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Ying Luo, South China University of Technology, China. Currently Assistant Professor in the Department of Automatic Control and Engineering, South China University of Technology, Dr Luo joined the? BS-Ph.D program? in 2004. He went to Utah State University, Logan, UT, USA, as a visiting scholar/exchange PhD?from September 2007 to February 2009 and received the PhD degree in Automation Science and Engineering from South China University of Technology, Guangzhou, China, in July 2009. Dr Luo has been based at Utah State University, Logan, UT, USA, as a postdoctoral fellow from September 2009. He has worked on several patents and has published research in some key journals.
YangQuan Chen, Utah State University, USA. Dr Chen is Associate Professor in the Department of Electrical & Computer Engineering as well as Director at the Center for Self-Organizing & Intelligent Systems (CSOIS) of Utah State University, Logan. Prior to this he has held positions at Seagate Tech. Int., Singapore, National University of Singapore and Nanyang Technological University. He is the owner of 14 patents and Senior Member of the IEEE.
Acronyms xix
Foreword xxiii
Preface xxv
Acknowledgments xxix
PART I FUNDAMENTALS OF FRACTIONAL CONTROLS
1 Introduction 3
1.1 Fractional Calculus 3
1.1.1 Definitions and Properties 4
1.1.2 Laplace Transform 6
1.1.3 Fractional Order Dynamic Systems 6
1.1.4 Stability of LTI Fractional Order Systems 8
1.2 Fractional Order Controls 9
1.2.1 Why Fractional Order Control? 9
1.2.2 Basic Fractional Order Control Actions 9
1.2.3 A Historical Review of Fractional Order Controls 10
1.3 Fractional Order Motion Controls 20
1.4 Contributions 22
1.5 Organization 22
PART II FRACTIONAL ORDER VELOCITY SERVO
2 Fractional Order PI Controller Designs for Velocity Servo Systems
25
2.1 Introduction 25
2.2 FOPTD Systems and Three Controllers Considered 27
2.3 Design Specifications 27
2.4 Fractional Order PI and [PI] Controller Designs 28
2.4.1 Integer Order PID Controller Design 28
2.4.2 Fractional Order PI Controller Design 30
2.4.3 Fractional Order [PI] Controller Design 34
2.5 Simulation 38
2.6 Chapter Summary 39
3 Tuning Fractional Order PI Controllers for Fractional Order Velocity
Systems with Experimental Validation 41
3.1 Introduction 41
3.2 Three Controllers to Be Designed and Tuning Specifications 42
3.3 Tuning Three Controllers for FOVS 42
3.4 Illustrative Examples and Design Procedure Summaries 43
3.4.1 Fractional Order [PI] Controller Design Procedures 44
3.4.2 Fractional Order PI Controller Design Procedures 44
3.4.3 Integer Order PID Controller Design Procedures 45
3.5 Simulation Illustration 45
3.6 Experimental Validation 49
3.6.1 Experimental Setup 50
3.6.2 HIL Emulation of the FOVS 51
3.6.3 Experimental Results 51
3.7 Chapter Summary 54
4 Relay Feedback Tuning of Robust PID Controllers 59
4.1 Introduction 59
4.2 Slope Adjustment of the Phase Bode Plot 62
4.3 The New PID Controller Design Formulae 65
4.4 Phase and Magnitude Measurement Via Relay Feedback Tests 66
4.5 Illustrative Examples 67
4.5.1 High-order Plant P2(s) 67
4.5.2 Plant with an Integrator P5(s) 69
4.5.3 Plant with a Time Delay P6(s) 69
4.5.4 Plant with an Integrator and a Time Delay P7(s) 70
4.6 Chapter Summary 72
5 Auto-Tuning of Fractional Order Controllers with Iso-Damping 73
5.1 Introduction 73
5.2 FOPI and FO[PI] Controllers Design Formulae 75
5.2.1 FOPI Controller Auto-Tuning 75
5.2.2 FO[PI] Controller Auto-Tuning 78
5.3 Measurements for Auto-Tuning 80
5.4 Simulation Illustration 80
5.4.1 High-Order Plant P2(s) 80
5.4.2 Plant with an Integrator P5(s) 83
