- Email: [email protected]
Fuzzy stability and synchronization of hyperchaos systems
Chaos, Solitons and Fractals 35 (2008) 922–930 www.elsevier.com/locate/chaos
Fuzzy stability and synchronization of hyperchaos systems Junwei Wang a
a,*
, Xiaohua Xiong
a,b
, Meichun Zhao
a,c
, Yanbin Zhang
a
School of Mathematics and Computational Science, Zhongshan University Guangzhou 510275, PR China b Department of Computer Science, Jiangxi Normal University, Nanchang 330027, PR China c Department of Mathematics, Guangdong University of Finance, Gunangzhou 510521, PR China Accepted 26 May 2006
Communicated by Prof. G. Iovane
Abstract This paper studies stability and synchronization of hyperchaos systems via a fuzzy-model-based control design methodology. First, we utilize a Takagi–Sugeno fuzzy model to represent a hyperchaos system. Second, we design fuzzy-model-based controllers for stability and synchronization of the system, based on so-called ‘‘parallel distributed compensation (PDC)’’. Third, we reduce a question of stabilizing and synchronizing hyperchaos systems to linear matrix inequalities (LMI) so that convex programming techniques can solve these LMIs efficiently. Finally, the generalized Lorenz hyperchaos system is employed to illustrate the effectiveness of our designing controller. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction Since Ott, Grebogi and Yorke proposed the famous OGY method for chaos control [1,2], various control methods have been proposed, including adaptive method [3–5], back-stepping design [6,7], time-delay feedback control [8], active control [9], etc. In recent years, chaos control and synchronization have become one of the most focusing research topics in light of their potential applications in laser physics, chemical reactor, secure communication, biomedical science, and so on [10]. In particular, there has been some significant progress in the studies of identification, control and utilization of chaos by artificial intelligence technologies such as fuzzy logic and neural networks. Generally speaking, artificial intelligence systems are envisioned to be adaptive, robust, and fault-tolerant. Expert systems, fuzzy systems, and neural networks are among the most important artificial intelligence technologies that have emerged. Among these intelligent control technologies, fuzzy control has enjoyed remarkable success in many applications [11]. Moreover, recent advances in the study of fuzzy control have laid foundation for intelligent control of various nonlinear processes (including chaotic cases) [12,13]. Regarding fuzzy control, there have existed some works. In Ref. [12], so-called Takagi–Sugeno (T–S) fuzzy models on the Chua’s circuit with two types of nonlinear resistors are given, respectively. Tanaka et al. [13] proposed a fuzzy *
Corresponding author. E-mail address: [email protected] (J. Wang).
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.05.087
J. Wang et al. / Chaos, Solitons and Fractals 35 (2008) 922–930
923
model that can be applied to various chaotic systems, and designed a fuzzy controller by using the idea of ‘‘parallel distributed compensation (PDC)’’ [14], to control chaotic systems. In Refs. [15–18], the authors proposed a few adaptive fuzzy schemes for control of chaotic systems with unknown parameters. In this paper, we extend the fuzzy-model-based control technique to stability and synchronization of hyperchaos systems. For convenience, the generalized Lorenz hyperchaos system (GLHS)[19] is employed to illustrate methodology of the fuzzy control design. Section 2 presents some concepts of the Takagi–Sugeno (T–S) fuzzy model. The fuzzy model of the GLHS and the design of its fuzzy controller by using the idea of PDC are explored in Section 3. In particular, in Section 3, some stability conditions on fuzzy control design are reduced to the linear matrix inequalities, which can be solved efficiently by convex programming techniques. Section 4 describes the fuzzy synchronization of GLHS. Conclusive remarks are collected in Section 5.
