- Email: [email protected]
Synchronization of Arneodo chaotic system via backstepping fuzzy adaptive control
Accepted Manuscript Title: Synchronization of Arneodo chaotic system via backstepping fuzzy adaptive control Author: Wen-qing Wang Yong-qing Fan PII: DOI: Reference:
S0030-4026(15)00548-3 http://dx.doi.org/doi:10.1016/j.ijleo.2015.06.071 IJLEO 55711
To appear in: Received date: Accepted date:
8-5-2014 26-6-2015
Please cite this article as: W.-q. Wang, Synchronization of Arneodo chaotic system via backstepping fuzzy adaptive control, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.06.071 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.
Synchronization of Arneodo chaotic system via backstepping fuzzy adaptive control Wen-qing Wang, Yong-qing Fan
ip t
(School of Automation, Xi’an University of Posts and Telecommunications, Xi’an 710121 P. R. China)
Abstract: This paper deals with the synchronization for Arneodo chaotic system and response system with unknown nonlinear function.
Based on backstepping method, a fuzzy adaptive control scheme
cr
by combining fuzzy logic system with parameter is presented to achieve synchronization. Contrasting
us
to fuzzy adaptive controller in the previous literature, the number of adaptive laws reduced greatly by using the proposed controller in this paper. Numerical simulation has been validated the backstepping design and analysis.
an
Keywords: arenodo system; synchronization; fuzzy adaptive control; backstepping.
1. Introduction
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The problem of chaos synchronization is a common and widespread phenomenon in many science and engineering fields[1]. In recent years, many methods of
d
driver-response synchronization have been proposed. In [2-6], sliding mode variable
te
structure controllers are designed to achieve the synchronization of chaotic systems. For example, to study the transition of phase synchronization and complete
Ac ce p
synchronization with the unknown four parameters in the drive system, adaptive schemes are proposed in [7]. For a class of fractional-order chaotic and hyperchaotic systems with unknown Lipschitz constant, the adaptive impulsive synchronization is investigated in [8]. The backstepping methods of control can be used to obtain the synchronization process [9-12]. For nonlinear fractional-order hyperchaotic systems, a nonlinear control technique is proposed for synchronization in [13]. Besides, more results on driver-response synchronization can be found in [14,15]. To the best of our knowledge, no matter what method authors uses for investigating synchronization of the driver-response systems, the control schemes mainly associated with the obtainable information from the considered or chaotic systems. --------------------------------------Mailing Address: Box No. 494, School of Automation of Xi’an University of Posts and Telecommunications, Weiguo Road, Xi’an Higher Education Mega Center, Chang’an District, Xi’an, P. R. China. e-mail: [email protected],
e-mail: [email protected]
TEL: (029)88166461
Page 1 of 14
However, for the dynamical systems with unmodeled dynamics, which will motivates us for introducing of the fuzzy logic systems that may codify the meanings of expert or intuitive with linguistically expressible knowledge. On the anther hand, it is difficult to obtain the synchronization in the drive and
ip t
response systems with uncertainties or unknown nonlinearities. Form [16], we know that fuzzy logic systems are universal approximations, so fuzzy logic systems can be
cr
employed to approximate the uncertainties or unknown nonlinearities in chaotic systems. More recently, this approach was applied to synchronization of drive and
us
response systems in [17-19]. In these references, it should be mentioned that the synchronization schemes are based on that the outputs of Mamdani fuzzy logic
an
systems can be represented as a linear combination of certain fuzzy basic functions, and then the combinatorial coefficients can be estimated via adaptive laws. The
M
problem is that the number of adaptive laws depends on the number of fuzzy rules by using this method for system parameterization, which results in a large number of
d
adaptive laws to estimate the parameters. The disadvantage of such fuzzy adaptive controllers requires long on-line computation time and creates a large delay that will
te
violate the extreme sensitivity of chaotic systems. From this analysis, it is necessary
Ac ce p
to explore a new adaptive fuzzy control method for synchronization of chaotic systems.
In this paper, we consider use fuzzy logic system with parameter to solve
synchronization problems of drive and response systems. The parameter in fuzzy logic system is relation to the approximation accuracies, and has nothing to do with the logic structures of the fuzzy logic systems, which may reduce the on-line computational burden and avoid time delay. In this way, the number of the adaptive laws will reduce in corresponding. This paper is organized into the following sections. The dynamic models of 3-D Arneodo dynamics and slave systems are given in Section 2, and a common time-varying parameter is introduced into the outputs of fuzzy logic system to synthesize the adaptive fuzzy controllers for synchronization of chaotic systems. In Section 3, fuzzy adaptive controller synthesized based on the updated laws of
Page 2 of 14
parameters. An illustrative example is presented to demonstrate the proposed design procedure in Section 4. Finally, the conclusion is given in Section 5. 2. Model description and assumptions The following 3-D Arneodo dynamics as the drive system is considered
ip t
x1 x2 x2 x3 x ax bx x f ( x) 1 2 3 3
cr
(1)
where x ( x1 , x2 , x3 )T R 3 is the state vector, a and b are two known parameters
us
of the system, f ( x) R 3 is nonlinear continuous function.
Now, we consider the controlled 3-D Arneodo response system as follows:
an
y1 y2 y 2 y3 y3 ay1 by2 y3 f ( y ) gu
M
(2)
in which, y ( y1 , y2 , y3 )T R 3 is the state vector in the response system, g is
d
control gain and satisfies g g g , u is controller to be designed, nonlinear
te
continuous function f ( y ) R 3 .
The synchronizaiton error definded as e1 y1 x1 , e2 y2 x2 , e3 y3 x3 , then the
Ac ce p
error dynamics is obtained as
e1 e2 e2 e3 e3 ae1 be2 e3 f ( y ) f ( x) gu
(3)
t
The design problem is to design a contoller u so that the error ei (t ) 0 for i 1, 2,3 .
