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Suppressing chaos for a class of fractional-order chaotic systems by adaptive integer-order and fractional-order feedback control
Accepted Manuscript Title: Suppressing chaos for a class of fractional-order chaotic systems by adaptive integer-order and fractional-order feedback control Author: Ruihong Li Wei Li PII: DOI: Reference:
S0030-4026(15)00576-8 http://dx.doi.org/doi:10.1016/j.ijleo.2015.07.024 IJLEO 55739
To appear in: Received date: Accepted date:
4-6-2014 7-7-2015
Please cite this article as: R. Li, W. Li, Suppressing chaos for a class of fractionalorder chaotic systems by adaptive integer-order and fractional-order feedback control, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.07.024 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Suppressing chaos for a class of fractional-order chaotic systems by adaptive integer-order and fractional-order feedback control
ip t
Ruihong Li , Wei Li
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School of Mathematics and Statistics, Xidian University, Xi’an 710071, PR China
Abstract. This paper is devoted to studying how to more effectively suppress chaos for a class
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of fractional-order nonlinear systems. By means of adaptive control theory for integer-order nonlinear system, we propose two simple and novel adaptive feedback methods to control chaos.
an
Rigorous theoretical proof is provided based on some essential properties of fractional calculus and Barbalat’s Lyapunov-like stability theorem. It is discovered that both fractional-order
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feedback controller and integer-order one can guide chaotic trajectories to the unstable equilibrium point. To display the feasibility and validity of presented methods, some typical fractional-order chaotic systems have been chosen as numerical illustration. Furthermore, by comparing two
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stable and more flexible.
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different control techniques, one can find the fractional-order feedback control algorithm is more
Keywords: Fractional-order chaotic system; Integer-order feedback; Fractional-order feedback;
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Suppress chaos; Barbalat’s lemma
1. Introduction
Fractional calculus is a mathematical subject with more than 300 years old history,
but its application in physics and engineering has just started [1-3]. One possible explanation for such unpopularity is that there are many different definitions of fractional derivative, another one is fractional derivative has no evident geometrical interpretation due to its non-local property. However, over the past decade, a growing body of research suggests that some practical systems could display fractional order dynamics, such as viscoelasticity [4], dielectric polarization [5], electrode-electrolyte
Corresponding author. E-mail address: [email protected] (R.-h. Li) 1
Page 1 of 22
polarization [6], electromagnetic waves [7], quantitative finance [8], and quantum evolution of complex systems [9]. The most significant advantage of fractional calculus is it can provide an excellent instrument for the description of various materials and processes with long memory and hereditary property [10]. In recent
ip t
years, research on the dynamics of fractional-order systems has attracted more and
more attention. According to the Poincaré-Bendixson theorem [11], an integer-order
cr
nonlinear system must have a minimum order of three, which could make chaos appear. Nevertheless, the fractional-order nonlinear system is not the case. Numerous
us
studies have shown that plenty of three-dimensional systems of order 2 + with 0 < < 1 can exhibit chaotic behavior with appropriate system parameter, such as
an
fractional-order Lorenz families system [12-14], fractional-order Rössler system [15], fractional-order Arneodo system [16], fractional-order Newton-Leipnik system [17],
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fractional-order Genesio-Tesi system [18], fractional-order Liu system [19], fractional-order Chua’s system [20] and so on. In addition, some non-autonomous
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fractional-order systems [21-23] and high-dimensional fractional-order systems [24-27] could also display chaotic or hyper-chaotic behavior.
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Normally, chaos is considered a troublemaker. To avoid troubles arising from
Ac ce p
unusual behaviors of a chaotic system, chaos control has gained increasing attention in the past 20 years. An important objective of a chaos controller is to suppress chaotic oscillations completely or reduce them to regular oscillations. For fractionalorder chaotic system, there are two main types of controller, integer-order controller [28-35] and fractional-order controller [36-37]. Up to now, there have been a variety of control methods in the case of integer-order controller, such as linear feedback [28-29], nonlinear feedback [30-31], backstepping technique [32-33], sliding mode [34-35] and so on. However, despite the large amount of efforts, many key issues remain unresolved. The most urgent problem is how to design a general, efficient and physically available controller to suppress chaos. Linear feedback method is simple and practical, but it is difficult to determine the suitable feedback coefficient. Thereby, numerical calculation methods have to be used, which lead to linear feedback is only applicable to individual systems. And nonlinear feedback approach and backstepping 2
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technique have the same problem. Sliding mode controller is successful for some fractional-order systems, but, chattering phenomenon is the inherent vice. On the other hand, application of fractional calculus to achieve control has just begun and there are lots of problems worthy to study. Motivated by above discussions, we
suppress chaos for a class of fractional-order nonlinear systems.
ip t
attempt to put forward new fractional-order controller and integer-order controller to
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Adaptive feedback control [38] is a mature and effective method to suppress chaos
for integer-order chaotic systems. Compared with linear feedback method, adaptive
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feedback approach does not need to know feedback coefficients in advance. Hence, we will extend adaptive feedback method to suppress chaos for fractional-order
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chaotic systems. The rest of this paper is organized as follows. In Section 2, basic definitions in fractional calculus, properties and numerical algorithms are introduced.
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In Section 3 and 4, two new controllers are proposed to suppress chaos for a class of fractional-order chaotic systems. The stability analysis of the closed-loop system and
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numerical simulations of some typical fractional-order chaotic systems are included simultaneously. Section 5 is devoted to comparing two control algorithms and some
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conclusions are drawn is Section 6.
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2. Brief introduction to fractional calculus 2.1 Definitions of fractional derivative and its properties The operator t D0 , called differ-integral operator, is commonly used in fractional
calculus as notation for taking both fractional derivative and fractional integral in a single expression. This operator is defined as follows: d 0, dt , 0, t D0 1, t 0 (d ) , 0.
(1)
At present, there exist many different definitions of fractional derivative [1-3]. Among them, Grünwald- Letnikov, Riemann-Liouville and Caputo definitions are
3
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more reputed than others. The simplest definition which is also the easiest one to use is the Riemann-Liouville definition. This definition is described as: t D0 f (t )
d n n [ J 0 f (t )], dt n
(2)
t 1 (t ) 1 f ( )d , 0, 0 ( )
(3)
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J 0 f (t )
cr
fractional integral operator, which is defined as follows:
ip t
where n is the smallest integer greater than , i.e., n 1 n. J 0 () is the
and () is the Gamma function.
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Riemann-Liouville derivative t D0 is a continuous operator on , i.e. t D0 can bridge all the gaps among the integer derivatives and the integer integrals. So
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Riemann-Liouville derivative is a good and nature generalization of classical calculus [39]. Thereupon, one will select Riemann-Liouville definition in subsequent research.
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Throughout this paper, we use the notation D as the simplified form t D0 and its
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related properties are described below:
(1) For 1, the operation D1 f (t ) coincides with the ordinary derivative
Ac ce p
df (t ) . dt
(2) For 0 1, the following equation holds: D f (t )
1 d t (t ) f ( ) d 0 (1 ) dt
(4)
(3) Similar to integer-order calculus, fractional calculus are linear operation D [af (t ) bg (t )] aD f (t ) bD g (t ).
(5)
2.2 Numerical methods of fractional derivative For calculating fractional derivative numerically, frequency domain approximation algorithm and Adams-Bashforth-Moulton algorithm are the two most commonly used methods. The standard definitions of fractional differential-integral equation do not allow direct implementation in time domain. As a result, a method for approximate 4
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solution of fractional differential equation in frequency domain was developed in Ref. [40]. This approximation method is simple and convenient, which has been adopted in Refs. [14,16,18,31,33]. However, as the extreme sensitivity to initial conditions of chaotic behavior, above method is not reliable in the study of fractional-order chaotic
ip t
system, which has been already verified in Ref. [41]. Therefore, in this paper, one will adopt the generalized Adams-Bashforth-Moulton method [42], which derives analytic
cr
expression of fractional differential equation then numerically iterates the formula.
With good numerical stability, it has been applied to the research of chaotic behavior
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for fractional-order systems.
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Remark 1. || || , || ||1 and || ||2 respectively denote the -norm, 1-norm and 2-norm of vector, which is defined as follows: n
n
|| x || max | xi |, || x ||1 | xi |, || x ||2 ( xi2 )1 2 , x R n . i =1
(6)
i =1
M
1i n
3. Adaptive fractional-order feedback control
te
system
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Consider the chaos suppression scheme for n-dimensional fractional-order chaotic
Ac ce p
D x(t ) f ( x, t ) u ( x, t ),
(7)
where 0 1. Here, x R n is the state vector and
f ( x, t ) : R n R n is a
nonlinear chaotic vector function, which satisfies the following inequality:
| f i ( x) f i ( y ) | l max | x j y j |,
i, j 1,, n.
