Small-Scale Helicopter Automatic Autorotation
Transcript Of Small-Scale Helicopter Automatic Autorotation
Small-Scale Helicopter Automatic Autorotation
Modeling, Guidance, and Control
Small-Scale Helicopter Automatic Autorotation
Modeling, Guidance, and Control
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties, in het openbaar te verdedigen op vrijdag 18 september 2015 om 10:00 uur
door
Skander Taamallah
Master of Science in Aeronautics & Astronautics, Stanford University, U.S.A., Diplôme d’Ingénieur en Génie Electrique, I.N.S.A. Toulouse, France, geboren te Tunis, Tunesië.
Dit proefschrift is goedgekeurd door de
promotor: prof. dr. ir. P.M.J. Van den Hof promotor: prof. dr. ir. X. Bombois
Samenstelling promotiecommissie:
Rector Magnificus, Prof. dr. ir. P.M.J. Van den Hof Prof. dr. ir. X. Bombois
voorzitter Technische Universiteit Delft CNRS, Ecole Centrale de Lyon, Frankrijk
Onafhankelijke leden: Prof. dr. R. Babuska Prof. dr. J. Bokor Prof. dr. ir. M. Mulder Prof. dr. H. Nijmeijer Prof. dr. G. Scorletti
Technische Universiteit Delft Hungarian Academy of Sciences, Hongarije Technische Universiteit Delft Technische Universiteit Eindhoven Ecole Centrale de Lyon, Frankrijk
The research described in this thesis has been supported by the National Aerospace Laboratory (NLR), Amsterdam, The Netherlands.
Keywords:
Printed by: Front & Back:
Unmanned Aerial Vehicles, Small-Scale Helicopter, Automatic Autorotation, Trajectory Planning, Trajectory Tracking, Linear Parameter Varying Systems.
Ipskamp Drukkers.
View of a small-scale unmanned helicopter.
Copyright c 2015 by S. Taamallah
ISBN/EAN: 978-94-6259-831-7
An electronic version of this dissertation is available at http://repository.tudelft.nl/.
Considerate la vostra origine: non siete nati per vivere come bruti, ma per praticare la virtù e apprendere la conoscenza.
Dante Alighieri Divina Commedia, Inferno, Canto XXVI
Contents
Summary
xi
Samenvatting
xiii
Preface
xv
1 Introduction
1
1.1 Unmanned Aerial Vehicles (UAVs) . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Candidate applications . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Development and acquisition programs . . . . . . . . . . . . . . . 3
1.1.4 Airworthiness and safety aspects . . . . . . . . . . . . . . . . . . 4
1.2 The helicopter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Helicopter mini-UAVs . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Helicopter main rotor hubs . . . . . . . . . . . . . . . . . . . . . 8
1.3 Helicopter autorotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 Autorotation: a three-phases maneuver . . . . . . . . . . . . . . . 10
1.4 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Analysis of available options . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5.1 Model-free versus model-based options. . . . . . . . . . . . . . . 12
1.5.2 Integrated versus segregated options . . . . . . . . . . . . . . . . 13
1.5.3 Summary of previous analysis . . . . . . . . . . . . . . . . . . . 16
1.6 Research objectives and limitations . . . . . . . . . . . . . . . . . . . . . 17
1.7 Solution strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.7.1 Modeling of the nonlinear helicopter dynamics . . . . . . . . . . . 19
1.7.2 The Trajectory Planning (TP) . . . . . . . . . . . . . . . . . . . . 21
1.7.3 The Trajectory Tracking (TT). . . . . . . . . . . . . . . . . . . . 24
1.8 Overview of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.8.1 Contributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2 High-Order Modeling of the Helicopter Dynamics
47
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2 Helicopter modeling: general overview . . . . . . . . . . . . . . . . . . . 49
2.3 Model evaluation and validation . . . . . . . . . . . . . . . . . . . . . . 51
2.3.1 Trim results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3.2 Dynamic results . . . . . . . . . . . . . . . . . . . . . . . . . . 61
vii
viii
Contents
