ANALYSIS AND CONTROL OF NON-AFFINE, NON-STANDARD, SINGULARLY PERTURBED

287 Pages·2012·4.66 MB·English

ANALYSIS AND CONTROL OF NON-AFFINE, NON-STANDARD,
SINGULARLY PERTURBED SYSTEMS
A Dissertation
by
ANSHU NARANG
Submitted to the Office of Graduate Studies of
Texas A&M University
in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY
Approved by:
Chair of Committee, John Valasek
Committee Members, Aniruddha Datta
Helen L. Reed
Srinivas Rao Vadali
Department Head, Rodney D. W. Bowersox
December 2012
Major Subject: Aerospace Engineering
Copyright 2012 Anshu Narang
ABSTRACT
This dissertation addresses the control problem for the general class of control
non-affine, non-standard singularly perturbed continuous-time systems. The prob-
lem of control for nonlinear multiple time scale systems is addressed here for the
first time in a systematic manner. Toward this end, this dissertation develops the
theory of feedback passivation for non-affine systems. This is done by generalizing
the Kalman-Yakubovich-Popov lemma for non-affine systems. This generalization
is used to identify conditions under which non-affine systems can be rendered pas-
sive. Asymptotic stabilization for non-affine systems is guaranteed by using these
conditions along with well known passivity-based control methods. Unlike previ-
ous non-affine control approaches, the constructive static compensation technique
derived here does not make any assumptions regarding the control influence on the
nonlinear dynamical model. Along with these control laws, this dissertation presents
novel hierarchical control design procedures to address the two major difficulties
in control of multiple time scale systems: lack of an explicit small parameter that
models the time scale separation and the complexity of constructing the slow man-
ifold. These research issues are addressed by using insights from geometric singular
perturbation theory and control laws are designed without making any assumptions
regarding the construction of the slow manifold. The control schemes synthesized
accomplish asymptotic slow state tracking for multiple time scale systems and si-
multaneous slow and fast state trajectory tracking for two time scale systems. The
control laws are independent of the scalar perturbation parameter and an upper
bound for it is determined such that closed-loop system stability is guaranteed.
Performance of these methods is validated in simulation for several problems
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from science and engineering including the continuously stirred tank reactor, mag-
netic levitation, six degrees-of-freedom F-18/A Hornet model, non-minimum phase
helicopter and conventional take-off and landing aircraft models. Results show that
the proposed technique applies both to standard and non-standard forms of singu-
larlyperturbedsystemsandprovidesasymptotictrackingirrespectiveofthereference
trajectory. This dissertation also shows that some benchmark non-minimum phase
aerospace control problems can be posed as slow state tracking for multiple time
scale systems and techniques developed here provide an alternate method for exact
output tracking.
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To dearest Ma, Papa, Amma and Appa
and my closest friend and husband, Siddarth
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ACKNOWLEDGEMENTS
I am very grateful to my advisor Dr. John Valasek for his constant motivation,
support and guidance during my doctorate study. Working with him I have had
the opportunity to learn and experience the different aspects of academia that have
helped shape my future career aspirations. I am indebted to Dr. Valasek for nomi-
nating me as an associate member of the AIAA Guidance, Navigation and Control
Technical Committee. Thank you Sir.
AspecialthankyoutoDr.SrinivasR.Vadali, Dr.AniruddhaDattaandDr.Helen
L. Reed for their critical reviews, feedbacks, and suggestions to improve the quality
of my research work.
I would like to thank Dr. Suman Chakravorty for sharing his time and expertise
through numerous group meetings and classroom discussions, and Dr. James D.
Turner for sharing his insights during our discussions in the hallways. I would also
like to thank Dr. John E. Hurtado for providing me a firm foundation in dynamical
systems theory and giving me the opportunity to work in Land, Air and Space
Robotics Laboratory.
I am thankful to my undergraduate advisor and mentor, Dr. S. N Saxena for his
constant support in all my endeavors and motivating me to perform to the best of
my capabilities.
My sincere thanks to Ms. Wanda Romero, Ms. Karen Knabe and Ms. Colleen
Leatherman for going out of their way to help me throughout my graduate study.
I would like to thank the Zonta International Foundation for awarding me the
Amelia Earhart Fellowship which was very helpful during my graduate study.
I am fortunate to have the friendship and support of Priyanka, Ryan and Neha
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who have made life in College Station memorable. I cherish our long chats filled with
crazy humor and serious discussions about everything under the sun.
It would have been impossible to come this far without the unconditional love
and unwavering support of my family. My parents have always been my inspiration
and without their motivation I would not have been able to work this far from home.
I feel extremely fortunate to have parents whose love has made me what I am today.
I am very grateful to Amma and Appa who have supported me and believed in me.
Nowords candescribethe influenceSiddarth’sloveand constant wordsofmotivation
have had in my life. He has inspired me to excel and stood by me through both good
and challenging times. I thank my sister Suruchi, Navin Jijaji, dearest Smridula
and Pulkit for their continuous support and love. I feel blessed to be part of such a
wonderful family and dedicate this work to them.