Before presenting the main result of this paper, the following useful assumptions are introduced. Assumption 1: f ( z ) satisfies Lipschitz condition on a compact set W R 3 , that is, there exists a positive constant L (maybe unknown ) such that f ( z1 ) f ( z2 ) L z1 z2 for z1 , z2 W .
Page 3 of 14
Assumption 2: (ⅰ) Under Assumption 1, there exists a FLS F1 with the following rules (4) and an unknown positive real constant 1 satisfy sup f ( x) F1 ( x) 1 . () xV1
There exists a FLS F2 with the following rules (4) and an unknown positive real
ip t
constant 2 such that sup f ( y ) F2 ( y ) 2 . yV2
discourse W R n with the following form rules.
cr
Now, consider the following Mamdani type fuzzy logic systems on the universe of
If x1 is A1l and x2 is A2l and x3 is Anl , Then z1 is Bl , l1 1, 2, , p 1
1
1
(4)
us
1
where Akl ( k 1, 2,3 ), Bl denote fuzzy sets, Akl ( x) , denotes the membership function 1
1
1
of Akl .
an
1
If singleton fuzzifier, product inference and center-average defuzzifier are utilized,
M
then the output of fuzzy systems (4) is of the following form: p
l1 1 p
l1
k 1
l1 k
( xk )
n
Akl1 ( xk )
d
z1 F1 ( z1 )
n
B A
.
(5)
te
l1 1 k 1
In this paper, we introduce a non-zero time-varying parameter (t ) in (5) to
Ac ce p
produce the following result.
z z1 F1 ( 1 )
p
n
Bl1 Akl1 ( l1 1 p
k 1
n
Akl1 ( l1 1 k 1
xk )
xk )
.
(6)
Remark 1. If Assumption 2 is satisfied, for simplicity, we denote 1 2 to
represent approximation error.
3. Backstepping fuzzy adaptive control design In many practical engineering, the parameters
and
L
are unknown.
Let ˆ ˆ (t ) and Lˆ Lˆ (t ) denote the estimation values of and L , respectively. ˆ and L Lˆ L denote the estimate errors, respectively. The controller u in
response system (2) will be designed by the following control goal.
Page 4 of 14
Control goal: Design the fuzzy adaptive controller such that the synchronization error vector e (e1 , e2 , e3 )T is lim e(t ) 0 . t
In this section, based on the fuzzy adaptive control and backstepping method, we
ip t
propose the backstepping fuzzy adaptive controller be designed as in Theorem 1. Theorem 1. The Arenodo chaotic systems (1) and (2) can be synchronized by the backstepping fuzzy adaptive controller
cr
z z
(7)
us
0, u ua ub ,
an
z3 [ g kz3 3 / 2 a e1 3 b e2 3 / 2 e3 ] , z0 g z3 ua kz3 v , v . z0 0,
M
x y ub F1 ( ) F2 ( )
Updated laws:
te
d
1 ˆ 2 [r z3 ( 3 / 2 a e1 3 b e2 3 / 2 e3 L e )], z [ Lˆ (1 ) e g ˆ], z
Ac ce p
z3 e , Lˆ (1 ) e ,
(8)
z
(9)
z
0, z ˆ g , z
(10)
where z ( z1 , z2 , z3 )T , k , , , , are adjustable positive constants. is a positive
designing constant such that {z z } W . Proof: step 1:
we define z1 e1 , and consider a Lyapunov function candidate as V1
1 2 z1 2
(11)
Defferentiating V1 yields
V1 z1 z1 e1e1 e1e2 z12 z1 (e1 e2 )
(12)
Page 5 of 14
Let e1 e2 z2 , from (12), we can obtain the following inequality z 2 z22 z2 z2 V1 z12 z1 z2 z12 1 1 2 2 2 2
z12 z22 z2 (e2 e3 ) 2 2
an
z12 z22 z2 ( z2 e2 e3 z2 ) 2 2
z12 z22 z2 (e1 2e2 e3 ) 2 2
M
us
V2 V1 z2 z2 V1 z2 (e1 e2 )
ip t
The time derivative of V2 is obtained by
(14)
cr
step 2: the following Lyapunov function is choose 1 1 V2 V1 z22 ( z12 z22 ) 2 2
(13)
(15)
d
If we let z3 e1 2e2 e3 , then the following inequality holds.
te
z2 z2 V2 1 2 z2 z3 2 2
Ac ce p
step 3: Choose a Lyapunov function candidate as follows: 1 1 V V2 z32 ( z12 z22 z32 ) 2 2 The time dericative of V is given as:
(16)
(17)
z2 z2 V V2 z3 z3 1 2 z3 ( z2 z3 ) 2 2
z12 z22 z32 z z3 ( 3 z2 z3 ) 2 2 2 2
3 3 z12 z22 z32 z3 [( a )e1 (3 b)e2 e3 f ( y ) f ( x) gu ] 2 2 2 2 2
Case (1): z
1 2 1 2 2 1 2 1 2 z L , It is 2 2 2 2 shown that s 0 , so we consider the following positive definition function about s, In this case, we adopt open-loop control. Let s
Page 6 of 14
V
1 2 s . The derivative of V about t along the (3) is obtained: 2
3 3 z2 z2 z2 V ss s{ 1 2 3 z3 [( a )e1 (3 b)e2 e3 2 2 2 2 2
ip t
ˆ} ˆ 1 LL f ( y ) f ( x)] 2 1
3 3 ˆ} ˆ 1 LL a e1 3 b e2 e3 f ( y ) f ( x) ] 2 1 2 2
s{ z3 [
3 3 ˆ} ˆ 1 LL a e1 3 b e2 e3 L e ] 2 1 2 2
s{ z3 [
3 3 ˆ 1 L ( Lˆ z3 e )} a e1 3 b e2 e3 Lˆ e ] 2 1 2 2
rs
an
us
cr
s{ z3 [
(18)
sliding surface s 0 in finite times. Case (2): z
M
From the result of [20], (18) implies that the error state of the (3) can reach on the
d
The controller (7) is adopted in this case, and the following Lyapunov candidate is considered:
Ac ce p
te
1 1 2 1 2 1 2 V ( z12 z22 z32 ) L 2 2 2 2
(19)
3 3 z2 z2 z2 V 1 2 3 z3 [( a)e1 (3 b)e2 e3 f ( y ) f ( x) gua gub ] 2 2 2 2 2 ˆ ˆ 1 LL 1 1
3 3 z3[( a )e1 (3 b)e2 e3 gua ] z3[ f ( y ) f ( x) gub ] 2 2 ˆ ˆ 1 LL 1 1
(20)
By employing the controller (7), we can obtain that 3 3 z3 [( a )e1 (3 b)e2 e3 gua ] 2 2
gz [ g kz3 3 / 2 a e1 3 b e2 3 / 2 e3 ] 3 3 z3{( a )e1 (3 b)e2 e3 gkz3 3 } g 2 2
Page 7 of 14
z3 [
3 3 g a e1 3 b e2 e3 g kz3 ] z3 [ g kz3 3 / 2 a e1 3 b e2 3 / 2 e3 ] 2 2 g 3 3 g z3 [ a e1 3 b e2 e3 g kz3 ](1 ) 2 2 g
