Levine Analysis and Control of Nonlinear Systems

A Flatness-based Approach
1. Auflage 2009
ISBN: 978-3-642-00839-9
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 320 Seiten

Reihe: Mathematical Engineering

This book examines control of nonlinear systems. Coverage ranges from mathematical system theory to practical industrial control applications. The author offers web-based videos illustrating some dynamical aspects and case studies in simulation.

Autoren/Hrsg.

Levine, Jean

Inhaltsverzeichnis

1;Preface;7
2;Contents;9
3;Introduction;14
3.1;Trajectory Planning and Tracking;15
3.2;Equivalence and Flatness;16
3.3;Equivalence in System Theory;18
3.4;Equivalence and Stability;18
3.5;What is a Nonlinear Control System?;19
3.5.1;Nonlinearity versus Linearity;19
3.5.2;Uncontrolled versus Controlled Nonlinearity;20
4;Part I THEORY;24
4.1;Introduction to Differential Geometry;25
4.1.1;Manifold, Diffeomorphism;26
4.1.2;Vector Fields;29
4.1.2.1;Tangent space, Vector Field;29
4.1.2.2;Flow, Phase Portrait;31
4.1.2.3;Lie Derivative;33
4.1.2.4;Image of a Vector Field;34
4.1.2.5;First Integral, Straightening Out of a Vector Field;35
4.1.2.6;Lie Bracket;37
4.1.2.7;Distribution of Vector Fields;40
4.1.2.8;Integral Manifolds;41
4.1.2.9;First Order Partial Differential Equations;42
4.1.3;Differential Forms;44
4.1.3.1;Cotangent Space, Differential Form, Duality;44
4.1.3.2;Exterior differentiation;47
4.1.3.3;Image of a Differential Form;48
4.1.3.4;Pfaffian System, Complete Integrability;49
4.1.3.5;Lie Derivative of a 1-Form;51
4.1.3.6;Back to Frobenius Theorem;53
4.2;Introduction to Dynamical Systems;54
4.2.1;Recalls on Flows and Orbits;56
4.2.1.1;Equilibrium Point, Variational Equation;56
4.2.1.2;Periodic Orbit;58
4.2.1.3;Poincaré's Map;61
4.2.2;Stability of Equilibrium Points and Orbits;64
4.2.2.1;Attractor;64
4.2.2.2;Lyapunov Stability;66
4.2.2.3;Remarks on the Stability of Time-Varying Systems;69
4.2.2.4;Lyapunov's and Chetaev's Functions;70
4.2.2.5;Hartman-Grobman's and Shoshitaishvili's Theorems, Centre Manifold;75
4.2.3;Singularly Perturbed Systems;84
4.2.3.1;Invariant Slow Manifold;86
4.2.3.2;Persistence of the Invariant Slow Manifold;87
4.2.3.3;Robustness of the Stability;89
4.2.3.4;An Application to Modelling;90
4.2.4;Application to Hierarchical Control;92
4.2.4.1;Controlled Slow Dynamics;92
4.2.4.2;Hierarchical Feedback Design;93
4.2.4.3;Practical Applications;94
4.3;Controlled Systems, Controllability;98
4.3.1;Linear System Controllability;98
4.3.1.1;Kalman's Criterion;98
4.3.1.2;Controllability Canonical Form;101
4.3.1.3;Motion Planning;104
4.3.1.4;Trajectory Tracking, Pole Placement;106
4.3.2;Nonlinear System Controllability;107
4.3.2.1;First Order and Local Controllability;107
4.3.2.2;Local Controllability and Lie Brackets;108
4.3.2.3;Reachability;112
4.3.2.4;Lie Brackets and Kalman's Criterion for Linear Systems;115
4.4;Jets of Infinite Order, Lie-Bäcklund's Equivalence;117
4.4.1;An Introductory Example of Crane;117
4.4.2;Description of the System Trajectories;119
4.4.3;Jets of Infinite Order, Change of coordinates, Equivalence;123
4.4.3.1;Jets of Infinite Order, Global Coordinates;124
4.4.3.2;Product Manifolds, Product Topology;124
4.4.3.3;Cartan Vector Fields, Flows, Control Systems;125
