Abstract
In the current study, a numerical scheme based on Chebyshev polynomials is proposed to solve the problem of fractional control system. The operational matrix of fractional derivative is derived and that is used to transform the original problem into a system of linear equations. Lastly, several numerical examples are presented to verify the effectiveness and feasibility of the given method.
1. Introduction
Fractional calculus has a long history and it has been widely used in various fields of engineering, sciences, applied mathematics, and economics [1–5]. Many real-world problems such as physics, chemistry, fluid mechanics, control, and mathematical biology can be modelled by building fractional constitutive models [6–9]. The typical fractional feedback control system is given in Figure 1. is the fractional controller, 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 , and 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 [15–19]. 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.
2. Preliminaries and Notations
Definition 1 (see [22]). The left-sided Riemann-Liouville fractional integral of order , for a function , is defined aswhere and denotes the gamma function.
Definition 2 (see [22]). The left-sided Caputo fractional-order derivative of order , is defined aswhere .
3. Chebyshev Polynomials
3.1. The Properties of Chebyshev Polynomials
The analytical form of the Chebyshev polynomials of degree is given by [23].where and .
The orthogonality iswhere the weight function and
3.2. Function Approximation
Suppose that ; it may be expanded in terms of the Chebyshev polynomials aswhere the coefficient is given byIf we consider the truncated series in (5), then we havewhereThen the derivative of vector can be expressed bywhere is the operational matrix of derivative given bySimilarly, the operational matrix of differentiation of can be expressed aswhere .
3.3. 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].
Lemma 3. Let be a Chebyshev polynomial, then
Theorem 4. Let be the Chebyshev vector defined in (13) and suppose , thenwhere is the differential operational matrix of order in the Caputo sense and it is defined as follows:where
4. 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 asandwhere and can be obtained from (12). Substituting (19)-(26) into (1), we have
Test Problem 4.1. Consider the following fractional Relaxation-Oscillation equation systemIf the input function of the system is , the analytical solution of this system is . When , 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.
Test Problem 4.2. Consider the following fractional control system:where , the analytical solution of this system is . When , the output solutions by analytical method and our proposed method are shown in Figure 4. Figure 4 shows that the numerical solutions approximate to analytical solutions as increases. As increases, the resulting coefficient matrix becomes large and may be singular [24].
Test Problem 4.3. Consider the following fractional control system:where . The analytical solution of this system is . When , the absolute errors for the numerical and analytical results are listed in Table 1. Table 1 shows that the numerical solutions are in agreement with the analytical solutions well as grows. With increases, the coefficient matrix may be ill conditioned. The discussion on the ill-conditioned matrix is presented in the literature [25–27].
5. 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.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors equally contribute to this paper.
Acknowledgments
This work was supported by Project Name: Research on automatic loading technology of mechanical anchorage agent; Project Number: 2018-TD-MS049; Project Name: Two-arm anchor cable anchor car; Project Number: 2018-TD-MS047; Project Name: Research on key technology of repeated application of mobile wind bridge in coal mine; Project Number: 2018-TD-QN035; Project Name: Study on supporting technology and key technology of non-repeated retaining roadway along goaf; Project Number: 2018-TD-QN040.