Adaptive Observer Design for Parabolic Partial Differential Equations
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Adaptive Observer Design for Parabolic Partial Differential Equations by Pedro A. Ascencio A Thesis submitted in fulfilment of requirements for the degree of Doctor of Philosophy of Imperial College London Power and Control Research Group Department of Electrical and Electronic Engineering Imperial College London 2017 Dedicated to my wife M´onica, our daughter Elena pem. 1 Declaration of Originality I hereby declare that this is an original thesis and it is entirely based on my own work. I acknowledge the sources in every instance where I have used the ideas of other writers. This thesis was not and will not be submitted to any other university or institution for fulfilling the requirements of a degree. Copyright Declaration The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work. Pedro A. Ascencio London, 2017. 2 Abstract This thesis addresses the observer design problem, for a class of linear one-dimensional parabolic Partial Differential Equations, considering the simultaneous estimation of states and parameters from boundary measurements. The design is based on the Backstepping methodology for Partial Differential Equations and extends its central idea, the “Volterra transformation”, to compensate for the parameters uncertainties. The design steps seek to reject time-varying parameter uncertainties setting forth a type of differential boundary value problems (Kernel-PDE/ODEs) to accomplish its objective, the solution of which is computed at every fixed sampling time and constitutes the observer gains for states and pa- rameters. The design does not include any pre-transformation to some canonical form and/or a finite-dimensional formulation, and performs a direct parameter estimation from the original model. The observer design problem considers two cases of parameter uncertainty, at the “boundary”: control gain coefficient, and “in-domain”: diffusivity and reactivity parameters, respectively. For a Luenberger-type observer structure, the problems associated to one and two points of measurement at the boundary are studied through the application of an intuitive modification of the Volterra-type and Fredholm-type transformations. The resulting Kernel-PDE/ODEs are addressed by means of a novel methodology based on polynomial optimization and Sum-of-Squares decomposition. This approach allows recasting these coupled differential equations as convex optimization problems readily implementable resorting to semidefinite programming, with no restrictions to the spectral characteristics of some integral operators or system’s coefficients. Additionally, for polynomials Kernels, uniqueness and invertibility of the Fredholm-type transfor- mation are proved in the space of real analytic and continuous functions. The direct and inverse Kernels are approximated as the optimal polynomial solution of a Sum-of-Squares and Moment problem with theoretically arbitrary precision. Numerical simulations illustrate the effectiveness and potentialities of the methodology proposed to manage a variety of problems with different structures and objectives. 3 Acknowledgments I would like to express, on behalf of my family, our deep appreciation to my supervisors Professor Alessandro Astolfi and Professor Thomas Parisini for their invaluable support and guidance. Our feelings of endless gratitude for making possible my research and our family life. We are indebted to Mr. Pedro Gonzalez, my former Manager in Chile. Many thanks don Pedro to believe in us, helps to our daughter’s education, and provide us your generous support. I would like to thank my colleagues from the Control and Power research group, in particular to Pedro Ram´ırez and his family for their unique friendship, generosity and innumerable advice. Many thanks to Felipe Tobar for his kind and prompt answers. My gratitude also goes to Mrs. Michelle Hennessy-Hammond for her diligent support. I also thanks to Dr. Riccardo Ferrari for his guidance in my initial steps of research. From Chile, I want to express my sincere gratitude to Professor Daniel Sbarbaro, Professor Jos´e Espinoza and Professor Freddy Paiva for their constant support in many opportunities. I am also grateful to Chile’s National Commission for Scientific and Technological Research (Conicyt) for funding my PhD studies at Imperial College. I would like to thank my examiners, Professor Richard Vinter and Dr. Christophe Prieur, for their valuable feedback on my thesis. Finally, I would like to thanks my wife M´onica Salazar and our daughter Elena Ascencio. There is no words to express all that they have done throughout these years, their braveness, endurance, love and joy of living. Pedro A. Ascencio London, 2017. 