s˙r (t)

= − βTx(r) (sr (t), t).

(6.49)

Solving the reference state and input for a given reference trajectory of the moving interface sr (t) is known as an “inverse Stefan problem,” or “motion planning,” as studied in literature (see [160] for instance). Since the reference states are variables we pursue to track, it is feasible to assume the model validity for the reference states regardless of the control design, which is given by T (r) (x, t) ≥Tm , Tx(r) (sr (t), t) ≤0, qc(r) (t) ≥0,

∀x ∈ [0, sr (t)], s˙r (t) ≥ 0, ∀t ≥ 0.

∀t ≥ 0,

∀t ≥ 0,

(6.50) (6.51) (6.52)

In the previous chapters, we consider the constant setpoint interface position sr (t) ≡ sr , which yields the constant uniform reference temperature T (r) (x, t) = Tm for all (r) x ∈ [0, sr ] and for all t ≥ 0, and zero reference input qc (t) = 0. A desired property of such a constant reference states lies in a fact that the reference temperature profile can be projected onto the domain (0, s(t)) of the model. However, the timedependency of the reference trajectory of the moving interface sr (t) does not enable us to perform such an identical projection, as we see the difference of the domains (0, s(t)) and (0, sr (t)).

168

6 Open Problems

Energy Conservation The Stefan model (6.42)–(6.45) and the reference model (6.46)–(6.49) obey the following energy conservation law:    d k s(t) k (T (x, t) − Tm )dx + s(t) =qc (t), dt α 0 β    d k sr (t) (r) k (T (x, t) − Tm )dx + sr (t) =qc(r) (t). dt α 0 β

(6.53)

(6.54)

Thus, by defining the energy of the reference error as k ˜ E(t) = α +



s(t)



sr (t)

(T (x, t) − Tm )dx −

0

 (T (r) (x, t) − Tm )dx

0

k (s(t) − sr (t)), β

(6.55)

it follows that ˙˜ E(t) = qc (t) − qc(r) (t).

(6.56)

Then, the energy-shaping control law is designed by ˜ qc (t) = qc(r) (t) − cE(t),

(6.57)

which yields the explicit solution of the reference error energy as −ct ˜ ˜ E(t) = E(0)e .

(6.58)

Hence, similar to the previous chapters, the energy-shaping control design ensures the convergence of the reference error energy to zero, but it does not guarantee con(r) vergence of the model states (T (x, t), s(t)) to the reference states (Tt (x, t), sr (t)).

Reference Error System Let u(x, t) and X(t) be the reference error variables defined by u(x, t) = T (x, t) − T (r) (x, t),

X(t) = s(t) − sr (t).

(6.59)

6.2 How to Track to a Desired Motion?

169

Applying the Stefan system (6.42)–(6.45) and the reference system (6.46)–(6.49), one can derive the following reference error system: ut (x, t) =αuxx (x, t),

0 < x < s(t)

(6.60)

−kux (0, t) =qc (t) − qc(r) (t),

(6.61)

u(s(t), t) =f1 (X(t), t),

(6.62)

˙ X(t) = − βux (s(t), t) + f2 (X(t), t),

(6.63)

where f1 (X(t), t) = − T (r) (X(t) + sr (t), t),

(6.64)

f2 (X(t), t) =β(Tx(r) (X(t) + sr (t), t) − Tx(r) (sr (t), t)).

(6.65)

Thus, the reference error system (6.60)–(6.63) contains strong nonlinearity with respect to X(t). As the PDE backstepping method has not been applied to a general nonlinear ODE system coupled with a parabolic PDE, it is challenging to treat the system (6.60)–(6.63).

Linearized Reference Error System Supposing s(t) is close to sr (t), linearization around s(t) = sr (t) (i.e., X(t) = 0) is applied. Owing to the boundary condition (6.48), the nonlinear functions (6.64) and (6.65) satisfy f1 (0, t) = 0,

f2 (0, t) = 0.

(6.66)

Hence, linearization of f1 and f2 around X(t) = 0 leads to f1 (X(t), t) ≈ C(t)X(t),

(6.67)

f2 (X(t), t) ≈ A(t)X(t),

(6.68)

where ∂f1 (0, t) = −Tx(r) (sr (t), t), ∂X ∂f2 (r) A(t) = (sr (t), t). (0, t) = Txx ∂X

C(t) =

By the condition (6.51) for the reference state, it follows

(6.69) (6.70)

170

6 Open Problems

C(t) > 0,

∀t ≥ 0.

(6.71)

Taking the time derivative of the condition (6.48) yields (r)

Tt (sr (t), t) + s˙r (t)Tx(r) (sr (t), t) = 0,

(6.72)

which, with the help of (6.49), leads to  2 Tt(r) (sr (t), t) = β Tx(r) (sr (t), t) .

(6.73)

Therefore, by (6.70), it turns out A(t) =

2 β  (r) Tx (sr (t), t) > 0, α

∀t ≥ 0.

(6.74)

0 < x < s(t)

(6.75)

Then, the linearized system is described as ut (x, t) =αuxx (x, t),

−kux (0, t) =qc (t) − qc(r) (t), u(s(t), t) =C(t)X(t), ˙ X(t) =A(t)X(t) − βux (s(t), t),

(6.76) (6.77) (6.78)

which is an unstable system due to the positivity of A(t) shown in (6.74). Since the coefficients A(t) and C(t) are time-dependent, the backstepping transformation should also incorporate the time-dependent gain kernel functions, which is described by 

s(t)

w(x, t) = u(x, t) −

k(x, y, t)u(y, t)dy − φ(x − s(t), t)X(t).

(6.79)

x

Further Challenges The time-dependency of the gain kernel functions makes the problem much more complicated and challenging, which we have not derived yet so far. Note that, even if we can derive the time-dependent backstepping transformation and its gain kernel functions, the stability result holds locally, because we have applied linearization. Moreover, guaranteeing the positivity of the control input is not ensured yet. There might be another better approach which figures out all the remaining problems. For those reasons, the trajectory tracking of the Stefan system remains as an open problem.

6.3 Two-Dimensional Disk Geometry with Nonuniformity

171

6.3 Two-Dimensional Disk Geometry with Nonuniformity In this section, we consider the Stefan system on a two-dimensional (2-D) disk, where both the temperature and the phase interface geometry are not uniform along the angular coordinate, unlike the problem in Sect. 4.3. The backstepping method for a parabolic PDE on a 2-D disk was developed in [204] first and the result was extended to n-dimensional ball in [205]. Though we have not derived a backstepping transformation and control law so far, we introduce the model description and an energy-shaping control, and address some further challenges (Fig. 6.3).

Model Description We consider a liquid–solid material on a 2-D disk with a radius R. The outer domain of the disk is assumed to be occupied by the liquid phase and the inner domain is in the solid phase. Then, referring to [172], the one-phase Stefan problem on a disk geometry is governed by α α ∂t T (r, θ, t) = ∂r (r∂r T (r, θ, t)) + 2 ∂θθ T (r, θ, t), r r S(t, θ ) < r < R,

0 < θ < 2π

−k∂r T (R, θ, t) =qc (t, θ ), T (S(θ, t), θ, t) =Tm ,



∂t S(θ, t) = − β 1 +

Fig. 6.3 Schematic of the one-phase Stefan problem on a 2-D disk. The phase interface geometry S(θ, t) is not uniform along the angular axis θ

(6.80) (6.81)



∂θ S(θ, t) S(θ, t)

(6.82)

2  ∂r T (S(θ, t), θ, t),

(6.83)

172

6 Open Problems

where T (r, θ, t) is the temperature profile in the liquid phase which is distributed over the radial coordinate r and the angular coordinate θ ∈ (0, 2π ), and S(θ, t) denotes the phase interface geometry. Since we consider a nonuniform shape of the liquid and solid phases, the interface geometry S(t, θ ) is varying in the angular coordinate θ .

Energy Conservation A unique characteristic of the system (6.80)–(6.83) lies in the dynamics of (6.83), 2  S(θ,t) which includes a highly nonlinear term ∂θS(θ,t) ∂r T (S(θ, t), θ, t). One might wonder that, with such a complicated term, whether an energy conservation of the Stefan system holds or not. Let us explore the energy conservation. Applying the surface integral over a cylindrical coordinate, the internal energy of the Stefan system is given by E(t) =

k α +

 0

k β

2π



R

r (T (r, θ, t) − Tm ) drdθ,

S(θ,t)

   1 2π π R2 − S(θ, t)2 dθ , 2 0

(6.84)

where the first term denotes the specific heat of the material and the second term denotes the latent heat of the material involved with area of the liquid phase. Taking the time derivative of (6.84) along the solution of (6.80)–(6.83), we get dE k = dt α

2π  R



−

0

k β



 r∂t T (r, θ, t)dr−∂t S(θ, t)S(θ, t)(T (S(θ, t), θ, t) − Tm ) dθ,

S(θ,t) 2π

S(θ, t)∂t S(θ, t)dθ 0

  1 ∂r (r∂r T (r, θ, t)) + ∂θθ T (r, θ, t) drdθ, =k r 0 S(θ,t)      2π ∂θ S(θ, t) 2 +k S(θ, t) 1 + ∂r T (S(θ, t), θ, t)dθ S(θ, t) 0   2π   R 1 R∂r T (R, θ, t)) + ∂θθ T (r, θ, t)dr dθ, =k 0 S(θ,t) r    2π ∂θ S(θ, t) 2 +k S(θ, t) ∂r T (S(θ, t), θ, t)dθ, S(θ, t) 0 

2π



R

6.3 Two-Dimensional Disk Geometry with Nonuniformity



173

2π

=R

qc (θ, t)dθ 0



2π

+k



0

R

S(θ,t)

 1 (∂θ S(θ, t))2 ∂θθ T (r, θ, t)dr + ∂r T (S(θ, t), θ, t) dθ. r S(θ, t) (6.85)

Let us define  F (θ, t) =

R

S(θ,t)

1 ∂θ T (r, θ, t)dr. r

(6.86)

Taking the derivative of (6.86) with respect to θ , we get  ∂θ F (θ, t) =

R

S(θ,t)

1 ∂θ S(θ, t) ∂θθ T (r, θ, t)dr − ∂θ T (S(θ, t), θ, t). r S(θ, t)

(6.87)

Moreover, taking the derivative of (6.82) with respect to θ on both sides, we get ∂θ S(θ, t)∂r T (S(θ, t), θ, t) + ∂θ T (S(θ, t), θ, t) = 0,

(6.88)

which yields ∂θ T (S(θ, t), θ, t) = −∂θ S(θ, t)∂r T (S(θ, t), θ, t).

(6.89)

Thus, substituting (6.89) into (6.87), one can obtain ∂θ F (θ, t) =

(∂θ S(θ, t))2 ∂r T (S(θ, t), θ, t) + S(θ, t)



R

S(θ,t)

1 ∂θθ T (r, θ, t)dr. r

(6.90)

Plugging (6.90) into (6.85), we get dE =R dt



2π



0



=R

2π

qc (θ, t)dθ + k

∂θ F (θ, t)dθ, 0

2π

qc (θ, t)dθ + k (F (2π, t) − F (0, t))

0

 =R

2π

qc (θ, t)dθ,

(6.91)

0

in which we used F (2π, t) = F (0, t) due to periodic property T (r, 2π, t) = T (r, 0, t) and S(2π, t) = S(0, t). As (6.91) physically means that the growth of the internal energy is equal to the external work performed by the heat flux

174

6 Open Problems

input qc (θ, t), the relation serves as the first law of thermodynamics, namely, the conservation of energy holds. Since the energy conservation law is derived, the energy-shaping control can be designed. In the case of multidimensional Stefan problem, the control objective is achieving a desired “shape” of the liquid (or solid) phase. Let S ∗ (θ ) denote the desired setpoint shape in polar coordinate, which is a function in the angle θ ∈ [0, 2π ]. Since an equilibrium temperature state is a uniform melting temperature Tm , the reference error energy is defined by k ˜ E(t) = α



2π 0



R

k r (T (r, θ, t) − Tm ) drdθ − 2β S(θ,t)



2π

(S(θ, t)2 − S ∗ (θ )2 )dθ.

0

(6.92) Then, the energy-shaping control design for the 2-D disk Stefan problem is given by 

k qc (θ, t) = −c α



 k  2 ∗ 2 S(θ, t) − S (θ ) , r (T (r, θ, t) − Tm ) dr − 2β S(θ,t) (6.93) R

which leads to d E˜ ˜ = −cE(t). dt

(6.94)

Hence, under the closed-loop system with the energy-shaping control (6.93), it follows that −ct ˜ ˜ E(t) = E(0)e .

(6.95)

Note that the control law (6.93) does not satisfy the same differential equation as (6.94), unlike 1-D problem considered in previous chapters, due to θ -dependency of the control law. Taking the partial derivative of the control law (6.93) with respect to t, we get ∂t qc (θ, t) = −cqc (θ, t) − ∂θ F (θ, t),

(6.96)

which does not enable us to ensure the positivity of the control input qc (t) ≥ 0 due to the term −∂θ F (θ, t) in (6.96).

6.4 Comments and Remarks

175

Further Challenges One of the further challenges is to ensure the positivity of the control input and the model validity of the temperature T (r, θ, t) ≥ Tm , under a designed control law, such as energy-shaping control or a backstepping control. Indeed, deriving the backstepping transformation for the Stefan system on a 2-D disk is also challenging mainly due to the nonlinear term arising in the phase interface dynamics (6.83). Moreover, proving stability of the closed-loop system is also challenging, since the Lyapunov analysis should be involved with the spatial norm of both the radial and angular coordinates.

6.4 Comments and Remarks There are several other open control-theoretic problems, some of which have been introduced in other chapters. We summarize them in the following list: • • • • • •

adaptive control of the Stefan system (see [24, 106, 185]), bilateral control of the two-phase Stefan system (see [28, 41, 225]), sampled-data observer for the Stefan system (see [3, 105]), event-triggered control for the Stefan system (see [67]), disturbance rejection for the Stefan system (see [12, 215]), extremum seeking of an unknown map cascaded with the Stefan system (see [71, 154]).

While problems in this list have all been considered for PDE and PDEODE systems on fixed domains, virtually all of them are unexplored for PDEs with moving boundaries. Solving such problems requires significant theoretical innovation in many methodologies—adaptive control, observer design, sampleddata systems, event-triggered control, and extremum seeking. Advancing so many subject is beyond the capacity of our small team and presents significant research opportunities for a larger community of researchers contributing to control and estimation of Stefan systems.

Part II

Applications and Experiment

Chapter 7

Sea Ice

7.1 Importance of the Arctic Sea Ice for Global Climate Modeling The Arctic sea ice has been studied intensively in the field of climate and geoscience [219]. One of the main reasons is due to ice-albedo feedback which influences climate dynamics through the high reflectivity of sea ice. The other reason is the rapid decline of the Arctic sea ice extent in the recent decade shown in several observations. These observations motivate the investigation of future sea ice amount. Several studies have developed a computational model of the Arctic sea ice and performed numerical simulations of the model with initial sea ice temperature profile [221]. However, the spatially distributed temperature in sea ice is difficult to recover in real-time using a limited number of thermal sensors. Hence, the online estimation of the sea ice temperature profile based on some available measurements is crucial for the prediction of the sea ice thickness. A thermodynamic model for the Arctic sea ice was first developed in [143] (hereafter MU71), in which the authors investigated the correspondence between the annual cycle pattern acquired from the simulation and empirical data of [202]. The model involves a temperature diffusion equation evolving on a spatial domain defined as the sea ice thickness. Due to melting or freezing phenomena, the aforementioned spatial domain is time-varying. Such a model is of the Stefan type [82] and involves a PDE with a state-dependent moving boundary driven by a Neumann boundary value. Refined models of MU71 have been suggested in literature. For instance, [176] proposed a numerical model to achieve faster and accurate computation of MU71 by discretizing the temperature profile into some layers and neglecting the salinity effect. An energy-conserving model of MU71 was introduced in [20] by taking into account an internal brine pocket melting on surface ablation and the vertically varying salinity profile. Their thermodynamic model was demonstrated © Springer Nature Switzerland AG 2020 S. Koga, M. Krstic, Materials Phase Change PDE Control & Estimation, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-58490-0_7

179

180

7 Sea Ice

by [19] using a global climate model with a Lagrangian ice thickness distribution. Combining these two models, [220] developed an energy-conserving three-layer model of sea ice by treating the upper half of the ice as a variable heat capacity layer. Remote sensing techniques have been employed to obtain the Arctic sea ice data in several studies. In [84], the authors suggested an algorithm to calculate sea ice surface temperature using the satellite measured brightness temperatures, which provided an excellent measurement of the actual surface temperature of the sea ice during the Arctic cold period. The Arctic sea ice thickness data were acquired in [135] through a satellite called “ICESat” during 2003–2008 and compared with the data in [170] observed by a submarine during 1958–2000. More recent data describing the evolution of the sea ice thickness have been collected between 2010 and 2014 from the satellite called “CryoSat-2” [134]. On the other hand, state estimation has been studied as a specific type of data assimilation which utilizes the numerical model along with the measured value. For finite-dimensional systems associated with noisy measurements, a well-known approach is the Kalman Filter. Another well-known method is the Luenberger type state observer, which reconstructs the state variable from partially measured variables. For the application to sea ice, [72] developed an adjoint-based method as an iterative state and parameter estimation for the coupled sea ice-ocean in the Labrador Sea and Baffin Bay to minimize an uncertainty-weighted modeldata misfit in a least-square sense as suggested in [222], using Massachusetts Institute of Technology general circulation model (MITgcm) developed in [142]. In [73], the same methodology was applied to reconstruct the global ocean and ice concentration. Their sea ice model is based on the zero-layer approximation of the numerical model in [176], which is a crude model lacking internal heat storage and promoting fast melting.

7.2 Thermodynamic Model of Arctic Sea Ice The thermodynamic model of MU71 describes the time evolution of the sea ice temperature profile in the vertical axis along with its thickness, which also evolves in time due to accumulation or ablation caused by energy balance. Figure 7.1 provides a schematic of the Arctic sea ice model. During the seasons other than summer (July and August), the sea ice is covered by snow, and the surface position of the snow also evolves in time. Let Ts (x, t), Ti (x, t) denote the temperature profile of snow and sea ice, and h(t) and H (t) denote the thickness of snow and sea ice. The total incoming heat flux from the atmosphere is denoted by Fa , and the heat flux from the ocean is denoted by Fw . The Arctic sea ice model suggested by MU71 gives governing equations of a Stefan-type free boundary problem formulated as

7.2 Thermodynamic Model of Arctic Sea Ice Fig. 7.1 Schematic of the vertical one-dimensional model of the Arctic sea ice

181

−h(t)

snow

0

sea-ice H(t)

x

Fa − I0 − σ (Ts (−h(t), t) + 273)4 + ks  = ρs c0

ocean

∂Ts (−h(t), t) ∂x

0, if Ts (−h(t), t) < Tm1 , ˙ −q h(t), if Ts (−h(t), t) = Tm1 ,

∂Ts ∂ 2 Ts (x, t) =ks 2 (x, t), ∀x ∈ (−h(t), 0), ∂t ∂x Ts (0, t) =Ti (0, t),

ks ρci (Ti , S)

(7.1) (7.2) (7.3)

∂Ts ∂Ti (0, t) =k0 (0, t), ∂x ∂x

(7.4)

∂Ti ∂ 2 Ti (x, t) =ki (Ti , S) 2 (x, t) + I0 κi e−κi x , ∀x ∈ (0, H (t)), ∂t ∂x

(7.5)

Ti (H (t), t) =Tm2 , ∂Ti (H (t), t) − Fw , q H˙ (t) =ki ∂x

(7.6) (7.7)

where I0 , σ , ks , ρs , c0 , k0 , ρ, Tm1 , Tm2 , and q are solar radiation penetrating the ice, Stefan-Boltzmann constant, thermal conductivity of snow, density of snow, heat capacity of pure ice, thermal conductivity of pure ice, density of pure ice, melting point of surface snow, melting point of bottom sea ice, and latent heat of fusion, respectively. The total heat flux from the air is given by Fa = (1 − α)Fr + FL + Fs + Fl ,

(7.8)

where Fr , FL , Fs , Fl , and α denote the incoming solar short-wave radiation, the long-wave radiation from the atmosphere and clouds, the flux of sensible heat, the latent heat in the adjacent air, and the surface albedo, respectively. The heat capacity and thermal conductivity of the sea ice are affected by the salinity as

182

7 Sea Ice

ci (Ti , S(x)) = c0 +

γ1 S(x) γ2 S(x) , , ki (Ti , S(x)) = k0 + Ti (x, t) Ti (x, t)2

(7.9)

where S(x) denotes the salinity in the sea ice. γ1 and γ2 represent the weight parameters. The thermodynamic model (7.1)–(7.7) allows us to predict the future thickness (h(t), H (t)) and the temperature profile (Ts , Ti ) given the accurate initial data. However, from a practical point of view, it is not feasible to obtain a complete temperature profile due to a limited number of thermal sensors. To deal with the problem, the estimation algorithm is designed so that the state estimation converges to the actual state starting from an initial estimate.

7.3 Annual Cycle Simulation of Sea Ice Thickness For the computation, we use boundary immobilization method and finite difference semi-discretization [132] with 100-point mesh in space, and the resulting approximated ODEs are calculated by using MATLAB ode15 solver.

Input Parameters The input parameters are taken from [143] in SI units and Table 7.1 shows the monthly averaged values of heat fluxes coming from the atmosphere for each month. Table 7.2 shows the physical parameters of snow and sea ice. Following [20], the salinity profile is described by Table 7.1 Average monthly values for the energy fluxes

Symbol Unit Jan. Feb. Mar. Apr. May. Jun. Jul. Aug. Sep. Oct. Nov. Dec.

Fr W/m2 0 0 30.7 160 286 310 220 145 59.7 6.46 0 0

FL W/m2 168 166 166 187 244 291 308 302 266 224 181 176

Fs W/m2 19.0 12.3 11.6 4.68 −7.26 −6.30 −4.84 −6.46 −2.74 1.61 9.04 12.8

Fl W/m2 0 −0.323 −0.484 −1.45 −7.43 −11.3 −10.3 −10.7 −6.30 −3.07 −0.161 −0.161

α ··· ··· 0.83 0.81 0.82 0.78 0.64 0.69 0.84 0.85 ··· ···

7.4 Temperature Profile Estimation

183

Table 7.2 Physical parameters of snow and sea ice Symbol ρs ks ρ c0 k0 γ1 γ2 I0 κi Tm1 Tm2

Meaning Density (snow) Conductivity (snow) Density (ice) Heat capacity (ice) Conductivity (ice) Weight of heat capacity Weight of conductivity Solar radiation Penetration rate Melting temperature of sea ice at surface Melting temperature of sea ice at bottom

Unit kg/m3 W/m/◦ C kg/m3 J/kg/◦ C W/m/◦ C kJ ◦ C/kg W/m W/m2 /m ◦C ◦C

   n x 

m+ x H (t) S(x) = A 1 − cos π , H (t)

Value 330 0.31 917 2110 2.034 18.0 0.117 1.59 1.5 −0.1 −1.8



(7.10)

where A = 1.6, n = 0.407, and m = 0.573.

Simulation Test of MU71 Using the given data, first the simulation of (7.1)–(7.7) is performed and showed in Fig. 7.2 to recover the evolution of h(t) and H (t) in the annual season as in [143]. The dynamic behavior of the snow surface and the bottom of sea ice are shown in Fig. 7.2a, and the time evolution of the temperature profile in sea ice is illustrated in Fig. 7.2b. We can see that both of Fig. 7.2a, b have a good agreement with the simulation results shown in [143].

7.4 Temperature Profile Estimation In this section, we derive the estimation algorithm utilizing some available measurements and show the exponential convergence of the designed estimation to a simplified sea ice model. The ice thickness and surface temperature are measured in several studies [84, 135, 170]. It is indeed typical to check observability before observer design, at least for systems on a constant domain (see [151] for instance). Here, we start with the observer design that is accompanied by a proof of exponential stability, which ensures the states’ detectability.

184

7 Sea Ice

−1 Air Snow

Vertical Axis [m]

0

1

Sea-Ice

2

3

Surface position of snow − h(t) Surface position of sea-ice s(t) Bottom position of sea-ice H (t)

Ocean

4 Jan.1 Feb.1Mar.1Apr.1May.1Jun.1 Jul.1 Aug.1Sep.1 Oct.1 Nov.1Dec.1

Month (a)

−18

Depth of Sea-Ice [m]

−16

0.5 1 1.5 2 2.5 3

−14

0 −2−4 −6 −12 −8 −10 −14 −16 −18

−12

−10 Sea-Ice −8 −6 −4 −2 Ocean

0

Sea-Ice Temperature T i (x, t) 3.5 Jan.1 Feb.1Mar.1Apr.1May.1Jun.1 Jul.1 Aug.1Sep.1 Oct.1Nov.1Dec.1

Month (b) Fig. 7.2 Simulation tests of the plant (7.1)–(7.7) on annual cycle. Both (a) and (b) are in good agreement with the simulation results in [143]. (a) Thickness evolution of the snow and sea ice. (b) Time evolution of temperature profile in sea ice

7.4 Temperature Profile Estimation

185

Simplification of the Model For the sake of the design and stability proof, we give a simplification on the system (7.1)–(7.7). The effect of the salinity profile on the physical parameters is assumed to be sufficiently small so that it can be negligible, i.e., S(x) = 0. Therefore, the heat equation of the sea ice temperature (7.5) is rewritten as ∂Ti ∂ 2 Ti (x, t) =Di 2 (x, t) + I¯0 κi e−κi x , ∀x ∈ (0, H (t)), ∂t ∂x

(7.11)

where the diffusion coefficient is defined as Di = k0 /ρc0 . Next, we impose the following assumptions. Assumption 7.1 The thickness H (t) is positive and upper bounded, i.e., there exists H¯ > 0 such that 0 < H (t) < H¯ , for all t ≥ 0. Assumption 7.2 H˙ (t) is bounded, i.e., there exists M > 0 such that |H˙ (t)| < M, for all t ≥ 0. According to [135], the observation data of the sea ice’s thickness from the 1950s to 2008 show that the maximum value including the uncertainty is less than 5 m. Moreover, the largest variation of the thickness in a snow-covered season of a year essentially happens from December to March as an accumulation, and most of the literature shows at most 20 cm accumulation per month. Hence, conservatively it is plausible to set H¯ = 10 m, and M = 50 cm/Month = 1.9 × 10−7 m/s. Mathematically, the existence of the classical solution of the simple Stefan problem given by (7.11) and (7.6)–(7.7) has been established in literature. We refer the readers to follow [82] for the detailed explanation. The solution of the original sea ice model (7.1)–(7.7) has not been studied due to its high complexity.

Observer Structure Suppose that the sea ice thickness and the ice surface temperature are obtained as measurements Y1 (t) and Y2 (t), i.e. Y1 (t) =H (t),

Y2 (t) = Ti (0, t).

(7.12)

The state estimate Tˆi of the sea ice temperature is governed by a copy of the plant (7.11) and (7.6)–(7.7) plus the error injection of H (t), namely, as follows:   ∂ Tˆi ∂ 2 Tˆi (x, t) =Di 2 (x, t) + I¯0 κi e−κi x − p1 (x, t) Y1 (t) − Hˆ (t) , ∀x ∈ (0, H (t)) ∂t ∂x (7.13)

186

7 Sea Ice

  Tˆi (0, t) =Y2 (t) − p2 (t) Y1 (t) − Hˆ (t) ,   Tˆi (H (t), t) =Tm2 − p3 (t) Y1 (t) − Hˆ (t) ,

(7.14) (7.15)

  ∂ Tˆi Fw (Y1 (t), t) − , H˙ˆ (t) = p4 (t) Y1 (t) − Hˆ (t) + β ∂x q

where β :=

ki q.

(7.16)

Next, we define the estimation error states as

T˜ (x, t) := −(Ti (x, t) − Tˆi (x, t)),

H˜ (t) := H (t) − Hˆ (t),

(7.17)

where the negative sign is added to be consistent with the description developed in Chap. 3 for the liquid phase. Subtraction of the observer system (7.13)–(7.16) from the system (7.11) and (7.6)–(7.7) yields the estimation error system as ∂ 2 T˜ ∂ T˜ (x, t) = Di 2 (x, t) − p1 (x, t)H˜ (t), ∀x ∈ (0, H (t)) ∂t ∂x T˜ (0, t) = − p2 (t)H˜ (t), T˜ (H (t), t) = − p3 (t)H˜ (t),

(7.18) (7.19) (7.20)

∂ T˜ H˙˜ (t) = − p4 (t)H˜ (t) − β (H (t), t). ∂x

(7.21)

Our goal is to design the observer gains p1 (x, t), p2 (t), p3 (t), p4 (t) so that the temperature error T˜ converges to zero. The main theorem is stated as follows. Theorem 7.1 Let Assumptions 7.1 and 7.2 hold. Consider the estimation error system (7.18)–(7.21) with the design of the observer gains cλx I1 (z) p1 (x, t) = + β z



εH (t) 3 − Di β

 λ2 x

I2 (z) λ3 x 3 I3 (z) + , Di β z 3 z2

p2 (t) =0,

(7.22) (7.23)

λ H (t) − ε, 2β   βλ λH (t)2 λ p4 (t) =c − + 1− εH (t), 2 8Di 2Di

p3 (t) = −

(7.24) (7.25)

where λ > 0, c > 0, and ε > 0 are positive free parameters, z is defined by  z := λ¯ (H (t)2 − x 2 ),

(7.26)

7.4 Temperature Profile Estimation

187

where λ¯ := there exist

λ Di , and Ij (·) denotes the modified Bessel function of the j th kind. Then, positive constants c∗ > 0 and M˜ > 0 such that, for all c > c∗ , the norm

 Φ(t) :=

H (t)

T˜ (x, t)2 dx + H˜ (t)2

(7.27)

0

satisfies the following exponential decay: − min{λ,c}t ˜ Φ(t) ≤ MΦ(0)e ,

(7.28)

namely, the origin of the estimation error system is exponentially stable in the spatial L2 norm. Remark 7.1 The observer gains (7.22)–(7.25) include the thickness H (t), so the gains are not precomputed offline, but are easily calculated online, along with the state estimation. Owing to the slow dynamics of the sea ice model, the computation time is much less than the time step size, which enables the real-time computation of the proposed observer. Remark 7.2 The measurements (7.12) are assumed to be noiseless; however, in practice, the measured data accompany with some noise. Preferably the observer needs pre-filtering to deal with the noisy measurements. To handle the discrete-time measurements in practice as in [164], the designed observer should be discretized in time such as Euler or Runge-Kutta methods so that the estimation can be computed at every sampling of the discrete-time measurements. The free parameters λ, c, and ε have their physical units [1/s], [1/s], and [◦ C/m], respectively. Hence we can see the consistency of the physical units in the estimation error system (7.18)–(7.21) together with (7.22)–(7.25).

Gain Derivation via State Transformation For the estimation error system (7.18)–(7.21), we apply the following invertible transformations: T˜ (x, t) =w(x, t) − w(x, t) =T˜ (x, t) −

 

H (t)

q(x, y)w(y, t)dy − ψ(x, H (t))H˜ (t),

(7.29)

r(x, y)T˜ (y, t)dy − φ(x, H (t))H˜ (t),

(7.30)

x H (t) x

which map the estimation error system (7.18)–(7.21) into the following target system:

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wt (x, t) =Di wxx (x, t) − λw(x, t) − H˙ (t)f (x, H (t))H˜ (t), ∀x ∈ (0, H (t)) (7.31) w(0, t) =0,

(7.32)

w(H (t), t) =εH˜ (t),

(7.33)

H˙˜ (t) = − cH˜ (t) − βwx (H (t), t),

(7.34)

where f (x, H (t)) is to be determined. Taking the first and second spatial derivatives of the transformation (7.29), we get T˜x (x, t) =wx (x, t) + q(x, x)w(x, t)  H (t) − qx (x, y)w(y, t)dy − ψx (x, H (t))H˜ (t),

(7.35)

x

  d ˜ Txx (x, t) =wxx (x, t) + q(x, x)wx (x, t) + qx (x, x) + q(x, x) w(x, t) dx  H (t) qxx (x, y)w(y, t)dy − ψxx (x, H (t))H˜ (t). (7.36) − x

Next, taking the time derivative of (7.29) along the solution of the target system (7.31)–(7.34), using integration by parts, and substituting the boundary condition (7.33), we get T˜t (x, t) =Di wxx (x, t) + Di q(x, x)wx (x, t) − (λ + Di qy (x, x))w(x, t) + (βψ(x, H (t)) − Di q(x, H (t)))wx (H (t), t) + (Di εqy (x, H (t)) + cψ(x, H (t)))H˜ (t)  H (t) + (λq(x, y) − Di qyy (x, y))w(y, t)dy x

− H˙ (t)H˜ (t) (εq(x, H (t)) + ψH (x, H (t))   H (t) +f (x, H (t)) − q(x, y)f (y, H (t))dy . x

Thus, by (7.36) and (7.37), we have T˜t (x, t) − Di T˜xx (x, t) + p1 (x, t)H˜ (t)   d = − λ + 2Di q(x, x) w(x, t) dx + (βψ(x, H (t)) − Di q(x, H (t)))wx (H (t), t)

(7.37)

7.4 Temperature Profile Estimation

189

  + Di εqy (x, H (t)) + Di ψxx (x, H (t)) + cψ(x, H (t)) + p1 (x, t) H˜ (t)  H (t) + (λq(x, y) + Di qxx (x, y) − Di qyy (x, y))w(y, t)dy x

− H˙ (t)H˜ (t) (εq(x, H (t)) + ψH (x, H (t))   H (t) +f (x, H (t)) − q(x, y)f (y, H (t))dy .

(7.38)

x

Substituting x = 0 and x = H (t) into (7.29), we get T˜ (0, t) + p2 (t)H˜ (t) = −



H (t)

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+ (p2 (t) − ψ(0, H (t)))H˜ (t), T˜ (H (t), t) + p3 (t)H˜ (t) =(ε − ψ(H, H ) + p3 (t))H˜ (t).

(7.39) (7.40)

Moreover, substituting x = H (t) into (7.35) yields H˙˜ (t) + p4 (t)H˜ (t) + β T˜x (H (t), t) =(p4 (t) − c + β(εq(H (t), H (t)) − ψx (H (t), H (t))))H˜ (t).

(7.41)

Therefore, for the Eqs. (7.18)–(7.21) to hold, the gain kernel functions must satisfy the following conditions: ¯ y), qxx (x, y) − qyy (x, y) = − λq(x, λ¯ d q(x, x) = − , dx 2

q(0, y) = 0,

βψ(x, H (t)) =Di q(x, H (t)),

(7.42) (7.43) (7.44)

and the observer gains must satisfy p1 (x, t) = − Di (εqy (x, H (t)) + ψxx (x, H )) − cψ(x, H ),

(7.45)

p2 (t) =ψ(0, H (t)),

(7.46)

p3 (t) =ψ(H (t), H (t)) − ε,

(7.47)

p4 (t) =c − β(εq(H (t), H (t)) − ψx (H (t), H (t))),

(7.48)

and the function f (x, H (t)) must satisfy

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 f (x, H ) + εq(x, H ) + ψH (x, H ) =

H

q(x, y)f (y, H )dy.

(7.49)

x

The solutions to (7.42)–(7.44) are uniquely given by q(x, y) = − λ¯ x ψ(x, H (t)) = −

I1

 

 ¯ 2 − x2) λ(y

¯ 2 − x2) λ(y

,

λ I1 (z) x , β z

(7.50) (7.51)

where z is defined by (7.26). Then, using (7.50)–(7.51), the conditions (7.45)–(7.48) are led to the explicit formulations of the observer gains given as (7.22)–(7.25). In the similar manner, the conditions for the gain kernel functions of the inverse transformation (7.30) are given by rxx (x, y) − ryy (x, y) =λ¯ r(x, y), d λ¯ r(x, x) = , dx 2

r(0, y) = 0,

βφ(x, H (t)) =Di r(x, H (t)),

(7.52) (7.53) (7.54)

and the function f (x, H (t)) is obtained by f (x, H (t)) = r(x, H (t))p3 (H (t)) + φH (x, H (t)).

(7.55)

The solutions to (7.52)–(7.54) are given by ¯ r(x, y) =λx

J1

  λ¯ (y 2 − x 2 ) λ J1 (z)  , , φ(x, H ) = x β z λ¯ (y 2 − x 2 )

(7.56)

where J1 is Bessel function of the first kind. Using the solutions (7.56), the function f (x, H (t)) is obtained explicitly by (7.55), which also satisfies the condition (7.49). Hence, the transformation from (T˜ , H˜ ) to (w, H˜ ) is invertible.

Stability Analysis We prove the exponential stability of the origin of the estimation error system (7.18)–(7.21) in the spatial L2 norm. First, we show the exponential stability of the origin of the target system (7.31)–(7.34). We consider the following Lyapunov functional:

7.4 Temperature Profile Estimation

191

V =

1 ε ˜ 2 H (t) . ||w||2 + 2 2β

(7.57)

Taking the time derivative of (7.57) together with the solution of (7.31)–(7.34) yields H˙ (t) 2 ˜ 2 εc ε H (t) V˙ = − Di ||wx ||2 − λ||w||2 − H˜ (t)2 + β 2  H (t) − H˙ (t)H˜ (t) w(x, t)f (x, H (t))dx.

(7.58)

0

Applying Young’s and Cauchy-Schwarz inequalities to the last term in (7.58) with the help of Assumption 7.2, and choosing the gain parameter c to satisfy c>

βM 2 f¯ + βMε, ελ

(7.59)

one can obtain the following inequality: V˙ ≤ − min{λ, c}V .

(7.60)

Applying comparison principle to the differential inequality (7.60), we get V (t) ≤ V (0)e− min{λ,c}t .

(7.61)

Hence, the target system (7.31)–(7.34) is exponentially stable at the origin. Due to the invertibility of the transformations (7.29) and (7.30), there exist positive constants M > 0 and M¯ > 0 such that for the norm Φ(t) defined in (7.27) the ¯ inequalities hold MΦ(t) ≤ V (t) ≤ MΦ(t). Hence, we obtain (7.28) by defining ˜ ¯ M = M/M, which completes the proof of Theorem 7.1. Note that the designed backstepping observer achieves faster convergence with a possibility of causing ¯ overshoot since the overshoot coefficient M/M is a monotonically increasing function in the observer gains’ parameters (λ, c). While we have focused on the simplified PDE (7.11) to derive a rigorous proof of the proposed state estimation design (7.13)–(7.16) with observer gains given by (7.22)–(7.25), simulation studies are performed by applying the estimation design to the original thermodynamic model (7.1)–(7.7) including salinity.

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7.5 Numerical Tests of the Sea Ice Estimation Initial Conditions The simulation results of temperature estimation Tˆi computed by (7.13)–(7.16) along with the available measurements obtained by the online calculation of (7.1)– (7.7) are shown in Fig. 7.3. Here the initial temperature profiles are formulated as Ts (x, 0) = Ti (x, 0) =

k0 (Tm1 − T0 ) x + T0 , ks H0

  4π x Tm1 − T0 , x + T0 + a sin H0 H0

(7.62) (7.63)

where T0 = Ti (0, 0) which is obtained by solving fourth order algebraic equation from (7.1) and the input data, and a is set as a = 1 ◦ C. The estimated initial temperature is chosen as Tm1 − T0 (x 2 − 2dH0 x) + T0 Tˆi (x, 0) = 2 H0 (1 − 2d)

(7.64)

with setting d = 1/4. Hence, the initial temperature estimate is lower than the actual temperature. This initial condition satisfies the boundary conditions (7.14) and (7.15). The initial state of the estimated ice thickness Hˆ (0) is set as that of the true thickness, i.e., Hˆ (0) = H (0), which is feasible because the thickness is actually measured.