2.4 Preliminary analysis of the rigid-body dynamics . . . . . . . . . . . . . . 64 2.4.1 Linearizing the nonlinear helicopter model . . . . . . . . . . . . . 65 2.4.2 The engine ON case . . . . . . . . . . . . . . . . . . . . . . . . 69 2.4.3 The engine OFF case . . . . . . . . . . . . . . . . . . . . . . . . 70
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.6 Appendix A: Nomenclature. . . . . . . . . . . . . . . . . . . . . . . . . 71 2.7 Appendix B: Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.8 Appendix C: Rigid-body equations of motion . . . . . . . . . . . . . . . . 76 2.9 Appendix D: Main rotor . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.10 Appendix E: Tail rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.11 Appendix F: Fuselage . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 2.12 Appendix G: Vertical and horizontal tails . . . . . . . . . . . . . . . . . . 98 2.13 Appendix H: Problem data . . . . . . . . . . . . . . . . . . . . . . . . . 99 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3 Off-line Trajectory Planning
107
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.2.1 Cost functional . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.2.2 Boundary conditions and trajectory constraints . . . . . . . . . . . 111
3.3 The optimal control problem . . . . . . . . . . . . . . . . . . . . . . . . 111
3.4 Direct optimal control and discretization methods. . . . . . . . . . . . . . 113
3.5 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.5.1 The Height-Velocity (H-V) diagram . . . . . . . . . . . . . . . . 117
3.5.2 Evaluation of cost functionals. . . . . . . . . . . . . . . . . . . . 118
3.5.3 Optimal autorotations: effect of initial conditions . . . . . . . . . . 121
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4 On-line Trajectory Planning and Tracking: System Design
141
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.1.1 Main contributions . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.2 General control architecture . . . . . . . . . . . . . . . . . . . . . . . . 144
4.3 Flatness-based Trajectory Planning (TP) . . . . . . . . . . . . . . . . . . 144
4.3.1 Flat outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.3.2 Flat output parametrization . . . . . . . . . . . . . . . . . . . . . 147
4.3.3 Optimal trajectory planning for the engine OFF case . . . . . . . . 147
4.4 Robust control based Trajectory Tracking (TT) . . . . . . . . . . . . . . . 151
4.4.1 Linear multivariable µ control design . . . . . . . . . . . . . . . . 153
4.4.2 Controller assessment metrics . . . . . . . . . . . . . . . . . . . 155
4.5 Design of the engine OFF inner-loop controller . . . . . . . . . . . . . . . 157
4.5.1 Choice of nominal plant model for the inner-loop control design . . 157
4.5.2 Selection of weights . . . . . . . . . . . . . . . . . . . . . . . . 158
4.5.3 Controller synthesis and analysis . . . . . . . . . . . . . . . . . . 159
Contents
ix
4.6 Design of the engine OFF outer-loop controller . . . . . . . . . . . . . . . 162 4.6.1 Selection of weights . . . . . . . . . . . . . . . . . . . . . . . . 162 4.6.2 Controller synthesis and analysis . . . . . . . . . . . . . . . . . . 163 4.6.3 Adapting the engine OFF outer-loop controller . . . . . . . . . . . 165
4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.8 Appendix A: Optimal trajectory planning for the engine ON case . . . . . . 166 4.9 Appendix B: Design of the inner-loop controller for the engine ON case . . 167 4.10 Appendix C: Design of the outer-loop controller for the engine ON case . . 170 4.11 Appendix D: Maximum roll (or pitch) angle for safe (i.e. successful) land-
ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 4.12 Appendix E: Proof of Lemma 1. . . . . . . . . . . . . . . . . . . . . . . 176 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
5 On-line Trajectory Planning and Tracking: Simulation Results
183
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
5.2 Setting up the trajectory planning for the engine ON cases . . . . . . . . . 184
5.3 Setting up the trajectory planning for the engine OFF cases . . . . . . . . . 187