2. Takagi–Sugeno fuzzy model In this section, we review Takagi–Sugeno fuzzy model [20]. It can be used to design an oriented fuzzy controller, and is both simple and natural. The main feature of the Takagi–Sugeno fuzzy model is that it can describe the local dynamics of each fuzzy rule by a linear system model, and that the overall fuzzy model is achieved by fuzzy blending of all the linear system models. Generally, the Takagi–Sugeno fuzzy model is of the following form: Rule i: IF x1(t) is Mi1. . . and xn(t) is Min THEN xðtÞ ¼ Ai xðtÞ þ Bi uðtÞ;
ð1Þ
where xðtÞ ¼ ½x1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞT ; uðtÞ ¼ ½u1 ðtÞ; u2 ðtÞ; . . . ; um ðtÞT ; i = 1, 2, . . . , r (r is the number of IF-THEN rules), Mij are fuzzy sets, and x(t) = Aix(t) + Biu(t) is the output from the ith IF-THEN rule. Given a pair of (x(t), u(t)), the final output of the fuzzy system is inferred as follows: Pr xi ðxðtÞÞfAi xðtÞ þ Bi uðtÞg x_ ¼ i¼1 Pr ; ð2Þ i¼1 xi ðxðtÞÞ where xi ðxðtÞÞ ¼
n Y
M ij ðxðtÞÞ
ð3Þ
j¼1
for all t, and Mij(x(t)) is the grade of membership of xj(t) in Mij. The open-loop system of (2) is Pr x ðxðtÞÞAi xðtÞ Pr i x_ ¼ i¼1 ; i¼1 xi ðxðtÞÞ
ð4Þ
where it is assumed that r X xi ðxðtÞÞ > 0 i¼1
xi ðxðtÞÞ P 0;
i ¼ 1; 2; . . . ; r:
P By introducing hi ðxðtÞÞ ¼ xi ðxðtÞÞ= ri¼1 xi ðxðtÞÞ instead of xi(x(t)), (2) and (4) can be rewritten as r X x_ ¼ hi ðxðtÞÞfAi xðtÞ þ Bi uðtÞg; x_ ¼
i¼1 r X i¼1
hi ðxðtÞÞAi xðtÞ:
ð5Þ ð6Þ
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J. Wang et al. / Chaos, Solitons and Fractals 35 (2008) 922–930
Note that r X
hi ðxðtÞÞ ¼ 1;
i¼1
hi ðxðtÞÞ P 0;
i ¼ 1; 2; . . . ; r
for all t. hi(x(t)) can be regarded as the normalized weight of the IF-THEN rules. For convenience, we only consider the case of Bi = B = E (unite matrix), i = 1, 2, . . . , r for the fuzzy system in the following sections. Next, we show the applications of the fuzzy model (6) to the generalized Lorenz hyperchaos system (GLHS).
3. Fuzzy model for GLHS and design of its PDC controller 3.1. Fuzzy modeling of GLHS In this subsection, we will express the GLHS with the T–S fuzzy model. The GLHS [19] is derived by adding one more stable variable into the original generalized Lorenz system [21]. The dynamical behavior of GLHS is governed by 8 x_ ¼ a11 x þ a12 y; > > > < y_ ¼ a x þ a y þ w xz; 21 22 ð7Þ > z þ xy; z _ ¼ a 33 > > : w_ ¼ kx: Note that Eq. (7) covers a large class of hyperchaos systems, e.g., the Lorenz, Chen and Lu¨ hyperchaotic systems [22–24]. Taking a11 = a12 = 10, a21 = 28, a22 = 1, a33 ¼ 83, k = 10, we obtain the hyperchaotic Lorenz attractor as depicted in Fig. 1. Assume that x(t) 2 [d, d] and d > 0. Then, we have the following fuzzy model which exactly represents the nonlinear equation (7), i.e., GLHS, under x(t) 2 [d, d]: Rule 1 : If xðtÞ is M 1 ; then X_ ¼ A1 X ðtÞ; ð8Þ Rule 2 : If xðtÞ is M 2 ; then X_ ¼ A2 X ðtÞ; where X(t) = [x(t), y(t), z(t), w(t)]T 0 1 a11 a12 0 0 Ba C B 21 a22 d 1 C A1 ¼ B C @ 0 d a33 0 A k
0
0
0
and 0
1 a12 0 0 a22 d 1 C C C; d a33 0 A 0 0 0 1 xðtÞ M 1 ðxðtÞÞ ¼ 1þ ; 2 d 1 xðtÞ 1 : M 2 ðxðtÞÞ ¼ 2 d a11 Ba B 21 A2 ¼ B @ 0 k
In this paper, a11 = a12 = 10, a21 = 28, a22 = 1, a33 ¼ 83, k = 10 and d = 30. So, the final output of the fuzzy model of GLHS is given by X_ ðtÞ ¼
2 X
M i Ai X ðtÞ:
ð9Þ
i¼1
For any region of interest, GLHS can be modelled exactly by the fuzzy system via of appropriate choice of d. Moreover, the corresponding fuzzy control system has the following form:
J. Wang et al. / Chaos, Solitons and Fractals 35 (2008) 922–930
925
Fig. 1. Projection of the Lorenz hyperchaotic attractor in the GLHS, (a) Projection on x y z; (b) Projection on x z w; (c) Projection on y z w. Where a11 = a12 = 10, a21 = 28, a22 = 1, a33 ¼ 83, k = 10.
