Numerical Simulation of Fractional Control System Using Chebyshev Polynomials

1School of Mechanical Engineering, Taiyuan University of Science and Technology, Taiyuan 030024, China 2Collaborative Innovation Center of Taiyuan Heavy Machinery Equipment, Taiyuan 030024, China 3Taiyuan Institute of China Coal Technology Engineering Group, Taiyuan 03006, China 4School of Material Science and Engineering, Taiyuan University of Science and Technology, Taiyuan 030024, China 5College of Mechanical Engineering, Taiyuan University of Technology, Taiyuan 030024, China


Introduction
Fractional calculus has a long history and it has been widely used in various fields of engineering, sciences, applied mathematics, and economics [1][2][3][4][5].Many real-world problems such as physics, chemistry, fluid mechanics, control, and mathematical biology can be modelled by building fractional constitutive models [6][7][8][9].The typical fractional feedback control system is given in Figure 1.  () is the fractional controller,  0 () is the transfer function of fractional controller system, and   () is the feedback loop transfer function of fractional system.() and () are the input and output of the system.
The above fractional control system is a continuous system when the switch is always closed, and its time domain model can be established by the following formula [10]: where   =  0    , and   >  −1 > ⋅ ⋅ ⋅ >  0 ≥ 0,   >  −1 > ⋅ ⋅ ⋅ >  0 ≥ 0,   ,   are arbitrary real numbers.The  field is described by Laplace transform of (1) as So far, various numerical methods are presented to solve fractional differential equations.These methods include wavelets method [11,12], Chebyshev and Legendre polynomials [13,14], and collocation method [15][16][17][18][19].In [20], N. I. Mahmudov utilized an approximate method to study partial-approximate controllability of semilinear nonlocal fractional evolution equations.In [21], Ali Lotf used Epsilon penalty and an extension of the Ritz method for solving a class of fractional optimal control problems with mixed boundary conditions.In this paper, we get the numerical solutions of fractional control system using Chebyshev polynomials.The paper is organized as follows: in the next section, the definitions about fractional calculus are introduced.In Section 3, some relevant properties of Chebyshev polynomials are given.Numerical methods together with numerical examples are illustrated in Section 4. A conclusion is drawn in Section 5.

Chebyshev Polynomials
. .e Properties of Chebyshev Polynomials.The analytical form of the Chebyshev polynomials   () of degree  is given by [23].
The orthogonality is where the weight function () = 1/ √  −  2 and . .Function Approximation.Suppose that () ∈  2 [0, 1]; it may be expanded in terms of the Chebyshev polynomials as where the coefficient   is given by If we consider the truncated series in ( 5), then we have where Then the derivative of vector Φ() can be expressed by Φ ()  = P (1) Φ () , where P (1) is the ( + 1) × ( + 1) operational matrix of derivative given by Similarly, the operational matrix P  of − differentiation of Φ() can be expressed as where P  = (P (1) )  .
. .Operational Matrix of Fractional-Order Derivative.The main objective of this section is to prove the following theorem for the fractional derivatives of the Chebyshev polynomials [23].
Theorem 4. Let Φ() be the Chebyshev vector defined in ( ) and suppose  > 0, then where P () is the ( + 1) × ( + 1) differential operational matrix of order  in the Caputo sense and it is defined as follows:

Numerical Experiments
In this section, we utilize the Chebyshev polynomials to carry out the numerical simulation of fractional control system.Firstly, each term of (1) can be expressed by the Chebyshev polynomials basis as . . . and . . .
where  and  can be obtained from (12).Substituting ( 19)-( 26) into (1), we have Test Problem . .Consider the following fractional Relaxation-Oscillation equation system If the input function of the system is () =  2 + 2 1.5 /Γ(2.5), the analytical solution of this system is () =  2 .When  = 5, the output solutions by analytical method and our proposed method are shown in Figure 2, and the absolute errors for the analytical and numerical solutions are shown in Figure 3.

Conclusions
This paper presents a numerical approach for solving the fractional control system using Chebyshev polynomials.The derived operational matrix of fractional derivative is used to transfer the original problem into a system of linear algebra equations which can be easily solved.Numerical results show that the numerical solutions converge to the analytical solutions well as  grows.

Figure 2 :Figure 3 :
Figure 2: The output solutions by analytical method and our proposed method.

Figure 4 :
Figure 4: The analytical and numerical solutions for some different values of .

Table 1 :
Absolute errors for the numerical and analytical results.