0 From (20) and (21), the following inequality is hold.
1 1 1 x 1 y 1 y y x 1 x f ( y ) f ( x) f ( ) f ( ) f ( ) F2 ( ) F1 ( ) f ( )] g g g g g g
an
gz3[
1 1 x y ˆ ˆ 1 LL f ( y ) f ( x) F1 ( ) F2 ( )] 1 1 g g
us
gz3 [
cr
ˆ ˆ 1 LL V z3 [ f ( y ) f ( x) gub ] 1 1
ip t
(21)
ˆ ˆ 1 LL 1 1
1 ˆ ˆ 1 LL L e z3 g 1 1
M
z3 L e z3
d
Lˆ (1 ) e g ˆ 1 1 (ˆ g ) 1L[ Lˆ (1 ) e ] (22)
te
0
The above inequality (22) means that the error state of (3) is bounded, and thus t
Ac ce p
e 0 by employing Barbalat's Lemma [20]. Finally, Case 1 and Case 2 complete the
proof of Theorem 1.
4. Simulation Examples
In this section, we present 3-D Arneodo dynamics (1) and (2) to illustrate the
effectiveness of the proposed method, the parameters in (1) and (2) are chosen as a 7.5 , b 3.8 . The initial values of the drive system (1) are chosen as x1 (0) 14 ,
x2 (0) 5 , x3 (0) 6 , and initial values of the response system (2) are chosen as y1 (0) 1 , y2 (0) 12 , y3 (0) 1.6 . The simulation results without any controller are shown as Figure 1.
Page 8 of 14
20 10 0 -10
0
5
10
15
20
25 Time(sec) (a)
30
0
5
10
15
20
25 Time(sec) (b)
30
0
5
10
15
20
35
40
45
cr
20 0
-20
0
M
-20 -40
40
45
50
25 Time(sec) (c)
30
35
40
45
50
d
y3 d n a 3 x
35
an
20
us
y2 d n a 2 x
50
ip t
y1 d n a 1 x
te
Figure 1. Time response of x and y without any controller In order to obtain synchronization between drive system and response system by
Ac ce p
using controller (7), the parameters are selected as
40 ,
k 500 ,
0.0001 , 0.0005 , 0.0001 , 0.0002 . Now, we approximate the two
continuous unknown nonlinear functions f ( x) and f ( y ) . The FLSs F1 ( x) and F2 ( y )
are constructed as (23)-(24) by choosing 6 fuzzy rules A1l A2l B l ,
A1l A2l Bl ( l 1, 2,3 ),respectively: RF(1l ) : If x1 is Al ( x1 ) and x2 is Al ( x2 ) and x3 is Al ( x3 ) , 1
2
3
Then 1 ( x ) is B ( y1 ) l
(23)
RF(l2) : If y1 is A l ( y1 ) and y2 is A l ( y2 ) and y3 is Al ( y3 ) , 1
2
3
Then 2 ( y ) is B ( y2 ) l
(24)
Page 9 of 14
2
2
in (23), the membership functions are selected as A e h ( x 40) , A e h ( x 40) , 1
1 1
2
2
2
2
1 2
2
2
A e h ( x 40) , A e h ( x 0.0001) , A e h ( x 0.0001) , A e h ( x 0.0001) , A e h ( x 40) , 3
1 3
1
2 1
2
2
2 2
2
3
2 3
2
1
3 1
2
2
A e h ( x 40) , A e h ( x 40) , B e ( y 160) , B e ( y 0.0001) , B e ( y 160) , where 2
3 2
3
3 3
1
1
1
2
1
3
2
ip t
parameter h 10 . We select the membership functions in (24) as: A e h ( y 40) , 1
1 1
2
1 2
2
2
A e h ( y 40) , 1
3 1
3
1 3
2
A e h ( y
2 40)
3 2
1
2 1
2
2 2
2
2 0.0001)
3
3 3
2
1
2
10
0
5
10
0
5
10
15
20
te
0
5
10
15
20
0
-20
Ac ce p
y2 d n a 2 x
2
2
2
20
25 Time(sec) (a)
30
35
40
45
50
25 Time(sec) (b)
30
35
40
45
50
25 Time(sec) (c)
30
35
40
45
50
d
20
15
M
0 -10
2
an
20 y1 d n a 1 x
3
2 3
us
2
2
, A e h ( y 0.0001) ,
, A e h ( y 40) , B e ( y 160) , B e ( y 0.0001) ,
B e ( y 160) . The simulation results are shown in Figure 2-4. 3
2
cr
2
A e h ( y 40) , A e h ( y 40) , A e h ( y 0.0001) , A e h ( y
20
y3 d n a 3 x
0
-20 -40
Figure 2. Time response of x and y
Page 10 of 14
15 10
e
1
5 0 -5
0
5
10
15
20
25 Time(sec) (a)
30
35
40
0
5
10
15
20
25 Time(sec) (b)
30
35
40
0
5
10
15
20
25 Time(sec) (c)
30
45
50
0 2
-10 -20
10 e
3
0
35
50
40
45
50
an
-10
us
20
45
cr
e
ip t
10
Ac ce p
te
d
M
Figure 3. Time response of errors between drive system and response system
Figure 4. Time response of errors between drive system and response system In Figure 2, the blue line denotes the states responses of drive system, the red line represent the states responses of response system by using fuzzy adaptive controller (7). From Figure 3, it is clear that states of the drive and response systems can be synchronized. Figure 3 shows that synchronization errors can converged to zero or
Page 11 of 14
near zero between the drive system (1) and response system (2). Figure 4 is the simulation result of the parameter time response in controller (7). 5. Conclusions A novel design fuzzy adaptive control based on backstepping method is proposed
ip t
for 3-D systems’ chaotic synchronization. The main advantage of the control scheme is that only three common parameters are needed to be adjusted automatically, which
cr
enable the number of adaptive laws reduced greatly such that the on-line
computational burden is reduced. In addition, fuzzy logic systems have general form
us
and the number of update laws has nothing to do with the number of IF_THEN rules in this paper, which make designer pay his or her attention to construct the fuzzy logic
an
systems to approximate the unknown functions in the dynamic equations of the master and slave systems. Therefore, the methods based on intuition inferences can be employed to generate the fuzzy systems with fewer fuzzy rules and high
M
interpretability. The methods fuzzy adaptive synchronization controller is a most advantage which will benefit to the engineering application.
