Fractional Order Motion Controls

Acronyms xix
Foreword xxiii
Preface xxv
Acknowledgments xxix

PART I FUNDAMENTALS OF FRACTIONAL CONTROLS

1 Introduction 3
1.1 Fractional Calculus 3
1.2 Fractional Order Controls 9
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
2.5 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
3.5 Simulation Illustration 45
3.6 Experimental Validation 49
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
4.6 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
5.3 Measurements for Auto-Tuning 80
5.4 Simulation Illustration 80
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
6.3 Design Procedures 94
6.4 Simulation Example 95
6.5 Experiments 99
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
7.4 Parameter Design Examples and Bode Plot Validations 108
7.5 Implementation of Two Fractional Order Operators 110
7.6 Simulation 111
7.7 Experiment 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
8.4 The Solution Existence Range and An Online Computation Method 127
8.5 Experiment 135
8.6 Chapter Summary 136

9 Experimental Study of Fractional Order PD Controller 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
9.4 Simulation Illustration 144
9.5 Experimental Study 148
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
10.4 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
11.2 Stability Region and Flat Phase Tuning Rule for the Robust PID Controller Design 168
11.3 PID Controller Design with Pre-Specifications on Ám and !c 171
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
12.3 Design Procedures Summary with An Illustrative Example 194
12.4 Complete Information Collection for Achievable Region of w_c and F_m 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
14.4 Frequency-domain analysis of the FOAFC performance for the periodic disturbance 231
14.5 Simulation Illustration 233
14.6 Experiment Validation 237
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
15.3 Simulation Illustration 252
15.4 Experiments on A Lab Testbed - Dynamometer 258
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
16.3 Simulation Illustrations 282
16.4 Experimental Validation 284
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
17.5 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
18.3 Static and Dynamic Models of Friction and Describing Functions for Friction Models 316
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
19.5 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
20.3 Integer/Fractional Order Controller Designs 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
20.9 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
21.4 Readjustment for the Designed FOPD Controller 377
21.5 Experiment 380
21.6 Chapter Summary 383

References 385
Index 403