X_ ðtÞ ¼
2 X
M i Ai X ðtÞ þ BuðtÞ:
ð10Þ
i¼1
3.2. Design of fuzzy controller and its stability analysis We employ the idea of PDC to design the fuzzy control law for stabilization of the GLHS. Each rule is constructed from the corresponding rule of the T–S fuzzy model of GLHS. The designed fuzzy controller shares the same fuzzy sets as in the fuzzy model of the GLHS in the previous subsection. The PDC provides the following structure of the fuzzy control rule for the T–S fuzzy model of the GLHS: Rule 1: IF x(t) is M1, THEN uðtÞ ¼ F 1 X ðtÞ;
ð11Þ
Rule 2: IF x(t) is M2, THEN uðtÞ ¼ F 2 X ðtÞ:
ð12Þ
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J. Wang et al. / Chaos, Solitons and Fractals 35 (2008) 922–930
Thus, the overall fuzzy controller can be represented by uðtÞ ¼
2 X
M i F i X ðtÞ:
ð13Þ
i¼1
By substituting designed fuzzy controller (13) into the controlled GLHS (10), we obtain X_ ðtÞ ¼
2 X
M i ðAi BF i ÞX ðtÞ:
ð14Þ
i¼1
Using Lyapunov stability theorem, we get the following theorem for the stabilization of the GLHS. Theorem 1. The equilibrium of the fuzzy control system (14) of GLHS is asymptotically stable in the large if there exist a common positive definite matrix P such that ðAi F i ÞT P þ P ðAi F i Þ < 0;
i ¼ 1; 2:
ð15Þ
Proof. Define the Lyapunov function as V ðX Þ ¼ X ðtÞT PX ðtÞ: Then, the derivative of V along trajectories (14) is given by V_ ðX Þ ¼ X_ T ðtÞPX ðtÞ þ X ðtÞT P X_ ðtÞ ¼
2 X
M i X T ðtÞ½ðAi F i ÞT P þ P ðAi F i ÞX ðtÞ
i¼1
Under the conditions (15), we have V_ ðX Þ < 0. Therefore, by Lyapunov asymptotic stability theorem, we get the asymptotic stability of the equilibrium of the fuzzy control system (14). After multiplying the inequality by P1 on the two sides of (15) and defining Mi = PFi, we rewrite the above condition as ATi P þ PAi M i M Ti < 0;
i ¼ 1; 2:
ð16Þ
According to the condition, we can find a positive definite matrix P and Mi satisfying the LMIs, or determine that no such P and Mi exist. This is a convex feasibility problem and can be solved efficiently by the recently-developed interiorpoint methods. From the solutions P and Mi, the feedback gains Fi can be given by F i ¼ P 1 M i ;
i ¼ 1; 2:
ð17Þ
The MATLAB is employed to solve the LMIs (16). Numerical simulations depicted in Fig. 2 show that the stable fuzzy controller design for the GLHS fuzzy model is feasible. In particular, it can be seen from Fig. 2 that the designed fuzzy controller with feedback gains F1, F2 stabilizes the GLHS system, i.e., x(t) ! 0, y(t) ! 0, z(t) ! 0, and w(t) ! 0 as t ! +1. Finally, the overall fuzzy controller to stabilize the GLHS is shown as follows: uðtÞ ¼ M 1 F 1 X ðtÞ M 2 F 2 X ðtÞ:
ð18Þ
where 0
9:5000 B 19:0000 B F1 ¼ B @ 0:0000 5:0000
19:0000 0:5000 0:0000 0:5000
1 0:0000 5:0000 0:0000 0:5000 C C C; 2:1667 0:0000 A 0:0000
0:5000
0
1 19:0000 0:5000 0:0000 0:5000 B 9:5000 19:0000 0:0000 5:0000 C B C F 2B C: @ 0:0000 0:0000 2:1667 0:0000 A 5:0000
0:5000
0:0000
0:5000
4. Fuzzy synchronization of GLHS In this section, we deal with synchronization problem of GLHS. The controller to guarantee synchronization is derived through LMI-based design method.
J. Wang et al. / Chaos, Solitons and Fractals 35 (2008) 922–930
927
2.5 2
z
1.5 1 0.5 0 2 4
1 2
0
0
-1 -2
y
x
1.5 x(t) y(t) z(t) w(t)
x,y,z,w
1
0.5
0
- 0.5 -1 -1.5 0
5
10
15
t
20
25
30
Fig. 2. Numerical results of the controlled GLHS with controller (18).