d
Acknowledgment (61305098),
the
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This research was supported in part by the Natural Science Foundation of China Youth
Foundation
of
Xi'an
University
of
Posts
and
Ac ce p
Telecommunications (110-0435). References
[1]. A. Pikovsky, M. Roseblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, New York: Cambridge University Press, 2003
[2]. A. Bagheri, J. J. Moghaddam, Decoupled adaptive neuro-fuzzy(DANF) sliding mode control system for a chaotic problem, Expert System with Applications,
36(3-2): 6062-6068, 2009
[3]. M. Yahyazadeh, A. R. Noei, R. Ghaderi, Synchronization of chaotic systems with known and unknown parameters using a modified active sliding mode control, ISA Transactions, 50(2): 262–267, 2011 [4]. D. I. R. Almeida, J. Alvarez, J. G. Barajas, Robust synchronization of Sprott circuits using sliding mode control. Chaos, Solitons & Fractals 30(1): 11–18,
Page 12 of 14
2006 [5]. M. C. Pai, Global synchronization of uncertain chaotic systems via discrete-time sliding mode control, Applied Mathematics and Computation, 227(5): 663-671, 2014
ip t
[6]. C. Mou, C. S. Jiang, J. Bin, Q. X. Wu, Sliding mode synchronization controller
design with neural network for uncertain chaotic systems, Chaos, Solitons &
cr
Fractals, 39(4):1856–1863, 2009
[7]. J. Ma, F. Li, L. Huang, W. Y. Jin, Complete synchronization, phase
us
synchronization and parameters estimation in a realistic chaotic system. Communications in Nonlinear Science and Numerical Simulation, 16(9):
an
3770–3785, 2011
[8]. H. Xi, S. Yu, R. Zhang, L. Xu, Adaptive impulsive synchronization for a class of
M
fractional-order chaotic and hyperchaotic systems, Optic-International Journal for Light and Electron Optics, 125(9): 2036-2040, 2014
d
[9]. G. Li, A. Khajepour, Robust control of a hydraulically driven flexible arm using
2005
J. H. Park, Synchronization of Genesio chaotic system via backstepping
Ac ce p
[10].
te
backstepping technique, Journal of Sound and Vibration, 280(3-5): 759-775,
approach, Chaos, Solitons & Fractals 27(5): 1369–1375, 2006
[11].
F. X. Chen, L. Chen, W. D. Zhang, Stabilization of parameters perturbation
chaotic system via adaptive backstepping technique, Applied Mathematics and Computation, 200(1):101-109, 2008
[12].
K. B. Deng, J. Li, S. M. Yu, Dynamics analysis and synchronization of a new
chaotic attractor, Optic, (In Press, Available online 20 March), 2014
[13].
A. E. Matouk, A.A. Elsadany, Achieving synchronization between the
fractional-order hyperchaotic Novel and Chen systems via a new nonlinear control technique, Applied Mathematics Letters, 29: 30-35, 2014 [14].
H. Y. Du, Function projective synchronization in drive-response dynamical
networks with non-identical nodes, Chaos, Solitons & Fractals, 44(7): 510-514, 2011
Page 13 of 14
[15].
L. Lü, C. Li, L. Chen, L. Wei, Lag projective synchronization of a class of
complex network constituted nodes with chaotic behavior, Commun Nonlinear Sci Numer Simulat, 19(8): 2843-2849, 2014 [16].
L. X. Wang, J. M. Mendel, Fuzzy basis functions, universal approximation,
ip t
and orthogonal least squares learning, IEEE Transactions on Neural Networks, 3(5): 807–814, 1992
B. Chen, X. Liu, S. Tong, Adaptive fuzzy approach to control unified chaotic
cr
[17].
systems, Chaos, Solitons & Fractals 34(4): 1180–1187, 2007
S. Y. Li, Z. M. Ge, Generalized synchronization of chaotic systems with
us
[18].
Applications, 38(3): 2302–2310, 2011 [19].
an
different orders by fuzzy logic constant controller, Expert Systems with
A. Poursamad, A. H. D. Markazi, Adaptive fuzzy sliding mode control for
M
multi-input multi-output chaotic systems. Chaos, Solitons & Fractals 42(5): 3100–3109, 2009
J.J.E. Slotine, W. Li, Applied Nonlinear Control, Prentice Hall, Englewood
Ac ce p
te
Cliffs, 1991
d
[20].
