Mathematical Problems in Engineering
Volume 2018 (2018), Article ID 4270764, 5 pages
https://doi.org/10.1155/2018/4270764

Research Article

Numerical Simulation of Fractional Control System Using Chebyshev Polynomials

Jun Zhang,1,2,3 Yugui Li,2,4 and Jiaquan Xie⁵

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

Correspondence should be addressed to Yugui Li; liyugui2018@126.com

Received 1 October 2018; Revised 7 November 2018; Accepted 2 December 2018; Published 13 December 2018

Academic Editor: Ivo Petras

Copyright © 2018 Jun Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

Figure 1: Fractional discrete control 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 as

where and denotes the gamma function.

Definition 2 (see [22]). The left-sided Caputo fractional-order derivative of order , is defined as

where .

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 is

where the weight function and

3.2. Function Approximation

Suppose that ; 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 can be expressed by

where is the operational matrix of derivative given by

Similarly, the operational matrix of differentiation of can be expressed as

where .

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 , then

where 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 as

and

where and can be obtained from (12). Substituting (19)-(26) into (1), we have

Test Problem 4.1. Consider the following fractional Relaxation-Oscillation equation system

If 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.

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

Figure 3: The absolute errors between the analytical and numerical results.

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].

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

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].

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

Time (seconds)
Analytical results

0.1
-0.9900
4.2619269e-4
8.7631391e-6
1.9898239e-7

0.2
-0.9600
5.3938108e-4
7.8927198e-6
1.7497499e-7

0.3
-0.9100
6.8319833e-4
5.3619996e-6
2.4727493e-7

0.4
-0.8400
5.87172528-4
3.6391397e-6
2.8979387e-7

0.5
-0.7500
6.8719873e-5
2.3701310e-5
4.9839739e-7

0.6
-0.6400
7.3264873e-4
2.8748927e-5
5.3773098e-6

0.7
-0.5100
7.7838719e-4
3.9210200e-5
7.3937910e-6

0.8
-0.3600
8.3719731e-4
3.7101706e-5
5.3793700e-7

0.9
-0.1900
9.3871937e-4
4.2171077e-5
6.3897193e-6

1.0
0
8.6152715e-4
5.8973427e-5
7.6386329e-7

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.

References

H. G. Sun, “A new collection of real world applications of fractional calculus in science and engineering,” Communications in Nonlinear Science & Numerical Simulation, vol. 64, pp. 213–231, 2018.

A. D. Obembe, H. Y. Al-Yousef, M. E. Hossain et al., “Fractional derivatives and their applications in reservoir engineering problems: A review,” Journal of Petroleum Science & Engineering, vol. 157, pp. 312–327, 2017.

L.-L. Huang, “Fractional discrete-time diffusion equation with uncertainty: applications of fuzzy discrete fractional calculus,” Physica A: Statistical Mechanics and its Applications, vol. 508, pp. 166–175, 2018.

R. R. Nigmatullin, “On fractional filtering versus conventional filtering in economics,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 4, pp. 979–986, 2010.

N. Haldrup, “Long memory, fractional integration, and cross-sectional aggregation,” Journal of Econometrics, vol. 199, no. 1, pp. 1–11, 2017.

V. E. Tarasov, “Time-dependent fractional dynamics with memory in quantum and economic physics,” Annals of Physics, vol. 383, pp. 579–599, 2017.

Y. Liang, “Stretching chemical heterogeneities by melt migration in an upwelling mantle: An analysis based on time-dependent batch and fractional melting models,” Earth and Planetary Science Letters, vol. 498, pp. 275–287, 2018.

B. Sikora, “Constrained controllability of fractional linear systems with delays in control,” Systems & Control Letters, vol. 106, pp. 9–15, 2017.

D. Kumar, “Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology,” Chinese Journal of Physics, vol. 56, no. 1, pp. 75–85, 2018.

D. Y. Xue, Fractional calculus and fractional order control, Science Press, Beijing, 2018.

F. Mohammadi, “A generalized fractional-order Legendre wavelet Tau method for solving fractional differential equations,” Journal of Computational & Applied Mathematics, vol. 339, pp. 306–316, 2017.

J. Xie, “A two-dimensional Chebyshev wavelets approach for solving the Fokker-Planck equations of time and space fractional derivatives type with variable coefficients,” Applied Mathematics and Computation, vol. 332, pp. 197–208, 2018.

F. Zhao, “Chebyshev polynomials approach for numerically solving system of two-dimensional fractional PDEs and convergence analysis,” Applied Mathematics and Computation, vol. 313, pp. 321–330, 2017.

Z. Meng, “Numerical solutions of nonlinear fractional differential equations by alternative Legendre polynomials,” Applied Mathematics and Computation, vol. 336, pp. 454–464, 2018.

L. Pezza, “A multiscale collocation method for fractional differential problems,” Mathematics and Computers in Simulation, vol. 147, pp. 210–219, 2018.

Z. J. Fu, “Boundary particle method for Laplace transformed time fractional diffusion equations,” Journal of Computational Physics, vol. 235, pp. 52–66, 2013.

Z.-J. Fu, “Method of approximate particular solutions for constant- and variable-order fractional diffusion models,” Engineering Analysis with Boundary Elements, vol. 57, pp. 37–46, 2015.

W. Chen, “Fractional diffusion equations by the Kansa method,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1614–1620, 2010.

J. Lin, “A novel meshless method for fully nonlinear advection-diffusion-reaction problems to model transfer in anisotropic media,” Applied Mathematics and Computation, vol. 339, pp. 459–476, 2018.

N. I. Mahmudov, “Partial-approximate controllability of nonlocal fractional evolution equations via approximating method,” Applied Mathematics and Computation, vol. 334, pp. 227–238, 2018.

A. Lotfi, “Epsilon penalty method combined with an extension of the Ritz method for solving a class of fractional optimal control problems with mixed inequality constraints,” Applied Numerical Mathematics, vol. 135, pp. 497–509, 2018.

I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.

E. H. Doha, “A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order,” Computers & Mathematics with Applications, vol. 62, no. 5, pp. 2364–2373, 2011.

J. Lin, “Fast simulation of multi-dimensional wave problems by the sparse scheme of the method of fundamental solutions,” Computers & Mathematics with Applications, vol. 72, no. 3, pp. 555–567, 2016.

Z.-J. Fu, “A boundary-type meshless solver for transient heat conduction analysis of slender functionally graded materials with exponential variations,” Computers & Mathematics with Applications. An International Journal, vol. 76, no. 4, pp. 760–773, 2018.

E. J. Kansa, “On the ill-conditioned nature of C∞ RBF strong collocation,” Engineering Analysis with Boundary Elements, vol. 78, pp. 26–30, 2017.

Z. Fu, “Singular boundary method for wave propagation analysis in periodic structures,” Journal of Sound and Vibration, vol. 425, pp. 170–188, 2018.