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Control Chaos in System with Fractional Order

Yamin Wang1, Xiaozhou Yin2, Yong Liu3* 1Basis Course of Lianyungang Technical College, Lianyungang, China

2Lianyungang Technical College, Lianyungang, China

3School of Mathematical Science, Yancheng Teachers University, Yancheng, China Email: *yongliumath@163.com

Received February 26,2012; revised March 18, 2012; accepted March 28, 2012

ABSTRACT

In this paper, by utilizing the fractional calculus theory and computer simulations, dynamics of the fractional order sys- tem is studied. Further, we have extended the nonlinear feedback control in ODE systems to fractional order systems, in order to eliminate the chaotic behavior. The results are proved analytically by stability condition for fractional order system. Moreover numerical simulations are shown to verify the effectiveness of the proposed control scheme.

Keywords: Chaos; Fractional Order System; Nonlinear Feedback Control

1. Introduction

Fractional calculus is a classical mathematical concept, with a history as long as calculus itself. It is a genera- lization of ordinary differentiation and integration to ar- bitrary order, and is the fundamental theories of frac- tional order dynamical systems. Fractional-order diffe- rential/integral has been applied in physics and engi- neering, such as viscoelastic system [1], dielectric pola- rization [2], electrode-electrolyte polarization [3] and electromagnetic wave [4], and so on.

The fractional order system and its potential appli- cation in engineering field become promising and attrac- tive due to the development of the fractional order cal- culus. Typically, chaotic systems remain chaotic when their equations become fractional. For example, it has been shown that the fractional order Chua’s circuit with an appropriate cubic nonlinearity and with an order as low as 2.7 can produce a chaotic attractor [5].

However, there are essential differences between ordi- nary differential equation systems and fractional order differential systems. Most properties and conclusions of ordinary differential equation systems cannot be extend- ed to that of the fractional order differential systems. Therefore, the fractional order systems have been paid more attention. Recently, many investigations were de- voted to the chaotic dynamics and chaotic control of fractional order systems [6-12].

In this paper, practical scheme is proposed to eliminate the chaotic behaviors in fractional order system by ex- tending the nonlinear feedback control in ODE systems to fractional-order systems. This paper is organized as

follows. In Section 2, the numerical algorithm for the fractional order system is briefly introduced. In Section 3, Dynamics of the fractional order system is numerically studied. In Section 4, general approach to feedback con- trol scheme is given, and then we have extended this control scheme to fractional order system, numerical results are shown. Finally, in Section 5, concluding com- ments are given.

2. Fractional Derivative and Numerical

Algorithm

There are two approximation methods for solving frac- tional differential equations. The first one is an improved version of the Adams-Bashforth-Moulton algorithm, and the rest one is the frequency domain approximation. The Caputo derivative definition involves a time-domain com- putation in which nonhomogenous initial conditions are needed, and those values are readily determined. In this paper, the Caputo fractional derivative defined in [13] is often described by

 

=  n

 

, > 0,

q n q

D f t Jf t q

n q J

when is the first integer that is not less than ,  is the α-order Riemann-Liouville integral operator which defined by

 

  

1

 

0 1

= t d ,

J f tt   f   

where  is the Gamma function, 0 < 1.

Now we consider the fractional order system [14] which is given by

*Corresponding author.

(2)

1 1 2 2 3 3 d d d d d d q q q q q q x t y t z t             = , = , = , x yz

y z a x

z xy

 

  

q q 1, = 1, 2,3 .

i

(1)

where i is the fractional order,

By exploiting the Adams-Bashforth-Moulton scheme [15], the fractional order system (1) can be discretized as followings:

0 < i

1 1 1

,

n n

j j j

y z

1

1 0

1 1

1, , 1 =0 1 = 2 2 q n n q n j n j h

x x x

q

h x y z

         q            

1 1 , n n j j a x 2

1 0 1

2 2

2, , 1 =0 2 = 2 2 q n n q n

j n j

j

h

y y y z

q h

y z a x

       q                

3

1 1 1

1 ,

q

n n

j j j

h

x y

z x y

      1 0 3 3 3, , =0 3 = 2 2 n n q n j n j

z z z

q h q           

 

1

= n n j j j ,

1 0 1, , 1

=0 1

n j

j

xx

 xy z q

 

1 0 2, , 1

1

= n ,

n j n j j j

y y

yza x =0

2 j

q

 

1

= n ,

n j n j j j

z z z x y

q

 

 

1

1 = 0,

1) 1 ,

1 = 1,

n n q n j

j n j n

 

1 0 3, , 1

=0 3 j



, , 1 1

=

i j n

q

qi i

1

1

2 2(

i

qi qi qi

n jn jn j

    



 

, , 1=

1

,

qi

qi qi

i j n i

h

n j n j

q

   

=

q

D X AX

0 j n i, = 1, 2,3.

