control of a fractional-order arneodo system
TRANSCRIPT
Control of A Fractional-Order Arneodo System
Kun Zhang1, a, Hua Wang2,b and Huitao Wang3,c 1,2,3Engineering Research Center of Metallurgical Energy Conservation and Emission Reduction,
Ministry of Education Kunming University of Science and Technology, Yunnan, China
[email protected], [email protected], [email protected]
Keywords: Fractional-order Arneodo system; the fractional Routh-Hurwitz; Chaos; Feedback control
Abstract. In this work, stability analysis of the fractional-order Arneodo system is studied by using
the fractional Routh-Hurwitz criteria. Furthermore, the fractional Routh-Hurwitz conditions are
used to control chaos in the proposed fractional-order system to its equilibria. Based on the
fractional Routh-Hurwitz conditions and using specific choice of linear feedback controllers, it is
shown that the Arneodo system is controlled to its equilibrium points. Numerical results show the
effectiveness of the theoretical analysis.
Introduction
Studying chaos in fractional-order dynamical systems is an interesting topic. Many fractional-order dynamical systems behave chaotically such as the fractional-order Chua system [1], the fractional-order Lorenz system [2], the fractional-order Chen system [3], the fractional-order Rössler system [4], and the fractional-order Liu system [5]. Chaos control and synchronization in integer-order differential systems have been studied in depth [6, 7]. Recently, chaos control and synchronization of fractional differential systems start to attract increasing attention due to its potential applications in secure communication [8] and control processing [9], and many investigations have been devoted to achieve chaos control and synchronization in fractional-order chaotic and hyperchaotic systems [10−15].
In 1985, Arneodo et al. introduced the Arneodo chaotic model [16]. In 2005, Jiang discussed the control of Arneodo chaotic system by the Routh-Hurwitz theory [17]. And Lu investigated the chaotic behaviors of the fractional-order Arneodo system and studied synchronization of this system by using the linear and nonlinear control methods [18].
In this paper, we propose to eliminate the chaotic behaviors from the fractional-order Arneodo system by using the linear feedback control. The paper will be outlined as follows. In Section 2, the fractional derivatives and the fractional-order Arneodo system are introduced, and chaos in the fractional-order Arneodo system is presented. In Section 3, we are going to use the fractional Routh-Hurwitz conditions given in [9, 13, 20] to study the stability conditions in the fractional-order Arneodo system. Conditions for linear feedback control are obtained as well. The effect of fractional order on chaos control is shown. Finally, the conclusions are given in Section 4.
Fractional derivatives and chaos in the fractional-order Arneodo system
Fractional derivatives and the fractional- order Arneodo system. Although there are several
definitions for the fractional differential operator, the following definition is often used:
( )
*D ( ) ( )m my x J y xα α−= , α > 0, (1)
where m = [α], i.e., m is the first integer which is not less than α, y(m)
is the general m-order derivative, J
β is the β-order Riemann-Liouville integral operator which is expressed as follows:
1
0
1( ) ( ) ( ) ( )
( )
x
J z x x t z t d tβ β
β−= −
Γ ∫ , β ) (2)
The operator α*D is generally called “α-order Caputo differential operator” [19].
