Robust Adaptive Control for Fractional-Order Systems with Disturbance and Saturation

Chapter 2Fractional Calculus and Fractional-Order Systems

2.1 Fractional Calculus

In this section, several important functions of fractional calculus, the definition of the fractional integral, three main fractional derivatives, and some important lemmas are introduced, which will be used throughout the book.

2.1.1 Several Important Functions of Fractional Calculus

In this subsection, three important functions of fractional calculus will be introduced: the gamma function, the beta function, and the Mittag–Leffler (ML) function.

Definition 2.1 (Gamma Function [154] [163–165])

The gamma function is defined by the following integral form:

2.1

where belongs to the field of complex numbers.

According to the definition of the gamma function (2.1), it has the following property:

2.2

On the basis of Equation (2.2), we have

2.3

Definition 2.2 (Beta Function [165])

The beta function is defined as

2.4

where and belong to the field of complex numbers.

Combining Equations (2.1 ...