Analysis and Control of Nonlinear Systems (eBook)

Preface 7
Contents 9
Introduction 14
Trajectory Planning and Tracking 15
Equivalence and Flatness 16
Equivalence in System Theory 18
Equivalence and Stability 18
What is a Nonlinear Control System? 19
Nonlinearity versus Linearity 19
Uncontrolled versus Controlled Nonlinearity 20
Part I THEORY 24
Introduction to Differential Geometry 25
Manifold, Diffeomorphism 26
Vector Fields 29
Tangent space, Vector Field 29
Flow, Phase Portrait 31
Lie Derivative 33
Image of a Vector Field 34
First Integral, Straightening Out of a Vector Field 35
Lie Bracket 37
Distribution of Vector Fields 40
Integral Manifolds 41
First Order Partial Differential Equations 42
Differential Forms 44
Cotangent Space, Differential Form, Duality 44
Exterior differentiation 47
Image of a Differential Form 48
Pfaffian System, Complete Integrability 49
Lie Derivative of a 1-Form 51
Back to Frobenius Theorem 53
Introduction to Dynamical Systems 54
Recalls on Flows and Orbits 56
Equilibrium Point, Variational Equation 56
Periodic Orbit 58
Poincaré's Map 61
Stability of Equilibrium Points and Orbits 64
Attractor 64
Lyapunov Stability 66
Remarks on the Stability of Time-Varying Systems 69
Lyapunov's and Chetaev's Functions 70
Hartman-Grobman's and Shoshitaishvili's Theorems, Centre Manifold 75
Singularly Perturbed Systems 84
Invariant Slow Manifold 86
Persistence of the Invariant Slow Manifold 87
Robustness of the Stability 89
An Application to Modelling 90
Application to Hierarchical Control 92
Controlled Slow Dynamics 92
Hierarchical Feedback Design 93
Practical Applications 94
Controlled Systems, Controllability 98
Linear System Controllability 98
Kalman's Criterion 98
Controllability Canonical Form 101
Motion Planning 104
Trajectory Tracking, Pole Placement 106
Nonlinear System Controllability 107
First Order and Local Controllability 107
Local Controllability and Lie Brackets 108
Reachability 112
Lie Brackets and Kalman's Criterion for Linear Systems 115
Jets of Infinite Order, Lie-Bäcklund's Equivalence 117
An Introductory Example of Crane 117
Description of the System Trajectories 119
Jets of Infinite Order, Change of coordinates, Equivalence 123
Jets of Infinite Order, Global Coordinates 124
Product Manifolds, Product Topology 124
Cartan Vector Fields, Flows, Control Systems 125
Lie-Bäcklund Equivalence 131
Properties of the L-B Equivalence 135
Endogenous Dynamic Feedback 137
Differentially Flat Systems 141
Flat System, Flat Output 141
Examples 143
Mass-Spring System 143
Robot Control 144
Pendulum 145
Non Holonomic Vehicle 148
Vehicle with Trailers 149
Flatness and Controllability 151
Flatness and Linearization 153
Mass-Spring System (followed) 154
Robot Control (followed) 155
Pendulum (followed) 155
Non Holonomic Vehicle (followed) 156
Flat Output Characterization 156
The Ruled Manifold Necessary Condition 158
Variational Characterization 161
The Polynomial Matrix Approach 162
Practical Computation of the Smith Decomposition 165
Flatness Necessary and Sufficient Conditions 167
The Operator d 170
Strong Closedness Necessary and Sufficient Conditions 173
Flat Outputs of Linear Controllable Systems 177
Examples 180
Flatness and Motion Planning 190
Motion Planning Without Constraint 191
The General Case 191
Rest-to-Rest Trajectories 193
Motion Planning With Constraints 194
Geometric Constraints 195
Quantitative Constraints 198
Application to Predictive Control 199
Flatness and Tracking 201
The Tracking Problem 201
Pendulum (conclusion) 202
Non Holonomic Vehicle (conclusion) 203
Control of the Clock 205
Part II APPLICATIONS 208
DC Motor Starting Phase 209
Tracking of a Step Speed Reference 210
Flatness Based Tracking 212
Displacements of a Linear Motor With Oscillating Masses 216
Single Mass Case 217
Displacement Without Taking Account of the Auxiliary Mass 218
Displacements Taking Account of the Auxiliary Mass 219
Comparisons 221
Displacements With Two Auxiliary Masses 225
Synchronization of a Pair of Independent Windshield Wipers 230
Introduction 230
The Model of a Single Wiper 233
Open Loop Synchronization of the Pair of Wipers by Motion Planning 234
Trajectory Tracking 238
Synchronization by Clock Control 240
Control of Magnetic Bearings 248
Analysis and Control of a Ball 250
Modelling 250
Current Control 251
Voltage Control 258
Hierarchical Control 260
The General Shaft 270
Modelling 271
Current Control 272
Hierarchical Control 274
Implementation 274
Observer Design 275
Digital Control 278
Experimental Results 278
Platform Description 278
Experiments 279
Crane Control 283
Orientation 283
Straight Line Displacement 286
Approximate Tracking of Straight Line by Hierarchical PID Control 286
Straight Line Tracking Without Small Angle Approximation 290
Obstacle Avoidance 294
Tracking With Small Angle Approximation 295
Tracking Without Small Angle Approximation 297
Automatic Flight Control Systems 299
Generic Aircraft Model 300
Flatness Based Autopilot Design 303
References 311
Index 320