Advanced Materials Research Vols. 383-390 (2012) pp 4405-4412Online available since 2011/Nov/22 at www.scientific.net© (2012) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMR.383-390.4405
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The fractional-order Arneodo system is described by the following nonlinear differential equations:
1
1
2
2
3
3
3
,
,
,
q
q
q
q
q
q
d xy
dt
d yz
dt
d zbx cy dz ex
dt
=
=
= − − − +
(3)
where the fractional differential operator is the Caputo differential operator; i.e., */i iq q qd dt D= , i =
1,2,3; 0 < q1, q2, q3 ≤ 1; its order is denoted by q = (q1, q2, q3) here. Chaos in the fractional-order Arneodo system. The following differential equation:
* ( ) ( , ( ))qD y t f t y t= , 0 ≤ t ≤ T, ( ) ( )
0(0)k ky y= , k = 0, ···, m − 1, where m = [q],
is equivalent to the Volterra integral equation:
1( ) 1
00
0
1( ) ( ) ( , ( ))
! ( )
km tk q
k
ty t y t s f s y s ds
k q
−−
=
= + −Γ∑ ∫ . (4)
Diethelm et al. used the predictor-correctors scheme [22–25], based on the Adams-Bashforth-Moulton algorithm to integrate (4). By applying this scheme to the fractional-order Arneodo system,
and setting h = T/N, tn = nh, n = 0, 1, ···, N ∈ Z+, then (3) can be discretized as follows:
1 1
1 0 1 1, , 1
01 1
( ) ( )( 2) ( 2)
q q np
n n j n j
j
h hx x y y
q qα+ + +
=
= + +Γ + Γ + ∑ ,
2 2
1 0 1 2, , 1
02 2
( ) ( )( 2) ( 2)
q q np
n n j n j
j
h hy y z z
q qα+ + +
=
= + +Γ + Γ + ∑ ,
3
3
3
1 0 1 1 1 1
3
3
3, , 1
03
( )( 2)
( ),( 2)
qp p p p
n n n n n
q n
j n j j j j
j
hz z bx cy dz ex
q
hbx cy dz x
qα
+ + + + +
+=
= + − − − + +Γ +
− − − +Γ + ∑
where
1 0 1, , 1
01
1( )
( )
np
n j n j
j
x x yq
β+ +=
= +Γ ∑ ,
1 0 2, , 1
02
1( )
( )
np
n j n j
j
y y zq
β+ +=
= +Γ ∑ ,
3
1 0 3, , 1
03
1( )
( )
np
n j n j j j j
j
z z bx cy dz exq
β+ +=
= + − − − +Γ ∑ ,
1
1 1 1
, , 1
( )( 1) ,
( 2) ( ) 2( 1) ,
1,
i i
i i i
q q
i
q q q
i j n
n n q n
n j n j n jα
+
+ + ++
− − +
= − + + − − − +
0,
1 ,
1,
j
j n
j n
=
≤ ≤
= +
, , 1 (( 1) ( ) ), 0 , 1, 2, 3.i
i i
qq q
i j n
i
hn j n j j n and i
qβ + = − + − − ≤ ≤ =
The error estimate of the above scheme is maxj = 0, 1, ···, N |y (tj) – yh (tj)| = O (hp), in which p = min (2, 1 + qi) and qi > 0, i = 1, 2, 3.
4406 Manufacturing Science and Technology, ICMST2011
The fractional-order Arneodo system is studied in [18] for varying the derivative order q and the parameter d. It has been shown for q1 = q2 = q3 = q, the system has chaotic behavior when (q, d) = (0.9, 0.4), (q, d) = (0.8, −0.9) and (q, d) = (0.7, −1.73). And for q = 06, 0.5, no chaotic behavior is found which indicates that the lowest limit of the fractional order q for this fractional-order Arneodo system (3) to be chaotic is q = 0.6 ~ 0.7.
In this paper, let (b, c, d, e) = (−5.5, 3.5, 0.4, −1), q = (0.90, 0.90, 0.90), and initial condition is chosen as (−2, 1, 1) for (3), respectively. For theses parameters, system (3) exhibits chaotic behavior in Fig. 1.
