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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 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 Controllers 9.4.1 Optimal PI λ D controller design 9.4.2 Optimal PI λ D 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

Overall, the book will be useful for the one interested in the theoretical and numerical tools to design control systems. Krishnan Balachandran in: Zentralblatt MATH 1406.33001

Table of ContentsForewordPrefaceChapter 1 Introduction to Fractional Calculus and Fractional-order Control1.1 Historical Review of Fractional Calculus1.2 Fractional Modelling of the Real World1.3 Introduction to Fractional-order Control1.4 Structures of the BookChapter 2 Mathematical Prerequisites2.1 Elementary Special Functions2.1.1 Error and complementary error functions2.1.2 Gamma functions2.1.3 Beta functions2.2 Dawson Functions and Hypergeometric Functions2.2.1 Dawson function2.2.2 Hypergeometric functions2.3 Mittag-Leffler Functions2.3.1 Mittag-Leffler function with one parameter2.3.2 Mittag-Leffler functions with two parameters2.3.3 Mittag-Leffler functions with more parameters2.3.4 Derivatives of Mittag-Leffler functions2.3.5 Numerical evaluation of Mittag-Leffler functions2.4 Some Linear Algebra Techniques2.4.1 Kronecker product and Kronecker sum2.4.2 Matrix inverse2.4.3 Arbitrary matrix function evaluations2.5 Numerical Optimisation Problems and Solutions2.5.1 Unconstrained optimisation problems and solutions2.5.2 Constrained optimisation problems and solutions2.5.3 Global optimisation solutions2.6 Laplace Transform2.6.1 Definitions and properties2.6.2 Computer solutions to Laplace transform problemsChapter 3 Definitions and Computation Algorithms of Fractional-order Derivatives and Integrals3.1 Fractional-order Cauchy Integral Formula3.1.1 Cauchy Integrals3.1.2 Fractional-order derivative and integral formula for commonly used functions3.2 Gr]unwald-Letnikov Definition3.2.1 Deriving high-order derivatives3.2.2 Gr]unwald-Letnikov definition of fractional-order derivatives3.2.3 Numerical computation of Gr]unwald-Letnikov derivatives3.2.4 Podlubny's matrix algorithm3.2.5 Studies on short-memory effect3.3 Riemann-Liouville Definition3.3.1 High-order integrals3.3.2 Riemann-Liouville fractional-order definition3.3.3 Riemann-Liouville formula of commonly used functions3.3.4 Properties of initial time translation3.3.5 Numerical implementation of Riemann-Liouville definition3.4 High-precision Computation Algorithms of Fractional-order Deriva- tives and Integrals3.4.1 Construction of generating functions with arbitrary orders3.4.2 FFT-based algorithm3.4.3 A recursive formula3.4.4 A better fitting at initial instances3.4.5 Revisit to the matrix algorithm3.5 Caputo Definition3.6 Relationships among Different Definitions3.6.1 Relationship between G-L and R-L definitions3.6.2 Relationships between Caputo and R-L definitions3.6.3 Computation of Caputo fractional-order derivatives3.6.4 High-precision computation of Caputo derivatives3.7 Properties of Fractional-order Derivatives and IntegralsChapter 4 Solutions of Linear Fractional-order Differential Equations4.1 Introduction to Linear Fractional-order Differential Equations4.1.1 General form of linear fractional-order differential equations4.1.2 Initial value problems of fractional-order derivatives under different definitions4.1.3 An important Laplace transform formula4.2 Analytical Solutions of Some Fractional-order Differential Equations4.2.1 One-term differential equations4.2.2 Two-term differential equations4.2.3 Three-term differential equations4.2.4 General n-term differential equations4.3 Analytical Solutions of Commensurate-order Differential Equations4.3.1 General form of commensurate-order differential equations4.3.2 Some commonly used Laplace transforms in linear fractional- order systems4.3.3 Analytical solutions of commensurate-order equations4.4 Closed-form Solutions of Fractional-order Differential Equations with Zero Initial Conditions4.4.1 Closed-form solution4.4.2 High-precision closed-form algorithm4.4.3 Matrix approach for linear differential equations4.5 Numerical Solutions to Caputo Differential Equations with Nonzero Initial Conditions4.5.1 Mathematical description of Caputo equations4.5.2 Taylor auxiliary algorithm4.5.3 Exponential auxiliary algorithm4.5.4 Modified exponential auxiliary algorithm4.6 Numerical Solutions of Irrational Fractional-order Equations4.6.1 Irrational transfer function expression4.6.2 Numerical inverse Laplace transforms4.6.3 Stability assessment of irrational systems4.6.4 Numerical Laplace transformChapter 5 Approximation