Choose controlled GLHS (10) as drive system, where u(t) is the controller to be determined to achieve synchronization. Consider the following fuzzy model which has different initial conditions with (10) as response system: Rule 1 : If xs ðtÞ is M 1 ; then X_ s ¼ A1 X s ðtÞ; Rule 2 : If xs ðtÞ is M 2 ; then X_ s ¼ A2 X s ðtÞ:
ð19Þ
where Xs(t) = [xs(t), ys(t), zs(t), ws(t)]T. The defuzzification process is given as X_ s ðtÞ ¼
2 X
hi ðxs ðtÞÞAi X s ðtÞ:
ð20Þ
i¼1
Assume that e(t) = X(t) Xs(t). Then, from (20) and (10), we obtain the following error system: e_ ðtÞ ¼
2 X
hi ðxðtÞÞAi X ðtÞ
i¼1
2 X
hi ðxs ðtÞÞAi X s ðtÞ þ BuðtÞ:
ð21Þ
i¼1
We design two subcontrollers to realize the synchronization. Subcontroller A: Control Rule i: Rule i : If xðtÞ is M i ; then uA ðtÞ ¼ F i X ðtÞ;
i ¼ 1; 2:
ð22Þ
i ¼ 1; 2:
ð23Þ
Subcontroller B: Control Rule i: Rule i : If xs ðtÞ is M i ; then uB ðtÞ ¼ F i X s ðtÞ;
The overall fuzzy controller is constructed by the parallel connection, i.e., uðtÞ ¼ uA ðtÞ þ uB ðtÞ ¼
2 X i¼1
hi ðxðtÞF i X ðtÞÞ þ
2 X i¼1
hi ðxs ðtÞF i X s ðtÞÞ:
ð24Þ
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J. Wang et al. / Chaos, Solitons and Fractals 35 (2008) 922–930
The design is to determine the feedback gains Fi. After substituting (24) into (21), we have e_ ðtÞ ¼
2 X
hi ðxðtÞÞðAi BF i ÞX ðtÞ
i¼1
2 X
hi ðxs ðtÞÞðAi BF i ÞX s ðtÞ:
ð25Þ
i¼1
Theorem 2 (Linearization of T–S fuzzy system). The error system (25) is exactly linearized via the LMI-based fuzzy controller (24) as e_ ðtÞ ¼ HeðtÞ, where H ¼ Ai BF i ;
i ¼ 1; 2
if there exist the feedback gains Fi(i = 1, 2) such that A1 BF 1 ¼ A2 BF 2 :
ð26Þ
The proof is simple and hence omitted here. In this paper, we choose a stable matrix H as follows: 0 1 1 0 0 0 B 0 1 0 0 C B C H ¼B C: @ 0 0 1 0 A 0 0 0 1
20
30
15
20
10 10
ys(t)
xs(t)
5 0
-5 -
- 10
10
-15 -20 20 15 10 5 - - - -
(a)
- 20 0 x(t)
5
10
15
20
(b)
100
45
80
40
60
-20
-10
0 y(t)
10
20 w(t)
40
20
30
40
ws(t)
30
zs(t)
-30 -30
50
35
25 20 15 10 5 0 0
(c)
0
10
20
30 z(t)
40
50
(d)
20 0
- 20 - 40 - 60 - 80 -80 - 60 -40 -20
0
60
80
Fig. 3. Synchronization of the GLHS: (a) x(t) vs xs(t), (b) y(t) vs ys(t), (c) z(t) vs zs(t), (d) w(t) vs ws(t).
100
J. Wang et al. / Chaos, Solitons and Fractals 35 (2008) 922–930
929
Then we can get the following fuzzy controller to achieve synchronization for two coupled GLHS: uðtÞ ¼
2 X
hi ðxðtÞÞF i X ðtÞ þ
i¼1
2 X
hi ðxs ðtÞÞF i X s ðtÞ
ð27Þ
i¼1
with feedback gains 0
9 11
B 29 B F 1 ¼ B1 ðA1 H Þ ¼ B @ 1
0 1 11
B 29 B F 1 ¼ B1 ðA1 H Þ ¼ B @ 1
0
9
1
29
31 5=3
9 9
0
1
1
2C C C; 1A 1 1
31
2C C C: 1A
1
ð28Þ
1
1
29 5=3 1
1
ð29Þ
1
Fig. 3 shows the synchronization results, where the initial conditions of X(0) are slightly different from those of Xs(0). It can be seen from Fig. 3 that the fuzzy controller design to realize synchronization for GLHS is feasible, i.e., e(t) ! 0 as t ! +1.
