Model-based tracking control of nonlinear systems

Model-Based Control of Nonlinear Systems presents model-based

control techniques for nonlinear, constrained systems. It covers constructive

control design methods with an emphasis on modeling constrained

systems, generating dynamic control models, and designing tracking

control algorithms for the models.

The book’s interdisciplinary approach illustrates how system modeling and

control theory are essential to control design projects. Organized according

to the steps in a control design project, the text first discusses kinematic and

dynamic modeling methods, including programmed constraints, Lagrange’s

equations, Boltzmann−Hamel equations, and generalized programmed

motion equations. The next chapter describes basic control concepts and

the use of nonlinear control theory. After exploring stabilization strategies

for nonlinear systems, the author presents existing model-based tracking

control algorithms and path-following strategies for nonlinear systems.

The final chapter develops a new model reference tracking strategy for

programmed motion.

Throughout the text, two examples of mechanical systems are used to

illustrate the theory and simulation results. The first example is a unicycle

model (nonholonomic system) and the second is a two-link planar

manipulator model (holonomic system). With a focus on constructive

modeling and control methods, this book provides the tools and techniques

to support the control design process.

Features

• Addresses dynamic modeling and control design methods for

nonlinear mechanical systems

• Covers kinematics and dynamic model formulation

• Describes the design of control algorithms

• Presents a new tracking control strategy architecture

• Includes problems and references in each chapter

K11043

Elżbieta Jarzębowska

Model-Based Tracking

Control of Nonlinear Systems

Mechanical Engineering CRC SERIES: MODERN MECHANICS AND MATHEMATICS Model-Based Tracking Control of Nonlinear Systems Jarzębowska

K11043_Cover.indd 1 4/16/12 1:48 PM

Model-Based Tracking

Control of Nonlinear Systems

CRC SERIES: MODERN MECHANICS AND MATHEMATICS

Series Editors: David Gao and Martin Ostoja-Starzewski

PUBLISHED TITLES

BEYOND PERTURBATION: INTRODUCTION TO THE HOMOTOPY ANALYSIS METHOD

by Shijun Liao

ClassiCal and Generalized Models of elastiC rods

by Dorin Ie¸san

ConfiGurational forCes: therMoMeChaniCs, PhysiCs, MatheMatiCs, and nuMeriCs

by Gérard A. Maugin

CONTINUUM MECHANICS AND PLASTICITY

by Han-Chin Wu

HYBRID AND INCOMPATIBLE FINITE ELEMENT METHODS

by Theodore H.H. Pian and Chang-Chun Wu

INTRODUCTION TO ASYMPTOTIC METHODS

by Jan Awrejcewicz and Vadim A. Krysko

MECHANICS OF ELASTIC COMPOSITES

by Nicolaie Dan Cristescu, Eduard-Marius Craciun, and Eugen Soós

Model-Based traCkinG Control of nonlinear systeMs

by El·

zbieta Jarz˛ebowska

ModelinG and analytiCal Methods in triBoloGy

by Ilya I. Kudish and Michael J. Covitch

MICROSTRUCTURAL RANDOMNESS AND SCalinG IN MECHANICS OF MATERIALS

by Martin Ostoja-Starzewski

CRC SERIES: MODERN MECHANICS AND MATHEMATICS

Elżbieta Jarzębowska

Warsaw University of Technology

Warsaw, Poland

Model-Based Tracking

Control of Nonlinear Systems

CRC Press

Taylor & Francis Group

6000 Broken Sound Parkway NW, Suite 300

Boca Raton, FL 33487-2742

© 2012 by Taylor & Francis Group, LLC

CRC Press is an imprint of Taylor & Francis Group, an Informa business

No claim to original U.S. Government works

Version Date: 20120504

International Standard Book Number-13: 978-1-4398-1982-1 (eBook - PDF)

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Visit the Taylor & Francis Web site at

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To my Parents, who equipped me with passion and courage for life, and to

the memory of Prof. Roman Gutowski, who introduced me to mechanics.

vii

Contents

Preface......................................................................................................................xi

The Author........................................................................................................... xiii

1. Introduction......................................................................................................1

1.1 Scope and Outline..................................................................................3

1.2 Mechanics and Nonlinear Control......................................................6

1.3 Role of Modeling in a Control Design Process................................20

References........................................................................................................ 21