3. Dynamic Analysis of the Fractional Order

System

Theorem 1: The fractional linear autonomous system

 

0 = 0

X X

 

is locally asymptotically stable if and only if

π

min > , = 1, 2, , .

2

i i

q

argin

*

=

Theorem 2: Suppose x x

 

= ,

q

D x f x

be an equilibrium point of a fractional nonlinear system

If the eigenvalues of the Jacobian matrix

* = = x x f A x  

 

satisfy π

min > , = 1, 2, , ,

2

i i

q

argin

*

= .

then the system is locally asymptotically stable at the equilibrium point x x

The system (1) has five equilibrium points:

0 1 2

3 4

= 0,0,0 , = , , , = , , ,

= , , , = , , .

S S S

S S                                              

2 4 2 2 4 2

= , = .

2 2

a aa a

     

= 2, = 5,a

where

 we obtain When

0 1 2 3 4

= 0,0,0 , = 4.0317,1.4142,5.7016 , = 4.0317, 1.4142,5.7016 ,

= 0.4961,1.4142, 0.7016 , = 0.4961, 1.4142, 0.7016 .

S S S S S      

First, we choose 0

to study, the eigen-

values of the Jacobian matrix are

= 0, 0, 0 S

1= 2= 2 and

 

3= 1.

 We can obtain arg

 

1 =arg

 

2 =π and

 

= 0. arg

S 1, 2

q q q3

3 According to Theorem 2, we can easily

conclude that the equilibrium 0 of system (1) is un- stable when and are all greater than zero.

1 = 4.0317,1.4142,5.7016

S and

We choose

= 4.0317, 1.4142,5.7016

S  

1= 4.0000, = 0.50000 4.243162 i

2 to study, the eigen-

values of the Jacobian matrix are

and

   

3= 0.50000 4.24316 .i We can obtain

 

 

1 =π, = 1.4535

 

argarg2 and

 

= 1.4535. arg  

1, 2

q q q3

3 According to Theorem 2, we can

easily conclude that when and are all less than 0.9253 1.4535 2,

π

S S1, 2

(3)

3

q 0.9253 S S1, 2

142, 0.7016

42, 0.7016

 

0 1.4

are all great than , the equilibrium of system (1) is unstable.

Finally , when choose

and

to study, the eigenva- lues of the Jacobian matrix are

3= .4

S

4 = 0 1.41

S

= 4 =

0.4961,1 

.4961,

1 .0000, 2 0.5000 1284i

 

.50000 1

  and

3= 0 .41284 .i

 

 

1 =π,arg

 

= 1.23

7834

S

q

0.7834, 0.92

= =q =

We can obtain and

3 According to Theorem 2, we can

easily conclude that when 1 and 3 are all great than , the equilibrium of system (1) is unstable.

arg  

06. arg  0.

= 2 1.2306 ,

q q q

, , S q q, q

0.7834

53 2

3, 4 S S

or

S S

= 0.911 q

In sum, there exists at least one stable equilibrium 1 2 3 and 4 of system (1), when 1 2 and 3 are all less than , i.e., the system (1) will be stabilized at one point

1 2 3 4 finally; when

1 2 and 3 are all greater than 0. , all the equilibriums of system (1) are unstable, the system (1) will exhibit a chaotic behaviour; when

i the problem will be complicated,

the system (1) may be convergent, periodic or chaotic. For example, when 1 2 3 the value of the largest Lyapunov exponent is 0.1653. Obviously, the fractional order system (1) is chaotic. When

1 2 3 the fractional order system (1) is not chaotic, but periodic orbits appear.

