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2017 IEEE 6th Data Driven Control and Learning Systems Conference
May 26-27, 2017, Chongqing, China
Modified Function Projective Synchronization of Fractional-order
Hyperchaotic Systems Based on Active Sliding Mode Control
Yuan Gao1,2,3, Hangfang Hu1, Ling Yu1, Haiying Yuan1,3, Xisheng Dai1,3
1.College of Electrical and Information Engineering, Guangxi University of science and technology, Liuzhou 545006,China
E-mail: [email protected]
2. Guangxi Key Laboratory of Automobile Components and Vehicle Technology, Liuzhou 545006, China
3. Key Laboratory of Industrial Process Intelligent Control Technology of Department of Guangxi Education, Liuzhou 545006, China
Abstract: Considering the time-varying scaling function matrix and system disturbances, a new sliding mode control strategy is
proposed to realize modified function projective synchronization (MFPS) of two different fractional-order hyperchaotic systems,
meanwhile improve the control robustness of synchronization system. From the MFPS error equations, combining a proper
fractional-order exponential reaching raw, an active controller for MFPS is derived out via sliding mode control technology. By
mean of the stability theorem, the asymptotic stability of synchronization error system is proved. Simulation results of the MFPS
between fractional-order hyperchaoticLorenz system and Chen system demonstrate the validity of the presented method.
Key Words: Fractional-order, Hyperchaotic system, Scaling function matrix, Modified function projective synchronization, Sliding
mode control
(GPS) of the fractional-order hyperchaotic Chen system.
Ref. [15] studied the MPS problem of two different
fractional-order hyperchaotic systems with system
parameter uncertainty. Ref. [16] presented a MFPS scheme
for a class of partially linear fractional-order chaotic systems,
in which the controlleris designed out by Routh-Hurwitz
On the other hand, there are many uncertain factors and
disturbances in the practical systems. Therefore, it is
necessary to overcome the influence of disturbances on the
chaos synchronization. The sliding mode control (SMC) is
well known as a kind of robust control techniques [17], which
has been widely used for controlling the disturbed nonlinear
system. Nevertheless, most of the existing SMC methods for
chaos (or hyperchaos) control or synchronization focus on
the integral-order systems[18,19]. In recent years, there are a
few results about the fractional-order chaotic systems[20, 21,
. Based on the principle of active control, Ref. [21]
designed a new SMC controller with a fractional-order
sliding mode surface to realize the PS of two
fractional-order hyperchaotic systems. For a class of
different structure synchronization problem of uncertain
fractional-order chaotic systems, ref. [22] proposed an
active feedback controller based on adaptive neural network,
where the sliding mode controller is designed to compensate
the influence of uncertain disturbances. In order to enhance
the control robustness, according to the stability theorem of
fractional-order system, ref. [15] and [23] studied active
SMC method to obtain the MPS of two different
fractional-order chaotic and hyperchaotic system,
Inspired by the discussions above, in order to obtain more
complex relationships of chaos synchronization, we
dedicate to study the MFPS of two non-identical
fractional-order hyperchaotic systems with disturbances,
and propose a new active SMC method. In this paper,
considering the time-varying scaling function matrix,we
introduce a reaching raw of sliding mode surface, and then a
proper sliding mode controller is derivedby mean of the
stability theory of the fractional-order system. At last, the
numerical results of the MFPS between the fractional-order
Chaos synchronization has always been a hot issue
because of great application prospect in the field of secure
communication[1],since this interesting phenomenon was
proved through a practical circuit experiment in 1990[2]. By
now, complete synchronization[3], phase synchronization[4],
generalized synchronization[5], projective synchronization
(PS)[6], and other types of chaos synchronization have been
studied[7]. PS is one of important chaos synchronization
types, whose main characteristic is that the state variables of
the drive system and response system are synchronized
according to a certain proportion[6]. In recent years, inspired
with the PS, people have further investigated a new scheme
named MFPS[8]. The MFPS scheme is an extension of many
existing synchronization schemes including modified
projective synchronization(MPS)[9], function projective
synchronization(FPS), et al.[10], where the drive system and
the response system are synchronized up to a scaling
function matrix instead of constant matrix, namely, state
variables of two systems correlate each other in pairs by a
scaling function. Due to the more unpredictability of the
scaling function, the MFPS can further improve the
anti-decryption of chaos secure communication, and attracts
more attention on it. By now, some research results about
the MFPS have been reported[11, 12, 13], however, most of
these works focused on integral-order system.
Chaos synchronization for the fractional-order system,
due to its great potentials in the fields of automatic control
andsecure communication, which has attracted increasing
attention in the last few years. In [14], using the stability
theorem of fractional-order system, Wu et al. provided a
method for realizing generalized projective synchronization
This work was supported by the Natural Science Foundation of Guangxi
Province of China (Grant No.2014GXNSFBA118284), the Scientific
Research Fund of Guangxi Education Department (Grant
No.KY2016YB244),the Opening Project of Guangxi Key Laboratory of
AutomobileComponents and Vehicle Technology (Grant No.