Tuning Method for Gain Parameters The design parameters (λ, c, ε) are selected as follows: 1. Choose ε ≈ β for the norm (7.57) to be similarly weighted. 2. Select λ to be the inverse of a desired time constant (i.e., the time at 63% decay of the estimation error is achieved): here we set as one day, leading to 1 λ ≈ 24×3600 = 1.2 · 10−5 . 3. Select c sufficiently larger than λ so that the decay rate min{λ, c} is not reduced and (7.59) is satisfied. Finally, these parameters are varied around these reference values until we observe a smooth and sufficiently fast convergence. Throughout the simulation, we see that the minimum value of the time step size in ode solver is more than 1 min, while the computation time of each time update is less than 0.1 s, which shows its real-time implementability as addressed in Remark 7.1.

7.5 Numerical Tests of the Sea Ice Estimation

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Numerical Simulation of State Estimation The contour plot of the simulation results of Ti (x, t) and Tˆi (x, t) for open-loop estimation by setting all the observer gain to be zero is depicted in Fig. 7.3a, and those for the proposed estimation are depicted in Fig. 7.3b and Fig. 7.4a, b

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(b) Fig. 7.4 Simulation results of the plant (7.1)–(7.7) and the backstepping estimator (7.13)–(7.16) with some chosen free parameters. (a) The proposed estimation with larger value of λ than Fig. 7.3b. The overshoot beyond the true temperature is observed during the first 2 days. (b) The proposed estimation with smaller value of λ than Fig. 7.3b. The convergence speed gets slower than the result of Fig. 7.3b

with observer gains (7.22)–(7.25), respectively, by using input data on January. For the proposed estimation, we fix the parameters of c =3.0 × 10−5 and ε = 1.0 × 10−8 , and use the parameter of λ =5.0 × 10−6 in Fig. 7.3b, λ =1.0 × 10−5 in Fig. 7.4a, and λ =5.0 × 10−7 in Fig. 7.4b. The figures show that the backstepping observer gain makes the convergence speed of the estimation to the actual value approximately five to ten times faster at every point in sea ice. As seen in Fig. 7.4, while the larger value of λ makes the convergence speed faster, it causes more overshoot beyond the actual temperature. Hence, the tradeoff between

7.6 Comments and Remarks

195

the convergence speed and overshoot can be handled by tuning the gain parameter λ appropriately, thereby the parameters used in (b) achieve the desired performance. The overshoot behavior is noted at the end of Sect. 7.4 from a theoretical perspective. Consequently, the stability properties stated in Theorem 7.1 for the simplified model can be observed in numerical results of the proposed estimation applied to the original model (7.1)–(7.7). To visualize the convergence of the estimated temperature profile used in 7.3b more clearly, Fig. 7.5 illustrates the profiles of both true temperature (black solid) and estimated temperature (red dash) on January 1st to 3rd in (a)–(c), respectively. We observe that the estimated temperature profile becomes almost the same as the true temperature profile on January 3rd, which is 2 days after the estimation algorithm runs. Moreover, Fig. 7.5d depicts the time evolution of H˜ (t), which is an estimation error of the ice’s thickness. We observe that the error is “enlarged” from H˜ (0) = 0 due to the error of temperature profile, and returns to zero after the temperature profiles become almost indistinguishable on January 3rd, from which the necessity of the estimator of the ice’s thickness is ensured while the thickness is actually measured. Finally, we have studied the robustness of the proposed observer by varying the parameters Di , β, and Fw in the observer (7.13)–(7.16) and the gains (7.22)–(7.25) to Di (1 + δ1 ), β(1 + δ2 ), and Fw (1 + δ3 ) with setting δ1 = 0.3, δ2 = −0.3, and δ3 = 0.4. Figure 7.6a shows the contour plots of estimated and true temperature profiles and Fig. 7.6b shows the evolution of H˜ (t). From both figures, we can see that the observer states converge and stay around the true states with a modest error after 5 days, which illustrates robust performance of the proposed observer under the parameters’ uncertainties.

7.6 Comments and Remarks In this chapter, we develop the estimation algorithm for temperature profile in the Arctic sea ice via backstepping observer design [113]. The observer gains are derived so that the convergence of the state estimate to the actual state is guaranteed theoretically for a simplified model. Numerical simulation is employed to investigate the performance of the observer design with the original thermodynamic model, which illustrates ten times faster convergence of state estimation to the actual temperature than the straightforward open-loop estimation. While we have assumed the online availability of the measurements, these data acquired by satellites typically accompany a time-delay due to the communication. Such a time-delay can be compensated by extending the method developed in [118] for control design under actuator delay to the estimator design under sensor delay following the procedure in [124]. In addition, the physical parameters used in this chapter are uncertain variables in practice, where the uncertain parameters can be assumed to be constants at each month, and hence it is significant to design

196 Fig. 7.5 Simulation result of the plant (7.1)–(7.7) and the estimator (7.13)–(7.16) with parameters used in Fig. 7.3b. (a) Temperature profile of both true and estimate on January 1st. (b) Temperature profile of both true and estimate on January 2nd. (c) Temperature profile of both true and estimate on January 3rd. (d) The time evolution of thickness estimation error H˜ (t)

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a simultaneous state and parameter estimation algorithm such as [151] using a reduced-order model via Pade-approximation and [17] using data-driven extremum seeking as an iterative learning method. Instead of adaptive estimation, interval observers for the state estimation of uncertain parabolic PDEs have been proposed in [110]. Moreover, applying the optimal control of the Stefan problem developed in [5, 18, 88] to the estimation of the sea ice model is also an interesting direction for future work.

Chapter 8

Lithium-Ion Batteries

8.1 Battery Management Systems Battery management is crucial for safe and efficient use of numerous kinds of electronics such as smartphones and laptops, and electric vehicles. Among several chemical materials used for electrodes of lithium-ion batteries, Lithium Iron Phosphate (LFP) has several attractive features as an active material in lithium-ion batteries such as thermal safety, high energy, and power density [155]. LFP and other common active materials show unique charge–discharge characteristics due to an underlying crystallographic solid–solid phase transition. Electrochemical models for lithium-ion batteries with single phase materials do not allow to capture these unique characteristics and thus a mathematical description of phase transitions needs to be added to these models. Electrochemical models are of interest for the design of accurate estimation algorithms in battery management systems. Estimation algorithms based on these models provide visibility into operating regimes that induce degradation enabling a larger domain of operation, therefore increasing the performance of the battery in terms of energy capacity, power capacity, and fast charge rates [38, 159]. Electrochemical model-based estimation is challenging for several reasons. First, measurements of lithium concentrations outside specialized laboratory environments are impractical. Second, the concentration dynamics are governed by coupled and nonlinear partial differential algebraic equations (PDAE) [201]. Finally, the only measurable quantities (voltage and current) are related to dynamic states through a nonlinear function. Electrochemical models describe the relevant dynamic phenomena in lithiumion cells: diffusion, intercalation, and electrochemical kinematics (see Fig. 8.1). These models predict accurately the internal states of the battery; however, their complexity renders a challenging problem for estimation algorithms. For this reason, most approaches develop estimation algorithms based on simplified models. Among the various simplified models, the single particle model (SPM) has been © Springer Nature Switzerland AG 2020 S. Koga, M. Krstic, Materials Phase Change PDE Control & Estimation, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-58490-0_8

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8 Lithium-Ion Batteries

Voltage: V

Separater

Negative Electrode

Positive Electrode

Li+

r

r

Fig. 8.1 Schematic of lithium-ion battery and the description of particles in electrochemical models. The concentration dynamics of lithium-ion is governed on the geometry of each particle

broadly used in the observer design problem, see [54, 55, 150, 151, 158, 200, 212]. The main characteristic of the SPM is the use of a single spherical particle to represent diffusion of lithium ions in the intercalation sites of the porous active materials in the electrodes. LFP has been extensibility used in lithium ion cells due to its thermal stability, cost effectiveness, nontoxic nature, and long cycle life [155]. An electrochemical model for LFP batteries was proposed in [190] based on a core-shell model, where the concentration at the core is assumed constant and diffusion is allowed for the phase in the shell. The LFP model with phase transition electrode was revisited in [230] with a more complete core-shell model, allowing diffusion in both phases of an LiCoO2 cathode. The estimation problem for batteries with LFP electrodes has been relatively less studied; a particle filter was derived in [173] and a Sequential Monte Carlo filter was derived in [139]. The core-shell model proposed for phase transition electrodes is described by a parabolic PDE with a state-dependent moving boundary.

8.2 Electrochemical Model with Phase Change Electrode

201

8.2 Electrochemical Model with Phase Change Electrode The electrochemical model for lithium-ion cells with a phase transition material in the positive electrode follows [190]. We restrict the problem to particular initial conditions of the concentration of lithium ions in the particles (i.e., intercalation sites) of the positive electrode and consider only discharge processes. The initial concentration of lithium ions in the particles of the positive electrode follows a coreshell configuration where the core has a constant distribution of lithium ions in a low concentration phase (the α phase), and the shell has a constant distribution of lithium ions in a high concentration phase (the β phase). During discharge, the fluxes of lithium ions at the surface of the particles in the positive electrode are positive, thus increasing the concentration of lithium ions in the shell and the phase boundary is moving to the center, i.e., a shrinking core process, as depicted in Fig. 8.2.

Single Particle Model The single particle model is a simple electrochemical model that accounts for some phenomena in lithium-ion cells. The main simplification in this model comes from the assumption that a single diffusion equation in an spherical particle can be used to model the diffusion of lithium ions in all the intercalation sites of the active material of each electrode. In the SPM, the ionic molar fluxes jn,± (t) on both electrodes are proportional to the current density I (t) applied to the cell jn,± (t) = ∓

I (t) , as,± F L±

(8.1)

where as,± = 3s,± /Rp,± is the interfacial area (per unit volume), s,± is the volume fraction of active material in each electrode, Rp,± is the averaged radius of the intercalation sites (particles) in the electrodes, F is the Faraday constant, and L± is the thickness of each electrode. Throughout this chapter, the subscripts

β α

β

α

Fig. 8.2 Phase transition in the positive particle during discharge. The particle starts with a large core of low concentration phase α and a small shell of high concentration phase β. During discharge there is a positive flux of lithium ion in the surface of the positive particle, increasing the concentration and increasing the size of the β-phase shell

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+ and − indicate that the variable corresponds to the positive or negative particle. The concentration dynamics of lithium ions in the negative electrode (single phase) follow the Fick’s law for diffusion   Ds,− ∂ ∂cs,− 2 ∂cs,− (r, t) = 2 (r, t) , (8.2) r ∂t ∂r r ∂r for r ∈ (0, Rp,− ), t > 0 with boundary conditions ∂cs,− (0, t) = 0, ∂r Ds,−

∂cs,− (Rp,− , t) = −jn,− (t), ∂r

(8.3) (8.4)

and initial condition c0,− ∈ C (0, Rp,− ). Diffusion in the positive particle follows a core-shell model. In the core of the particle, i.e., for r ∈ (0, rp (t)), lithium ions are in the α-phase. The concentration in the core is assumed to be constant and equal to the equilibrium value of the α-phase, i.e., cs,+ (r) = cs,α for all r ∈ (0, rp (t)). In the shell of the spherical particle, i.e., for r ∈ (rp (t), Rp,+ ), the concentration of lithium ions is in β-phase. The concentration dynamics of lithium-ions in the shell of the positive particle follows the Fick’s law for diffusion   Ds,+ ∂ ∂cs,+ 2 ∂cs,+ (r, t) = 2 (r, t) , r ∂t ∂r r ∂r

(8.5)

for r ∈ (rp (t), Rp,+ ) with boundary conditions cs,+ (rp (t), t) = cs,β , Ds,+

∂cs,+ (Rp,+ , t) = −jn,+ (t), ∂r

(8.6) (8.7)

and initial conditions c0,+ ∈ C (rp (0), Rp,+ ). The time-evolution of the moving interface rp (t) is not given explicitly. Instead, mass balance at the moving interface yields the following state-dependent dynamics: (cs,β − cs,α )

drp (t) ∂cs,+ = −Ds,+ (rp (t), t). dt ∂r

(8.8)

Overpotentials η± (t) are found by solving the nonlinear algebraic equation . αc F i0,± (t) - αa F η± (t) e RT − e− RT η± (t) , F   α  α i0,± (t) = F k± css,± (t) c ce,0 cs,max,± − css,± (t) a ,

jn,± (t) =

(8.9) (8.10)

where css,± (t) := cs,± (Rp,± , t). The electric potential in each electrode is given by

8.2 Electrochemical Model with Phase Change Electrode

φs,± (t) = η± (t) + U± (css,± (t)) + Rf,± Fjn,± (t).

203

(8.11)

Finally, output voltage is computed as the difference between the electric potential in each electrode V (t) = φs,+ (t) − φs,− (t).

(8.12)

Equations (8.5)–(8.12) form a complete description of the single particle model with a phase transition electrode, and it provides the following property on the moving interface during the discharge process. Remark 8.1 During the single discharge process, the current density I (t) maintains positive, i.e., I (t) > 0 for ∀t > 0. This current positivity ensures the moving interface being shrinking. Furthermore, the initial interface position is less than the cell radius. Hence, drp (t) < 0, dt

(8.13)

0 ≤ rp (t) < Rp,+ .

(8.14)

Mass Conservation In this model, the total amount of lithium ions is conserved. The mathematical description of this property is given in the following lemma. Lemma 8.1 The total amount of lithium nLi in solid phase (moles per unit area) defined as nLi (t) = s,− L− cs,− (t) + s,+ L+ cs,+ (t),

(8.15)

where cs,− (t) and cs,+ (t) are the volumetric averages of the concentrations cs,− (t) = cs,+ (t) =

3 3 Rp,−

3 3 Rp,+



Rp,−

cs,− (r, t)r 2 dr,

(8.16)

cs,+ (r, t)r 2 dr,

(8.17)

0



Rp,+

0

is conserved, namely dnLi (t)/dt = 0. Lemma 8.1 was derived in [112] for electrodes with a single phase, and we can show that this result extends to electrodes with phase transition materials.

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8 Lithium-Ion Batteries

Proof In our problem formulation there is a single phase in the negative particle and there are two phases in the positive particle, i.e., α-phase in the core and β-phase in the shell. The concentration in α-phase at the core is assumed to be constant (at its equilibrium value cs,α ). Under these assumptions, the time derivative of (8.15) is given by dnLi 3s,+ L+ 2 (t) = − as,− L− jn,− (t) − as,+ L+ jn,+ (t) − rp (t) 3 dt Rp,+    drp  ∂cs,+ (t) cs,β − cs,α + Ds,+ (rp (t), t) . × dt ∂r

(8.18)

Hence, the molar flux equations in (8.1) and the dynamics of the moving interface in (8.8) lead to dnLi (t)/dt = 0. In a more general formulation introduced in [108, 109], i.e., when both electrodes have multiple phase transitions not necessarily at the equilibrium, mass conservation of lithium ions is guaranteed with the following interface dynamics:   dri[a,b] 1 ∂c [a,b] − ∂c [a,b] + Da (ri (t) , t) − Db (ri (t) , t) , (t) = dt cb − ca ∂r ∂r

(8.19)

where ri[a,b] is the interface radius between any two phases (phase a and phase b) in any electrode. Each phase has a distinct equilibrium ca , cb and diffusion coefficient Da , Db .

8.3 State-of-Charge Estimation Now, a state estimation algorithm for concentration of lithium ions, in both negative and positive electrodes, is provided in this section from the single particle model. The state observer for the positive electrode is derived via the backstepping method for moving boundary PDEs, and the observer for the negative electrode is derived from the mass conservation property.

Observer for Phase Transition Positive Electrode The state observer is a copy of the diffusion system (8.5)–(8.7) in the positive electrode together with output error injection   ∂5 cs,+ ∂5 cs,+ Ds,+ ∂ (r, t) = 2 (r, t) r2 ∂t ∂r r ∂r

8.3 State-of-Charge Estimation

205

  + P (5 rp (t), r) css,+ (t) − 5 cs,+ (Rp,+ , t) ,

(8.20)

for r ∈ (5 rp (t), Rp,+ ) with boundary conditions rp (t), t) =cβ , 5 cs,+ (5 Ds,+

(8.21)

∂5 cs,+ (Rp,+ , t) = − jn,+ (t) ∂r

  cs,+ (Rp,+ , t) , + Q(5 rp (t)) css,+ (t) − 5

(8.22)

rp (0), Rp,+ ) and r5p (0) ∈ (0, Rp,+ ). Observer and initial conditions 5 c0,+ ∈ L 2 (5 gains are given by 2 Rp,+

P (5 rp (t), r) = Ds,+ λ Ds,+ Q(5 rp (t)) = Rp,+



r

l(t)s(t)

I2 (z(t)) , z(t)

 λ s(t) + 1 , 2

(8.23) (8.24)

where I2 (·) is a modified Bessel function of the second kind and λ=

λ , Ds,+

s(t) = Rp,+ − r5p (t), l(t) = r − r5p (t),    z(t) = λ s(t)2 − l(t)2 .

(8.25) (8.26) (8.27)

The parameter λ > 0 is designed to achieve faster convergence of the estimated concentration to true concentration. Moreover, the estimator for the moving interface position is given by the following dynamics: (cs,β − cs,α )

  d r5p (t) = −κ css,+ (t) − 5 cs,+ (Rp,+ , t) dt ∂5 cs,+ −Ds,+ (ˆrp (t), t), ∂r

(8.28)

where the parameter κ > 0 is designed to achieve fast convergence of the estimated interface position to the true value. The stability of the estimation error system is theoretically proven for the PDE observer (8.20)–(8.22) with gains (8.23), (8.24) under the assumption r5p (t) ≡ rp (t) for all t ≥ 0 in the next section. As the moving interface position rp (t) is unknown in practice, we construct the estimator (8.28), and use the estimated interface position r5p (t) in the gains (8.23), (8.24) of PDE observer.

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8 Lithium-Ion Batteries

The sign of the observer gain in (8.28) (first term in the right-hand side) is determined based on the monotonic relation, namely, as the surface concentration css,+ (t) is increased the moving interface position rp (t) is decreased. Physically, as the battery is discharged, the domain of the lithium rich β-phase in the positive electrode is expanded from the outer region. Hence, the observer (8.28) is designed so that if the measured surface concentration is larger than the estimated surface concentration, the battery is discharged more than estimated, and the domain of β-phase for the estimator is driven to be expanded.

Stability Analysis of the Estimation Error System with Known Interface Position Let 2 cs,+ (r, t) be an estimation error defined by 2 cs,+ (r, t) := cs,+ (r, t) − 5 cs,+ (r, t). The stability analysis of the estimation error system is presented in the following theorem. Theorem 8.1 Consider the plant PDE (8.5)–(8.7) and the PDE observer (8.20)– (8.22) with observer gains (8.23) and (8.24) under the properties of (8.13), (8.14), and the assumption r5p (t) ≡ rp (t) for all t ≥ 0. Then, for any initial estimation error c6 s,+ (r, 0), the estimation error is exponentially stable at the origin in the sense of the norm 

Rp,+

r 22 cs,+ (r, t)2 dr.

(8.29)

rp (t)

Note that subtracting (8.20)–(8.22) from (8.5)–(8.7) under r5p (t) ≡ rp (t) yields the estimation error dynamics   Ds,+ ∂ ∂2 cs,+ ∂2 cs,+ (r, t) = 2 (r, t) − P (rp (t), r)2 r2 cs,+ (Rp,+ , t), ∂t ∂r r ∂r (8.30) 2 cs,+ (rp (t), t) =0, Ds,+

(8.31)

  ∂2 cs,+ (Rp,+ , t) = − Q rp (t) 2 cs,+ (Rp,+ , t). ∂r

(8.32)

Change of Coordinate First, we introduce the following change of coordinate and state variable to simplify the structure of the estimation error dynamics in a cartesian coordinate: x = Rp,+ − r,

(8.33)

8.3 State-of-Charge Estimation

207

2 u(x, t) = r2 cs,+ (r, t), s(t) = Rp,+ − r5p (t).

(8.34) (8.35)

The estimation error dynamics (8.30)–(8.32) is rewritten by the new coordinate and state as ∂ 22 u ∂2 u (x, t) = Ds,+ 2 (x, t) − P (s(t), x)2 u(0, t), ∂t ∂x 2 u(s(t), t) = 0,

(8.36) (8.37)

∂2 u (0, t) = −Q(s(t))2 u(0, t), ∂x

(8.38)

where P (s(t), x) = Q(s(t)) =

r P (rp (t), r), Rp,+ 1

−

Rp,+

1 Q(rp (t)). Ds,+

(8.39) (8.40)

With respect to the variable (8.35), the properties (8.13) and (8.14) presented in Remark 8.1 are equivalent to s˙ (t) > 0,

(8.41)

0 < s(t) ≤ Rp,+ .

(8.42)

Derivation of Observer Gains Consider the following invertible transformation from the estimation error 2 u(x, t) to the transformed state w 2(x, t): 

x

w 2(x, t) = 2 u(x, t) +

q(x, y)2 u(y, t)dy,

(8.43)

p(x, y)2 w (y, t)dy,

(8.44)

0

 2 u(x, t) = w 2(x, t) +

x

0

where x = s(t) − x, y = s(t) − y. Similar to observer design in Sect. 3.4, we can show that if the gain kernel functions and the observer gains satisfy the following conditions: ∂ 2p ∂ 2p ( x, ¯ y) ¯ − (x, ¯ y) ¯ = − λ¯ p(x, ¯ y), ¯ ∂ x¯ 2 ∂ y¯ 2

(8.45)

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8 Lithium-Ion Batteries

λ¯ p(x, ¯ x) ¯ = x, ¯ 2

(8.46)

p(0, y) ¯ =0,

(8.47)

∂ 2q ∂ 2q (x, ¯ y) ¯ − 2 (x, ¯ y) ¯ =λ¯ q(x, ¯ y), ¯ 2 ∂ x¯ ∂ y¯ q(x, ¯ x) ¯ =−

λ¯ x, ¯ 2

q(0, y) ¯ =0,

(8.48) (8.49) (8.50)

P (s(t), x) =Ds,+ py¯ (x, ¯ s(t)),

(8.51)

Q(s(t)) = − p(s(t), s(t)),

(8.52)

then the following target w 2-system is obtained:  x ∂w 2 ∂ 2w 2 (x, t) = Ds,+ 2 (x, t) − λ2 w(x, t) + s˙ (t) q  (x, y) ∂t ∂x 0    y × w 2(y, t) + p(y, z)2 w(z, t)dz dy,

(8.53)

0

w 2(s(t), t) = 0,

(8.54)

∂w 2 (0, t) = 0, ∂x

(8.55)

where q  (x, y) = ∂q ∂x (x, y) + explicit solutions:

∂q ∂y (x, y).

The Eqs. (8.45)–(8.50) lead to the following

    2 2 I1 λ y −x p(x, y) =λx    , λ y2 − x2     J1 λ y2 − x2 q(x, y) = − λx    , λ y2 − x2

(8.56)

(8.57)

with a modified Bessel function I1 (·) and a Bessel function J1 (·) of the first kind, respectively. Substituting the solution (8.56) to the conditions (8.51), (8.52) (note 1 (z) that dIdz = I2z(z) for all z), and taking back to the original coordinate and variables, the observer gains are derived as (8.23) and (8.24).

8.3 State-of-Charge Estimation

209

Stability Proof As done in Sect. 3.4, we consider the time evolution of the following Lyapunov function: W (t) =

1 2



s(t)

w 2(x, t)2 dx.

(8.58)

0

Taking the time derivative of (8.58) along with (8.53)–(8.55) yields W˙ (t) = − Ds,+

s(t)  ∂ w 2



 + s˙ (t)

∂x

0 s(t)

2  (x, t) dx − λ 

w 2(x, t)

0

w 2(x, t)2 dx

0 x

q  (x, y)

0

  w 2(y, t) +

s(t)

y





P (y, z)2 w(z, t)dz dy dx.

(8.59)

0

Applying Young’s, Cauchy Schwartz, and Poincare’s inequalities with the help of the properties (8.41) and (8.42), one can show that there exists a constant a > 0 such that the following inequality holds (refer to Sect. 3.4 for the detailed steps): W˙ (t) ≤ − bW (t) + a s˙ (t)W (t), where b =

Ds,+ 2 4Rp,+

(8.60)

+ λ. With the help of (8.41) and (8.42), it yields the exponential

decay of W (t) as W (t) ≤ eaRp,+ W (0)e−bt .

(8.61)

Hence, the origin of w 2-system is shown to be exponentially stable, from which we conclude Theorem 8.1.

Observer for Negative Electrode The observer design for lithium ion concentration in the negative electrode is constructed by the copy of the dynamics (8.2)–(8.4) together with the output injection of the positive electrode   Ds,− ∂ ∂5 cs,− ∂5 cs,− (r, t) = 2 (r, t) + P− (rp (t))2 r2 cs,+ (Rp,+ , t), ∂t ∂r r ∂r for r ∈ (0, Rp,− ), t > 0 with boundary conditions

(8.62)

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8 Lithium-Ion Batteries

∂5 cs,− (0, t) =0, ∂r Ds,−

(8.63)

  ∂5 cs,− cs,+ (Rp,+ , t). (Rp,− , t) = − jn,− (t) + Q− rp (t) 2 ∂r

(8.64)

Observer gains in the negative electrode are computed to conserve the total amount of lithium ions in the state observer defined as 5 nLi (t) =

3s,+ L+ 3 Rp,+



Rp,+

5 cs,+ (r, t)r 2 dr +

3s,− L−

0

3 Rp,−



Rp,−

5 cs,− (r, t)r 2 dr.

0

(8.65) Taking the time derivative of (8.65) along with the dynamics (8.20)–(8.28) and (8.62)–(8.64) leads to d5 nLi,+ = − as,+ L+ jn,+ (t) − as,− L− jn,− (t) + F2 cs,+ (Rp,+ , t), dt

(8.66)

where F is defined by  F = as,+ L+

κ



5 rp (t) + Q(5 rp (t)) + as,− L− Q− (5 rp (t)) 2 Rp,+  3s,+ L+ Rp,+ 2 r P (5 rp (t), r)dr + s,− L− P− (5 rp (t)). + 3 Rp,+ 5 rp (t) 2

(8.67)

By the balance of the ionic molar fluxes given in (8.1), the first line in the right-hand side of (8.66) is canceled. Therefore, by designing the observer gains as as,+ L+ Q− (rp (t)) = − as,− L−

 Q(rp (t)) +

s,+ L+ 3 P− (rp (t)) = − 3 s,− L− Rp,+ d5 n



Rp,+

5 rp (t)

κ 2 Rp,+

 5 rp (t)

2

,

(8.68)



P (rp (t))r 2 dr ,

(8.69)

one can show that dtLi,+ = 0 from (8.66). Hence, the observer error in the negative electrode approaches to zero uniformly in space with the help of Theorem 8.1.

8.4 Numerical Simulation

211

8.4 Numerical Simulation Description of the Proposed Algorithm For the spatial discretization of the diffusion equations in the single particle model and the observer, we use a finite volume method with nonuniform grid [229]. The reason to use a finite volume method is to guarantee the mass conservation property after discretization. The grid is a set of N points r(t) = [r1 (t), r2 (t), . . . , rN −1 (t), rN (t)] that divide the domain (rp (t), Rp ) (or (0, Rp ) for the negative electrode) in a finite set of nonuniform intervals, as depicted in Fig. 8.3, and allow us to define a fine discretization near the boundaries of the domain without increasing the size of the finite-dimensional state significantly. Briefly speaking, in the finite volume method, one defines a finite-dimensional state cs (t) = [cs,2 , . . . , cs,N −1 , cs,N ], where each element corresponds to the concentration at one of the spatial discretization points.. Then, solving the diffusion equation within each of the intervals ri−12+ri , ri +r2i+1 and using a linear interpolation, a set of ordinary differential equations is obtained, that define the dynamics of the finite-dimensional state. For example, in the positive electrode, the dynamic equation for the finite-dimensional approximation in β-phase is M(t)

dcs (t) = A(t)cs (t) + B(t)u(t), dt

(8.70)

css,+ = Ccs (t),

(8.71)

with   u(t) = jn (t), cs,β ,

(8.72)

and continuous matrices M(t), A(t) ∈ RN −1×N −1 , B1 (t) ∈ RN −1×2 , C ∈ R1×N −1 defined as

r2

r3

r1 = rp (t) Fig. 8.3 Nonuniform grid for spatial discretization

rN −2 rN −1

rN = Rp

212

8 Lithium-Ion Batteries

⎡3 ⎢ ⎢ ⎢ ⎢ M(t) = ⎢ ⎢ ⎢ ⎢ ⎣

1 4 v2 (t) 8 v3 (t) 1 3 4 v2 (t) 4 v3 (t) 0 18 v3 (t)

.. . 0 0

.. . 0 0

⎡

0 ··· 1 8 v4 (t) · · · 3 4 v4 (t) · · · .. .

0 0 0 .. .

⎤

⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ · · · 14 vN (t) ⎦ · · · 34 vN (t)

0 0

(8.73)

⎤ a2,2 (t) a2,3 (t) 0 · · · 0 0 ⎢ a (t) a (t) a (t) · · · ⎥ 0 0 ⎢ 3,2 ⎥ 3,3 3,4 ⎢ ⎥ 0 0 ⎢ 0 a4,3 (t) a4,4 (t) · · · ⎥ ⎢ ⎥, A(t) = ⎢ . .. .. .. .. ⎥ . ⎢ . ⎥ . . . . ⎢ ⎥ ⎣ 0 0 0 · · · aN −1,N −1 (t) aN −1,N (t) ⎦ 0 0 0 · · · aN,N −1 (t) aN,N (t) ⎡ ⎤ 0 b1,2 (t) ⎢ 0 0 ⎥ ⎢ ⎥ ⎢ .. .. ⎥ . B(t) = ⎢ . . ⎥ ⎢ ⎥ ⎣ 0 0 ⎦ bN,1

(8.74)

(8.75)

0

In (8.73), the volumes vi (t) are computed based on the geometry of the grid, that is  vi (t) =

ri (t) + ri+1 (t) 2

3

 −

ri (t) + ri−1 (t) 2

3  ,

(8.76)

.

(8.77)

for i ∈ {2, 3, . . . , N − 1}, and  vN,N (t) =



Rp3 (t) −

Rp (t) + rN −1 (t) 2

3 

The diagonal entries of the matrix A(t) in 8.74 are aii (t) = −

Ds (ri (t) + ri−1 (t))2 Ds (ri+1 (t) + ri (t))2 − , 4 ri+1 (t) − r1 (t) 4 ri (t) − ri−1 (t)

(8.78)

for i ∈ {2, 3, . . . , N − 1}, and aN,N (t) =

Ds (Rp (t) + rN −1 (t))2 . 4 Rp (t) − rN −1 (t)

The entries in the line above the diagonal are

(8.79)

8.4 Numerical Simulation

213

ai,i+1 (t) =

Ds (ri+1 (t) + ri (t))2 , 4 ri+1 (t) − ri (t)

(8.80)

for i ∈ {3, 4, . . . , N }, and the entries in the line below the diagonal are ai,i−1 (t) =

Ds (ri (t) + ri−1 (t))2 , 4 ri (t) − ri−1 (t)

(8.81)

for i ∈ {2, 3, 4, . . . , N − 1}. The nonzero elements of the matrix B(t) in (8.75) are b1,2 (t) =

Ds (r2 (t) + rp (t))2 , 4 r2 − rp (t)

(8.82)

bN,1 = −Rp2 .

(8.83)

The initial conditions for (8.70)–(8.71) are the evaluation of the initial concentration profile at each point of the grid r. For the discretization of the continuous-time finitedimensional system we use the backward Euler method, which leads to M[k+1]

cs,[k+1] − cs,[k] = A[k+1] cs,[k+1] + B[k+1] jk+1 , t[k+1] − t[k] css,k+1 = Ccs,k+1 ,

(8.84) (8.85)

where the quantities with subscript [k] ∈ {0, 1, . . . , K} correspond to the quantities at the discrete times {t0 , t1 , . . . , tNT }. The discretization of the boundary dynamics follows a discrete version of (8.8), derived properly to guarantee mass conservation, that is 3 3 − rp,[k]    rp,[k+1]  =− t[k+1] − t[k] Ds cs,β − cs,α 1



r2,[k] − r1,[k] 2

2

cs,1,[k+1] − cs,β . r2,[k] − r1,[k] (8.86)

The time update of the spatial grid corresponding to the increase of domain in time is performed carefully to ensure mass conversation while keeping the size of the grid constant over time. For this reason a new point in the discretization grid at every time step according to the dynamics of the moving boundary is introduced, if a threshold in surpass. If the threshold is not exceeding, the corresponding mass change is added to a mass memory variable for consideration in the next time step. To the new point in the grid we associate a new state equal to cβ . Then, to avoid an increase of the number of states at every time step, an interpolation is performed from the grid with the additional point, that is, with size N +1, to a new grid defined in the larger domain but with N point. The interpolation is linear and is corrected for mass conservation (see the description in Algorithm 6).

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8 Lithium-Ion Batteries

Algorithm 6: Time update for increasing domain (shrinking core) Data: Provided  some spatial grid r[k] = [r1,[k] , r2,[k] , . . . , rN,[k] ], state value cs,[k] = cs,1,[k] , cs,2,[k] , . . . , cs,N,[k] , at time t[k] , and memory of mass difference m[k−1] (the right-hand side of (8.86)) compute the mass change m[k] due to nonzero flux at the interface, i.e., the right-hand side of (8.86) and add it to the mass change memory m[k] ← m[k] + m[k−1] , if m[k] > mthreshold then compute a new for the boundary from the interface dynamics (8.86): 1/3  m[k] 3 − cs,β −c , rp,[k] ← rp,[k] s,α add this value as a new point to the grid:   r[k] ← rp,[k] , r1,[k] , r2,[k] , . . . , rN,[k] add a new entry to the state concentration with the value cβ : cs,[k] ← [cβ , cs,1,[k] , cs,2,[k] , . . . , cs,N,[k] ],  = [r     define a new nonuniform grid r[k] 1,[k] , r2,[k] , . . . , rN,[k] ] such that r1,[k] = rp,[k]  and rN,[k] = Rp interpolate (linearly) the concentration state to the new grid:  cs ← interp r  , r, cs compute average concentrations cs and cs over the spherical volume correct for mass conservation with the scaling factor cs /cs : cs cs ← cs cs  , updated state value c return updated grid r[k] s,[k] , and set memory m[k] = 0. else return unchanged spatial grid r[k] , unchanged state value cs,[k] , and updated mass memory m[k] . end

Test with Constant Discharge Input To test the observer we run a numerical example with a constant discharge current of 5 [C-rate] (Figs. 8.4 and 8.5). We are assuming css,+ is available directly from measurements to be used as output error injection in the observer. In practice, this quantity could be estimated from measurements. Figure 8.6 shows the estimated concentration of lithium ions in β-phase in the positive particle; one can compare this to the true concentration in Fig. 8.5. Figure 8.7 shows the averaged concentration in the positive particle, both true value (black) and estimated value (blue). Convergence of the estimate to the true value is achieved within 0.8 min, a relatively short time. Furthermore, Fig. 8.8 shows the time evolution of the moving interface of the both true value (black) and estimated value (blue), which also illustrates the convergence of the estimate to the true value. Note that SoC is directly proportional to the averaged concentrations; then the importance to evaluate the estimation of this quantity.

8.4 Numerical Simulation

215

3.6 [A/m 2 ]

2.8

2

9 [A/m ]

2.6

2

18 [A/m ]

2.4 25 [A/m 2 ]

2.2 36 [A/m 2 ]

2 0

2

4

6

8

10

Fig. 8.4 Voltage plot for different (constant) current discharge inputs, which shows the analogous behavior to [190]

Fig. 8.5 Normalized concentration of lithium ions in a growing β-phase region. The plot corresponds to a 5 min simulation of constant 5[C-rate] discharge. The plot does not show the α-phase portion of the concentration since it is assumed to be constant

Comparison with the Extended Kalman Filter Since the spatial discretization of the diffusion equations is performed for computation of the electrochemical model, we can apply the Extended Kalman Filter (EKF) to the reduced-order model as another approach for SoC estimation. The estimator design by the EKF for the Stefan system has been developed in Sect. 3.6. Here, we

216

8 Lithium-Ion Batteries

Fig. 8.6 Estimate of the concentration of lithium ions in the positive particle. Starting from the initial error, the estimated profile converges to the true profile in Fig. 8.5

0.5

0.4

0.3

0.2 0

0.2

0.4

0.6

0.8

1

Fig. 8.7 Averaged concentration of true value (black solid) and estimated averaged concentration (blue dashed) in the positive particle normalized by the maximum concentration

compare the performance of SoC estimation between the proposed backstepping (BKS) observer and the EKF with incorporating a measurement noise. Figure 8.9 shows a simulation result of SoC estimation via the BKS observer (blue) and the EKF (red). The initial SoC in the model is around 66 %, while the initial SoC in the estimator is around 46 % for both BKS and EKF. Figure 8.9a depicts the result under the noise-free measurement, which illustrates that the BKS estimate converges and almost stays at the true value within 5 min, while the estimate by the EKF converges quickly first but does not stay at the true value even after 10 min. Next, we incorporate the Gaussian noise in the measured value of the surface concentration, and compare the performance in Fig. 8.9b. The result illustrates that the BKS estimate still converges to the true value but it accompanies a noisy signal, while the EKF estimate has less noisy signal. Hence, in this one sample

8.4 Numerical Simulation

217

1 0.9 0.8 0.7 0

0.5

1

1.5

2

Fig. 8.8 The estimated interface position becomes the same value as the true interface position after 0.5 min Fig. 8.9 Comparison of SoC estimation between the proposed BKS observer and the EKF. This sample simulation illustrates that the proposed BKS is superior in convergence speed, while the EKF is superior in noise attenuation. Note that each method can reduce the drawback by appropriately tuning the free parameters. (a) Estimation without noise. (b) Estimation with measurement noise

70 60 50 40 0

5

10

(a)

70 60 50 40 0

5

10

(b)

simulation, we observe that the BKS estimator is superior in convergence speed, while the EKF estimator is superior in noise attenuation. However, by lowing the gain parameters (λ, κ) in the BKS estimator, we observe that the amplitude of the noisy estimate can be reduced in exchange for the convergence speed. Moreover,

218

8 Lithium-Ion Batteries

the EKF estimate also can be improved to accelerate the convergence speed by appropriately tuning the free parameters. Hence, there is an essential tradeoff between the convergence speed and the noise attenuation for both methods, and it is not appropriate to address which method is superior in general. Nevertheless, one of the advantages of the proposed BKS estimator is to have only two free parameters to tune for any given discretization number, while the EKF algorithm requires a tuning of covariance matrices in which the number of free parameters is increased as the discretization number is increased.