5. Conclusions In this paper, we have represented the hyperchaos system with a Takagi–Sugeno fuzzy model. The controller for stability and synchronization of hyperchaos system based on T–S fuzzy model and PDC control design is given. Of course, the design procedure is conceptually simple and nature. Moreover, the sufficient conditions for hyperchaos stability and synchronization are reduced to linear matrix inequalities (LMI’s) and so are solved very efficiently in practice by convex programming techniques for LMI’s. Numerical simulations on GLHS are given to illustrate the effectiveness of the fuzzy model-based controller for hyperchaos stability and synchronization. To the best of our knowledge, this is the first report on fuzzy stability and synchronization of hyperchaos system. Finally, we note that this fuzzy modelbased controller design for stability and synchronization can be applied to other hyperchaos systems.
Acknowledgement This work was partially supported by the Natural Science Foundation of Guangdong Province, PR China (No. 50033047).
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[13] Tanaka K, Ikeda T, Wang HO. A unified approach to controlling chaos via an LMI-based fuzzy control system design. IEEE Trans Circ Syst – I 1998;45:1021–40. [14] Wang H. Tanaka K, Griffin M. Parallel distributed compensation of nonlinear systems by Takagi and Sugeno’s fuzzy model. In: Proceedings of FUZZ-IEEE’95. p. 531–8. [15] Spooner JT, Passino KM. Stable adaptive control using fuzzy systems and neural networks. IEEE Trans Fuzzy Syst 1996;4(3):339–59. [16] H.J. Kang, H. Son, C. Kwon, M. Park, A new approach to adaptive fuzzy control. In: Proceedings of the IEEE international conference on fuzzy systems, Anchorage, AK, 1998. p. 264–7. [17] Fischle K, Schroder D. An improved stable adaptive fuzzy control method. IEEE Trans Fuzzy Syst 1999;7(1):27–40. [18] Leu YG, Wang WY, Lee TT. Robust adaptive fuzzy-neural controllers for uncertain nonlinear systems. IEEE Trans Robot Automat 1999;15(5):805–17. [19] Li Y, Tang W, Cheng G. Hyperchaos evolved from the generalized Lorenz equation. Int J Circ Theor Appl 2005;33:235–51. [20] Takagi T, Sugeno M. Fuzzy identification of systems and its applications to modeling and control. IEEE Trans Syst, Man, Cyber 1985;15:116–32. [21] Celikovsky S, Chen G. On a generalized Lorenz canonical form of chaotic systems. Int J Bifurc Chaos 2002;12:1789–812. [22] Lorenz EN. Deterministic nonperiodic flow. J Atmos Sci 1963;20:130–41. [23] Chen G, Ueta T. Yet another chaotic attractor. Int J Bifurc Chaos 1999;9(7):1465–6. [24] Lu¨ J, Chen G. A new chaotic attractor coined. Int J Bifurc Chaos 2002;12(3):659–61.
Fuzzy stability and synchronization of hyperchaos systems Junwei Wang a
a,*
, Xiaohua Xiong
a,b
, Meichun Zhao
a,c
, Yanbin Zhang
a
School of Mathematics and Computational Science, Zhongshan University Guangzhou 510275, PR China b Department of Computer Science, Jiangxi Normal University, Nanchang 330027, PR China c Department of Mathematics, Guangdong University of Finance, Gunangzhou 510521, PR China Accepted 26 May 2006
Communicated by Prof. G. Iovane
Abstract This paper studies stability and synchronization of hyperchaos systems via a fuzzy-model-based control design methodology. First, we utilize a Takagi–Sugeno fuzzy model to represent a hyperchaos system. Second, we design fuzzy-model-based controllers for stability and synchronization of the system, based on so-called ‘‘parallel distributed compensation (PDC)’’. Third, we reduce a question of stabilizing and synchronizing hyperchaos systems to linear matrix inequalities (LMI) so that convex programming techniques can solve these LMIs efficiently. Finally, the generalized Lorenz hyperchaos system is employed to illustrate the effectiveness of our designing controller. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction Since Ott, Grebogi and Yorke proposed the famous OGY method for chaos control [1,2], various control methods have been proposed, including adaptive method [3–5], back-stepping design [6,7], time-delay feedback control [8], active control [9], etc. In recent years, chaos control and synchronization have become one of the most focusing research topics in light of their potential applications in laser physics, chemical reactor, secure communication, biomedical science, and so on [10]. In particular, there has been some significant progress in the studies of identification, control and utilization of chaos by artificial intelligence technologies such as fuzzy logic and neural networks. Generally speaking, artificial intelligence systems are envisioned to be adaptive, robust, and fault-tolerant. Expert systems, fuzzy systems, and neural networks are among the most important artificial intelligence technologies that have emerged. Among these intelligent control technologies, fuzzy control has enjoyed remarkable success in many applications [11]. Moreover, recent advances in the study of fuzzy control have laid foundation for intelligent control of various nonlinear processes (including chaotic cases) [12,13]. Regarding fuzzy control, there have existed some works. In Ref. [12], so-called Takagi–Sugeno (T–S) fuzzy models on the Chua’s circuit with two types of nonlinear resistors are given, respectively. Tanaka et al. [13] proposed a fuzzy *
Corresponding author. E-mail address: [email protected] (J. Wang).