2015KFZD03), the Project of Key laboratoryof Industrial Process
Intelligent Control Techno- logyof Guangxi Higher Education Institutes
(GrantNo. IPICT-2016-01 and IPICT-2016-05)
978-1-5090-5461-9/17/$31.00 ©2017 IEEE
hyperchaotic Chen system and Lorenz system verify the
presented method.
DD x
Consider the following fractional-order hyperchaotic
drive system
A1 x f1 ( x, t ) d x (t )
DD x
and the unidirectionally coupled response system
Dq y
A2 y f 2 ( y, t ) d y (t ) u (2)
matrix Q
f1 ( x, t ), f 2 ( y, t ) : R o R are the continuous nonlinear
(u1 , u2 ,}, un )T is a control input vector,
d x (t ) (d x1 , d x 2 ,}, d xn )  R
and d y (t )
(d y1 , d y 2 ,
t of
where e(t )
(e1 (t ), e2 (t ),..., en (t ))  R , and ei
Dq e w(t ) d y (t ) H (t )d x (t ) A2e (13)
In order to decrease the design complexity of sub-controller
w (t ) , the sliding surface is directly defined as follows:
s(e) Ce
yi (14)
where C diag(c1 , c2 ,}, cn ) is a constant matrix.
To design the sliding mode controller, an exponential
reaching law is selected as
Dq s H sgn( s) Ps
Active sliding mode control scheme of MFPS
where P diag( p1 , p2 ,}, pn ) denotes a positive-definite
matrix, sgn(.) represents the sign function, the gain factor
H !0.
From (13) and (14), one can have
Stability theory for fractional-order system
Lemma 1 For the following fractional-order system
DD x
where 0 D 1 , x  R and A  R . If and only if the
Dq s C[ Kw(t ) d y (t ) H (t )d x (t ) A2e]
Neglecting the disturbances in (16), and according to the
fractional-order reaching law (15), we can derive the SMC
input w (t ) as follows:
following relationship is satisfied
,then system (5) is asymptotically stable [25].
A2 y f 2 ( y, t ) d y (t ) u H (t )( A1 x
the control parameters, w (t )  R is a SMC input that need
to be designed.
Substituting (12) into (10), we have
arg(Oi ( A)) ! 0.5DS , i 1, 2,..., n
Design of controller
where K = diag(k1 , k2 ,}, kn ) is a constant gain matrix for
hi (t ) xi (i 1, 2,}, n) . From (3), MFPS means that e(t)
satisfies the relationship lim e
Considering the advantages of better stability and robustness
of the SMC system,therefore, the controller can be written
u Kw(t ) F ( x , y ,t )
2,}, n) are constants, MFPS will be simplified into GPS.
H (t ) f1 ( x, t ) Dtq ( H (t ) x)
where Н (t ) diag(h1 (t ), h2 (t ),..., hn (t )) is called the
scaling function matrix, and diag(.) means diagonal matrix,
scaling function factors hi (t ) z 0(i 1, 2, }, n) are
differentiable bounded functions, ||.||denotes the matrix
Obviously, the MFPS scheme becomes FPS when.
h1 (t ) h2 (t ) } hn (t ) Furthermore, while hi (t )(i 1,
xT QDD x d 0
F ( x , y ,t )= f 2 ( y, t ) H (t )( A1 A2 ) x
Definethe synchronization error vector as
e (t ) = y H (t ) x
Q ! 0 , and the following relationship is
According to the principle of active control, F ( x, y, t ) is
introduced as following form
Definition1 For arbitrary initial values x (0) and y (0) ,
system(1) and system(2) are said to be MFPS if there exists
control vector u such that
. If there exists a
f1 ( x, t ) d x (t )) D q ( H (t ) x )
disturbance of drive system and response system,
lim y H (t ) x
From (1), (2) and (4), we can obtain the following
dynamical model of the MFPS error
}, d yn )  R represent the bounded and differentiable
,then system (8) is asymptotically stable
A( x) x
) ( x)
y ( y1 , y2 ,}, yn )T  Rn denote
system state variables, A1 and A2 represent constant matrix,
functions, u
F ( x)
where 0 D 1 , x  R and A  R
where D indicates the fractional derivative of Caputo
definition[24], fractional order number q  (0,1) , x ( x1 ,
x2 ,}, xn )T  R n and
where 0 D 0 , x  R and F  R If all the
eigenvalues O of the Jacobian matrix J wF ( x) / wx
calculated at the equilibrium points satisfy | arg(O ) |! DS / 2 ,
then system (7) is locally asymptotically stable.
