Cover image for Adaptive approximation based control : unifying neural, fuzzy and traditional adaptive approximation approaches
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Adaptive approximation based control : unifying neural, fuzzy and traditional adaptive approximation approaches
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Publication Information:
Hoboken, NJ : Wiley-Interscience, 2006
ISBN:
9780471727880
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Summary

Summary

A highly accessible and unified approach to the design and analysisof intelligent control systems

Adaptive Approximation Based Control is a tool every controldesigner should have in his or her control toolbox.

Mixing approximation theory, parameter estimation, and feedbackcontrol, this book presents a unified approach designed to enablereaders to apply adaptive approximation based control to existingsystems, and, more importantly, to gain enough intuition andunderstanding to manipulate and combine it with other control toolsfor applications that have not been encountered before.

The authors provide readers with a thought-provoking framework forrigorously considering such questions as:
* What properties should the function approximator have?
* Are certain families of approximators superior to others?
* Can the stability and the convergence of the approximatorparameters be guaranteed?
* Can control systems be designed to be robust in the face ofnoise, disturbances, and unmodeled effects?
* Can this approach handle significant changes in the dynamics dueto such disruptions as system failure?
* What types of nonlinear dynamic systems are amenable to thisapproach?
* What are the limitations of adaptive approximation basedcontrol?

Combining theoretical formulation and design techniques withextensive use of simulation examples, this book is a stimulatingtext for researchers and graduate students and a valuable resourcefor practicing engineers.


Author Notes

Jay A. Farrell, PhD, is Professor and former chair of the Department of Electrical Engineering at the University of California at Riverside
Marios M. Polycarpou, PhD, is Professor and Interim Head of the Department of Electrical and Computer Engineering at the University of Cyprus