5.4 Discussion of closed-loop simulation results for the engine ON cases . . . . 189
5.5 Discussion of closed-loop simulation results for the engine OFF cases . . . 192
5.5.1 System energy: the engine ON versus engine OFF cases . . . . . . 193
5.5.2 Closed-loop response with respect to sensors noise and wind dis-
turbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
6 Affine LPV Modeling
217
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
6.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
6.3 Step 1: Identifying the central model (A0, B0) . . . . . . . . . . . . . . . . 227
6.4 Step 2: Identifying the basis functions {Ls, Rs}Ss=1 . . . . . . . . . . . . . . 227 6.5 Step 3: Identifying the basis functions {Tw, Zw}Ww=1 . . . . . . . . . . . . . 228
6.6
Step 4.1: Identifying the parameters
ηi
N i=1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
229
6.7 Step 4.2: Obtaining the mapping η(x(t), u(t)) . . . . . . . . . . . . . . . . 231
6.8
Steps 5.1 and 5.2: Identifying the parameters
ζi
N i=1
and
obtaining
the
map-
ping ζ(x(t), u(t)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
6.9 Application to the modeling and control of a modified pointmass pendulum . 232
6.9.1 Building the LPV models. . . . . . . . . . . . . . . . . . . . . . 233
6.9.2 Open-Loop analysis . . . . . . . . . . . . . . . . . . . . . . . . 235
6.9.3 Closed-Loop analysis. . . . . . . . . . . . . . . . . . . . . . . . 238
6.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
6.11 Appendix A: Kalman-Yakubovich-Popov (KYP) Lemma with spectral mask
constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
6.11.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
x
Contents
6.12 Appendix B: Identifying the set of parameters
η1(ti),
...,
ηS
(ti)
N i=1
for
a
specific
case
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
248
6.13 Appendix C: Problem data . . . . . . . . . . . . . . . . . . . . . . . . . 250
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
7 Conclusions and future research
261
7.1 Contribution of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . 262
7.2 Recommendations for future research. . . . . . . . . . . . . . . . . . . . 264
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
List of Abbreviations
281
Curriculum Vitæ
285
List of Publications
287
Modeling, Guidance, and Control
Small-Scale Helicopter Automatic Autorotation
Modeling, Guidance, and Control
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties, in het openbaar te verdedigen op vrijdag 18 september 2015 om 10:00 uur
door
Skander Taamallah
Master of Science in Aeronautics & Astronautics, Stanford University, U.S.A., Diplôme d’Ingénieur en Génie Electrique, I.N.S.A. Toulouse, France, geboren te Tunis, Tunesië.
Dit proefschrift is goedgekeurd door de
promotor: prof. dr. ir. P.M.J. Van den Hof promotor: prof. dr. ir. X. Bombois
Samenstelling promotiecommissie:
Rector Magnificus, Prof. dr. ir. P.M.J. Van den Hof Prof. dr. ir. X. Bombois
voorzitter Technische Universiteit Delft CNRS, Ecole Centrale de Lyon, Frankrijk
Onafhankelijke leden: Prof. dr. R. Babuska Prof. dr. J. Bokor Prof. dr. ir. M. Mulder Prof. dr. H. Nijmeijer Prof. dr. G. Scorletti
Technische Universiteit Delft Hungarian Academy of Sciences, Hongarije Technische Universiteit Delft Technische Universiteit Eindhoven Ecole Centrale de Lyon, Frankrijk
The research described in this thesis has been supported by the National Aerospace Laboratory (NLR), Amsterdam, The Netherlands.
Keywords:
Printed by: Front & Back:
Unmanned Aerial Vehicles, Small-Scale Helicopter, Automatic Autorotation, Trajectory Planning, Trajectory Tracking, Linear Parameter Varying Systems.
Ipskamp Drukkers.
View of a small-scale unmanned helicopter.
Copyright c 2015 by S. Taamallah
ISBN/EAN: 978-94-6259-831-7
An electronic version of this dissertation is available at http://repository.tudelft.nl/.
Considerate la vostra origine: non siete nati per vivere come bruti, ma per praticare la virtù e apprendere la conoscenza.