5.4.3 Plant with a Time Delay P6(s) 83
5.5 Chapter Summary 87
PART III FRACTIONAL ORDER POSITION SERVO
6 Fractional Order PD Controller Tuning for Position Systems 91
6.1 Introduction 91
6.2 Fractional Order PD Controller Design for Position Servos 92
6.2.1 Traditional PD Controller 92
6.2.2 Fractional Order PD¹ Controller 93
6.3 Design Procedures 94
6.4 Simulation Example 95
6.4.1 Step Response Comparison 96
6.4.2 Ramp Response Comparison 97
6.5 Experiments 99
6.5.1 Introduction of the Experimental Platform 99
6.5.2 Experimental model simulation 99
6.5.3 Experiments on the Dynamometer 100
6.6 Chapter Summary 101
7 Fractional Order [PD] Controller Synthesis for Position Servo
Systems 105
7.1 Introduction 105
7.2 Position Control Plants and Design Specifications 106
7.3 Fractional Order [PD] Controller Design 106
7.3.1 Numerical Computation of the Controller Parameters 107
7.3.2 Summary of the Design Procedures 108
7.4 Parameter Design Examples and Bode Plot Validations 108
7.4.1 FO[PD] Controller Design 108
7.4.2 FOPD Controller Design 109
7.4.3 IOPID Controller Design 109
7.5 Implementation of Two Fractional Order Operators 110
7.5.1 Implementation of s¸ for FOPD 110
7.5.2 Implementation of (1 + ¿s)¹ for FO[PD] 111
7.6 Simulation 111
7.6.1 Case-I: Step Response Comparison with T = 0:4s 111
7.6.2 Case-II: Step Response Comparison with T = 0:04s 113
7.6.3 Step Response Comparison with Time Delay 118
7.6.4 Step Response Comparison with Backlash Nonlinearity 119
7.7 Experiment 120
7.7.1 Introduction to the Experimental Platform 120
7.7.2 Experimental Tests on Dynamometer Platform 120
7.8 Chapter Summary 122
8 Time-Constant Robust Analysis and Design of Fractional Order
[PD] Controller 123
8.1 Introduction 123
8.2 Problem Statement 124
8.3 FO[PD] Tuning Specifications and Rules 125
8.3.1 FO[PD] Robustness to Time-Constant Variations 126
8.3.2 Numerical Computation Process 127
8.4 The Solution Existence Range and An Online Computation
Method 127
8.4.1 The Solution Existence Range 128
8.4.2 Numerical Computation Example and Simulation Test 129
8.4.3 Comparison 132
8.4.4 Online Computation 133
8.5 Experiment 135
8.6 Chapter Summary 136
9 Experimental Study of Fractional OrderPDController Synthesis
for Fractional Order Position Servo Systems 139
9.1 Introduction 139
9.2 Fractional Order Systems and Fractional Order Controller
Considered 140
9.3 FOPD Controller Design Procedure for the Fractional Order
Position Servo Systems 141
9.3.1 Preliminary and Design Specifications 141
9.3.2 Numerical Computation Process 142
9.3.3 Summary of Design Procedure 143
9.4 Simulation Illustration 144
9.4.1 Case-1: IOS Based Design for IOS 145
9.4.2 Case-2: IOS Based Design for FOS 147
9.4.3 Case-3: FOS Based Design for FOS 148
9.5 Experimental Study 148
9.5.1 HIL Experimental Setup 148
9.5.2 HIL Emulation of the FOS 149
9.5.3 Experimental Results 150
9.6 Chapter Summary 153
10 Fractional Order [PD] Controller Design and Comparison for
Fractional Order Position Servo Systems 155
10.1 Introduction 155
10.2 Fractional Order Position Servo Systems and Fractional Order
Controllers 156
10.3 Fractional Order [PD] Controller Design 156
10.3.1 Numerical Computation Process 157
10.3.2 Design Procedure Summary 158
10.3.3 Design Example and Bode Plot Validation of FO[PD]
Design 158
10.4 Integer Order PID Controller and Fractional Order PD Controller
Designs 159
10.5 Simulation Comparisons 160
10.6 Chapter Summary 162
PART IV STABILITY AND FEASIBILITY FOR FOPID DESIGN
11 Stability and Design Feasibility of Robust PID Controllers for
FOPTD Systems 165
11.1 Introduction 165
11.1.1 Research Questions 165
11.1.2 Previous Work 166