8.5 Comments and Remarks This chapter develops the estimation algorithm for SoC via electrochemical modelbased moving-boundary PDE observer and provides the numerical study illustrating the desired performance of the proposed method [115]. Towards a complete SoC estimation algorithm for lithium ion batteries with phase transition materials, an extension the existing SoC estimation algorithms from SPM to complex electrode settings will be considered, as was already achieved for electrodes with multiple active material [35]. It was noted in [191] that two different particles sizes are needed to correctly model LFP electrodes, and this correction can be added to our results following [35] (Table 8.1). One of the main assumptions for the model Table 8.1 Parameters of LFP used in the simulation

Negative Separator Parameters L[m]a 50 × 10−6 25 × 10−6 max 3 a cs [mol/m ] 27,760 cs,α [mol/m3 ]b cs,β [mol/m3 ]b Rp [m]a 11 × 10−6 2 a Ds [m /s] 9 × 10−14 a s [−] 0.33 Rf [Ωm2 ]b 1 × 10−5 2 b Rc [Ωm ] 0 k [m2.5 /mol0.5 s]a 3 × 10−5 Other parameters and physical constants A [m]b 1 F [As/mol] 96,487 R [J/Kmol] 8.314472 T [K]b 298 ce [mol/m3 ]a 1 × 103 a αa , αc [−] 0.5 a Borrowed b Assumed

from [191]

Positive 74 × 10−6 20,950 max 0.0480×cs,+ max 0.8920×cs,+ −9 52 × 10 8 × 10−18 0.27 0 6.5 × 10−3 3 × 10−17

8.5 Comments and Remarks

219

in this chapter is the restriction to only two coexisting phases in a single particle reduced further to a single phase problem by assuming a constant core phase. The relaxation of this assumption could be achieved through designing the state observer of concentration of lithium-ions in two phases together with the estimation of the interface position, by extending the method for control design in Chap. 5 to the observer design. Furthermore, the robustness of the estimator’s performance under some additive measurement noise can be studied in terms of input-to-state stability (ISS) following [34].

Chapter 9

Polymer 3D-Printing via Screw Extrusion

9.1 Emergence of 3D-Printing Additive manufacturing stands out among manufacturing technologies as a versatile tool for high flexibility and fast adaptability in production. It is applicable in a variety of producing industries, ranging from tissue engineering [147], thermoplastics [208], metal [136] and ceramic [175] fabrication. One of the most popular types of 3D printing is Fused Deposition Modeling (FDM) [148], which uses filaments as raw material, which have to be precisely manufactured to achieve a good final product quality [29].

9.2 Screw Extrusion Process From the polymer processing and extrusion cooking industry, screw extruders are well-known devices. Results stated in [130, 138, 149, 197] give an in-depth description of screw geometries and extruder setups. Then also describe the dynamics of the extrusion process, which consists of a conveying zone, a melting zone, and a mixing zone. A mathematical description of such a model is derived by mass, momentum, and energy balances and appears as a system of coupled transport equations coupled through a moving interface. This model is used in [138] to describe an extrusion cooking process. The boundary control of a similar model is developed in [56, 57] under the assumption of constant viscosity. More recent contributions consider screw extrusion as a useful technology for 3D printing applications [58, 61, 208], allowing to manufacture a wider variety of materials than FDM, while using polymer granules as raw material [208]. In [58], a time-delay control was developed for a model consisting of two phases, similar to [138]. In both cases, stabilization of the moving interface separating a © Springer Nature Switzerland AG 2020 S. Koga, M. Krstic, Materials Phase Change PDE Control & Estimation, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-58490-0_9

221

222

9 Polymer 3D-Printing via Screw Extrusion

conveying and a melting zone is achieved with a fast convergence rate. Another approach which enables to control screw extruders in 3D printing is proposed in [61], where an energy-based model is established, simplifying the implementation of the control law and circumventing difficulties with state measurement. In other words, the control of the outflow rate at the nozzle only relies on the measurement of the heater current and the screw speed. In the screw extrusion process, the solid material is convected from the feed to the nozzle located at the end of a heating chamber. The solid raw material is melted and mixed before being expelled through the nozzle as a thin filament. For this process, the thermal behavior is an important factor which characterizes the final product quality. Indeed, heat is supplied into the system by the heaters surrounding the extruder’s barrel on the one hand and by the viscous heat generation due to a shearing effect [138] on the other hand. The process of the phase transition from the solid to the liquid polymer can be described as the Stefan problem. In this context, the dynamics of the solid–liquid phase interface is derived from the energy conservation in which the latent heat required for melting is driven by the internal heat of the liquid phase, resulting in the interface velocity to be proportional to the temperature gradients of the adjacent phase. For instance, in [64], the Stefan problem for a polymer crystallization process is described, and the analytical crystallization time is derived.

9.3 Thermodynamic Modeling We focus on the thermodynamic model of the screw extrusion process in coordinate one-dimensional, along the vertical axis, motivated by [197], which developed a thermodynamic phase change model for polymer processing. The model provides the time evolution of the temperature profile of the extruded material and the interface position between the fed polymer granules and the molten polymer. The granular pellets are conveyed by the screw rotation at a given speed b along the vertical axis while the barrel temperature is uniformly maintained at Tb . Defining Ts (x, t) and Tl (x, t) as the temperature profiles of solid phase (polymer granules) over the spatial domain x ∈ (0, s(t)) and liquid phase (molten polymer) over the spatial domain x ∈ (s(t), L), respectively, the following thermodynamical model ∂Ts ∂ 2 Ts ∂Ts (x, t) =αs 2 (x, t) − b (x, t) + hs (Tb − Ts (x, t)) , ∂t ∂x ∂x

0 < x < s(t), (9.1)

∂Tl ∂ 2 Tl ∂Tl (x, t) =αl 2 (x, t) − b (x, t) + hl (Tb − Tl (x, t)) , ∂t ∂x ∂x

s(t) < x < L (9.2)

9.3 Thermodynamic Modeling

223

is derived from the energy conservation and heat conduction laws. In this chapter, we consider the temperature distribution in the liquid to be static as stated in (9.13) and in Assumption 9.1 (see Sect. 9.4). Here, αi =

ki , ρi ci

(9.3)

hi =

h¯ i , ρi ci

(9.4)

where ρi , ci , ki , and h¯ i for i ∈ {s, l} are the density, the heat capacity, the thermal conductivity, and the heat transfer coefficient, respectively and the subscripts s and l are associated with the solid or liquid phase, respectively. Referring to [209] which introduces a model of spatially averaged temperature for screw extrusion, we incorporate the convective heat transfer through the barrel temperature in (9.1) (9.2). The boundary conditions at x = 0 and x = L follow the heat conduction law, and the temperature at the interface x = s(t) is maintained at the melting point Tm , described as qf (t) ∂Ts (0, t) = − , ∂x ks

Ts (s(t), t) = Tm ,

(9.5)

∂Tl q∗ (L, t) = m , ∂x kl

Tl (s(t), t) = Tm ,

(9.6)

∗ > 0 is a heat flux at the where qf (t) < 0 is a freezing controller at the inlet and qm nozzle which is assumed to be constant in time. The interface dynamics are derived by the energy balance at the interface as

ρs ΔH s˙ (t) = ks

∂Ts ∂Tl (s(t), t) − kl (s(t), t). ∂x ∂x

(9.7)

The Eqs. (9.1)–(9.7) are the solid–liquid phase change model known as the “twophase” Stefan problem. Such a phase change model was developed for polymer processing. Here, we give the following remark to emphasize the conditions for the model (9.1)–(9.7) to be physically validated. Remark 9.1 In this chapter, we assume the pressure in the chamber to be static, and the melting temperature is constant to avoid supercooling. Then, to keep the physical state of each phase, the following conditions must hold: Ts (x, t) ≤Tm ,

∀x ∈ (0, s(t)),

∀t > 0,

(9.8)

Tl (x, t) ≥Tm ,

∀x ∈ (s(t), L),

∀t > 0,

(9.9)

which represent the model validity conditions (Fig. 9.1).

224

9 Polymer 3D-Printing via Screw Extrusion

faucet

screw

qf (t)

0

polymer granules

s(t)

melt polymer

heater ∗ qm

L

x

Fig. 9.1 Schematic of screw extruder for original description (left) and model description (right)

Remark 9.2 We assume the existence of a heating/cooling system that keeps the pellets at a controlled temperature, as stated in (9.5), which describes the heat flux control at the inlet. Extruders can be equipped with raw material preconditioners as intermediate unit operators, which for instance help to pre-heat ingredients before they enter the extruder chamber by adding steam. The preconditioners are usually located between the inlet and the extruder chamber, and a continuous flow of material from the feeder to the preconditioner is maintained [68, 168].

9.4 Ink Production Control Based on Screw Speed To ensure a continuous extrusion process, the control of the quantity of molten polymer that remains in the extruder chamber at any given time is crucial. By definition, the volume of fully melted material contained in the chamber is directly related to the position of the solid–liquid interface that needs to be controlled.

Steady-State Solution An analytical solution of the steady-state temperature profile denoted as   Ts,eq (x), Tl,eq (x) for any given setpoint value of the interface position defined as sr can be computed by setting the time derivative of the system (9.1)–(9.7) to zero. Hence, from (9.1) and (9.2) the following set of ordinary differential equations in space is obtained:

9.4 Ink Production Control Based on Screw Speed



225

   (x) − bT  (x) + h T − T 0 = αs Ts,eq s b s,eq (x) , s,eq    (x) − bT  (x) + h T − T 0 = αl Tl,eq l b l,eq (x) , l,eq

(9.10)

and the boundary values are given as 

q∗

 (0) = Ts,eq  (L) Tl,eq

− kfs , ∗ qm kl ,

=

Ts,eq (sr ) = Tm , Tl,eq (sr ) = Tm .

(9.11)

At equilibrium, the interface equation (9.7) satisfies the following equality:   0 =ks Ts,eq (sr ) − kl Tl,eq (sr ).

(9.12)

The solution to the set of differential equations (9.10) has the following form: 

Tl,eq (x) =

p1 eq1 (x−sr ) + p2 eq2 (x−sr ) + Tb ,

Ts,eq (x) = p3 eq3 (x−sr ) + p4 eq4 (x−sr ) + Tb ,

(9.13)

where  b2 + 4αl hl q1 = q2 = , 2αl   b + b2 + 4αs hs b − b2 + 4αs hs q3 = , q4 = . 2αs 2αs b+



b2 + 4αl hl , 2αl

b−

(9.14) (9.15)

Let r = Tb − Tm .

(9.16)

Substituting (9.13) into the boundary conditions (9.11) and (9.12), we obtain p1 =

∗ /k rq2 eq2 (L−sr ) + qm l , q1 eq1 (L−sr ) − q2 eq2 (L−sr )

p2 = − p3 =

∗ /k rq1 eq1 (L−sr ) + qm l , q1 eq1 (L−sr ) − q2 eq2 (L−sr )

rq4 + K/ks , q3 − q4

−rq3 − K/ks , q3 − q4   ∗ kl r(−q1 q2 ) eq1 (L−sr ) − eq2 (L−sr ) + (q1 − q2 )qm , K= q1 eq1 (L−sr ) − q2 eq2 (L−sr )

p4 =

(9.17) (9.18) (9.19) (9.20) (9.21)

226

9 Polymer 3D-Printing via Screw Extrusion

and the steady-state input is given by qf∗ =p3 q3 e−q3 sr + p4 q4 e−q4 sr .

(9.22)

∗ ) are prescribed, the steady-state input is Hence, once the parameters (sr , Tb , qm uniquely obtained.

Barrel Temperature Condition for a Valid Steady-State For the model validity, the steady-state must satisfy (9.8) and (9.9), which restricts the barrel temperature to some physically admissible values. Lemma 9.1 If the barrel temperature satisfies −q ≤ Tb − Tm ≤ q, ¯

(9.23)

where ∗ ∗ (q1 − q2 )qm qm , q¯ = − , (9.24) qden kl q2 eq2 (L−sr )     = − kl q1 q2 eq1 (L−sr ) − eq2 (L−sr ) + ks q3 q1 eq1 (L−sr ) − q2 eq2 (L−sr ) ,

q= qden

(9.25) then the steady-state solution satisfies (9.8) and (9.9).  (s ) ≥ 0 which yields Proof Since Tl,eq (sr ) = Tm , it is necessary to have Tl,eq r

p1 q1 + p2 q2 ≥ 0.

(9.26)

Substituting (9.17) and (9.18) into (9.26), we get T b − Tm ≥

∗ (q1 − q2 ) qm  , kl q1 q2 eq1 (L−sr ) − eq2 (L−sr )

(9.27)

knowing that q1 q2 < 0. With the help of (9.26) and from (9.13) the derivative of Tl,eq (x) satisfies    Tl,eq (x) ≥ p1 q1 eq1 (x−sr ) − eq2 (x−sr ) .

(9.28)

9.4 Ink Production Control Based on Screw Speed

227

 (x) ≥ 0 for all x ∈ (s , L) is p q ≥ 0 which Thus, the sufficient condition of Tl,eq r 1 1 yields

Tb − Tm ≤ −

∗ qm . q kl q2 e 2 (L−sr )

(9.29)

Next, the solid steady-state satisfies Ts,eq (sr ) = Tm , so it is necessary to have  (s ) ≥ 0 leading to p q + p q ≥ 0 which trivially holds under condition Ts,eq r 3 3 4 4 of (9.26). Hence, from (9.13), the derivative of Ts,eq (x) satisfies    Ts,eq (x) ≥ p4 q4 −eq3 (x−sr ) + eq4 (x−sr ) .

(9.30)

 (x) ≥ 0 is p q ≥ 0, which yields Then, the sufficient condition for Ts,eq 4 4

Tb − T m ≥ −

∗ (q1 − q2 )qm . qden

(9.31)

One can notice that condition (9.31) is less conservative than condition (9.27). Hence, combining (9.29) and (9.31), we conclude Lemma 9.1.

Estimator Design for Temperature Profile Generally, the full-state feedback control law is designed by assuming that the spatially distributed temperature profile can be measured. Some imaging-based thermal sensors, such as the IR camera, enable capturing the entire profile of temperature. However, these sensors include high noise and detect the temperature of the chamber, which contains a nominal error from the temperature of the polymer inside. Instead, single point thermal sensors such as thermocouples enable accurate measurement of the surface temperature at the inlet of the extruder. Moreover, the interface position between the polymer granules and the melt polymer can be detected by cameras via image signal processing. Thus, we build an observer to estimate the temperature profile by utilizing these two available measurements. Let Tˆs (x, t) be the estimated temperature profile. The observer design for Tˆs (x, t) is stated in the following theorem. Theorem 9.1 Consider the plant model (9.1), (9.5) with the two available measurements of Y1 (t) = s(t), and the following PDE observer

Y2 (t) = Ts (0, t),

(9.32)

228

9 Polymer 3D-Printing via Screw Extrusion

∂ Tˆs ∂ 2 Tˆs ∂ Tˆs (x, t) =αs 2 (x, t) − b (x, t) ∂t ∂x ∂x   + hs Tb − Tˆs (x, t) , 0 < x < Y1 (t),   ∂ Tˆs qf (t) (0, t) = − − γ Y2 (t) − Tˆs (0, t) , ∂x ks

(9.33) (9.34)

Tˆs (s(t), t) =Tm ,

(9.35)

where γ = 2αb s . Assume that s(t) ∈ (0, L) and s˙ (t) ≥ 0 for all t ≥ 0. Then, the observer error system is exponentially stable at the origin in the sense of the norm ˜ Φ(t) := ||Ts (x, t) − Tˆs (x, t)||H1 .

(9.36)

More precisely, there exists a positive constant M˜ > 0 such that the following inequality holds: −2 ˜ ˜ Φ(t) ≤ M˜ Φ(0)e



2

b hs + 4α + s

αs 4L2

 t

.

(9.37)

Remark 9.3 As stated in Lemma 9.3 (Sect. 9.4), the assumption s˙ (t) ≥ 0 for all t ≥ 0 holds under the closed-loop control law proposed in Sect. 9.4. Proof Let u˜ be the estimation error state defined by u˜ := Ts − Tˆs .

(9.38)

Subtraction of the observer system (9.33)–(9.35) from the plant (9.1) and (9.5) yields the following estimation error system: ∂ u˜ ∂ 2 u˜ ∂ u˜ (x, t) =αs 2 (x, t) − b (x, t) − hs u(x, ˜ t), ∂t ∂x ∂x ∂ u˜ (0, t) =γ u(0, ˜ t), ∂x u(s(t), ˜ t) =0.

(9.39) (9.40) (9.41)

Let us introduce the following change of variable z˜ (x, t) = u(x, ˜ t)e−γ x .

(9.42)

Then, u-system ˜ in (9.39)–(9.41) is converted into the following z˜ -system: ∂ z˜ ∂ 2 z˜ =α 2 − λ˜z, ∂t ∂x

(9.43)

9.4 Ink Production Control Based on Screw Speed

229

∂ z˜ (0, t) =0, ∂x

(9.44)

z˜ (s(t), t) =0,

(9.45)

2

b . To study the stability of the estimation error state at the origin, where λ = hs + 4α s we consider the Lyapunov functional

1 V˜ = 2

 0

s(t)

1 z˜ (x, t) dx + 2



2

s(t)

0

∂ z˜ (x, t)2 dx. ∂x

(9.46)

Taking the time derivative of (9.46) along the solution of (9.39)–(9.41) leads to // 2 //2 // //2 // ∂ z˜ // // ∂ z˜ // s˙ (t) ∂ z˜ ˙ / / / / ˜ (s(t), t)2 . V = − αs // 2 // − (αs + λ)//// //// − λ||˜z||2L2 − ∂x L2 2 ∂x ∂x L2

(9.47)

∂ z˜ (s(t), t) derived from the time derivative Note that we used ∂∂tz˜ (s(t), t) = −˙s (t) ∂x of the boundary condition (9.45). With the help of s(t) ∈ (0, L), Poincare’s ∂ z˜ 2 ∂ z˜ 2 ∂ 2 z˜ 2 inequality gives ||˜z||2L2 ≤ 4L2 || ∂x ||L2 and || ∂x ||L2 ≤ 4L2 || ∂x 2 ||L2 . Applying these inequalities and s˙ (t) ≥ 0 to (9.47) leads to the following differential inequality:

 αs  ˜ V. V˙˜ ≤ − 2 λ + 4L2

(9.48)

Applying the comparison principle to (9.48) yields −2 V˜ (t) ≤ V˜ (0)e

 λ+

αs 4L2

 t

.

(9.49)

By the definition of z˜ given in (9.42), for the norm of u-system, ˜ the following upper and lower bounds hold: ˜ 2L2 ≤ e2γ L ||˜z||2L2 , ||˜z||2L2 ≤ ||u|| // //2 // //2 // // // ∂ z˜ // // // ≤ 2// ∂ u˜ // + 2γ 2 ||u|| ˜ 2L2 , // ∂x // // ∂x // L2 L2 // //  // //2 // ∂ u˜ // // ∂ z˜ //2 2γ L 2 2 // // ≤ 2e // // + γ ||˜z|| L2 . // ∂x // // ∂x // L2 L2

(9.50) (9.51)

(9.52)

˜ Hence, by defining Φ(t) = ||u|| ˜ 2H , the following inequalities hold: 1

M˜ 1 V˜ ≤ Φ˜ ≤ M˜ 2 V˜ , where

(9.53)

230

9 Polymer 3D-Printing via Screw Extrusion

M˜ 1 =

1 , max{3, 2γ 2 }

(9.54)

M˜ 2 =e2γ L max{3, 2γ 2 }.

(9.55)

Applying (9.49) to (9.53) with defining M˜ = M˜ 2 /M˜ 1 leads to the conclusion in Theorem 9.1. In addition, the estimated temperature can maintain not greater value than the true temperature in the plant, as stated in the following lemma. Lemma 9.2 If u(x, ˜ 0) ≥ 0, ∀x ∈ (0, s0 ), then u(x, ˜ t) ≥0, ∂ u˜ (s(t), t) ≤0, ∂x

∀x ∈ (0, s(t)),

∀t ≥ 0,

∀t ≥ 0.

(9.56) (9.57)

Proof Applying Maximum principle to z˜ -system governed by (9.43)–(9.45) leads to the statement that if z˜ (0, t) ≥ 0, ∀x ∈ (0, s0 ), then z˜ (x, t) ≥ 0, ∀x ∈ (0, s(t)), ∀t ≥ 0. By the relation between z˜ and u˜ given in (9.42), we prove Lemma 9.2, with the help of Hopf’s lemma. The properties in Lemma 9.2 are required to guarantee the positivity of the boundary heat input under the output feedback control design which is given in the later sections. Remark 9.4 The convergence speed of the designed observer is characterized by αs b2 hs + 4α + 4L 2 as seen in the estimate of the norm (9.37), which cannot be s chosen arbitrarily fast for given physical constants and the manufacturing speed. The performance improvement to fasten the observer’s convergence can be achieved by adding the measurement error injection to the observer PDE formulated by   ∂ 2 Tˆs ∂ Tˆs ∂ Tˆs (x, t) =αs 2 (x, t) − b (x, t) + hs Tb − Tˆs (x, t) ∂t ∂x ∂x + p(x, t)(Y2 (t) − Tˆs (0, t)), 0 < x < Y1 (t),

(9.58)

where the distributed observer gain p(x, t) can be designed using backstepping method as developed in Chap. 3. However, with the PDE observer (9.58), it is challenging to ensure the positivity of the output feedback control law. Since this chapter’s primary focus is on control design, we use the PDE observer given in (9.33)–(9.35).

9.4 Ink Production Control Based on Screw Speed

231

Control Design of Boundary Heat When the solid pellets are injected and heated into the extruder chamber, the amount of the molten polymer expands, reducing the quantity of solid material into the chamber. Thus a cooling effect arising from the continuous feeding of cooler pellets enables to maintain the interface at the desired setpoint. The setpoint open-loop boundary heat control qf (t) = qf∗ [see (9.11)] is not sufficient to drive the solid– liquid interface position to the desired setpoint. In this section, we develop the control design of the boundary heat at the inlet to drive the interface to the setpoint while stabilizing the temperature profile at the steady-state.

Reference Error System for Dynamics Reduced to a Single Phase First, we impose the following assumption on the liquid temperature. Assumption 9.1 The liquid temperature is at steady-state profile, i.e., Tl (x, t) = Tl,eq (x). Assumption 9.1 reasonably describes the case where the entire extruder chamber is filled with molten polymer at equilibrium temperature at the initial time. Thus, under the setup introduced later in Sect. 9.4, the L2 -norm of the reference error temperature v(x, t)L2 = Tl (x, t) − Tl,eq (x)L2 converges to zero according to a straightforward Lyapunov analysis. Under Assumption 9.1, the twophase dynamics governed by (9.1)–(9.7) are reduced to a single-phase model. Let (u(x, t), u(x, ˆ t), X(t)) be the reference error variables defined by u(x, t) = − ks (Ts (x, t) − Ts,eq (x)),

(9.59)

u(x, ˆ t) = − ks (Tˆs (x, t) − Ts,eq (x)),

(9.60)

X(t) =s(t) − sr .

(9.61)

Note that the negative signs are included in (9.59) and (9.60) to make the states (u, u) ˆ have positivity properties for the model validity conditions to hold. Then the estimation error state u˜ defined by (9.38) yields u(x, ˜ t) = u(x, ˆ t) − u(x, t).

(9.62)

We rewrite the original system (9.1)–(9.7) using the reference and estimation error states (u, ˆ X, u). ˜ Substituting x = s(t) into (9.60) with the help of (9.35), we get u(s(t), ˆ t) =ks (Ts,eq (s(t)) − Tm ).

(9.63)

In addition, rewriting (9.7) in term of u(x, ˆ t) with u(x, ˜ t) leads to the following equation of interface dynamics:

232

9 Polymer 3D-Printing via Screw Extrusion

˙ X(t) = − β¯



   ∂ u˜ ∂ uˆ   (s(t), t) − (s(t), t) + β¯ ks Ts,eq (s(t)) − kl Tl,eq (s(t)) , ∂x ∂x (9.64)

where β¯ = (ρs ΔH )−1 .

(9.65)

Taking a linearization of the right-hand side of (9.63) and (9.64) with respect to s(t) around the setpoint sr and by the steady-state solutions in (9.13), the dynamics of the reference error system is obtained by ∂ uˆ ∂ 2 uˆ ∂ uˆ (x, t) =αs 2 (x, t) − b (x, t) − hs u(x, ˆ t), ∂t ∂x ∂x ∂ uˆ (0, t) = − U (t) + γ u(0, ˜ t), ∂x u(s(t), ˆ t) =CX(t), ∂ u˜ ∂ uˆ ˙ X(t) =AX(t) − β¯ (s(t), t) + β¯ (s(t), t), ∂x ∂x

(9.66) (9.67) (9.68) (9.69)

where U (t) = − (qf (t) − qf∗ ), C =ks (p3 q3 + p4 q4 ) ,   A =β¯ ks (p3 q32 + p4 q42 ) − kl (p1 q12 + p2 q22 ) .

(9.70) (9.71) (9.72)

Backstepping Transformation A well-known design method of the output feedback control for PDEs is achieved by introducing the backstepping transformation which maps the observer PDE by using the gain kernel function derived for the full-state feedback control. Therefore, we consider the following transformation: w(x, ˆ t) =u(x, ˆ t) −

 β¯ s(t) φ(x − y)u(y, ˆ t)dy − φ(x − s(t))X(t), αs x

(9.73)

where φ is the gain kernel function derived in Sect. 4.1, which satisfies the following differential equation with the initial condition:   ¯ βb  ¯ αs φ (x)−(b + βC)φ (x) − A − C + hs φ(x) = 0, αs 

(9.74)

9.4 Ink Production Control Based on Screw Speed

φ(0) =0,

φ  (0) =

c , β¯

233

(9.75)

where c > 0 is a control gain. The solution to (9.74) with (9.75) is uniquely given by φ(x) =

  c e d1 x − e d2 x , ¯ 1 − d2 ) β(d

(9.76)

where d1 , d2 are defined by √ b¯ + D , d1 = 2αs ¯ b¯ =b + βC,

√ b¯ − D d2 = , 2αs   ¯ βb 2 ¯ D = b + 4αs A − C + hs . αs

(9.77) (9.78)

The full-state feedback control law is designed by Ufull (t) = − γ u(0, t) − where γ =

b 2αs ,

 β¯ s(t) f (x)u(x, t)dx − f (s(t))X(t), αs 0

(9.79)

and

f (x) =φ  (−x) − γ φ(−x),   c = (d1 − γ )e−d1 x − (d2 − γ )e−d2 x . ¯ 1 − d2 ) β(d

(9.80) (9.81)

The associated output feedback control law is generally designed by replacing the plant state in the full-state feedback control law with the observer state. Since X(t) in (9.79) can be directly measured and its observer state is not constructed, we keep the term X(t). Moreover, for the sake of proving the positivity of the designed control law later, we also hold the boundary value term u(0, t) in (9.79), which can also be directly measured. Hence, the resulting observer-based output feedback control law is designed by U (t) = − γ u(0, t) −

 β¯ s(t) f (x)u(x, ˆ t)dx − f (s(t))X(t). αs 0

(9.82)

Then, taking the derivatives of (9.73) in x and t along the solution of (9.66)–(9.69) with the gain kernel function (9.76), the transformed (w, ˆ X)-system (the so-called “target system”) is described by the following dynamics: ∂ wˆ ∂ 2 wˆ ∂ wˆ (x, t) =αs 2 (x, t) − b (x, t) − hs w(x, ˆ t) + s˙ (t)g(x − s(t))X(t) ∂t ∂x ∂x

234

9 Polymer 3D-Printing via Screw Extrusion

¯ − βφ(x − s(t))

∂ u˜ (s(t), t), 0 < x < s(t) ∂x

∂ wˆ (0, t) =γ w(0, ˆ t), ∂x

(9.83) (9.84)

w(s(t), ˆ t) =CX(t),

(9.85)

∂ u˜ ∂ wˆ ˙ (s(t), t) + β¯ (s(t), t), X(t) = (A−c)X(t) − β¯ ∂x ∂x

(9.86)

where g(x) = φ  (x) −

β¯ Cφ(x). αs

(9.87)

Rewriting the control law (9.82) with respect to the boundary heat control qf (t), the estimated temperature Tˆs , the reference steady-state Ts,eq , and the measured variables Y1 (t) and Y2 (t), the resulting output feedback control is described by qf (t) =qf∗ − γ ks (Y2 (t) − Ts,eq (0)) −

¯ s  Y1 (t) βk f (x)(Tˆs (x, t) − Ts,eq (x))dx αs 0

+ f (Y1 (t))(Y1 (t) − sr ).

(9.88)

Theoretical Analysis for a Specific Setup While the controller is designed through the backstepping method, the stability of the target system is not proven theoretically. Moreover, the condition of model validity needs to be satisfied under the control law. To achieve a theoretical result, in this section, we impose the following assumptions. Assumption 9.2 The initial condition of the estimated temperature profile is not higher than that of the true temperature profile, i.e., Tˆs (x, 0) ≤ Ts (x, 0), for all x ∈ (0, s0 ), where s0 := s(0). Assumption 9.3 The barrel temperature is set as melting temperature and the ∗ = 0. external heat input is zero, i.e., Tb = Tm , qm Under Assumption 9.2, it holds u(x, ˜ t) ≥ 0,

∂ u˜ (s(t), t) ≤ 0, ∂x

∀x ∈ (0, s(t)),

t ≥ 0,

(9.89)

as proven in Lemma 9.2. Under Assumption 9.3, the steady-state profiles (9.13), and steady-state input (9.22) becomes

9.4 Ink Production Control Based on Screw Speed

Tl,eq (x) = Tm ,

Ts,eq (x) = Tm ,

235

qf∗ = 0.

(9.90)

Also, it follows that C = 0,

A = 0.

(9.91)

In addition, the following setpoint restriction is given. Assumption 9.4 The setpoint is chosen to satisfy sr > s 0 +

¯ s  s0 f (x) βk (Tm − Tˆs (x, 0))dx. αs 0 f (s0 )

(9.92)

The physical meaning of Assumption 9.4 is that the user needs to choose the setpoint position sr sufficiently close to the outlet of the extruder as compared to the initial interface position s0 , depending on the initial temperature profile of the solid polymer granules. Such a choice of the setpoint position becomes more restrictive as the initial temperature profile in the solid polymer decreases. The main theorem is stated as follows. Theorem 9.2 Let Assumptions 9.1–9.4 hold. Then, the closed-loop system consisting of the plant (9.1)–(9.7), the measurements (9.32), the observer (9.33)–(9.35), and the control law (9.88) satisfies the conditions for model validity (9.8), (9.9), and is exponentially stable at the origin in the norm ˆ Φ(t) :=||Ts (x, t) − Ts,eq (x)||H1 + ||Ts (x, t) − Tˆs (x, t)||H1 + |s(t) − sr |, (9.93) namely, there exists a positive constant Mˆ > 0 such that −dt ˆ ˆ Φ(t) ≤ Mˆ Φ(0)e

holds, where d = min

0

αs 16sr

+

b2 4αs

(9.94)

1 + hs , c .

The proof of Theorem 9.2 is established by showing that (9.8) and (9.9) are satisfied and employing a Lyapunov analysis through the remaining of this section.

Model Validity Condition Let Z(t) be defined as Z(t) =U (t) + γ u(0, t)  β¯ s(t) f (x)u(x, ˆ t)dx − f (s(t))X(t). =− αs 0

(9.95)

236

9 Polymer 3D-Printing via Screw Extrusion

The following lemma is stated. Lemma 9.3 The following properties hold: Z(t) >0, u(x, t) >0,

∀t ≥ 0,

(9.96)

s˙ (t) > 0

s(0) 0. We prove (9.96) by contradiction approach. Assume that (9.96) is not valid, which implies ∃t ∗ > 0 such that Z(t) > 0,

∀t ∈ (0, t ∗ ),

Z(t ∗ ) = 0.

(9.102)

Similar to Lemma 9.2, by Maximum principle and Hopf’s lemma, we get u(x, t) > 0,

s˙ (t) > 0,

∀x ∈ (0, s(t)),

∀t ∈ (0, t ∗ ),

(9.103)

∀t ∈ (0, t ∗ ).

(9.104)

which, with the help of Lemma 9.2, leads to u(x, ˆ t) >0,

s(t) > 0,

∀x ∈ (0, s(t)),

Applying (9.102) and (9.104) to (9.95) with f (x) > 0 leads to X(t) < 0, for all t ∈ (0, t ∗ ). Therefore, applying this inequality and (9.103) to (9.101) leads to ˙ Z(t) > −cZ(t),

∀t ∈ (0, t ∗ ).

(9.105)

Applying Gronwall’s inequality to (9.105) leads to Z(t) ≥ Z(0)e−ct , ∀t ∈ ∗ (0, t ∗ ]. Thus, we have Z(t ∗ ) ≥ Z(0)e−ct > 0, which contradicts with the assumption (9.102). Hence, (9.96) is proved. Then, by Maximum principle, (9.97) holds. Imposing (9.96) and (9.97) on (9.95), we obtain X(t) < 0 which leads to (9.98).

9.4 Ink Production Control Based on Screw Speed

237

Stability Analysis Taking into account A = C = 0, we study the stability of the target (w, X)-system governed by (9.83)–(9.86). Let zˆ be a variable defined by zˆ (x, t) = w(x, ˆ t)e−γ x .

(9.106)

z˜ := ue ˜ −γ x ,

(9.107)

Applying (9.106) and

in (9.42), the target (w, X)-system in (9.83)–(9.86) leads to the following (ˆz, X)system: ∂ zˆ ∂ 2 zˆ (x, t) =αs 2 (x, t) − λˆz(x, t) + s˙ (t)g(x − s(t))X(t)e−γ x ∂t ∂x ∂ z˜ ¯ − βφ(x − s(t)) (s(t), t), 0 < x < s(t) ∂x ∂ zˆ (0, t) =0, ∂x

(9.108)

zˆ (s(t), t) =0,

(9.110)

˙ ¯ γ s(t) X(t) = −cX(t) − βe





∂ zˆ ∂ z˜ (s(t), t) − (s(t), t) , ∂x ∂x

(9.109)

(9.111)

where λ := hs +

b2 . 4αs

(9.112)

Consider the following functional 1 Vˆ = 2



s(t)

zˆ (x, t)2 dx +

0

1 2

 0

s(t)  ∂ zˆ

2 p (x, t) dx + X(t)2 , ∂x 2

(9.113)

where p > 0 is to be determined. Taking the time derivative of (9.113) along with the solution of (9.108)–(9.111), and applying Young’s, Cauchy-Schwarz, and Agmon’s inequalities, we get //  // αs 4pβ¯ 2 sr e2γ sr //// ∂ 2 zˆ ////2 − // ∂x 2 // 2 c // //2 // ∂ zˆ // pc X(t)2 − (αs + λ)//// //// − λ||ˆz||2 − ∂x 2

V˙ˆ ≤ −



238

9 Polymer 3D-Printing via Screw Extrusion

    1 pβ¯ 2 e2γ sr ∂ z˜ 1 2 2 ¯ + (s(t), t)2 + β ||φ|| + 2hs αs c ∂x  // //2  // ∂ zˆ // s˙ (t) 2 2 2 + + ||g|| ||ˆz|| + //// //// , (1 + g)X(t) ¯ 2 ∂x

(9.114)

where g¯ :=

max (g(0)2 + g(−s(t))2 + ||g  ||2 ).

s(t)∈(0,sr )

(9.115)

Choosing p=

cαs e−2γ sr , 16β¯ 2 sr

(9.116)

the inequality (9.114) is led to // // // //2 // ∂ zˆ // αs //// ∂ 2 zˆ ////2 pc ˙ ˆ V ≤ − // 2 // − (αs + λ)//// //// − λ||ˆz||2 − X(t)2 4 ∂x ∂x 2 // 2 //2 // ∂ z˜ // + a s˙ (t)Vˆ + M1 //// 2 //// , ∂x

(9.117)

where 

 (1 + g) ¯ , ||g||2 , 1 , p     1 αs 1 2 2 ¯ + . + M1 =4sr β ||φ|| 2hs αs 16sr a = max

(9.118) (9.119)

Thus, using the Lyapunov function V˜ in (9.46) for the estimation error z˜ -system (9.43)–(9.45), we define the Lyapunov function for the total (ˆz, X, z˜ )-system as V = Vˆ +

2M1 ˜ V. αs

(9.120)

Combining the inequalities (9.47) and (9.117) leads to V˙ ≤ − dV + a s˙ (t)V ,

(9.121)

where 

 αs d = min + λ, c . 16sr

(9.122)

9.5 Simulation Results

239

Following the procedure in Chap. 2, the inequality (9.121) with (9.97) and (9.98) leads to the exponential norm estimate V (t) ≤ ea(s(t)−s0 ) V (0)e−dt ≤ easr V (0)e−dt .

(9.123)

Let Ψ (t) = ||w||2H1 + X(t)2 .

(9.124)

¯ , MV ≤ Ψ (t) ≤ MV

(9.125)

Then, we have

where 1 M¯ =2 max{e2γ sr (1 + γ 2 ), }, p  p −1 M = max{2(1 + γ 2 ), } . 2

(9.126) (9.127)

Therefore, Ψ (t) ≤

M¯ asr e Ψ (0)e−dt , M

(9.128)

which proves the exponential stability of the target w-system in H1 -norm. Since the u-system in (9.66)–(9.69) and the target w-system in (9.83)–(9.86) have equivalent stability property due to the invertibility of the backstepping transformation (9.73), the exponential estimate in H1 -norm is also guaranteed for the u-system, which concludes the proof of Theorem 9.2.

9.5 Simulation Results Setup and Method In order to numerically investigate the controller’s performance in different operating conditions, we have employed the simulation of the original “two-phase” model governed by (9.1)–(9.7) without assuming that the liquid phase is at steady-state, run the PDE observer given in (9.33)–(9.35), and implemented the associated output feedback controller (9.88). We used the boundary immobilization method to obtain a fixed boundary system and discretized the system with finite differences to construct a finite-dimensional representation of the model and the estimate.

240 Table 9.1 HDPE parameters obtained by [197]

9 Polymer 3D-Printing via Screw Extrusion Melting point Specific heat solid Specific heat melt Therm. conduct. solid Therm. conduct. melt Solid density Melt density Heat of fusion

Tm cs cl ks kl ρs ρl ΔH

135 ◦ C 1895 Jkg−1 K−1 2640 Jkg−1 K−1 0.373 Wm−1 K−1 0.324 Wm−1 K−1 955 kgm−3 780 kgm−3 39, 000 Jkg−1

Using Matlab’s ode23s solver, we simulated the setup with three different advection speeds b ranging from 2 mm/s to 50 mm/s, to cover a wide spectrum of operating modes. The material parameters are chosen from [197], in which distinct values for high-density polyethylene in solid and liquid state were experimentally derived (see Table 9.1). The extruder length is given by a physical device, and here we used L =10 cm. The initial conditions of the true temperature profile and the estimated temperature profiles are set as linear profiles with the boundary temperature T and Tˆ , namely, Ts (x, 0) = (T − Tm )(1 − x/s0 ) + Tm , Tˆs (x, 0) = (Tˆ − Tm )(1 − x/s0 ) + Tm . In the simulation, we set T = 125 ◦ C and T = 105 ◦ C to satisfy Assumption 9.2. The free parameters are the constant barrel temperature Tb , the auxiliary heat ∗ at the outlet, and the control gain c. The barrel temperature and the input qm auxiliary heat input are chosen so that they neutralize the cooling effect of the initial temperature in the solid phase within a range close to the conditions imposed in Sect. 9.4, namely, the barrel temperature is chosen close to the melting temperature Tb and the auxiliary heat input is chosen as a positive value close to zero. Since the initial profile of the estimated temperature is slightly below the melting temperature, the cooling effect of the initial temperature is limited, and thereby we set the free ∗ = 100 W/m2 . parameters sufficiently close to the specific setup, Tb = 145 ◦ C and qm

Gain Tuning For a given advection speed, the control gain c is adjusted so that the following two properties are observed: • The convergence of the interface position to the setpoint position is achieved sufficiently fast. • The temperature in the solid phase maintains a reasonable value during the process. As the control gain gets larger, the convergence becomes faster; however, the temperature in the solid phase can reach an unreasonably low value due to the large amount of the cooling input. Hence, we aim to choose a suitable value of the control gain c. First, we test three simulations under the small advection speed b =2 mm/s

9.5 Simulation Results

241

1 2 3 4 5 0

2

4

6

8

10

(a) 140 120 100 80 60 40 20

0

0.5

1

1.5

2

(b) Fig. 9.2 The closed-loop responses under the control gains c = 0.05/s (dash), c = 0.2/s (solid), and c = 0.4/s (dotted). The convergence of the interface position is sufficiently fast and the boundary temperature remains a reasonable range for c = 0.2/s (solid). (a) The time evolution of the interface position. (b) The time evolution of the boundary temperature

by setting the control gain as c =0.05/s, c =0.2/s, and c =0.4/s, respectively. The closed-loop responses of the interface position s(t) and the boundary temperature Ts (0, t) are depicted in Fig. 9.2a, b, respectively. From Fig. 9.2a, we can observe that the convergence of the interface position with c =0.05/s takes approximately 10 min, while those with c =0.2/s and c =0.4/s take 6 min. Additionally, from Fig. 9.2b, the boundary temperature with c =0.4/s reaches a value around 20 ◦ C, which is a relatively low temperature, while the boundary temperatures with c =0.05/s and c =0.2/s remain the reasonable range 80–135 ◦ C. Therefore, the control gain c =0.2/s is a suitable value which satisfies the two desired properties.