S S , q q q  , , S S

= 92 , 53 , = q q 0.91, q q

4. Feedback Control

Let us consider the fractional order system

 

= f x

,u t, ,

q

D x t (2) where x t

 

is the system state vector, and u t

 

the control input vector. Given a reference signal x t

 

,

 

the problem is to design a controller in the state feedback form:

 

t ,

= ,

u t g x

where g x t

 

, is the vector-valued function, so that the controlled system

 

q

D x t f

 

x t t,

 

= x g, ,

 

can be driven by the feedback control g(x, t) to achieve the goal of target tracking so we must have

limtt f x tx t = 0. Let x t

 

= e x

, = ,

be a periodic orbit or fixed point of the given system (2) with , then we obtain the system error

= 0 u

 

t =F e t

 

, , x

,

,0, .

q

D e

where and

 

 

,

F e t g x t tf xt

Theorem 3: If 0 is a fixed poi of the system (2) an

f x

nt

d the eigenvalues of the Jacobian matrix at the equili- brium point 0 satisfies the condition

 

π

m ni i > , = 1, 2, q

argi

i , ,

2 n

then the trajectory x t

 

of system (2) converge to x

 

t .

e fractional o ), we Let us consider th rder system (2 propose to stabilize unstable periodic orbit or fixed point, the controlled system is as follows:

1

dq x= x yz u     

1 1 2 2 2 3 3 3 , d d = , d d = , d q q q q q t y

y z a x u t

z

z xy u t             (3)

Since

x y z  , ,

is solution of system (1), then we have:

1

dq x

 

1 2 2 3 3 = , d d = , d

d = .

d q q q q q x yz t y

y z a x t z z xy t                       (4)

Subtracting (4) from (3) with notation, 1

e =x x e, 2= y y e, 3= z z, we obtain the system error 1 1 1 1 1 2 2

2 1 2

2 3 3 3 3 3 d = , d d = , d d = . d q q q q q q e

e yz yz u t

e

e ae zx zx u t

e

e xy xy u t                       (5)

We define the control function as follow

2

3 3

= ,

= .

u zx zx

u xy xy e

1= ,

u yz yz

             (6)

So the system error (5) becomes

1 1 2 2 2 1 2 3 3 3 3 d d = , d d

= 1 .

d q q q q q t e e ae t e e t           (7) 1 1 d = , q e e    

(4)

The Jacobian matrix of system (7) is

0 0

0 ,

0 0 1

a

 

 

 

so we have the eigenvalues 1=2= and 3= 1 .

  When > 1 all eigenvalues are real nega- tives, one has arg

 

i =π, therefore

 

> π , = 2,3, 2

i i

argi for all qi satisfies

2, = 1, 2,3,

q i it follows from Theorem 3 that the

1, q

0 < i <

trajectory x t

 

of system (2) converges to x t

 

and

5. Nume

ul

we give num ich prove the performance of the propo e oned in Section 2 we have

Mou ati

20, t

= 0.93, = 0.95

q q

0 S

= , = , = .

u yz u zx u xy z

The control can be started at any time according to our needs, so we choose to activate the control when

in order to make a comparison between the behavior before activation of control and after it.

For 1 2 and q3 = 0.98, unstable point has been stabilized, as shown in Figure 1, note that

1  2  3 

20,

t The control is activated

 

when and the evolution of x t y t z t,

   

,

= 22.5 t 0

S

= 0.93, = 0.95

q q q = 0,98,

1 S

= 0.92

q q =q = 0,97,

2 S

= = = 0,94,

q q q S3

= = = 0,96,

q q q S4

t 20

is chaotic, then when the control is started at we see that is rapidly stabilized.

For 1 2 and 3 the un-

stable point has been stabilized, as shown in Figure 2.

the control is completed.

rical Sim ation

In this section erical results wh

For 1 and 2 3 the unstable point has been stabilized, as shown in Figure 3.

For 1 2 3 the unstable point has been stabilized, as shown in Figure 4.

For 1 2 3 the unstable point has been stabilized, as shown in Figure 5.

sed schem . As menti

implemented the improved Adams-

Bashforth- lton algorithm for numerical simul on. When is less than , there is a chaotic behavior,

the equilibrium point S0 for q1 = 0.93, q2 = 0.95 and q3 = 0.98.

Figure 1. Stabilizing

[image:4.595.69.526.333.727.2]

Figure 2. Stabilizing the equilibrium point S1 for q1 = 0.93, q2 = 0.95 and q3 = 0.98.

[image:4.595.72.523.343.448.2]
(5)
[image:5.595.72.524.96.356.2]

Figure 4. Stabilizing the equilibrium point S3 for q1 = q2 = q3 = 0.94.

Figure 5. Stabilizing the equilibrium point S for q = q = q = 0.96. 4 1 2 3

but when the control is activated at = 20t , the five 2, 3

S S S S and pidly stabilized.

6. Conclusions

Chaotic phenomenon makes prediction impossible in the real world; then the deletion of this phenomenon from fractional order system is very useful, the main con- tribution of this paper is to this end.

