Fractional-Order Control Systems: Fundamentals and Numerical Implementations

Fractional-Order Control Systems: Fundamentals and Numerical Implementations

by Dingyu Xue

Hardcover

$126.99
View All Available Formats & Editions
Use Standard Shipping. For guaranteed delivery by December 24, use Express or Expedited Shipping.

Product Details

ISBN-13: 9783110499995
Publisher: De Gruyter
Publication date: 07/10/2017
Series: Fractional Calculus in Applied Sciences and Engineering Series , #1
Pages: 388
Product dimensions: 6.69(w) x 9.45(h) x (d)
Age Range: 18 Years

About the Author

Dingyu Xue, Northeastern University China, China

Table of Contents

Foreword v

Preface vii

1 Introduction to fractional calculus and fractional-order control 1

1.1 Historical review of fractional calculus 1

1.2 Fractional modelling in the real world 3

1.3 Introduction to fractional-order control tools 5

1.4 Structure of the book 6

1.4.1 Outlines of the book 6

1.4.2 A guide of reading the book 7

2 Mathematical prerequisites 9

2.1 Elementary special functions 9

2.1.1 Error and complementary error functions 10

2.1.2 Gamma functions 11

2.1.3 Beta functions 15

2.2 Dawson function and hypergeometric functions 18

2.2.1 Dawson function 18

2.2.2 Hypergeometric functions 20

2.3 Mittag-Leffler functions 23

2.3.1 Mittag-Leffler functions with one parameter 23

2.3.2 Mittag-Leffler functions with two parameters 26

2.3.3 Mittag-Leffler functions with more parameters 30

2.3.4 Derivatives of Mittag-Leffler functions 30

2.3.5 Numerical evaluation of Mittag-Leffler functions and their derivatives 33

2.4 Some linear algebra techniques 34

2.4.1 Kronecker product and Kronecker sum 35

2.4.2 Matrix inverse 35

2.4.3 Arbitrary matrix function evaluations 38

2.5 Numerical optimisation problems and solutions 40

2.5.1 Unconstrained optimisation problems and solutions 40

2.5.2 Constrained optimisation problems and solutions 41

2.5.3 Global optimisation solutions 44

2.6 Laplace transform 47

2.6.1 Definitions and properties 47

2.6.2 Computer solutions to Laplace transform problems 48

3 Definitions and computation algorithms of fractional-order derivatives and integrals 51

3.1 Fractional-order Cauchy integral formula 52

3.1.1 Cauchy integral formula 52

3.1.2 Fractional-order derivative and integral formula for commonly used functions 53

3.2 Grünwald-Letnikov definition 53

3.2.1 Deriving high-order derivatives 53

3.2.2 Grünwald-Letnikov definition in fractional calculus 54

3.2.3 Numerical computation of Grünwald-Letnikov derivatives and integrals 54

3.2.4 Podlubny's matrix algorithm 61

3.2.5 Studies on short-memory effect 62

3.3 Riemann-Liouville definition 66

3.3.1 High-order integrals 66

3.3.2 Riemann-Liouville fractional-order definition 66

3.3.3 Riemann-Liouville formula of commonly used functions 67

3.3.4 Properties of initial time translation 68

3.3.5 Numerical implementation of the Riemann-Liouville definition 69

3.4 High-precision computation algorithms of fractional-order derivatives and integrals 70

3.4.1 Construction of generating functions with arbitrary orders 71

3.4.2 FFT-based algorithm 74

3.4.3 A new recursive formula 76

3.4.4 A better fitting at initial instances 80

3.4.5 Revisit to the matrix algorithm 85

3.5 Caputo definition 86

3.6 Relationships between different definitions 87

3.6.1 Relationship between Grünwald-Letnikov and Riemann-Liouville definitions 88

3.6.2 Relationships between Caputo and Riemann-Liouvilie definitions 88

3.6.3 Computation of Caputo fractional-order derivatives 89

3.6.4 High-precision computation of Caputo derivatives 91

3.7 Properties of fractional-order derivatives and integrals 93

3.7.1 Properties 93

3.7.2 Geometric and physical interpretations of fractional-order integrals 95

4 Solutions of linear fractional-order differential equations 99

4.1 Introduction to linear fractional-order differential equations 99

