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  • Nonlinear Model Predictive Control: A Sampled-Data Feedback Perspective

    Von der Fakultät Maschinenbau der Universität Stuttgart

    zur Erlangung der Würde eines

    Doktor–Ingenieurs (Dr.–Ing.) genehmigte Abhandlung

    Vorgelegt von

    Rolf Findeisen geboren in Nürtingen

    Hauptberichter: Prof. Dr.-Ing. F. Allgöwer Mitberichter: Prof. Dr.ing. B. A. Foss

    Tag der mündlichen Prüfung: 9. Dezember 2004

    Institut für Systemtheorie technischer Prozesse der Universität Stuttgart

    2004

  • III

    Acknowledgements

    This work was developed during my employment as scientific coworker at the Institute for Systems Theory in Engineering (IST) of the Universität Stuttgart.

    I would like to especially thank my supervisor, Prof. Dr.-Ing. Frank Allgöwer, for his support, encour- agement, and confidence throughout my work. He provided an open and stimulating environment at the Institute, which encouraged me to pursue my track of research. Many thanks go also to the mem- bers of the dissertation committee, Prof. Dr.-Ing. h.c. M. Zeitz, and especially to the co-referent Prof. Bjarne Foss from the NTNU in Trondheim, Norway.

    Many people have significantly contributed to this work. Firstly, I would like to mention Moritz Diehl from the Institute for Scientific Computing of the University of Heidelberg for the long years of cooperation’s and fruitful discussions, be it be professional or personal, which hopefully will continue as successful as by now in the future. Special thanks go also to Lars Imsland, who is now with SINTEF ICT, in Trondheim. Major parts of the work on the output feedback problem where developed during Lars stay at the IST and my enjoyable stay at the NTNU in Trondheim.

    I do not want to forget my colleagues and by now good friends from the IST. Firstly there are Eric Bullinger and Patrick Menold, who came together with me from the ETH Zürich to Stuttgart. Both of them, as well as Ansgar Rehm, where always there once I had to discuss something, be it personal or professional. I especially remember the frequent coffee breaks, which sometimes ended up in excellent research ideas. I also do not want to forget Christian Ebenbauer, who joined our group in an early stage and whom one can always approach having something to discuss.

    There are many more people here to mention and to thank for their continuing support and help. I just want to name a few, knowing that there will be always some left out or forgotten, please do not take it personal in this case: all current and former members of the IST, Doris Köhler, Zoltan Nagy, Stefan Schwarzkopf, Flynn Marquardt, Tobias Raff, Alejandor Vargas, Chen Hong, Fan Hui, Renato Lepore, Andrey Yonchev, Ilknur Disli Uslu, Hans-Georg Bock, all students whom I supervised during their student thesis, and many more ...

    Finally, but not last möchte ich meinen Eltern und meinen Brüdern Peter und Jürgen für ihre Unter- stützung, Geduld und Verständniss danken.

    Rolf Findeisen Stuttgart, Dezember 2004

  • IV

    MEINEN ELTERN UND MEINEN BRÜDERN

  • V

    Contents

    Summary VIII

    Deutsche Kurzfassung IX

    List of Symbols VIII

    1 Introduction 1

    1.1 NMPC and Sampled-data Open-loop Feedback . . . . . . . . . . . . . . . . . . . . 2

    1.2 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

    1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

    2 A Brief Review of Nonlinear Model Predictive Control 7

    2.1 Basic Principle of Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . 7

    2.2 Basic Mathematical Formulation of NMPC . . . . . . . . . . . . . . . . . . . . . . 8

    2.2.1 Instantaneous NMPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

    2.2.2 Sampled-data Open-loop NMPC . . . . . . . . . . . . . . . . . . . . . . . . 10

    2.3 Properties, Advantages and Drawbacks of NMPC . . . . . . . . . . . . . . . . . . . 11

    2.4 Numerical Aspects of NMPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    2.5 System Theoretical Aspects of NMPC . . . . . . . . . . . . . . . . . . . . . . . . . 14