In this paper, we investig rder applying the fractiona

al ord

h the r-

dback control scheme has been ex- actional order system. The results are

effectiveness of

to th

Theory of Viscoelasticity,” Journal of Applie, Vol. 51, No. 2, 1984, pp. 299-307. doi:10.1115/1.3167616

points 0, ,1 S4 are ra

[2] H. H. Sun, A. A. Abdelwahad and B. Onaral, “Linear Approximation of Transfer Function with a Pole of Frac- tional Order,” IEEE Transactions on Automatic Control, Vol. 29, No. 5, 1984, pp. 441-444.

doi:10.1109/TAC.1984.1103551

[3] M. Ichise, Y. Nagayanagi and T. Kojima, “An Analog Simulation of Noninteger Order Transfer Functions for Analysis of Electrode Process,” Journal of Electroana- cial Electrochemistry, Vol. 33, No. 2, 1971, pp. 253-265.

do ate the system with fractional

l calculus technique. Accord- o

ing to the stability theory of the fraction er system, dynamical behaviors of the fractional order system are analyzed, bot oretically and numerically. Furthe more, nonlinear fee

tended to control fr

proved analytically by stability condition for fractional order system. Numerically the unstable fixed points have been successively stabilized for different values of q q1, 2 and q3. Numerical results have verif

the proposed scheme.

7. Acknowledgements

This work is supported by the Qing Lan Project of Jiangsu Province under the Grant Nos. 2010 and the 333 Project of Jiangsu Province under the Grant Nos. 2011.

REFERENCES

[1] R. C. Koeller, “Application of Fractional Calculus ied the

e

lytical Chemistry and Interfa

i:10.1016/S0022-0728(71)80115-8

[4] O. Heaviside, “Electromagnetic Theory,” Chelsea, New York, 1971.

zo and H. K. Qammer, “Chaos

f the Fractional Lorenz System,” Physical Review Letters,

Vol. 91, No. 3 1.

doi:10.1103/Ph

[5] T. T. Hartley, C. F. Loren

in a Fractional Order Chua’s System,” IEEE Transactions on Circuits and Systems I, Vol. 42, No. 8, 1995, pp. 485- 490.

[6] I. Grigorenko and E. Grigorenko, “Chaotic Dynamics o

, 2003, Article ID: 03410 ysRevLett.91.034101

[7] C. Li and G. Chen, “Chaos in the Fractional Order Chen System and Its Control,” Chaos, Solitons and Fractals, Vol. 22, No. 3, 2004, pp. 549-554.

doi:10.1016/j.chaos.2004.02.035

[8] C. P. Li and G. J. Peng, “Chaos in Chen’s System with a Fractional Order,” Chaos, Solitons and Fractals, Vol. 22, No. 2, 2004, pp. 443-450.

doi:10.1016/j.chaos.2004.02.013

[9] Z. M. Ge and A. R. Zhang, “Chaos in a Modified van der

(6)

Pol System and in Its Fractional Order Systems,” Chaos,

Solitons and Fractals, Vol. 35, No. 2, 2007, pp. 1791- 1822 . doi:10.1016/j.chaos.2005.12.024

[10] C. G. Li and G. R. Chen, “Chaos and H

tional Order Rossler Equation,” Physica Ayperchaos in Frac-, Vol. 341, No. 10, 2004, pp. 55-61. doi:10.1016/j.physa.2004.04.113 [11] W. Zhang, S. Zhou, H. Li and H. Zhu, “Chaos in a Frac-

tional-Order Rossler System,” Chaos, Solitons and Frac- tals, Vol. 42, No. 3, 2009, pp. 1684-1691.

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[12] D. G. Varsha and B. Sachin, “Chaos in Fractional Or- dered Liu System,” Computers & Mathematics with Ap- plications, Vol. 59, No. 3, 2010, pp. 1117-1127.

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[13] M. Caputo, “Linear Models of Dissipation Whose Q Is Almost Frequency Independent-II,” Geophysical Journal of the Royal Astronomical Society, Vol. 13, No. 5, 1967, pp. 529-539. doi:10.1111/j.1365-246X.1967.tb02303.x

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[15] K. Diethelm, N. J. Ford and A. D. Freed, “A Predic- tor-Corrector Approach for the Numerical Solution,”

Figure

Figure 3. Stabilizing the equilibrium point S2 for q1 = 0.92 and q2 = q3 = 0.97.
Figure 4. Stabilizing the equilibrium point S3 for q1 = q2 = q3 = 0.94.

References

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