4.1.1 General form of linear fractional-order differential equations 100

4.1.2 Initial value problems of fractional-order derivatives under different definitions 100

4.1.3 An important Laplace transform formula 102

4.2 Analytical solutions of some fractional-order differential equations 103

4.2.1 One-term differential equations 103

4.2.2 Two-term differential equations 103

4.2.3 Three-term differential equations 104

4.2.4 General n-term differential equations 105

4.3 Analytical solutions of commensurate-order differential equations 106

4.3.1 General form of commensurate-order differential equations 106

4.3.2 Some commonly used Laplace transforms in linear fractional-order systems 107

4.3.3 Analytical solutions of commensurate-order equations 109

4.4 Closed-form solutions of linear fractional-order differential equations with zero initial conditions 113

4.4.1 Closed-form solution 113

4.4.2 The matrix-based solution algorithm 117

4.4.3 High-precision closed-form algorithm 119

4.5 Numerical solutions to Caputo differential equations with nonzero initial conditions 121

4.5.1 Mathematical description of Caputo equations 121

4.5.2 Taylor auxiliary algorithm 121

4.5.3 High-precision algorithm 124

4.6 Numerical solutions of irrational system equations 130

4.6.1 Irrational transfer function expression 130

4.6.2 Simulation with numerical inverse Laplace transforms 131

4.6.3 Closed-loop irrational system response evaluation 133

4.6.4 Stability assessment of irrational systems 134

4.6.5 Numerical Laplace transform 138

5 Approximation of fractional-order operators 141

5.1 Some continued fraction-based approximations 142

5.1.1 Continued fraction approximation 142

5.1.2 Carlson's method 145

5.1.3 Matsuda-Fujii filter 147

5.2 Oustaloup fitter approximations 149

5.2.1 Ordinary Oustaloup approximation 149

5.2.2 A modified Oustaloup filter 155

5.3 Integer-order approximations of fractional-order transfer functions 157

5.3.1 High-order approximations 157

5.3.2 Low-order approximation via optimal model reduction techniques 161

5.4 Approximations of irrational models 165

5.4.1 Frequency response fitting approach 166

5.4.2 Charef approximation -168

5.4.3 Optimised Charef filters for complicated irrational models 173

6 Modelling and analysis of multivariable fractional-order transfer function matrices 179

6.1 FOTF - Creation of a MATLAB class 180

6.1.1 Defining FOTF class 180

6.1.2 Display function programming 182

6.1.3 Multivariable FOTF matrix support 184

6.2 Interconnections of FOTF blocks 185

6.2.1 Multiplications of FOTF blocks 185

6.2.2 Adding FOTF blocks 187

6.2.3 Feedback function 188

6.2.4 Other supporting functions 190

6.2.5 Conversions between FOTFs and commensurate-order models 193

6.3 Properties of linear fractional-order systems 195

6.3.1 Stability analysis 195

6.3.2 Partial fraction expansion and stability assessment 198

6.3.3 Norms of fractional-order systems 199

6.4 Frequency domain analysis 201

6.4.1 Frequency domain analysis of SISO systems 201

6.4.2 Stability assessment with Nyquist plots 203

6.4.3 Diagonal dominance analysis 204

6.4.4 Frequency response evaluation under complicated structures 207

6.4.5 Singular value plots in multivariable systems 210

6.5 Time domain analysis 211

6.5.1 Step and impulse responses 211

6.5.2 Time responses under arbitrary input signals 215

6.6 Root locus for commensurate-order systems 217

7 State space modelling and analysis of linear fractional-order systems 221

7.1 Standard representation of state space models 221

7.2 Descriptions of fractional-order state space models 222

7.2.1 Class design of FOSS 222

7.2.2 Conversions between FOSS and FOTF objects 224

7.2.3 Model augmentation under different base orders 226

7.2.4 Interconnection of FOSS blocks 227

7.3 Properties of fractional-order state space models 231

7.3.1 Stability assessment 231

7.3.2 State transition matrices 232

7.3.3 Controllability and observability 235

7.3.4 Controllable and observable staircase forms 235

7.3.5 Norm measures 236

7.4 Analysis of fractional-order state space models 237

7.5 Extended fractional state space models 238

7.5.1 Extended linear fractional-order state space models 238

7.5.2 Nonlinear fractional-order state space models 240

8 Numerical solutions of nonlinear fractional-order differential equations 243

8.1 Numerical solutions of nonlinear Caputo equations 243

8.1.1 Numerical solutions of single-term equations 244

8.1.2 Solutions of multi-term equations 249

8.1.3 Numerical solutions of extended fractional-order state space equations 252

8.1.4 Equation solution-based algorithm 256

8.2 Efficient high-precision algorithms for Caputo equations 258

8.2.1 Predictor equation 258

8.2.2 Corrector equation 261

8.2.3 High-precision matrix algorithm for implicit Caputo equations 263

8.3 Simulink block library for typical fractional-order components 265

8.3.1 FOTF block library 265

8.3.2 Implementation of FOTF matrix block 270

8.3.3 Numerical solutions of control problems with Simulink 272

8.3.4 Validations of Simulink results 275

8.4 Block diagram-based solutions of fractional-order differential equations with zero initial conditions 275

8.5 Block diagram solutions of nonlinear Caputo differential equations 282

8.5.1 Design of a Caputo operator block 282

8.5.2 Typical procedures in modelling Caputo equations 284

8.5.3 Simpler block diagram-based solutions of Caputo equations 286

8.5.4 Numerical solutions of implicit fractional-order differential equations 291

9 Design of fractional-order PID controllers 293

9.1 Introduction to Fractional-order PID controllers 293

9.2 Optimum design of integer-order PID controllers 295

9.2.1 Tuning rules for FOPDT plants 295

9.2.2 Meaningful objective functions for servo control 297

9.2.3 OptimPID: An optimum PID controller design interface 300

9.3 Fractional-order PID controller design with frequency response approach 302

9.3.1 General descriptions of the frequency domain design specifications 303

9.3.2 Design of PIλDμ controllers for FOPDT plants 304

9.3.3 Controller design with FOIDPT plants 308

9.3.4 Design of PIλDμ for typical fractional-order plants 310

9.3.5 Design of PIDμ controllers 311

9.3.6 Design of FO-[PD] controllers 311

9.3.7 Other considerations on controller design 312

9.4 Optimal design of PIλDμ controllers 312

9.4.1 Optimal PIλDμ controller design 312

9.4.2 Optimal PIλDμ controller design for plants with delays 317

9.4.3 OptimFOPID: An optimal fraction a I-order PID controller design interface 320

9.5 Design of fuzzy fractional-order PID controllers 322

9.5.1 Fuzzy rules of the controller parameters 322

9.5.2 Design and implementation with Simulink 323

10 Frequency domain controller design for multivariable fractional-order systems 329

10.1 Pseudodiagonalisation of multivariable systems 329

10.1.1 Pseudodiagonalisation and implementations 330

10.1.2 Individual channel design of the controllers 333

10.1.3 Robustness analysis of controller design through examples 337

10.2 Parameter optimisation design for multivariable fractional-order systems 340

10.2.1 Parameter optimisation design of integer-order controller 340

10.2.2 Design procedures of parameter optimisation controllers 343

10.2.3 Investigations on the robustness of the controllers 347

10.2.4 Controller design for plants with time delays 349

A Inverse Laplace transforms involving fractional and irrational operations 353

A.1 Special functions for Laplace transform 353

A.2 Laplace Transform tables 353

B FOTF Toolbox functions and models 357

B.1 Computational MATLAB functions 357

B.2 Object-oriented program design 359

B.3 Simulink models 361

B.4 Functions and models for the examples 361

C Benchmark problems for the assessment of fractional-order differential equation algorithms 363

Bibliography 365

Index 369

Customer Reviews

Most Helpful Customer Reviews

See All Customer Reviews