Tracking Control of Nonlinear Mechanical Systems

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Transcript of Tracking Control of Nonlinear Mechanical Systems

  • TRACKING CONTROL OF

    NONLINEAR MECHANICAL SYSTEMS

    PROEFSCHRIFT

    ter verkrijging vande graad van doctor aan de Universiteit Twente,

    op gezag van de rector magnificus,prof.dr. F.A. van Vught,

    volgens besluit van het College voor Promotiesin het openbaar te verdedigen

    op vrijdag 14 april 2000 te 15.00 uur.

    door

    Adriaan Arie Johannes Lefeber

    geboren op 7 december 1972

    te Beverwijk

  • Dit proefschrift is goedgekeurd door de promotoren

    prof. dr. ir. H. Kwakernaak,

    prof. dr. H. Nijmeijer.

  • Ik droomde eens en zie ik liepaan t strand bij lage tij.Ik was daar niet alleen, want ookde Heer liep aan mijn zij.We liepen saam het leven dooren lieten in het zand,een spoor van stappen, twee aan twee;de Heer liep aan mijn hand.Ik stopte en keek achter mijen zag mijn levensloop,in tijden van geluk en vreugdvan diepe smart en hoop.Maar als ik goed het spoor bekeek,zag ik langs heel de baan,daar waar het juist het moeilijkst was,maar een paar stappen staan : : :Ik zei toen: Heer, waarom dan toch?Juist toen k U nodig had,juist toen ik zelf geen uitkomst zagop t zwaarste deel van t pad : : :De Heer keek toen vol liefd mij aanen antwoordd op mijn vragen:Mijn lieve kind, toen t moeilijk was,toen heb Ik jou gedragen : : :

  • Erjen LefeberSystems EngineeringDept. of Mechanical EngineeringEindhoven University of TechnologyP.O. Box 5135600 MB EindhovenThe Netherlands

    CIP-DATA KONINKLIJKE BIBLIOTHEEK, DEN HAAG

    Tracking control of nonlinear mechanical systems / Adriaan Arie Johannes Lefeber. En-schede : Universiteit Twente, 2000. vi, 165 p.: ill.; 24.5 cmAuteursnaam op omslag: Erjen Lefeber. Proefschrift Universiteit Twente, Enschede. Lit.opg.: p.153159 Met samenvatting in het Nederlands.ISBN 90-365-1426-6

    Ook verschenen als Online Resource

  • Preface

    How to start a preface to a thesis that should give an overview of what I have been doing foralmost four years? That is not an easy question to answer. Paradoxically, by writing downthis question I also found an answer to it.

    If one comes to think about it, this preface is full of paradoxes. It is funny to realize thatthis probably is the first page that people read (except from the title, and for Dutch readersalso the poem, I presume), whereas it is one of the last pages I wrote. I could elaborate moreon these paradoxes, but that would make this first page so philosophical and paradoxical thateven this first page will not be read : : :

    At a time like this, as a period of my life is about to finish, I can not avoid looking back at it,and looking forward too. While typing this sentence, my thoughts went out to several people.These include Henk, Guido, Anders, Kristin, Tono & Lena, my committee, colleagues, par-ents and not least of all Wieke (thinking about periods of my life : : : ). They all are peoplewho, amongst others, deserve credit for their contributions (in one way or the other) to thisthesis.

    First of all I would like to thank my supervisor and promotor Henk Nijmeijer. During the lastfive years we have come to know each other quite well. I am grateful for the opportunity ofworking with him, for the possibilities of visiting several colleagues, and for making me putthings in the right perspective. I am really looking forward to our collaboration in Eindhovenin the near future.

    I would like to thank Guido Blankenstein for being more than just a colleague. I have ben-efited from our (lively) discussions and his willingness to listen to me when I tried to settlemy thoughts. Furthermore, I enjoyed the moments we spent together outside of office and hisspecial sense of humor.

