# Stabilization of the FO-BLDCM Chaotic System in the Sense of Lyapunov

Authors: Ping Zhou; Rongji Bai; Hao Cai
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

Based on an integer-order Brushless DC motors (IO-BLDCM) system, we give a fractional-order Brushless DC motors (FO-BLDCM) system in this paper. There exists a chaotic attractor for fractional-order0.95<q≤1 in the FO-BLDCM system. Furthermore, using the Lyapunov direct method for fractional-order system, a control scheme is proposed to stabilize the FO-BLDCM chaotic system in the sense of Lyapunov. Numerical simulation shows that the control scheme in this paper is valid for the FO-BLDCM chaotic system.


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## Body

## 1. Introduction

Brushless DC motors (BLDCM) system [1–4] has several advantages over brushed DC motors, like elimination of ionizing sparks, overall reduction of electromagnetic interference, reduced noise, longer lifetime, increased efficiency and reliability, and so forth; BLDCM has been widely applied in positioning and actuation systems, motion control systems, radio controlled cars, and industrial automation design. Recently, some results have shown that the chaotic motion can be presented in BLDCM system. However, the chaotic motion in BLDCM system is not acceptable in practical situations, because it can destroy the stable operation of the BLDCM system and can lead to system malfunction in practical applications. So, BLDCM system stability is usually a prerequisite of practical application. Up to now, in order to control the chaotic motion in BLDCM system, some schemes have been presented [2–4].On the other hand, it has been recognized that many real-world physical systems can be described by fractional-order differential equations, such as the fractional-order telegraph system [5], the fractional-order heat conduction system [6], the fractional-order diffusion and superdiffusion system [7, 8], the fractional-order Chua system [9], the fractional-order Duffing system [10], the fractional-order cellular neural network [11], the fractional-order gyroscopes system [12], and the fractional-order microelectromechanical system [13]. Meanwhile, it is well known that many fractional-order systems exhibit chaotic behavior, such as the fractional Lorenz chaotic system [14], the fractional-order Chua chaotic system [9], the fractional-order Duffing chaotic system [10], the fractional-order Volta chaotic system [15], the fractional-order gyroscopes chaotic system [12], the fractional-order microelectromechanical chaotic system [13], and the fractional-order chaotic electronic circuit [16]. Furthermore, control of the fractional-order chaotic systems has been attracting more attention in recent years [17–20].Motivated by the above considerations, a FO-BLDCM system under loading conditions is presented. By numerical calculation, we find that the FO-BLDCM system exhibits a chaotic attractor, and we obtain the largest Lyapunov exponent of the FO-BLDCM system. Furthermore, based on the Lyapunov direct method for fractional-order system [21–23], we propose a control scheme to stabilize the FO-BLDCM chaotic system. The result in this paper shows that the FO-BLDCM chaotic system can be stable in the sense of Lyapunov.