Dante Alighieri Divina Commedia, Inferno, Canto XXVI
Contents
Summary
xi
Samenvatting
xiii
Preface
xv
1 Introduction
1
1.1 Unmanned Aerial Vehicles (UAVs) . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Candidate applications . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Development and acquisition programs . . . . . . . . . . . . . . . 3
1.1.4 Airworthiness and safety aspects . . . . . . . . . . . . . . . . . . 4
1.2 The helicopter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Helicopter mini-UAVs . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Helicopter main rotor hubs . . . . . . . . . . . . . . . . . . . . . 8
1.3 Helicopter autorotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 Autorotation: a three-phases maneuver . . . . . . . . . . . . . . . 10
1.4 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Analysis of available options . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5.1 Model-free versus model-based options. . . . . . . . . . . . . . . 12
1.5.2 Integrated versus segregated options . . . . . . . . . . . . . . . . 13
1.5.3 Summary of previous analysis . . . . . . . . . . . . . . . . . . . 16
1.6 Research objectives and limitations . . . . . . . . . . . . . . . . . . . . . 17
1.7 Solution strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.7.1 Modeling of the nonlinear helicopter dynamics . . . . . . . . . . . 19
1.7.2 The Trajectory Planning (TP) . . . . . . . . . . . . . . . . . . . . 21
1.7.3 The Trajectory Tracking (TT). . . . . . . . . . . . . . . . . . . . 24
1.8 Overview of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.8.1 Contributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2 High-Order Modeling of the Helicopter Dynamics
47
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2 Helicopter modeling: general overview . . . . . . . . . . . . . . . . . . . 49
2.3 Model evaluation and validation . . . . . . . . . . . . . . . . . . . . . . 51
2.3.1 Trim results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3.2 Dynamic results . . . . . . . . . . . . . . . . . . . . . . . . . . 61
vii
viii
Contents
2.4 Preliminary analysis of the rigid-body dynamics . . . . . . . . . . . . . . 64 2.4.1 Linearizing the nonlinear helicopter model . . . . . . . . . . . . . 65 2.4.2 The engine ON case . . . . . . . . . . . . . . . . . . . . . . . . 69 2.4.3 The engine OFF case . . . . . . . . . . . . . . . . . . . . . . . . 70
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.6 Appendix A: Nomenclature. . . . . . . . . . . . . . . . . . . . . . . . . 71 2.7 Appendix B: Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.8 Appendix C: Rigid-body equations of motion . . . . . . . . . . . . . . . . 76 2.9 Appendix D: Main rotor . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.10 Appendix E: Tail rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.11 Appendix F: Fuselage . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 2.12 Appendix G: Vertical and horizontal tails . . . . . . . . . . . . . . . . . . 98 2.13 Appendix H: Problem data . . . . . . . . . . . . . . . . . . . . . . . . . 99 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3 Off-line Trajectory Planning
107
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.2.1 Cost functional . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.2.2 Boundary conditions and trajectory constraints . . . . . . . . . . . 111
3.3 The optimal control problem . . . . . . . . . . . . . . . . . . . . . . . . 111
3.4 Direct optimal control and discretization methods. . . . . . . . . . . . . . 113
3.5 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.5.1 The Height-Velocity (H-V) diagram . . . . . . . . . . . . . . . . 117
3.5.2 Evaluation of cost functionals. . . . . . . . . . . . . . . . . . . . 118
3.5.3 Optimal autorotations: effect of initial conditions . . . . . . . . . . 121
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4 On-line Trajectory Planning and Tracking: System Design
141
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.1.1 Main contributions . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.2 General control architecture . . . . . . . . . . . . . . . . . . . . . . . . 144
4.3 Flatness-based Trajectory Planning (TP) . . . . . . . . . . . . . . . . . . 144