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S0030-4026(15)00548-3 http://dx.doi.org/doi:10.1016/j.ijleo.2015.06.071 IJLEO 55711
To appear in: Received date: Accepted date:
8-5-2014 26-6-2015
Please cite this article as: W.-q. Wang, Synchronization of Arneodo chaotic system via backstepping fuzzy adaptive control, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.06.071 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.
Synchronization of Arneodo chaotic system via backstepping fuzzy adaptive control Wen-qing Wang, Yong-qing Fan
ip t
(School of Automation, Xi’an University of Posts and Telecommunications, Xi’an 710121 P. R. China)
Abstract: This paper deals with the synchronization for Arneodo chaotic system and response system with unknown nonlinear function.
Based on backstepping method, a fuzzy adaptive control scheme
cr
by combining fuzzy logic system with parameter is presented to achieve synchronization. Contrasting
us
to fuzzy adaptive controller in the previous literature, the number of adaptive laws reduced greatly by using the proposed controller in this paper. Numerical simulation has been validated the backstepping design and analysis.
an
Keywords: arenodo system; synchronization; fuzzy adaptive control; backstepping.
1. Introduction
M
The problem of chaos synchronization is a common and widespread phenomenon in many science and engineering fields[1]. In recent years, many methods of
d
driver-response synchronization have been proposed. In [2-6], sliding mode variable
te
structure controllers are designed to achieve the synchronization of chaotic systems. For example, to study the transition of phase synchronization and complete
Ac ce p
synchronization with the unknown four parameters in the drive system, adaptive schemes are proposed in [7]. For a class of fractional-order chaotic and hyperchaotic systems with unknown Lipschitz constant, the adaptive impulsive synchronization is investigated in [8]. The backstepping methods of control can be used to obtain the synchronization process [9-12]. For nonlinear fractional-order hyperchaotic systems, a nonlinear control technique is proposed for synchronization in [13]. Besides, more results on driver-response synchronization can be found in [14,15]. To the best of our knowledge, no matter what method authors uses for investigating synchronization of the driver-response systems, the control schemes mainly associated with the obtainable information from the considered or chaotic systems. --------------------------------------Mailing Address: Box No. 494, School of Automation of Xi’an University of Posts and Telecommunications, Weiguo Road, Xi’an Higher Education Mega Center, Chang’an District, Xi’an, P. R. China. e-mail: [email protected],
e-mail: [email protected]
TEL: (029)88166461
Page 1 of 14
However, for the dynamical systems with unmodeled dynamics, which will motivates us for introducing of the fuzzy logic systems that may codify the meanings of expert or intuitive with linguistically expressible knowledge. On the anther hand, it is difficult to obtain the synchronization in the drive and
ip t
response systems with uncertainties or unknown nonlinearities. Form [16], we know that fuzzy logic systems are universal approximations, so fuzzy logic systems can be
cr
employed to approximate the uncertainties or unknown nonlinearities in chaotic systems. More recently, this approach was applied to synchronization of drive and
us
response systems in [17-19]. In these references, it should be mentioned that the synchronization schemes are based on that the outputs of Mamdani fuzzy logic
an
systems can be represented as a linear combination of certain fuzzy basic functions, and then the combinatorial coefficients can be estimated via adaptive laws. The
M
problem is that the number of adaptive laws depends on the number of fuzzy rules by using this method for system parameterization, which results in a large number of
d
adaptive laws to estimate the parameters. The disadvantage of such fuzzy adaptive controllers requires long on-line computation time and creates a large delay that will
te
violate the extreme sensitivity of chaotic systems. From this analysis, it is necessary
Ac ce p
to explore a new adaptive fuzzy control method for synchronization of chaotic systems.
In this paper, we consider use fuzzy logic system with parameter to solve
synchronization problems of drive and response systems. The parameter in fuzzy logic system is relation to the approximation accuracies, and has nothing to do with the logic structures of the fuzzy logic systems, which may reduce the on-line computational burden and avoid time delay. In this way, the number of the adaptive laws will reduce in corresponding. This paper is organized into the following sections. The dynamic models of 3-D Arneodo dynamics and slave systems are given in Section 2, and a common time-varying parameter is introduced into the outputs of fuzzy logic system to synthesize the adaptive fuzzy controllers for synchronization of chaotic systems. In Section 3, fuzzy adaptive controller synthesized based on the updated laws of
Page 2 of 14
parameters. An illustrative example is presented to demonstrate the proposed design procedure in Section 4. Finally, the conclusion is given in Section 5. 2. Model description and assumptions The following 3-D Arneodo dynamics as the drive system is considered
ip t
x1 x2 x2 x3 x ax bx x f ( x) 1 2 3 3
cr
(1)
where x ( x1 , x2 , x3 )T R 3 is the state vector, a and b are two known parameters
us
of the system, f ( x) R 3 is nonlinear continuous function.
Now, we consider the controlled 3-D Arneodo response system as follows:
an
y1 y2 y 2 y3 y3 ay1 by2 y3 f ( y ) gu
M
(2)
in which, y ( y1 , y2 , y3 )T R 3 is the state vector in the response system, g is
d
control gain and satisfies g g g , u is controller to be designed, nonlinear
te
continuous function f ( y ) R 3 .
The synchronizaiton error definded as e1 y1 x1 , e2 y2 x2 , e3 y3 x3 , then the
Ac ce p
error dynamics is obtained as
e1 e2 e2 e3 e3 ae1 be2 e3 f ( y ) f ( x) gu
(3)
t
The design problem is to design a contoller u so that the error ei (t ) 0 for i 1, 2,3 .
Before presenting the main result of this paper, the following useful assumptions are introduced. Assumption 1: f ( z ) satisfies Lipschitz condition on a compact set W R 3 , that is, there exists a positive constant L (maybe unknown ) such that f ( z1 ) f ( z2 ) L z1 z2 for z1 , z2 W .