## 2. A FO-BLDCM Chaotic System

Ge and Chang [2] proposed a mathematical model for a BLDCM system under loading conditions and found that the BLDCM system creates chaotic behavior. Based on the mathematical model in [2], we present a FO-BLDCM system model which is described as follows:(1)Dtq0Cx1=Vq-x1-x2x3+ρx3,Dtq0Cx2=Vd-δx2+x1x3,Dtq0Cx3=-TL+σx1-x3+ηx1x2,where 0<q≤1 is the fractional order and Dtq0Cxi=(1/Γ(1-q))∫0t(dxi(τ)/(t-τ)q)(i=1,2,3). Constants Vq, Vd are related to the fictitious inductance on the quadrature-axis and direct-axis, respectively; constant TL is related to the external load, Coulomb friction, cogging effect, and so forth; constants δ, η are related to the fictitious inductance on the quadrature-axis and direct-axis; constant σ is related to the inertia of rotator and the viscous damping coefficient; constant ρ is a free parameter. In this paper, we choose system parameters in FO-BLDCM system (1) as follows: Vq=0.168, Vd=20.66, TL=0.53, δ=0.875, η=0.26, σ=4.56, and ρ=60.By the improved version of the Adams-Bashforth-Moulton numerical algorithm [24] for fractional-order nonlinear system, the fractional-order system (1) can be discretized as follows:(2)x1n+1=x10+hqΓq+2Vq-x1pn+1-x1pn+1x3p+ρy3pn+1+∑j=0ncj,n+1Vq-x1j-x1jx3j+ρx3j,x2n+1=x20+hqΓq+2Vd-δx2pn+1+x1pn+1x3pn+1+∑j=0ncj,n+1Vd-δx2j+x1jx3j,x3n+1=x30+hqΓq+2-TL+σx1pn+1-x3pn+1+ηx1pn+1x2pn+1+∑j=0ncj,n+1-TL+σx1j-x3j+ηx1jx2j,x1pn+1=x10+1Γq·∑j=0nbj,n+1Vq-x1j-x1jx3j+ρx3j,x2pn+1=x20+1Γq∑j=0nbj,n+1Vd-δx2j+x1jx3j,x3pn+1=y30+1Γq·∑j=0nbj,n+1-TL+σx1j-x3j+ηx1jx2j,cj,n+1=nq+1-n-qn+1q,j=0n-j+2q+1+n-jq+1-2n-j+1q+1,1≤j≤n1,j=n+1,bj,n+1=hqqn-j+1q-n-jq,0≤j≤n.The error is(3)xitn-xin=ohβ,β=min⁡2,1+q,where h is step size.Recently, Zhou and Huang [25] introduced one numerical method to calculate the largest Lyapunov exponent (LLE) for the fractional-order nonlinear system. Now, we can calculate the largest Lyapunov exponent (LLE) of FO-BLDCM system (1) for difference fractional-order q. Figure 1 displays the LLE of FO-BLDCM system (1) with respect to the fractional-order q.Figure 1
The largest Lyapunov exponent of FO-BLDCM system (1) with respect to the fractional-order q.According to Figure1, the LLE of FO-BLDCM system (1) is larger than zero for 0.95<q≤1, which implies that chaotic behavior will emerge for 0.95<q≤1. Figure 2 shows a chaotic attractor of the FO-BLDCM system (1) for q=0.96, in which the LLE is 1.4972. Conversely, the LLE of FO-BLDCM system (1) is less than zero for q≤0.95, which implies that FO-BLDCM system (1) is stable. For example, the LLE is -0.4246 for q=0.94, and the evolution of state of x1, x2, and x3 is shown in Figure 3.Figure 2
A chaotic attractor in FO-BLDCM system (1) for q=0.96.Figure 3
Evolution of the state ofx1, x2, and x3 in FO-BLDCM system (1) with q=0.94.