(8)
where l is a positive constant. Remark 2. Such condition may be called as uniform Lipschitz condition, which can
ensure the existence and uniqueness of fractional-order dynamical system [43]. It is easy to verify that most fractional-order chaotic systems [12-20, 24-27] satisfy this condition. Suppose that x* R n is an unstable equilibrium point of uncontrolled fractionalorder chaotic system and u ( x, t ) is the undetermined controller. For this purpose,
5
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one defines control error as e x x* , then error system can be obtained as follows: D e D x D x* f ( x, t ) f ( x* , t ) u ( x, t ).
(9)
3.1 Adaptive control algorithm
ip t
Theorem 1. If the controller in (9) is given by
ui (t ) ki ei ki ( xi xi* ), i 1,, n.
(10)
cr
where ki (1, , n) are feedback strength which are updated according to the
D ki (t ) i ei2 i ( xi xi* ) 2 , i 1, , n
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following rules:
(11)
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where i (i 1, , n) are arbitrary positive constants. Then, the controlled fractional -order chaotic system (7) could be guided to the unstable equilibrium point x* .
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Remark 3. Because state variables of chaotic system are bounded, here we suppose control function k (e, t ) meets the following condition: mi | ki (e, t ) | M i , i 1,, n.
d
(12)
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Lemma 1. (Barbalat lemma [44]). If (t ) : R R is a uniformly positive function t
for t 0 and if the integral lim ( ) d exists and is finite, then lim (t ) 0.
Ac ce p
t
t
0
Proof. Applying the controller (10) to the error system (9) results in the following
system:
D e D x D x* f ( x, t ) f ( x* , t ) ke.
(13)
Now, we introduce the following Lyapunov function:
V (t , e(t ))
where L1
1 n 1 2 L1 n 1 1 2 J 0 ki J 0 ei 2 2 i 1 i 1 i
(14)
( M l )n , M max{| M i |}, m min{| mi |}. Taking the time derivative of 1i n 1 i n m
V along the trajectories of error system (13), one has
6
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n n 1 V ( J 01- ei )( D0 ei ) L1 ( J 01- ki )( D0 ki ) i 1 i 1 i n t t 1 n - - 2 ( 0 (t ) ei ( ) d )( f i ( y ) fi ( x) ki ei ) L1 ( 0 (t ) ki ( ) d )ei (1 ) i 1 i 1
n t 1 n t - - 2 ( 0 (t ) | ei ( ) | d ) | ( f i ( y ) - f i ( x)) ki ei | L1 ( 0 (t ) ki ( )d )ei (1 ) i 1 i 1
n t t 1 n - - 2 || e || ( 0 (t ) d ) | ( fi ( y ) f i ( x)) ki ei | L1 mi ( 0 (t ) d )ei (1 ) i 1 i 1
n n n t1- [ || e || | f i ( y ) f i ( x) | || e || | ki ei | L1 mi ei2 ] (2 ) i 1 i 1 i 1
t1- [l || e || (2 )
i 1
1 j n
j
x j | M || e ||
n
| e i 1
i
n
| L1 m ei2 ] i 1
1-
t1- [nl || e ||2 Mn || e ||2 L1 m || e ||2 ] (2 )
t 1- [( M l )n L1 m] || e ||2 0, (2 )
t1- [( M l )n L1 m] || e ||22 (2 )
M
an
= eT P (t )e (t ) 0,
t1- [ L1m ( M l )n]I (2 )
d
P(t )
where
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t [nl || e ||2 M || e || || e ||1 L1 m || e ||22 ] (2 )
(15)
cr
n
max | y
ip t
is a positive definite matrix. Now
te
integrating (15) from zero to t yields t
V (t ) ( ) d V (0)
Ac ce p
0
(16)
and above equation means that
t
0
( ) d V (0)
(17)
Due to V 0, as t approach to infinite, the above integral is always less than or t
equal to V (0), so lim ( ) d exists and is finite. Based on Lemma 1, one has t
0
lim (t ) lim eT P(t )e 0 lim e(t ) 0. t
t
t
(18)
Then, state trajectories of error system (13) will converge to the zero point. That is, fractional-order chaotic system (7) could be controlled to the unstable equilibrium point x* . 3.2 Numerical simulation In this section, two groups of fractional-order chaotic systems are concretely 7
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presented to show the correctness and effectiveness of proposed scheme. Example 1. Consider the fractional-order Lorenz system as follows: d x1 a ( x2 x1 ), dt d x2 cx1 x2 x1 x3 , dt d x3 bx3 x1 x2 , dt
cr
ip t
(19)
8 where a, b, c 0, 0 1. Especially, when (a, b, c) (10, , 28) and 0.995, the 3
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fractional-order Lorenz system has a chaotic attractor [12] as depicted in Fig. 1.
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System (19) has three equilibrium points E1 (0, 0, 0), E2,3 (± 6 2, ± 6 2, 27), which are all unstable with 0.995. Initial values are X 0 = (0.001, - 0.004,0.002) such
from k0 = (0.005, - 0.003, 0.006)
M
that fractional-order Lorenz system is capable of chaos. The adaptive control starts and feedback intensity are i 1.0(i 1, 2,3).
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Numerical results are shown in Fig. 2, from which one can see the chaotic orbits of
te
system (19) will be stabilized to E1 , E2 and E3 respectively in a very short time. Example 2. Consider the fractional-order Rössler system as follows:
Ac ce p
d x1 x2 x4 , dt d x2 x1 ax2 x3 , dt d x3 bx3 cx4 , dt d x4 d x1 x4 , dt
(20)
where a, b, c, d 0, 0 1. Set a = 0.32, b = 0.05, c = 0.5, d = 3.0 and 0.95, the fractional-order Rössler system has a hyper-chaotic attractor [15] as depicted in Fig. 3. System (20) has two equilibrium points E1,2 (±
11 6 5 6 25 6 5 6 ,± , , ), 5 22 11 22
which are all unstable with 0.95. Here, we only consider the stabilization of 8
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fixed point E1. Initial values are X 0 = (- 25.0,5.0, 20.0, - 8.0) such that fractionalorder Rössler system is capable of hyper-chaos. The adaptive control starts from k0 = (- 5.0,7.0, - 8.0,6.0) and feedback intensity are i 1.0(i 1, 2,3, 4). Numerical
ip t
results are shown in Fig. 4, from which one can see the chaotic orbits of system (20)
cr
will be stabilized to E1 in a very short time.
4. Adaptive integer-order feedback control
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In this section, adaptive integer-order feedback control technique will be extended to suppress chaos for fractional-order chaotic system (7). Thereupon, one has the
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following conclusion: 4.1 Adaptive control algorithm
strength are changed as follows:
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Theorem 2. If the controller (10) is still adopted and the update rules of feedback
are arbitrary positive constants. Then, the controlled
te
i (i 1, , n)
where
(21)
d
D1ki (t ) i ei2 i ( xi xi* ) 2 , i 1, , n
fractional-order chaotic system (7) could still be guided to the unstable equilibrium
Ac ce p
point x* .
Proof. Here, fractional-order stability theory is employed as usual. We introduce
another Lyapunov function as follows:
V (t , e(t ))
(2 ) n 1 2 L2 J 0 ei 2 2t1 i 1
n
1
i 1
ki2
(22)
i
where L2 is a constant bigger than ( M l )n, and the definition of M is the same
as last section. Taking the time derivative of V , one has n (2 ) L V 1 ( J 01 ei )( D0 ei ) 2 t 2 i 1
n
1
i =1
ki2
i
n (2 ) 1 1 ( (t ) ei ( ) d )( f i ( x) f i ( x* ) ki ei ) L2 ei2 (1 ) 0 t i 1 i 1 n
(23)
t
Similar to the derivation in previous section, we have the following conclusion: 9
Page 9 of 22
V [( M l )n L2 ] || e ||22 0.
(24)
Then, based on the Lyapunov stability theory [45], state trajectories of error system (13) will converge to the zero point. That is, fractional-order chaotic system (7) will
ip t
be controlled to the unstable equilibrium point x* . 4.2 Numerical simulation
cr
In this section, the other two groups of fractional-order chaotic systems are concretely presented to show the correctness and effectiveness of proposed scheme.
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Example 3. Consider the fractional-order Chua’s system as follows:
(25)
M
where
an
d x1 a ( x2 x1 f ( x1 )), dt d x2 x1 x2 x3 , dt d x1 bx2 , dt
1 f ( x) m1 x(t ) (m0 m1 )(| x(t ) 1| | x (t ) 1|). Here, all parameters are 2
d
positive and 0 1. Especially, when (a, b, m0 , m1 ) (10,14.87, 1.27, 0.68) and
te
0.975, the fractional-order Chua’s system has a chaotic attractor [20] as depicted
Ac ce p
in Fig. 5.
System (25) only has one equilibrium point E (0, 0,0), which is unstable with
0.975. Initial values are X 0 = (0.006,0.002, - 0.005) such that fractional-order
Chua’s
system
is
capable
of
chaos.