of Fractional-order Operators5.1 Some of the Continued Fraction based Approximations5.1.1 Continued fraction approximation5.1.2 Carlson's method5.1.3 Matsuda's method5.2 Oustaloup Filter Approximations5.2.1 Ordinary Oustaloup approximation5.2.2 A modified Oustaloup filter5.3 Integer-order Approximations of Fractional-order Transfer Functions5.3.1 High-order approximations5.3.2 Low-order approximation via optimal model reduction tech- niques5.4 Approximations of Irregular Fractional-order Models5.4.1 Frequency response fitting approach5.4.2 Charef approximation5.4.3 Optimised Charef filters for complicated irrational modelsChapter 6 Modelling and Analysis of Multivariable Fractional-order Transfer Function Matrices 1596.1 FOTF -- Creation of a MATLAB Object6.1.1 Defining FOTF class6.1.2 Display function programming6.1.3 Multivariable FOTF support6.1.4 Other fundamental facilities6.2 Interconnections of FOTF Blocks6.2.1 Multiplications of FOTF blocks6.2.2 Adding FOTF blocks6.2.3 Feedback function6.2.4 Other supporting functions6.2.5 Conversions between FOTFs and commensurate-order models6.3 Properties of Linear Fractional-order Systems6.3.1 Stability analysis6.3.2 Norms of fractional-order systems6.4 Frequency Domain Analysis6.4.1 Frequency domain analysis of SISO systems6.4.2 Diagonal dominance analysis6.4.3 Frequency response evaluation under complicated structures6.4.4 Singular value plots in multivariable systems6.5 Time Domain Analysis6.6 Root Locus for Commensurate-order SystemsChapter 7 State Space Modelling and Analysis of Linear Commensurate-order Systems7.1 Standard Representation of State Space Models7.2 Modelling of Fractional-order State Space Models7.2.1 Class design of FOSS7.2.2 Conversions between FOSS and FOTF objects7.2.3 Model augmentation with different base orders7.2.4 Interconnection of FOSS blocks7.3 Properties of Fractional-order State Space Models7.3.1 Stability assessment7.3.2 State space equations and state transition matrices7.3.3 Controllability and observability7.3.4 Norm measures7.4 Analysis of Fractional-order State Space Models7.5 Extended Linear State Space ModelsChapter 8 Numerical Solutions of Nonlinear Fractional-order Differential Equations8.1 Numerical Solutions of Class of Nonlinear Explicit Caputo Equations8.2 High-precision Numerical Solutions of Nonlinear Fractional-order Differential Equations8.3 Simulink Block Library for Typical Fractional-order Components8.3.1 A FOTF block library8.3.2 Implementation of FOTF matrix block8.3.3 Numerical solutions of control problems with Simulink8.4 Solutions of Fractional-order Differential Equations with Zero Initial Conditions8.5 Block Diagram Solutions of Caputo Different Equations8.5.1 Caputo differentiator block8.5.2 Block diagram based solutions of Caputo equations8.5.3 Design of Caputo operator blocksChapter 9 Fractional-order PID Controller Design9.1 Introduction to Fractional-order PID Controllers9.2 Optimum Design of Integer-order PID Controllers9.2.1 Tuning rules for FOPDT plants9.2.2 Meaningful objective functions for servo control9.2.3 OptimPID: an optimum PID controller design interface9.3 Fractional-order PID Controller Tuning Rules for Integer-order Plant Templates9.3.1 Tuning rules for FOPDT plants9.3.2 PI λ D controller design for FOPDT plants9.3.3 FO-[PD] controller for FOPDT plants9.3.4 FO-[PD] controller for FOLIDT plants with integrators9.4 Optimal Design of PI λ D Controllers9.4.1 Optimal PI λ D controller design9.4.2 Optimal PI λ D controller design for plants with delays9.4.3 OptimFOPID: an optimal fractional-order PID controller design interface9.5 Design of Fuzzy Fractional-order PID ControllersChapter 10 Controller Design for Multivariable Fractional-order Systems10.1 Pseudodiagonalisation of multivariable systems10.2 Parameter Optimisation Design for Multivariable Systems10.2.1 Parameter optimisation with integer-order controller10.2.2 Parameter optimisation under fractional-order controllers10.3 Controller Design with Quantitative Feedback Theory

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 D 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 D Controllers 9.4.1 Optimal PI D controller design 9.4.2 Optimal PI D 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

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 � 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 � Controllers 9.4.1 Optimal PI λ D � controller design 9.4.2 Optimal PI λ D � 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

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Dingyu Xue, Northeastern University China, China

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