Chaos control of the fractional-order Arneodo system
Fractional order Routh-Hurwitz conditions. Consider a three-dimensional fractional-order
system:
1
1
2
2
3
3
( )( , , ),
( )( , , ),
( )( , , ),
q
q
q
q
q
q
d x tf x y z
dt
d y tg x y z
dt
d z th x y z
dt
=
=
=
(5)
where q1, q2, q3 ∈ [0, 1). The Jacobian matrix of the system (5) at the equilibrium points is
/ / /
/ / /
/ / /
f x f y f z
g x g y g z
h x h y h z
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
(6)
The eigenvalue equation of the equilibrium point is given by the following polynomial:
( ) 3 2
1 2 3P a a aλ λ λ λ= + + + , (7)
and its discriminant D(P) is given as:
2 3 3 2
1 2 3 1 2 3 1 2 3( ) 18 ( ) 4 4 27D P a a a a a a a a a= + − − − , (8)
Theorem 1. If the eigenvalues of the Jacobian matrix (6) satisfy
|arg (λ)| ≥ πα/2, α = max (q1, q2, q3), (9)
namely all the roots of the polynomial equation (7) satisfy (9), and then the system is asymptotically stable at the equilibrium points.
The stability region of the fractional-order system with α is illustrated in Fig. 2 (in which σ, ω
refer to the real and imaginary parts of the eigenvalues, respectively, and j= 1− ). Using the results of [9, 20, 21], we have the following fractional Routh-Hurwitz conditions:
a) If D(P) > 0, then the necessary and sufficient condition for the equilibrium points to be locally asymptotically stable, is a1 > 0, a3 > 0, a1a2 − a3 > 0.
b) If D(P) < 0, a1 ≥ 0, a2 ≥ 0, a3 > 0, then the equilibrium point is locally asymptotically stable for α < 2/3. However, if D(P) < 0, a1 < 0, a2 < 0, α > 2/3, then all roots of (7) satisfy the condition |arg( λ )| < πα/2.
c) If D(P) < 0, a1 > 0, a2 > 0, a1a2 − a3 =0, then the equilibrium point is locally asymptotically
stable for all α ∈ [0, 1). d) The necessary condition for the equilibrium point, to be locally asymptotically stable, is a3 >
0. The proof for above conditions can be seen in detail from [9, 15, 22].
Advanced Materials Research Vols. 383-390 4407
The stability of equilibrium points. Let (b, c, d, e) = (−5.5, 3.5, 0.4, −1) and
3
0,
0,
0.
y
z
bx cy dz ex
=
=− − − + =
(10)
Solving (10) for its roots, we can get that (3) has three equilibrium points, which are respectively
described as follows: E1 (0, 0, 0), E2 ( 5.5 , 0, 0), E3 (− 5.5 , 0, 0). From system (3), E2 and E3 are
shown in Fig. 1 when initial condition is chosen as (−2, 1, 1). If the equilibrium point is E ( x� , y� , z� ), the Jacobian matrix of (3) is
0 1 0
0 0 1
3
J
b ex c d
= − + − − �
,
The characteristic equation of the Jacobian matrix J is
3 2 23 0d d b exλ λ λ+ + + − =� . (11)
Lemma 1. When the parameters (b, c, d, e) = (−5.5, 3.5, 0.4, −1), the equilibrium point E1 of (3)
is unstable for all α ∈ [0, 1). Proof. When (b, c, d, e) = (−5.5, 3.5, 0.4, −1), substituting coordinate of E1 into (11), we can get
the characteristic polynomial as follows
3 20.4 3.5 5.5 0λ λ λ+ + − = . (12)
The eigenvalues of (12) is λ1 = 1.0792, λ2 = −0.7396+2.1329i, λ3 = −0.7396−2.1329i. Here λ1 is a
positive real number. Therefore, the equilibrium point E1 is unstable for all α ∈ [0, 1). Lemma 2. When the parameters (b, c, d, e) = (−5.5, 3.5, 0.4, −1), if α < 0.8146, equilibrium
points E2 and E3 of (3) are stable. Proof. When (b, c, d, e) = (−5.5, 3.5, 0.4, −1), substituting coordinate of E2 or E3 into (11), we
can get the same characteristic polynomial as follows
3 20.4 3.5 11 0λ λ λ+ + + = . (13)
The eigenvalues of (13) are λ1 = −1.8138, λ2 = 0.7069+2.359i and λ3 = 0.7069−2.359i, λ1 is a negative real number, and that the arguments of λ2, 3 are 1.2796 and −1.2796, respectively. Therefore, if α < 0.8146, then system (3) is stable at equilibrium points E2 and E3.
