Efficient numerical solution of large scale algebraic matrix equations in PDE control and model order reduction [Elektronische Ressource] / presented by: Jens Saak
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Efficient numerical solution of large scale algebraic matrix equations in PDE control and model order reduction [Elektronische Ressource] / presented by: Jens Saak

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Ecient Numerical Solution of Large ScaleAlgebraic Matrix Equations in PDE Controland Model Order ReductionDissertationsubmitted to Faculty of Mathematicsat Chemnitz University of Technologyin accordance with the requirements for the degreeDr. rer. nat.x˙(t)= x(t)+ u(t);M N By(t)= x(t)Cˆ ˆ ˆM N B˙= xˆ(t)+ u(t);xˆ(t)=y(t) ˆ xˆ(t)Cpresented by: Dipl. Math. Jens SaakAdvisor: Prof. Dr. Peter BennerReviewers: Prof. Dr. Enrique S. Quintana-Ort´ıProf. Dr. Ekkehard W. SachsChemnitz July 6, 2009iiiiito S¸enayivACKNOWLEDGEMENTSFinancial Support. Large parts of this research have been refined in the projectsParallele numerische Losung¨ von Optimalsteuerungsproblemen fur¨ instationar¨ e Diusions-Konvektions-Reaktionsgleichungen (project A15 in SFB393 Parallele Numerische Simulationfur¨ Physik und Kontinuumsmechanik), Numerische Losung¨ von Optimalsteuerungsproblemenfur¨ instationar¨ e Diusions-Konvektions- und Diusions-Reaktionsgleichungen and IntegrierteSimulation des Systems “Werkzeugmaschine-Antrieb-Zerspanprozess” auf der Grundlage ord-nungsreduzierter FEM-Strukturmodelle supported by the German Research Foundation(DFG) over the past years.

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Published 01 January 2009
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Ecient Numerical Solution of Large Scale
Algebraic Matrix Equations in PDE Control
and Model Order Reduction
Dissertation
submitted to Faculty of Mathematics
at Chemnitz University of Technology
in accordance with the requirements for the degree
Dr. rer. nat.
x˙(t)= x(t)+ u(t);M N B
y(t)= x(t)C
ˆ ˆ ˆM N B
˙ = xˆ(t)+ u(t);xˆ(t)
=y(t) ˆ xˆ(t)C
presented by: Dipl. Math. Jens Saak
Advisor: Prof. Dr. Peter Benner
Reviewers: Prof. Dr. Enrique S. Quintana-Ort´ı
Prof. Dr. Ekkehard W. Sachs
Chemnitz July 6, 2009iiiii
to S¸enayivACKNOWLEDGEMENTS
Financial Support. Large parts of this research have been refined in the projects
Parallele numerische Losung¨ von Optimalsteuerungsproblemen fur¨ instationar¨ e
DiusionsKonvektions-Reaktionsgleichungen (project A15 in SFB393 Parallele Numerische Simulation
fur¨ Physik und Kontinuumsmechanik), Numerische Losung¨ von Optimalsteuerungsproblemen
fur¨ instationar¨ e Diusions-Konvektions- und Diusions-Reaktionsgleichungen and Integrierte
Simulation des Systems “Werkzeugmaschine-Antrieb-Zerspanprozess” auf der Grundlage
ordnungsreduzierter FEM-Strukturmodelle supported by the German Research Foundation
(DFG) over the past years. Besides these, the integration and exchange project Parallele
Algorithmen fur¨ hochdimensionale, dunnbes¨ etzte algebraische Riccatigleichungen und
Anwendungen in der Regelungstheorie inside the Acciones Integradas Hispano-Alemanas program
of the German Academic Exchange Service (DAAD) has enabled me to undertake some
very helpful and inspiring trips to Universitat Jaume I in Castellon´ (Spain).
