# Numerical Simulation of Fractional Control System Using Chebyshev Polynomials

Authors: Jun Zhang; Yugui Li; Jiaquan Xie
Journal: Mathematical Problems in Engineering
 (2018)

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## 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.


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## Body

## 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. Gct is the fractional controller, G0t is the transfer function of fractional controller system, and Gft is the feedback loop transfer function of fractional system. Ut and Yt 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]:(1)anDαnyt+an-1Dαn-1yt+⋯+a0Dα0yt=bmDβmut+bm-1Dβm-1ut+⋯+b0Dβ0utwhere Dα=0cDtα, and αn>αn-1>⋯>α0≥0,βn>βn-1>⋯>β0≥0,ak,bk are arbitrary real numbers. The S field is described by Laplace transform of (1) as (2)Gs=YsUs=bmsβm+bm-1sβm-1+⋯+b0sβ0ansαn+an-1sαn-1+⋯+a0sα0So 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 Section3, 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 orderμ,μ∈R+, for a function yt, is defined as(3)I0tμyt=1Γμ∫0tyςt-ςu-1dς,t>0,where ς∈R+ and Γ· denotes the gamma function.Definition 2 (see [22]).
The left-sided Caputo fractional-order derivative of orderμ,μ∈R+, is defined as(4)D0ctμyt=1Γm-μ∫0tymςdςt-ςμ+1-m,m-1<μ≤m,where m∈N+.


## 3. Chebyshev Polynomials

### 3.1. The Properties of Chebyshev Polynomials

The analytical form of the Chebyshev polynomialsTit of degree i is given by [23].(5)Tit=i∑k=0i-1i-ki+k-1!22ki-k!2k!tk,t∈0,1where Ti0=-1i and Ti1=1.The orthogonality is(6)∫01TjtTktwtdt=hk,where the weight function wt=1/t-t2 and(7)hk=bk2π,k=j,0,k≠j,b0=2,bk=1,k≥1.


### 3.2. Function Approximation

Suppose thatyt∈L20,1; it may be expanded in terms of the Chebyshev polynomials as(8)yt=∑i=0∞ciTit,where the coefficient ci is given by(9)ci=1h1∫01ytTitwtdt,i=0,1,2,…If we consider the truncated series in (5), then we have(10)yt≈yMt=∑i=0MciTit=CTΦt,where(11)C=c0,c1,…,cMT,Φt=T0t,T1t,…,TMtT.Then the derivative of vector Φt can be expressed by(12)dΦtdt=P1Φt,where P1 is the M+1×M+1 operational matrix of derivative given by(13)P1=pij=4ibj,j=0,1,…,i=j+k,k=1,3,5,…,M,ifMis  odd,k=1,3,5,…,M-1,ifMis  even,0,otherwiseSimilarly, the operational matrix Pn of n-times differentiation of Φt can be expressed as(14)dnΦtdtn=PnΦt,where Pn=P1n.


### 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.
LetTit be a Chebyshev polynomial, then (15)DμTit=0,i=0,1,2,…,μ-1,μ>0.Theorem 4.
LetΦt be the Chebyshev vector defined in (13) and suppose μ>0, then(16)D0cμΦt≈PμΦt,where Pμ is the M+1×M+1 differential operational matrix of order μ in the Caputo sense and it is defined as follows:(17)Pμ=000⋯0⋮⋮⋮⋯⋮000⋯0Sμμ,0Sμμ,1Sμμ,2⋯Sμμ,M⋮⋮⋮⋯⋮Sμi,0Sμi,1Sμi,2⋯Sμi,M⋮⋮⋮⋯⋮SμM,0SμM,1SμM,2⋯SμM,Mwhere(18)Sμi,j=∑k=μi-1i-k2ii+k-1!Γk-μ+1/2bjΓk+1/2i-k!Γk-μ-j+1Γk-μ+j+1i=μ,μ+1,…,M.


## 3.1. The Properties of Chebyshev Polynomials

The analytical form of the Chebyshev polynomialsTit of degree i is given by [23].(5)Tit=i∑k=0i-1i-ki+k-1!22ki-k!2k!tk,t∈0,1where Ti0=-1i and Ti1=1.The orthogonality is(6)∫01TjtTktwtdt=hk,where the weight function wt=1/t-t2 and(7)hk=bk2π,k=j,0,k≠j,b0=2,bk=1,k≥1.