4.3.1 Flat outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.3.2 Flat output parametrization . . . . . . . . . . . . . . . . . . . . . 147
4.3.3 Optimal trajectory planning for the engine OFF case . . . . . . . . 147
4.4 Robust control based Trajectory Tracking (TT) . . . . . . . . . . . . . . . 151
4.4.1 Linear multivariable µ control design . . . . . . . . . . . . . . . . 153
4.4.2 Controller assessment metrics . . . . . . . . . . . . . . . . . . . 155
4.5 Design of the engine OFF inner-loop controller . . . . . . . . . . . . . . . 157
4.5.1 Choice of nominal plant model for the inner-loop control design . . 157
4.5.2 Selection of weights . . . . . . . . . . . . . . . . . . . . . . . . 158
4.5.3 Controller synthesis and analysis . . . . . . . . . . . . . . . . . . 159
Contents
ix
4.6 Design of the engine OFF outer-loop controller . . . . . . . . . . . . . . . 162 4.6.1 Selection of weights . . . . . . . . . . . . . . . . . . . . . . . . 162 4.6.2 Controller synthesis and analysis . . . . . . . . . . . . . . . . . . 163 4.6.3 Adapting the engine OFF outer-loop controller . . . . . . . . . . . 165
4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.8 Appendix A: Optimal trajectory planning for the engine ON case . . . . . . 166 4.9 Appendix B: Design of the inner-loop controller for the engine ON case . . 167 4.10 Appendix C: Design of the outer-loop controller for the engine ON case . . 170 4.11 Appendix D: Maximum roll (or pitch) angle for safe (i.e. successful) land-
ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 4.12 Appendix E: Proof of Lemma 1. . . . . . . . . . . . . . . . . . . . . . . 176 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
5 On-line Trajectory Planning and Tracking: Simulation Results
183
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
5.2 Setting up the trajectory planning for the engine ON cases . . . . . . . . . 184
5.3 Setting up the trajectory planning for the engine OFF cases . . . . . . . . . 187
5.4 Discussion of closed-loop simulation results for the engine ON cases . . . . 189
5.5 Discussion of closed-loop simulation results for the engine OFF cases . . . 192
5.5.1 System energy: the engine ON versus engine OFF cases . . . . . . 193
5.5.2 Closed-loop response with respect to sensors noise and wind dis-
turbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
6 Affine LPV Modeling
217
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
6.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
6.3 Step 1: Identifying the central model (A0, B0) . . . . . . . . . . . . . . . . 227
6.4 Step 2: Identifying the basis functions {Ls, Rs}Ss=1 . . . . . . . . . . . . . . 227 6.5 Step 3: Identifying the basis functions {Tw, Zw}Ww=1 . . . . . . . . . . . . . 228
6.6
Step 4.1: Identifying the parameters
ηi
N i=1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
229
6.7 Step 4.2: Obtaining the mapping η(x(t), u(t)) . . . . . . . . . . . . . . . . 231
6.8
Steps 5.1 and 5.2: Identifying the parameters
ζi
N i=1
and
obtaining
the
map-
ping ζ(x(t), u(t)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
6.9 Application to the modeling and control of a modified pointmass pendulum . 232
6.9.1 Building the LPV models. . . . . . . . . . . . . . . . . . . . . . 233
6.9.2 Open-Loop analysis . . . . . . . . . . . . . . . . . . . . . . . . 235
6.9.3 Closed-Loop analysis. . . . . . . . . . . . . . . . . . . . . . . . 238
6.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
6.11 Appendix A: Kalman-Yakubovich-Popov (KYP) Lemma with spectral mask
constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
6.11.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
x
Contents
6.12 Appendix B: Identifying the set of parameters
η1(ti),
...,
ηS
(ti)
N i=1
for
a
specific
case
.
.
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.
.
.
.
.
.
.
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.
.
.
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248
6.13 Appendix C: Problem data . . . . . . . . . . . . . . . . . . . . . . . . . 250
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
7 Conclusions and future research
261
7.1 Contribution of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . 262
7.2 Recommendations for future research. . . . . . . . . . . . . . . . . . . . 264
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
List of Abbreviations
281
Curriculum Vitæ
285
List of Publications
287