The adaptive
control
starts
from
k0 = (- 0.003, 0.004, - 0.006) and feedback intensity is i 1.0(i 1, 2,3). Numerical
results are shown in Fig. 6, from which one can see the chaotic orbits of system (25) will be stabilized to E in a very short time. Example 4. Consider the fractional-order Genesio-Tesi system as follows:
10
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d x1 x2 , dt d x2 x3 , dt d x3 ax1 bx2 cx3 dx12 , dt
ip t
(26)
where a, b, c, d 0, 0 1. Set a = 1.0, b = 1.1, c = 0.44, d = 1.0 and 0.996, the
cr
fractional-order Genesio-Tesi system has a chaotic attractor [18] as depicted in Fig. 7.
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System (26) has two equilibrium points E1 (0, 0, 0), E2 (1, 0, 0), which are all unstable with 0.996. Initial values are X 0 = (- 0.001, 0.005, 0.002) such that fractional-
an
order Genesio-Tesi system is capable of chaos. The adaptive control starts from k0 = (6.0, - 4.0, - 3.0) and feedback intensity is i 1.0(i 1, 2,3). Numerical results
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are shown in Fig. 8, from which one can see the chaotic orbits of system (26) will be
d
stabilized to E1 and E2 respectively in a very short time.
te
5. Comparison between two control schemes Through above studies, it is easy to find both integer-order control approach and
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fractional-order control technique can suppress chaos successfully. In this section, we try to explore the differences between these two feedback methods in control performance.
In the following numerical calculations, two control methods are simultaneously
applied to fractional-order chaotic systems in Examples 1-4, where system parameters are selected as the same as Figs .2, 4, 6 and 8. Take state variable x1 for example, the
comparison diagrams are plotted in Fig. 9. Let denotes the distance for state
variable x1 converges x1* at t = 20 s, which measures the convergence speed of control algorithm. For different fractional-order chaotic systems, 1 indicates the integer-order controller and 2 indicates the fractional-order controller. From Fig .9 (a, b, d) and Table 1 (1, 2, 4), it is clear that two control methods make little difference 11
Page 11 of 22
in control effect. But from Fig. 9 (c) and Table 1 (3), one can see the controlled fractional-order chaotic system could more quickly attain control target with integer-order feedback controller than fractional-order one. It seems that the integer-order feedback controller has better control performance.
ip t
But as a new emerging research field, the fractional-order controller has greater researching significance and more application prospects. For instance, in integer-order
cr
controller, only feedback coefficient can be used as adjustable parameter. However, for fractional-order case, there exists another adjustable parameter .
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Take suppressing chaos for Chua’s system as an example, Fig. 10 exhibits time
an
histories for state variable x1 with different fractional order . Obviously as increases, the convergence speed is faster and faster. On the other hand, fractional-order system is at least as stable as their integer-order counterpart. Hence,
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the controlled system is more stable with fractional-order feedback controller than
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6. Summary and discussion
d
integer-order one.
Even though chaos dynamics on fractional-order systems has been extensively
Ac ce p
discussed, the study on suppressing chaos for fractional-order chaotic system is still at its initial stage. To the best of our knowledge, there have been only a few literatures focusing on chaos suppression for a class of fractional-order chaotic systems. In contrast, our work proposed a general method which can guide chaotic orbits of fractional-order chaotic system to the unstable equilibrium. Based on Lyapunov stability theorem of fractional-order dynamical system, adaptive integer-order feedback and fractional-order feedback controllers are designed. Meanwhile, four groups of numerical examples are provided to show the effectiveness of presented methods. It is worth noting that two new control techniques are simple and easy to implement, and what’s more they could be applied to almost all of known fractionalorder chaotic systems. Further, it is demonstrated that fractional-order controller has the same control effect with integer order one in most cases. And, fractional-order
12
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controller has more parameters to adjust control performance. Acknowledgement This work was supported by the National Natural Science Foundation of China
ip t
(Grant No. 11202155, 11302157) and the Fundamental Research Funds for the
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Central University (Grant No. JB140710) References
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[1] I. Podlubny, Fractional differential equations, Academic Press, New York, 1999. [2] P.L. Butzer, U. Westphal, An introduction to fractional calculus, World Scientific, Singapore, 2000.
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[3] R. Hilfer, Applications of fractional calculus in physics, World Scientific, New Jersey, 2001. [4] R.L. Bagley, R.A. Calico, Fractional order state equations for the control of viscoelastically damped structures, J. Guid. Control Dyn. 14 (1991) 304-311.
[5] H.H. Sun, A.A. Abdelwahad, B. Onaral, Linear approximation of transfer function with a
M
pole of fractional order, IEEE Trans. Automat. Control. 29 (1984) 441-444.
[6] M. Ichise, Y. Nagayanagi, T. Kojima, An analog simulation of noninteger order transfer functions for analysis of electrode process, J. Electro-Anal. Chem. 33 (1971) 253-265.
te
d
[7] O. Heaviside, Electromagnetic Theory, Chelsea, New York, 1971. [8] N. Laskin, Fractional market dynamics, Phys. A 287 (2000) 482-492. [9] D. Kusnezov, A. Bulgac, G.D. Dang, Quantum Lévy processes and fractional kinetics, Phys. Rev. Lett. 82 (1999) 1136-1139.
[10] L.P. Chen, Y. Chai, R.C. Wu, Linear matrix inequality criteria for robust synchronization of
Ac ce p
uncertain fractional-order chaotic systems, Chaos 21 (2011) 043107-13.
[11] M.W. Hirsch, S. Smale, Differential equations: dynamical systems and linear algebra, Academic Press, New York, 1974.
[12] I. Grigorenko, E. Grigorenko, Chaotic dynamics of the fractional Lorenz system, Phys. Rev. Lett. 91 (2003) 034101-4.
[13] C.G. Li, G.R. Chen, Chaos in the fractional order Chen system and its control, Chaos Soliton Fract. 22 (2004) 549-554.
[14] J.G. Lu, Chaotic dynamics of the fractional-order Lü system and its synchronization, Phys. Lett. A 354 (2006) 305-311.
[15] C.G. Li, G.R. Chen, Chaos and hyperchaos in the fractional order Rössler equations, Phys. A 341 (2004) 55-61.
[16] J.G. Lu, Chaotic dynamics and synchronization of fractional-order Arneodo’s systems, Chaos Soliton Fract. 26 (2005) 1125-1133.
[17] L.J. Sheu, H.K. Chen, J.H. Chen, L.M. Tam, W.C. Chen, K.T. Lin, Y. Kang, Chaos in the Newton-Leipnik system with fractional order, Chaos Soliton Fract. 36 (2008) 98-103.
[18] J.G. Lu, Chaotic dynamics and synchronization of fractional-order Genesio-Tesi systems, Chin. Phys. 14 (2005) 1517-1521.
[19] X.Y. Wang, M.J. Wang, Dynamic analysis of the fractional-order Liu system and its 13
Page 13 of 22
synchronization, Chaos 17 (2007) 033106-6.
[20] C.P. Li, W.H. Deng, D. Xu, Chaos synchronization of the Chua system with a fractional order, Phys. A 360 (2006) 171-185.
[21] L.J. Sheu, H.K. Chen, J.H. Chen, L.M. Tam, Chaotic dynamics of the fractionally damped Duffing equation, Chaos Soliton Fract. 32 (2007) 1459-1468. Chaos Soliton Fract. 35 (2008) 188-198.
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[22] J.H. Chen, W. C. Chen, Chaotic dynamics of the fractionally damped van der Pol equation, [23] Z.M. Ge, C.X. Yi, Chaos in a nonlinear damped Mathieu system, in a nano resonator system and in its fractional order systems, Chaos Soliton Fract. 32 (2007) 42-61.
cr
[24] W.M. Ahmad, Hyperchaos in fractional order nonlinear systems, Chaos Soliton Fract. 26 (2005) 1459-1465.
[25] C.X. Liu, L. Liu, Circuit implementation of a new hyperchaos in fractional-order system,
us
Chin. Phys. B 17 (2008) 2829-2836.
[26] X.Y. Wang, J.M. Song, Synchronization of the fractional order hyperchaos Lorenz systems
an
with activation feedback control, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 3351 -3357.
[27] A.K. Matouk, Stability conditions, hyperchaos and control in a novel fractional order hyperchaotic system, Phys. Lett. A 373 (2009) 2166-2173.
[28] X.Y. Wang, Y.J. He, M.J. Wang, Chaos control of a fractional order modified coupled
M
dynamos system, Nonlinear Dyn. 71 (2009) 6126-6134.
[29] A.E. Matouk, Chaos, feedback control and synchronization of a fractional-order modified
d
autonomous Van der Pol-Duffing circuit, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 975-986.
[30] Z.M. Odibat, Adaptive feedback control and synchronization of non-identical chaotic
te
fractional order systems, Nonlinear Dyn. 60 (2010) 479-487.
[31] J. Yang, D.L. Qi, The feedback control of fractional order unified chaotic system, Chin. Phys.