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.05.087
J. Wang et al. / Chaos, Solitons and Fractals 35 (2008) 922–930
923
model that can be applied to various chaotic systems, and designed a fuzzy controller by using the idea of ‘‘parallel distributed compensation (PDC)’’ [14], to control chaotic systems. In Refs. [15–18], the authors proposed a few adaptive fuzzy schemes for control of chaotic systems with unknown parameters. In this paper, we extend the fuzzy-model-based control technique to stability and synchronization of hyperchaos systems. For convenience, the generalized Lorenz hyperchaos system (GLHS)[19] is employed to illustrate methodology of the fuzzy control design. Section 2 presents some concepts of the Takagi–Sugeno (T–S) fuzzy model. The fuzzy model of the GLHS and the design of its fuzzy controller by using the idea of PDC are explored in Section 3. In particular, in Section 3, some stability conditions on fuzzy control design are reduced to the linear matrix inequalities, which can be solved efficiently by convex programming techniques. Section 4 describes the fuzzy synchronization of GLHS. Conclusive remarks are collected in Section 5.
2. Takagi–Sugeno fuzzy model In this section, we review Takagi–Sugeno fuzzy model [20]. It can be used to design an oriented fuzzy controller, and is both simple and natural. The main feature of the Takagi–Sugeno fuzzy model is that it can describe the local dynamics of each fuzzy rule by a linear system model, and that the overall fuzzy model is achieved by fuzzy blending of all the linear system models. Generally, the Takagi–Sugeno fuzzy model is of the following form: Rule i: IF x1(t) is Mi1. . . and xn(t) is Min THEN xðtÞ ¼ Ai xðtÞ þ Bi uðtÞ;
ð1Þ
where xðtÞ ¼ ½x1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞT ; uðtÞ ¼ ½u1 ðtÞ; u2 ðtÞ; . . . ; um ðtÞT ; i = 1, 2, . . . , r (r is the number of IF-THEN rules), Mij are fuzzy sets, and x(t) = Aix(t) + Biu(t) is the output from the ith IF-THEN rule. Given a pair of (x(t), u(t)), the final output of the fuzzy system is inferred as follows: Pr xi ðxðtÞÞfAi xðtÞ þ Bi uðtÞg x_ ¼ i¼1 Pr ; ð2Þ i¼1 xi ðxðtÞÞ where xi ðxðtÞÞ ¼
n Y
M ij ðxðtÞÞ
ð3Þ
j¼1
for all t, and Mij(x(t)) is the grade of membership of xj(t) in Mij. The open-loop system of (2) is Pr x ðxðtÞÞAi xðtÞ Pr i x_ ¼ i¼1 ; i¼1 xi ðxðtÞÞ
ð4Þ
where it is assumed that r X xi ðxðtÞÞ > 0 i¼1
xi ðxðtÞÞ P 0;
i ¼ 1; 2; . . . ; r:
P By introducing hi ðxðtÞÞ ¼ xi ðxðtÞÞ= ri¼1 xi ðxðtÞÞ instead of xi(x(t)), (2) and (4) can be rewritten as r X x_ ¼ hi ðxðtÞÞfAi xðtÞ þ Bi uðtÞg; x_ ¼
i¼1 r X i¼1
hi ðxðtÞÞAi xðtÞ:
ð5Þ ð6Þ
924
J. Wang et al. / Chaos, Solitons and Fractals 35 (2008) 922–930
Note that r X
hi ðxðtÞÞ ¼ 1;
i¼1
hi ðxðtÞÞ P 0;
i ¼ 1; 2; . . . ; r
for all t. hi(x(t)) can be regarded as the normalized weight of the IF-THEN rules. For convenience, we only consider the case of Bi = B = E (unite matrix), i = 1, 2, . . . , r for the fuzzy system in the following sections. Next, we show the applications of the fuzzy model (6) to the generalized Lorenz hyperchaos system (GLHS).