    I would like to express my gratitude to Anders Robertsson, not only for our working togetherand his hospitality during my visit to Lund, but especially for all the playing on words inEnglish during our conversations, e-mails and phone calls. In addition I am indebted toKristin Pettersen, Antonio Lora, Elena Panteley, Romeo Ortega, Zhong-Ping Jiang, JanuszJakubiak and Rafael Kelly, as they all contributed to this thesis.

    I am grateful to the members of my promotion committee for thorough reading my manu-script: Prof. Huibert Kwakernaak, Prof. Henk Nijmeijer, Prof. Claude Samson of Sophia An-tipolis (France), Prof. Koos Rooda of Eindhoven, and Prof. Ben Jonker, Prof. Arun Bagchi,and Prof. Arjan van der Schaft of Twente. I would also like to thank Prof. Guy Campion of

    v

  • vi Preface

    Louvain-la-Neuve (Belgium) for his feedback during the meetings of my advisory committee.

    I would like to take the opportunity to thank all my (former) colleagues at the Faculty ofMathematical Sciences in Twente, and especially those of the Systems, Signals and Controlgroup for having provided me with such a creative and friendly atmosphere to work in. Spe-cial thanks also go to the Systems Engineering group in Eindhoven (my current colleagues)for giving me the opportunity to finish my thesis quietly and showing me a glimpse of thechallenges I will face in the near future.

    A special word of thanks goes to Henk Ernst Blok, Paul Huijnen, Kristin Pettersen, and PhilChimento who sacrificed themselves for going through (parts of) a draft version of this thesis.They all contributed to this thesis with their valuable comments.

    I would finally like to thank all my friends, my beloved family, and all the people whose nameI did not mention explicitly. A special word of thanks goes to my fiancee Wieke Fikse for allher support and help.

    Erjen Lefeber

  • Contents

    Preface v

    1 Introduction 5

    1.1 Formulation of the tracking control problem . . . . . . . . . . . . . . . . . . 6

    1.2 Non-holonomic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    1.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

    2 Preliminaries 13

    2.1 Mathematical preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    2.2 Lyapunov stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    2.3 Linear time-varying systems . . . . . . . . . . . . . . . . . . . . . . . . . . 18

    2.4 Cascaded systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

    2.5 Backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

    I A cascaded approach to tracking 27

    3 Introduction to Part I 29

    3.1 Cascaded design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    3.2 An introductory example: tracking of a rotating rigid body . . . . . . . . . . 31

    4 Tracking of a mobile robot 35

    4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

    4.2 State-feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

    4.2.1 Previous results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

    4.2.2 A cascaded design . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

    1

  • 2 Contents

    4.3 Dynamic output-feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

    4.3.1 Unmeasured xe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

    4.3.2 Unmeasured ye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

    4.4 Saturated control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

    4.4.1 A Lyapunov design . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

    4.4.2 A cascaded design . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

    4.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

    4.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

    5 Tracking of non-holonomic systems in chained form 57

    5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

    5.2 The search for a cascaded structure . . . . . . . . . . . . . . . . . . . . . . . 59

    5.3 State-feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

    5.4 Dynamic output-feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

    5.5 Saturated control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

    5.6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

    5.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

    6 Tracking control of an under-actuated ship 73

    6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

    6.2 The tracking error dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 75

    6.3 State-feedback: a cascaded approach . . . . . . . . . . . . . . . . . . . . . . 76

    6.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

    6.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

    II Robot manipulators and adaptive control 89

    7 Introduction to Part II 91

    7.1 Paper I: Global asymptotic stability of robot manipulators with linear PIDand PI2D control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

    7.2 Paper II: Adaptive and filtered visual servoing of planar robots . . . . . . . . 92

    7.3 Paper III: Adaptive tracking control of non-holonomic systems: an example . 92

    7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

  • Contents 3

    8 GAS of robot manipulators with linear PID and PI2D control 95

    8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

    8.1.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

    8.1.2 Mode