Ac ce p
B 19 (2010) 020508-4.
[32] W.M. Ahmad, A.M. Harb, On nonlinear control design for autonomous chaotic systems of integer and fractional orders, Chaos Soliton Fract. 18 (2003) 693-701.
[33] X. Gao, J.B. Yu, Chaos and chaotic control in a fractional-order electronic oscillator, Chin. Phys. 14 (2005) 908-913.
[34] C. Yin, S.M. Zhong, W.F. Chen, Design of sliding mode controller for a class of fractionalorder chaotic systems, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 356-366.
[35] Z. Wang, X. Xia, H. Shen, Control of an uncertain fractional order economic system via adaptive sliding mode, Neurocomputing 83 (2012) 83-88.
[36] M.S. Tavazoei, M. Haeri, Chaos control via a simple fractional-order controller, Phys. Lett. A 372 (2008) 798-807.
[37] M.S. Tavazoei, M. Haeri, S. Bolouki, M. Siami, Using fracional-order integrator to control chaos in single-input chaotic systems, Nonlinear Dyn. 55 (2009) 179-190.
[38] D.B. Huang, Adaptive-feedback control algorithm, Phys. Rev. E 73 (2006) 066204-8. [39] W.H. Deng, C.P. Li, The evolution of chaotic dynamics for fractional unified system, Phys. Lett. A 372 (2008) 401-407.
[40] A. Charef, H.H. Sun, Y.Y. Tsao, B. Onaral, Fractal system as represented by singularity function, IEEE Trans. Automat. Control. 37 (1992) 1465-1470.
14
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[41] M.S. Tavazoei, M. Haeri, Limitations of frequency domain approximation for detecting chaos in fractional order systems, Nonlinear Anal. TMA 69 (2008) 1299-1320.
[42] K. Diethelm, N.J. Ford, A.D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn. 29 (2009) 3-22.
[43] K. Diethelm, N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002) 229-248.
ip t
[44] J. E. Slotine, W.P. Li, Applied Nonlinear Control, Prentice Hall, New Jersey, 1990. [45] Y. Li, Y.Q. Chen, I. Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic
Ac ce p
te
d
M
an
us
cr
systems, Automatica 45 (2009) 1965-1969.
15
Page 15 of 22
Fig. 1. Chaotic attractor of the fractional-order Lorenz system. Fig. 2. The controlled fractional-order Lorenz system converges to one of its equilibriums by the adaptive fractional-order feedback controller. (a) E1 ; (b) E2 ; (c) E3 .
ip t
Fig. 3. Hyper-chaotic attractor of the fractional-order Rössler system in different projection coordinate surface. (a) x1 - x2 - x3 ; (b) x1 - x2 - x4 ; (c) x1 - x3 - x4 ; (d) x2 - x3 - x4 .
cr
Fig. 4. The controlled fractional-order Rössler system converges to E1 by the
us
adaptive fractional- order feedback controller. (a) x1 (t ), x3 (t ) ; (b) x2 (t ), x4 (t ) . Fig. 5. Chaotic attractor of the fractional-order Chua’s system.
adaptive integer- order feedback controller.
an
Fig. 6. The controlled fractional-order Chua’s system converges to E by the
M
Fig. 7. Chaotic attractor of the fractional-order Genesio-Tesi system. Fig. 8. The controlled fractional-order Genesio-Tesi system converges to one of its
d
equilibriums by the adaptive integer-order feedback controller. (a) E1 ; (b) E2 .
te
Fig. 9. The controlled fractional-order chaotic system converges to one of its equilibriums by the adaptive integer-order (denote by ‘blue –’) and the
Ac ce p
fractional-order (denote by ‘red –’) feedback controller. (a) Lorenz system; (b) Rössler system; (c) Chua’s system; (d) Genesio-Tesi system. Fig. 10. The controlled fractional-order Chua’s system converges to zero equilibrium by the adaptive fractional-order controller with different fractional order .
16
Page 16 of 22
Fig.1
ip t
50 40 30 20
cr
)t ( 3 x
us
10 0 40 20
20
10
0
an
0
-20
-10
-40
x (t)
-20
x 1(t)
M
2
Fig.2 5
d
0 ) (t 1 x
-5 0
2
4
6
te
-10
8
10
12
14
16
18
20
0
-5
)t ( 2 x
-10
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10 t(sec.)
12
14
16
18
20
Ac ce p
-15
10
)t ( 3 x
5
0
(a)
10
)t ( 1 x
0 )t ( 1 x
5
0
0
2
4
6
8
10
12
14
16
18
-10
20
10 )t ( 2 x
)t ( 2 x 0
2
4
6
8
10
12
14
16
18
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10 t(sec.)
12
14
16
18
20
30
20
20 ) (t 3 x
10 0
2
-5
-10
20
30 ) (t 3 x
0
0
5
0
-5
0
2
4
6
8
10 t(sec.)
12
14
16
18
10 0
20
(b)
(c)
17
Page 17 of 22
Fig.3
200
115 110
0 105 100
-200
) (t 4 x
95
ip t
) (t 3 x
-400 90 85 100
-600 100 100
100
0
0
0
-100
-100
-100
-100
-200 -200
x 2(t)
-300
-200
-200
x 2(t)
x 1(t)
-300
x 1(t)
(b)
200
0
an
200
us
(a)
)t ( 4 x
cr
0
0
-200
)t ( 4 x
-400
-600 120
M
-400
-200
-600 120
110
100
0
100
-100
90
-200 -300
100 50
100
0 -50
90 80
x 3(t)
x 1(t)
d
80
x 3(t)
110
x 2(t)
(d)
te
(c)
-100 -150
Ac ce p
Fig.4
20
10 x 1(t)
15
x 2(t)
x 3(t)
x 4(t) 5
10 5
)t ( 3 x ， ) (t 1 x
)t ( 4 x ， ) (t 2 x
0
-5
0
-5
-10 -15
-10
-20 -25
0
2
4
6
8
10 t(sec.)
12
14
16
18
-15
20
(a)
0
2
4
6
8
10 t(sec.)
12
14
16
18
20
(b)
18
Page 18 of 22
Fig.5
cr
0
us
)t ( 3 x
ip t
5
-5 -4
-1
-0.5
-2
0
0
an
0.5
2
4
1
x (t) 2
M
x 1(t)
d
Fig.6
)t ( 1 x
te
2
1
Ac ce p
0
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10 t(sec.)
12
14
16
18
20
0.4
)t ( 2 x
0.2
0 0
-1
)t ( 3 x
-2 -3
19
Page 19 of 22
Fig.7
ip t
1
0.5
cr
0
)t ( 3 x
us
-0.5
-1 -0.5 0
0
0.5
an
0.5
1
d
5 0
0
2
4
6
8
10
12
14
Ac ce p
0.5
te
-5
1 )t ( 2 x
0
x (t) 2
M
Fig.8
-10
1
1.5
x 1(t)
)t ( 1 x
-1
-0.5
0
2
4
6
8
10
12
14
16
16
18
18
2 0 )t ( 1 x
-2 -4 -6
20
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10 t(sec.)
12
14
16
18
20
1 )t ( 2 x
0.5
0
20
10
5
5
) (t 3 x
) (t 3 x
0
-5
0
2
4
6
8
10 t(sec.)
12
14
16
18
0
-5
20
(a)
(b)
20
Page 20 of 22
Fig.9 1
1.5
0 1 -1
))t ( 1 x( gl
-3
ip t
0.5
-2 ))t ( 1 x( gl
0
-4 -0.5
-6 0
2
4
6
8
10 t(sec.)
12
14
16
18
-1
20
0
2
10 t(sec.)
12
14
16
18
20
1
0
0
-0.5
-1
16
18
20
an
0.5
-2
)) (t 1 x( gl
-1.5
-3 -4
M
-2 -2.5
-5
-3
-6
0
2
4
6
8
10 t(sec.)
12
14
16
18
-7
20
0
2
4
6
8
d
-3.5
8
(b)
-1 )) (t 1 x( gl
6
us
(a)
4
cr
-5
12
14
(d)
te
(c)
10 t(sec.)
Ac ce p
Fig.10 0.5
0
=0.975 =0.98
-0.5
=0.985 =0.99
-1
)) (t 1 x( gl
-1.5
-2 -2.5 -3 -3.5
0
2
4
6
8
10 t(sec.)
12
14
16
18
20
21
Page 21 of 22
[46] Table. 1. Comparative data of convergence speed with integer-order and fractional-order controller for different fractional-order chaotic systems.