## 3. Control of the FO-BLDCM Chaotic System

In this section, control of the FO-BLDCM chaotic system will be discussed.Theorem 1.
Give the controlled FO-BLDCM system as(4)Dtq0Cx1=Vq-x1-x2x3+ρx3,Dtq0Cx2=Vd-δx2+x1x3,Dtq0Cx3=-TL+σx1-x3+ηx1x2+ux1,x2and u(x1,x2)=[m1-ρ-σ-(η-1)x-2](x1-x-1)+(m2-x-1-ηx1)(x2-x-2) is a scalar controller, x-i(i=1,2,3) is the equilibrium point of FO-BLDCM system (1), and mi(i=1,2) is suitable constant. If m1=-2l1l2, m2=-2l3l4, and li(i=1,2,3,4) satisfies l1<1, l3<δ, l22+l42<σ, then the equilibrium point x-i(i=1,2,3) in controlled system (4) is stable in the sense of Lyapunov.Proof.
First, letyi(t)=(xi-x-i)(i=1,2,3). Then, the controlled system (4) can be changed as(5)Dtq0Cy1t=-y1t-x-3y2t+ρ-x-2y3t-y2ty3t,Dtq0Cy2t=x-3y1t-δy2t+x-1y3t+y1ty3t,Dtq0Cy3t=m1-ρ+x-2y1t+m2-x-1y2t-σy3t.Obviously, yi(t)=0(i=1,2,3) is the origin of controlled system (5).
Second, lety(t)=y1ty2ty3tT, whereT denotes the transpose for a matrix. Then, we have the following:(6)0.5Dtq0CytTyt-ytTDtq0Cyt=0.5∑i=13Dtq0Cyit2-∑i=13yitDtq0Cyit=∑i=131Γ1-q∫0tyiτt-τqdyiτ-∑i=131Γ1-q∫0tyitt-τqdyiτ=∑i=131Γ1-q∫0tyiτ-yitt-τqdyiτ=∑i=131Γ1-q∫0t0.5t-τqdyiτ-yit2=∑i=130.5Γ1-qyiτ-yit2t-τqτ=t-yi0-yit2tq-∫0tqyiτ-yit2t-τq+1dτ.Hence,(7)0.5Dtq0CytTyt-ytTDtq0Cyt=∑i=130.5Γ1-qyiτ-yit2t-τqτ=t-yi0-yit2tq-∫0tqyiτ-yit2t-τq+1dτ.Now, let us consider the first term in formula (7); we have(8)∑i=130.5Γ1-qyiτ-yit2t-τqτ=t=∑i=130.5Γ1-qlimτ→t⁡yiτ-yit2t-τq=∑i=130.5Γ1-qlimτ→t⁡2yiτ-yitdyiτ/dτqt-τq-1=∑i=131qΓ1-qlimτ→t⁡t-τ1-qyiτ-yit·dyiτdτ=0.According to (7)-(8), one can yield to(9)0.5Dtq0CytTyt-ytTDtq0Cyt=-∑i=130.5Γ1-qyi0-yit2tq+∫0tqyiτ-yit2t-τq+1dτ≤0.From inequality (9), one has(10)0.5Dtq0CytTyt≤ytTDtq0Cyt.
Finally, consider a positive definite Lyapunov function as follows:(11)Vt=0.5ytTyt.According to system (5) and inequality (10), we can obtain the fractional derivative of the Lyapunov function as follows:(12)Dtq0CVt≤ytTDtq0Cyt=∑i=13yitDtq0Cyit=-y12t-δy22t-σy32t+m1y1ty3t+m2y2ty3t.
Based on inequality (12), using the assumptions m1=-2l1l2, m2=-2l3l4, l1<1, l3<δ, and l22+l42<σ, we can obtain the following:(13)Dtq0CVt≤-l1y1t+l2y3t2-l3y2t+l4y3t2-1-l12y12t-δ-l32y22t-σ-l22-l42y32t≤0,∀yiti=1,2,3.
So, according to the results in [22, 23], inequality (13) implies that the origin of controlled system (5) is stable in the sense of Lyapunov. Therefore, the equilibrium point x-i(i=1,2,3) in controlled system (4) is stable in the sense of Lyapunov, which allows concluding the proof.Remark 2.
Using the assumptionsm1=-2l1l2, m2=-2l3l4, l1<1, l3<δ, and l22+l42<σ, we can obtain that the suitable constant mi(i=1,2) in Theorem 1 satisfies m12+δ-1m22<4σ.Remark 3.
According to controlled system (4), if controlled system (4) reached the equilibrium point x-i(i=1,2,3), then the scalar controller u(x1,x2) is removed.Next, we apply Theorem1 to stably control the equilibrium point x-i(i=1,2,3). It is easy to obtain that the fractional-order BDCM chaotic system (1) with q=0.96 has three real equilibrium points: p1=(-2.6658,59.7612,-11.8655), p2=(0.0481,23.6112,-0.00329), and p3=(2.6926,59.7852,11.7551). Taking the control parameters m1=1 and m2=2 and initial conditions x1=10, x2=20, and x3=30 for controlled fractional-order system (4), the simulation results are shown in Figures 4–6. Here, constant li(i=1,2,3,4) is l1=1/2, l2=-1, l3=2, and l4=-0.5. Figure 4 shows the evolution of the states in system (4), where (x-1,x-2,x-3)=(-2.6658,59.7612,-11.8655). This result indicates that system (4) can be stable in the equilibrium point p1; Figure 5 shows the evolution of the states in system (4), where (x-1,x-2,x-3)=(0.0481,23.6112,-0.00329). This result indicates that system (4) can be stable in the equilibrium point p2. Figure 6 shows the evolution of the states in system (4), where (x-1,x-2,x-3)=(2.6926,59.7852,11.7551). This result indicates that system (4) can be stable in the equilibrium point p3.Figure 4
Evolution of the state ofx1, x2, and x3 in controlled system (4) with q=0.96 for equilibrium point p1.Figure 5
Evolution of the state ofx1, x2, and x3 in controlled system (4) with q=0.96 for equilibrium point p2.Figure 6
Evolution of the state ofx1, x2, and x3 in controlled system (4) with q=0.96 for equilibrium point p3.According to Figures4–6, we can obtain that the proposed theorem is valid for FO-BLDCM chaotic system.


## 4. Conclusions

A FO-BLDCM system is proposed in this paper. The chaotic motion can be presented in the FO-BLDCM system for0.95<q≤1. The chaotic phase portraits for q=0.96 and the largest Lyapunov exponent with varying the fractional order are obtained by numerical calculation. Based on the Lyapunov direct method for fractional-order system, we proposed a control scheme to stabilize the FO-BLDCM chaotic system in the sense of Lyapunov. The simulation results show that the proposed scheme is effective.



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