Personal Thanks. My primary thanks go to my advisor and teacher Peter Benner for
the introduction to and the guidance in this highly fascinating field of research with all
its interesting types of applications. On the other hand the best mentor still depends on
the many people working in the background. Therefore my thanks go to the colleagues
and friends that I had the pleasure to work with at TU Chemnitz during the past almost
6 years. I cannot mention single persons without forgetting other important ones. So I
will only pick the three most important ones, Ulrike Baur, Sabine Hein and Hermann
Mena. As for Hermann I can only cite himself. I am particularly grateful to my dear
friend Hermann Mena together with whom (among the most interesting topics in life) large
parts of this work were discussed in our mid-afternoon coee breaks. The countless inspiring
discussions with Sabine have given me an increasingly deeper insight to many aspects
of LQR and LQG design for parabolic PDEs. Also, I would know hardly half as much
as I do today about model order reduction, if there had not been Ulrike providing me
with endless advice and numerous suggestions regarding the field.
I also want to thank Enrique S. Quintana-Ort´ı and his workgroup at Universitat Jaume
I for the warm welcomes in Spain and for providing me a quiet place to work outvi
the foundation of this document undisturbedly. Especially Alfredo Remon´ and Sergio
Barrachina have always been more than helpful in organizing my journeys and made
the visits to Spain some of the most enjoyable time of the past years.
My special thanks are expressed to the student assistant Martin Kohler¨ for the massive
work in implementing the upcomingC.M.E.S.S. library and performing the extensive
testing that made large parts of Section 4.4.3 and the corresponding numerical results
in Chapter 8 possible.
My family and many friends have supported me in hundreds of ways over the years
and I hope that I have outweighed their help adequately every now and then.
Finally I want to thank the most important people in the current period of my life and
the best friends that I can think of. Although both of them may have been hundreds to
thousands of kilometers away almost all the time, they have been closer than anyone.
Lars Fischer, whom I cannot thank enough for constantly pushing me a few steps further
than I actually wanted to go – it has always been a source of personal progress. Last
and definitely most importantly I thank S¸enay Yavuz, who opened the door to all of
this. I owe her more than I can ever tell or repay.CONTENTS
List of Figures xi
List of Tables xv
List of Algorithms xvii
List of Acronyms xix
List of Symbols xxi
1. Introduction 1
2. Basic Concepts 5
2.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2. Finite Dimensional Systems and Control Theory Basics . . . . . . . . . . 7
2.2.1. LTI Systems in State Space Representation . . . . . . . . . . . . . 7
2.2.2. Generalized State Space Form and Descriptor Systems . . . . . . 9
2.2.3. Second Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.4. Linear-Quadratic Optimal Control in Finite Dimensions . . . . . 12
2.3. LQR Optimal Control of Parabolic PDEs . . . . . . . . . . . . . . . . . . . 17
2.3.1. Approximation Theory . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4. Balanced Truncation Model Order Reduction . . . . . . . . . . . . . . . . 23
3. Model Problems and Test Examples 25
3.1. An Academic Model Example: FDM Semi-Discretized Heat Equation . . 26
3.2. An Artificial Test Case with Prescribed Spectrum . . . . . . . . . . . . . . 27
3.3. Selective Cooling of Steel Profiles: Cooling a Rail in a Rolling Mill . . . . 27
3.3.1. Model Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.2. Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.3. Boundary Conditions and Boundary Control . . . . . . . . . . . . 29
3.3.4. Choice of State Weighting Operator Q and Output Operator C . . 31
3.3.5. Units of Measurement and Scaling . . . . . . . . . . . . . . . . . . 32viii Contents
3.4. Chemical Reactors: Controling the Temperature at Inflows . . . . . . . . 33
3.5. The SLICOT CD-Player . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.6. The Spiral Inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.7. A Scalable Oscillator Example . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.8. The Butterfly Gyro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.9. Fraunhofer/Bosch Acceleration Sensor . . . . . . . . . . . . . . . . . . . . 37
4. Efficient Solution of Large Scale Matrix Equations 39