    2.5.1 Nominal Stability of NMPC . . . . . . . . . . . . . . . . . . . . . . . . . . 14

    2.5.1.1 Stabilizing Instantaneous NMPC Schemes . . . . . . . . . . . . . 15

    2.5.1.2 Stabilizing Sampled-data Open-loop NMPC Schemes . . . . . . . 17

    2.5.2 Robustness and Robust Design of NMPC . . . . . . . . . . . . . . . . . . . 20

    2.5.3 Output-Feedback and NMPC . . . . . . . . . . . . . . . . . . . . . . . . . . 21

    2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

  • VI CONTENTS

    3 Computational Issues in Sampled-data NMPC 25

    3.1 NMPC Formulations Facilitating Efficient Solutions . . . . . . . . . . . . . . . . . . 26

    3.1.1 Use of Short Horizon Lengths and Non-stringent Stability Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

    3.1.2 Use of Suboptimal NMPC Strategies, Feasibility Implies Stability . . . . . . 27

    3.2 Solution of the NMPC Optimal Control Problem . . . . . . . . . . . . . . . . . . . 27

    3.2.1 Solution by Direct Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 28

    3.3 Efficient Solution by Direct Multiple Shooting . . . . . . . . . . . . . . . . . . . . . 31

    3.3.1 Basics of Direct Multiple Shooting . . . . . . . . . . . . . . . . . . . . . . 31

    3.3.2 Solution and Properties of the Direct Multiple Shooting NLP . . . . . . . . . 33

    3.3.3 Further Twists to Achieve Fast Solutions in the Case of NMPC . . . . . . . . 35

    3.4 Control of a High-Purity Distillation Column . . . . . . . . . . . . . . . . . . . . . 35

    3.4.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

    3.4.2 Experimental Verifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

    3.5 Efficient Solution via the Real-time Iteration Scheme . . . . . . . . . . . . . . . . . 42

    3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

    4 Stability of Sampled-data Open-loop State-feedback 45

    4.1 Sampled-data Feedback and Sampled-data Open-loop Feedback . . . . . . . . . . . 46

    4.2 Basic Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

    4.3 Convergence of Sampled-data Control . . . . . . . . . . . . . . . . . . . . . . . . . 49

    4.3.1 Expansions and Generalizations . . . . . . . . . . . . . . . . . . . . . . . . 53

    4.4 Suitable Sampled-data Feedbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

    4.4.1 Instantaneous Feedbacks and Sampled-data Control . . . . . . . . . . . . . . 54

    4.4.2 Stability of Sampled-data NMPC . . . . . . . . . . . . . . . . . . . . . . . 56

    4.4.3 Control of a CSTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

    4.5 Consideration of Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

    4.5.1 Measurement Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

    4.5.2 Computational Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

    4.5.3 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

    4.6 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

  • CONTENTS VII

    5 Inherent Robustness Properties of Sampled-data Open-loop Feedbacks 72

    5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

    5.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

    5.3 Considered Type of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

    5.4 Robustness to Additive Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . 78

    5.5 Robustness to Input Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

    5.6 Robustness to Measurement and State Estimation Errors . . . . . . . . . . . . . . . 83

    5.7 Inherent Robustness of Sampled-data Open-loop NMPC . . . . . . . . . . . . . . . 85

    5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

    6 Sampled-data Open-loop Output-feedback 87

    6.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

    6.2 Semi-regional Practical Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

    6.3 Suitable Observer Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

    6.3.1 High Gain Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

    6.3.2 Moving Horizon Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

    6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

    6.4.1 Example I: Control of a Bioreactor . . . . . . . . . . . . . . . . . . . . . . . 94

    6.4.2 Example II: Control of a Pendulum-cart System . . . . . . . . . . . . . . . . 97

    6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

    7 Conclusions and Outlook 102

    7.1 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

    A Proof of Lemma 4.1 105

    Bibliography 107

  • VIII

    Summary