4.4.3.4;Lie-Bäcklund Equivalence;131
4.4.3.5;Properties of the L-B Equivalence;135
4.4.3.6;Endogenous Dynamic Feedback;137
4.5;Differentially Flat Systems;141
4.5.1;Flat System, Flat Output;141
4.5.2;Examples;143
4.5.2.1;Mass-Spring System;143
4.5.2.2;Robot Control;144
4.5.2.3;Pendulum;145
4.5.2.4;Non Holonomic Vehicle;148
4.5.2.5;Vehicle with Trailers;149
4.5.3;Flatness and Controllability;151
4.5.4;Flatness and Linearization;153
4.5.4.1;Mass-Spring System (followed);154
4.5.4.2;Robot Control (followed);155
4.5.4.3;Pendulum (followed);155
4.5.4.4;Non Holonomic Vehicle (followed);156
4.5.5;Flat Output Characterization;156
4.5.5.1;The Ruled Manifold Necessary Condition;158
4.5.5.2;Variational Characterization;161
4.5.5.3;The Polynomial Matrix Approach;162
4.5.5.4;Practical Computation of the Smith Decomposition;165
4.5.5.5;Flatness Necessary and Sufficient Conditions;167
4.5.5.6;The Operator d;170
4.5.5.7;Strong Closedness Necessary and Sufficient Conditions;173
4.5.5.8;Flat Outputs of Linear Controllable Systems;177
4.5.5.9;Examples;180
4.6;Flatness and Motion Planning;190
4.6.1;Motion Planning Without Constraint;191
4.6.1.1;The General Case;191
4.6.1.2;Rest-to-Rest Trajectories;193
4.6.2;Motion Planning With Constraints;194
4.6.2.1;Geometric Constraints;195
4.6.2.2;Quantitative Constraints;198
4.6.3;Application to Predictive Control;199
4.7;Flatness and Tracking;201
4.7.1;The Tracking Problem;201
4.7.1.1;Pendulum (conclusion);202
4.7.1.2;Non Holonomic Vehicle (conclusion);203
4.7.2;Control of the Clock;205
5;Part II APPLICATIONS;208
5.1;DC Motor Starting Phase;209
5.1.1;Tracking of a Step Speed Reference;210
5.1.2;Flatness Based Tracking;212
5.2;Displacements of a Linear Motor With Oscillating Masses;216
5.2.1;Single Mass Case;217
5.2.1.1;Displacement Without Taking Account of the Auxiliary Mass;218
5.2.1.2;Displacements Taking Account of the Auxiliary Mass;219
5.2.1.3;Comparisons;221
5.2.2;Displacements With Two Auxiliary Masses;225
5.3;Synchronization of a Pair of Independent Windshield Wipers;230
5.3.1;Introduction;230
5.3.2;The Model of a Single Wiper;233
5.3.3;Open Loop Synchronization of the Pair of Wipers by Motion Planning;234
5.3.4;Trajectory Tracking;238
5.3.5;Synchronization by Clock Control;240
5.4;Control of Magnetic Bearings;248
5.4.1;Analysis and Control of a Ball;250
5.4.1.1;Modelling;250
5.4.1.2;Current Control;251
5.4.1.3;Voltage Control;258
5.4.1.4;Hierarchical Control;260
5.4.2;The General Shaft;270
5.4.2.1;Modelling;271
5.4.2.2;Current Control;272
5.4.2.3;Hierarchical Control;274
5.4.3;Implementation;274
5.4.3.1;Observer Design;275
5.4.3.2;Digital Control;278
5.4.4;Experimental Results;278
5.4.4.1;Platform Description;278
5.4.4.2;Experiments;279
5.5;Crane Control;283
5.5.1;Orientation;283
5.5.2;Straight Line Displacement;286
5.5.2.1;Approximate Tracking of Straight Line by Hierarchical PID Control;286
5.5.2.2;Straight Line Tracking Without Small Angle Approximation;290
5.5.3;Obstacle Avoidance;294
5.5.3.1;Tracking With Small Angle Approximation;295
5.5.3.2;Tracking Without Small Angle Approximation;297
5.6;Automatic Flight Control Systems;299
5.6.1;Generic Aircraft Model;300
5.6.1.1;Flatness Based Autopilot Design;303
5.6.2;References;311
5.7;Index;320