4.1. The ADI Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2. Lyapunov Equations: An ADI Model Problem . . . . . . . . . . . . . . . 41
4.3. ADI Shift Parameter Selection . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3.1. Review of Existing Parameter Selection Methods . . . . . . . . . . 43
4.3.2. Suboptimal Parameter Computation . . . . . . . . . . . . . . . . . 47
4.3.3. Dominant Pole Based Shifts for Balancing Based MOR . . . . . . 49
4.4. Acceleration of the LRCF-ADI Method for Lyapunov Equations . . . . . 51
4.4.1. Column Compression for the LRCFs . . . . . . . . . . . . . . . . . 51
4.4.2. Hybrid Krylov-ADI Solvers for the Lyapunov Equation . . . . . . 52
4.4.3. Software Engineering Aspects . . . . . . . . . . . . . . . . . . . . 59
4.5. Algebraic Riccati Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.5.1. Newtons Method for Algebraic Riccati Equations . . . . . . . . . 60
4.5.2. Ecient Computation of Feedback Gain Matrices . . . . . . . . . 63
4.5.3. Modified Variants of the LRCF-NM . . . . . . . . . . . . . . . . . 65
4.5.4. The Relationship of LRCF-NM and the QADI Iteration . . . . . . 66
4.5.5. Does CFQADI Allow Low-Rank Factor Computations? . . . . . . 67
4.6. Stopping Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5. Generalized Systems and Generalized Matrix Equations 71
5.1. Avoiding the Mass Matrix by Matrix Decomposition . . . . . . . . . . . . 72
5.1.1. Algebraic Riccati Equations and Feedback Computations . . . . . 74
5.1.2. Lyapunov Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2. Implicit Handling of the Inverse Mass Matrix . . . . . . . . . . . . . . . . 75
5.2.1. Algebraic Riccati Equations and Feedback Computations . . . . . 76
5.2.2. Lyapunov Equations and Balancing Based Model Order Reduction 77
6. Application in Optimal Control of Parabolic PDEs 81
6.1. Tracking Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2. Suboptimality Estimation from Approximation Error Results . . . . . . . 83
6.3. Adaptive-LQR for quasilinear Parabolic PDEs . . . . . . . . . . . . . . . 85
6.3.1. Relation to Model Predictive Control . . . . . . . . . . . . . . . . 86
6.3.2. Identification of Nonlinear MPC Building Blocks . . . . . . . . . 88
7. Application in MOR of First and Second Order Systems 89
7.1. First Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7.1.1. Standard State Space Systems . . . . . . . . . . . . . . . . . . . . . 90Contents ix
7.1.2. Generalized State Space Systems . . . . . . . . . . . . . . . . . . . 93
7.2. Second Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.2.1. Ecient Computation of Reduced First Order Models . . . . . . 96
7.2.2. Regaining the Second Order Structure for the Reduced Order Model 98
7.2.3. Adaptive Choice of Reduced Model Order . . . . . . . . . . . . . 100
8. Numerical Tests 101
8.1. Numerical Tests for the ADI Shift Parameter Selections . . . . . . . . . . 102
8.1.1. FDM Semi-Discretized Convection-Diusion-Reaction Equation . 102
8.1.2. FDM Heat Equation . . . . . . . . . . . . . . . . 103
8.1.3. FEM Semi-Discretizedusion Equation . . . . . . 103
8.1.4. Dominant Pole Shifts and LR-SRM . . . . . . . . . . . . . . . . . . 104
8.2. Accelerating Large Scale Matrix Equation Solvers . . . . . . . . . . . . . 110
8.2.1. Accelerated Solution of large scale LEs . . . . . . . . . . . . . . . 110
8.2.2. of large scale AREs . . . . . . . . . . . . . . 110
8.3. Model Order Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
8.3.1. Reduction of First Order Systems . . . . . . . . . . . . . . . . . . . 116
8.3.2. of Second Order to First Order ROMs . . . . . 121
8.3.3. Reduction of Order Systems to Second Order ROMs . . . 122
8.4. Comparison of the Matlab and C Implementations . . . . . . . . . . . . 126
8.4.1. Shared Memory Parallelization . . . . . . . . . . . . . . . . . . . . 127
8.4.2. TimingsC.M.E.S.S. vs. M.E.S.S. . . . . . . . . . . . . . . . . . . . 130
9. Conclusions and Outlook 133
9.1. Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 133
9.2. Future Research Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . 135
A. Selective Cooling of Steel Profiles: Exponential Stabilization and
Discretization 139
A.1. Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
A.1.1. Linear-Quadratic Regulator Problems in Hilbert Spaces . . . . . . 140
A.1.2. Weak Formulation and Abstract Cauchy Problem . . . . . . . . . 140
A.1.3. Approximation by Finite Dimensional Systems . . . . . . . . . . . 143
A.2. Approximation of Abstract Cauchy Problems . . . . . . . . . . . . . . . . 143
A.3. Implementation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
B. Theses 149
Bibliography 151
Index 163x Contents