2 (fractional-order)
(1) Lorenz system
4.8803 106
5.1951 106
(2) Rössler system
3.1999 103
3.9145 103
(3) Chua’s system
7.8222 104
1.2734 103
(4) Genesio-Tesi system
6.1829 106
cr
us
6.4438 106
Ac ce p
te
d
M
an
[47]
ip t
1 (integer-order)
fractional-order chaotic system
22
Page 22 of 22
S0030-4026(15)00576-8 http://dx.doi.org/doi:10.1016/j.ijleo.2015.07.024 IJLEO 55739
To appear in: Received date: Accepted date:
4-6-2014 7-7-2015
Please cite this article as: R. Li, W. Li, Suppressing chaos for a class of fractionalorder chaotic systems by adaptive integer-order and fractional-order feedback control, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.07.024 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Suppressing chaos for a class of fractional-order chaotic systems by adaptive integer-order and fractional-order feedback control
ip t
Ruihong Li , Wei Li
cr
School of Mathematics and Statistics, Xidian University, Xi’an 710071, PR China
Abstract. This paper is devoted to studying how to more effectively suppress chaos for a class
us
of fractional-order nonlinear systems. By means of adaptive control theory for integer-order nonlinear system, we propose two simple and novel adaptive feedback methods to control chaos.
an
Rigorous theoretical proof is provided based on some essential properties of fractional calculus and Barbalat’s Lyapunov-like stability theorem. It is discovered that both fractional-order
M
feedback controller and integer-order one can guide chaotic trajectories to the unstable equilibrium point. To display the feasibility and validity of presented methods, some typical fractional-order chaotic systems have been chosen as numerical illustration. Furthermore, by comparing two
te
stable and more flexible.
d
different control techniques, one can find the fractional-order feedback control algorithm is more
Keywords: Fractional-order chaotic system; Integer-order feedback; Fractional-order feedback;
Ac ce p
Suppress chaos; Barbalat’s lemma
1. Introduction
Fractional calculus is a mathematical subject with more than 300 years old history,
but its application in physics and engineering has just started [1-3]. One possible explanation for such unpopularity is that there are many different definitions of fractional derivative, another one is fractional derivative has no evident geometrical interpretation due to its non-local property. However, over the past decade, a growing body of research suggests that some practical systems could display fractional order dynamics, such as viscoelasticity [4], dielectric polarization [5], electrode-electrolyte
Corresponding author. E-mail address: [email protected] (R.-h. Li) 1
Page 1 of 22
polarization [6], electromagnetic waves [7], quantitative finance [8], and quantum evolution of complex systems [9]. The most significant advantage of fractional calculus is it can provide an excellent instrument for the description of various materials and processes with long memory and hereditary property [10]. In recent
ip t
years, research on the dynamics of fractional-order systems has attracted more and
more attention. According to the Poincaré-Bendixson theorem [11], an integer-order
cr
nonlinear system must have a minimum order of three, which could make chaos appear. Nevertheless, the fractional-order nonlinear system is not the case. Numerous
us
studies have shown that plenty of three-dimensional systems of order 2 + with 0 < < 1 can exhibit chaotic behavior with appropriate system parameter, such as
an
fractional-order Lorenz families system [12-14], fractional-order Rössler system [15], fractional-order Arneodo system [16], fractional-order Newton-Leipnik system [17],
M
fractional-order Genesio-Tesi system [18], fractional-order Liu system [19], fractional-order Chua’s system [20] and so on. In addition, some non-autonomous
d
fractional-order systems [21-23] and high-dimensional fractional-order systems [24-27] could also display chaotic or hyper-chaotic behavior.
te
Normally, chaos is considered a troublemaker. To avoid troubles arising from
Ac ce p
unusual behaviors of a chaotic system, chaos control has gained increasing attention in the past 20 years. An important objective of a chaos controller is to suppress chaotic oscillations completely or reduce them to regular oscillations. For fractionalorder chaotic system, there are two main types of controller, integer-order controller [28-35] and fractional-order controller [36-37]. Up to now, there have been a variety of control methods in the case of integer-order controller, such as linear feedback [28-29], nonlinear feedback [30-31], backstepping technique [32-33], sliding mode [34-35] and so on. However, despite the large amount of efforts, many key issues remain unresolved. The most urgent problem is how to design a general, efficient and physically available controller to suppress chaos. Linear feedback method is simple and practical, but it is difficult to determine the suitable feedback coefficient. Thereby, numerical calculation methods have to be used, which lead to linear feedback is only applicable to individual systems. And nonlinear feedback approach and backstepping 2
Page 2 of 22
technique have the same problem. Sliding mode controller is successful for some fractional-order systems, but, chattering phenomenon is the inherent vice. On the other hand, application of fractional calculus to achieve control has just begun and there are lots of problems worthy to study. Motivated by above discussions, we
suppress chaos for a class of fractional-order nonlinear systems.
ip t
attempt to put forward new fractional-order controller and integer-order controller to
cr
Adaptive feedback control [38] is a mature and effective method to suppress chaos
for integer-order chaotic systems. Compared with linear feedback method, adaptive
us
feedback approach does not need to know feedback coefficients in advance. Hence, we will extend adaptive feedback method to suppress chaos for fractional-order
an
chaotic systems. The rest of this paper is organized as follows. In Section 2, basic definitions in fractional calculus, properties and numerical algorithms are introduced.
M
In Section 3 and 4, two new controllers are proposed to suppress chaos for a class of fractional-order chaotic systems. The stability analysis of the closed-loop system and
d
numerical simulations of some typical fractional-order chaotic systems are included simultaneously. Section 5 is devoted to comparing two control algorithms and some
te
conclusions are drawn is Section 6.
Ac ce p
2. Brief introduction to fractional calculus 2.1 Definitions of fractional derivative and its properties The operator t D0 , called differ-integral operator, is commonly used in fractional
calculus as notation for taking both fractional derivative and fractional integral in a single expression. This operator is defined as follows: d 0, dt , 0, t D0 1, t 0 (d ) , 0.
(1)
At present, there exist many different definitions of fractional derivative [1-3]. Among them, Grünwald- Letnikov, Riemann-Liouville and Caputo definitions are
3
Page 3 of 22
more reputed than others. The simplest definition which is also the easiest one to use is the Riemann-Liouville definition. This definition is described as: t D0 f (t )
d n n [ J 0 f (t )], dt n
(2)
t 1 (t ) 1 f ( )d , 0, 0 ( )
(3)
us
J 0 f (t )
cr
fractional integral operator, which is defined as follows:
ip t
where n is the smallest integer greater than , i.e., n 1 n. J 0 () is the
and () is the Gamma function.
an
Riemann-Liouville derivative t D0 is a continuous operator on , i.e. t D0 can bridge all the gaps among the integer derivatives and the integer integrals. So
M
Riemann-Liouville derivative is a good and nature generalization of classical calculus [39]. Thereupon, one will select Riemann-Liouville definition in subsequent research.
d
Throughout this paper, we use the notation D as the simplified form t D0 and its
te
related properties are described below:
(1) For 1, the operation D1 f (t ) coincides with the ordinary derivative
Ac ce p
df (t ) . dt
(2) For 0 1, the following equation holds: D f (t )
1 d t (t ) f ( ) d 0 (1 ) dt
(4)
(3) Similar to integer-order calculus, fractional calculus are linear operation D [af (t ) bg (t )] aD f (t ) bD g (t ).
(5)
2.2 Numerical methods of fractional derivative For calculating fractional derivative numerically, frequency domain approximation algorithm and Adams-Bashforth-Moulton algorithm are the two most commonly used methods. The standard definitions of fractional differential-integral equation do not allow direct implementation in time domain. As a result, a method for approximate 4
Page 4 of 22
solution of fractional differential equation in frequency domain was developed in Ref. [40]. This approximation method is simple and convenient, which has been adopted in Refs. [14,16,18,31,33]. However, as the extreme sensitivity to initial conditions of chaotic behavior, above method is not reliable in the study of fractional-order chaotic
ip t
system, which has been already verified in Ref. [41]. Therefore, in this paper, one will adopt the generalized Adams-Bashforth-Moulton method [42], which derives analytic
cr
expression of fractional differential equation then numerically iterates the formula.
With good numerical stability, it has been applied to the research of chaotic behavior
us
for fractional-order systems.
an
Remark 1. || || , || ||1 and || ||2 respectively denote the -norm, 1-norm and 2-norm of vector, which is defined as follows: n
n
|| x || max | xi |, || x ||1 | xi |, || x ||2 ( xi2 )1 2 , x R n . i =1
(6)
i =1
M
1i n
3. Adaptive fractional-order feedback control
te
system
d
Consider the chaos suppression scheme for n-dimensional fractional-order chaotic
Ac ce p
D x(t ) f ( x, t ) u ( x, t ),
(7)
where 0 1. Here, x R n is the state vector and
f ( x, t ) : R n R n is a
nonlinear chaotic vector function, which satisfies the following inequality:
| f i ( x) f i ( y ) | l max | x j y j |,
i, j 1,, n.
(8)
where l is a positive constant. Remark 2. Such condition may be called as uniform Lipschitz condition, which can
ensure the existence and uniqueness of fractional-order dynamical system [43]. It is easy to verify that most fractional-order chaotic systems [12-20, 24-27] satisfy this condition. Suppose that x* R n is an unstable equilibrium point of uncontrolled fractionalorder chaotic system and u ( x, t ) is the undetermined controller. For this purpose,
5
Page 5 of 22
one defines control error as e x x* , then error system can be obtained as follows: D e D x D x* f ( x, t ) f ( x* , t ) u ( x, t ).