Page 3 of 14
Assumption 2: (ⅰ) Under Assumption 1, there exists a FLS F1 with the following rules (4) and an unknown positive real constant 1 satisfy sup f ( x) F1 ( x) 1 . () xV1
There exists a FLS F2 with the following rules (4) and an unknown positive real
ip t
constant 2 such that sup f ( y ) F2 ( y ) 2 . yV2
discourse W R n with the following form rules.
cr
Now, consider the following Mamdani type fuzzy logic systems on the universe of
If x1 is A1l and x2 is A2l and x3 is Anl , Then z1 is Bl , l1 1, 2, , p 1
1
1
(4)
us
1
where Akl ( k 1, 2,3 ), Bl denote fuzzy sets, Akl ( x) , denotes the membership function 1
1
1
of Akl .
an
1
If singleton fuzzifier, product inference and center-average defuzzifier are utilized,
M
then the output of fuzzy systems (4) is of the following form: p
l1 1 p
l1
k 1
l1 k
( xk )
n
Akl1 ( xk )
d
z1 F1 ( z1 )
n
B A
.
(5)
te
l1 1 k 1
In this paper, we introduce a non-zero time-varying parameter (t ) in (5) to
Ac ce p
produce the following result.
z z1 F1 ( 1 )
p
n
Bl1 Akl1 ( l1 1 p
k 1
n
Akl1 ( l1 1 k 1
xk )
xk )
.
(6)
Remark 1. If Assumption 2 is satisfied, for simplicity, we denote 1 2 to
represent approximation error.
3. Backstepping fuzzy adaptive control design In many practical engineering, the parameters
and
L
are unknown.
Let ˆ ˆ (t ) and Lˆ Lˆ (t ) denote the estimation values of and L , respectively. ˆ and L Lˆ L denote the estimate errors, respectively. The controller u in
response system (2) will be designed by the following control goal.
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Control goal: Design the fuzzy adaptive controller such that the synchronization error vector e (e1 , e2 , e3 )T is lim e(t ) 0 . t
In this section, based on the fuzzy adaptive control and backstepping method, we
ip t
propose the backstepping fuzzy adaptive controller be designed as in Theorem 1. Theorem 1. The Arenodo chaotic systems (1) and (2) can be synchronized by the backstepping fuzzy adaptive controller
cr
z z
(7)
us
0, u ua ub ,
an
z3 [ g kz3 3 / 2 a e1 3 b e2 3 / 2 e3 ] , z0 g z3 ua kz3 v , v . z0 0,
M
x y ub F1 ( ) F2 ( )
Updated laws:
te
d
1 ˆ 2 [r z3 ( 3 / 2 a e1 3 b e2 3 / 2 e3 L e )], z [ Lˆ (1 ) e g ˆ], z
Ac ce p
z3 e , Lˆ (1 ) e ,
(8)
z
(9)
z
0, z ˆ g , z
(10)
where z ( z1 , z2 , z3 )T , k , , , , are adjustable positive constants. is a positive
designing constant such that {z z } W . Proof: step 1:
we define z1 e1 , and consider a Lyapunov function candidate as V1
1 2 z1 2
(11)
Defferentiating V1 yields
V1 z1 z1 e1e1 e1e2 z12 z1 (e1 e2 )
(12)
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Let e1 e2 z2 , from (12), we can obtain the following inequality z 2 z22 z2 z2 V1 z12 z1 z2 z12 1 1 2 2 2 2
z12 z22 z2 (e2 e3 ) 2 2
an
z12 z22 z2 ( z2 e2 e3 z2 ) 2 2
z12 z22 z2 (e1 2e2 e3 ) 2 2
M
us
V2 V1 z2 z2 V1 z2 (e1 e2 )
ip t
The time derivative of V2 is obtained by
(14)
cr
step 2: the following Lyapunov function is choose 1 1 V2 V1 z22 ( z12 z22 ) 2 2
(13)
(15)
d
If we let z3 e1 2e2 e3 , then the following inequality holds.
te
z2 z2 V2 1 2 z2 z3 2 2
Ac ce p
step 3: Choose a Lyapunov function candidate as follows: 1 1 V V2 z32 ( z12 z22 z32 ) 2 2 The time dericative of V is given as:
(16)
(17)
z2 z2 V V2 z3 z3 1 2 z3 ( z2 z3 ) 2 2
z12 z22 z32 z z3 ( 3 z2 z3 ) 2 2 2 2
3 3 z12 z22 z32 z3 [( a )e1 (3 b)e2 e3 f ( y ) f ( x) gu ] 2 2 2 2 2
Case (1): z
1 2 1 2 2 1 2 1 2 z L , It is 2 2 2 2 shown that s 0 , so we consider the following positive definition function about s, In this case, we adopt open-loop control. Let s
Page 6 of 14
V
1 2 s . The derivative of V about t along the (3) is obtained: 2
3 3 z2 z2 z2 V ss s{ 1 2 3 z3 [( a )e1 (3 b)e2 e3 2 2 2 2 2
ip t
ˆ} ˆ 1 LL f ( y ) f ( x)] 2 1
3 3 ˆ} ˆ 1 LL a e1 3 b e2 e3 f ( y ) f ( x) ] 2 1 2 2
s{ z3 [
3 3 ˆ} ˆ 1 LL a e1 3 b e2 e3 L e ] 2 1 2 2
s{ z3 [
3 3 ˆ 1 L ( Lˆ z3 e )} a e1 3 b e2 e3 Lˆ e ] 2 1 2 2
rs
an
us
cr
s{ z3 [
(18)
sliding surface s 0 in finite times. Case (2): z
M
From the result of [20], (18) implies that the error state of the (3) can reach on the
d
The controller (7) is adopted in this case, and the following Lyapunov candidate is considered:
Ac ce p
te
1 1 2 1 2 1 2 V ( z12 z22 z32 ) L 2 2 2 2
(19)
3 3 z2 z2 z2 V 1 2 3 z3 [( a)e1 (3 b)e2 e3 f ( y ) f ( x) gua gub ] 2 2 2 2 2 ˆ ˆ 1 LL 1 1
3 3 z3[( a )e1 (3 b)e2 e3 gua ] z3[ f ( y ) f ( x) gub ] 2 2 ˆ ˆ 1 LL 1 1
(20)
By employing the controller (7), we can obtain that 3 3 z3 [( a )e1 (3 b)e2 e3 gua ] 2 2
gz [ g kz3 3 / 2 a e1 3 b e2 3 / 2 e3 ] 3 3 z3{( a )e1 (3 b)e2 e3 gkz3 3 } g 2 2
Page 7 of 14
z3 [
3 3 g a e1 3 b e2 e3 g kz3 ] z3 [ g kz3 3 / 2 a e1 3 b e2 3 / 2 e3 ] 2 2 g 3 3 g z3 [ a e1 3 b e2 e3 g kz3 ](1 ) 2 2 g