Table of Contents

Prefacep. xiii
1 Introductionp. 1
1.1 Systems and Control Terminologyp. 1
1.2 Nonlinear Systemsp. 3
1.3 Feedback Control Approachesp. 4
1.3.1 Linear Designp. 4
1.3.2 Adaptive Linear Designp. 6
1.3.3 Nonlinear Designp. 9
1.3.4 Adaptive Approximation Based Designp. 11
1.3.5 Example Summaryp. 13
1.4 Components of Approximation Based Controlp. 15
1.4.1 Control Architecturep. 15
1.4.2 Function Approximatorp. 16
1.4.3 Stable Training Algorithmp. 17
1.5 Discussion and Philosophical Commentsp. 18
1.6 Exercises and Design Problemsp. 19
2 Approximation Theoryp. 23
2.1 Motivating Examplep. 24
2.2 Interpolationp. 29
2.3 Function Approximationp. 30
2.3.1 Offline (Batch) Function Approximationp. 31
2.3.2 Adaptive Function Approximationp. 33
2.4 Approximator Propertiesp. 39
2.4.1 Parameter (Non) Linearityp. 39
2.4.2 Classical Approximation Resultsp. 43
2.4.3 Network Approximatorsp. 46
2.4.4 Nodal Processorsp. 48
2.4.5 Universal Approximatorp. 50
2.4.6 Best Approximator Propertyp. 52
2.4.7 Generalizationp. 54
2.4.8 Extent of Influence Function Supportp. 56
2.4.9 Approximator Transparencyp. 65
2.4.10 Haar Conditionsp. 66
2.4.11 Multivariable Approximation by Tensor Productsp. 67
2.5 Summaryp. 68
2.6 Exercises and Design Problemsp. 69
3 Approximation Structuresp. 71
3.1 Model Typesp. 72
3.1.1 Physically Based Modelsp. 72
3.1.2 Structure (Model) Free Approximationp. 73
3.1.3 Function Approximation Structuresp. 74
3.2 Polynomialsp. 75
3.2.1 Descriptionp. 75
3.2.2 Propertiesp. 77
3.3 Splinesp. 78
3.3.1 Descriptionp. 78
3.3.2 Propertiesp. 83
3.4 Radial Basis Functionsp. 84
3.4.1 Descriptionp. 84
3.4.2 Propertiesp. 86
3.5 Cerebellar Model Articulation Controllerp. 87
3.5.1 Descriptionp. 88
3.5.2 Propertiesp. 89
3.6 Multilayer Perceptronp. 93
3.6.1 Descriptionp. 93
3.6.2 Propertiesp. 95
3.7 Fuzzy Approximationp. 96
3.7.1 Descriptionp. 96
3.7.2 Takagi-Sugeno Fuzzy Systemsp. 104
3.7.3 Propertiesp. 105
3.8 Waveletsp. 106
3.8.1 Multiresolution Analysis (MRA)p. 108
3.8.2 MRA Propertiesp. 110
3.9 Further Readingp. 112
3.10 Exercises and Design Problemsp. 112
4 Parameter Estimation Methodsp. 115
4.1 Formulation for Adaptive Approximationp. 116
4.1.1 Illustrative Examplep. 116
4.1.2 Motivating Simulation Examplesp. 118
4.1.3 Problem Statementp. 124
4.1.4 Discussion of Issues in Parametric Estimationp. 125
4.2 Derivation of Parametric Modelsp. 127
4.2.1 Problem Formulation for Full-State Measurementp. 128
4.2.2 Filtering Techniquesp. 129
4.2.3 SPR Filteringp. 131
4.2.4 Linearly Parameterized Approximatorsp. 131
4.2.5 Parametric Models in State Space Formp. 133
4.2.6 Parametric Models of Discrete-Time Systemsp. 134
4.2.7 Parametric Models of Input-Output Systemsp. 136
4.3 Design of Online Learning Schemesp. 138
4.3.1 Error Filtering Online Learning (EFOL) Schemep. 138
4.3.2 Regressor Filtering Online Learning (RFOL) Schemep. 140
4.4 Continuous-Time Parameter Estimationp. 141
4.4.1 Lyapunov-Based Algorithmsp. 143
4.4.2 Optimization Methodsp. 148
4.4.3 Summaryp. 154
4.5 Online Learning: Analysisp. 154
4.5.1 Analysis of LIP EFOL Scheme with Lyapunov Synthesis Methodp. 155
4.5.2 Analysis of LIP RFOL Scheme with the Gradient Algorithmp. 158
4.5.3 Analysis of LIP RFOL Scheme with RLS Algorithmp. 160
4.5.4 Persistency of Excitation and Parameter Convergencep. 161
4.6 Robust Learning Algorithmsp. 163
4.6.1 Projection Modificationp. 165
4.6.2 [epsilon]-Modificationp. 168
4.6.3 [sigma]-Modificationp. 169
4.6.4 Dead-Zone Modificationp. 170
4.6.5 Discussion and Comparisonp. 172
4.7 Concluding Summaryp. 173
4.8 Exercises and Design Problemsp. 173
5 Nonlinear Control Architecturesp. 179
5.1 Small-Signal Linearizationp. 180
5.1.1 Linearizing Around an Equilibrium Pointp. 181