Control of the fractional-order Arneodo system via linear feedback control. The controlled
fractional-order Arneodo system is given by:
1
1
2
2
3
3
1
2
3
3
( ),
( ),
( ),
q
s
q
q
s
q
q
s
q
d xy k x x
dt
d yz k y y
dt
d zbx cy dz ex k z z
dt
= − −
= − −
= − − − + − −
�
�
�
(14)
where E ( x� , y� , z� ) represents an arbitrary equilibrium point of system (3). k1, k2, k3 are feedback
control gains which can make the eigenvalues of the linearized equation of the controlled system (14) satisfy one of the above-mentioned Routh-Hurwitz conditions, then the trajectory of the controlled system (14) asymptotically approaches the unstable equilibrium point E ( x� , y� , z� ).
4408 Manufacturing Science and Technology, ICMST2011
Substituting the coordinate of E ( x� , y� , z� ) into (14), we get the Jacobian matrix as follows
1
2
2
3
1 0
0 1
3
k
J k
b ex c k d
− = − − + − − − �
, (15)
and the corresponding characteristic equation is
3 2
1 2 3 1 2 2 3 1 3
2
1 2 1 2 3 1 2 1
( ) ( ) (
) ( 3 )
0,
P k k k d k k k k k k
dk dk c k k k dk k ck b ex
λ λ λ
λ
= + + + + + + + +
+ + + + + + −
=
� (16)
and its discriminant is 2 3 3 2
1 2 3 1 2 3 1 2 3( ) 18 ( ) 4 4 27D P a a a a a a a a a= + − − − ,
where
1 1 2 3
2 1 2 2 3 1 3 1 2
2
3 1 2 3 1 2 1
,
,
3 .
a k k k d
a k k k k k k dk dk c
a k k k dk k ck b ex
= + + +
= + + + + + = + + + − �
(17)
Stabilizing the equilibrium point E1. When (b, c, d, e) = (−5.5, 3.5, 0.4, −1), k1 = 1, k2 = 2, and
k3 > 6.1286, we have D (P) > 0, a1 > 0, a3 > 0, a1 a2 > a3, all the eigenvalues are real and located in
the left half of the s-plane. So Routh-Hurwitz conditions are the necessary and sufficient conditions
for the existence of Theorem 1; When k1 = 2, −0.3125 < k2 < 7.0509, and k3 = 2, we have D(P) < 0,
a1 ≥ 0, a2 ≥ 0, a3 > 0, the eigenvalues satisfy (9).
Stabilizing equilibrium points E2 and E3 if any α ∈∈∈∈ [0, 1). When (b, c, d, e) = (−5.5, 3.5, 0.4,
−1), and k1 = k2 = k3 = k, k ≥ −0.1333, we can obtain a1 ≥ 0, a2 ≥ 0, a3 > 0 and D (P) < 0 from (17).
So all the real eigenvalues of (16) are negative, and the real parts of complex conjugate eigenvalues
are negative. Therefore, namely for any α ∈ [0, 1), all the eigenvalues satisfy (9), and the trajectory
of the controlled fractional-order system (14) is asymptotically stable at the equilibrium point E2.
Similarly, when k1 = k2 = k3 = k and k ≥ −0.1333 the controlled fractional-order system (14) is
asymptotically stable at the equilibrium point E3 for any α ∈ [0, 1).
