Fractional Order Motion Controls Book

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