ABSTRACT
Chapter 1 is the backbone of the book, as it contains all of the mathematical postulations used in the subsequent chapters. The special mathematical functions that play a vital role in establishing the proper definition of operators of fractional calculus are briefly reviewed. The most widely used definitions and properties of fractional-order integrals and fractional-order derivatives are provided. To analyze the behavior of linear lumped fractional-order systems, the Laplace transforms of fractional-order operators are derived, and then fractional-order systems are categorized along with a discussion about their stability. Different types of fractional-order controllers as well as advantages and disadvantages of classical types are discussed. Triangular orthogonal functions are presented along with approximation formulae for estimating functions and fractional-order integrals. Finally, triangular strip operational matrices, which are the basis for the proposed fractional-order controller tuning technique described in Chapter 8, are derived from the classical finite difference formula and the Grunwald-Letnikov fractional-order derivative. Source codes developed in MATLAB for triangular function operational matrices and triangular strip operational matrices are provided.