3. Fuzzy model for GLHS and design of its PDC controller 3.1. Fuzzy modeling of GLHS In this subsection, we will express the GLHS with the T–S fuzzy model. The GLHS [19] is derived by adding one more stable variable into the original generalized Lorenz system [21]. The dynamical behavior of GLHS is governed by 8 x_ ¼ a11 x þ a12 y; > > > < y_ ¼ a x þ a y þ w xz; 21 22 ð7Þ > z þ xy; z _ ¼ a 33 > > : w_ ¼ kx: Note that Eq. (7) covers a large class of hyperchaos systems, e.g., the Lorenz, Chen and Lu¨ hyperchaotic systems [22–24]. Taking a11 = a12 = 10, a21 = 28, a22 = 1, a33 ¼ 83, k = 10, we obtain the hyperchaotic Lorenz attractor as depicted in Fig. 1. Assume that x(t) 2 [d, d] and d > 0. Then, we have the following fuzzy model which exactly represents the nonlinear equation (7), i.e., GLHS, under x(t) 2 [d, d]: Rule 1 : If xðtÞ is M 1 ; then X_ ¼ A1 X ðtÞ; ð8Þ Rule 2 : If xðtÞ is M 2 ; then X_ ¼ A2 X ðtÞ; where X(t) = [x(t), y(t), z(t), w(t)]T 0 1 a11 a12 0 0 Ba C B 21 a22 d 1 C A1 ¼ B C @ 0 d a33 0 A k
0
0
0
and 0
1 a12 0 0 a22 d 1 C C C; d a33 0 A 0 0 0 1 xðtÞ M 1 ðxðtÞÞ ¼ 1þ ; 2 d 1 xðtÞ 1 : M 2 ðxðtÞÞ ¼ 2 d a11 Ba B 21 A2 ¼ B @ 0 k
In this paper, a11 = a12 = 10, a21 = 28, a22 = 1, a33 ¼ 83, k = 10 and d = 30. So, the final output of the fuzzy model of GLHS is given by X_ ðtÞ ¼
2 X
M i Ai X ðtÞ:
ð9Þ
i¼1
For any region of interest, GLHS can be modelled exactly by the fuzzy system via of appropriate choice of d. Moreover, the corresponding fuzzy control system has the following form:
J. Wang et al. / Chaos, Solitons and Fractals 35 (2008) 922–930
925
Fig. 1. Projection of the Lorenz hyperchaotic attractor in the GLHS, (a) Projection on x y z; (b) Projection on x z w; (c) Projection on y z w. Where a11 = a12 = 10, a21 = 28, a22 = 1, a33 ¼ 83, k = 10.
X_ ðtÞ ¼
2 X
M i Ai X ðtÞ þ BuðtÞ:
ð10Þ
i¼1
3.2. Design of fuzzy controller and its stability analysis We employ the idea of PDC to design the fuzzy control law for stabilization of the GLHS. Each rule is constructed from the corresponding rule of the T–S fuzzy model of GLHS. The designed fuzzy controller shares the same fuzzy sets as in the fuzzy model of the GLHS in the previous subsection. The PDC provides the following structure of the fuzzy control rule for the T–S fuzzy model of the GLHS: Rule 1: IF x(t) is M1, THEN uðtÞ ¼ F 1 X ðtÞ;
ð11Þ
Rule 2: IF x(t) is M2, THEN uðtÞ ¼ F 2 X ðtÞ:
ð12Þ
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J. Wang et al. / Chaos, Solitons and Fractals 35 (2008) 922–930
Thus, the overall fuzzy controller can be represented by uðtÞ ¼
2 X
M i F i X ðtÞ:
ð13Þ
i¼1
By substituting designed fuzzy controller (13) into the controlled GLHS (10), we obtain X_ ðtÞ ¼
2 X
M i ðAi BF i ÞX ðtÞ:
ð14Þ
i¼1
Using Lyapunov stability theorem, we get the following theorem for the stabilization of the GLHS. Theorem 1. The equilibrium of the fuzzy control system (14) of GLHS is asymptotically stable in the large if there exist a common positive definite matrix P such that ðAi F i ÞT P þ P ðAi F i Þ < 0;
i ¼ 1; 2:
ð15Þ
Proof. Define the Lyapunov function as V ðX Þ ¼ X ðtÞT PX ðtÞ: Then, the derivative of V along trajectories (14) is given by V_ ðX Þ ¼ X_ T ðtÞPX ðtÞ þ X ðtÞT P X_ ðtÞ ¼
2 X
M i X T ðtÞ½ðAi F i ÞT P þ P ðAi F i ÞX ðtÞ
