Verified methods for state and parameter estimators for nonlinear uncertain systems with applications in engineering [Elektronische Ressource] / von Marco Kletting

Informations

• Publié par: universitat_ulm
• Publié le: 01 janvier 2009
• Nombre de lectures: 18
• Langue: English
• Poids de l'ouvrage: 6 Mo

Extrait

Verified Methods for State and Parameter Estimators for
Nonlinear Uncertain Systems with Applications in
Engineering

DISSERTATION


zur Erlangung des akademischen Grades eines



DOKTOR-INGENIEURS

(DR.-ING.)


der Fakultät für Ingenieurwissenschaften
und Informatik der Universität Ulm


von


MARCO KLETTING
AUS DORNSTADT-TEMMENHAUSEN




Gutachter: Prof. Dr. Eberhard P. Hofer
Dr. Èric Walter

Amtierender Dekan: Prof. Dr. Ing. Michael Weber



Ulm, 15.05.2009 Preface
This dissertation resulted from my work as a research assistant at the Institute of Measure-
ment, Control and Micro Technology of the Faculty of Engineering and Computer Science
at the University of Ulm in Germany.
First of all, I would like to thank sincerely my doctor father Prof. Dr. Eberhard P. Hofer for
his support, his encouragement, and guidance during my research. I am also very grateful
to Dr. Eric Walter for being supervisor of my doctoral thesis and for his careful reading of
the manuscript and for his valuable advices. Furthermore I would like to thank my former
colleague Dr.-Ing. Andreas Rauh for the many precious discussions and suggestions and for
the great collaboration during my time at the Institute. I would also like to thank Prof.-Dr.-
Ing. Harald Aschemann for his support. Special thanks to my friend Dr.-Ing. Felix Antritter
for a very fruitful cooperation and making it possible to extend my field of research. Also
thanks to all of my students for their excellent work.
My deepest gratitude belongs to my beloved wife Jane. Her constant support helped me
also through difficult times. And I also thank my three little children for being patient with
their Dad, when he was too busy to play with them.For my beloved wife Jane
”Many women have done excellently,
but you surpass them all.”(Prov. 31:29)Contents
1 Introduction 2
1.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Goals of this Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Applications 5
2.1 Non-Isothermal Stirred Tank Reactor . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Mechanical Positioning System . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Double Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Biological Waste Water Treatment Plant . . . . . . . . . . . . . . . . . . . . 11
2.5 Magnetic Levitation System . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Interval Analysis and Taylor Models 16
3.1 Interval Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Rounding in Interval Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Interval Newton Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Dependency Problem and Wrapping Effect . . . . . . . . . . . . . . . . . . . 22
3.4.1 Dependency Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4.2 Wrapping Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.5 Optimized Interval Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5.1 Taylor Inclusion Functions . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5.2 Monotonicity Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5.3 Iterative Improvement of Infimum and Supremum . . . . . . . . . . . 30
3.6 Taylor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.6.1 Definition of Taylor Models . . . . . . . . . . . . . . . . . . . . . . . 30
3.6.2 Taylor Model Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.6.3 Range Bounding of Taylor Models . . . . . . . . . . . . . . . . . . . . 35
4 Verified Simulation of Nonlinear Uncertain Systems 36
4.1 Verified Techniques Based on Interval Enclosures . . . . . . . . . . . . . . . 37
4.1.1 Basic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1.2 Mean value form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1.3 Monotonicity Test and Iterative Range Computation . . . . . . . . . 39
4.1.4 Implicit Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.1.5 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . 40
4.1.6 Interval Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.1.7 Combination of Interval Splitting and Coordinate Transformation . . 50
4.2 Consistency Techniques for Reduction of Overestimation . . . . . . . . . . . 52
4.2.1 One Step Consistency Tests . . . . . . . . . . . . . . . . . . . . . . . 54Contents ii
4.2.2 Merging Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2.3 Multi-Step Consistency Techniques . . . . . . . . . . . . . . . . . . . 57
4.2.4 Application: Non-Isothermal Stirred Tank Reactor . . . . . . . . . . 62
4.3 Verified Integration of Uncertain Systems with State-Dependant Switching
Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3.1 Basic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3.2 Optimized Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3.3 Application: Mechanical Positioning System . . . . . . . . . . . . . . 69
4.4 Verified Techniques Based on Taylor Models . . . . . . . . . . . . . . . . . . 74
4.4.1 Basic Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4.2 Reduction of the Wrapping Effect in Taylor Model based Verified In-
tegrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5 Verified State and Parameter Estimators 94
5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2 Observability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.2.1 Observability in Linear Systems . . . . . . . . . . . . . . . . . . . . . 98
5.2.2 Observability in Nonlinear Systems . . . . . . . . . . . . . . . . . . . 98
5.2.3 Verified Observability Analysis. . . . . . . . . . . . . . . . . . . . . . 101
5.3 Interval Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.3.1 Prediction Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.3.2 Correction Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Non-Isothermal Stirred Tank Reactor . . . . . . . . . . . . . . 111
Double Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . 119
Biological Waste Water Treatment Plant . . . . . . . . . . . . . 121
Mechanical Positioning System . . . . . . . . . . . . . . . . . . 126
5.4 Taylor Model Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.4.1 Prediction Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.4.2 Correction Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.4.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Non-Isothermal Stirred Tank Reactor . . . . . . . . . . . . . . 141
Double Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . 146
Biological Waste Water Treatment Plant . . . . . . . . . . . . . 150
5.5 Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.5.1 Non-Isothermal Stirred Tank Reactor . . . . . . . . . . . . . . . . . . 155
5.5.2 Double Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.5.3 Biological Waste Water Treatment Plant . . . . . . . . . . . . . . . . 165
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6 Verified Methods for Guaranteed Robust Tracking with Flatness Based Con-
trollers 171
6.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.2 Flatness Based Controller Design . . . . . . . . . . . . . . . . . . . . . . . . 172
6.2.1 Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.2.2 Flatness Based Feedforward Controller . . . . . . . . . . . . . . . . . 173Contents iii
6.2.3 Flatness Based Tracking Controller design . . . . . . . . . . . . . . . 173
6.2.4 Tracking using a Nonlinear Tracking Observer . . . . . . . . . . . . . 174
6.3 Robustness Analysis of the Tracking Controller . . . . . . . . . . . . . . . . 175
6.4 Application: Magnetic Levitation system . . . . . . . . . . . . . . . . . . . . 178
7 Verified State and Parameter Estimators in Closed Loop Control 182
7.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.2 Application: Magnetic Levitation System . . . . . . . . . . . . . . . . . . . . 183
8 Conclusions and Outlook on Future Research 190
A Examples for Taylor Model Based Verified Integration of ODEs 193
A.1 Nonlinear Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
A.2 Nonlinear Example with Preconditioning . . . . . . . . . . . . . . . . . . . . 199
Bibliography 202List of Figures
2.1