4 Contents Declaration of Originality 2 Copyright Declaration 2 Abstract 3 Acknowledgements 4 Contents 5 List of Tables 9 List of Figures 10 1 Introduction 11 1.1 Motivation . 11 1.2 A brief review of Distributed Parameter Systems theory . 12 1.2.1 Control . 13 1.2.2 State Estimation . 14 1.2.3 Parameter Estimation . 16 1.2.3.1 Adaptive Observers . 17 1.3 ThesisOutline .................................... 18 1.3.1 Objectives...................................... 18 1.3.2 Organization and Contributions . 19 1.3.2.1 Publications . 21 1.4 Preliminary Terminology . 21 1.4.1 Notation . 21 1.4.2 Definition...................................... 22 1.4.3 Abbreviations . 22 2 Formulation of Differential Equations as Convex Optimization Problems 23 2.1 Sum-of-Squares Approach for PDEs . 24 5 6 2.1.1 Compact Basic Semi-algebraic Sets . 24 2.1.2 Minimax Approximation . 26 2.1.3 Least Squares Approximation . 28 2.1.4 Polytopic Domain Decomposition . 30 2.2 Moment Approach for PDEs . 31 2.3 Computational Examples . 32 2.3.1 One-Dimensional Differential Equation-BVP . 32 2.3.1.1 Steady Convection-dominated Problem . 33 2.3.1.2 Steady Diffusion-Reaction Problem . 35 2.3.1.3 Sturm–Liouville Eigenvalue Problem . 38 2.3.2 Two-Dimensional PDE-BVP . 42 2.3.2.1 Poisson 2-Dimensional Equation . 42 2.3.3 Rational Polynomial Functions . 42 2.3.3.1 Reciprocal Function Problem . 42 2.3.4 Nonlinear Differential Equations . 44 2.3.4.1 Algebraic Riccati Differential Equation . 44 3 Convex Optimization Approach for Backstepping PDE Design 47 3.1 Introduction...................................... 48 3.1.1 Integral Compact Operators . 49 3.2 Parabolic PDE and the Volterra-type Operator . 51 3.2.1 ProblemSetting .................................. 51 3.2.2 Kernel-PDE as a Convex Optimization Problem . 52 3.2.3 Approximate Inverse Transformation . 55 3.3 Hyperbolic PIDE and the Fredholm-type Operator . 59 3.3.1 ProblemSetting .................................. 59 3.3.2 Existence,UniquenessandInvertibility . 61 3.3.3 Kernel-PIDE as a Convex Optimization Problem . 66 3.3.4 Approximate Inverse Transformation . 70 3.4 NumericalResults ................................... 71 3.4.1 Parabolic PDE with constant reactivity term . 71 3.4.2 Parabolic PDE with spatially varying reactivity term . 73 3.4.3 Hyperbolic PIDE . 75 4 Adaptive Observer Design for a Class of Parabolic PDEs 77 4.1 Introduction...................................... 78 4.2 Observer Design for one Uncertain Boundary Parameter . 79 4.2.1 ProblemSetting .................................. 79 7 4.2.2 Adaptive Observer . 80 4.2.3 Design via the Volterra-type Transformation . 80 4.2.3.1 Kernel-PDE/ODE as a Convex Optimization Problem . 84 4.3 Design for an Uncertain Reactivity Parameter . 85 4.3.1 ProblemSetting .................................. 85 4.3.2 Target System . 86 4.3.3 Design for one boundary measurement: Volterra-type Transformation . 87 4.3.3.1 Kernel-PDE/ODE as a Convex Optimization Problem . 91 4.3.4 Design for two boundary measurements: Fredholm-type Transformation . 92 4.3.4.1 Kernel-PDE/ODE as a Convex Optimization Problem . 95 4.4 NumericalResults ................................... 96 4.4.1 System with an Uncertain Boundary Parameter . 96 4.4.2 System with an Uncertain Reactivity Parameter . 97 5 Adaptive Observer for a Model of Lithium-Ion Batteries 100 5.1 Introduction...................................... 101 5.2 Single Particle Model of the Lithium-Ion Batteries . 103 5.3 AdaptiveObserverDesign .............................. 106 5.3.1 SPM Formulation for Observer Design . 106 5.3.2 Target System . 106 5.3.3 Design via the Volterra-type Transformation . 108 5.3.4 Coupled PDE-ODE as a Convex Optimization Problem . 111 5.3.5 Uncoupled Kernel-PDE/ODE via Convex Optimization . 112 5.4 OutputMappingInversion. 114 5.4.1 Inversion via Moment Approach . 115 5.5 NumericalResults ................................... 116 6 Conclusions 119 6.1 Differential Equations as Convex Optimization Problems . 119 6.2 Convex Formulation of Backstepping Design for PDEs . 120 6.3 AdaptiveObserverDesign .............................. 121 6.3.1 Adaptive Observer for Lithium-Ion Batteries . 122 6.4 FutureResearchDirections . 123 7 Bibliography 124 Appendix A Polynomial Optimization via Sum-of-Squares 147 A.1 Polynomial Optimization . 147 A.1.1 Sum-of-Squares . 149 8 A.1.2 Moments . 150 A.1.2.1 Multi-dimensional Notation . 153 A.1.3 Primal-DualPerspective. 154 Appendix B Backstepping Design for PDEs 156 B.1 TheFundamentalIdea...............................