242

9 Polymer 3D-Printing via Screw Extrusion

Simulation Results Following the gain tuning, the simulation results for higher advection speed b =10 mm/s and b =50 mm/s are performed, and the control gain is adjusted as c =1.0/s and c =5.0/s, respectively. For these advection speeds and the adjusted control gains, the closed-loop responses of the interface position s(t), the boundary control input qf (t), and the boundary temperature Ts (0, t) are shown in Fig. 9.3a– c, respectively. The interface responses, depicted in Fig. 9.3a, have quite similar behaviors in all three setups. However, the control input, shown in Fig. 9.3b, appears to act faster for higher advection speeds but exhibits similar qualitative behavior. Similar properties were observed in the boundary temperature response in Fig. 9.3c. Note that all three figures have different time ranges. Moreover, for the fast operating condition b =50 mm/s, the comparison of the estimated temperature profile and the true temperature profile at t =0 s, 0.2 s, 0.4 s are shown in Fig. 9.4a–c, respectively. We can observe that the estimated temperature profile gets almost the same as the true temperature profile at 0.4 s, associated with the expansion of the solid granules’ region. Hence, the convergence of the designed observer to the true temperature profile is approximately 1000 times faster than the convergence of the interface position to the setpoint position, which is a sufficiently quick performance of the temperature estimation.

Comparison with PI Control For comparison, we also tested a closed-loop setup with PI control given by qf (t) =

qf∗

 + KP (s(t) − sr ) + KI

t

(s(τ ) − sr )dτ,

(9.129)

t0

where KP and KI are gain parameters to be tuned in order to achieve the desired performance. However, for any choice of the parameters we have tried, the closedloop response of the interface position does not stabilize at the setpoint sr . Figure 9.5 depicts the responses under PI control with a relatively suitable choice of the gains. The plot in Fig. 9.5b shows that the temperature at the inlet of the extruder gets above the melting temperature at approximately 2.9 min, which violates the validity condition (9.9) of the solid polymer temperature, while our proposed output feedback control guarantees to satisfy the condition under the closed-loop system. Such an overshoot behavior beyond the melting temperature might be reduced by PID control; however, the velocity of the interface position is nearly impossible to measure online, and the differentiator generally causes high noise. Overall, the proposed output feedback control law illustrates superior performance to PI control in terms of both convergence to the setpoint and the validity condition.

9.5 Simulation Results

243

1 2 3 4 5 0

2

4

6

8

10

(a)

0

105

-5

-10

-15

0

0.5

1

1.5

2

(b) 140 120 100 80 60

0

10

20

30

40

50

60

(c) Fig. 9.3 The closed-loop responses under the proposed output feedback control law for each operating speed. (a) For each operating speed, the interface position is stabilized after 6 min. (b) The transient of the control input gets shorter as the operation gets faster. (c) The boundary temperature maintains reasonable value for the material and safe operation

244

9 Polymer 3D-Printing via Screw Extrusion

140

Temperature profiles at 0 [sec]

130 120 110 100 0

140

0.5

1

1.5

2

Temperature profiles at 0.2 [sec]

120 100 80 0

140

0.5

1

1.5

2

Temperature profiles at 0.4 [sec]

130 120 110 100 90 0

0.5

1

1.5

2

Fig. 9.4 The comparison of the true and estimated temperature profiles at t = 0 s, t = 0.2 s, and t = 0.4 s

From the simulations, we conclude that our control design achieves a stable interface position, even with very fast advection speeds 50 mm/s, with which a particle inserted in the inlet will travel in two seconds through the extruder, when assuming a 10 cm extruder.

9.6 Comments and Remarks In this chapter, we designed an observer and the associated output feedback control to stabilize a filament production process of the screw extrusion-based polymer 3Dprinting [119]. The steady-state analysis is provided by setting the setpoint as a given value, and the control design to stabilize the interface position is derived.

9.6 Comments and Remarks

245

0

2

4

6

8 0

1

2

3

4

5

6

4

5

6

(a) 160 150 140 130 120 0

1

2

3

(b) Fig. 9.5 The closed-loop response under PI control. The performance is bad due to the violation of the physical condition. (a) The interface position causes a huge overshoot and is not stabilized. (b) The boundary temperature is heated up after 1 min and gets above the melting temperature after 2.9 min, which violates the condition for the solid phase temperature

The simulation results illustrate the effectiveness of the boundary feedback control law for some given screw speeds. While the theoretical analysis in this chapter is established under the assumptions on the liquid phase temperature maintaining the steady-state temperature and on the heat flux at the outlet being zero, it is of interest to further develop the control law for the two-phase system following Chap. 5, and to guarantee the robustness of the control with respect to the nonzero heat flux at the outlet following Chap. 4, which could relax the two aforementioned assumptions.

Chapter 10

Metal 3D-Printing via Selective Laser Sintering

10.1 Selective Laser Sintering Metal Additive Manufacturing (AM) is a state-of-the-art manufacturing technology which has emerged rapidly in the last decade as observed from the growth in its global market. AM’s impact relies on products and supply chains in numerous industries such as automobiles, consumer electronics, aerospace, and medical devices [47]. While industrial AM systems for polymer materials can produce reasonable quality for customers, AM for metallic materials still has room for quality improvement. Selective Laser Sintering (SLS) is the most common technique of the powder-bed fusion AM processes that fabricate structurally sound three-dimensional products from computer-aided design (CAD) models [2]. Using a high-power laser, a thin layer of the metal powder at the surface of the bed is fused to produce a desired geometry. A melt pool created by the laser solidifies to a solid metal component. Such a layer-by-layer process to fabricate the entire object enables a relatively fast process speed together with complex geometry. As the phase transformation of the metal powder occurs in a short time scale while operating a fast scanning speed of the laser, SLS yields an inhomogeneous temperature field which leads to a complex computational prediction of the geometry of the melt pool (see for example [120]). The large thermal gradient inside the metal can lead to brittle parts, and thus the temporal evolution of the temperature field has a significant role to guarantee the quality of the fabrication. Using a thermodynamic model for phase transformations, several research articles have studied the evolution of the melt pool in SLS by means of the Stefan problem [82], and employed Finite Element (FE) methods to obtain computational models [4, 43, 50, 169]. As in the applications in the previous chapters, the Stefan problem is governed by a parabolic PDE for the temperature field, defined on the time-varying spatial domain of the melting front, whose evolution is given by the Neumann © Springer Nature Switzerland AG 2020 S. Koga, M. Krstic, Materials Phase Change PDE Control & Estimation, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-58490-0_10

247

248

10 Metal 3D-Printing via Selective Laser Sintering

boundary value of the PDE state at the front position. A comprehensive review of the thermal modeling of melt pool dynamics in SLS can be found in [228]. Process control for SLS has been developed to guarantee sufficient mechanical properties of the fabricated three-dimensional object. For instance, in [231] a control system to eliminate thermal gradients in the post-sintering temperature is designed using an IR camera as a sensor and laser power density as an actuator. Repetitive control methods for SLS-based AM have been developed in [210, 216] by utilizing three-dimensional finite element simulation for the melt-pool evolution. As a powder deposition process, [37] proposed a control design for laser power and scanning speed to drive the solid–liquid interface position in the melt pool to some predetermined setpoint geometry using an adjoint-based optimization for the Stefan problem developed in [88].

10.2 Physical Model SLS is a common AM technique as a layer-by-layer process to fabricate a 3D-object through repetitive melting and solidification. At each layer, first the solid object under the process of fabrication is covered by a thin granular metal powder layer at the surface. Next, a laser beam is injected through reflection by scanner mirrors to heat up and fuse the metal powder at selective areas of the surface. A local melt pool is developed by the laser power in the metal powder. Through a phase change, the domain of the melt pool is limited by the position of the melting front, which is varying in time. This configuration is depicted in Fig. 10.1. The physical modeling of the melt pool dynamics has been developed in the literature by means of the

Fig. 10.1 Schematic of the powder-bed metal AM via Selective Laser Sintering (SLS). The melt pool is generated due to the emission of the laser energy

10.2 Physical Model

249

Stefan problem. Following the work in [4] for a one-dimensional approximation of the Stefan problem in the vertical direction, and incorporating the in-domassin effect of the laser power to the temperature dynamics, we consider the following governing equations: ∂T ∂ 2T (x, t) =α 2 (x, t) + g(x)qc (t), ∂t ∂x ∂T (0, t) =qc (t), −k ∂x

x ∈ (0, s(t)),

(10.2)

T (s(t), t) =Tm , s˙ (t) = − β

(10.1)

(10.3) ∂T (s(t), t), ∂x

(10.4)

where T (x, t) denotes the temperature profile in the melt pool along the vertical coordinate x ∈ (0, s(t)), α :=

k ρcp

(10.5)

is the diffusion coefficient, k is the thermal conductivity, ρ is the density, cp is the specific heat capacity, Tm is the constant melting temperature, β :=

k ρHf

(10.6)

with the latent heat of fusion Hf , and qc (t) is the controlled laser power. By Beer’s law for optical penetration of the energy, in [169] the spatially varying function g(x) is given by g(x) =

 x 1 , exp − ρcp δ δ

(10.7)

where δ is called optical penetration rate. Here, we consider a broader class of the spatial function g(x) satisfying the following assumption. Assumption 10.1 The spatially varying function g(x) is positive, i.e., g(x) ≥ 0, ∀x ≥ 0. The condition to validate the physical model (10.1)–(10.4) is given in the following remark. Remark 10.1 The model (10.1)–(10.4) is physically valid if and only if T (x, t) ≥ Tm ,

∀x ∈ (0, s(t)),

∀t ≥ 0.

(10.8)

250

10 Metal 3D-Printing via Selective Laser Sintering

Based on the above condition, we impose the following assumption on the initial data. Assumption 10.2 s0 := s(0) > 0, T0 (x) := T (x, 0) ≥ Tm for all x ∈ [0, s0 ], and T0 (x) is continuously differentiable in x ∈ [0, s0 ]. A sufficient condition to guarantee (10.8) is given by the following lemma. Lemma 10.1 If qc (t) > 0 for all t ∈ [0, t1 ) for some t1 > 0, then T (x, t) > Tm for all x ∈ (0, s(t)) and for all t ∈ [0, t1 ). Moreover, if qc (t) > 0 for all t ≥ 0, then T (x, t) > Tm for all x ∈ (0, s(t)) and for all t ≥ 0. Lemma 10.1 is proven by applying the maximum principle and Hopf’s lemma for parabolic PDEs as shown in [70, p. 26, Corollary 2]. Therefore, the condition qc (t) > 0 for all t ≥ 0 stands as a constraint of the laser power input, which needs to be ensured after the feedback control law is designed.

10.3 State Feedback Control Problem Statement and Main Result The steady-state solution (Teq (x), seq ) of the system (10.1)–(10.4) with zero laser power qc (t) = 0 yields a uniform melting temperature Teq (x) = Tm and a constant interface position given by the initial data. As proposed in [37], driving the depth of the melt pool to the setpoint is desired in AM, and thus we design qc (t) > 0 such that the interface position s(t) converges to the setpoint sr . A restriction on the choice of the setpoint is given in the following assumption. Assumption 10.3 Given the initial conditions T0 (x) and s0 , the setpoint sr is chosen to satisfy the following inequality: sr > s0 +

β α



s0

(T0 (x) − Tm )dx.

(10.9)

0

The necessity of Assumption 10.3 can be derived by considering the energy conservation law described by d dt



k α

 0

s(t)

    k s(t) k (T (x, t) − Tm )dx + s(t) = 1 + g(x)dx qc (t). β α 0 (10.10)

Imposing the constraint qc (t) > 0 for all t ≥ 0 and taking the time integration of (10.10) from the initial time to infinity, the condition given in Assumption (10.3) is obtained. Under these assumptions, we design the control law and state our main result as follows.

10.3 State Feedback Control

251

Theorem 10.1 Under Assumptions 10.1–10.3, the closed-loop system consisting of the plant (10.1)–(10.4) and the control law  qc (t) = −c

k α



s(t) 0

 k (T (x, t) − Tm )dx + (s(t) − sr ) , β

(10.11)

where c > 0 is a control gain, satisfies the model validity condition (10.8), and there exists a positive constant M > 0 such that the norm Φ(t) := ||T (x, t) − Tm ||2H1 + (s(t) − sr )2

(10.12)

satisfies the following exponential decay: Φ(t) ≤ MΦ(0)e−bt ,

(10.13)

0 1 where b = min 4sα2 , c , namely, the origin of the closed-loop system is exponenr tially stable in the spatial H1 norm. Remark 10.2 The control law (10.11) is equivalent to the design developed in Sect. 2.3 for the system without in-domain effect, as we can see that (10.11) is not dependent on the spatially varying function g(x). Hence, the stability of the closed-loop system is robust with respect to the uncertainty of g(x) as far as Assumption 10.1 holds. The proof of Theorem 10.1 is established through the remainder of this section by following the steps in Chap. 2.

Reference Error and Target System We define the reference error states as follows: u(x, t) = T (x, t) − Tm ,

X(t) = s(t) − sr .

(10.14)

Using these variables, the original system (10.1)–(10.4) is led to the following reference error system: ut (x, t) =αuxx (x, t) + g(x)qc (t), −kux (0, t) =qc (t), u(s(t), t) =0, ˙ X(t) = − βux (s(t), t).

x ∈ (0, s(t)),

(10.15) (10.16) (10.17) (10.18)

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10 Metal 3D-Printing via Selective Laser Sintering

Following the procedure in Chap. 2, we introduce the following backstepping transformation w(x, t) =u(x, t) −

c α



s(t)

(x − y)u(y, t)dy +

x

c (s(t) − x)X(t). β

(10.19)

Taking the time and spatial derivatives of (10.19) along the solution of (10.15)– (10.18) yields the following target system wt (x, t) =αwxx (x, t) +

c s˙ (t)X(t) + g(x, ¯ s(t))qc (t), β

(10.20)

w(s(t), t) =0,

(10.21)

wx (0, t) =0,

(10.22)

˙ X(t) = − cX(t) − βwx (s(t), t),

(10.23)

where c g(x, ¯ s(t)) :=g(x) − α



s(t)

(x − y)g(y)dy.

(10.24)

x

The boundary condition (10.16) leads to the control design  qc (t) = −ck

1 α



s(t)

0

 1 u(x, t)dx + X(t) , β

(10.25)

which is equivalent to (10.11).

Model Validity Conditions To guarantee the condition (10.8) for the model validity to hold under the closedloop system, we provide the following lemma. Lemma 10.2 The closed-loop system of (10.15)–(10.18) under the control law (10.25) satisfies the following properties: qc (t) >0, u(x, t) >0,

∀t ≥ 0, ∀x ∈ (0, s(t)),

(10.26) ∀t ≥ 0,

(10.27)

s˙ (t) >0,

∀t ≥ 0,

(10.28)

s0 < s(t) 0.

(10.29)

10.3 State Feedback Control

253

Proof First, we use contradiction approach to prove (10.26), namely, we assume that there exists a finite time t ∗ > 0 such that qc (t) > 0, ∀t ∈ [0, t ∗ ) and qc (t ∗ ) = 0. Then, by Lemma 10.1, we have u(x, t) > 0, ∀x ∈ (0, s(t)), ∀t ∈ [0, t ∗ ), and s˙ (t) > 0, ∀t ∈ [0, t ∗ ). Then, by the control law (10.25), we deduce 0 < s0 < s(t) < sr , ∀t ∈ (0, t ∗ ). Taking the time derivative of the control law (10.25) leads to 

k q˙c (t) = − c 1 + α





s(t)

g(x)dx qc (t).

(10.30)

0

Applying s(t) < sr ∀t ∈ (0, t ∗ ) with the help of g(x) ≥ 0 yields the following inequality:    k sr q(t) ˙ > −c 1 + g(x)dx qc (t), α 0

∀t ∈ (0, t ∗ ).

(10.31)

Applying comparison principle to (10.31), we get the following inequality: 

qc (t) > qc (0)e

  s −c 1+ αk 0 r g(x)dx t

,

∀t ∈ (0, t ∗ ).

(10.32)

s     Hence, qc (t ∗ ) ≥ qc (0) exp −c 1 + αk 0 r g(x)dx t ∗ . However, Assumption 10.3 for setpoint restriction ensures qc (0) > 0, and hence qc (t ∗ ) > 0, which contradicts with the imposed assumption qc (t ∗ ) = 0. Thus, the positivity (10.26) is proven. Moreover, the properties (10.27)–(10.29) are shown by applying Lemma 10.1 for infinite time domain, and extending the manner we presented at the beginning of this proof for finite time domain to the infinite time domain. Moreover, we have the following lemma. Lemma 10.3 The control law (10.25) under the closed-loop system satisfies the following inequalities: 

qc (0)e

  s −c 1+ αk 0 r g(x)dx t

< qc (t) < qc (0)e−ct ,

∀t ≥ 0.

(10.33)

Applying the conditions (10.26) and (10.29) to (10.30) with the use of comparison principle directly leads to (10.33).

Stability Proof In this section, we prove the exponential stability of the closed-loop system by Lyapunov method with the help of the properties shown in Lemmas 10.2 and 10.3. First, we prove the stability of the target (w, X)-system given in (10.20)–(10.23). Note that for this target system, Poincare’s and Agmon’s inequalities are given by

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10 Metal 3D-Printing via Selective Laser Sintering

||w||2 ≤4sr2 ||wx ||2 ,

||wx ||2 ≤ 4sr2 ||wxx ||2 ,

wx (s(t), t)2 ≤4sr ||wxx ||2 .

(10.34) (10.35)

Let V be the Lyapunov function defined by V (t) =

1 1 p ||w||2 + ||wx ||2 + X(t)2 . 2 2 2sr2

(10.36)

Taking the time derivative of (10.36) along the solution of (10.20)–(10.23) leads to α c V˙ (t) = − 2 ||wx ||2 + 2 s˙ (t)X(t) sr βsr − α||wxx ||2 − 

s(t)

−



s(t)

w(x, t)dx +

0

1 sr2



s(t)

g(x, ¯ s(t))w(x, t)dxqc (t)

0

c 1 s˙ (t)X(t)wx (s(t), t) − s˙ (t)wx (s(t), t)2 β 2

g(x, ¯ s(t))wxx (x, t)dxqc (t) − pcX(t)2 − pβX(t)wx (s(t), t).

(10.37)

0

Applying the Cauchy-Schwarz, Young, and Poincare inequalities to the term on the second line in (10.37), for a positive constant γ1 > 0, we get 

s(t) 0

g(x, ¯ s(t))w(x, t)dxqc (t) ≤ 2γ1 sr2 ||wx ||2 +

1 ||g|| ¯ 2 qc (t)2 . 2γ1

(10.38)

Applying Young and Cauchy-Schwarz inequalities to the term on the fifth line in (10.37), for a positive constant γ2 > 0, we get 

s(t)

−

g(x, ¯ s(t))wxx (x, t)dxqc (t) ≤

0

||g|| ¯ 2 γ2 ||wxx ||2 + qc (t)2 . 2γ2 2

(10.39)

Applying Young’s and Agmon’s inequalities, we get −pβX(t)wx (s(t), t) ≤

pc 2pβ 2 sr X(t)2 + ||wxx ||2 . 2 c

(10.40)

Therefore, by applying (10.38)–(10.40) to (10.37) with setting γ1 =

α , 4sr2

γ2 =

2||g|| ¯ 2 , α

p=

cα , 4β 2 sr

we arrive at α V˙ (t) ≤ − 2 8sr



 pc 1 2 ||wx || + 2 ||w|| − X(t)2 2 sr 2

(10.41)

10.3 State Feedback Control

255

  1 3 8sr c p 2 2 2 2 ¯ qc (t) + s˙ (t) X(t) + ||g|| ||w|| + α α 2 2sr3 ≤ − bV + a s˙ (t)V +

3 ||g|| ¯ 2 qc (t)2 , α

(10.42)

where  1 8sr c , , a = max sr α 



 α b = min ,c . 4sr2

(10.43)

Let W be the functional defined by W (t) = V (t)e−as(t) .

(10.44)

With the use of (10.42), the time derivative of (10.44) is shown to satisfy 3 ¯ 2 qc (t)2 e−as(t) . W˙ (t) ≤ − bW (t) + ||g|| α

(10.45)

Recalling the definition of g¯ in (10.24) and applying Young’s and Cauchy Schwarz inequalities, the spatial L2 norm is bounded by ⎛



s(t)

||g|| ¯ 2 ≤2 0



⎝g(x)2 +

c α



s(t)

2 ⎞ (x − y)g(y)dy ⎠ dx

x

2  2  s(t)  s(t) c (x − y)g(y)dy dx ≤2||g||2 + 2 2 α 0 x   c2 sr4 ||g||2L2 (0,sr ) , ≤2 1 + 12α 2

(10.46)

which is time-independent. By Lemma 10.3, the square of the controller is bounded by qc (t)2 ≤qc (0)2 e−2ct .

(10.47)

Applying (10.46) and (10.47) to (10.45), we get W˙ (t) ≤ − bW (t) + Nqc (0)2 e−2ct ,

(10.48)

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10 Metal 3D-Printing via Selective Laser Sintering

where N is a positive constant defined by 6 N= α

  c2 sr4 ||g||2L2 (0,sr ) . 1+ 12α 2

(10.49)

Applying comparison principle to (10.48) leads to W (t) ≤W0 e−bt + Nqc (0)2 e−bt



t

e−(2c−b)τ dτ.

(10.50)

0

0 1 Noting b = min 4sα2 , c given in (10.43), it is easy to see that 2c > b. Thus, r through the calculation of the integration, the inequality (10.50) is led to   Nqc (0)2 −bt e . W ≤ W0 + 2c − b

(10.51)

By V = W eas(t) , (10.51) leads to the inequality of V as   Nqc (0)2 −bt e , V ≤eas(t) W0 + 2c − b   Nqc (0)2 −bt e . ≤easr V0 + 2c − b

(10.52)

By the invertibility of the transformation, there exist positive constants M¯ > 0 and M > 0 such that for the norm of the original (u, X)-system and the norm of the target (w, X)-system, it holds ¯ MΦ(t) ≤ V (t) ≤ MΦ(t),

(10.53)

Φ(t) := ||u||2H1 + X(t)2 .

(10.54)

where

   s(t) Moreover, since the control law is qc (t) = −ck α1 0 u(x, t)dx + β1 X(t) , taking the square of the control law and applying Young’s and Cauchy-Schwarz inequalities lead to the following inequality: qc (0)2 ≤ LΦ(0), where L = 2c2 k 2 max{ αsr2 , β12 }. Combining these, we arrive at

(10.55)

10.4 Observer and Output Feedback Control Design

easr Φ(t) ≤ M

257

  NL ¯ Φ0 e−bt , M+ 2c − b

(10.56)

from which the origin of the closed-loop system of (u, X) is shown to be exponentially stable.

10.4 Observer and Output Feedback Control Design Here, we construct a state observer to estimate the temperature profile with measuring only the position of the melting front s(t), and design an output-feedback control law for the actuated laser power by utilizing the estimated temperature profile. Therefore, the measured output y(t) is given by y(t) = s(t).

(10.57)

Throughout this section, we assume that the output (10.57) does not include a disturbance for the purpose of proving the stability of the closed-loop system. In a later section, we perform a numerical simulation with a disturbance included in the measurement (10.57).

Observer for the Temperature Profile Let u(x, ˆ t) be the estimate of the reference error of the temperature u(x, t) = T (x, t) − Tm . We design the PDE observer for uˆ as a copy of the plant (10.15)– (10.17) with utilizing the measurement y(t) as the domain of the PDE estimator as follows: uˆ t (x, t) =α uˆ xx (x, t) + g(x)q ˆ c (t),

x ∈ (0, y(t)),

−k uˆ x (0, t) =qc (t), u(y(t), ˆ t) =0,

(10.58) (10.59) (10.60)

where g(x) ˆ is the guess of the spatial function g(x) in the plant, which is essentially uncertain. Let u˜ be the estimation error state defined by u˜ := u − u. ˆ Then, the dynamics of ODE (10.18) can be rewritten with respect to uˆ and u˜ as follows: ˙ X(t) = − β uˆ x (s(t), t) − β u˜ x (s(t), t).

(10.61)

Subtracting the observer PDE (10.58)–(10.60) from the plant PDE (10.15)–(10.17), we get the dynamics of the estimation error as follows:

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10 Metal 3D-Printing via Selective Laser Sintering

u˜ t (x, t) =α u˜ xx (x, t) + g(x)q ˜ c (t),

x ∈ (0, s(t)),

(10.62)

u˜ x (0, t) =0,

(10.63)

u(s(t), ˜ t) =0,

(10.64)

where g(x) ˜ := g(x) − g(x). ˆ

Output Feedback Control Design and Stability Proof The output feedback control law is designed by replacing the plant state u in the full-state feedback control law (10.25) with the observer state u, ˆ resulting in the following description:  qc (t) = −c

k α



y(t) 0

 k u(x, ˆ t)dx + (y(t) − sr ) . β

(10.65)

To prove the stability of the closed-loop system, we require the following assumptions: Assumption 10.4 u(x, ˆ 0) ≥ u(x, 0) ≥ 0 for all x ∈ (0, s0 ). Assumption 10.5 g(x) ˆ ≥ g(x) ≥ 0 for all x ∈ (0, sr ). Assumption 10.6 The setpoint sr is chosen to satisfy sr > y(0) +

β α



y(0)

u(x, ˆ 0)dx.

(10.66)

0

Remark 10.3 g(x) ˆ can be chosen so that Assumption 10.5 holds. In laser sintering, x g(x) is given by g(x) = ρc1p δ e− δ , and the penetration rate coefficient δ is highly uncertain. However, it is possible to know the upper bound δ and lower bound δ, i.e., 0 < δ ≤ δ ≤ δ < ∞. Thus, a conservative choice of g(x) ˆ to satisfy Assumption 10.5 is g(x) ˆ =

  x 1 . exp − ρcp δ δ

(10.67)

The theorem for the observer-based output feedback control is given below. Theorem 10.2 Under Assumptions 10.4–10.6, consider the closed-loop system consisting of the plant (10.15)–(10.18), the measurement (10.57), the observer (10.58)–(10.60), and the output feedback control (10.65). Then, for any g(x) ˆ satisfying

10.4 Observer and Output Feedback Control Design



sr

2 (g(x) − g(x)) ˆ dx ≤

0

259

α , 80k 2 sr

(10.68)

the closed-loop system satisfies the model validity condition (10.8), and there exists a positive constant M > 0 such that the norm ˜ 2H1 Φ(t) := ||u||2H1 + (s(t) − sr )2 + ||u||

(10.69)

satisfies the following exponential decay: Φ(t) ≤ MΦ(0)e−bt ,

(10.70)

0 1 where b = 14 min 4sα2 , c , namely, the origin of the closed-loop system is r exponentially stable in the spatial H1 norm. The proof of Theorem 10.2 is provided in the remainder of this section.

Model Validity Conditions First, we prove the following lemma that is analogous to Lemma 10.2. Lemma 10.4 The closed-loop system satisfies the following properties: qc (t) >0, u(x, ˜ t) 0, u(x, t) >0, s0 < s(t) 0.

(10.74) (10.75)

Proof We use the contradiction approach. Assume that there exists a finite time t1 > 0 such that (10.71) is violated, namely, qc (t) > 0 for all t ∈ (0, t1 ) and qc (t1 ) = 0. Then, owing to g(x) ˜ < 0 given by Assumption 10.5, the estimation error PDE (10.62) satisfies u˜ t (x, t) < α u˜ xx , and therefore, applying maximum principle to u˜ system leads to the negativity of the solution, i.e., u(x, ˜ t) < 0, for all x ∈ (0, s(t)), for all t ∈ (0, t1 ), and u˜ x (s(t), t) ≥ 0,

∀t ∈ (0, t1 ).

(10.76)

Taking the time derivative of the control law (10.65) along the solution of (10.58)– (10.61) leads to the following differential equation:

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10 Metal 3D-Printing via Selective Laser Sintering



k q˙c (t) = −c 1 + α



s(t)

 g(x)dx ˆ qc (t) + ck u˜ x (s(t), t).

(10.77)

0

Applying (10.76) to (10.77) leads to 

k q˙c (t) > −c 1 + α



s(t)

 g(x)dx ˆ qc (t),

∀t ∈ (0, t1 ).

(10.78)

0

Using the same procedure as the derivation in Lemma 10.2, the differential inequality (10.78) leads to the following inequality of the solution: 

qc (t) > qc (0)e

  s ˆ t −c 1+ αk 0 r g(x)dx

,

∀t ∈ (0, t1 ).

(10.79)

Thus, we have qc (t1 ) > 0, which contradicts the assumption qc (t1 ) = 0. Hence, we deduce (10.71), and, again, applying the maximum principle to (10.62)–(10.64) leads to the properties (10.72) and (10.73) for all t ≥ 0. The conditions (10.74) and (10.75) are derived by same procedure as in Lemma 10.2.

Stability Analysis To study the stability of the plant states (u, X) under the output feedback design (10.65), we prove the stability of coupled u-system ˜ (10.62)–(10.64) and (u, ˆ X)system (10.58)–(10.61). As in full-state feedback design, we introduce the following backstepping transformation from (u, ˆ X)-system to (w, ˆ X)-system: c w(x, ˆ t) =u(x, ˆ t) − α



s(t)

(x − y)u(y, ˆ t)dy +

x

c (s(t) − x)X(t). β

(10.80)

Taking the time and spatial derivatives of (10.80) together with (10.58)–(10.61) yields the following target system: wˆ t (x, t) =α wˆ xx (x, t) +

c s˙ (t)X(t) + g(x, ¯ s(t))qc (t) β

− c(s(t) − x)u˜ x (s(t), t),

(10.81)

w(s(t), ˆ t) =0,

(10.82)

wˆ x (0, t) =0,

(10.83)

˙ X(t) = − cX(t) − β wˆ x (s(t), t) − β u˜ x (s(t), t).

(10.84)

10.4 Observer and Output Feedback Control Design

261

First, we prove the stability of the coupled u-system ˜ (10.62)–(10.64) and (w, ˆ X)system (10.81)–(10.84). Consider the functional V˜ defined by 1 1 V˜ = 2 ||u|| ˜ 2 + ||u˜ x ||2 . 2 2sr

(10.85)

Taking the time derivative of (10.85) along the solution of (10.62)–(10.64), we get  s(t) s˙ (t) α u˜ x (s(t), t)2 − α||u˜ xx ||2 u(x, ˜ t)g(x)dxq ˜ V˙˜ = − 2 ||u˜ x ||2 + c (t) − 2 sr 0  s(t) − u˜ xx (x, t)g(x)dxq ˜ (10.86) c (t). 0

Applying Young’s and Cauchy-Schwarz inequalities to (10.86) leads to α 5 α ||g|| ˜ 2 |qc (t)|2 . V˙˜ ≤ − ||u˜ xx ||2 − 2 ||u˜ x ||2 + 2 2α 2sr

(10.87)

In the stability proof of the full-state feedback system, the square norm of the control law was shown to be bounded by an exponential function in time prior to Lyapunov analysis with the help of the closed-form of the differential equation of the controller. However, under the output feedback control design, as observed from the differential equation (10.77) including the extra term of u˜ x (s(t), t), it is hard to apply the same approach. To deal with the problem, we additionally consider the functional Q(t) defined by Q(t) =

1 qc (t)2 . 2

(10.88)

The time derivative of (10.88) with the help of (10.77) yields the following: 

k ˙ Q(t) =−c 1+ α



s(t)

 g(x)dx qc (t)2 + ck u˜ x (s(t), t)qc (t)

0

c ≤ − qc (t)2 + 2ck 2 sr ||u˜ xx ||2 , 2

(10.89)

where we used Young’s, Cauchy-Schwarz, and Agmon’s inequalities for the derivation from the first line to the second line. We consider the following functional: Θ(t) = V˜ +

α Q. 8ck 2 sr

(10.90)

Taking time derivative of (10.90) and applying the inequalities (10.87) and (10.89) with the help of (10.68) yield

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10 Metal 3D-Printing via Selective Laser Sintering

α α α ˙ |qc (t)|2 . Θ(t) ≤ − ||u˜ xx ||2 − 2 ||u˜ x ||2 − 4 2sr 32k 2 sr

(10.91)

Additionally, as defined by (10.36), we consider the following functional: 1 1 p Vˆ (t) = 2 ||w|| ˆ 2 + ||wˆ x ||2 + X(t)2 , 2 2 2sr

(10.92)

where p=

cα . 4β 2 sr

(10.93)

Referring to the derivation of (10.42) with additional terms involving u˜ x (s(t), t) in (10.81) and (10.84), one can deduce that the time derivative of (10.92) is bounded by     pc α 18sr3 c2 1 2 2 2 ||wx || + 2 ||w|| − u˜ x (s(t), t)2 X(t) + + 4 3α 4sr sr   1 3 8sr c p 2 2 2 2 + ||g|| ||w|| + ¯ qc (t) + s˙ (t) X(t) α α 2 2sr3   b ˆ 72sr4 c2 3 2 2 ˆ ≤ − V + a s˙ (t)V + ||g|| (10.94) ¯ qc (t) + + α ||u˜ xx ||2 . 2 α 3α

α V˙ˆ (t) ≤ − 16sr2

Finally, by defining V (t) = Vˆ (t) + rΘ(t),

(10.95)

for sufficiently large r > 0, taking the time derivative of (10.95) and applying (10.94) and (10.91) lead to b rα rα ˜ Q(t), V˙ ≤ − Vˆ + a s˙ (t)Vˆ − V − 2 2 16sr 32k 2 sr ¯ + a s˙ (t)V , ≤ − bV

(10.96)

where   α 1 , c . b¯ = min 4 4sr2

(10.97)

Applying comparison principle to (10.96) with the help of s˙ (t) > 0 and s0 < s(t) < sr leads to the exponential decay of the norm as ¯

V (t) ≤ easr V (0)e−bt .

(10.98)

10.5 Numerical Simulation

263

Let ˜ 2H1 , Ψ (t) =||u|| ˆ 2H1 + X(t)2 + ||u|| ˜ 2H1 + qc (t)2 . Ψ¯ (t) =||u|| ˆ 2H1 + X(t)2 + ||u||

(10.99) (10.100)

Due to the invertibility of the transformation from (u, ˆ X) and (w, ˆ X), the norm equivalence between V and Ψ¯ holds, i.e., there exist positive constants M > 0 and M > 0 such that M Ψ¯ (t) ≤ V (t) ≤ M Ψ¯ (t)

(10.101)

holds. Furthermore, using the bound of qc (t)2 derived in (10.55) (qc (t)2 ≤ LΨ (t) for some L > 0), we obtain Ψ (t) ≤ Ψ¯ (t) ≤ (1 + L)Ψ (t).

(10.102)

Therefore, we get Ψ (t) ≤ easr

M (1 + L)Ψ (0)e−bt . M

(10.103)

Finally, the norm equivalence between (u, X, u)-system ˜ and (u, ˆ X, u)-system ˜ holds due to the relation u = uˆ + u, ˜ and therefore the exponential decay of the norm holds for the functional ˜ 2H1 , Φ(t) = ||u||2H1 + X(t)2 + ||u||

(10.104)

from which we conclude Theorem 10.2.

10.5 Numerical Simulation In this section we provide two illustrations. First, we show that the controller is effective even when applied to a considerably more complex and realistic model than the one for which the design was conducted and the theorems proven. Second, we push the model mismatch to the point of the controller failing, identifying the size of the modeling error which is intolerable for the controller.

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10 Metal 3D-Printing via Selective Laser Sintering

Incorporating the Freezing Effect from the Solid Phase The dynamics of the moving interface (10.4) is given under the assumption that the freezing effect from the solid metal part is negligible; however, we incorporate the freezing effect in the numerical simulation by modifying the dynamics as s˙ (t) = − βl

∂Tl ∂Ts (s(t), t) + βs (s(t), t), ∂x ∂x

(10.105)

where the variables with subscript l and s denote those of the liquid phase (melt pool) and the solid phase (metal part), respectively. Similar to PDE for the liquid phase, the governing equation of the solid phase is given by ∂ 2 Ts ∂Ts (x, t) =αs 2 (x, t), ∂t ∂x ∂Ts (L, t) =0, ∂x

x ∈ (s(t), L),

(10.106) (10.107)

Ts (s(t), t) =Tm ,

(10.108)

where L is the thickness of the metal part. For computation of the Stefan problem (10.1)–(10.3), (10.105)–(10.108), we use boundary immobilization method combined with finite difference semi-discretization [132] for both the liquid and solid PDEs. The resulting approximated ODEs are calculated by using MATLAB ode15 solver.

Input Parameters We use the physical parameters of Ti6Al4V which is a popular composite material for metal AM, as given in Table 10.1. The initial values are set as s0 = 50 µm, and Table 10.1 Physical properties of Ti6Al4V alloy given by [146]

Description Density (liquid) Density (solid) Latent heat of fusion Heat capacity (liquid) Heat capacity (solid) Thermal conductivity (liquid) Thermal conductivity (solid) Melting temperature

Symbol ρl ρs Hf cp,l cp,s kl ks Tm

Value 3920 kg · m−3 4200 kg · m−3 2.86 × 105 J · kg−1 830 J · kg−1 · K−1 730 J · kg−1 · K−1 32.5 W · m−1 26.0 W · m−1 1650 ◦ C

10.5 Numerical Simulation

Tl (x, 0) = T

265

1 + cos



πx s0



+ Tm , ∀x ∈ [0, s0 ], 2   x + Tm , ∀x ∈ [0, s0 ], Tˆl (x, 0) = Tˆ 1 − s0   L−x + Tm , ∀x ∈ [s0 , L], Tˆs (x, 0) = T 1 − L − s0

(10.109) (10.110) (10.111)

where T = 10 ◦ C, Tˆ = 50 ◦ C, and T = −100 ◦ C. Note that the profiles have boundary values Tl (0, 0) = T +Tm , Tˆl (x, 0) = Tˆ +Tm , and Tˆs (x, 0) = T +Tm . The setpoint is chosen as sr = 200 µm, which is a reasonable value for layer thickness of the SLS-based AM. Then, the setpoint restriction (10.9) is satisfied. The control gain c is set to have a reasonable value for the laser power at initial time, and here x we choose c = 10,000/s. The spatially varying function is set as g(x) = ρc1p δ e− δ following [169], where δ = 10 µm. The thickness L of the metal part is set as L = 2 cm.