11.1.3 Contributions in This Chapter 166
11.2 Stability Region and Flat Phase Tuning Rule for the Robust PID
Controller Design 168
11.2.1 Preliminary 168
11.2.2 Stability Region of PID Controller for FOPTD Plants 169
11.3 PID Controller Design with Pre-Specifications on Ám and !c 171
11.3.1 Design Scheme 171
11.3.2 Flat Phase Tuning Rule for the Robust PID Controller
Design 172
11.3.3 Design Procedures Summary with An Example 174
11.3.4 How to Find the Achievable Region of the Two
Specifications? 177
11.4 Simulation Illustration 180
11.5 Chapter Summary 185
12 Stability and Design Feasibility of Robust FOPI Controllers for
FOPTD Systems 187
12.1 Introduction 187
12.2 Stabilizing and Robust FOPI Controller Design for FOPTD
Systems 188
12.2.1 The Plant and Controller Considered 188
12.2.2 Stability Region Analysis of the FOPI Controller 188
12.2.3 FOPI Parameters Design with Two Specifications 191
12.2.4 FOPI Parameters Design with An Additional Flat
Phase Constraint 192
12.2.5 Achievable Region of Two Design Indexes for FOPI
Controller Design 194
12.3 Design Procedures Summary with An Illustrative Example 194
12.4 Complete Information Collection for Achievable Region of !c
and Ám 197
12.5 Simulation Illustration 201
12.6 Chapter Summary 207
PART V FRACTIONAL ORDER DISTURBANCE COMPENSATORS
13 Fractional Order Disturbance Observer 211
13.1 Introduction 211
13.2 Disturbance Observer (DOB) 212
13.3 Actual Design Parameters In DOB and Their Effects 213
13.4 Loss of The Phase Margin With DOB 215
13.5 Solution One: Rule-Based Switched Low Pass Filtering With
Varying Relative Degree 216
13.6 The Proposed Solution: Guaranteed Phase Margin Method
Using Fractional Order Low Pass Filtering 216
13.7 Implementation Issues: Stable Minimum-Phase Frequency
Domain Fitting 218
13.8 Chapter Summary 222
14 Fractional Order Adaptive Feed-forward Cancellation 223
14.1 Introduction 223
14.2 Fractional Order Adaptive Feed-forward Cancellation 225
14.3 Equivalence Between Fractional Order Internal Model Principle
and Fractional Order Adaptive Feed-Forward Cancellation 229
14.3.1 Single-Frequency Disturbance Cancellation 229
14.3.2 Generalization to Multi-Frequency Disturbance
Cancellation 230
14.4 Frequency-domain analysis of the FOAFC performance for the
periodic disturbance 231
14.5 Simulation Illustration 233
14.6 Experiment Validation 237
14.6.1 Introduction to the Experiment Platform 237
14.6.2 Experiments on the Dynamometer 240
14.7 Chapter Summary 241
15 Fractional Order Robust Control for Cogging Effect 243
15.1 Introduction 243
15.2 Fractional Order Robust Control of Cogging Effect
Compensation 244
15.2.1 Cogging Effect Analysis 244
15.2.2 Motivations and Problem Formulation 244
15.2.3 IO Robust Control Stability Analysis 246
15.2.4 FO Robust Control Stability Analysis 248
15.3 Simulation Illustration 252
15.3.1 Case-1: IORC with constant reference speed 252
15.3.2 Case-2: FORC with constant reference speed 255
15.3.3 Case-3: IO/FORC with varying reference speed 258
15.4 Experiments on A Lab Testbed - Dynamometer 258
15.4.1 Introduction to The Experimental Platform 258
15.4.2 Experiments on the Dynamometer 259
15.5 Chapter Summary 264
16 Fractional Order Periodic Adaptive Learning Compensation 275
16.1 Introduction 275
16.2 Fractional Order Periodic Adaptive learning Compensation for
the State-dependent Periodic Disturbance 276
16.2.1 The General Form of the State-Dependent Periodic
Disturbance 276
16.2.2 Problem Formulation 276
16.2.3 Stability Analysis 278
16.3 Simulation Illustrations 282
16.3.1 Case-1: Integer Order PALC 283
16.3.2 Case-2: Fractional Order PALC 284
16.4 Experimental Validation 284