2. Dynamics Modeling of Constrained Systems.........................................25

2.1 Introduction—Art of Modeling.........................................................25

2.1.1 Selection of Coordinates........................................................26

2.1.2 Generalized Velocities and Quasi-Velocities......................29

2.2 Constrained Systems........................................................................... 31

2.2.1 Holonomic Constraints.......................................................... 32

2.2.2 Nonholonomic Constraints...................................................33

2.2.3 Programmed Constraints......................................................35

2.3 Equations of Motion for Systems with

First Order Constraints..................................................................... 37

2.3.1 D’Alembert Principle..............................................................38

2.3.2 Lagrange’s Equations for Holonomic Systems...................45

2.3.3 Lagrange’s Equations for First Order Nonholonomic

Systems.....................................................................................50

2.3.4 Maggi’s Equations................................................................... 52

2.3.5 Nielsen’s Equations.................................................................55

2.3.6 Equations of Motion in Quasi-Coordinates........................58

2.4 Equations of Motion for Systems with High Order

Constraints........................................................................................... 67

2.4.1 An Extended Concept of Constraints—Programmed

Constraints............................................................................... 67

2.4.2 Generalized Programmed Motion Equations

Specified in Generalized Coordinates................................. 76

2.4.3 Generalized Programmed Motion Equations

Specified in Quasi-Coordinates............................................88

Problems...........................................................................................................94

References........................................................................................................94

viii Contents

3. Introduction to Nonlinear Control Theory..............................................99

3.1 Stability Properties of Nonlinear Systems.......................................99

3.1.1 State-Space Representation of Nonlinear Systems.............99

3.1.2 Stability Theorems of the Lyapunov Direct Method....... 101

3.1.3 Special Formulations of Stability Theorems..................... 103

3.2 Classification of Control Problems.................................................. 111

3.2.1 Stabilization........................................................................... 112

3.2.2 Trajectory and Motion Tracking......................................... 115

3.2.3 Path Following...................................................................... 117

3.3 Control Properties of Nonlinear Systems....................................... 118

3.3.1 Classification of Constrained Control Systems................ 118

3.3.2 Accessibility and Controllability........................................122

3.3.3 Stabilizability......................................................................... 131

3.3.4 Differential Flatness............................................................. 135

3.4 Kinematic Control Models................................................................ 136

3.5 Dynamic Control Models................................................................. 144

3.6 Feedback Linearization of Nonlinear Systems.............................. 147

3.7 Model-Based Control Design Methods........................................... 152

3.8 Flatness-Based Control Design Methods....................................... 155

3.8.1 Basic Notions of Equivalence and Flatness....................... 155

3.8.2 Flatness in Control Applications........................................ 159

3.8.3 Flatness-Based Control Design—Examples...................... 161

3.8.4 Concluding Remarks—Verifying Flatness........................ 167

3.9 Other Control Design Techniques for Nonlinear Systems.......... 167

3.9.1 Backstepping......................................................................... 169

3.9.2 Sliding Mode Control........................................................... 173

Problems......................................................................................................... 175

References...................................................................................................... 176

4. Stabilization Strategies for Nonlinear Systems.................................... 183

Problems......................................................................................................... 189

References...................................................................................................... 189

5. Model-Based Tracking Control of Nonlinear Systems........................ 191

5.1 A Unified Control-Oriented Model for Constrained Systems.... 191

5.2 Tracking Control of Holonomic Systems........................................ 196

5.3 Tracking Control of First Order Nonholonomic Systems............200

5.4 Tracking Control of Underactuated Systems.................................206

5.5 Tracking Control Algorithms Specified in

Quasi-Coordinates............................................................................ 212

Problems.........................................................................................................222

References......................................................................................................222

Contents ix

6. Path Following Strategies for Nonlinear Systems................................225

6.1 Path Following Strategies Based on Kinematic

Control Models...................................................................................226

6.2 Path Following Strategies Based on Dynamic

Control Models..................................................................................229

Problems......................................................................................................... 231

References...................................................................................................... 231

7. Model Reference Tracking Control of High Order

Nonholonomic Systems..............................................................................233

7.1 Model Reference Tracking Control Strategy for

Programmed Motion.........................................................................234

7.1.1 A Reference Dynamic Model

for Programmed Motion.....................................................234

7.1.2 Architecture of the Model Reference Tracking

Control Strategy for Programmed Motion........................235

7.1.3 A Controller Design for Programmed

Motion Tracking................................................................... 237