Practical Setup for the Observer Design As presented in Remark 10.5, the spatially varying function g(x) ˆ in the observer (10.58)–(10.60) is chosen as (10.67), where the upper bound and the lower bound of the penetration rate are chosen as δ = 8 µm and δ = 12 µm. Moreover, we incorporate the measurement uncertainty as the constant bias d, namely, the measured value y(t) for the interface position s(t) is given by y(t) = s(t) + d.

(10.112)

In the simulation study, we investigate the results with noise-free d = 0, the positive bias d > 0, and the negative bias d < 0.

Simulation Results Robustness of the Performance Under Small Perturbations The simulation results of the interface position, the laser power controller, and the surface temperature are given in Fig. 10.2a–c, respectively, for the cases of the measurements under noise-free d = 0 µm (red), the positive bias d = 5 µm (blue), and the negative bias d = −5 µm (green). Figure 10.2a shows that under the noise-free measurement (red) the interface position s(t) converges to the setpoint

266 Fig. 10.2 The responses of the system (10.1)–(10.3) and (10.105)–(10.108), under the output feedback control law (10.65) associated with the observer (10.58)–(10.60). The proposed method is successful: the convergence of the interface, the positivity of input, and the required condition for the liquid temperature are all achieved for both positive and negative measurement biases, namely, for d = 0, 5, −5 µm, and in the presence of the solid phase. (a) In spite of the interface measurement bias d in (10.112), the regulation near the interface setpoint is achieved. (b) Positivity of the laser power control is maintained. (c) The model validity of the boundary liquid temperature is maintained, i.e., T (0, t) ≥ Tm , in spite of the presence of the solid phase and of the interface measurement bias

10 Metal 3D-Printing via Selective Laser Sintering 200 150 100 50 0

5

10

15

20

15

20

15

20

(a) 0.4 0.3 0.2 0.1 0 0

5

10

(b) 2000 1900 1800 1700 1600 0

5

10

(c) sr without overshooting in a short time scale 10 ms, which illustrates sufficiently fast process of the melting each layer. On the other hand, in the presence of the measurement bias, a modest error of the converging position of the interface from the setpoint position is observed in both positive and negative bias. From Fig. 10.2b we observe that the implemented output feedback control maintains a positive value under noise-free measurement and even in the presence of the measurement bias, which satisfies the constraint for the input laser power. Owing to the positive-valued input, Fig. 10.2c shows that the temperature at the surface position remains above the melting temperature Tm , which ensures the condition (10.8) for the validity of the model addressed in Remark 10.1 under both noise-free measurement and the biased measurement. Therefore, the numerical results illustrate that the proposed

10.5 Numerical Simulation

1700

267 Temperature profiles at 0 [msec]

1690 1680 1670 1660 1650 0

1950

50

100

150

200

Temperature profiles at 1 [msec]

1900 1850 1800 1750 1700 1650 0

1680

50

100

150

200

Temperature profiles at 5 [msec]

1670

1660

1650 0

50

100

150

200

Fig. 10.3 The snapshots of the true temperature profile (solid line) and the estimated temperature profile (dash line) at t = 0, 1, and 5 ms under the positive bias

observer-based output feedback control law performs robustly even in the presence of the measurement uncertainty. Figure 10.3 depicts the snapshots of the true temperature profile (solid line) and the estimated temperature profile (dash line) at t = 0, 1, 5 ms in the presence of the positive bias. From Fig. 10.3, we observe that the estimated temperature profile gradually converges to the true temperature profile, albeit the convergence speed is not fast. Nevertheless, the control objective and the model validity conditions are well satisfied as observed in Fig. 10.2, which shows the sufficient performance of the observer for the purpose of stabilization of the melt pool in SLS.

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10 Metal 3D-Printing via Selective Laser Sintering

Limitation of the Performance Under Large Perturbations However, under large uncertainty in model caused by the cold (the negative heat) in the solid metal, or by the measurement uncertainties, the proposed method is shown to, expectedly, violate the required conditions for the physical model. Figure 10.4 depicts the interface response under an initial very low temperature in the solid phase (10.111), namely, T = −1630 ◦ C that results in the boundary value Ts (L, 0) = T + Tm = 20 ◦ C, which is still physically possible. As we observe from Fig. 10.4, the interface disappears and with it the molten metal phase, at around 14 ms due to the complete solidification of the melt pool. This is caused by an insufficient amount of the laser power input for the given “deeply” frozen solid metal initial state. The limitation of the proposed control law can be relaxed by designing a “two-phase”-based control law proposed in the absence of radiation in Chap. 5, which will be considered with radiation in future work. Next, we investigate the closed-loop response under d = −30 µm, namely, a large negative bias in the measurement. Figure 10.5 shows the response of the control input and the boundary temperature. Figure 10.5a illustrates that the controlled laser power reaches negative value after t = 2 ms, which violates the input constraint. Due to the negative input, Fig. 10.5b illustrates that the boundary temperature of the melt pool reaches below the melting temperature, which physically causes the solidification of the melt pool from the controlled boundary and therefore the condition of the model is violated. Hence, there is a limitation of the performance of the proposed control law with respect to the level of the measurement uncertainty in the case of negative bias. Figure 10.6 shows the plot of minimum value of the control input qc (t) over the time interval t ∈ [0, tf ] where tf = 20 ms under the bias d ranging from −5 µm to −15 µm. From Fig. 10.6, we observe that the critical value of the bias violating the positivity of the input is between −7 and −8 µm, which is approximately 15 % of the initial interface position s0 = 50 µm. On the other hand, under the positive bias, as long as Assumption 10.6 holds, we observe that the performance of the control law is robust as we have seen in Fig. 10.2. Fig. 10.4 The interface response of the closed-loop system under very cold initial temperature of the solid phase. The proposed method fails, in this caricatured scenario, and the melt pool gets entirely frozen, as observed from the disappearance of the interface position around t = 14 ms

50 40 30 20 10 0

0

5

10

15

10.5 Numerical Simulation

269

0.1

0.05

0 0

5

10

15

20

15

20

(a) 2200

2000

1800

1600 0

5

10

(b) Fig. 10.5 The response of the closed-loop system under a large negative bias d = −30 µm. The proposed method fails due to the violation of the positivity of the input and of the condition of the liquid temperature. (a) The input of the laser power reaches negative value, which cannot be implemented in practice. (b) The liquid temperature can be below the melting temperature, which causes the solidification of the melt pool from the controlled boundary

10-4 0

-5

-10 -15

-10

-5

Fig. 10.6 Plot of the minimum value of the control input over time through varying the bias d from −5 µm to −15 µm. Positivity of the input is violated under the bias d ≤ −8 µm

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10 Metal 3D-Printing via Selective Laser Sintering

10.6 Comments and Remarks In this chapter we have developed the control design of laser power in SLS for metal AM by using the PDE backstepping method in a form of both full-state and output feedback. The governing equation is given by the one-phase Stefan problem with in-domain effect of the controlled laser power to the PDE dynamics. The closedloop system is shown to satisfy some required conditions for the physical model to be valid, and the exponential stability at the origin is proven. Numerical simulation is performed by computing the full “two-phase” Stefan model incorporating the cooing from the solid metal part and adding the measurement uncertainty. The simulation results illustrate that the proposed output feedback control design enables the laser melting to drive the depth of the melt pool to the desired setpoint sufficiently fast, and the performance is robust under perturbations of the model and the measurements. We push the proposed control law to its failure limit by exhibiting the closed-loop responses that violate the required conditions of the physical states under the large perturbations of the model and the measurement. The limitation of the controller’s robustness might be exploited analytically in the sense of Input-to-State Stability (ISS), by using the Lyapunov method for the perturbed system including the model and measurement uncertainties, similar to ISS analysis in Chap. 4. One challenge lies in the fact that the model and the estimator are defined on a distinct range of the spatial domains, (0, s(t)) and (0, s(t) + d(t)), under the measurement uncertainty of the moving interface. Such a discrepancy of the domains makes the analysis of the estimation error system much harder. Moreover, while we have focused only on the stabilization of the melt pool’s depth in this chapter, the stabilization of the surface area of the melt pool is also a significant task for the scanning process. This motivates us to develop the control design for the Stefan problem along the surface geometry of the powder bed in metal AM process, which is also challenging due to the movement of the scanner mirror.

Chapter 11

Experimental Study with Paraffin Melting

There are still relatively few, but a growing number, of results on the experimental application of the backstepping design for boundary control and observer of PDEs. In [157], the tracking control for flexible articulated wings on a robotic aircraft is designed by combining PDE backstepping for feedback stabilization and feedforward trajectory planning, and the performance of the designed boundary controller is demonstrated by conducting the experiment of bending a long thin beam. The validation of the backstepping boundary observer design using experimental data have been studied in [111] for microfluidic systems, in [85] for oil drilling, in [52] for thermoacoustic instability in Rijke tube, and in [226] for congested freeway traffic following their design in [223, 224]. In this chapter we add to this list our experimental results on the boundary control and observer for the Stefan system which is governed by a parabolic PDE with statedependent moving boundaries “(a nonlinear system).” The fidelity of the Stefan model has been validated in several experimental studies. Among the various materials in the aforementioned applications, phase change materials (PCMs) in latent heat thermal energy storage systems for numerous applications (e.g., heat pumps, solar engineering, and spacecraft thermal controls) have been extensively used to investigate the correspondence of the experimental data with the numerical model of the Stefan problem (see [63, 227] for a detailed review on simulations of PCMs). While there are several materials that are considered PCM, paraffins are frequently utilized owing to the attractive features of a safe temperature range for melting, low cost, noncorrosiveness, and predictable thermal and chemical behaviors [177].

© Springer Nature Switzerland AG 2020 S. Koga, M. Krstic, Materials Phase Change PDE Control & Estimation, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-58490-0_11

271

272

11 Experimental Study with Paraffin Melting

11.1 Modeling of PCM We consider a cylindrical paraffin with the diameter R and the total length L, which is enclosed with an acrylic container serving as a thermal insulation. Hence, the geometry of the model is represented by a cylindrical coordinate (r, θ, x) which denotes the radial distance from the center, the angular degree from a base, and a displacement from the upper side of the container, respectively (see Fig. 11.1). In the phase change material, the liquid–solid phase boundary exists as a domain inside the coordinate. We establish the physical model under the following assumptions. Assumption 11.1 The temperature profile of the paraffin is uniformly distributed along the circle-shaped cross section of the cylinder. Mathematically, we can describe the property as T (r, θ, x, t) ≡ T (x, t),

∀r ∈ [0, R], ∀θ ∈ [0, 2π ).

(11.1)

Moreover, the domain of the phase boundary is uniform along the cross section of the cylinder, which enables to describe the location as x = s(t). Assumption 11.2 There is no convection in the liquid phase. Owing to the cylindrical geometry and the thermal insulation along the side, if Assumption 11.1 holds at the initial time t = t0 , then it holds for all time t > t0 . By Assumption 11.1, the geometry of the physical model can be described by one-dimensional coordinate in x. In addition, by Assumption 11.2, the governing equation is only given by the energy conservation without imposing a mass and

qc (t)

Tm

0

T (x, t)

0

liquid s(t)

s(t)

solid L

L

x

x

Fig. 11.1 Schematic of one-dimensional model of paraffin as a Phase Change Material (PCM) in vertical coordinate

11.1 Modeling of PCM

273

momentum balance. Combining the local energy conservation law inside the liquid phase domain x ∈ (0, s(t)) and Fourier’s thermal conduction law, the time evolution of the temperature profile is given by the following parabolic PDE: ∂T ∂ 2T (x, t) = α 2 (x, t), ∂t ∂x

0 < x < s(t),

(11.2)

k where α := ρC with a density ρ, heat capacity Cp , and the thermal conductivity p k for liquid phase, respectively. At the surface x = 0, there is a heat loss due to the convective heat transfer through the surrounding air, which yields the following energy balance:

−k

∂T (0, t) =qc (t) − h(T (0, t) − Ta ), ∂x

(11.3)

where qc (t) is a manipulated heat flux per unit area, h denotes the heat transfer coefficient, and Ta denotes the ambient temperature (room temperature). As a fundamental physical condition of the thermal phase change, the temperature at the liquid-solid phase boundary x = s(t) maintains the constant melting temperature Tm , which renders the boundary condition as T (s(t), t) = Tm .

(11.4)

Moreover, the local energy balance at the position of the liquid–solid phase boundary x = s(t) leads to the Stefan condition defined as the following nonlinear ODE: ρΔH ∗ s˙ (t) = −k

∂T (s(t), t) − qlos , ∂x

(11.5)

where ΔH ∗ and qlos denote the latent heat of fusion and heat loss at the interface, respectively. Remark 11.1 To ensure that the model (11.2)–(11.5) remains physically valid, the following conditions must hold: T (x, t) ≥Tm ,

∀x ∈ (0, s(t)),

0 0.

∀t > 0,

(11.6) (11.7)

The conditions (11.6) and (11.7) are proven to hold after the design of the heat input qc (t).

274

11 Experimental Study with Paraffin Melting

11.2 Nominal Feedback Control Design Control Objective and Steady-State Solution The objective is to drive the phase boundary location s(t) to a desired setpoint sr by controlling the heat flux qc (t). As a desired state, the steady-state solution of the temperature profile Tr (x) at s(t) = sr needs to be considered. By setting the time derivative of the physical model to zero, the steady-state of the temperature is obtained by Tr (x) =

qlos (sr − x) + Tm . k

(11.8)

Then, at the steady-state, the heat flux input must have a balance with the heat loss at the surface and the interface, which is described as   hsr ∗ qlos + h (Tm − Ta ) . qc = 1 + (11.9) k

Continuous-Time Full-State Feedback Control Design While the governing equations (11.2)–(11.5) are given by the local energy balance law at each location in the domain x ∈ [0, s(t)], in order to prescribe the growth of the internal energy through the heat input and heat loss, the macroscopic energy conservation should be considered. The internal energy of the system is composed of the specific heat and the latent heat given by  k s(t) k (11.10) E(t) = (T (x, t) − Tm )dx + s(t), α 0 β k where β := ρΔH ∗ . Taking the time derivative of (11.10), we obtain the macroscopic energy conservation law as

˙ E(t) = qc (t) − h(T (0, t) − Ta ) − qlos .

(11.11)

The setpoint energy is given by substituting the steady-state solution (Tr (x), sr ) into (11.10), which yields  k k sr k qlos sr2 Er = + sr . (Tr (x) − Tm )dx + sr = (11.12) α 0 β 2α β To achieve the control objective driving the system states (T , s) to the reference setpoint (Tr , sr ), it is necessary that the internal energy E(t) grows to the setpoint internal energy Er . The idea of our control design originates from parabolic PDEODE backstepping in [122] but in this section we provide a simplified exposition

11.3 Implementable Control Algorithm Using Sensors and Software

275

based on energy shaping. Namely, we define the reference error of the internal energy as ˜ E(t) = E(t) − Er ,

(11.13)

and design the control law as follows: ˜ + h(T (0, t) − Ta ) + qlos qc (t) = − cE(t)    k s(t) k =−c (T (x, t) − Tm )dx + (s(t) − sr ) α 0 β +

cqlos sr2 + h(T (0, t) − Ta ) + qlos . 2α

(11.14)

(11.15)

Here, we impose the following restriction on the setpoint position sr . Assumption 11.3 The setpoint sr is chosen to satisfy s0 +

β α



s0

(T0 (x) − Tm )dx < sr +

0

βqlos 2 s < L. 2kα r

(11.16)

Then, the control (11.15) ensures the conditions (11.6) and (11.7), and we state the following theorem. Theorem 11.1 Consider the closed-loop system consisting of the plant (11.2)– (11.5) with the control law (11.15). Then, the conditions (11.6) and (11.7) for the ∗ > 0 such that for all model validity hold, and there exists a positive constant qlos ∗ qlos ∈ (0, qlos ) the closed-loop system is exponentially stable in the sense of the norm 

s(t)

(T (x, t) − Tr (x))2 dx + (s(t) − sr )2 .

(11.17)

0

The proof of Theorem 11.1 is given at the end of this chapter. For accelerated convergence, the backstepping PDE-ODE control design in Theorem 2 in [122] would have to be pursued.

11.3 Implementable Control Algorithm Using Sensors and Software This section presents the feedback control algorithm that we actually implement in the experiment. Note that the full-state feedback control design developed in Sect. 11.2 requires the following three assumptions:

276

11 Experimental Study with Paraffin Melting

Fig. 11.2 Block diagram of the observer-based output feedback control. The interface position s(t) and the surface temperature T (0, t) are available as two measurements

Controller

qc (t)

Tˆ (x, t)

Phase Change Model Measurements

Estimator

y(t) = (s(t), T (0, t))

s(t)

• the spatial profile of the temperature is available, • the spatial integration of the temperature profile is computed, • and the measurements are obtained and the controller is manipulated continuously in time. The first assumption is relaxed by introducing a state observer governed by a PDE to estimate the entire temperature profile under measured temperature only at the surface and the measured position of the phase interface, and redesigning the controller by associated output feedback control law. The block diagram is depicted in Fig. 11.2. Next, the PDE observer is approximated by an ODE observer through the truncation of the observer state and then the spatial integration in the output feedback control law is approximated by the trapezoidal rule, which removes the second assumption. Finally, the third assumption is relaxed by further improving the observer and the output feedback control by sampled-data design implemented under the measurements obtained at each discrete sampling time by following the idea of the sampled-data observer [97].

PDE Observer and Output Feedback Design Suppose that we have the following two measurements: y (1) (t) =s(t),

(11.18)

y (2) (t) =T (0, t),

(11.19)

for all t ≥ 0. The state observer for the PDE system (11.2)–(11.4) is designed by ∂ 2 Tˆ ∂ Tˆ (x, t) =α 2 (x, t), 0 < x < y (1) (t), ∂t ∂x ˆ ∂T −k (0, t) =qc (t) − h(y (2) (t) − Ta ) + κ1 (y (2) (t) − Tˆ (0, t)), ∂x Tˆ (y (1) (t), t) =Tm ,

(11.20) (11.21) (11.22)

for all t ≥ 0, where κ1 > 0 is the observer gain tuned by the user. To study the performance of the observer, we introduce the estimation error variable T˜ (x, t) defined by

11.3 Implementable Control Algorithm Using Sensors and Software

T˜ (x, t) := T (x, t) − Tˆ (x, t).

277

(11.23)

Subtracting the observer system (11.20)–(11.22) from the plant (11.2)–(11.5) leads to the estimation error system as follows: ∂ T˜ ∂ 2 T˜ (x, t) =α 2 (x, t), ∂t ∂x ∂ T˜ κ1 (0, t) = T˜ (0, t), ∂x k ˜ T (s(t), t) =0.

0 < x < s(t),

(11.24) (11.25) (11.26)

Then, the performance of the PDE observer (11.20)–(11.22) is guaranteed by the following lemma. Lemma 11.1 The estimation error  system (11.24)–(11.26) is exponentially stable s(t) ˜ in the spatial L2 -norm ||T˜ || = T (x, t)2 dx. 0 In addition to the PDE observer given in (11.20)–(11.22) to estimate the temperature profile, we introduce the following observer reconstructing the interface position as a copy of (11.5) plus the measurement injection of the interface position: β ∂ Tˆ (s(t), t) − qlos + κ2 (s(t) − sˆ (t)), s˙ˆ (t) = − β ∂x k

(11.27)

where κ2 > 0 is an observer gain. The observer (11.27) is not essential to estimate the interface position since we suppose that we can accurately measure the interface position in continuous time. However, in the next section we propose the redesign of the observer under the sampled-data measurements, and the observer (11.27) is required to reconstruct the unmeasured interface position during the sampling time period. By defining s˜ (t) := s(t) − sˆ (t), the dynamics of s˜ (t) is given by ∂ T˜ s˙˜ (t) = −β (s(t), t) − κ2 s˜ (t). ∂x

(11.28)

The stability of (T˜ , s˜ )-system in (11.24)–(11.26), (11.28) is addressed in the following lemma. Lemma 11.2 Assume that there exists s¯ > 0 such that 0 < s(t) < s¯ for all t ≥ 0, and 0 ≤ qlos < 2βα s¯ holds. The estimation error system (11.24)–(11.26), (11.28) is ˜

˜ exponentially stable in the norm Φ(t) := ||T˜ ||2 + || ∂∂xT ||2 + s˜ (t)2 .

˜ Lemma 11.2 is proved by analyzing the time derivative of Φ(t) and applying Lyapunov’s method. The details are omitted here. The associated output-feedback control is given by replacing the true temperature profile in the full-state feedback

278

11 Experimental Study with Paraffin Melting

control law (11.15) with the estimated temperature Tˆ (x, t) calculated by the observer (11.20)–(11.22), resulting in the following form: 

k qc (t) = − c α +

 0

s(t)

k (Tˆ (x, t) − Tm )dx + (s(t) − sr ) β

cqlos sr2 + h(T (0, t) − Ta ) + qlos . 2α



(11.29)

Then, owing to the separation principle, it is shown that the output feedback control law stabilizes the plant state (T (x, t), s(t)) at the desired reference (Tr (x), sr ). Theorem 11.2 Assume that Tˆ (x, 0) ≥ T (x, 0) for all x ∈ [0, s0 ]. Consider the closed-loop system consisting of the plant (11.2)–(11.5), the observer (11.20)– (11.22), and the output feedback control law (11.29). Then, the conditions (11.6) and (11.7) for the model validity and T˜ (x, t) ≤ 0 hold for all x ∈ (0, s(t)) and ∗ > 0 such that for all for all t ≥ 0, and there exists a positive constant qlos ∗ qlos ∈ (0, qlos ) the closed-loop system is exponentially stable in the sense of the norm ||T − Tr ||2 + (s(t) − sr )2 + ||T˜ ||2 . The proof of Theorem 11.2 is presented at the end of this chapter.

ODE Observer Derived from Discretized PDE To implement the designed observer via numerical computation, we derive the spatially discretized model of (11.2)–(11.5). Let N ∈ N be the number of grid points for the spatial discretization, Δx be the width defined by Δx = N1 , and φ (i) (t) be defined by φ (i) (t) =T (iΔxs(t), t) − Tm ,

i = 0, 1, 2, · · · , N,

φ(t) =[φ (1) (t), φ (2) (t), φ (3) (t), · · · , φ (N ) (t)]T .

(11.30) (11.31)

Note that taking the total time derivative of (11.30) yields φ˙ (i) (t) =



∂T ∂T + iΔx s˙ (t) ∂t ∂x

/ / / /

.

(11.32)

x=iΔxs(t)

Then, the spatially discretized model of (11.2)–(11.5) is governed by the following coupled nonlinear ODEs of the states φ(t) and s(t): φ˙ (0) (t) =a(s(t))φ (0) (t) + p(s(t))φ(t) + b(s(t))q¯c (t), ˙ =q(s(t))φ (0) (t) + R(s(t))φ(t) + f (φ(t), s(t)), φ(t)

(11.33) (11.34)

11.3 Implementable Control Algorithm Using Sensors and Software

s˙ (t) =g(s(t))φ(t) −

β qlos , k

279

(11.35)

where q¯c (t) := qc (t) − h(φ (0) (t) + Tm − Ta ),

(11.36)

2α , (s(t)Δx)2

(11.37)

and a(s(t)) = − b(s(t)) =

4α , ks(t)Δx

2α (s(t)Δx)2 α q(s(t)) = (s(t)Δx)2

p(s(t)) =

g(s(t)) = −

(11.38) 

 1 01,N −1 ,

(11.39)



 1 01,N −1 ,

(11.40)

  β 01,N −3 1 −4 , 2s(t)Δx

(11.41)

0i,j ∈ Ri×j denotes a matrix in which all the elements are zero, and R(s(t)) ∈ RN −1×N −1 has its elements ri,j at ith row and j th column given by 2α , ∀i = 1, 2, · · · , N − 1 (s(t)Δx)2 α =ri,i+1 = , ∀i = 1, 2, · · · , N − 1, (s(t)Δx)2

ri,i = − ri+1,i

(11.42) (11.43)

and all other elements are zero. The function f (φ(t), s(t)) is a nonlinear function of the dynamics derived from the last term in (11.32), which has its ith element  (i+1)  φ − φ (i−1) β fi = iΔx g(s(t))φ(t) − qlos . k 2Δxs(t)

(11.44)

Hence, by defining the state vector ψ ∈ RN +1 T  ψ = φ (0) φ s

(11.45)

the coupled dynamics (11.33)–(11.35) can be described by the following ODE on the state ψ: ψ˙ = A(s)ψ + B(s)q¯c + F (ψ) + θ,

(11.46)

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11 Experimental Study with Paraffin Melting

where ⎤ a(s) p(s) 0 A(s) = ⎣ q(s) R(s) 0 ⎦ , 0 g(s) 0 T  B(s) = b(s) 01,N −1 0 ,  T F (ψ) = 0 f (φ, s) 0 , .T θ = 0 01,N −1 − βk qlos . ⎡

(11.47)

(11.48) (11.49) (11.50)

Let y ∈ R2 be the vector associated with the measurements (11.18) and (11.19) defined by  T y = y (1) y (2) .

(11.51)

The measurement vector is described by y = Cψ + d,

(11.52)

where  0 01,N −1 1 , C= 1 01,N −1 0  T d = 0 Tm . 

(11.53) (11.54)

Following the same procedure, the continuous-time PDE observer designed in (11.24)–(11.26) is implemented by the following ODE observer: ˆ + θ + K(y − y) ˆ ψ˙ˆ = A(y (1) )ψˆ + B(y (1) )q¯c + F (ψ)

(11.55)

where yˆ = C ψˆ + d, and K ∈ RN +2,2 is defined by ⎡

0

K = ⎣ 0N −1,1 κ2

⎤ b(s)κ1 0N −1,1 ⎦ . 0

(11.56)

11.3 Implementable Control Algorithm Using Sensors and Software

281

Sampled-Data Design of ODE Observer and Output Feedback The measured data are obtained not continuously in time but at each sampling time {tj : j = 0, 1, 2, · · · }. Here, we consider the sampling scheduling as periodic sampling with period τ , which leads to the sequence of the sampling time as tj = t0 + j τ,

j = 1, 2, · · ·

(11.57)

Hence, the sampled-data measurements are obtained by y (1) (tj ) =s(tj ),

(11.58)

y (2) (tj ) =T (0, tj ).

(11.59)

We employ the methods proposed in [97], namely, we introduce the so-called “InterSample-Predictor” (ISP) which serves as an estimate of the measured variables during the sampling periods given the measurements at each sampling time as an initial state, and compute the continuous-time observer coupled with ISP as an estimate of the state variables. Let w (1) (t) and w (2) (t) be the ISP states reconstructing y (1) (t) = s(t) and (2) y (t) = T (0, t), respectively. At every sampling time t = tj , we set w (1) (tj ) = y (1) (tj ),

w (2) (tj ) = y (2) (tj ).

(11.60)

For t ∈ [tj , tj +1 ), by referring to (11.33) and (11.35), the dynamics of ISP is given by β qlos , k

(11.61)

ˆ + b(ˆs (t))q¯c (t). w˙ (2) (t) =a(ˆs (t))φˆ (0) (t) + p(ˆs (t))φ(t)

(11.62)

ˆ − w˙ (1) (t) =g(ˆs (t))φ(t)

The dynamics of the continuous-time observer state ψˆ is given by the copy of the observer (11.55) with replacing the measurement states by ISP states as follows: ˆ + θ + K(w − y), ˆ ψ˙ˆ =A(w (1) )ψˆ + B(w (1) )q¯c + F (ψ)

(11.63)

T  where w = w(1) w (2) . Furthermore, the states w (1) , w (2) , ψˆ are discretized in time with the time step Δt, and ODEs (11.61)–(11.63) are computed numerically by forward Euler method, which leads to Algorithm 1 providing ISP-based sampleddata observer at each sampling time. Using the sampled-data observer states, the associated output feedback control law given in (11.29) under the availability of the continuous-time PDE observer is redesigned by

282

11 Experimental Study with Paraffin Melting

ky (1) (tj ) qc (tj ) = − c αN



 N  1 (2) kεsr2 (i) ˆ φ (tj ) + c (y (tj ) − Tm ) + 2 2α i=2

k − c (y (1) (tj ) − sr ) + h(y (2) (tj ) − Ta ) + qlos , β

(11.64)

where we use trapezoidal rule for approximating the spatial integration.

11.4 Experimental Setup and Calibration of Unknown Parameters This section presents the experimental setup and the results under a constant heat input to validate the sampled-data observer in Algorithm 1.

Sample Preparation and Heating Chamber PCM-37 (Microtek laboratories, Inc., Dayton, OH, USA) is chosen as the phase change material for our experiment. Its melting temperature is 37 ◦ C. Thermal properties of PCM-37 are summarized in Table 11.1. Cylindrical rod is prepared by casting of molten PCM-37 with an acrylic container with 63.5 mm diameter and a flat bottom. Molten PCM-37 is poured through a paper filter to remove particles and casting can be done by keeping container in a room temperature for 12 h. The rod of PCM-37 is pushed out from the mold once it becomes solid, and then inserted into another acrylic chamber for a heating experiment.

Algorithm 7: ISP-based sampled-data observer at sampling time tj for j ∈ {0, 1, · · · } Input : y (1) (tj ), y (2) (tj ), ψˆ j τ ; (1) (2) wj τ ← y (1) (tj ); wj τ ← y (2) (tj ); for l = 0, 1, · · · , I , do i ← j τ + l; yˆi ← C ψˆ i + d;   (1) ← wi(1) + Δt g(ˆsi )φˆ i − βk qlos ; wi+1   (2) (2) (0) wi+1 ← wi + Δt a(ˆsi )φˆ i + p(ˆsi )φˆ i + b(ˆsi )q¯c,i ; (1) (1) ψˆ i+1 ← ψˆ i + Δt (A(wi )ψˆ i + B(wi )q¯c,i + F (ψˆ i ) + θ + K(wi − yˆi )); end for Output : ψˆ (j +1)τ

11.4 Experimental Setup and Calibration of Unknown Parameters Table 11.1 Thermophysical parameters of PCM-37

Description Density Latent heat of fusion Heat capacity Melting temperature Thermal conductivity

283 Symbol ρ ΔH ∗ cp Tm k

Value 790 kg · m−3 210 J · g−1 2.38 J · g−1 · K−1 37 ◦ C 0.22 W · m−1

Figure 11.3 illustrates a structure of the chamber to heat the cast rod from top side. Our experiments show that this configuration has least influence of convection. A removable lid on top of chamber has a heater and thermocouple sensor and the PCM-37 rods are inserted so that it contacts with the heater firmly. The space above the heater is prepared for thermal insulation and avoids an accumulation of small bubbles on the heater, which are generated when PCM-37 is melted. Due to the transparency change of PCM-37 upon a phase change, the position of the boundary is measured by using a digital camera with interval shuttering.

Experiment Under a Constant Input The boundary heat actuator qc (t) is controlled by an electric current ic (t) connected with the film heater under the following relation: ic (t) ∼

qc (t)π R 2 , Res

(11.65)

where R is the radius of the cylinder, and Res is the resistance of the film heater at room temperature. In this experiment, we had R = 3.175 cm and Res = 13.9 . Due to the limitation of the equipment, the electric current is bounded by the constant imax , i.e., 0 ≤ ic (t) ≤ imax .

(11.66)

We conducted an open-loop melting test by keeping the current input at the maximum value imax = 0.79 A for 2 h after the phase interface reaches 0.5 cm. Then, we observed that the phase interface position was evolving uniformly along the vertical coordinate of the cylinder, which validates Assumption 11.1. The image of the experiment is shown in Fig. 11.3b.

284

11 Experimental Study with Paraffin Melting

Fig. 11.3 The images of the experimental apparatus and setup using PCM-37. (a) Schematic of the apparatus for melting paraffin. (b) The real experiment of melting paraffin with sensors and an actuator

Calibration of Unknown Parameters Let t0 be the time at which we observe that s(t0 ) = 0.5 cm and fix it at t0 = 0. Let tf = 2 h be the process time. We measured the phase boundary position and the surface temperature at every 10 min as a sampling time period, namely, the sampling scheduling is described as tj = t0 + j τ for j = 1, 2, · · · m with τ = 10 min and m = tf /τ = 12. Let e be the normalized estimation error vector defined by . (1) (1) (1) (2) (2) (2) , , e0 , e1 , · · · , em e = e0 , e1 , · · · , em

(11.67)

11.5 Experiment with Closed-Loop Control

285

where (i)

ej =

y (i) (tj ) − yˆ (i) (tj ) , y (i) (tj )

i = 1, 2,

j = 0, 1, · · · , m.

(11.68)

Using the measured data, the heat transfer coefficient h and the freezing heat from the solid phase qlos are calibrated to minimize the estimation error. However, for the sake of sustaining the robustness of the control algorithm, the estimated temperature profile should be higher than the true temperature profile, for the condition shown in Theorem 11.2 holds. Since both the measured surface temperature and the measured interface position are monotonically increasing as the temperature profile gets larger, both the estimated surface temperature and interface position should be higher than the measured values. Taking these into account, the unknown parameters are calibrated so that min eT e,

h,qlos

subject to e  0,

(11.69)

where  denotes an element-wise inequality. We varied the parameters in a range 0 ≤ h ≤ 30 and 0 ≤ qlos ≤ 500 with the step sizes Δh = 1 and Δqlos = 20, respectively. Then, we observed that the problem (11.69) is solved with the parameters h =20 W/m2 K and qlos =400 W/m2 . Figure 11.4 shows the comparison of the measured data with the estimated values of the interface position and the surface temperature under the obtained parameters. We can observe that the estimated values have a good agreement with the measured data and satisfy the constraint (11.69). A reference value of the convective heat transfer coefficient h for plastic is reported in [45] as h = 21 ± 2 W/m2 K, which also shows a good agreement with the identified value.

11.5 Experiment with Closed-Loop Control In this section, we present our main result on experimental validation of the proposed feedback control algorithm. The paraffin was completely solidified at the initial time of the experiment.

Gain Tuning The control gain c > 0 is an essential free parameter for the input current ic (t) to satisfy the constraint (11.66). Here we provide how to tune the gain. First, the current input is kept as imax while the paraffin starts to be molten from the top and the liquid–solid interface position is less than s0 := 0.5 cm from the top. At the

286

11 Experimental Study with Paraffin Melting 2.5 2

Estimated value Measured data

1.5 1 0.5 0 0

0.5

1

1.5

2

1.5

2

(a) 120

Estimated value Measured data

100

80

60 0

0.5

1

(b) Fig. 11.4 The estimated values (blue dash) with h = 20 W/m2 K and qlos = 400 W/m2 , which are in good agreement with the measured data (green dots) and satisfy (11.69). (a) The interface position. (b) The surface temperature

time when the interface position reaches to s0 , we measured the surface temperature y (2) (t0 ), and compute k ks0 (2) qlos sr2 (y (t0 ) − Tm ) − + (s0 − sr ), E˜ 0 = 2α 2α β cmax =

2 Res imax π R2

− h(y (2) (t0 ) − Ta ) − qlos . −E˜ 0

(11.70)

(11.71)

Then, at least we require c < cmax , since the input becomes ic (t0 ) = imax when c = cmax by (11.64) and (11.65). Moreover, from the results in Sect. 4.5, for the sampled-data state feedback control of the Stefan problem, given a sampling time period τ > 0, the control gain needs to be chosen to satisfy c < τ1 to ensure the conditions of model validity and the closed-loop stability [117]. Considering these two conditions, we take the gain tuning as

11.5 Experiment with Closed-Loop Control

  1 , c = δ min cmax , τ

287

(11.72)

where δ ∈ (0, 1) is a free parameter. In this experiment, we used δ =0.8.

Proposed Control Law The control algorithm in the experiment is applied as follows. 1. The input current ic (t) is injected at the maximum value imax (0.79 A). 2. Once we observe that the liquid–solid interface arrives at 0.5 cm, the surface temperature is measured and only the observer is computed by Algorithm 1 with keeping the maximum input current. 3. After that, at every sampling time 10 min, both the surface temperature and the interface position are measured, and the observer is computed by Algorithm 1 and the heat controller is obtained by (11.64). 4. Given the value of the controller, the current input is given by (11.65). We repeat (3) and (4) for 5 h.

Experimental Results We conducted the experiment with melting paraffin by implementing the control algorithm above. The setpoint position is chosen as sr = 2 cm, and the time step size in the observer is Δt = 0.05 s. Figure 11.5 depicts the results of the experiment by showing measured data of the phase interface position, Fig. 11.6 shows the input current and the surface temperature, and Fig. 11.7 depicts the estimated temperature profiles of the liquid paraffin and the measured temperature profile of the acrylic chamber obtained by IR camera, respectively. From Fig. 11.5a, we can observe that the experiment was a success: the phase interface position reached to the value s0 = 0.5 cm at t0 = 25 min and converged to the chosen setpoint position sr = 2 cm asymptotically and stays at the setpoint after 4 h. This result can be also visually seen in Fig. 11.5b which are snapshots of the melting paraffin at every hour. A ruler attached on the acrylic chamber shows the distance from the position of the heat actuator, which gives the measured value of the phase interface position depicted in the left plot, and hence the convergence of the interface position is visually observed. Figure 11.6a shows that the input current starts from the maximum value imax = 0.79 A under the constraint and the feedback control is implemented at every sampling time 10 min from t = 35 min which is 10 min after t0 = 25 min. After 4 h, the current input stays at the steadystate input calculated by (11.9) and (11.65). From 11.6b, we can observe that the estimated surface temperature has similar behavior to the measured surface

11 Experimental Study with Paraffin Melting

Interface position [cm]

288

2.5 2 1.5 1 0.5 0

0

1

2

3

4

5

Time [hour] (a)

0h

1h

2h

3h

4h

5h

heater interface

(b) Fig. 11.5 The experimental result of the time evolution of the interface position under the proposed feedback control algorithm. The experiment was successful: the liquid–solid interface position converged to the setpoint position sr = 2 cm. (a) The plot is depicted at every 0.1 cm increase until the interface reaches 2 cm and after that depicted at every 20 min. (b) The snapshots of the melting paraffin at every hour is given right, which shows the interface evolution by a ruler attached on the acrylic chamber

temperature together with a nominal error around 5–10 ◦ C, of which the cause is discussed later. Figure 11.7a illustrates that the estimated temperature profile converges to the reference profile given by (11.8) and almost corresponds to the reference after 4 h. The thermography included in Fig. 11.7b is obtained by IR camera taken at t = 2 h, which illustrates that the temperature is the highest (white color) at the position of the heat controller and is monotonically decreasing as the vertical position goes towards the bottom. The temperature profiles of the acrylic in the plot are given by referring to the temperature along the white arrow in the thermography. We observe that the profiles are almost linearly distributed in the space at every hour, of which the property is also observed in the estimated temperature profiles of the liquid paraffin shown in Fig. 11.7a, though the material of the focus is distinct and the temperature value is different. Moreover, the slope of

11.5 Experiment with Closed-Loop Control Fig. 11.6 The experimental result of the proposed feedback control algorithm and the surface temperature. (a) The input current started from the maximum value imax of the input constraint, and the feedback control was implemented from 35 min. After 4 h, the current input stayed at the steady-state input calculated by (11.9). (b) The estimated surface temperature has a similar behavior to the measured surface temperature together with a nominal error around 5–10 ◦ C

289

0.8

0.7

0.6 0

1

2

3

4

5

3

4

5

(a) 100 90 80 70 0

1

2

(b) the profiles is dropped from t = 1 h to t = 2 h in Fig. 11.7b, which is also similarly observed in Fig. 11.7a. Thus, while it is not accurate to refer to the thermography of the acrylic chamber as a temperature profile of the paraffin inside, we see some similar behavior of the evolution of the temperature profiles.