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TABLE OF CONTENTS
Page
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Standard and Non-Standard Forms of Singularly Perturbed Systems . 6
1.2 Review of Stabilization Methods for Singularly Perturbed Systems . . 8
1.3 Review of Control Methodologies for Non-Affine in Control Nonlinear
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2. RESEARCH OBJECTIVES . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1 System Class Description . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1 System Properties . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Objectives and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Research Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3. A CONSTRUCTIVE STABILIZATION APPROACH FOR OPEN-LOOP
UNSTABLE NON-AFFINE SYSTEMS . . . . . . . . . . . . . . . . . . . . 29
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Passive Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 Feedback Equivalence to a Passive System/Feedback Passivation . . . 36
3.4 Control Synthesis for Stabilization . . . . . . . . . . . . . . . . . . . . 41
3.4.1 Control Synthesis for Multi-Input Non-Affine Systems . . . . . 41
3.4.2 Construction of Control for Single-Input Non-Affine Systems . 44
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3.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5.1 Purpose and Scope . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5.2 One-Dimensional Non-Affine Unstable Dynamics . . . . . . . . 47
3.5.3 Continuously Stirred Tank Reactor . . . . . . . . . . . . . . . 53
3.5.4 Magnetic Levitation System . . . . . . . . . . . . . . . . . . . 59
3.6 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.6.1 Benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.6.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4. ASYMPTOTIC STABILIZATION AND SLOW STATE TRACKING OF
CONTROL-AFFINE, TWO TIME SCALE SYSTEMS . . . . . . . . . . . 68
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2 Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 Approach I: Modified Composite Control . . . . . . . . . . . . . . . . 71
4.3.1 Center Manifold and Its Computation . . . . . . . . . . . . . 71
4.3.2 Control Law Development . . . . . . . . . . . . . . . . . . . . 75
4.3.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 88
4.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.4 Approach II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.4.1 Control Law Development . . . . . . . . . . . . . . . . . . . . 104
4.4.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.4.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 109
4.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.5 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5. SIMULTANEOUS TRACKING OF SLOW AND FAST TRAJECTORIES
FOR CONTROL-AFFINE, TWO TIME SCALE SYSTEMS . . . . . . . . 122
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.2 Control Formulation and Stability Analysis . . . . . . . . . . . . . . . 122
5.2.1 Control Law Development . . . . . . . . . . . . . . . . . . . . 123
5.2.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.3.1 Purpose and Scope . . . . . . . . . . . . . . . . . . . . . . . . 129
5.3.2 Generic Two Degrees-of-Freedom Nonlinear Kinetic Model . . 129
5.3.3 Combined Longitudinal and Lateral/Directional Maneuver for
a F/A-18 Hornet . . . . . . . . . . . . . . . . . . . . . . . . . 133
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5.4 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.4.1 Benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.4.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6. CONTROL OF NONLINEAR, NON-AFFINE, NON-STANDARD MUL-
TIPLE TIME SCALE SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . 140
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.2 Background: Reduced-Order Models . . . . . . . . . . . . . . . . . . 140
6.3 Control Formulation and Stability Analysis . . . . . . . . . . . . . . . 144
6.3.1 Control Formulation . . . . . . . . . . . . . . . . . . . . . . . 145
6.3.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.4.1 Purpose and Scope . . . . . . . . . . . . . . . . . . . . . . . . 155
6.4.2 Standard Two Time Scale Model . . . . . . . . . . . . . . . . 155
6.4.3 Non-Standard Multiple Time Scale System . . . . . . . . . . . 157
6.5 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7. SOME APPLICATIONS TO CONTROL OF WEAKLY MINIMUM AND
NON-MINIMUM PHASE, NONLINEAR DYNAMICAL SYSTEMS . . . . 163
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.2 The Beam and Ball Experiment . . . . . . . . . . . . . . . . . . . . . 163
7.2.1 Dynamical Model . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.2.2 Time Scale Separation Analysis . . . . . . . . . . . . . . . . . 169
7.2.3 Control Formulation . . . . . . . . . . . . . . . . . . . . . . . 171
7.2.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 173
7.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.3 HoverControlforanUnmannedThreeDegrees-of-FreedomHelicopter
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.3.1 Model Description and Open-Loop Analysis . . . . . . . . . . 181
7.3.2 Time Scale Analysis of the Helicopter Model . . . . . . . . . . 189
7.3.3 Control Formulation and Stability Analysis . . . . . . . . . . . 191
7.3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 203
7.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
7.4 Nap-of-the-Earth Maneuver Control for Conventional Take-off and
Landing Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
7.4.1 Dynamical Model and Open-Loop Analysis . . . . . . . . . . . 212
7.4.2 Time Scale Analysis of the Aircraft Model . . . . . . . . . . . 217
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7.4.3 Control Formulation . . . . . . . . . . . . . . . . . . . . . . . 219
7.4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 222
7.4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
7.5 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
7.5.1 Benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
7.5.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
8. CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . . . . . . . . 234
8.1 Contributions of Research . . . . . . . . . . . . . . . . . . . . . . . . 234
8.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
8.3 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
APPENDIX A. REVIEW OF GEOMETRIC SINGULAR PERTURBATION
THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
APPENDIX B. COMPOSITE LYAPUNOV APPROACH FOR STABILITY
ANALYSIS OF SINGULARLY PERTURBED SYSTEMS . . . . . . . . . 262
APPENDIX C. NONLINEAR F/A-18 HORNET AIRCRAFT MODEL . . . 269
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