16.4.1 Introduction to the Experiment Platform 284
16.4.2 Experiments on the Dynamometer 285
16.5 Chapter Summary 288
PART VI EFFECTS OF FRACTIONAL ORDER CONTROLS ON
NONLINEARITIES
17 Fractional Order PID Control of A DC-Motor with Elastic Shaft 293
17.1 Introduction 293
17.2 The Benchmark Position Servo System 294
17.3 A Modified Approximate Realization Method 295
17.4 Comparative Simulations 297
17.4.1 Best IOPID vs. Best FOPID 297
17.4.2 How to Decide ¸ and ¹? 298
17.4.3 Which N Is Good Enough? 299
17.4.4 Robustness Against Load Variations 300
17.4.5 FOPI Controllers 301
17.4.6 Robustness to Mechanical Nonlinearities 304
17.4.7 Robustness to Elasticity Parameter Change 304
17.5 Chapter Summary 305
18 Fractional Order Ultra Low-Speed Position Servo 313
18.1 Introduction 313
18.2 Ultra Low-Speed Position Tracking using Designed FOPD and
Optimized IOPI 314
18.2.1 FOPD Design for the Position Tracking without
Considering the Friction Effect 314
18.2.2 Ultra Low-Speed Position Tracking Performance with
Designed FOPD and Optimized IOPI 315
18.3 Static and Dynamic Models of Friction and Describing
Functions for Friction Models 316
18.3.1 Static and Dynamic Models of Friction 316
18.3.2 Describing Functions for Friction Models and Two
Uncoupling Methods of Linear and Nonlinear Parts 318
18.4 Simulation Analysis with IOPI and FOPD Controllers Using
Describing Function 321
18.5 Extended Experimental Demonstration 324
18.6 Chapter Summary 325
19 Optimized Fractional Order Conditional Integrator 329
19.1 Introduction 329
19.2 Clegg Conditional Integrator 330
19.3 Intelligent Conditional Integrator 331
19.4 The Optimized Fractional Order Conditional Integrator 332
19.4.1 Fractional Order Conditional Integrator 333
19.4.2 Optimality Criteria 335
19.4.3 Optimization of the FOCI 336
19.5 Simulation Validation 340
19.6 Chapter Summary 342
PART VII FRACTIONAL ORDER CONTROL APPLICATIONS
20 Lateral Directional Fractional Order Control of A Small Fixed-
Wing UAV 345
20.1 Introduction 345
20.2 Flight Control System of Small Fixed-Wing UAV 346
20.2.1 Dynamics of Small Fixed-Wing UAV 346
20.2.2 The ChangE Small Fixed-Wing UAV Flight Control
Platform 347
20.2.3 Closed-Loop System Identification 348
20.3 Integer/Fractional Order Controller Designs 351
20.3.1 Integer/Fractional Controllers Considered and Design
Rules 351
20.4 Modified Ziegler-Nichols PI Controller Design 352
20.5 Fractional Order (PI)¸ Controller Design 353
20.6 Fractional Order PI Controller Design 355
20.7 Integer Order PID Controller Design 356
20.8 Simulation Illustration 357
20.8.1 Fractional Order Controllers Implementation 357
20.8.2 Simulation Results 361
20.9 Flight Experiments 363
20.10 Chapter Summary 367
21 Fractional Order PD Controller Synthesis and Implementation
for HDD Servo System 369
21.1 Introduction 369
21.2 Fractional Order Controller Design with “Flat Phase” 370
21.3 Implementation of the Fractional Order Controller 372
21.3.1 Phase Loss from the Sampling Delay 372
21.3.2 Gain Boosting from Discretization 375
21.4 Readjustment for the Designed FOPD Controller 377
21.4.1 Phase Margin Readjustment with Phase Loss Prediction 377
21.4.2 Gain Crossover Frequency Readjustment with Gain
Boosting Prediction 377
21.4.3 Phase Slope Readjustment with the Phase Loss Slope
Prediction 377
21.4.4 FO Controller Design and Implementation Procedures
Summary 379
21.5 Experiment 380
21.5.1 Original Integer Order Controller Design 381
21.5.2 Implementation of Fractional Order Controller 382
21.5.3 Track Following Performance 382
21.5.4 Throughput Performance 382
21.6 Chapter Summary 383
References 385
Index 403
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