(9)
3.1 Adaptive control algorithm
ip t
Theorem 1. If the controller in (9) is given by
ui (t ) ki ei ki ( xi xi* ), i 1,, n.
(10)
cr
where ki (1, , n) are feedback strength which are updated according to the
D ki (t ) i ei2 i ( xi xi* ) 2 , i 1, , n
us
following rules:
(11)
an
where i (i 1, , n) are arbitrary positive constants. Then, the controlled fractional -order chaotic system (7) could be guided to the unstable equilibrium point x* .
M
Remark 3. Because state variables of chaotic system are bounded, here we suppose control function k (e, t ) meets the following condition: mi | ki (e, t ) | M i , i 1,, n.
d
(12)
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Lemma 1. (Barbalat lemma [44]). If (t ) : R R is a uniformly positive function t
for t 0 and if the integral lim ( ) d exists and is finite, then lim (t ) 0.
Ac ce p
t
t
0
Proof. Applying the controller (10) to the error system (9) results in the following
system:
D e D x D x* f ( x, t ) f ( x* , t ) ke.
(13)
Now, we introduce the following Lyapunov function:
V (t , e(t ))
where L1
1 n 1 2 L1 n 1 1 2 J 0 ki J 0 ei 2 2 i 1 i 1 i
(14)
( M l )n , M max{| M i |}, m min{| mi |}. Taking the time derivative of 1i n 1 i n m
V along the trajectories of error system (13), one has
6
Page 6 of 22
n n 1 V ( J 01- ei )( D0 ei ) L1 ( J 01- ki )( D0 ki ) i 1 i 1 i n t t 1 n - - 2 ( 0 (t ) ei ( ) d )( f i ( y ) fi ( x) ki ei ) L1 ( 0 (t ) ki ( ) d )ei (1 ) i 1 i 1
n t 1 n t - - 2 ( 0 (t ) | ei ( ) | d ) | ( f i ( y ) - f i ( x)) ki ei | L1 ( 0 (t ) ki ( )d )ei (1 ) i 1 i 1
n t t 1 n - - 2 || e || ( 0 (t ) d ) | ( fi ( y ) f i ( x)) ki ei | L1 mi ( 0 (t ) d )ei (1 ) i 1 i 1
n n n t1- [ || e || | f i ( y ) f i ( x) | || e || | ki ei | L1 mi ei2 ] (2 ) i 1 i 1 i 1
t1- [l || e || (2 )
i 1
1 j n
j
x j | M || e ||
n
| e i 1
i
n
| L1 m ei2 ] i 1
1-
t1- [nl || e ||2 Mn || e ||2 L1 m || e ||2 ] (2 )
t 1- [( M l )n L1 m] || e ||2 0, (2 )
t1- [( M l )n L1 m] || e ||22 (2 )
M
an
= eT P (t )e (t ) 0,
t1- [ L1m ( M l )n]I (2 )
d
P(t )
where
us
t [nl || e ||2 M || e || || e ||1 L1 m || e ||22 ] (2 )
(15)
cr
n
max | y
ip t
is a positive definite matrix. Now
te
integrating (15) from zero to t yields t
V (t ) ( ) d V (0)
Ac ce p
0
(16)
and above equation means that
t
0
( ) d V (0)
(17)
Due to V 0, as t approach to infinite, the above integral is always less than or t
equal to V (0), so lim ( ) d exists and is finite. Based on Lemma 1, one has t
0
lim (t ) lim eT P(t )e 0 lim e(t ) 0. t
t
t
(18)
Then, state trajectories of error system (13) will converge to the zero point. That is, fractional-order chaotic system (7) could be controlled to the unstable equilibrium point x* . 3.2 Numerical simulation In this section, two groups of fractional-order chaotic systems are concretely 7
Page 7 of 22
presented to show the correctness and effectiveness of proposed scheme. Example 1. Consider the fractional-order Lorenz system as follows: d x1 a ( x2 x1 ), dt d x2 cx1 x2 x1 x3 , dt d x3 bx3 x1 x2 , dt
cr
ip t
(19)
8 where a, b, c 0, 0 1. Especially, when (a, b, c) (10, , 28) and 0.995, the 3
us
fractional-order Lorenz system has a chaotic attractor [12] as depicted in Fig. 1.
an
System (19) has three equilibrium points E1 (0, 0, 0), E2,3 (± 6 2, ± 6 2, 27), which are all unstable with 0.995. Initial values are X 0 = (0.001, - 0.004,0.002) such
from k0 = (0.005, - 0.003, 0.006)
M
that fractional-order Lorenz system is capable of chaos. The adaptive control starts and feedback intensity are i 1.0(i 1, 2,3).
d
Numerical results are shown in Fig. 2, from which one can see the chaotic orbits of
te
system (19) will be stabilized to E1 , E2 and E3 respectively in a very short time. Example 2. Consider the fractional-order Rössler system as follows:
Ac ce p
d x1 x2 x4 , dt d x2 x1 ax2 x3 , dt d x3 bx3 cx4 , dt d x4 d x1 x4 , dt
(20)
where a, b, c, d 0, 0 1. Set a = 0.32, b = 0.05, c = 0.5, d = 3.0 and 0.95, the fractional-order Rössler system has a hyper-chaotic attractor [15] as depicted in Fig. 3. System (20) has two equilibrium points E1,2 (±
11 6 5 6 25 6 5 6 ,± , , ), 5 22 11 22
which are all unstable with 0.95. Here, we only consider the stabilization of 8
Page 8 of 22
fixed point E1. Initial values are X 0 = (- 25.0,5.0, 20.0, - 8.0) such that fractionalorder Rössler system is capable of hyper-chaos. The adaptive control starts from k0 = (- 5.0,7.0, - 8.0,6.0) and feedback intensity are i 1.0(i 1, 2,3, 4). Numerical
ip t
results are shown in Fig. 4, from which one can see the chaotic orbits of system (20)
cr
will be stabilized to E1 in a very short time.
4. Adaptive integer-order feedback control
us
In this section, adaptive integer-order feedback control technique will be extended to suppress chaos for fractional-order chaotic system (7). Thereupon, one has the
an
following conclusion: 4.1 Adaptive control algorithm
strength are changed as follows:
M
Theorem 2. If the controller (10) is still adopted and the update rules of feedback
are arbitrary positive constants. Then, the controlled
te
i (i 1, , n)
where
(21)
d
D1ki (t ) i ei2 i ( xi xi* ) 2 , i 1, , n
fractional-order chaotic system (7) could still be guided to the unstable equilibrium
Ac ce p
point x* .
Proof. Here, fractional-order stability theory is employed as usual. We introduce
another Lyapunov function as follows:
V (t , e(t ))
(2 ) n 1 2 L2 J 0 ei 2 2t1 i 1
n
1
i 1
ki2
(22)
i
where L2 is a constant bigger than ( M l )n, and the definition of M is the same
as last section. Taking the time derivative of V , one has n (2 ) L V 1 ( J 01 ei )( D0 ei ) 2 t 2 i 1
n
1
i =1
ki2
i
n (2 ) 1 1 ( (t ) ei ( ) d )( f i ( x) f i ( x* ) ki ei ) L2 ei2 (1 ) 0 t i 1 i 1 n
(23)
t
Similar to the derivation in previous section, we have the following conclusion: 9
Page 9 of 22
V [( M l )n L2 ] || e ||22 0.
(24)
Then, based on the Lyapunov stability theory [45], state trajectories of error system (13) will converge to the zero point. That is, fractional-order chaotic system (7) will
ip t
be controlled to the unstable equilibrium point x* . 4.2 Numerical simulation
cr
In this section, the other two groups of fractional-order chaotic systems are concretely presented to show the correctness and effectiveness of proposed scheme.
us
Example 3. Consider the fractional-order Chua’s system as follows:
(25)
M
where
an
d x1 a ( x2 x1 f ( x1 )), dt d x2 x1 x2 x3 , dt d x1 bx2 , dt
1 f ( x) m1 x(t ) (m0 m1 )(| x(t ) 1| | x (t ) 1|). Here, all parameters are 2
d
positive and 0 1. Especially, when (a, b, m0 , m1 ) (10,14.87, 1.27, 0.68) and
te
0.975, the fractional-order Chua’s system has a chaotic attractor [20] as depicted
Ac ce p
in Fig. 5.
System (25) only has one equilibrium point E (0, 0,0), which is unstable with
0.975. Initial values are X 0 = (0.006,0.002, - 0.005) such that fractional-order
Chua’s
system
is
capable
of
chaos.
