### 2.1. Fractional Calculus

Fractional calculus is a topic in mathematics that is more than 300 years old. The idea of fractional calculus was suggested early in the development of regular (integer-order) calculus, with the first literature reference being associated with a letter, from Leibniz to L'Hospital in 1695. In this letter the half-order derivative was first mentioned.

There are several definitions of the fractional derivative/integral as a one common operator known as "differintegral" (see, e.g., [4–6]):

The Riemann-Liouville (RL) definition is given as

for and where is the *Gamma* function.

The Caputo's definition of fractional derivatives can be written as

for .

If we consider , where is a real constant and means the integer part, we can write the Grünwald-Letnikov (GL) definition as

where and are the bounds of operation for . Usually, we assume lower boundary .

For many engineering applications the Laplace transform methods are often used. The Laplace transform of the RL, the GL, and Caputo's fractional derivative/integral, under zero initial conditions for order is given by [5]:

A function, which plays a very important role in the fractional calculus, was in fact introduced by Humbert and Agarwal [7]. It is a two-parameter function of the *Mittag-Leffler* type defined as [4]:

Note that fractional calculus holds many important and interesting properties, which were described for instance in [3–5].

### 2.2. Fractional-Order Systems

There are several possible interpretations of the fractional-order systems. Here are mentioned three of them.

A general fractional-order linear system can be described by a fractional differential equation of the form [4]:

where denotes the Riemann-Liouville, Caputo's or Grünwald-Letnikov fractional derivative depending on initial conditions and their physical meaning or by the corresponding transfer function of incommensurate real orders of the following form [4]:

where , are constants, and , are arbitrary real or rational numbers and without loss of generality they can be arranged as , and .

The fractional-order linear time-invariant system can also be represented by the following state-space model:

where , , and are the state, input and output vectors of the system and , , , and are the fractional orders. If , system (2.8) is called a commensurate-order system, otherwise it is an incommensurate-order system.

In this paper, we will also consider the general incommensurate fractional-order nonlinear system represented as follows:

where are nonlinear functions and are initial conditions. The vector representation of (2.9) is:

where for , and .

The equilibrium points of system (2.10) are calculated via solving the following equation

and we suppose that is an equilibrium point of the fractional-order nonlinear system (2.10).

### 2.3. Discrete Time Approximation of Fractional Calculus: Numerical Method

In general, if a function is approximated by a grid function, , where is the grid size, the approximation for its fractional derivative of order can be expressed as [8]:

where is the backward shift operator and is a generating function. This generating function and its expansion determine both the form of the approximation and the coefficients [9]. In this way, the discretization of continuous fractional-order differentiator/integrator
can be expressed as . It is known that the forward difference rule is not suitable for applications to causal problems [8, 9].

As a generating function, can be used in generally the following formula [10]:

where and are denoted the gain and phase tuning parameters, respectively, and is sampling period. For example, when and , the generating function (2.13) becomes the forward Euler, the Tustin, the Al-Alaoui, the backward Euler, the implicit Adams rules, respectively. In this sense the generating formula can be tuned more precisely.

The expansion of the generating functions can be done by power series expansion (PSE). It is very important to note that PSE scheme leads to approximations in the form of polynomials of degree , that is, the discretized fractional order derivative is in the form of finite impulse response (FIR) filters, which have only zeros [11].

In this paper, for directly discretizing , , we will concentrate on the FIR form of discretization where as a generating function we will adopt a backward Euler rule. The mentioned operator, obtained from (2.13) for , raised to power , has the form

Then, the resulting transfer function, approximating the fractional-order operators, can be obtained by applying the relationship [12]:

where is the transform of the output sequence , is the transform of the input sequence , and denotes the expression, which results from the power series expansion of the function .

Doing so gives [13]:

where denotes the discrete equivalent of the fractional-order operator, considered as processes, and is the polynomial with degree of variable .

By using the short memory principle [4], the discrete equivalent of the fractional-order integrodifferential operator, , is given by

where is the memory length and are binomial coefficients where [14]

For practical numerical calculation of the fractional derivative and integral we can derive the formula from relation (2.17), where the sampling period is in numerical evaluation replaced by the time step of calculation , then we get

where for or for in the relation (2.19). By using a relation (2.14) we obtained a first-order approximation of the fractional derivative of order . Another possibility for the approximation is use, the trapezoidal rule, that is, the use of the generating function (2.13) for and then the PSE, which is convergent of order 2. Other forms of generation functions for higher-order approximation of the fractional order derivative are presented in [9].

Obviously, for this simplification we pay a penalty in the form of some inaccuracy. If , we can easily establish the following estimate for determining the memory length , providing the required accuracy [4]:

An evaluation of the short memory effect and convergence relation of the error between short and long memory were clearly described and also proved in [4].

For general numerical solution of the fractional differential equation, let us consider the following initial value problem

with initial conditions , where . Using approximation (2.19), we obtain the numerical solution, which can be expressed as

where . For the memory term expressed by sum, a "short memory" principle can be used or without using "short memory" principle, we put for all in (2.22).