Fractional-Order Control Systems

Beschreibung

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. ContentsIntroduction to fractional calculus and fractional-order controlMathematical prerequisitesDefinitions and computation algorithms of fractional-order derivatives and IntegralsSolutions of linear fractional-order differential equationsApproximation of fractional-order operatorsModelling and analysis of multivariable fractional-order transfer function MatricesState space modelling and analysis of linear fractional-order SystemsNumerical solutions of nonlinear fractional-order differential EquationsDesign of fractional-order PID controllersFrequency domain controller design for multivariable fractional-order SystemsInverse Laplace transforms involving fractional and irrational OperationsFOTF Toolbox functions and modelsBenchmark problems for the assessment of fractional-order differential equation algorithms

Autor*in

Dingyü Xue

Stimmen zu »Fractional-Order Control Systems«

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 µ 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 µ Controllers9.4.1 Optimal PI λ D µ controller design9.4.2 Optimal PI λ D µ 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
()

Details

Link teilen