7.2 Non-Adaptive Tracking Control Algorithms for

Programmed Motions....................................................................... 240

7.2.1 Programmed Motion Tracking for a Unicycle.................. 240

7.2.2 Programmed Motion Tracking

for a Planar Manipulator..................................................... 242

7.2.3 Programmed Motion Tracking for a Two-Wheeled

Mobile Robot......................................................................... 246

7.3 Adaptive Tracking Control Algorithms for

Programmed Motions....................................................................... 249

7.3.1 Adaptive Programmed Motion Tracking

for a Planar Manipulator.....................................................250

7.3.2 Adaptive Programmed Motion Tracking

for a Unicycle.........................................................................254

7.4 Learning Tracking Control Algorithms

for Programmed Motions.................................................................258

7.5 Tracking Control Algorithms for Programmed Motions

Specified in Quasi-Coordinates....................................................... 261

7.5.1 Tracking Control of the Unicycle Model

Specified in Quasi-Coordinates.......................................... 262

7.5.2 Tracking Control of the Planar Manipulator Model

Specified in Quasi-Coordinates.......................................... 262

7.6 Tracking Control Algorithms for Programmed Motions

with the Velocity Observer...............................................................264

7.7 Other Applications of the Model Reference Tracking Control

Strategy for Programmed Motion................................................... 270

7.7.1 Hybrid Programmed Motion-Force Tracking................... 270

x Contents

7.7.2 Application of a Kinematic Model

as a Reference Model for Programmed Motions..............277

7.7.3 Robot Formation Control..................................................... 281

Problems.........................................................................................................290

References......................................................................................................290

8. Concluding Remarks.................................................................................. 293

xi

Preface

The book presents model-based control methods and techniques for non￾linear, specifically constrained, systems. It focuses on constructive control

design methods with an emphasis on modeling constrained systems, gen￾erating dynamic control models, and designing tracking control algorithms

for them.

Actually, an active research geared by applications continues on dynam￾ics and control of constrained systems. It is reflected by numerous research

papers, monographs, and research reports. Many of them are listed at the

end of each book chapter, but it is impossible to make the list complete.

The book is not aimed at the survey of existing modeling, tracking, and

stabilization design methods and algorithms. It offers some generalization

of a tracking control design for constrained mechanical systems for which

constraints can be of the programmed type and of arbitrary order. This

generalization is developed throughout the book in accordance with the

three main steps of a control design project, i.e., model building, control￾ler design, and a controller implementation. The book content focuses on

model building and, based upon this model that consists of the generalized

programmed motion equations, on a presentation of new tracking control

strategy architecture.

The author would like to thank the editors at Taylor & Francis for their

support in the book edition; Karol Pietrak, a Ph.D. candidate at Warsaw

University of Technology, Warsaw, Poland, for excellent figure drawings in

the book, and Maria Sanjuan-Janiec for the original book cover design.

xiii

The Author

Elz˙bieta M. Jarze˛bowska is currently with the Institute of Aeronautics

and Applied Mechanics at the Power and Aeronautical Engineering

Department, Warsaw University of Technology, Warsaw, Poland. She

received the B.S., M.S., and Ph.D., D.Sc. degrees in mechanical engineering,

control and mechanics of constrained systems from the Warsaw University

of Technology.

Her fields of research expertise and teaching include dynamics modeling

and analysis of multibody systems, nonlinear control of multibody systems

including nonholonomic, underactuated, unmaned aerial vehicles (UAV),

wheeled robotic systems, and geometric control theory.

Professor Jarze˛bowska was involved in research projects for Automotive

Research Center and Engineering Research Center for Reconfigurable

Machining Systems at the University of Michigan, Ann Arbor, Michigan.

She also gained valuable experience when working for Ford Motor Company

Research Laboratories, Dearborn, Michigan.

She is a member of American Society of Mechanical Engineers (ASME),

Institute of Electrical and Electronics Engineers (IEEE), Gesellschaft für

Angewandte Mathematik und Mechanik (GAMM), International Federation

for the Promotion of Mechanism and Machine (IFToMM), Science Technical

Committee of Mechatronics, and International Society for Advanced Research

(SAR).

Her hobbies are psychology, swimming, yachting, and traveling.