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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-Lef er Functions2.3.1 Mittag-Lef er function with one parameter2.3.2 Mittag-Lef er functions with two parameters2.3.3 Mittag-Lef er functions with more parameters2.3.4 Derivatives of Mittag-Lef er functions2.3.5 Numerical evaluation of Mittag-Lef er 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 defin