0 From (20) and (21), the following inequality is hold.
1 1 1 x 1 y 1 y y x 1 x f ( y ) f ( x) f ( ) f ( ) f ( ) F2 ( ) F1 ( ) f ( )] g g g g g g
an
gz3[
1 1 x y ˆ ˆ 1 LL f ( y ) f ( x) F1 ( ) F2 ( )] 1 1 g g
us
gz3 [
cr
ˆ ˆ 1 LL V z3 [ f ( y ) f ( x) gub ] 1 1
ip t
(21)
ˆ ˆ 1 LL 1 1
1 ˆ ˆ 1 LL L e z3 g 1 1
M
z3 L e z3
d
Lˆ (1 ) e g ˆ 1 1 (ˆ g ) 1L[ Lˆ (1 ) e ] (22)
te
0
The above inequality (22) means that the error state of (3) is bounded, and thus t
Ac ce p
e 0 by employing Barbalat's Lemma [20]. Finally, Case 1 and Case 2 complete the
proof of Theorem 1.
4. Simulation Examples
In this section, we present 3-D Arneodo dynamics (1) and (2) to illustrate the
effectiveness of the proposed method, the parameters in (1) and (2) are chosen as a 7.5 , b 3.8 . The initial values of the drive system (1) are chosen as x1 (0) 14 ,
x2 (0) 5 , x3 (0) 6 , and initial values of the response system (2) are chosen as y1 (0) 1 , y2 (0) 12 , y3 (0) 1.6 . The simulation results without any controller are shown as Figure 1.
Page 8 of 14
20 10 0 -10
0
5
10
15
20
25 Time(sec) (a)
30
0
5
10
15
20
25 Time(sec) (b)
30
0
5
10
15
20
35
40
45
cr
20 0
-20
0
M
-20 -40
40
45
50
25 Time(sec) (c)
30
35
40
45
50
d
y3 d n a 3 x
35
an
20
us
y2 d n a 2 x
50
ip t
y1 d n a 1 x
te
Figure 1. Time response of x and y without any controller In order to obtain synchronization between drive system and response system by
Ac ce p
using controller (7), the parameters are selected as
40 ,
k 500 ,
0.0001 , 0.0005 , 0.0001 , 0.0002 . Now, we approximate the two
continuous unknown nonlinear functions f ( x) and f ( y ) . The FLSs F1 ( x) and F2 ( y )
are constructed as (23)-(24) by choosing 6 fuzzy rules A1l A2l B l ,
A1l A2l Bl ( l 1, 2,3 ),respectively: RF(1l ) : If x1 is Al ( x1 ) and x2 is Al ( x2 ) and x3 is Al ( x3 ) , 1
2
3
Then 1 ( x ) is B ( y1 ) l
(23)
RF(l2) : If y1 is A l ( y1 ) and y2 is A l ( y2 ) and y3 is Al ( y3 ) , 1
2
3
Then 2 ( y ) is B ( y2 ) l
(24)
Page 9 of 14
2
2
in (23), the membership functions are selected as A e h ( x 40) , A e h ( x 40) , 1
1 1
2
2
2
2
1 2
2
2
A e h ( x 40) , A e h ( x 0.0001) , A e h ( x 0.0001) , A e h ( x 0.0001) , A e h ( x 40) , 3
1 3
1
2 1
2
2
2 2
2
3
2 3
2
1
3 1
2
2
A e h ( x 40) , A e h ( x 40) , B e ( y 160) , B e ( y 0.0001) , B e ( y 160) , where 2
3 2
3
3 3
1
1
1
2
1
3
2
ip t
parameter h 10 . We select the membership functions in (24) as: A e h ( y 40) , 1
1 1
2
1 2
2
2
A e h ( y 40) , 1
3 1
3
1 3
2
A e h ( y
2 40)
3 2
1
2 1
2
2 2
2
2 0.0001)
3
3 3
2
1
2
10
0
5
10
0
5
10
15
20
te
0
5
10
15
20
0
-20
Ac ce p
y2 d n a 2 x
2
2
2
20
25 Time(sec) (a)
30
35
40
45
50
25 Time(sec) (b)
30
35
40
45
50
25 Time(sec) (c)
30
35
40
45
50
d
20
15
M
0 -10
2
an
20 y1 d n a 1 x
3
2 3
us
2
2
, A e h ( y 0.0001) ,
, A e h ( y 40) , B e ( y 160) , B e ( y 0.0001) ,
B e ( y 160) . The simulation results are shown in Figure 2-4. 3
2
cr
2
A e h ( y 40) , A e h ( y 40) , A e h ( y 0.0001) , A e h ( y
20
y3 d n a 3 x
0
-20 -40
Figure 2. Time response of x and y
Page 10 of 14
15 10
e
1
5 0 -5
0
5
10
15
20
25 Time(sec) (a)
30
35
40
0
5
10
15
20
25 Time(sec) (b)
30
35
40
0
5
10
15
20
25 Time(sec) (c)
30
45
50
0 2
-10 -20
10 e
3
0
35
50
40
45
50
an
-10
us
20
45
cr
e
ip t
10
Ac ce p
te
d
M
Figure 3. Time response of errors between drive system and response system
Figure 4. Time response of errors between drive system and response system In Figure 2, the blue line denotes the states responses of drive system, the red line represent the states responses of response system by using fuzzy adaptive controller (7). From Figure 3, it is clear that states of the drive and response systems can be synchronized. Figure 3 shows that synchronization errors can converged to zero or
Page 11 of 14
near zero between the drive system (1) and response system (2). Figure 4 is the simulation result of the parameter time response in controller (7). 5. Conclusions A novel design fuzzy adaptive control based on backstepping method is proposed
ip t
for 3-D systems’ chaotic synchronization. The main advantage of the control scheme is that only three common parameters are needed to be adjusted automatically, which
cr
enable the number of adaptive laws reduced greatly such that the on-line
computational burden is reduced. In addition, fuzzy logic systems have general form
us
and the number of update laws has nothing to do with the number of IF_THEN rules in this paper, which make designer pay his or her attention to construct the fuzzy logic
an
systems to approximate the unknown functions in the dynamic equations of the master and slave systems. Therefore, the methods based on intuition inferences can be employed to generate the fuzzy systems with fewer fuzzy rules and high
M
interpretability. The methods fuzzy adaptive synchronization controller is a most advantage which will benefit to the engineering application.