Discussion While we observe that the control objective is successfully achieved in the experiment, the temperature estimation accompanies a nominal error from the measured value as illustrated in Fig. 11.6b. Since the estimated surface temperature is lower than the measured one, the incorporated heat loss in the observer is higher than the true heat loss in paraffin during the closed-loop experiment. This might be caused by overestimating the calibrated heat transfer coefficient h. To investigate the validity, the numerical simulation of the closed-loop system of the model (11.46), the measurement (11.51), the observer in Algorithm 1, and the output feedback control law (11.64) is studied, where the heat transfer coefficient in the observer is

290

11 Experimental Study with Paraffin Melting 90 80 70 60 50 40 30 0

0.5

1

1.5

2

1.5

2

(a) 60 55 50 45 40 35 30 0

Depth

0.5

1

(b) Fig. 11.7 The time evolution of the estimated temperature profile and the measured temperature profile of the cylinder by IR camera. (a) The estimated temperature profile of the liquid paraffin at every hour. The profile gradually converged to the reference profile given by (11.8) and almost corresponded to the reference after 4 h. (b) The measured temperature profile of the acrylic chamber obtained by IR camera at every hour. The profile is given along the white arrow in the thermography

set as h =20 W/m2 K while the one in the model is set as h =16 W/m2 K. Figure 11.8 depicts the evolution of the measured surface temperature (green dots) at every sampling time and the estimated surface temperature (blue line), respectively. We observe that the plot in Fig. 11.8 is in good agreement with Fig. 11.6b, which leads us to conjecture that the cause of the estimation error lies in the parametric error of the calibrated heat loss h. Nevertheless, the control’s performance was robust as we see in Fig. 11.5.

11.6 Proof of Theoretical Results

291

100

90

80

70 0

1

2

3

4

5

Fig. 11.8 Simulation of the closed-loop system with setting h =16 W/m2 K in the model while h =20 W/m2 K in the observer. The plot is similar to Fig. 11.6b, by which we conjecture that the estimation error of the surface temperature in Fig. 11.6b is caused by the parameter error of h

11.6 Proof of Theoretical Results Hereafter we define ε=

qlos . k

(11.73)

Proof of Theorem 11.1 Guaranteeing Conditions of Model Validity First, we prove the following lemma to guarantee the conditions of model validity. Lemma 11.3 Under Assumption 11.3, consider the closed-loop system consisting of the plant (11.2)–(11.5) with the control law (11.15). Then, the following properties hold for all t ≥ 0: T (x, t) >Tm ,

∀x ∈ (0, s(t)),

∂T (s(t), t) qlos , 0 < s(t) 0 for all t ≥ 0. With the help of this inequality, we can apply the theorem in [178, page 3] to the governing equations (11.2)–(11.5), and thereby for any t¯ ≤ σ where 0 < σ ≤ ∞, there is a unique solution of the system (11.2)–(11.5) with satisfying the properties (11.74) and 0 < s(t) < L for all t ∈ (0, t¯), and if σ = ∞ then s(σ ) = 0 or s(σ ) = L. However, by (11.11), (11.14), and (11.79) with the help of E(0) > 0, we obtain E(t) > 0, which at least ensures that s(σ ) = 0. In addition, by (11.79), we have β α



s(t)

(T (x, t) − Tm )dx < −s(t) + sr +

0

βεsr2 , 2α

(11.80)

for all t ∈ (0, t¯). Applying (11.74) to (11.80) with the help of Assumption 11.3 yields (11.77) for all t ∈ (0, t¯). Thus, we also derive s(σ ) = L. Therefore, σ = ∞, and the properties (11.74), (11.75), and (11.77) hold for all t ≥ 0. Finally, applying (11.79) and (11.74) to (11.14) leads to (11.76), from which we additionally have the following property:  |s(t) − sr | ≤ M := max

 βεsr2 , sr . 2α

(11.81)

Backstepping Transformation Next, we define the reference error states (u, X) as u(x, t) = T (x, t) − Tr (x),

X(t) = s(t) − sr .

(11.82)

Then, rewriting the system’s dynamics (11.2)–(11.5) with respect to (u, X) leads to the following reference error system: ∂u ∂ 2u (x, t) =α 2 (x, t), ∂t ∂x ∂u (0, t) = − q˜c (t)/k, ∂x

0 < x < s(t),

(11.83) (11.84)

11.6 Proof of Theoretical Results

293

u(s(t), t) =εX(t), ∂u ˙ X(t) = − β (s(t), t), ∂x

(11.85) (11.86)

where ˜ q˜c (t) :=qc (t) − h(T (0, t) − Ta ) − qlos = −cE(t).

(11.87)

Referring to the procedure in Chap. 2, we introduce the following backstepping transformation: w(x, t) =u(x, t) −

β α



s(t)

φ(x − y)u(y, t)dy − φ(x − s(t))X(t),

(11.88)

x

where the gain kernel function φ is given by c φ(x) = x. β

(11.89)

We derive the transformed (w, X)-system. Taking the time and spatial derivatives of (11.88) together with the solution of (11.83)–(11.86), and substituting the control law (11.15), the target (w, X)-system is derived as follows:   ∂w ∂ 2w c c (x, t) =α 2 (x, t) − cεX(t) − s˙ (t) (x − s(t))ε − X(t), ∂t α β ∂x

(11.90)

cε ∂w (0, t) = − X(t)2 , ∂x 2α

(11.91)

w(s(t), t) =εX(t),

(11.92)

∂w ˙ (s(t), t). X(t) = − cX(t) − β ∂x

(11.93)

Stability Analysis We prove the stability of (w, X)-system governed by (11.90)–(11.93) using Lyapunov’s method. Let V be the Lyapunov function defined by V =

1 ε ||w||2 + X(t)2 . 2α 2β

(11.94)

Note that Poincare’s and Agmon’s inequalities for the system (11.90)–(11.93) with 0 < s(t) < s¯ lead to

294

11 Experimental Study with Paraffin Melting

// // // ∂w //2 // , ||w||2 ≤2¯s ε2 X(t)2 + 4¯s 2 //// ∂x // // // // ∂w //2 // . w(0, t)2 ≤2ε2 X(t)2 + 4¯s //// ∂x //

(11.95) (11.96)

Taking the time derivative of (11.94) along the solution of (11.90)–(11.93) yields // //  // ∂w //2 cε s(t) s˙ (t) cε 2 / / / / ˙ V = w(s(t), t) − // w(0, t)X(t)2 − w(x, t)dxX(t) + 2α ∂x // α 0 2α  c˙s (t) s(t) cε − f (x)w(x, t)dxX(t) − X(t)2 , (11.97) α β 0 where f =

1 1 (x − s(t))ε − . α β

(11.98)

Applying Young’s inequality to the two terms in the second line of (11.97), we get cε − α

 0

s(t)

w(x, t)dxX(t) ≤

cε βcεs¯ X(t)2 + ||w||2 , 2β 2α 2

cε 1 s¯ c2 ε2 X(t)4 . w(0, t)X(t)2 ≤ w(0, t)2 + 2α 8¯s 2α 2

(11.99) (11.100)

In addition, applying Young’s and Cauchy-Schwarz inequalities to the term in third line of (11.97), we get c˙s (t) − α

 0

s(t)

c|˙s (t)| f (x)w(x, t)dxX(t) ≤ 2α



 √ 1 2 2 2 √ ||f || ||w|| + ε|X(t)| . ε (11.101)

Applying (11.99)–(11.101), and (11.95)–(11.96) to (11.97) with the help of |X| ≤ M derived in (11.81) leads to the following inequality: 

   1 ε2 s¯ c2 ε2 M 2 cε βcεs¯ |˙s (t)| 2 2 ||w|| X(t)2 + − − ε X(t)2 − − 2 2 2 2β 2¯s 2α 8¯s 2α 2α   √ c|˙s (t)| 1 2 2 2 (11.102) √ ||f || ||w|| + ε|X(t)| . 2α ε

V˙ ≤ −

Noting the property (11.75), the dynamics (11.5) yields the following bound:

11.6 Proof of Theoretical Results

295

|˙s (t)| ≤ −β

∂T (s(t), t) + βε. ∂x

(11.103)

Let z(t) be a variable defined by z(t) = s(t) + 2βεt.

(11.104)

The time derivative of (11.104) is given by z˙ (t) = −β

∂T (s(t), t) + βε > 0. ∂x

(11.105)

Therefore, |˙s (t)| ≤ z˙ (t) holds. Applying this inequality to (11.102), and supposing that the following inequalities hold: 1>

8βcεs¯ 3 , α2

(11.106)

c 2ε 2¯s c2 εM 2 > + , β s¯ α2

(11.107)

V˙ ≤ −bV + a z˙ (t)V ,

(11.108)

we get

where      β 2α ε2 s¯ 3 s¯ βε c , max √ , + + √ α α β2 ε 3α 2 ε 0 α c1 b = min . , 8¯s 2 2

a=

(11.109) (11.110)

Consider the functional W defined by W = V e−az(t) .

(11.111)

Then, the time derivative is shown to satisfy W˙ ≤ (V˙ − a z˙ (t)V )e−az(t) ≤ −bW (t),

(11.112)

which leads to W (t) ≤ W (0)e−bt , and hence V (t) ≤ea(z(t)−z(0)) V (0)e−bt = ea(s(t)−s(0)) e2aβεt V (0)e−bt b

≤ea s¯ V (0)e− 2 t ,

(11.113)

296

11 Experimental Study with Paraffin Melting

under the condition 2aβε < b2 , which is equivalent to     0 α 1 √ √ ε2 s¯ 3 s¯ βε2 8βc < min , β + ε + ,c . max 2α ε 2 2 2 α α 3α β 4¯s

(11.114)

Finally, all the conditions (11.106), (11.107), (11.114) introduced in the stability proof hold for sufficiently small ε > 0, i.e., there exists a positive constant ε∗ > 0 such that for all ε ∈ (0, ε∗ ) the conditions hold and therefore the decay of the norm (11.113) is satisfied, from which we complete the proof of Theorem 11.1.

Proof of Theorem 11.2 Since the procedure in the proof of Theorem 11.2 is analogous to the proof of Theorem 11.1, we omit here. We show only the proof of the properties in Lemma 11.3. Here, the reference error of the energy (11.13) is redefined by k ˜ E(t) = α



s(t) 

0

 εs 2 Tˆ (x, t) − Tm dx − r 2

 +

k (s(t) − sr ). β

(11.115)

Taking the time derivative of (11.115) with the help of (11.20)–(11.22) and (11.29) leads to ˜ ˜ + κ T˜ (0, t) − k ∂ T (s(t), t). E˙˜ = −cE(t) ∂x

(11.116)

Applying the maximum principle and Hopf’s lemma to (11.24)–(11.26) yields the following properties: T˜ (x, t) ≤0, ∂ T˜ (s(t), t) ≥0, ∂x

∀x ∈ [0, s(t)],

∀t ≥ 0,

(11.117)

∀t ≥ 0.

(11.118)

˙˜ ˜ E(t) ≤ −cE(t),

(11.119)

Thus, (11.116) yields

and applying the comparison principle, one obtains −ct ˜ ˜ . E(t) ≤ E(0)e

(11.120)

11.7 Comments and Remarks

297

Finally, applying the same steps from (11.79) to (11.81), we deduce that all the properties in Lemma 11.3 hold under the output feedback control system. Then, using the same procedure as in the proof of Theorem 11.1 leads to Theorem 11.2.

11.7 Comments and Remarks This chapter has shown the experimental validation of a boundary feedback control algorithm developed for the phase change process. The physical model is formulated by a Stefan system governed by a parabolic PDE with a state-dependent moving boundary described by an ODE, with unknown heat losses at both the surface and the phase interface. The nominal continuous-time full-state feedback control has been presented by means of energy-shaping and the closed-loop stability is proven by applying the backstepping-based state transformation and Lyapunov method. Then, an implementable control algorithm is developed by further designing an observer-based output feedback with finite-dimensional approximation, under the sampled-data measurements of the surface temperature and the interface position. The experiment was conducted by melting the paraffin with a cylindrical shape. The unknown parameters of the heat losses are calibrated using the experimental data under a constant input. Finally, the proposed feedback control was implemented in the experiment, which provided a successful result of the convergence of the phase interface position to a priori chosen setpoint position. This chapter has provided the first experimental result of the boundary feedback control for the phase change process modeled by the Stefan problem. Therefore, there are several potential future direction in experimental validation of the extended models such as the two-phase Stefan problem in Chap. 5, the Stefan problem under materials’ convection modeled for the polymer 3D-printing in Chap. 9, and the delay-compensated control under the actuator delay in Chap. 4. Developing event-triggered control is also an interesting problem, which has been designed for both linear and nonlinear ODEs [196], networked control systems [152], model predictive control systems [86], delay systems [153], and PDE dynamics for both hyperbolic and parabolic systems using backstepping approach [65–67]. Another direction is designing an adaptive control to simultaneously regulate the input and learn the unknown parameters following [102, 185].

Chapter 12

Open Problems

In this chapter, we introduce several open application problems for control of the Stefan system. As indicated in Chap. 6, the readers are welcome to pursue the topics.

12.1 Tumor Growth Modeling the dynamics of tumor cell is crucial for enhancing the capability of predicting the tumor growth in patients’ body. Byrne and Chaplain [30] proposed a model of the growth of nonnecrotic (living cells) and vascularized (receiving blood supply through vessels) tumor in the absence and presence of inhibitors. The tumor’s evolution is modeled by a Stefan system where the growth is driven by a diffusing nutrient concentration. The analysis of their model was extended and rectified by [76], specifically on the existence and uniqueness of the system when the birth rate of cells exceeds their death rate at the tumor’s boundary in the absence of the inhibitor. The analysis was extended by [49] to the system in the presence of inhibitor. For a practical antiangiogenic therapy, [83] introduced a theory for tumor growth under angiogenic inhibitor control which is both explanatory and clinically implementable.

© Springer Nature Switzerland AG 2020 S. Koga, M. Krstic, Materials Phase Change PDE Control & Estimation, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-58490-0_12

299

300

12 Open Problems

Model Description Inhibitor tumor

r 0

¯ R(t) R

The model of the tumor growth proposed by [30] is described by the following dynamics: D1 ∂ ∂σ (r, t) = 2 ∂t r ∂r

 r2

 ∂σ (r, t) + Γ (σB − σ (r, t)) + g1 (σ, β), ∂r 0 < r < R(t),

(12.1)

where σ (r, t) is the nutrient concentration of the tumor, D1 is the diffusion coefficient, σB is a constant nutrient concentration in vasculature (blood vessel), and Γ is the rate of blood-tissue transfer per unit length (assumed constant). When the tumor growth is in the avascular phase, namely, when the growth is rapidly promoted, we have Γ = 0. The domain of PDE is governed by the tumor radius R(t) which evolves in time. The boundary conditions of the nutrient concentration are described as ∂σ (0, t) =0, ∂r

(12.2)

σ (R(t), t) =σ¯ .

(12.3)

Consider the process of injecting drugs as an inhibitor from the outside the tumor, as proposed in [189]. Assuming that similar effects govern the evolution of the inhibitor in the tumor as developed in [30, 49], the following reaction-diffusion equation is also obtained: D2 ∂ ∂β (r, t) = 2 ∂t r ∂r

  2 ∂β (r, t) + g2 (σ, β), r ∂r

¯ 0 < r < R,

(12.4)

where β(r, t) is the inhibitor concentration, D2 is the diffusion coefficient of inhibitor, and R¯ is the outer radius where the inhibitor is injected. We consider the boundary control of the inhibitor concentration from the outside the tumor radius. The boundary conditions of the inhibitor concentration are described as

12.1 Tumor Growth

301

¯ t) =U (t), β(R,

(12.5)

∂β (0, t) =0. ∂r

(12.6)

Let S(σ, β) denote the cell proliferation rate at a point inside the tumor. Then, mass conservation yields the evolution of the tumor radius governed by 1 ˙ = R(t)2 R(t) 3



R(t)

S(σ, β)r 2 dr.

(12.7)

0

Referring to [49], the cell proliferation rate is formulated by S(σ, β) = μ(σ (r, t) − σ˜ ) − νβ(r, t),

(12.8)

where σ˜ < σ¯ . To simplify the problem, we impose the following assumptions. Assumption 12.1 The tumor is in an avascular phase, i.e., Γ = 0. Assumption 12.2 Referring to [49], the reaction terms are simplified as g1 (σ, β) = − λ0 σ (r, t) − γ1 β(r, t), g2 (σ, β) = − γ2 β(r, t).

(12.9) (12.10)

Under Assumptions 12.1 and 12.2, the tumor and inhibitor dynamics (12.1)–(12.7) are described by the following system: D1 ∂ ∂σ (r, t) = 2 ∂t r ∂r

  2 ∂σ (r, t) − λ0 σ (r, t) − γ1 β(r, t), r ∂r

  D2 ∂ ∂β 2 ∂β (r, t) = 2 (r, t) − γ2 β(r, t), r ∂t ∂r r ∂r

0 < r < R(t), (12.11)

¯ 0 < r < R,

(12.12)

∂σ (0, t) =0, ∂r

(12.13)

σ (R(t), t) =σ¯ ,

(12.14)

¯ t) =U (t), β(R, ∂β (0, t) =0, ∂r  R(t) 1 ˙ = R(t)2 R(t) (μ(σ (r, t) − σ˜ ) − νβ(r, t))r 2 dr. 3 0

(12.15) (12.16) (12.17)

The objective of our work is to design the injecting inhibitor concentration U (t) to prevent the tumor’s evolution by achieving the desired size Rr .

302

12 Open Problems

Steady-State Solution For a given desired size Rr of the tumor, we solve the steady-state profiles (σeq (r), βeq (r)). Setting the time derivatives in (12.11)–(12.17) to zero, one can derive that the steady-state solution must satisfy the following conditions: D1  σ (r) − λ0 σeq (r) − γ1 βeq (r), 2r eq D2   β (r) − γ2 βeq (r), (r) + 0 =D2 βeq 2r eq  0 =D1 σeq (r) +

(12.18) (12.19)

 σeq (0) =0,

(12.20)

σeq (Rr , t) =σ¯ ,

(12.21)

¯ =U ∗ , βeq (R)

(12.22)

 βeq (0)

=0,  0=

(12.23) Rr

(μ(σeq (r) − σ˜ ) − νβeq (r))r 2 dr.

(12.24)

0

Then, by all the conditions (12.18)–(12.24), the steady-state solution (σeq (r), βeq (r)) are obtained by      1 λ¯ 0 r + K2 sinh K1 sinh γ¯2 r , r   √  K2 γ¯2 − λ¯ 0 sinh γ¯2 r βeq (r) = , γ¯1 r σeq (r) =

(12.25) (12.26)

where γ¯1 =

γ1 , D1

γ¯2 =

γ2 , D2

λ¯ 0 =

λ0 , D1

(12.27)

and K1 and K2 satisfy     ¯ λ0 Rr + K2 sinh γ¯2 Rr =σ¯ Rr , K1 sinh

(12.28)



  γ¯2 − λ¯ 0 μσ˜ Rr3 , μg(λ¯ 0 )K1 + μ − ν g(γ¯2 )K2 = γ¯1 3 where we define

(12.29)

12.1 Tumor Growth

1 g(x) = √ x

303

   √   √  1 Rr cosh Rr x − √ sinh Rr x . x

(12.30)

Reference Error System Let v(x, t), u(x, t), and X(t) be the reference error variables defined by v(r, t) =σ (r, t) − σeq (r),

(12.31)

u(r, t) =β(r, t) − βeq (r),

(12.32)

1 X(t) = (R(t)3 − Rr3 ). 3

(12.33)

Then, by subtracting (12.18)–(12.24) from the model (12.11)–(12.17), the reference error system is obtained as D1 ∂ ∂v (r, t) = 2 ∂t r ∂r

  ∂v r 2 (r, t) − λ0 v(r, t) − γ1 u(r, t), ∂r

  D2 ∂ ∂u ∂u (r, t) = 2 r 2 (r, t) − γ2 u(r, t), ∂t ∂r r ∂r

0 < r < R(t), (12.34)

¯ 0 < r < R,

∂v (0, t) =0, ∂r

(12.35) (12.36)

v(R(t), t) =h(R(t)),

(12.37)

¯ t) =U (t), u(R,

(12.38)

∂u (0, t) =0, ∂r

(12.39)

˙ X(t) =f (R(t)) +



R(t)

(μv(r, t) − νu(r, t))r 2 dr,

(12.40)

0

where h(R(t)) =σ¯ − σeq (R(t)),  R(t) f (R(t)) = (μ(σeq (r) − σ˜ ) − νβeq (r))r 2 dr. Rr

(12.41) (12.42)

304

12 Open Problems

Linearized Reference Error System Taking the linearization of the nonlinear function h(R(t)) defined by (12.41) around X = 0, one can obtain / ∂h // h(R(t)) ≈h|X=0 + X(t), ∂X /X=0 / ∂R ∂h // X(t), =h(Rr ) + ∂X ∂R /X=0 1/3 /  / ∂ 3X + Rr3 / h (Rr )X(t), =σ¯ − σeq (Rr ) + / ∂X X=0 −2/3 //  3  / σeq (Rr )X(t), = − 3X + Rr / X=0

=−

 (R ) σeq r X(t), 2 Rr

(12.43)

where (12.21) is applied. In a similar manner, the linearization of f (R(t)) given in (12.42) leads to f (R(t)) ≈f (Rr ) + Rr−2 f  (Rr )X(t), =Rr−2 (μ(σeq (Rr ) − σ˜ ) − νβeq (Rr ))Rr2 X(t) =(μ(σ¯ − σ˜ ) − νβeq (Rr ))X(t).

(12.44)

Thus, by defining C := −

 (R ) σeq r

Rr2

(12.45)

,

A :=μ(σ¯ − σ˜ ) − νβeq (Rr ),

(12.46)

the linearized system of (12.34)–(12.40) is obtained as ∂v D1 ∂ (r, t) = 2 ∂t r ∂r

  ∂v r 2 (r, t) − λ0 v(r, t) − γ1 u(r, t), ∂r

  D2 ∂ ∂u ∂u (r, t) = 2 r 2 (r, t) − γ2 u(r, t), ∂t ∂r r ∂r ∂v (0, t) =0, ∂r

0 < r < R(t), (12.47)

¯ 0 < r < R,

(12.48) (12.49)

12.2 Axonal Growth

305

v(R(t), t) =CX(t),

(12.50)

¯ t) =U (t), u(R,

(12.51)

∂u (0, t) =0, ∂r ˙ X(t) =AX(t) +

(12.52) 

R(t)

(μv(r, t) − νu(r, t))r 2 dr.

(12.53)

0

An important characteristic of the system (12.47)–(12.53) lies in the dynamics of ODE (12.53), which has a distributed effect of both the nutrient and inhibitor concentration profiles. The backstepping method for such a kind of system has been developed in [13] to stabilize a linear ODE driven by a distributed effect of a diffusion-advection PDE with a fixed domain. The extension of the approach to the tumor growth process governed by (12.47)–(12.53) is an exciting open problem.

12.2 Axonal Growth Neuroscience has become one of the most influential areas of biology in the last decades, as scientists pursue the understanding of the functionality of perception and brain as nervous systems [92]. Each neuron transmits electric signals propagated through the axon. Computational modeling for describing the dynamics of such a neuron and axon has been studied in the literature. A first work on the Stefantype PDE model of the axonal growth has been employed in [145], and the broader computational modeling for finally simulating the morphology of neurons has been developed in [80] with showing the experimental validation. In this section, we show a more recent work on the Stefan model for axonal growth developed in [59, 60] and suggest some potential methodology for boundary control.

Model Description In neuronal physiology, it has been verified that the growth of a newborn axon is directly driven by the presence of the group of proteins called “tubulin.” There are two assumptions for modeling the axonal growth given here. One is that the tubulin is the only substance involved in the growth of an axon. Another assumption is that the molecules of free tubulin are so small that we can model it as a homogeneous continuum. The one-dimensional coordinate x is given along the axon, and the length of the axon is denoted by l(t). Let c(x, t) be the concentration of tubulin along the axon, cs (t) be the concentration of tubulin in the soma, and cc (t) be the concentration of tubulin in the cone. The degradation of the tubulin is caused along the axon at the constant rate g. The flux of tubulin is determined by active

306

12 Open Problems

transport by motor proteins having the constant velocity a and diffusion of free tubulin with diffusivity D. For a given volume Vc of the growth cone and the cross-sectional area A, the ratio lc := VAc stands as a length parameter involved with the size of the growth cone. In the cone, consumption of tubulin is caused by the degradation at the rate g and by assembly of dimers to microtubules, which elongates the axon at a constant rate r˜g , where the constant r˜g is the reaction rate of polymerization of GTP (guanosine triphosphate) bound tubulin dimers to microtubule bound GDP (guanosine diphosphate). The assembled microtubules in the growth cone is assumed to disassemble at the constant rate s˜g . The axonal growth model driven by a tubulin is described as ∂c ∂ 2c ∂c (x, t) =D 2 (x, t) − a (x, t) − gc(x, t), ∂t ∂x ∂x

x ∈ (0, l(t)),

(12.54)

c(0, t) =cs (t),

(12.55)

c(l(t), t) =cc (t),

(12.56)

lc

dcc ∂c (t) =(a − glc )cc (t) − D (l(t), t) dt ∂x − (rg cc (t) + r˜g lc )(cc (t) − c∞ ), dl (t) =rg (cc (t) − c∞ ), dt

(12.57) (12.58)

where rg is a lumped parameter involved with concentration-rate constant, defined r˜ V

g c with the effective area of polymerization growth Ag and the density by rg := ρA g of the assembled microtubules ρ. c∞ is an equilibrium concentration in the cone to stop the axonal growth.

Steady-State Solution For a given desired length of the axon ls , we consider the steady-state solution (ceq (x), c∞ , ls ) of the axonal growth model governed by (12.54)–(12.58). The steady-state solution satisfies the conditions in (12.54)–(12.58) with setting the time derivative to zero, which leads to   0 =Dceq (x) − aceq (x) − gceq (x),

(12.59)

ceq (0) =cs∗ ,

(12.60)

ceq (ls ) =c∞ ,

(12.61)

0 =(a

 − glc )c∞ − Dceq (ls ).

(12.62)

12.2 Axonal Growth

307

By the Eqs. (12.59), (12.61), and (12.62), the steady-state solution is obtained by   ceq (x) = c∞ K+ eλ+ (x−ls ) + K− eλ− (x−ls ) ,

(12.63)

where λ+ = K+ =

  a 2 + 4Dg a − a 2 + 4Dg , λ− = , 2D 2D a − 2glc a − 2glc 1 +  , K− = −  . 2 2 2 a + 4Dg 2 a 2 + 4Dg

a+ 1 2

(12.64) (12.65)

Using these values, by (12.60), the steady-state concentration in the soma is given by   cs∗ = c∞ K+ e−λ+ ls + K− e−λ− ls .

(12.66)

Reference Error System Let u(x, t), z1 (t), and z2 (t) be the reference error states defined by u(x, t) =c(x, t) − ceq (x),

(12.67)

z1 (t) =cc (t) − c∞ ,

(12.68)

z2 (t) =l(t) − ls .

(12.69)

By the original system (12.54)–(12.58) and the steady-state solution satisfying (12.59)–(12.62), one can derive the following reference error system: ∂u ∂u ∂ 2u (x, t) =D 2 (x, t) − a (x, t) − gu(x, t), ∂t ∂x ∂x u(0, t) =U (t), u(l(t), t) =cc (t) − ceq (l(t)), ∂u dz1 (t) =az ˜ 1 (t) − β (l(t), t) − κz1 (t)2 , dt ∂x dl (t) =rg z1 (t), dt where

(12.70) (12.71) (12.72) (12.73) (12.74)

308

12 Open Problems

a˜ =

a − rg c∞ − g − r˜g , lc

β=

D , lc

κ=

rg . lc

(12.75) (12.76)

By using the steady-state solution (12.63), the condition (12.72) can be rewritten by ˜ 2 (t)), cc (t) − ceq (l(t)) = z1 (t) + h(z

(12.77)

where   ˜ 2 (t)) = c∞ 1 − K+ eλ+ z2 (t) − K− eλ− z2 (t) . h(z

(12.78)

Let X ∈ R2 be ODE state vector defined by X(t) = [z1 (t) z2 (t)]T .

(12.79)

Then, the reference error system (12.70)–(12.74) can be described by the following nonlinear coupled PDE-ODE system: ∂u ∂ 2u ∂u (x, t) =D 2 (x, t) − a (x, t) − gu(x, t), ∂t ∂x ∂x u(0, t) =U (t),

(12.80) (12.81)

˜ 2 (t)), u(l(t), t) =z1 (t) + h(z

(12.82)

∂u d X(t) =AX(t) + f (X(t)) + B (l(t), t), dt ∂x

(12.83)

where 

   a˜ 0 −β A= , B= , rg 0 0 2  f (X(t)) = − κ e1T X(t) , ˜ 2T X(t)), h(X(t)) =e1T X(t) + h(e e1 =[1 0],

e2 = [0 1].

(12.84) (12.85) (12.86) (12.87)

12.2 Axonal Growth

309

Linearized Reference Error System Applying linearization of X(t) around zero states, we get the following linearized reference error system: ∂u ∂u ∂ 2u (x, t) =D 2 (x, t) − a (x, t) − gu(x, t), ∂t ∂x ∂x

(12.88)

u(0, t) =U (t),

(12.89)

u(l(t), t) =C T X(t),

(12.90)

∂u d X(t) =AX(t) + B (l(t), t), dt ∂x

(12.91)

where  C= 1

(a − glc )c∞ − D

T .

(12.92)

Backstepping Transformation Following the procedure in [198], we can obtain the backstepping transformation given by 

l(t)

w(x, t) = u(x, t) −

k(x, y)u(y, t)dy − φ(x − l(t))T X(t),

(12.93)

x

where k(x, y) ∈ R and φ(x − l(t)) ∈ R2 are the gain kernel functions to be determined, which transforms onto the target system ∂w ∂ 2w ∂w (x, t) =D 2 (x, t) − a (x, t) − gw(x, t) ∂t ∂x ∂x   ˙ − l(t) k(x, l(t))u(l(t), t) − φ  (x − l(t))T X(t) ,

(12.94)

w(0, t) =0,

(12.95)

w(l(t), t) =0,

(12.96)

d ∂w X(t) =(A + BK)X(t) + B (l(t), t), dt ∂x

(12.97)

where K ∈ R2 is a control gain chosen to make A + BK Hurwitz. In detail, by setting

310

12 Open Problems

K = [k1

k2 ],

(12.98)

one can obtain the conditions for (k1 , k2 ) to make A + BK Hurwitz as k1 >

a˜ , β

k2 > 0.

(12.99)

Moreover, by the condition (12.74) and (12.90), the nonlinear target PDE (12.94) is rewritten by ∂w ∂ 2w ∂w (x, t) =D 2 (x, t) − a (x, t) − gw(x, t) + F (x, X(t)), ∂t ∂x ∂x

(12.100)

where   F (x, X(t)) = −rg e1T X(t) k(x, X(t) + ls )C T − φ  (x − X(t) − ls )T X(t). (12.101)

Further Challenges One of the remaining challenges is to guarantee the stability of the nonlinear target PDE-ODE governed by (12.100), (12.95)–(12.97). A good property of the nonlinear target PDE (12.100) is that the additional nonlinear term (12.101) is a function of only the ODE state X(t), which enhances a prospect for proving a local stability of the nonlinear target system (12.100), (12.95)–(12.97) by Lyapunov analysis, as performed in [46]. This is not the case of the Stefan system studied in the previous chapters. Namely, the target system of the Stefan system in the previous chapters contains a nonlinear term involved with the Neumann boundary value of the PDE state, which renders a further challenge to prove a local stability through an analysis of a high order norm. Another problem we have not focused on, but consider possibly important, is a practical demand of desired length of axon, and practical methods for controlling the tubulin in the soma and sensing the tubulin concentration.

12.3 Comments and Remarks As introduced in the Preface, there are several other applications in which the model is described by the Stefan system studied in literature, which are provided in the following list.

12.3 Comments and Remarks

• • • • •

311

cancer treatment via cryosurgery [131, 166, 167], spreading of invasive species in ecology [62], information diffusion on online social networks [137, 218], domain walls in ferroelectric thin films [144], Black-Scholes model of American option pricing [39].

Designing controllers and estimators, guaranteeing stability and model validity, and investigating the performance in numerical simulation are significant tasks for the applications of PDE control to the aforementioned subjects. We heartily recommend them to the readers. The book provides supporting technical materials. We wish we had the time to pursue some of these applications, as this is where the true impact lies, even in expanding the theoretical frontier.

Appendix A

Bessel Functions

Bessel function of the first kind is a solution to the following Bessel’s differential equation: x2

d 2y dy + (x 2 − n2 )y = 0, +x dx dx 2

(A.1)

where n is generally an arbitrary complex number but here we consider a positive integer n ∈ {1, 2, · · · }. A series representation of the solution y = Jn (x) is described by Jn (x) =

∞  (−1)m (x/2)n+2m . m!(m + n)!

(A.2)

m=0

Modified Bessel function of the first kind is a solution to the following modified Bessel’s differential equation: x2

d 2y dy + (x 2 − n2 )y = 0. +x dx dx 2

(A.3)

A series representation of the solution y = In (x) is described by In (x) =

∞  (x/2)n+2m . m!(m + n)!

(A.4)

m=0

© Springer Nature Switzerland AG 2020 S. Koga, M. Krstic, Materials Phase Change PDE Control & Estimation, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-58490-0

313

314

A Bessel Functions

Some properties of the functions are given by 2nJn (x) = x(Jn−1 (x) + Jn+1 (x)),

(A.5)

Jn (−x) = (−1) Jn (x),

(A.6)

n

In (x) = i −n Jn (ix),

In (ix) = i n Jn (x),

(A.7)

2nIn (x) = x(In−1 (x) − In+1 (x)),

(A.8)

In (−x) = (−1) In (x).

(A.9)

n

Derivatives are given by d 1 n Jn (x) = (Jn−1 (x) − Jn+1 (x)) = Jn (x) − Jn+1 (x), dx 2 x d n d −n (x Jn (x)) =x n Jn−1 (x), (x Jn (x)) = −x −n Jn+1 (x), dx dx d 1 n In (x) = (In−1 (x) + In+1 (x)) = In (x) + In+1 (x), dx 2 x d n d (x In (x)) =x n In−1 (x), (x −n In (x)) = x −n In+1 (x). dx dx

(A.10) (A.11) (A.12) (A.13)

Appendix B

Some Inequalities

B.1 Cauchy-Schwarz Inequality



D

 f (x)g(x)dx ≤

0

D

f (x)2 dx

1/2  ·

0

D

1/2 g(x)2 dx

.

(B.1)

0

B.2 Poincare’s Inequality 

D

 w(x) dx ≤2Dw(D) + 4D 2

2

0



D

 w(x) dx ≤2Dw(0) + 4D 2

2

0

D

2

wx (x)2 dx

(B.2)

wx (x)2 dx.

(B.3)

0

2

D

0

Proof 

D 0

 w(x) dx = 2

xw(x)2 |D 0

D

−2

xw(x)wx (x)dx 0



D

= Dw(D) − 2 2

xw(x)wx (x)dx 0

© Springer Nature Switzerland AG 2020 S. Koga, M. Krstic, Materials Phase Change PDE Control & Estimation, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-58490-0

315

316

B Some Inequalities

1 2

≤ Dw(D)2 +



D



D

w(x)2 dx + 2

0

x 2 wx (x)2 dx.

0

Thus, we arrived at (B.2). 

D

 w(x) dx = (x 2

0

− D)w(x)2 |D 0 

D

= Dw(0) − 2 2

D

−2

(x − D)w(x)wx (x)dx

0

(x − D)w(x)wx (x)dx

0

1 ≤ Dw(0) + 2



2

D



D

w(x) dx + 2 2

0

(x − D)2 wx (x)2 dx.

0

Thus, we arrived at (B.3).

B.3 Agmon’s Inequality

Agmon’s Inequality (Case 1) ||w||2∞ ≤ w(0)2 + 2||w||2 ||wx ||2

(B.4)

Proof 

x

1 (w(x)2 − w(0)2 ). 2

w(x)wx (x)dx =

0

Taking absolute values and triangle inequality, 

x

w(x) ≤ w(0) + 2 2

2

|w(x)||wx (x)|dx

0



D

≤ w(0) + 2 2

|w(x)||wx (x)|dx.

0

Because the left-hand side does not depend on x, we arrive at (B.5).

Agmon’s Inequality (Case 1: Extended) ||w||2∞ ≤ 2w(0)2 + 4D||wx ||2 .

(B.5)

B.3 Agmon’s Inequality

317

Proof By (B.5) and applying Young’s inequality with γ D, we have 1 ||w||22 + γ D||wx ||2 γD     4 2 2 + γ D||wx ||2 . w(0) + ≤ 1+ γ γ

||w||2∞ ≤ w(0)2 +

(B.6)

Setting γ = 2 leads to the inequality.

Agmon’s Inequality (Case 2) w(0)2 ≤

D+1 ||w||2 + ||wx ||2 . D

(B.7)

Proof Taking integral, we have 

x

−

w(x)wx (x)dx =

0

1 (w(0)2 − w(x)2 ). 2

In addition, by Young’s inequality, we have 

x

−



 1 w(y)2 + wy (y)2 dy 0 2  1 ≤ ||w||2 + ||wx ||2 . 2

w(x)wx (x)dx ≤

0

x

(B.8)

Taking integration, we have 

D 0

1 (w(0)2 − w(x)2 )dx ≤ 2



D 0

 1 ||w||2 + ||wx ||2 dx. 2

(B.9)

For non-x-dependent terms, we obtain   Dw(0)2 − ||w||2 ≤ D ||w||2 + ||wx ||2 .

(B.10)

Therefore, we arrive at w(0)2 ≤

D+1 ||w||2 + ||wx ||2 . D

(B.11)

318

B Some Inequalities

In the same way, we have 

D

1 w(x)wx (x)dx = (w(D)2 − w(x)2 ) 2 x  D  D   1 w(y)2 + wy (y)2 dy w(x)wx (x)dx ≤ x x 2  1 ≤ ||w||2 + ||wx ||2 . 2

(B.12)

Therefore, we arrive at w(D)2 ≤

D+1 ||w||2 + ||wx ||2 . D

(B.13)

Appendix C

Stable Systems and Their Proofs

C.1 One-Phase Stefan Problem with Monotonic Interface Consider the system wt (x, t) =αwxx (x, t) + s˙ (t)φ  (x − s(t))X(t),

(C.1)

wx (0, t) =0,

(C.2)

w(s(t), t) =0,

(C.3)

˙ X(t) = − cX(t) − βwx (s(t), t).

(C.4)

Lemma C.1 With the conditions s˙ (t) > 0,

0 < s(t) < s¯ ,

(C.5)

for some positive constant s¯ > 0, (w, X)-system in (C.1)–(C.4) is exponentially stable at the origin in the sense of the spatial H1 -norm defined by  Φ(t) :=

s(t)



s(t)

w(x, t) dx + 2

0

wx (x, t)2 dx + X(t)2 .

(C.6)

0

Proof 1 V1 = 2



s(t)

w(x, t)2 dx.

(C.7)

0

Taking the time derivative of (C.7), we have

© Springer Nature Switzerland AG 2020 S. Koga, M. Krstic, Materials Phase Change PDE Control & Estimation, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-58490-0

319

320

C Stable Systems and Their Proofs



s(t)

1 w(x, t)wt (x, t)dx + s˙ (t)w(s(t), t)2 2 0  s(t)  s(t) =α w(x, t)wxx (x, t)dx + s˙ (t)X(t) φ  (x − s(t))w(x, t)dx

V˙1 =

0 y=s(t)

= αw(x, t)wx (x, t)|y=0 

s(t)

+ s˙ (t)X(t)

0



s(t)

−α

φ  (x − s(t))w(x, t)dx

0



s(t)

= −α

wx (x, t)2 dx

0



wx (x, t) dx + s˙ (t)X(t) 2

0

s(t)

φ  (x − s(t))w(x, t)dx.