## 3.2. Function Approximation

Suppose thatyt∈L20,1; it may be expanded in terms of the Chebyshev polynomials as(8)yt=∑i=0∞ciTit,where the coefficient ci is given by(9)ci=1h1∫01ytTitwtdt,i=0,1,2,…If we consider the truncated series in (5), then we have(10)yt≈yMt=∑i=0MciTit=CTΦt,where(11)C=c0,c1,…,cMT,Φt=T0t,T1t,…,TMtT.Then the derivative of vector Φt can be expressed by(12)dΦtdt=P1Φt,where P1 is the M+1×M+1 operational matrix of derivative given by(13)P1=pij=4ibj,j=0,1,…,i=j+k,k=1,3,5,…,M,ifMis  odd,k=1,3,5,…,M-1,ifMis  even,0,otherwiseSimilarly, the operational matrix Pn of n-times differentiation of Φt can be expressed as(14)dnΦtdtn=PnΦt,where Pn=P1n.


## 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.
LetTit be a Chebyshev polynomial, then (15)DμTit=0,i=0,1,2,…,μ-1,μ>0.Theorem 4.
LetΦt be the Chebyshev vector defined in (13) and suppose μ>0, then(16)D0cμΦt≈PμΦt,where Pμ is the M+1×M+1 differential operational matrix of order μ in the Caputo sense and it is defined as follows:(17)Pμ=000⋯0⋮⋮⋮⋯⋮000⋯0Sμμ,0Sμμ,1Sμμ,2⋯Sμμ,M⋮⋮⋮⋯⋮Sμi,0Sμi,1Sμi,2⋯Sμi,M⋮⋮⋮⋯⋮SμM,0SμM,1SμM,2⋯SμM,Mwhere(18)Sμi,j=∑k=μi-1i-k2ii+k-1!Γk-μ+1/2bjΓk+1/2i-k!Γk-μ-j+1Γk-μ+j+1i=μ,μ+1,…,M.


## 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(19)Dαnyt≈0cDtαnCTΦt≈CTPαnΦt,(20)Dαn-1yt≈0cDtαn-1CTΦt≈CTPαn-1Φt,(21)⋮(22)Dα1yt≈0cDtα1CTΦt≈CTPα1Φt,and(23)Dβnut≈0cDtβnUTΦt≈UTPβnΦt,(24)Dβn-1ut≈0cDtβn-1UTΦt≈UTPβn-1Φt,(25)⋮(26)Dβ1ut≈0cDtβ1UTΦt≈UTPβ1Φt,where C and U can be obtained from (12). Substituting (19)-(26) into (1), we have(27)anCTPαnΦt+an-1CTPαn-1Φt+⋯+a0CTPα1Φt=bmUTPβnΦt+bm-1UTPβn-1Φt+⋯+b0UTPβ1ΦtTest Problem 4.1. Consider the following fractional Relaxation-Oscillation equation system(28)D0.5yt+yt=ut,y0=0,t∈0,50If the input function of the system is ut=t2+2t1.5/Γ2.5, the analytical solution of this system is yt=t2. When M=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.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:(29)y′′t+D1.8yt+yt=ut,y′0=y0=0,t∈0,4where ut=t2t-3+6t1.2/Γ2.2-6t0.2/Γ1.2+6t-6, the analytical solution of this system is yt=t2t-3. When M=3,4,5, 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 M increases. As M increases, the resulting coefficient matrix becomes large and may be singular [24].Figure 4
The analytical and numerical solutions for some different values ofM.Test Problem 4.3. Consider the following fractional control system:(30)D1.8yt+D1.5yt+y′t+yt=ut,y′0=0,y0=-1,t∈0,1where ut=t2+2/Γ1.2t0.2+2/Γ1.5t0.5+2t-1. The analytical solution of this system is yt=t2-1. When M=4,6,8, 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 M grows. With M 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 M = 4 M = 6 M = 8 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 asM grows.



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