Lemma 3 Consider an autonomous system described as
MFPS of fractional-order hyperchaotic system
Dq x
F ( x)
w(t )= (CK )1[H sgn( s) Ps CA2e]
wherethe switching part plays a role of suppressing
disturbances. Substituting (17) into (12), thus, the controller
can be expressed as follows
Lemma 2 For a commensurate fractional-order nonlinear
dynamical system[26]
u K (CK )1[H sgn( s) Ps CA2e ] f 2 ( y, t )
­ Dtq y1 b1 ( y2 y1 ) y4 d y1 u1
° q
° Dt y2 b2 y1 y1 y3 y2 d y 2 u2
® q
° Dt y3 y1 y2 b3 y3 d y 3 u3
°Dq y y y b y d u
1 3
4 4
¯ t 4
H (t )( A1 A2 ) x H (t ) f1 ( x, t ) D q ( H (t ) x )
Stability analysis
Selecting a candidate function is selected as follows
If q 0.95, b1 10, b2 28, b3 8 / 3, b4 1 , the Lorenz
system displays the hyperchaotic behavior[29].
eT D q e
,substituting Eqs.(10),(18) into (19), and using commutative
law of multiplication between diagonal matrixes, we have
From (23) and (24), one can see that x
and y
e T { A2 e K (CK ) 1 C [H sgn(e ) ( P A2 )e ]
u2 , u3 , u4 )T is control input vector, corresponding system
d y (t ) H (t )d x (t )}
matrixes and nonlinear functionvectors are described as
e T ˜ [ H (t )d x (t )]
H ¦ ei e T Pe e T d y (t ) e T [ H (t )d x (t )]
i 1
i 1
i 1
d H ¦ ei r e (a b)¦ ei
max{| h1 (t )d x1 (t ) |, | h2 (t )d x 2 (t ) |,},| hn (t )d xn (t ) |} ,
b max{| d y1 (t ) , d y 2 (t ) , }, d y n (t ) |} , r
min{ p1 ,
Let us presume the following conditions satisfied
, then we get
\ d (H a b)¦ ei r e d 0
d y (t )
y(0) (12, 9,15,3)
>0.1sin(t ),0.5sin(t ),0.2sin(t ),0.3sin(t )@
scaling function matrix H (t )
, the
diag(2sin(t ),sin(t ) 4,
cos(3t ),cos(t ) 0.2) , the parameters and gain matrixes of
the controller r 1.2 , K = diag(1,1,1,1) , P diag(50,
70,42,35) and C diag(5,5,5,5) , step size is set as
i 1
From the Lemma 3, the error system is asymptotically
Simulation results
0.002. For indicating the reliability of the controller, we use
the predictor-corrector algorithm to obtain numerical
solution of the fractional-order systems[30], and implement
the control operation after t 4s .
From figure 1 and 2, we see that synchronization errors
ei (i 1,2,3,4) converge to zeros quickly after control, and
In this section,an example of the MFPS is provided to
verify the effectiveness of our method, in which
fractional-order hyperchaotic Chen system and Lorenz
system are chosen asthe drive system and response system,
The fractional-order Chen system with external
disturbances is expressed as
­ Dtq x1 a1 ( x2 x1 ) x4 d x1
° q
° Dt x2 a2 x1 x1 x3 a3 x2 d x 2
® q
° Dt x3 x1 x2 a4 x3 d x 3
° D q x x x a x d
¯ t 4 2 3 5 4
d x (t ) [0.8sin(0.2t ),0.6sin(0.5t ),0.4cos(2t ),0.5cos(0.6t )]T ,
(8, 5,
In the simulation, initial values of system x(0)
p2 ,}, pn } .
H t ab
( y1 , y2 , y3 , y4 )T are system state vectors, u (u1 ,
1 ·
§ 0 ·
§ 35 35 0
x x
7 12 0
0 ¸
, f1 ( x ,t ) ¨ 1 3 ¸
A1 = ¨
¨ x1 x2 ¸
¨ 0
0 3 0 ¸
0 0 0.5 ¹
© 0
© x2 x3 ¹
§ 0 ·
§ 10 10
y1 y3 ¸
28 1
, f 2 ( y ,t )
A2 =
¨ y1 y2 ¸
¨ 0
0 8 / 3 0 ¸
© 0
© y1 y3 ¹
H e sgn(e ) re ˜ e e ˜ d y (t )
( x1 , x2 , x3 , x4 )T
eT Dqe
e T [H sgn(e ) Pe d y (t ) H (t )d x (t )]
the relationship yi hi (t ) xi (i 1,2,3, 4) can be satisfied,
which means the MFPS between system (23) and system (24)
is obtained.
when q 0.95, a1 35, a2 7, a3 12, a4 3, a5 0.5 ,
the Chen system shows hyperchaos[28].
The fractional-order Lorenz system with disturbances is
describedas follows
Fig. 2: The state trajectories of the drive and response systems
In this paper, a new active SMC method for realizing
MFPS of two different fractional-order hyperchaotic
systems with external disturbances is presented. Based on
the stability theory of fraction-order system, the active
sliding mode controller is designed. An example of MFPS
and its simulation results show the validity of the proposed
method. The research results can provide a useful reference
for achieving more complex and effective synchronization
relationship of the fractional-order hyperchaotic systems
with disturbances.
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