Fractional-Order Control Systems: Fundamentals and Numerical Implementations (Fractional Calculus in Applied Sciences and Engineering, 1) - Hardcover

Xue, Dingyü

Hardcover

ISBN 10: 3110499991 ISBN 13: 9783110499995

Publisher: De Gruyter, 2017

Synopsis

This book explains the essentials of fractional calculus and demonstrates its application in control system modeling, analysis and design. It presents original research to improve the solution of fractional differential equations and to analyze fractional-order control systems with multi-variables. Through high-quality Matlab programs, it provides engineers and applied mathematicians with theoretical and numerical tools to design control systems.

About this title

About the Author

Dingyu Xue, Northeastern University China, China

Review

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 Solutio

Edition details

• Publisher: De Gruyter
• Publication date: 2017
• ISBN 10: 3110499991
• ISBN 13: 9783110499995
• Binding: Hardcover
• Language: English
• Edition number: 1
• Number of pages: 388