d
Acknowledgment (61305098),
the
te
This research was supported in part by the Natural Science Foundation of China Youth
Foundation
of
Xi'an
University
of
Posts
and
Ac ce p
Telecommunications (110-0435). References
[1]. A. Pikovsky, M. Roseblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, New York: Cambridge University Press, 2003
[2]. A. Bagheri, J. J. Moghaddam, Decoupled adaptive neuro-fuzzy(DANF) sliding mode control system for a chaotic problem, Expert System with Applications,
36(3-2): 6062-6068, 2009
[3]. M. Yahyazadeh, A. R. Noei, R. Ghaderi, Synchronization of chaotic systems with known and unknown parameters using a modified active sliding mode control, ISA Transactions, 50(2): 262–267, 2011 [4]. D. I. R. Almeida, J. Alvarez, J. G. Barajas, Robust synchronization of Sprott circuits using sliding mode control. Chaos, Solitons & Fractals 30(1): 11–18,
Page 12 of 14
2006 [5]. M. C. Pai, Global synchronization of uncertain chaotic systems via discrete-time sliding mode control, Applied Mathematics and Computation, 227(5): 663-671, 2014
ip t
[6]. C. Mou, C. S. Jiang, J. Bin, Q. X. Wu, Sliding mode synchronization controller
design with neural network for uncertain chaotic systems, Chaos, Solitons &
cr
Fractals, 39(4):1856–1863, 2009
[7]. J. Ma, F. Li, L. Huang, W. Y. Jin, Complete synchronization, phase
us
synchronization and parameters estimation in a realistic chaotic system. Communications in Nonlinear Science and Numerical Simulation, 16(9):
an
3770–3785, 2011
[8]. H. Xi, S. Yu, R. Zhang, L. Xu, Adaptive impulsive synchronization for a class of
M
fractional-order chaotic and hyperchaotic systems, Optic-International Journal for Light and Electron Optics, 125(9): 2036-2040, 2014
d
[9]. G. Li, A. Khajepour, Robust control of a hydraulically driven flexible arm using
2005
J. H. Park, Synchronization of Genesio chaotic system via backstepping
Ac ce p
[10].
te
backstepping technique, Journal of Sound and Vibration, 280(3-5): 759-775,
approach, Chaos, Solitons & Fractals 27(5): 1369–1375, 2006
[11].
F. X. Chen, L. Chen, W. D. Zhang, Stabilization of parameters perturbation
chaotic system via adaptive backstepping technique, Applied Mathematics and Computation, 200(1):101-109, 2008
[12].
K. B. Deng, J. Li, S. M. Yu, Dynamics analysis and synchronization of a new
chaotic attractor, Optic, (In Press, Available online 20 March), 2014
[13].
A. E. Matouk, A.A. Elsadany, Achieving synchronization between the
fractional-order hyperchaotic Novel and Chen systems via a new nonlinear control technique, Applied Mathematics Letters, 29: 30-35, 2014 [14].
H. Y. Du, Function projective synchronization in drive-response dynamical
networks with non-identical nodes, Chaos, Solitons & Fractals, 44(7): 510-514, 2011
Page 13 of 14
[15].
L. Lü, C. Li, L. Chen, L. Wei, Lag projective synchronization of a class of
complex network constituted nodes with chaotic behavior, Commun Nonlinear Sci Numer Simulat, 19(8): 2843-2849, 2014 [16].
L. X. Wang, J. M. Mendel, Fuzzy basis functions, universal approximation,
ip t
and orthogonal least squares learning, IEEE Transactions on Neural Networks, 3(5): 807–814, 1992
B. Chen, X. Liu, S. Tong, Adaptive fuzzy approach to control unified chaotic
cr
[17].
systems, Chaos, Solitons & Fractals 34(4): 1180–1187, 2007
S. Y. Li, Z. M. Ge, Generalized synchronization of chaotic systems with
us
[18].
Applications, 38(3): 2302–2310, 2011 [19].
an
different orders by fuzzy logic constant controller, Expert Systems with
A. Poursamad, A. H. D. Markazi, Adaptive fuzzy sliding mode control for
M
multi-input multi-output chaotic systems. Chaos, Solitons & Fractals 42(5): 3100–3109, 2009
J.J.E. Slotine, W. Li, Applied Nonlinear Control, Prentice Hall, Englewood
Ac ce p
te
Cliffs, 1991
d
[20].
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