i¼1
Under the conditions (15), we have V_ ðX Þ < 0. Therefore, by Lyapunov asymptotic stability theorem, we get the asymptotic stability of the equilibrium of the fuzzy control system (14). After multiplying the inequality by P1 on the two sides of (15) and defining Mi = PFi, we rewrite the above condition as ATi P þ PAi M i M Ti < 0;
i ¼ 1; 2:
ð16Þ
According to the condition, we can find a positive definite matrix P and Mi satisfying the LMIs, or determine that no such P and Mi exist. This is a convex feasibility problem and can be solved efficiently by the recently-developed interiorpoint methods. From the solutions P and Mi, the feedback gains Fi can be given by F i ¼ P 1 M i ;
i ¼ 1; 2:
ð17Þ
The MATLAB is employed to solve the LMIs (16). Numerical simulations depicted in Fig. 2 show that the stable fuzzy controller design for the GLHS fuzzy model is feasible. In particular, it can be seen from Fig. 2 that the designed fuzzy controller with feedback gains F1, F2 stabilizes the GLHS system, i.e., x(t) ! 0, y(t) ! 0, z(t) ! 0, and w(t) ! 0 as t ! +1. Finally, the overall fuzzy controller to stabilize the GLHS is shown as follows: uðtÞ ¼ M 1 F 1 X ðtÞ M 2 F 2 X ðtÞ:
ð18Þ
where 0
9:5000 B 19:0000 B F1 ¼ B @ 0:0000 5:0000
19:0000 0:5000 0:0000 0:5000
1 0:0000 5:0000 0:0000 0:5000 C C C; 2:1667 0:0000 A 0:0000
0:5000
0
1 19:0000 0:5000 0:0000 0:5000 B 9:5000 19:0000 0:0000 5:0000 C B C F 2B C: @ 0:0000 0:0000 2:1667 0:0000 A 5:0000
0:5000
0:0000
0:5000
4. Fuzzy synchronization of GLHS In this section, we deal with synchronization problem of GLHS. The controller to guarantee synchronization is derived through LMI-based design method.
J. Wang et al. / Chaos, Solitons and Fractals 35 (2008) 922–930
927
2.5 2
z
1.5 1 0.5 0 2 4
1 2
0
0
-1 -2
y
x
1.5 x(t) y(t) z(t) w(t)
x,y,z,w
1
0.5
0
- 0.5 -1 -1.5 0
5
10
15
t
20
25
30
Fig. 2. Numerical results of the controlled GLHS with controller (18).
Choose controlled GLHS (10) as drive system, where u(t) is the controller to be determined to achieve synchronization. Consider the following fuzzy model which has different initial conditions with (10) as response system: Rule 1 : If xs ðtÞ is M 1 ; then X_ s ¼ A1 X s ðtÞ; Rule 2 : If xs ðtÞ is M 2 ; then X_ s ¼ A2 X s ðtÞ:
ð19Þ
where Xs(t) = [xs(t), ys(t), zs(t), ws(t)]T. The defuzzification process is given as X_ s ðtÞ ¼
2 X
hi ðxs ðtÞÞAi X s ðtÞ:
ð20Þ
i¼1
Assume that e(t) = X(t) Xs(t). Then, from (20) and (10), we obtain the following error system: e_ ðtÞ ¼
2 X
hi ðxðtÞÞAi X ðtÞ
i¼1
2 X
hi ðxs ðtÞÞAi X s ðtÞ þ BuðtÞ:
ð21Þ
i¼1
We design two subcontrollers to realize the synchronization. Subcontroller A: Control Rule i: Rule i : If xðtÞ is M i ; then uA ðtÞ ¼ F i X ðtÞ;
i ¼ 1; 2:
ð22Þ
i ¼ 1; 2:
ð23Þ
Subcontroller B: Control Rule i: Rule i : If xs ðtÞ is M i ; then uB ðtÞ ¼ F i X s ðtÞ;
The overall fuzzy controller is constructed by the parallel connection, i.e., uðtÞ ¼ uA ðtÞ þ uB ðtÞ ¼
2 X i¼1
hi ðxðtÞF i X ðtÞÞ þ
2 X i¼1
hi ðxs ðtÞF i X s ðtÞÞ:
ð24Þ
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J. Wang et al. / Chaos, Solitons and Fractals 35 (2008) 922–930
The design is to determine the feedback gains Fi. After substituting (24) into (21), we have e_ ðtÞ ¼
2 X
hi ðxðtÞÞðAi BF i ