This book explains the essentials of fractional calculus and demonstrates its application in control system modeling, analysis and design. It presents original research to find high-precision solutions to fractional-order differentiations and differential equations. Numerical algorithms and their implementations are proposed to analyze multivariable fractional-order control systems. Through high-quality MATLAB programs, it provides engineers and applied mathematicians with theoretical and numerical tools to design control systems.
Contents
Introduction to fractional calculus and fractional-order control
Mathematical prerequisites
Definitions and computation algorithms of fractional-order derivatives and Integrals
Solutions of linear fractional-order differential equations
Approximation of fractional-order operators
Modelling and analysis of multivariable fractional-order transfer function Matrices
State space modelling and analysis of linear fractional-order Systems
Numerical solutions of nonlinear fractional-order differential Equations
Design of fractional-order PID controllers
Frequency domain controller design for multivariable fractional-order Systems
Inverse Laplace transforms involving fractional and irrational Operations
FOTF Toolbox functions and models
Benchmark problems for the assessment of fractional-order differential equation algorithms
Table of Contents
Foreword
Preface
Chapter 1 Introduction to Fractional Calculus and Fractional-order Control
1.1 Historical Review of Fractional Calculus
1.2 Fractional Modelling of the Real World
1.3 Introduction to Fractional-order Control
1.4 Structures of the Book
Chapter 2 Mathematical Prerequisites
2.1 Elementary Special Functions
2.1.1 Error and complementary error functions
2.1.2 Gamma functions
2.1.3 Beta functions
2.2 Dawson Functions and Hypergeometric Functions
2.2.1 Dawson function
2.2.2 Hypergeometric functions
2.3 Mittag-Leffler Functions
2.3.1 Mittag-Leffler function with one parameter
2.3.2 Mittag-Leffler functions with two parameters
2.3.3 Mittag-Leffler functions with more parameters
2.3.4 Derivatives of Mittag-Leffler functions
2.3.5 Numerical evaluation of Mittag-Leffler functions
2.4 Some Linear Algebra Techniques
2.4.1 Kronecker product and Kronecker sum
2.4.2 Matrix inverse
2.4.3 Arbitrary matrix function evaluations
2.5 Numerical Optimisation Problems and Solutions
2.5.1 Unconstrained optimisation problems and solutions
2.5.2 Constrained optimisation problems and solutions
2.5.3 Global optimisation solutions
2.6 Laplace Transform
2.6.1 Definitions and properties
2.6.2 Computer solutions to Laplace transform problems
Chapter 3 Definitions and Computation Algorithms of Fractional-order Derivatives and Integrals
3.1 Fractional-order Cauchy Integral Formula
3.1.1 Cauchy Integrals
3.1.2 Fractional-order derivative and integral formula for commonly used functions
3.2 Gr¨unwald–Letnikov Definition
3.2.1 Deriving high-order derivatives
3.2.2 Gr¨unwald–Letnikov definition of fractional-order derivatives
3.2.3 Numerical computation of Gr¨unwald–Letnikov derivatives
3.2.4 Podlubny’s matrix algorithm
3.2.5 Studies on short-memory effect
3.3 Riemann–Liouville Definition
3.3.1 High-order integrals
3.3.2 Riemann–Liouville fractional-order definition
3.3.3 Riemann–Liouville formula of commonly used functions
3.3.4 Properties of initial time translation
3.3.5 Numerical implementation of Riemann–Liouville definition
3.4 High-precision Computation Algorithms of Fractional-order Deriva- tives and Integrals
3.4.1 Construction of generating functions with arbitrary orders
3.4.2 FFT-based algorithm
3.4.3 A recursive formula
3.4.4 A better fitting at initial instances
3.4.5 Revisit to the matrix algorithm
3.5 Caputo Definition
3.6 Relationships among Different Definitions
3.6.1 Relationship between G-L and R-L definitions
3.6.2 Relationships between Caputo and R-L definitions
3.6.3 Computation of Caputo fractional-order derivatives
3.6.4 High-precision computation of Caputo derivatives
3.7 Properties of Fractional-order Derivatives and Integrals
Chapter 4 Solutions of Linear Fractional-order Differential Equations
4.1 Introduction to Linear Fractional-order Differential Equations
4.1.1 General form of linear fractional-order differential equations
4.1.2 Initial value problems of fractional-order derivatives under different definitions
4.1.3 An important Laplace transform formula
4.2 Analytical Solutions of Some Fractional-order Differential Equations
4.2.1 One-term differential equations
4.2.2 Two-term differential equations
4.2.3 Three-term differential equations
4.2.4 General n-term differential equations
4.3 Analytical Solutions of Commensurate-order Differential Equations
4.3.1 General form of commensurate-order differential equations
4.3.2 Some commonly used Laplace transforms in linear fractional- order systems
4.3.3 Analytical solutions of commensurate-order equations