Simulation results. The parameters are chosen as (b, c, d, e) = (−5.5, 3.5, 0.4, −1), (q1, q2, q3) =
(0.90, 0.90, 0.90) to ensure the existence of chaos in the absence of control. The initial state is taken as
(−2, 1, 1), the time step is 0.1 (s), and the control is active when t ≥ 80 (s) in order to make a
comparison between the behavior before activation of control and after it. We simulate the process of which system (3) stabilizes to the equilibrium point E1 (0, 0, 0) using the
feedback method, as Fig. 3 shows. When k1 = 1, k2 = 2 and k3 = 7, we have D (P) > 0, a1 > 0, a3 > 0, and a1 a2 > a3, and the three roots are λ1 = –0.5823, λ2 = –6.3626, and λ3 = –3.4551, so the controlled fractional-order system (14) is asymptotically stable at E1 (0, 0, 0), Fig. 3(a) displays the simulation result. When k1 = 2, k2 = 3, and k3 = 2, we have D(P) < 0, a1 ≥ 0, a2 ≥ 0, and a3 > 0, and the roots of P(λ) are λ1 = –1.0857, λ2 = –3.1571 + 2.1626i, and λ3 = –3.1571 – 2.1626i. And all the eigenvalues are located in the stable region of the fractional order system as given in (3). Hence, the fractional-order system (14) is asymptotically stable at E1 (0, 0, 0), as shown in Fig. 3(b).
And we also simulate the process of which system (3) stabilizes to equilibrium points E2 and E3 using the feedback method. Fig. 4(a) shows the stabilization of the equilibrium point E2 for k1 = k2 = k3 = 2. Fig. 4(b) shows the stabilization of the equilibrium point E3 for k1 = k2 = k3 = 3.
Conclusions
In this paper, we have studied the local stability of the equilibria using the fractional Routh-Hurwitz conditions. Analytical conditions for linear feedback control have been implemented, showing the effect of the fractional order on controlling chaos in this system. Simulation results have illustrated the effectiveness of the proposed chaos control method.
Advanced Materials Research Vols. 383-390 4409
Acknowledgments
The authors wish to thank the reviewers for their careful reading and providing some pertinent suggestions. K. Zhang wishes to thank Professor Fang for his help and constant encouragement. The work is partially supported by the National Natural Science Foundation of China (U0937604).
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-4 -2 0 2 4-5
0
5
Xm
Ym
(a)
-10 -5 0 5 10-4
-2
0
2
4
Zm
Xm
(b)
-10 -5 0 5 10-5
0
5
Zm
Ym
(c)
-5
0
5
-10
0
10-5
0
5
Xm
Zm
Ym
(d)
Fig. 1 Attractor of Arneodo system with a fractional order. Here the time step-length is 0.04, and
the first 100 points are removed (at plotting attractors). (a) Portrait of x vs y. (b) Portrait of x vs z.
(c) Portrait of y vs z. (d) Portrait of x vs y vs z.
Advanced Materials Research Vols. 383-390 4411
Fig. 2 Stability region of fractional system for q1 = q2 = q3.
0 50 100 150 200 250 300
-10
-5
0
5
10
15
t (Sec.)
x, y, z
X
Y
Z
(a)
0 50 100 150 200 250 300-10
-5
0
5
10
15
t (Sec.)
x, y, z
X
Y
Z
(b)
Fig. 3 Time responses for the states x, y and z of the controlled system (14) stabilizing the
equilibrium point E1.
0 50 100 150 200 250 300
-6
-4
-2
0
2
4
6
8
t (Sec.)
x, y, z
X
Y
Z
(a) Stabilizing the equilibrium point
E2.
0 50 100 150 200 250 300-6
-4
-2
0
2
4
6
8
t (Sec.)
x, y, z
X
Y
Z
(b) Stabilizing the equilibrium point E3.
Fig. 4 Time responses for the states x, y and z of the controlled system (14) stabilizing equilibrium
points E2 and E3
4412 Manufacturing Science and Technology, ICMST2011
Manufacturing Science and Technology, ICMST2011 10.4028/www.scientific.net/AMR.383-390 Control of a Fractional-Order Arneodo System 10.4028/www.scientific.net/AMR.383-390.4405