5.1.2 Linearizing Around a Trajectoryp. 183
5.1.3 Gain Schedulingp. 186
5.2 Feedback Linearizationp. 188
5.2.1 Scalar Input-State Linearizationp. 188
5.2.2 Higher-Order Input-State Linearizationp. 190
5.2.3 Coordinate Transformations and Diffeomorphismsp. 193
5.2.4 Input-Output Feedback Linearizationp. 196
5.3 Backsteppingp. 203
5.3.1 Second Order Systemp. 203
5.3.2 Higher Order Systemsp. 205
5.3.3 Command Filtering Formulationp. 207
5.4 Robust Nonlinear Control Design Methodsp. 211
5.4.1 Bounding Controlp. 211
5.4.2 Sliding Mode Controlp. 212
5.4.3 Lyapunov Redesign Methodp. 215
5.4.4 Nonlinear Dampingp. 219
5.4.5 Adaptive Bounding Controlp. 220
5.5 Adaptive Nonlinear Controlp. 222
5.6 Concluding Summaryp. 225
5.7 Exercises and Design Problemsp. 226
6 Adaptive Approximation: Motivation and Issuesp. 231
6.1 Perspective for Adaptive Approximation Based Controlp. 232
6.2 Stabilization of a Scalar Systemp. 236
6.2.1 Feedback Linearizationp. 237
6.2.2 Small-Signal Linearizationp. 238
6.2.3 Unknown Nonlinearity with Known Boundsp. 239
6.2.4 Adaptive Bounding Methodsp. 241
6.2.5 Approximating the Unknown Nonlinearityp. 243
6.2.6 Combining Approximation with Bounding Methodsp. 250
6.2.7 Combining Approximation with Adaptive Bounding Methodsp. 252
6.2.8 Summaryp. 252
6.3 Adaptive Approximation Based Trackingp. 253
6.3.1 Feedback Linearizationp. 253
6.3.2 Tracking via Small-Signal Linearizationp. 253
6.3.3 Unknown Nonlinearities with Known Boundsp. 256
6.3.4 Adaptive Bounding Designp. 258
6.3.5 Adaptive Approximation of the Unknown Nonlinearitiesp. 262
6.3.6 Robust Adaptive Approximationp. 264
6.3.7 Combining Adaptive Approximation with Adaptive Boundingp. 266
6.3.8 Advanced Adaptive Approximation Issuesp. 271
6.4 Nonlinear Parameterized Adaptive Approximationp. 278
6.5 Concluding Summaryp. 280
6.6 Exercises and Design Problemsp. 281
7 Adaptive Approximation Based Control: General Theoryp. 285
7.1 Problem Formulationp. 286
7.1.1 Trajectory Trackingp. 286
7.1.2 Systemp. 286
7.1.3 Approximatorp. 287
7.1.4 Control Designp. 288
7.2 Approximation Based Feedback Linearizationp. 288
7.2.1 Scalar Systemp. 289
7.2.2 Input-Statep. 294
7.2.3 Input-Outputp. 306
7.2.4 Control Design Outside the Approximation Region Dp. 308
7.3 Approximation Based Backsteppingp. 309
7.3.1 Second Order Systemsp. 309
7.3.2 Higher Order Systemsp. 316
7.3.3 Command Filtering Approachp. 323
7.3.4 Robustness Considerationsp. 328
7.4 Concluding Summaryp. 330
7.5 Exercises and Design Problemsp. 331
8 Adaptive Approximation Based Control for Fixed-Wing Aircraftp. 333
8.1 Aircraft Model Introductionp. 334
8.1.1 Aircraft Dynamicsp. 334
8.1.2 Nondimensional Coefficientsp. 335
8.2 Angular Rate Control for Piloted Vehiclesp. 336
8.2.1 Model Representationp. 337
8.2.2 Baseline Controllerp. 337
8.2.3 Approximation Based Controllerp. 338
8.2.4 Simulation Resultsp. 345
8.3 Full Control for Autonomous Aircraftp. 349
8.3.1 Airspeed and Flight Path Angle Controlp. 350
8.3.2 Wind-Axes Angle Controlp. 355
8.3.3 Body Axis Angular Rate Controlp. 359
8.3.4 Control Law and Stability Propertiesp. 362
8.3.5 Approximator Definitionp. 365
8.3.6 Simulation Analysisp. 367
8.3.7 Conclusionsp. 371
8.4 Aircraft Notationp. 371
Appendix A Systems and Stability Conceptsp. 377
A.1 Systems Conceptsp. 377
A.2 Stability Conceptsp. 379
A.2.1 Stability Definitionsp. 379
A.2.2 Stability Analysis Toolsp. 381
A.2.3 Strictly Positive Real Transfer Functionsp. 391
A.3 General Resultsp. 392
A.4 Trajectory Generation Filtersp. 394
A.5 A Useful Inequalityp. 397
A.6 Exercises and Design Problemsp. 398
Appendix B Recommended Implementation and Debugging Approachp. 399
Referencesp. 401
Indexp. 417