The adaptive
control
starts
from
k0 = (- 0.003, 0.004, - 0.006) and feedback intensity is i 1.0(i 1, 2,3). Numerical
results are shown in Fig. 6, from which one can see the chaotic orbits of system (25) will be stabilized to E in a very short time. Example 4. Consider the fractional-order Genesio-Tesi system as follows:
10
Page 10 of 22
d x1 x2 , dt d x2 x3 , dt d x3 ax1 bx2 cx3 dx12 , dt
ip t
(26)
where a, b, c, d 0, 0 1. Set a = 1.0, b = 1.1, c = 0.44, d = 1.0 and 0.996, the
cr
fractional-order Genesio-Tesi system has a chaotic attractor [18] as depicted in Fig. 7.
us
System (26) has two equilibrium points E1 (0, 0, 0), E2 (1, 0, 0), which are all unstable with 0.996. Initial values are X 0 = (- 0.001, 0.005, 0.002) such that fractional-
an
order Genesio-Tesi system is capable of chaos. The adaptive control starts from k0 = (6.0, - 4.0, - 3.0) and feedback intensity is i 1.0(i 1, 2,3). Numerical results
M
are shown in Fig. 8, from which one can see the chaotic orbits of system (26) will be
d
stabilized to E1 and E2 respectively in a very short time.
te
5. Comparison between two control schemes Through above studies, it is easy to find both integer-order control approach and
Ac ce p
fractional-order control technique can suppress chaos successfully. In this section, we try to explore the differences between these two feedback methods in control performance.
In the following numerical calculations, two control methods are simultaneously
applied to fractional-order chaotic systems in Examples 1-4, where system parameters are selected as the same as Figs .2, 4, 6 and 8. Take state variable x1 for example, the
comparison diagrams are plotted in Fig. 9. Let denotes the distance for state
variable x1 converges x1* at t = 20 s, which measures the convergence speed of control algorithm. For different fractional-order chaotic systems, 1 indicates the integer-order controller and 2 indicates the fractional-order controller. From Fig .9 (a, b, d) and Table 1 (1, 2, 4), it is clear that two control methods make little difference 11
Page 11 of 22
in control effect. But from Fig. 9 (c) and Table 1 (3), one can see the controlled fractional-order chaotic system could more quickly attain control target with integer-order feedback controller than fractional-order one. It seems that the integer-order feedback controller has better control performance.
ip t
But as a new emerging research field, the fractional-order controller has greater researching significance and more application prospects. For instance, in integer-order
cr
controller, only feedback coefficient can be used as adjustable parameter. However, for fractional-order case, there exists another adjustable parameter .
us
Take suppressing chaos for Chua’s system as an example, Fig. 10 exhibits time
an
histories for state variable x1 with different fractional order . Obviously as increases, the convergence speed is faster and faster. On the other hand, fractional-order system is at least as stable as their integer-order counterpart. Hence,
M
the controlled system is more stable with fractional-order feedback controller than
te
6. Summary and discussion
d
integer-order one.
Even though chaos dynamics on fractional-order systems has been extensively
Ac ce p
discussed, the study on suppressing chaos for fractional-order chaotic system is still at its initial stage. To the best of our knowledge, there have been only a few literatures focusing on chaos suppression for a class of fractional-order chaotic systems. In contrast, our work proposed a general method which can guide chaotic orbits of fractional-order chaotic system to the unstable equilibrium. Based on Lyapunov stability theorem of fractional-order dynamical system, adaptive integer-order feedback and fractional-order feedback controllers are designed. Meanwhile, four groups of numerical examples are provided to show the effectiveness of presented methods. It is worth noting that two new control techniques are simple and easy to implement, and what’s more they could be applied to almost all of known fractionalorder chaotic systems. Further, it is demonstrated that fractional-order controller has the same control effect with integer order one in most cases. And, fractional-order
12
Page 12 of 22
controller has more parameters to adjust control performance. Acknowledgement This work was supported by the National Natural Science Foundation of China
ip t
(Grant No. 11202155, 11302157) and the Fundamental Research Funds for the
cr
Central University (Grant No. JB140710) References
us
[1] I. Podlubny, Fractional differential equations, Academic Press, New York, 1999. [2] P.L. Butzer, U. Westphal, An introduction to fractional calculus, World Scientific, Singapore, 2000.
an
[3] R. Hilfer, Applications of fractional calculus in physics, World Scientific, New Jersey, 2001. [4] R.L. Bagley, R.A. Calico, Fractional order state equations for the control of viscoelastically damped structures, J. Guid. Control Dyn. 14 (1991) 304-311.
[5] H.H. Sun, A.A. Abdelwahad, B. Onaral, Linear approximation of transfer function with a
M
pole of fractional order, IEEE Trans. Automat. Control. 29 (1984) 441-444.
[6] M. Ichise, Y. Nagayanagi, T. Kojima, An analog simulation of noninteger order transfer functions for analysis of electrode process, J. Electro-Anal. Chem. 33 (1971) 253-265.
te
d
[7] O. Heaviside, Electromagnetic Theory, Chelsea, New York, 1971. [8] N. Laskin, Fractional market dynamics, Phys. A 287 (2000) 482-492. [9] D. Kusnezov, A. Bulgac, G.D. Dang, Quantum Lévy processes and fractional kinetics, Phys. Rev. Lett. 82 (1999) 1136-1139.
[10] L.P. Chen, Y. Chai, R.C. Wu, Linear matrix inequality criteria for robust synchronization of
Ac ce p
uncertain fractional-order chaotic systems, Chaos 21 (2011) 043107-13.
[11] M.W. Hirsch, S. Smale, Differential equations: dynamical systems and linear algebra, Academic Press, New York, 1974.
[12] I. Grigorenko, E. Grigorenko, Chaotic dynamics of the fractional Lorenz system, Phys. Rev. Lett. 91 (2003) 034101-4.
[13] C.G. Li, G.R. Chen, Chaos in the fractional order Chen system and its control, Chaos Soliton Fract. 22 (2004) 549-554.
[14] J.G. Lu, Chaotic dynamics of the fractional-order Lü system and its synchronization, Phys. Lett. A 354 (2006) 305-311.
[15] C.G. Li, G.R. Chen, Chaos and hyperchaos in the fractional order Rössler equations, Phys. A 341 (2004) 55-61.
[16] J.G. Lu, Chaotic dynamics and synchronization of fractional-order Arneodo’s systems, Chaos Soliton Fract. 26 (2005) 1125-1133.
[17] L.J. Sheu, H.K. Chen, J.H. Chen, L.M. Tam, W.C. Chen, K.T. Lin, Y. Kang, Chaos in the Newton-Leipnik system with fractional order, Chaos Soliton Fract. 36 (2008) 98-103.
[18] J.G. Lu, Chaotic dynamics and synchronization of fractional-order Genesio-Tesi systems, Chin. Phys. 14 (2005) 1517-1521.
[19] X.Y. Wang, M.J. Wang, Dynamic analysis of the fractional-order Liu system and its 13
Page 13 of 22
synchronization, Chaos 17 (2007) 033106-6.
[20] C.P. Li, W.H. Deng, D. Xu, Chaos synchronization of the Chua system with a fractional order, Phys. A 360 (2006) 171-185.
[21] L.J. Sheu, H.K. Chen, J.H. Chen, L.M. Tam, Chaotic dynamics of the fractionally damped Duffing equation, Chaos Soliton Fract. 32 (2007) 1459-1468. Chaos Soliton Fract. 35 (2008) 188-198.
ip t
[22] J.H. Chen, W. C. Chen, Chaotic dynamics of the fractionally damped van der Pol equation, [23] Z.M. Ge, C.X. Yi, Chaos in a nonlinear damped Mathieu system, in a nano resonator system and in its fractional order systems, Chaos Soliton Fract. 32 (2007) 42-61.
cr
[24] W.M. Ahmad, Hyperchaos in fractional order nonlinear systems, Chaos Soliton Fract. 26 (2005) 1459-1465.
[25] C.X. Liu, L. Liu, Circuit implementation of a new hyperchaos in fractional-order system,
us
Chin. Phys. B 17 (2008) 2829-2836.
[26] X.Y. Wang, J.M. Song, Synchronization of the fractional order hyperchaos Lorenz systems
an
with activation feedback control, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 3351 -3357.
[27] A.K. Matouk, Stability conditions, hyperchaos and control in a novel fractional order hyperchaotic system, Phys. Lett. A 373 (2009) 2166-2173.
[28] X.Y. Wang, Y.J. He, M.J. Wang, Chaos control of a fractional order modified coupled
M
dynamos system, Nonlinear Dyn. 71 (2009) 6126-6134.
[29] A.E. Matouk, Chaos, feedback control and synchronization of a fractional-order modified
d
autonomous Van der Pol-Duffing circuit, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 975-986.
[30] Z.M. Odibat, Adaptive feedback control and synchronization of non-identical chaotic
te
fractional order systems, Nonlinear Dyn. 60 (2010) 479-487.
[31] J. Yang, D.L. Qi, The feedback control of fractional order unified chaotic system, Chin. Phys.