(C.8)

0

Next, we consider V2 defined by 1 V2 = 2



s(t)

wx (x, t)2 dx.

(C.9)

0

Taking the time derivative of (C.9), we get 

s(t)

1 wx (x, t)wxt (x, t)dx + s˙ (t)wx (s(t), t)2 2 0  s(t) 1 x=s(t) =wx (x, t)wt (x, t)|x=0 − wxx (x, t)wt (x, t)dx + s˙ (t)wx (s(t), t)2 2 0  s(t) =wx (s(t), t)wt (s(t), t) − α wxx (x, t)2 dx

V˙2 =

 − s˙ (t)X(t) 0

0 s(t)

1 φ  (x − s(t))wxx (x, t)dx + s˙ (t)wx (s(t), t)2 . 2

(C.10)

Taking the total time derivative of (C.2) on both sides, we obtain the following: d w(s(t), t) = wt (s(t), t) + s˙ (t)wx (s(t), t) = 0, dt

(C.11)

which yields wt (s(t), t) = −˙s (t)wx (s(t), t).

(C.12)

Moreover, the integration by parts in first term in the last line in (C.10) with the help of (C.2) is given by

C.1 One-Phase Stefan Problem with Monotonic Interface



s(t)

φ  (x − s(t))wxx (x, t)dx = φ  (0)wx (s(t), t) −

321



0

s(t)

φ  (x − s(t))wx (x, t)dx.

0

(C.13) Therefore, plugging (C.12) and (C.13) into (C.10), we arrive at 

s(t)

1 wxx (x, t)2 dx − s˙ (t)wx (s(t), t)2 2 0    s(t) − s˙ (t)X(t) φ  (0)wx (s(t), t) − φ  (x − s(t))wx (x, t)dx .

V˙2 = − α

(C.14)

0

Next, we consider V3 defined by V3 =

1 X(t)2 . 2

(C.15)

Using (C.4), the time derivative of (C.15) is given by ˙ V˙3 =X(t)X(t) = − cX(t)2 − βX(t)wx (s(t), t).

(C.16)

Let V be the functional defined by V = V1 + V2 + pV3 .

(C.17)

By (C.8), (C.14), and (C.16), the time derivative of (C.17) is given by V˙ = −α



s(t)

0

 wxx (x, t)2 dx − α



+ s˙ (t)X(t)

s(t)

wx (x, t)2 dx − pcX(t)2 − pβX(t)wx (s(t), t)

0 s(t)

φ  (x − s(t))w(x, t)dx −

0







s(t)

− s˙ (t)X(t) φ (0)wx (s(t), t) −

s˙ (t) wx (s(t), t)2 2





φ (x − s(t))wx (x, t)dx .

(C.18)

0

Using the fact that s˙ (t) > 0 and applying Young’s inequality yield − pβX(t)wx (s(t), t) ≤

  p β2 wx (s(t), t)2 , cX(t)2 + 2 c

(C.19)

322

C Stable Systems and Their Proofs

 s˙ (t)X(t)

s(t)

φ  (x − s(t))w(x, t)dx

0

⎛ 2 ⎞  s(t) 1 s˙ (t) ⎝ φ  (x − s(t))w(x, t)dx ⎠ , ≤ γ1 X(t)2 + 2 γ1 0 − s˙ (t)X(t)φ  (0)wx (s(t), t) ≤

(C.20)

  s˙ (t) 1 γ2 φ  (0)2 X(t)2 + wx (s(t), t)2 , 2 γ2 (C.21)

for some positive constants γ1 > 0 and γ2 > 0. Here we choose γ1 = φ  (0)2 ,

γ2 = 1.

(C.22)

Also, by Cauchy-Schwarz inequality, we have  2  s(t)



φ (x − s(t))w(x, t)dx

s(t)

≤

0



 

φ (x − s(t)) dx 2

0

≤ φ¯ 



s(t)

 2

w(x, t) dx 0

s(t)

w(x, t)2 dx,

(C.23)

0

where φ¯  =



s¯

φ  (−x)2 dx.

(C.24)

0

Applying (C.19)–(C.23) to (C.18), the following inequality on V is derived:  s(t)  s(t) pc pβ 2 V˙ ≤ − α wx (s(t), t)2 wxx (x, t)2 dx − α wx (x, t)2 dx − X(t)2 + 2 2c 0 0    s(t) φ¯  2  2 2 + s˙ (t) w(x, t) dx + φ (0) X(t) . (C.25) 2φ  (0)2 0  s(t) Applying Poincare’s and Agmon’s inequalities which give 0 w(x, t)2 dx ≤  s(t)  s(t) 4¯s 2 0 wx (x, t)2 dx and wx (s(t), t)2 ≤ 4¯s 0 wxx (x, t)2 dx, the inequality (C.25) becomes   s(t)   s(t) pc 2pβ 2 s¯ X(t)2 wxx (x, t)2 dx − α wx (x, t)2 dx − V˙ ≤ − α − c 2 0 0    s(t) φ¯  2  2 2 + s˙ (t) w(x, t) dx + φ (0) X(t) . (C.26) 2φ  (0)2 0

C.2 One-Phase Stefan Problem with Convection and Heat Loss

323

Therefore, by choosing p=

cα , 4β 2 s¯

(C.27)

we arrive at  s(t) α pc X(t)2 wx (x, t) dx − 2 w(x, t)2 dx − 2 4¯s 0 0    s(t) φ¯  2  2 2 + s˙ (t) w(x, t) dx + φ (0) X(t) 2φ  (0)2 0

α V˙ ≤ − 2 8¯s



s(t)

2

≤ − bV + a s˙ (t)V ,

(C.28)

where  φ¯  2φ  (0)2 , , p φ  (0)2 0 α 1 b = min ,c . 2 4¯s 

a = max

(C.29) (C.30)

However, the second term of the right-hand side of (C.28) does not enable to directly conclude the exponential stability. To deal with it, we introduce a new Lyapunov function W defined by W = V e−as(t) .

(C.31)

The time derivative of (C.31) is written as   W˙ = V˙ − a s˙ (t)V e−as(t) ,

(C.32)

and using (C.28) the following estimate can be deduced: W˙ ≤ −bW.

(C.33)

Hence, W (t) ≤ W (0)e−bt , and using 0 < s(t) < s¯ and (C.31), we obtain V (t) ≤ ea s¯ V (0)e−bt .

C.2 One-Phase Stefan Problem with Convection and Heat Loss Consider the system

(C.34)

324

C Stable Systems and Their Proofs

wt (x, t) =αwxx (x, t) + bwx (x, t) − hw(x, t) + s˙ (t)φ  (x − s(t))X(t),

0 < x < s(t)

(C.35)

wx (0, t) =γ w(0, t),

(C.36)

w(s(t), t) =0,

(C.37)

˙ X(t) = − cX(t) − βwx (s(t), t),

(C.38)

where b is! an arbitral " parameter (can be positive or negative), h ≥ 0, c > 0, and b γ > max 0, − 2α . Lemma C.2 With the conditions s˙ (t) > 0,

0 < s(t) < sr ,

(C.39)

for some positive constant sr > 0, (w, X)-system in (C.35)–(C.38) is exponentially stable at the origin in the sense of the spatial H1 -norm defined by 

s(t)

Φ(t) :=



s(t)

w(x, t) dx + 2

0

wx (x, t)2 dx + X(t)2 .

(C.40)

0

Proof We consider a functional V1 defined by 1 V1 = 2



s(t)

w(x, t)2 dx.

(C.41)

0

Taking the time derivative of (C.41) along with (C.35)–(C.38), we have 

s(t)

1 w(x, t)wt (x, t)dx + s˙ (t)w(s(t), t)2 2 0  s(t)  s(t)  =α w(x, t)wxx (x, t)dx + b w(x, t)wx (x, t)dx − h

V˙1 =

0



+ s˙ (t)X(t)

0 s(t)

s(t)

w(x, t)2 dx

0

φ  (x − s(t))w(x, t)dx

0



s(t)

b wx (x, t)2 dx + (w(s(t), t)2 − w(0, t)2 ) 2 0  s(t)  s(t) −h w(x, t)2 dx + s˙ (t)X(t) φ  (x − s(t))w(x, t)dx x=s(t)

=αw(x, t)wx (x, t)|x=0

0

−α

0

  b 2 2 w(0, t)2 = − α||wx || − h||w|| − γ α + 2

C.2 One-Phase Stefan Problem with Convection and Heat Loss

 + s˙ (t)X(t)

s(t)

φ  (x − s(t))w(x, t)dx.

325

(C.42)

0

Applying Young’s inequality with the help of s˙ (t) > 0 and 0 < s(t) < sr , we have  s˙ (t)X(t)

s(t)

φ  (x − s(t))w(x, t)dx ≤

0

where φ := sups(t)∈(0,sr ) ity to (C.42), we get



s(t)  φ (x 0

 s˙ (t)  2 X(t)2 + φ  ||w||2 , 2

(C.43)

− s(t))2 dx. Thus, applying the above inequal-

   b s˙ (t)  2 X(t)2 + φ  ||w||2 . w(0, t)2 + V˙1 ≤ − α||wx ||2 − h||w||2 − γ α + 2 2 (C.44) Next, we consider V2 defined by V2 =

1 2



s(t)

wx (x, t)2 dx.

(C.45)

0

Taking the time derivative of (C.45), we get 

s(t)

1 wx (x, t)wxt (x, t)dx + s˙ (t)wx (s(t), t)2 2 0  s(t) 1 x=s(t) = wx (x, t)wt (x, t)|x=0 − wxx (x, t)wt (x, t)dx + s˙ (t)wx (s(t), t)2 2 0

V˙2 =

= wx (s(t), t)wt (s(t), t) − γ w(0, t)wt (0, t)  s(t) − α||wxx ||2 − b wxx (x, t)wx (x, t)dx 0



s(t)

+h



wxx (x, t)w(x, t)dx − s˙ (t)X(t)

0

s(t)

φ  (x − s(t))wxx (x, t)dx

0

1 + s˙ (t)wx (s(t), t)2 . 2

(C.46)

The boundary condition w(s(t), t) = 0 yields wt (s(t), t) = −˙s (t)wx (s(t), t). Moreover, we have

(C.47)

326

C Stable Systems and Their Proofs



s(t)

φ  (x − s(t))wxx (x, t)dx

0

=φ  (0)wx (s(t), t) − γ φ  (−s(t))w(0, t) −



s(t)

φ  (x − s(t))wx (x, t)dx.

0

(C.48) Therefore, plugging (C.47) and (C.48) into (C.46), we arrive at V˙2 = − α||wxx ||2 − b



s(t)

wxx (x, t)wx (x, t)dx − γ hw(0, t)2 − h||wx ||2

0

− γ w(0, t)wt (0, t) − s˙ (t)X(t)  





× φ (0)wx (s(t), t) − γ φ (−s(t))w(0, t) −

s(t)

 

φ (x − s(t))wx (x, t)dx

0

1 − s˙ (t)wx (s(t), t)2 . 2

(C.49)

Applying Young’s and Cauchy-Schwarz inequalities, we get  −b

s(t)

wxx (x, t)wx (x, t)dx ≤

0

α b2 ||wxx ||2 + ||wx ||. 2 2α

(C.50)

Moreover, applying Young’s inequality with the help of s˙ (t) ≥ 0, we get s˙ (t) (wx (s(t), t)2 + φ  (0)2 X(t)2 ), − s˙ (t)X(t)φ  (0)wx (s(t), t) ≤ 2    s(t)   s˙ (t)X(t) γ φ (−s(t))w(0, t) + φ (x − s(t))wx (x, t)dx

(C.51)

0

≤

 s˙ (t)   2 2 (φs + 1)X(t)2 + γ 2 w(0, t)2 + φ  ||wx ||2 , 2

(C.52)

where φs :=

sup

|φ  (−s(t))|, 

φ 

:=

(C.53)

s(t)∈(0,sr )

sup s(t)∈(0,sr )

s(t)

φ  (x − s(t))2 dx.

0

Applying these inequalities to (C.49), we have

(C.54)

C.2 One-Phase Stefan Problem with Convection and Heat Loss

327

  2 b α 2 ˙ − h ||wx ||2 − γ hw(0, t)2 − γ w(0, t)wt (0, t) V2 ≤ − ||wxx || + 2 2α  s˙ (t)   2 2 2 + (φ (0) + φs + 1)X(t)2 + γ 2 w(0, t)2 + φ  ||wx ||2 . (C.55) 2 Next, we consider V3 defined by V3 =

1 X(t)2 . 2

(C.56)

The time derivative of (C.56) and applying Young’s and Agmon’s inequalities, we get ˙ V˙3 =X(t)X(t) = − cX(t)2 − βX(t)wx (s(t), t) β2 c wx (s(t), t)2 ≤ − X(t)2 + 2 2c c β2 (2wx (0, t)2 + 4sr ||wxx ||2 ) ≤ − X(t)2 + 2 2c c β 2γ 2 2sr β 2 w(0, t)2 + ||wxx ||2 = − X(t)2 + 2 c c c 4sr β 2 γ 2 2sr β 2 ||wx ||2 + ||wxx ||2 . ≤ − X(t)2 + 2 c c

(C.57)

Let V ∗ be the functional defined by V ∗ = V2 +

γ w(0, t)2 + pV3 , 2

(C.58)

where p=

cα . 8β 2 sr

(C.59)

By (C.55) and (C.57), the time derivative of (C.58) satisfies   2 γ 2α b α pc + − h ||wx ||2 − γ hw(0, t)2 − X(t)2 V˙ ∗ ≤ − ||wxx ||2 + 4 2α 2 2  s˙ (t)   2 2 2 (φ (0) + φs + 1)X(t)2 + γ 2 w(0, t)2 + φ  ||wx ||2 . (C.60) + 2 By Poincare’s inequality, we have

328

C Stable Systems and Their Proofs

||wx ||2 ≤ 2sr wx (0, t)2 + 4sr2 ||wxx ||2 = 2sr γ 2 w(0, t)2 + 4sr2 ||wxx ||2 .

(C.61)

Applying this to (C.60), we get  2  b α γ 2α 2 ||w || + V˙ ∗ ≤ − + − h ||wx ||2 x 2α 2 16sr2  2  αγ pc X(t)2 + − γ h w(0, t)2 − 8sr 2  s˙ (t)   2 2 2 (φ (0) + φs + 1)X(t)2 + γ 2 w(0, t)2 + φ  ||wx ||2 . + 2

(C.62)

Finally, let V be defined by V = V ∗ + qV1 ,

(C.63)

where q > 0 is a positive parameter to be determined. Then, the time derivative of V satisfies     2  α α b qα γ 2α 2 V˙ ≤ − + h + || − q + h ||w||2 − + ||w x 2 2α 2 16sr2 8sr2     b pc αγ 2 − q γα + + γ h w(0, t)2 − X(t)2 − 2 8sr 2  s˙ (t)   2 2 2 2 (φ (0) + φs + 1 + q)X(t)2 + qφ  ||w||2 + γ 2 w(0, t)2 + φ  ||wx ||2 . + 2 (C.64) Therefore, by choosing   b , γ > max 0, − 2α   b2 2 αγ (2γ sr + 1) q = max +γ , , α2 16sr2 (γ α + b2 )

(C.65) (C.66)

there exists a positive constant a > 0 such that V˙ ≤ −dV + a s˙ V

(C.67)

holds, where  d = min

 α + h, c . 16sr2

(C.68)

C.3 One-Phase Stefan Problem with Delay

329

Using the same approach as Sect. C.1, we can deduce that it holds V (t) ≤ easr V (0)e−dt .

(C.69)

Φ = ||w||2 + ||wx ||2 + X(t)2

(C.70)

Recall the definition of

given in (C.40). Then, with the help of Agmon’s inequality, we can obtain the following bound: MΦ ≤ V ≤ MΦ,

(C.71)

where M=

1 min{q, 1, p}, 2

M=

1 max{q, 1 + 4γ sr , p}. 2

(C.72)

Finally, by combining (C.69) and (C.71), we obtain Φ(t) ≤

M asr e V (0)e−dt , M

(C.73)

by which we complete the proof of Lemma C.2.

C.3 One-Phase Stefan Problem with Delay Consider the system zt (x, t) = − zx (x, t),

−D < x < 0

(C.74)

z(−D, t) =0,

(C.75)

wx (0, t) = − z(0, t), wt (x, t) =αwxx (x, t) +

(C.76) c s˙ (t)X(t), β

0 < x < s(t)

w(s(t), t) =0,

(C.77) (C.78)

˙ X(t) = − cX(t) − βwx (s(t), t).

(C.79)

Lemma C.3 With the conditions s˙ (t) > 0,

0 < s(t) < sr ,

(C.80)

330

C Stable Systems and Their Proofs

for some positive constant sr > 0, (w, X)-system in (C.74)–(C.79) is exponentially stable at the origin in the sense of the spatial H1 -norm defined by  Π (t) :=

0

−D



s(t)

zx (x, t) dx + 2

 w(x, t) dx + 2

0

s(t)

wx (x, t)2 dx + X(t)2 .

0

(C.81) Proof Change of Variable Introduce a change of variable ω(x, t) = w(x, t) + (x − s(t)) z(0, t).

(C.82)

Using (C.82), the target (z, w, X)-system (C.74)–(C.79) is described by (z, ω, X)system as z(−D, t) =0,

(C.83)

zt (x, t) = − zx (x, t),

−D < x < 0

ωx (0, t) =0,

(C.84) (C.85)

ωt (x, t) =αωxx (x, t) − (x − s(t)) zx (0, t)   c X(t) − z(0, t) , 0 < x < s(t) + s˙ (t) β ω(s(t), t) =0,

(C.86) (C.87)

˙ X(t) = − cX(t) − β(ωx (s(t), t) − z(0, t)).

(C.88)

Stability Analysis of (z, ω, X)-System First, we prove the exponential stability of the (z, ω, X)-system. Let V1 be the functional defined by  V1 =

0

−D

e−mx zx (x, t)2 dx,

(C.89)

where m > 0 is a positive parameter. (C.89) satisfies ||zx ||2L2 (−D,0) ≤ V1 ≤ emD ||zx ||2L2 (−D,0) .

(C.90)

Note that (C.83) yields zx (−D, t) = 0 through taking the time derivative and applying PDE (C.84). With the help of it, taking the time derivative of (C.89) together with (C.83)–(C.84) leads to

C.3 One-Phase Stefan Problem with Delay

V˙1 = − 2



0 −D

331

e−mx zx (x, t)zxx (x, t)dx

x=0 = − e−mx zx (x, t)2 |x=−D +

 = − zx (0, t)2 − m

0 −D



0



−D

 d −mx zx (x, t)2 dx e dx

e−mx zx (x, t)2 dx.

(C.91)

Let V2 be the functional defined by  1 2 2 ||ω||L2 (0,s(t)) + ||ωx ||L2 (0,s(t)) sr2    1 s(t) 1 2 2 dx. ω(x, t) + ω (x, t) = x 2 0 sr2

1 V2 = 2



(C.92)

(C.92) satisfies max{sr2 , 1}||ω||2H1 (0,s(t)) ≤ 2V2 ≤ max{1/sr2 , 1}||ω||2H1 (0,s(t)) .

(C.93)

Note that taking the total time derivative of (C.87) yields ωt (s(t), t) = −˙s (t)ωx (s(t), t). Taking the time derivative of (C.92) together with (C.85)–(C.87), we obtain   s˙ (t) 1 2 + ω (s(t), t)2 ω(s(t), t) V˙2 = x 2 sr2   s(t)  1 ω(x, t)ω (x, t) + ω (x, t)ω (x, t) dx + t x xt sr2 0 =

s˙ (t) ωx (s(t), t)2 2     c 1 s(t) X(t) − z(0, t) dx + 2 ω(x, t) αωxx (x, t) − (x − s(t)) zx (0, t) + s˙ (t) β sr 0  s(t) + ωx (s(t), t)ωt (s(t), t) − ωx (0, t)ωt (0, t) − ωxx (x, t)ωt (x, t)dx  s(t)

0

α 1 = − 2 ||ωx ||2L2 (0,s(t)) − 2 zx (0, t) (x − s(t)) ω(x, t)dx sr sr 0   s(t)  s˙ (t) c X(t) − z(0, t) ω(x, t)dx − α||ωxx ||2L2 (0,s(t)) + zx (0, t)ω(0, t) + 2 β sr 0   c s˙ (t) − (C.94) ωx (s(t), t)2 − s˙ (t) X(t) − z(0, t) ωx (s(t), t). 2 β

332

C Stable Systems and Their Proofs

Applying Young’s and Cauchy Schwarz inequalities to the second terms on the first and second line of the (C.94) with the help of 0 < s(t) < sr yields /  / /zx (0, t) /

s(t) 0

/ / (x − s(t)) ω(x, t)dx //

γ1 1 ≤ zx (0, t)2 + 2 2γ1 ≤

γ1 zx (0, t)2 2 1 + 2γ1



s(t)



s(t)

2 (x − s(t)) ω(x, t)dx

  (x − s(t)) dx

2

ω(x, t) dx , 0

≤

s3 γ1 zx (0, t)2 + r ||ω||2L2 (0,s(t)) , 2 6γ1

≤

γ1 2s 5 zx (0, t)2 + r ||ωx ||2L2 (0,s(t)) , 2 3γ1

≤



s(t)

2

0

|zx (0, t)ω(0, t)| ≤

,

0

(C.95)

γ2 1 zx (0, t)2 + ω(0, t)2 , 2 2γ2 2sr γ2 zx (0, t)2 + ||ωx ||2L2 (0,s(t)) , 2 γ2

(C.96)

where we utilized Poincare’s inequality ||ω||2L2 (0,s(t)) ≤ 4sr2 ||ωx ||2L2 (0,s(t)) and Agmon’s inequality ω(0, t)2 ≤ 4sr ||ωx ||2L2 (0,s(t)) , and γ1 > 0 and γ2 > 0 are positive parameters to be determined. Hence, applying (C.95) and (C.96) to (C.94) with the choice of γ1 =

8sr5 , 3α

(C.97)

γ2 =

8sr3 , α

(C.98)

the following differential inequality is deduced: α α 16sr3 zx (0, t)2 V˙2 ≤ − ||ωxx ||2L2 (0,s(t)) − 2 ||ωx ||2L2 (0,s(t)) + 2 3α 2sr   2 c 1 2 2 2 + s˙ (t) 2 2 X(t) + 2z(0, t) + 3 ||ω||L2 (0,s(t)) . β 2sr

(C.99)

C.3 One-Phase Stefan Problem with Delay

333

Let V3 be the functional defined by V3 =

1 X(t)2 . 2

(C.100)

Taking the time derivative of (C.100) and applying Young’s and Agmon’s inequalities, we obtain V˙3 = − cX(t)2 − βX(t)(ωx (s(t), t) − z(0, t)) 4β 2 sr 4Dβ 2 c ||ωxx ||2L2 (0,s(t)) + ||zx ||2L2 (−D,0) . ≤ − X(t)2 + 2 c c

(C.101)

Let V be the functional defined by V = qV1 + V2 + pV3 ,

(C.102)

where q > 0 and p > 0 are positive parameters to be determined. Combining (C.91), (C.99), and (C.101), we get   8pβ 2 sr α α ˙ 1− ||ωxx ||2L2 (0,s(t)) − 2 ||ωx ||2L2 (0,s(t)) V ≤− 2 cα 2sr   16sr3 zx (0, t)2 − q− 3α   4Dβ 2 pc X(t)2 ||zx ||2L2 (−D,0) − −m q −p mc 2    s(t) c2 1 2 2 2 + s˙ (t) 2 2 X(t) + 2z(0, t) + 3 ω(x, t) dx . β 2sr 0

(C.103)

Hence, by choosing the parameters as p=

cα , 16β 2 sr

 q = max

16sr3 Dα , 3α 2msr

 ,

the inequality (C.103) leads to α α V˙ ≤ − ||ωxx ||2L2 (0,s(t)) − 2 ||ωx ||2L2 (0,s(t)) 4 2sr   4Dβ pc X(t)2 ||zx ||2L2 (−D,0) − −m q −p mc 2    s(t) c2 1 2 2 2 + s˙ (t) 2 2 X(t) + 2z(0, t) + 3 ω(x, t) dx , β 2sr 0

(C.104)

334

C Stable Systems and Their Proofs

  4c α mq −mD pc 1 2 e X(t) + s˙ (t) ≤ − 2 V2 − V1 − V3 + 8DV1 + V2 , 2 2 sr 8sr β2 (C.105) from which we obtain the form of V˙ ≤ − bV + a s˙ (t)V ,

(C.106)

where  b = min

 m −mD α e , 2,c , 2 8sr

 a = max

 8D 1 4c2 , , 2 . q sr pβ

(C.107)

Hence, applying 0 < s(t) < sr , the exponential stability of (z, ω, X)-system is shown as V (t) ≤ V (0)easr e−bt .

(C.108)

Stability Analysis of (z, w, X)-System Taking the square of (C.82) and applying Young’s and Cauchy Schwarz inequality, we obtain ||ω||2H1 (0,s(t)) ≤2||w||2H1 (0,s(t)) + K1 ||zx ||2L2 (−D,0) ,

(C.109)

||w||2H1 (0,s(t)) ≤2||ω||2H1 (0,s(t)) + K1 ||zx ||2L2 (−D,0) ,

(C.110)

where K1 =

8Dsr3 + 8Dsr . 3

(C.111)

Consider the following norm: Π (t) = ||zx ||2L2 (−D,0) + ||w||2H1 (0,s(t)) + X(t)2 .

(C.112)

Then, recalling ||zx ||2L2 (−D,0) ≤ V1 ≤ emD ||zx ||2L2 (−D,0) ,

(C.113)

K2 ||ω||2H1 (0,s(t)) ≤ 2V2 ≤ K3 ||ω||2H1 (0,s(t)) ,

(C.114)

where K2 = max{sr2 , 1},

K3 = max{1/sr2 , 1},

applying (C.110) to (C.112) yields the following bound:

(C.115)

C.4 One-Phase Stefan Problem with Non-monotonic Interface and Disturbances

335

Π ≤(1 + K1 )||zx ||2L2 (−D,0) + 2||ω||2H1 (0,s(t)) + X(t)2 , ≤(1 + K1 )V1 + 4K2 V2 + 2V3 .

(C.116)

Moreover, recalling V = qV1 + V2 + pV3

(C.117)

and applying the above inequalities, the following bound on V is derived: K3 p ||ω||2H1 (0,s(t)) + X(t)2 , V ≤qemD ||zx ||2L2 (−D,0) + 2 2   K1 K3 K3 p ||w||2H1 (0,s(t)) + X(t)2 . ||zx ||2L2 (−D,0) + ≤ qemD + 2 2 2

(C.118)

Therefore, (C.116) and (C.118) lead to the following equivalence of the norm V and Π: ¯ (t), δV (t) ≤ Π (t) ≤ δV

(C.119)

where 1 1, 0 mD max qe + K12K3 , K3 , p2   2 1 . δ¯ = max (K1 + 1) , 4K2 , q p

δ=

(C.120)

(C.121)

By (C.108) and (C.119), we have δ¯ Π (t) ≤ Π (0)easr e−bt , δ

(C.122)

which yields the exponential stability of (z, w, X)-system, and we complete the proof of Lemma C.3.

C.4 One-Phase Stefan Problem with Non-monotonic Interface and Disturbances Consider the system wt (x, t) = αwxx (x, t) +

c s˙ (t)X(t) + φ(x − s(t))d(t), β

(C.123)

336

C Stable Systems and Their Proofs



 β s(t) β wx (0, t) = f (t)− ε w(0, t)− ψ(−y)w(y, t)dy−ψ(−s(t))X(t) , α α 0 (C.124) w(s(t), t) = εX(t),

(C.125)

˙ X(t) = −cX(t) − βwx (s(t), t) − d(t),

(C.126)

where ε > 0, c > 0, and φ(x) and ψ(x) are bounded continuous functions in x. Lemma C.4 Suppose that there exists a positive constant s¯ > 0 such that 0 < s(t) < s¯ .

(C.127)

Let V (t) be a Lyapunov function defined by V (t) =

1 ε ||w||2 + X(t)2 . 2α 2β

(C.128)

Then, there exists a positive constant ε∗ > 0 such that for all ε ∈ (0, ε∗ ) the following inequality holds: V˙ (t) ≤ − bV (t) + Γ d(t)2 + 2¯s f (t)2 + a|˙s (t)|V (t),

(C.129)

where   αc2 s¯ 2βε max 1, 3 3 , a= α 2β ε  2 2¯s 3 c¯s ε + 2 +ε . Γ = βc β α

b=

0α 1 1 min 2 , c , 8 s¯

(C.130) (C.131)

Moreover, suppose that there exists a time-varying function z(t) which satisfies z˙ (t) ≥ |˙s (t)|,

z ≤ z(t) ≤ z¯ ,

(C.132)

for some constants z ∈ R and z¯ ∈ R. Then, the system (C.123)–(C.126) is exponentially ISS with respect to f (t) and d(t). Proof Note that Poincare’s and Agmon’s inequalities for the system (C.123)– (C.125) with 0 < s(t) < s¯ lead to ||w||2 ≤ 2¯s ε2 X(t)2 + 4¯s 2 ||wx ||2 ,

(C.133)

w(0, t)2 ≤ 2ε2 X(t)2 + 4¯s ||wx ||2 .

(C.134)

C.4 One-Phase Stefan Problem with Non-monotonic Interface and Disturbances

337

Taking the time derivative of (C.128) along with the solution of (C.123)–(C.126), we have ε β V˙ (t) = − ||wx ||2 − cX(t)2 + εw(0, t)2 − w(0, t)f (t) β α 

β s(t) β − εw(0, t) ψ(−y)w(y, t)dy + ψ(−s(t))X(t) α α 0  1 s(t) ε − X(t)d(t) + φ(x − s(t))w(x, t)dxd(t) β α 0    s˙ (t) ε2 c s(t) 2 + X(t) + w(x, t)dxX(t) . (C.135) α 2 β 0 Applying Young’s inequality to the last term in the first line, the second, third, and fourth lines of (C.135), we obtain − w(0, t)f (t) ≤

β − w(0, t) α



s(t)

1 w(0, t)2 + 2¯s f (t)2 , 8¯s

(C.136)

ψ(−y)w(y, t)dy + ψ(−s(t))X(t)

0

1 β2 ≤ w(0, t)2 + 2 2 α γ1



2

s(t)

ψ(−y)w(y, t)dy

+ γ1 (ψ(−s(t))X(t))2 ,

0

(C.137)  ε 1 s(t) − X(t)d(t) + φ(x − s(t))w(x, t)dxd(t) β α 0 2   2 s(t) ε 1 1 (γ2 + γ3 ) 2 X(t) + d(t) + 2 φ(x − s(t))w(x, t)dx , ≤ 2γ2 β 2 2α γ3 0 (C.138) where γi > 0 for i = {1, 2, 3}. Applying (C.136)–(C.138) and Cauchy Schwarz, 2ε Poincare, and Agmon’s inequalities to (C.135) with choosing γ1 = 18 , γ2 = βc , and   2 3 s c¯s γ3 = 4¯ β + ε , we have α2 V˙ (t) ≤ −



1 2β s¯ − 2 α



  64c¯s 2 + 3 ε ||wx ||2 α

338

C Stable Systems and Their Proofs

 c + g(ε) X(t)2 + Γ d(t)2 + 2¯s f (t)2 −ε 8β / /  / |˙s (t)| 2 2c // s(t) / 2 + w(x, t)dxX(t)/ , ε X(t) + / / 2α β / 0 

(C.139)

where Γ = g(ε) = Since g(0) =

c 8β

(γ2 + γ3 ) , 2

(C.140)

ε β c − − 8β 2¯s α

 64c¯s 2 + 3 ε2 . α

(C.141)

 64c¯s 2 + 3 < 0, α

(C.142)



> 0 and g  (ε) = −

2βε 1 − 2¯s α



for all ε > 0, there exists ε∗ such that g(ε) > 0 for all ε ∈ (0, ε∗ ) and g(ε∗ ) = 0. Thus, setting ⎧ ⎨

⎫ ⎬

α   , ε < min ε∗ , ⎩ ⎭ 64c¯s 2 8β s¯ + 3 α

(C.143)

the inequality (C.139) leads to V˙ (t) ≤ − bV (t) + Γ d(t)2 + 2¯s f (t)2 / /  / s(t) / |˙s (t)| 2 2c / / + w(x, t)dxX(t)/ , ε X(t)2 + / / 2α β / 0

(C.144)

where b=

0α 1 1 min 2 , c . 8 s¯

Applying Young’s inequality to (C.144), the inequality (C.129) is derived.

(C.145)

References

1. I. Abel, M. Jankovic, M. Krstic, Constrained stabilization of multi-input linear systems with distinct input delays. IFAC-PapersOnLine 52(2), 82–87 (2019) 2. M. Agarwala, D. Bourell, J. Beaman, H. Marcus, J. Barlow, Direct selective laser sintering of metals. Rapid Prototyp. J. 1(1), 26–36 (1995) 3. T. Ahmed-Ali, I. Karafyllis, F. Giri, M. Krstic, F. Lamnabhi-Lagarrigue, Exponential stability analysis of sampled-data ODE-PDE systems and application to observer design. IEEE Trans. Autom. Control 62(6), 3091–3098 (2017) 4. S. Ahn, J. Murphy, J. Ramos, J. Beaman, Physical modeling for dynamic control of melting process in direct-SLS, in Proceedings of the 12th Annual Solid Freeform Fabrication Symposium, Austin, TX (2001), pp. 591–598 5. A. Alessandri, P. Bagnerini, M. Gaggero, Optimal control of propagating fronts by using level set methods and neural approximations. IEEE Trans. Neural Netw. Learn. Syst. 30(3), 902–912 (2018) 6. B.N. Am, E. Fridman, Network-based H∞ filtering of parabolic systems. Automatica 50(12), 3139–3146 (2014) 7. H. Anfinsen, O.M. Aamo, Adaptive Control of Hyperbolic PDEs (Springer, New York, 2019) 8. M. Arcak, P. Kokotovic, Nonlinear observers: a circle criterion design and robustness analysis. Automatica 37, 1923–1930 (2001) 9. A. Armaou, P.D. Christofides, Robust control of parabolic PDE systems with time-dependent spatial domains. Automatica 37, 61–69 (2001) 10. A. Baccoli, A. Pisano, Y. Orlov, Boundary control of coupled reaction-advection-diffusion systems with spatially-varying coefficients. Automatica 54, 80–90 (2015) 11. M. Bagheri, P. Naseradinmousavi, M. Krstic, Feedback linearization based predictor for time delay control of a high-DOF robot manipulator. Automatica 108, 108485 (2019) 12. H.I. Basturk, M. Krstic, Adaptive sinusoidal disturbance cancellation for unknown LTI systems despite input delay. Automatica 58, 131–138 (2015) 13. N. Bekiaris-Liberis, M. Krstic, Compensating the distributed effect of diffusion and counterconvection in multi-input and multi-output LTI systems. IEEE Trans. Autom. Control 56(3), 637–643 (2010) 14. N. Bekiaris-Liberis, M. Krstic, Compensation of state-dependent input delay for nonlinear systems. IEEE Trans. Autom. Control 58(2), 275–289 (2013) 15. N. Bekiaris-Liberis, M. Krstic, Compensation of wave actuator dynamics for nonlinear systems. IEEE Trans. Autom. Control 59(6), 1555–1570 (2014)

© Springer Nature Switzerland AG 2020 S. Koga, M. Krstic, Materials Phase Change PDE Control & Estimation, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-58490-0

339

340

References

16. N. Bekiaris-Liberis, M. Krstic, Nonlinear Control Under Nonconstant Delays, vol. 25 (SIAM, Philadelphia, 2014) 17. M. Benosman, Learning-Based Adaptive Control: An Extremum Seeking Approach–Theory and Applications (Butterworth-Heinemann, Oxford, 2016) 18. M.K. Bernauer, R. Herzog, Optimal control of the classical two-phase Stefan problem in level set formulation. SIAM J. Sci. Comput. 33(1), 342–363 (2011) 19. C.M. Bitz, M.M. Holland, A.J. Weaver, M. Eby, Simulating the ice-thickness distribution in a coupled climate model. J. Geophys. Res. Oceans 106(C2), 2441–2463 (2001) 20. C.M. Bitz, W.H. Lipscomb, An energy-conserving thermodynamic model of sea ice, J. Geophys. Res. 104(C7), 15–669 (1999) 21. N. Boonkumkrong, S. Kuntanapreeda, Backstepping boundary control: an application to rod temperature control with Neumann boundary condition. Proc. Inst. Mech. Eng. I J. Syst. Control Eng. 228(5), 295–302 (2014) 22. D.M. Boskovic, M. Krstic, W. Liu, Boundary control of an unstable heat equation via measurement of domain-averaged temperature. IEEE Trans. Autom. Control 46(12), 2022– 2028 (2001) 23. S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (SIAM, Philadelphia, 1994) 24. D. Bresch-Pietri, M. Krstic, Adaptive compensation of diffusion-advection actuator dynamics using boundary measurements, in 54th IEEE Conference on Decision and Control (CDC) (IEEE, Osaka, 2015), pp. 1224–1229 25. D. Bresch-Pietri, M. Krstic, Adaptive trajectory tracking despite unknown input delay and plant parameters. Automatica 45(9), 2074–2081 (2009) 26. D. Bresch-Pietri, M. Krstic, Delay-adaptive predictor feedback for systems with unknown long actuator delay. IEEE Trans. Autom. Control 55(9), 2106–2112 (2010) 27. D. Bresch-Pietri, M. Krstic, Delay-adaptive control for nonlinear systems. IEEE Trans. Autom. Control 59(5), 1203–1218 (2014) 28. M. Buisson-Fenet, S. Koga, M. Krstic, Control of piston position in inviscid gas by bilateral boundary actuation, in IEEE Conference on Decision and Control (CDC) (IEEE, Miami Beach, 2018), pp. 5622–5627 29. S. Bukkapatnam, B. Clark, Dynamic modeling and monitoring of contour crafting - an extrusion-based layered manufacturing process. J. Manuf. Sci. Eng. 129(1), 135–142 (2007) 30. H.M. Byrne, M.A.J. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors. Math. Biosci. 130(2), 151–181 (1995) 31. C.I. Byrnes, A. Isidori, Local stabilization of minimum-phase nonlinear systems. Syst. Control Lett. 11(1), 9–17 (1988) 32. X. Cai, M. Krstic, Nonlinear control under wave actuator dynamics with time- and statedependent moving boundary. Int. J. Rob. Nonlinear Control 25(2), 222–251 (2015) 33. J. Caldwell, Y.Y. Kwan, Numerical methods for one-dimensional Stefan problems. Commun. Numer. Methods Eng. 20(7), 535–545 (2004) 34. L. Camacho-Solorio, S. Moura, M. Krstic, Robustness of boundary observers for radial diffusion equations to parameter uncertainty, in 2018 American Control Conference (ACC) (IEEE, Milwaukee, 2018), pp. 3484–3489 35. L. Camacho-Solorio, M. Krstic, R. Klein, A. Mirtabatabaei, S.J. Moura, State estimation for an electrochemical model of multiple-material lithium-ion batteries, in ASME 2016 Dynamic Systems and Control Conference (American Society of Mechanical Engineers Digital Collection, New York, 2016) 36. J.R. Cannon, M. Primicerio, A two phase Stefan problem with flux boundary conditions. Annali di Matematica Pura ed Applicata 88(1), 193–205 (1971) 37. X. Cao, B. Ayalew, Partial differential equation-based multivariable control input optimization for laser-aided powder deposition processes. J. Manuf. Sci. Eng. 138(3), 031001 (2016) 38. N.A. Chaturvedi, R. Klein, J. Christensen, J. Ahmed, A. Kojic, Algorithms for advanced battery-management systems. IEEE Control Syst. Mag. 30(3), 49–68 (2010)