4.4 Closed-form Solutions of Fractional-order Differential Equations with Zero Initial Conditions
4.4.1 Closed-form solution
4.4.2 High-precision closed-form algorithm
4.4.3 Matrix approach for linear differential equations
4.5 Numerical Solutions to Caputo Differential Equations with Nonzero Initial Conditions
4.5.1 Mathematical description of Caputo equations
4.5.2 Taylor auxiliary algorithm
4.5.3 Exponential auxiliary algorithm
4.5.4 Modified exponential auxiliary algorithm
4.6 Numerical Solutions of Irrational Fractional-order Equations
4.6.1 Irrational transfer function expression
4.6.2 Numerical inverse Laplace transforms
4.6.3 Stability assessment of irrational systems
4.6.4 Numerical Laplace transform
Chapter 5 Approximation of Fractional-order Operators
5.1 Some of the Continued Fraction based Approximations
5.1.1 Continued fraction approximation
5.1.2 Carlson’s method
5.1.3 Matsuda’s method
5.2 Oustaloup Filter Approximations
5.2.1 Ordinary Oustaloup approximation
5.2.2 A modified Oustaloup filter
5.3 Integer-order Approximations of Fractional-order Transfer Functions
5.3.1 High-order approximations
5.3.2 Low-order approximation via optimal model reduction tech- niques
5.4 Approximations of Irregular Fractional-order Models
5.4.1 Frequency response fitting approach
5.4.2 Charef approximation
5.4.3 Optimised Charef filters for complicated irrational models
Chapter 6 Modelling and Analysis of Multivariable Fractional-order Transfer Function Matrices 159
6.1 FOTF ― Creation of a MATLAB Object
6.1.1 Defining FOTF class
6.1.2 Display function programming
6.1.3 Multivariable FOTF support
6.1.4 Other fundamental facilities
6.2 Interconnections of FOTF Blocks
6.2.1 Multiplications of FOTF blocks
6.2.2 Adding FOTF blocks
6.2.3 Feedback function
6.2.4 Other supporting functions
6.2.5 Conversions between FOTFs and commensurate-order models
6.3 Properties of Linear Fractional-order Systems
6.3.1 Stability analysis
6.3.2 Norms of fractional-order systems
6.4 Frequency Domain Analysis
6.4.1 Frequency domain analysis of SISO systems
6.4.2 Diagonal dominance analysis
6.4.3 Frequency response evaluation under complicated structures
6.4.4 Singular value plots in multivariable systems
6.5 Time Domain Analysis
6.6 Root Locus for Commensurate-order Systems
Chapter 7 State Space Modelling and Analysis of Linear Commensurate-order Systems
7.1 Standard Representation of State Space Models
7.2 Modelling of Fractional-order State Space Models
7.2.1 Class design of FOSS
7.2.2 Conversions between FOSS and FOTF objects
7.2.3 Model augmentation with different base orders
7.2.4 Interconnection of FOSS blocks
7.3 Properties of Fractional-order State Space Models
7.3.1 Stability assessment
7.3.2 State space equations and state transition matrices
7.3.3 Controllability and observability
7.3.4 Norm measures
7.4 Analysis of Fractional-order State Space Models
7.5 Extended Linear State Space Models
Chapter 8 Numerical Solutions of Nonlinear Fractional-order Differential Equations
8.1 Numerical Solutions of Class of Nonlinear Explicit Caputo Equations
8.2 High-precision Numerical Solutions of Nonlinear Fractional-order Differential Equations
8.3 Simulink Block Library for Typical Fractional-order Components
8.3.1 A FOTF block library
8.3.2 Implementation of FOTF matrix block
8.3.3 Numerical solutions of control problems with Simulink
8.4 Solutions of Fractional-order Differential Equations with Zero Initial Conditions
8.5 Block Diagram Solutions of Caputo Different Equations
8.5.1 Caputo differentiator block
8.5.2 Block diagram based solutions of Caputo equations
8.5.3 Design of Caputo operator blocks
Chapter 9 Fractional-order PID Controller Design
9.1 Introduction to Fractional-order PID Controllers
9.2 Optimum Design of Integer-order PID Controllers
9.2.1 Tuning rules for FOPDT plants
9.2.2 Meaningful objective functions for servo control
9.2.3 OptimPID: an optimum PID controller design interface
9.3 Fractional-order PID Controller Tuning Rules for Integer-order Plant Templates
9.3.1 Tuning rules for FOPDT plants
9.3.2 PI λ D µ controller design for FOPDT plants
9.3.3 FO-[PD] controller for FOPDT plants
9.3.4 FO-[PD] controller for FOLIDT plants with integrators
9.4 Optimal Design of PI λ D µ Controllers
9.4.1 Optimal PI λ D µ controller design
9.4.2 Optimal PI λ D µ controller design for plants with delays
9.4.3 OptimFOPID: an optimal fractional-order PID controller design interface
9.5 Design of Fuzzy Fractional-order PID Controllers
Chapter 10 Controller Design for Multivariable Fractional-order Systems
10.1 Pseudodiagonalisation of multivariable systems
10.2 Parameter Optimisation Design for Multivariable Systems
10.2.1 Parameter optimisation with integer-order controller
10.2.2 Parameter optimisation under fractional-order controllers
10.3 Controller Design with Quantitative Feedback Theory