Ac ce p
B 19 (2010) 020508-4.
[32] W.M. Ahmad, A.M. Harb, On nonlinear control design for autonomous chaotic systems of integer and fractional orders, Chaos Soliton Fract. 18 (2003) 693-701.
[33] X. Gao, J.B. Yu, Chaos and chaotic control in a fractional-order electronic oscillator, Chin. Phys. 14 (2005) 908-913.
[34] C. Yin, S.M. Zhong, W.F. Chen, Design of sliding mode controller for a class of fractionalorder chaotic systems, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 356-366.
[35] Z. Wang, X. Xia, H. Shen, Control of an uncertain fractional order economic system via adaptive sliding mode, Neurocomputing 83 (2012) 83-88.
[36] M.S. Tavazoei, M. Haeri, Chaos control via a simple fractional-order controller, Phys. Lett. A 372 (2008) 798-807.
[37] M.S. Tavazoei, M. Haeri, S. Bolouki, M. Siami, Using fracional-order integrator to control chaos in single-input chaotic systems, Nonlinear Dyn. 55 (2009) 179-190.
[38] D.B. Huang, Adaptive-feedback control algorithm, Phys. Rev. E 73 (2006) 066204-8. [39] W.H. Deng, C.P. Li, The evolution of chaotic dynamics for fractional unified system, Phys. Lett. A 372 (2008) 401-407.
[40] A. Charef, H.H. Sun, Y.Y. Tsao, B. Onaral, Fractal system as represented by singularity function, IEEE Trans. Automat. Control. 37 (1992) 1465-1470.
14
Page 14 of 22
[41] M.S. Tavazoei, M. Haeri, Limitations of frequency domain approximation for detecting chaos in fractional order systems, Nonlinear Anal. TMA 69 (2008) 1299-1320.
[42] K. Diethelm, N.J. Ford, A.D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn. 29 (2009) 3-22.
[43] K. Diethelm, N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002) 229-248.
ip t
[44] J. E. Slotine, W.P. Li, Applied Nonlinear Control, Prentice Hall, New Jersey, 1990. [45] Y. Li, Y.Q. Chen, I. Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic
Ac ce p
te
d
M
an
us
cr
systems, Automatica 45 (2009) 1965-1969.
15
Page 15 of 22
Fig. 1. Chaotic attractor of the fractional-order Lorenz system. Fig. 2. The controlled fractional-order Lorenz system converges to one of its equilibriums by the adaptive fractional-order feedback controller. (a) E1 ; (b) E2 ; (c) E3 .
ip t
Fig. 3. Hyper-chaotic attractor of the fractional-order Rössler system in different projection coordinate surface. (a) x1 - x2 - x3 ; (b) x1 - x2 - x4 ; (c) x1 - x3 - x4 ; (d) x2 - x3 - x4 .
cr
Fig. 4. The controlled fractional-order Rössler system converges to E1 by the
us
adaptive fractional- order feedback controller. (a) x1 (t ), x3 (t ) ; (b) x2 (t ), x4 (t ) . Fig. 5. Chaotic attractor of the fractional-order Chua’s system.
adaptive integer- order feedback controller.
an
Fig. 6. The controlled fractional-order Chua’s system converges to E by the
M
Fig. 7. Chaotic attractor of the fractional-order Genesio-Tesi system. Fig. 8. The controlled fractional-order Genesio-Tesi system converges to one of its
d
equilibriums by the adaptive integer-order feedback controller. (a) E1 ; (b) E2 .
te
Fig. 9. The controlled fractional-order chaotic system converges to one of its equilibriums by the adaptive integer-order (denote by ‘blue –’) and the
Ac ce p
fractional-order (denote by ‘red –’) feedback controller. (a) Lorenz system; (b) Rössler system; (c) Chua’s system; (d) Genesio-Tesi system. Fig. 10. The controlled fractional-order Chua’s system converges to zero equilibrium by the adaptive fractional-order controller with different fractional order .
16
Page 16 of 22
Fig.1
ip t
50 40 30 20
cr
)t ( 3 x
us
10 0 40 20
20
10
0
an
0
-20
-10
-40
x (t)
-20
x 1(t)
M
2
Fig.2 5
d
0 ) (t 1 x
-5 0
2
4
6
te
-10
8
10
12
14
16
18
20
0
-5
)t ( 2 x
-10
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10 t(sec.)
12
14
16
18
20
Ac ce p
-15
10
)t ( 3 x
5
0
(a)
10
)t ( 1 x
0 )t ( 1 x
5
0
0
2
4
6
8
10
12
14
16
18
-10
20
10 )t ( 2 x
)t ( 2 x 0
2
4
6
8
10
12
14
16
18
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10 t(sec.)
12
14
16
18
20
30
20
20 ) (t 3 x
10 0
2
-5
-10
20
30 ) (t 3 x
0
0
5
0
-5
0
2
4
6
8
10 t(sec.)
12
14
16
18
10 0
20
(b)
(c)
17
Page 17 of 22
Fig.3
200
115 110
0 105 100
-200
) (t 4 x
95
ip t
) (t 3 x
-400 90 85 100
-600 100 100
100
0
0
0
-100
-100
-100
-100
-200 -200
x 2(t)
-300
-200
-200
x 2(t)
x 1(t)
-300
x 1(t)
(b)
200
0
an
200
us
(a)
)t ( 4 x
cr
0
0
-200
)t ( 4 x
-400
-600 120
M
-400
-200
-600 120
110
100
0
100
-100
90
-200 -300
100 50
100
0 -50
90 80
x 3(t)
x 1(t)
d
80
x 3(t)
110
x 2(t)
(d)
te
(c)
-100 -150
Ac ce p
Fig.4
20
10 x 1(t)
15
x 2(t)
x 3(t)
x 4(t) 5
10 5
)t ( 3 x ， ) (t 1 x
)t ( 4 x ， ) (t 2 x
0
-5
0
-5
-10 -15
-10
-20 -25
0
2
4
6
8
10 t(sec.)
12
14
16
18
-15
20
(a)
0
2
4
6
8
10 t(sec.)
12
14
16
18
20
(b)
18
Page 18 of 22
Fig.5
cr
0
us
)t ( 3 x
ip t
5
-5 -4
-1
-0.5
-2
0
0
an
0.5
2
4
1
x (t) 2
M
x 1(t)
d
Fig.6
)t ( 1 x
te
2
1
Ac ce p
0
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10 t(sec.)
12
14
16
18
20
0.4
)t ( 2 x
0.2
0 0
-1
)t ( 3 x
-2 -3
19
Page 19 of 22
Fig.7
ip t
1
0.5
cr
0
)t ( 3 x
us
-0.5
-1 -0.5 0
0
0.5
an
0.5
1
d
5 0
0
2
4
6
8
10
12
14
Ac ce p
0.5
te
-5
1 )t ( 2 x
0
x (t) 2
M
Fig.8
-10
1
1.5
x 1(t)
)t ( 1 x
-1
-0.5
0
2
4
6
8
10
12
14
16
16
18
18
2 0 )t ( 1 x
-2 -4 -6
20
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10 t(sec.)
12
14
16
18
20
1 )t ( 2 x
0.5
0
20
10
5
5
) (t 3 x
) (t 3 x
0
-5
0
2
4
6
8
10 t(sec.)
12
14
16
18
0
-5
20
(a)
(b)
20
Page 20 of 22
Fig.9 1
1.5
0 1 -1
))t ( 1 x( gl
-3
ip t
0.5
-2 ))t ( 1 x( gl
0
-4 -0.5
-6 0
2
4
6
8
10 t(sec.)
12
14
16
18
-1
20
0
2
10 t(sec.)
12
14
16
18
20
1
0
0
-0.5
-1
16
18
20
an
0.5
-2
)) (t 1 x( gl
-1.5
-3 -4
M
-2 -2.5
-5
-3
-6
0
2
4
6
8
10 t(sec.)
12
14
16
18
-7
20
0
2
4
6
8
d
-3.5
8
(b)
-1 )) (t 1 x( gl
6
us
(a)
4
cr
-5
12
14
(d)
te
(c)
10 t(sec.)
Ac ce p
Fig.10 0.5
0
=0.975 =0.98
-0.5
=0.985 =0.99
-1
)) (t 1 x( gl
-1.5
-2 -2.5 -3 -3.5
0
2
4
6
8
10 t(sec.)
12
14
16
18
20
21
Page 21 of 22
[46] Table. 1. Comparative data of convergence speed with integer-order and fractional-order controller for different fractional-order chaotic systems.
2 (fractional-order)
(1) Lorenz system
4.8803 106
5.1951 106
(2) Rössler system
3.1999 103
3.9145 103
(3) Chua’s system
7.8222 104
1.2734 103
(4) Genesio-Tesi system
6.1829 106
cr
us
6.4438 106
Ac ce p
te
d
M
an
[47]
ip t
1 (integer-order)
fractional-order chaotic system
22
Page 22 of 22