References

341

39. X. Chen, J. Chadam, L. Jiang, W. Zheng, Convexity of the exercise boundary of the American put option on a zero dividend asset. Math. Financ. Int. J. Math. Stat. Financ. Econ. 18(1), 185–197 (2008) 40. S. Chen, M. Krstic, R. Vazquez, Backstepping boundary control of a 1-D 2 × 2 unstable diffusion-reaction PDE system with distinct input delays, in American Control Conference (ACC) (IEEE, Philadelphia, 2019), pp. 2564–2569 41. S. Chen, M. Krstic, R. Vazquez, Folding backstepping approach to parabolic PDE bilateral boundary control. IFAC-PapersOnLine 52(2), 76–81 (2019) 42. P.D. Christofides, Robust control of parabolic PDE systems. Chem. Eng. Sci. 53, 2949–2965 (1998) 43. H. Chung, S. Das, Numerical modeling of scanning laser-induced melting, vaporization and resolidification in metals subjected to step heat flux input. Int. J. Heat Mass Transf. 47(19), 4153–4164 (2004) 44. F. Conrad, D. Hilhorst, T.I. Seidman, Well-posedness of a moving boundary problem arising in a dissolution-growth process. Nonlinear Anal. 31, 795–803 (2007) 45. R. Conti, A.A. Gallitto, E. Fiordilino, Measurement of the convective heat-transfer coefficient. Phys. Teach. 52(2), 109–111 (2014) 46. J.M. Coron, R. Vazquez, M. Krstic, G. Bastin, Local exponential H 2 stabilization of a 2×2 quasilinear hyperbolic system using backstepping. SIAM J. Control Optim. 51(3), 2005–2035 (2013) 47. M. Cotteleer, J. Joyce, 3D opportunity: additive manufacturing paths to performance, innovation, and growth. Deloitte Rev. 14, 5–19 (2014) 48. J. Crepeau, Josef Stefan: his life and legacy in the thermal sciences. Exp. Therm. Fluid Sci. 15, 445–465 (1990) 49. S. Cui, A. Friedman, Analysis of a mathematical model of the effect of inhibitors on the growth of tumors. Math. Biosci. 164(2), 103–137 (2000) 50. K. Dai, L. Shaw, Finite element analysis of the effect of volume shrinkage during laser densification. Acta Mater. 53(18), 4743–4754 (2005) 51. N. Daraoui, P. Dufour, H. Hammouri, A. Hottot, Model predictive control during the primary drying stage of lyophilisation. Control Eng. Pract. 18, 483–494 (2010) 52. G.A. De Andrade, R. Vazquez, D.J. Pagano, Backstepping-based estimation of thermoacoustic oscillations in a Rijke tube with experimental validation. IEEE Trans. Autom. Control, Early access (2020). https://doi.org/10.1109/TAC.2020.2970152 53. J. Deutscher, Backstepping design of robust output feedback regulators for boundary controlled parabolic PDEs. IEEE Trans. Autom. Control 61(8), 2288–2294 (2016) 54. S. Dey, B. Ayalew, P. Pisu, Nonlinear robust observers for state-of-charge estimation of lithium-ion cells based on a reduced electrochemical model. IEEE Trans. Control Syst. Technol. 23(5), 1935–1942 (2015) 55. D. Di Domenico, A. Stefanopoulou, G. Fiengo, Lithium-ion battery state of charge and critical surface charge estimation using an electrochemical model-based extended Kalman filter. J. Dyn. Syst. Meas. Control 132(6), 061302 (2010) 56. M. Diagne, P. Shang, Z. Wang, Feedback stabilization of a food extrusion process described by 1D PDEs defined on coupled time-varying spatial domains, in Proceedings of the 13th IFAC Workshop on Time-Delay Systems, vol. 48, no. 12 (2015), pp. 51–56 57. M. Diagne, P. Shang, Z. Wang, Feedback stabilization for the mass balance equations of an extrusion process. IEEE Trans. Autom. Control 61(3), 760–765 (2016) 58. M. Diagne, N. Bekiaris-Liberis, A. Otto, M. Krstic, Control of transport PDE/nonlinear ODE cascades with state-dependent propagation speed. IEEE Trans. Autom. Control 62(12), 6278– 6293 (2017) 59. S. Diehl, E. Henningsson, A. Heyden, S. Perna, A one-dimensional moving-boundary model for tubulin-driven axonal growth. J. Theor. Biol. 358, 194–207 (2014) 60. S. Diehl, E. Henningsson, A. Heyden, Efficient simulations of tubulin-driven axonal growth. J. Comput. Neurosci. 41(1), 45–63 (2016)

342

References

61. D. Drotman, M. Diagne, R. Bitmead, M. Krstic, Control-oriented energy-based modeling of a screw extruder used for 3D printing, in 2016 Dynamic Systems and Control Conference, vol. 1001 (ASME, New York, 2016), pp. 48109 62. Y. Du, Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J. Math. Anal. 42(1), 377–405 (2010) 63. Y. Dutil, D.R. Rousse, N.B. Salah, S. Lassue, L. Zalewski, A review on phase-change materials: mathematical modeling and simulations. Renew. Sustain. Energy Rev. 15(1), 112– 130 (2011) 64. R. Escobedo, L. Fernández, Classical one-phase Stefan problems for describing polymer crystallization processes, in SIAM Journal on Applied Mathematics (SIAM, Philadelphia, 2013), pp. 254–280 65. N. Espitia, A. Girard, N. Marchand, C. Prieur, Event-based control of linear hyperbolic systems of conservation laws. Automatica 70, 275–287 (2016) 66. N. Espitia, A. Girard, N. Marchand, C. Prieur, Event-based boundary control of a linear 2 × 2 hyperbolic system via backstepping approach. IEEE Trans. Autom. Control 63(8), 2686–2693 (2017) 67. N. Espitia, I. Karafyllis, M. Krstic, Event-triggered boundary control of constant-parameter reaction-diffusion PDEs: a small-gain approach, Preprint, available at https://arxiv.org/abs/ 1909.10472 (2019) 68. Q. Fang, M. Hanna, Y. Lan, Extrusion system components, in Encyclopedia of Agricultural, Food, and Biological Engineering, ed. by D.R. Heldman (CRC Press, Boca Raton, 2003), pp. 301–305 69. I. Fantoni, R. Lozano, M.W. Spong, Energy based control of the pendubot. IEEE Trans. Autom. Control 45(4), 725–729 (2000) 70. A. Fasano, M. Primicerio, General free-boundary problems for the heat equation, I. J. Math. Anal. Appl. 57(3), 694–723 (1977) 71. J. Feiling, S. Koga, M. Krstic, T.R. Oliveira, Gradient extremum seeking for static maps with actuation dynamics governed by diffusion PDEs. Automatica 95, 197–206 (2018) 72. I. Fenty, P. Heimbach, Coupled sea ice ocean-state estimation in the Labrador Sea and Baffin Bay. J. Phys. Oceanogr. 43(5), 884–904 (2013) 73. I. Fenty, D. Menemenlis, H. Zhang, Global coupled sea ice-ocean state estimation. Clim. Dyn. 49(3), 931–956 (2015) 74. R. Freeman, P.V. Kokotovic, Robust Nonlinear Control Design: State-Space and Lyapunov Techniques (Springer, New York, 2008) 75. A. Friedman, Free boundary problems for parabolic equations I. Melting of solids. J. Math. Mech. 8(4), 499–517 (1959) 76. A. Friedman, F. Reitich, Analysis of a mathematical model for the growth of tumors. J. Math. Biol. 38(3), 262–284 (1999) 77. E. Fridman, A. Blighovsky, Robust sampled-data control of a class of semilinear parabolic systems. Automatica 48(5), 826–836 (2012) 78. E. Fridman, Sampled-data distributed H∞ control of transport reaction systems. SIAM J. Control Optim. 51(2), 1500–1527 (2013) 79. P. Frihauf, M. Krstic, Leader-enabled deployment onto planar curves: a PDE-based approach. IEEE Trans. Autom. Control 56(8), 1791–1806 (2011) 80. B.P. Graham, A. Van Ooyen, Mathematical modelling and numerical simulation of the morphological development of neurons. BMC Neurosci. 7(S1), S9 (2006) 81. M.S. Grewal, A.P. Andrews, Applications of Kalman filtering in aerospace 1960 to the present [historical perspectives]. IEEE Control Syst. Mag. 30(3), 69–78 (2010) 82. S. Gupta, The Classical Stefan Problem. Basic Concepts, Modelling and Analysis (Applied Mathematics and Mechanics, North-Holland, 2003) 83. P. Hahnfeldt, D. Panigrahy, J. Folkman, L. Hlatky, Tumor development under angiogenic signaling. Cancer Res. 59(19), 4770–4775 (1999) 84. D.K. Hall, J.R. Key, K.A. Casey, G.A. Riggs, D.J. Cavalieri, Sea ice surface temperature product from MODIS. IEEE Trans. Geosci. Remote Sens. 42(5), 1076–1087 (2004)

References

343

85. A. Hasan, O.M. Aamo, M. Krstic, Boundary observer design for hyperbolic PDE-ODE cascade systems. Automatica 68, 75–86 (2016) 86. K. Hashimoto, S. Adachi, D.V. Dimarogonas, Event-triggered intermittent sampling for nonlinear model predictive control. Automatica 81, 148–155 (2017) 87. T. Hatanaka, N. Chopra, M. Fujita, M.W. Spong, Passivity-Based Control and Estimation in Networked Robotics (Springer, Cham, 2015) 88. M. Hinze, S. Ziegenbalg, Optimal control of the free boundary in a two-phase Stefan problem. J. Comput. Phys. 223(2), 657–684 (2007) 89. T. Hu, Z. Lin, Control Systems with Actuator Saturation: Analysis and Design (Springer, New York, 2001) 90. A. Isidori, Nonlinear Control Systems (Springer, New York, 2013) 91. M. Izadi, S. Dubljevic, Backstepping output feedback control of moving boundary parabolic PDEs. Eur. J. Control 21, 27 – 35 (2015) 92. E.M. Izhikevich, Dynamical Systems in Neuroscience (MIT Press, Cambridge, 2007) 93. Z.P. Jiang, A.R. Teel, L. Praly, Small-gain theorem for ISS systems and applications. Math. Control Signals Syst. 7(2), 95–120 (1994) 94. Z.P. Jiang, I.M. Mareels, Y. Wang, A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems. Automatica 32(8), 1211–1215 (1996) 95. R.E. Kalman, A new approach to linear filtering and prediction problems. ASME J. Fluids Eng. 82(1), 35–45 (1960) 96. I. Karafyllis, C. Kravaris, Global stability results for systems under sampled-data control. Int. J. Rob. Nonlinear Control. IFAC-Affiliated J. 19(10), 1105–1128 (2009) 97. I. Karafyllis, C. Kravaris, From continuous-time design to sampled-data design of observers. IEEE Trans. Autom. Control 54(9), 2169–2174 (2009) 98. I. Karafyllis, M. Krstic, Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zero-order hold. IEEE Trans. Autom. Control 57(5), 1141–1154 (2012) 99. I. Karafyllis, M. Krstic, ISS with respect to boundary disturbances for 1-D parabolic PDEs. IEEE Trans. Autom. Control, 61(12), 3712–3724 (2016) 100. I. Karafyllis, M. Krstic, ISS in different norms for 1-D parabolic PDEs with boundary disturbances. SIAM J. Control Optim. 55(3), 1716–1751 (2017) 101. I. Karafyllis, M. Krstic, Predictor Feedback for Delay Systems: Implementations and Approximations (Springer, New York, 2017) 102. I. Karafyllis, M. Krstic, Adaptive certainty-equivalence control with regulation-triggered finite-time least-squares identification. IEEE Trans. Autom. Control 63(10), 3261–3275 (2018) 103. I. Karafyllis, M. Krstic, Sampled-data boundary feedback control of 1-D parabolic PDEs. Automatica 87, 226–237 (2018) 104. I. Karafyllis, M. Krstic, Input-to-State Stability for PDEs. Communications and Control Engineering (Springer, London, 2019) 105. I. Karafyllis, T. Ahmed-Ali, F. Giri, Sampled-data observers for 1-D parabolic PDEs with non-local outputs. Syst. Control Lett. 133, 104553 (2019) 106. I. Karafyllis, M. Krstic, K. Chrysafi, Adaptive boundary control of constant-parameter reaction-diffusion PDEs using regulation-triggered finite-time identification. Automatica 103, 166–179 (2019) 107. H.K. Khalil, J.W. Grizzle, Nonlinear Systems, vol. 3 (Prentice Hall, Upper Saddle River, NJ, 2002) 108. A. Khandelwal, K.S. Hariharan, V.S. Kumar, P. Gambhire, S.M. Kolake, D. Oh, S. Doo, Generalized moving boundary model for charge–discharge of LiFePO4/C cells. J. Power Sources 248, 101–114 (2014) 109. A. Khandelwal, K.S. Hariharan, P. Gambhire, S.M. Kolake, T. Yeo, S. Doo, Thermally coupled moving boundary model for charge–discharge of LiFePO4/C cells. J. Power Sources 279, 180–196 (2015)

344

References

110. T. Kharkovskaia, D. Efimov, E. Fridman, A. Polyakov, J.P. Richard, On design of interval observers for parabolic PDEs, in Proceedings of 20th IFAC World Congress, vol. 50(1) (2017), pp. 4045–4050 111. R.B. Khosroushahi, H.J. Marquez, PDE backstepping boundary observer design for microfluidic systems. IEEE Trans. Control Syst. Technol. 23(1), 380–388 (2014) 112. R. Klein, N.A. Chaturvedi, J. Christensen, J. Ahmed, R. Findeisen, A. Kojic, Electrochemical model based observer design for a lithium-ion battery. IEEE Trans. Control Syst. Technol. 21(2), 289–301 (2012) 113. S. Koga, M. Krstic, Arctic sea ice state estimation from thermodynamic PDE model. Automatica 112, 108713 (2020) 114. S. Koga, M. Krstic, Single-boundary control of the two-phase stefan system. Syst. Control Lett. 135, 104573 (2020) 115. S. Koga, L. Camacho-Solorio, M. Krstic, State estimation for lithium ion batteries with phase transition materials, in ASME 2017 Dynamic Systems and Control Conference (American Society of Mechanical Engineers, New York, 2017) 116. S. Koga, M. Diagne, M. Krstic, Control and state estimation of the one-phase Stefan problem via backstepping design. IEEE Trans. Autom. Control 64(2), 510–525 (2019) 117. S. Koga, I. Karafyllis, M. Krstic, Sampled-data control of the stefan system. Preprint, available at https://arxiv.org/abs/1906.01434 (2019) 118. S. Koga, D. Bresch-Pietri, M. Krstic, Delay compensated control of the Stefan problem and robustness to delay mismatch. Int. J. Rob. Nonlinear Control 30(6), 2304–2334 (2020) 119. S. Koga, D. Straub, M. Diagne, M. Krstic, Stabilization of filament production rate for screw extrusion-based polymer 3D-printing. J. Dyn. Syst. Meas. Control 142(3), 031005 (2020) 120. S. Kolossov, E. Boillat, R. Glardon, P. Fischer, M. Locher, 3D FE simulation for temperature evolution in the selective laser sintering process. Int. J. Mach. Tools Manuf. 44(2–3), 117–123 (2004) 121. M. Krstic, Lyapunov tools for predictor feedbacks for delay systems: inverse optimality and robustness to delay mismatch. Automatica 44(11), 2930–2935 (2008) 122. M. Krstic, Compensating actuator and sensor dynamics governed by diffusion PDEs. Syst. Control Lett. 58, 372–377 (2009) 123. M. Krstic, Control of an unstable reaction diffusion PDE with long input delay. Syst. Control Lett. 58(10), 773–782 (2009) 124. M. Krstic, Delay Compensation for Nonlinear, Adaptive, and PDE Systems (Birkhauser, Boston, 2009) 125. M. Krstic, Input delay compensation for forward complete and strict-feedforward nonlinear systems. IEEE Trans. Autom. Control 55(2), 287–303 (2010) 126. M. Krstic, A. Smyshlyaev, Adaptive boundary control for unstable parabolic PDEs–Part I: Lyapunov design. IEEE Trans. Autom. Control 53(7), 1575–1591 (2008) 127. M. Krstic, A. Smyshlyaev, Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays. Syst. Control Lett. 57, 750–758 (2008) 128. M. Krstic, A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs (SIAM, Singapore, 2008) 129. M. Krstic, I. Kanellakopoulos, P.V. Kokotovic, Nonlinear and Adaptive Control Design (Wiley, New York, 1995) 130. M. Kulshreshtha, C. Zaror, D. Jukes, An unsteady state model for twin screw extruders. Trans. IChemE Part C 70, 21–28 (1995) 131. A. Kumar, S. Kumar, V.K. Katiyar, S. Telles, Phase change heat transfer during cryosurgery of lung cancer using hyperbolic heat conduction model. Comput. Biol. Med. 84, 20–29 (2017) 132. S. Kutluay, A. R. Bahadir, A. Özdes, The numerical solution of one-phase classical Stefan problem. J. Comput. Appl. Math. 81(1), 135–144 (1997) 133. F. Kuznik, J. Virgone, J. Noel, Optimization of a phase change material wallboard for building use. Appl. Therm. Eng. 28(11), 1291–1298 (2008)

References

345

134. R. Kwok, G.F. Cunningham, Variability of Arctic sea ice thickness and volume from CryoSat2. Philos. Trans. R. Soc. A 373(2045) (2015). https://doi.org/10.1098/rsta.2014.0157 135. R. Kwok, D.A. Rothrock, Decline in Arctic sea ice thickness from submarine and ICESat records: 1958–2008. Geophys. Res. Lett. 36(15) (2009). https://doi.org/10.1029/ 2009GL039035 136. C. Ladd, J.H. So, J. Muth, M.D. Dickey, 3D printing of free standing liquid metal microstructures. Adv. Mater. 25(36), 5081–5085 (2013) 137. C. Lei, Z. Lin, H. Wang, The free boundary problem describing information diffusion in online social networks. J. Differ. Equ. 254(3), 1326–1341 (2013) 138. C.-H. Li, Modelling extrusion cooking, in Mathematical and Computer Modelling, vol. 33 (PERGAMON, London, 2001), pp. 553–563 139. J. Li, J.K. Barillas, C. Guenther, M.A. Danzer, Sequential Monte Carlo filter for state estimation of LiFePO4 batteries based on an online updated model. J. Power Sources 247, 156–162 (2014) 140. D. Luenberger, An introduction to observers. IEEE Trans. Autom. Control 16(6), 596–602 (1971) 141. A. Maidi, J.-P. Corriou, Boundary geometric control of a linear Stefan problem. J. Process Control 24, 939–946 (2014) 142. J. Marshall, A. Adcroft, C. Hill, L. Perelman, C. Heisey, A finite volume, incompressible Navier Stokes model for studies of the ocean on parallel computers. J. Geophys. Res. Oceans 102(C3), 5753–5766 (1997) 143. G.A. Maykut, N. Untersteiner, Some results from a time dependent thermodynamic model of sea ice. J. Geophys. Res. 76, 1550–1575 (1971) 144. L.J. McGilly, P. Yudin, L. Feigl, A.K. Tagantsev, N. Setter, Controlling domain wall motion in ferroelectric thin films. Nat. Nanotechnol. 10(2), 145 (2015) 145. D.R. McLean, A. Van Ooyen, B.P. Graham, Continuum model for tubulin-driven neurite elongation. Neurocomputing 58, 511–516 (2004) 146. K.C. Mills, Recommended Values of Thermophysical Properties for Selected Commercial Alloys (Woodhead, Cambridge, 2002) 147. V. Mironov, T. Boland, T. Trusk, G. Forgacs, R.R. Markwald, Organ printing: computer-aided jet-based 3D tissue engineering. Trends Biotechnol. 21(4), 157–161 (2003) 148. O.A. Mohamed, S.H. Masood, J.L. Bhowmik, Optimization of fused deposition modeling process parameters: a review of current research and future prospects. Adv. Manuf. 3(1), 42–53 (2015) 149. D.H. Morton-Jones, Polymer Processing (Chapman and Hall, London, 1989) 150. S.J. Moura, F.B. Argomedo, R. Klein, A. Mirtabatabaei, M. Krstic, Battery state estimation for a single particle model with electrolyte dynamics. IEEE Trans. Control Syst. Technol. 25(2), 453–468 (2016) 151. S.J. Moura, N.A. Chaturvedi, M. Krstic, Adaptive partial differential equation observer for battery state-of-charge/state-of-health estimation via an electrochemical model. J. Dyn. Syst. Meas. Control 136(1), 011015-1–011015-11 (2014) 152. C. Nowzari, E. Garcia, J. Cortés, Event-triggered communication and control of networked systems for multi-agent consensus. Automatica 105, 1–27 (2019) 153. E. Nozari, P. Tallapragada, J. Cortés, Event-triggered stabilization of nonlinear systems with time-varying sensing and actuation delay. Automatica 113, 108754 (2020) 154. T.R. Oliveira, M. Krstic, D. Tsubakino, Extremum seeking for static maps with delays. IEEE Trans. Autom. Control 62(4), 1911–1926 (2017) 155. A.K. Padhi, K.S. Nanjundaswamy, J.B. Goodenough, Phospho-olivines as positive-electrode materials for rechargeable lithium batteries. J. Electrochem. Soc. 144(4), 1188–1194 (1997) 156. C.V. Pao, Nonlinear Parabolic and Elliptic Equations (Springer, New York, 1992) 157. A.A. Paranjape, J. Guan, S.J. Chung, M. Krstic, PDE boundary control for flexible articulated wings on a robotic aircraft. IEEE Trans. Robot. 29(3), 625–640 (2013) 158. H.E. Perez, S.J. Moura, Sensitivity-based interval PDE observer for battery SOC estimation, in 2015 American Control Conference (ACC) (2015), pp. 323-328

346

References

159. H. Perez, N. Shahmohammadhamedani, S. Moura, Enhanced performance of li-ion batteries via modified reference governors and electrochemical models. IEEE/ASME Trans. Mechatron. 20(4), 1511–1520 (2015) 160. N. Petit, Control problems for one-dimensional fluids and reactive fluids with moving interfaces, in Advances in the Theory of Control, Signals and Systems with Physical Modeling. Lecture Notes in Control and Information Sciences, vol. 407 (EPF, Lausanne, 2010), pp. 323– 337 161. B. Petrus, J. Bentsman, B.G. Thomas, Application of enthalpy-based feedback control methodology to the two-sided Stefan problem, in American Control Conference (ACC) (IEEE, Portland, 2014), pp. 1015–1020 162. B. Petrus, J. Bentsman, B.G. Thomas, Feedback control of the two-phase Stefan problem, with an application to the continuous casting of steel, in Conference on Decision and Control (CDC) (IEEE, Atlanta, 2010), pp. 1731–1736 163. B. Petrus, J. Bentsman, B.G. Thomas, Enthalpy-based feedback control algorithms for the Stefan problem, in Conference on Decision and Control (CDC) (IEEE, Maui, 2012), pp. 7037–7042 164. B. Petrus, Z. Chen, J. Bentsman, B.G. Thomas, Online recalibration of the state estimators for a system with moving boundaries using sparse discrete-in-time temperature measurements. IEEE Trans. Autom. Control 63(4), 1090–1096 (2017) 165. J. Qi, R. Vazquez, M. Krstic, Multi-agent deployment in 3-D via PDE control. IEEE Trans. Autom. Control 60(4), 891–906 (2015) 166. Y. Rabin, A. Shitzer, Numerical solution of the multidimensional freezing problem during cryosurgery. J. Biomech. Eng. 120(1), 32–37 (1998) 167. Y. Rabin, T.F. Stahovich, Cryoheater as a means of cryosurgery control. Phys. Med. Biol. 48(5), 619 (2003) 168. M.N. Riaz, Extruders in Food Applications (CRC Press, Boca Raton, 2000), 240 pp 169. A.A. Rostami, A. Raisi, Temperature distribution and melt pool size in a semi-infinite body due to a moving laser heat source. Numer. Heat Transf. Part A Appl. 31(7), 783–796 (1997) 170. D.A. Rothrock, D.B. Percival, M. Wensnahan, The decline in arctic sea-ice thickness: separating the spatial, annual, and interannual variability in a quarter century of submarine data. J. Geophys. Res. Oceans 113(C5) (2008). https://doi.org/10.1029/2007JC004252 171. C. Sagert, F. Di Meglio, M. Krstic, P. Rouchon, Backstepping and flatness approaches for stabilization of the stick-slip phenomenon for drilling. IFAC Proc. Vol. 46(2), 779–784 (2013) 172. T. Saitoh, Numerical method for multi-dimensional freezing problems in arbitrary domains. J. Heat Transf. 100(2), 294–299 (1978) 173. S. Schwunk, N. Armbruster, S. Straub, J. Kehl, M. Vetter, Particle filter for state of charge and state of health estimation for lithium-iron phosphate batteries. J. Power Sources 239, 705–710 (2013) 174. A. Selivanov, E. Fridman, Sampled-data relay control of diffusion PDEs. Automatica 82, 59–68 (2017) 175. H. Seitz, W. Rieder, S. Irsen, B. Leukers, C. Tille, Three-dimensional printing of porous ceramic scaffolds for bone tissue engineering. J. Biomed. Mater. Res. Part B Appl. Biomater. 74(2), 782–788 (2005) 176. A.J. Semtner Jr, A model for the thermodynamic growth of sea ice in numerical investigations of climate. J. Phys. Oceanogr. 6(3), 379–389 (1976) 177. A. Sharma, V.V. Tyagi, C.R. Chen, D. Buddhi, Review on thermal energy storage with phase change materials and applications. Renew. Sustain. Energy Rev. 13(2), 318–345 (2009) 178. B. Sherman, A free boundary problem for the heat equation with prescribed flux at both fixed face and melting interface. Q. Appl. Math. 25(1), 53–63 (1967) 179. D. Simon, Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches (Wiley, New York, 2006) 180. A. Smyshlyaev, M. Krstic, Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations. IEEE Trans. Autom. Control 49(12), 2185–2202 (2004)

References

347

181. A. Smyshlyaev, M. Krstic, Backstepping observers for a class of parabolic PDEs. Syst. Control Lett. 54(7), 613–625 (2005) 182. A. Smyshlyaev, M. Krstic, On control design for PDEs with space-dependent diffusivity or time-dependent reactivity. Automatica 41(9), 1601–1608 (2005) 183. A. Smyshlyaev, M. Krstic, Adaptive boundary control for unstable parabolic PDEs–Part II: estimation-based designs. Automatica 43(9), 1543–1556 (2007) 184. A. Smyshlyaev, M. Krstic, Adaptive boundary control for unstable parabolic PDEs–Part III: output feedback examples with swapping identifiers, Automatica 43(9), 1557–1564 (2007) 185. A. Smyshlyaev, M. Krstic, Adaptive Control of Parabolic PDEs (Princeton University Press, Princeton, 2010) 186. E.D. Sontag, Input to state stability: basic concepts and results, in Nonlinear and Optimal Control Theory (Springer, Berlin/Heidelberg, 2008), pp. 163–220 187. E.D. Sontag, Y. Wang, On characterizations of the input-to-state stability property. Syst. Control Lett. 24(5), 351–359 (1995) 188. E.D. Sontag, Y. Wang, New characterizations of input-to-state stability. IEEE Trans. Autom. Control 41(9), 1283–1294 (1996) 189. B. Spangler, S.D. Fontaine, Y. Shi, L.C. Sambucetti, A.N. Mattis, B. Hann, J.A. Wells, A.R. Renslo, A novel tumor-activated prodrug strategy targeting ferrous iron is effective in multiple preclinical cancer models. J. Med. Chem. 59(24), 11161–11170 (2016) 190. V. Srinivasan, J. Newman, Discharge model for the lithium iron-phosphate electrode. J. Electrochem. Soc. 151, A1517–A1529 (2004) 191. V. Srinivasan, J. Newman, Design and optimization of a natural graphite/iron phosphate lithium-ion cell. J. Electrochem. Soc. 151(10), A1530–A1538 (2004) 192. D. Steeves, M. Krstic, R. Vazquez, Prescribed-time H 1 -stabilization of reaction-diffusion equations by means of output feedback, in 18th European Control Conference (ECC) (IEEE, Naples, 2019), pp. 1932–1937 193. D. Steeves, M. Krstic, R. Vazquez, Prescribed-time estimation and output regulation of the linearized Schr´’ odinger equation by backstepping. Eur. J. Control (2020). https://doi.org/10. 1016/j.ejcon.2020.02.009 194. J. Stefan, Uber die Theorie der Eisbildung, insbesondere uber die Eisbildung im Polarmeere. Annalen der Physik 278, 269–286 (1891) 195. G.A. Susto, M. Krstic, Control of PDE–ODE cascades with Neumann interconnections. J. Franklin Inst. 347, 284–314 (2010) 196. P. Tabuada, Event-triggered real-time scheduling of stabilizing control tasks. IEEE Trans. Autom. Control 52(9), 1680–1685 (2009) 197. Z. Tadmor, C. Gogos, Principles of Polymer Processing, (Wiley, New York, 2006) 198. S. Tang, C. Xie, State and output feedback boundary control for a coupled PDE–ODE system. Syst. Control Lett. 60, 540–545 (2011) 199. S. Tang, C. Xie, Z. Zhou, Stabilization for a class of delayed coupled PDE-ODE systems with boundary control, in Control and Decision Conference (CCDC) (2011), pp. 320–324 [Chinese] 200. S.X. Tang, L. Camacho-Solorio, Y. Wang, M. Krstic, State-of-charge estimation from a thermal-electrochemical model of lithium-ion batteries. Automatica 83, 206–219 (2017) 201. K.E. Thomas, J. Newman, R.M. Darling, Mathematical modeling of lithium batteries, in Advances in Lithium-Ion Batteries (Springer, Boston, MA, 2002), pp. 345–392 202. N. Untersteiner, On the mass and heat budget of Arctic sea ice. Arch. Meteorol. Geophys. Bioklimatol. A 12(2), 151–182 (1961) 203. R. Vazquez, M. Krstic, Control of Turbulent and Magnetohydrodynamic Channel Flows: Boundary Stabilization and State Estimation (Birkhauser, Boston, 2008) 204. R. Vazquez, M. Krstic, Explicit boundary control of a reaction-diffusion equation on a disk. IFAC Proc. Vol. 47(3), 1562–1567 (2014) 205. R. Vazquez, M. Krstic, Explicit output-feedback boundary control of reaction-diffusion PDEs on arbitrary-dimensional balls. ESAIM: Control Optim. Calc. Var. 22(4), 1078–1096 (2016)

348

References

206. R. Vazquez, M. Krstic, Boundary control of coupled reaction-advection-diffusion systems with spatially-varying coefficients. IEEE Trans. Autom. Control 62(4), 2026–2033 (2016) 207. R. Vazquez, M. Krstic, Boundary control and estimation of reaction-diffusion equations on the sphere under revolution symmetry conditions. Int. J. Control 92(1), 2–11 (2019) 208. H. Valkenaers, F. Vogeler, E. Ferraris, A. Voet, J. Kruth, A novel approach to additive manufacturing: screw extrusion 3D-printing, in Proceedings of the 10th International Conference on Multi-Material Micro Manufacture (Research Publishing, Singapore, 2013), pp. 235–238 209. B. Vergnes, G.D. Valle, L. Delamare, A global computer software for polymer flows in corotating twin screw extruders. Polym. Eng. Sci. 38(11), 1781–1792 (1998) 210. D. Wang, X. Chen, A multirate fractional-order repetitive control for laser-based additive manufacturing. Control Eng. Pract. 77, 41–51 (2018) 211. J. Wang, M. Krstic, Delay-compensated control of sandwiched ODE-PDE-ODE hyperbolic systems for oil drilling and disaster relief. Preprint available at https://arxiv.org/abs/1910. 05948 (2019) 212. Y. Wang, H. Fang, Z. Sahinoglu, T. Wada, S. Hara, Adaptive estimation of the state of charge for lithium-ion batteries: nonlinear geometric observer approach. IEEE Trans. Control Syst. Technol. 23(3), 948–962 (2014) 213. J. Wang, S. Koga, Y. Pi, M. Krstic, Axial vibration suppression in a partial differential equation model of ascending mining cable elevator. J. Dyn. Syst. Meas. Control 140(11), 111003 (2018) 214. J. Wang, Y. Pi, M. Krstic, Balancing and suppression of oscillations of tension and cage in dual-cable mining elevators. Automatica 98, 223–238 (2018) 215. J. Wang, S.X. Tang, Y. Pi, M. Krstic, Exponential regulation of the anti-collocatedly disturbed cage in a wave PDE-modeled ascending cable elevator. Automatica 95, 122–136 (2018) 216. D. Wang, T. Jiang, X. Chen, Control-oriented modeling and repetitive control in In-layer and cross-layer thermal interactions in selective laser sintering, in Dynamic Systems and Control Conference, vol. 59155 (American Society of Mechanical Engineers, New York, 2019), p. V002T27A001 217. J. Wang, S.X. Tang, M. Krstic, Adaptive output-feedback control of torsional vibration in off-shore rotary oil drilling systems. Automatica 111, 108640 (2020) 218. H. Wang, F. Wang, K. Xu, Modeling Information Diffusion in Online Social Networks with Partial Differential Equations, vol. 7 (Springer, New York, 2020) 219. J.S. Wettlaufer, Heat flux at the ice-ocean interface. J. Geophys. Res. Oceans 96, 7215–7236 (1991) 220. M. Winton, A reformulated three-layer sea ice model. J. Atmos. Oceanic Technol. 17(4), 525–531 (2000) 221. C. Wunsch, The Ocean Circulation Inverse Problem (Cambridge University Press, Cambridge, 1996) 222. C. Wunsch, P. Heimbach, Practical global oceanic state estimation. Phys. D Nonlinear Phenom 230(1–2), 197–208 (2007) 223. H. Yu, M. Krstic, Traffic congestion control for Aw-Rascle-Zhang model. Automatica 100, 38–51 (2019) 224. H. Yu, A.M. Bayen, M. Krstic, Boundary observer for congested freeway traffic state estimation via Aw-Rascle-Zhang model. IFAC-PapersOnLine 52(2), 183–188 (2019) 225. H. Yu, M. Diagne, L. Zhang, M. Krstic, Bilateral boundary control of moving shockwave in LWR model of congested traffic. IEEE Trans. Autom. Control, Early access (2020). https:// doi.org/10.1109/TAC.2020.2994031 226. H. Yu, Q. Gan, A.M. Bayen, M. Krstic, PDE traffic observer validated on freeway data. IEEE Trans. Control Syst. Technol., Early access (2020). https://doi.org/10.1109/TCST.2020. 2989101 227. B. Zalba, J.M. Marin, L.F. Cabeza, H. Mehling, Review on thermal energy storage with phase change: materials, heat transfer analysis and applications. Appl. Therm. Eng. 23, 251–283 (2003)

References

349

228. K. Zeng, D. Pal, B. Stucker, A review of thermal analysis methods in laser sintering and selective laser melting, in Proceedings of Solid Freeform Fabrication Symposium, Austin, TX, vol. 60 (2012), pp. 796–814 229. Y. Zeng, P. Albertus, R. Klein, N. Chaturvedi, A. Kojic, M.Z. Bazant, J. Christensen, Efficient conservative numerical schemes for 1d nonlinear spherical diffusion equations with applications in battery modeling. J. Electrochem. Soc. 160(9), A1565-A1571 (2013) 230. Q. Zhang, R.E. White, Moving boundary model for the discharge of a LiCoO2 electrode. J. Electrochem. Soc. 154(6), A587–A596 (2007) 231. L. Zhang, T. Phillips, A. Mok, D. Moser, J. Beaman, Automatic laser control system for selective laser sintering. IEEE Trans. Industr. Inform. 15(4), 2177–2185 (2018)

Index

Symbols 2-D disk, 171 3D printing, 221

A Adaptive control, 297 Additive manufacturing, 221, 247 Advection speed, 240 Agmon’s inequality, 36, 316 Arctic sea ice, 179

B Backstepping, 20 Backstepping transformation, 22, 30, 232 Backward Euler method, 213 Battery management systems, 199 Beer’s law, 249 Bessel functions, 78 Boundary control, 22

C Cauchy-Schwarz inequality, 36, 315 Class-K L function, 120 Class-K function, 120 Comparison principle, 136 Controllable pair, 25 Convective heat transfer, 93 Core-shell model, 202 Coupled PDE-ODE system, 4, 17

D Damping coefficient, 161 Delay compensation, 99 Detectability, 183 Diffusion equations, 21 Disturbance rejection, 175

E Electrochemical models, 199 Energy conservation law, 8, 101 Energy shaping, 19, 142 Explicit Euler method, 11 Exponential ISS, 121 Exponential stability, 34, 44, 99, 143 Extended Kalman filter, 85, 215 Extremum seeking, 175

F Fick’s law, 202 Finite difference, 9 Finite volume, 211

G Gain kernel, 22 Gain tuning, 38, 286 Global climate model, 180 Global exponential stability, 156 Gronwall’s inequality, 99

© Springer Nature Switzerland AG 2020 S. Koga, M. Krstic, Materials Phase Change PDE Control & Estimation, Systems & Control: Foundations & Applications, https://doi.org/10.1007/978-3-030-58490-0

351

352 H Heat transfer coefficient, 273 Hopf’s lemma, 5 Hurwitz matrix, 25

I Input-output normal form, 144 Input saturation, 38 Input-to-state stability, 118, 219 Inter-sample-predictor, 281 Inverse Stefan problem, 167 Inverse transformation, 31

J Jacobian, 88

K Kalman filter, 59

L Lithium-ion batteries, 199 Lithium Iron Phosphate, 199 Luenberger observer, 59 Lyapunov analysis, 261 Lyapunov method, 34, 99

M Mass balance, 202 Maximum principle, 8, 99, 141, 250 Modified Bessel function, 205 Motion planning, 167 Moving boundary, 3

O Observability, 183 Observer, 60 one-phase Stefan problem, 4 Open-loop estimation, 193 Optimal filter, 88 Output feedback control, 66

P Phase change materials, 271 PI control, 242

Index Poincare’s inequality, 315 Predictor-based feedback, 105

R Reaction-diffusion PDE, 21, 59 Reduced-order model, 85 Reference error system, 20 Reference trajectory, 166 Robustness to delay mismatch, 107 Robustness to parameter uncertainty, 50 Runge-Kutta method, 187

S Sampled-data control, 128 Sampling scheduling, 129 Screw extrusion, 221 Selective laser sintering, 247 Sensor delay, 195 Separation principle, 68 Setpoint regulation, 20 Settling time, 156 Single particle model, 199 SoC estimation, 216 Spherical coordinates, 112 State observer, 204 State-of-charge, 204 Stefan condition, 4, 273 Stefan number, 6 Stefan problem, 1 Stefan system, 1

T Target system, 22, 63, 187, 233 Time-varying backstepping transformation, 166 Trajectory tracking, 100, 166 Transport equations, 221 Transport PDE, 102 Two-phase Stefan system, 139

W White Gaussian noise, 87

Z Zero-order-hold, 128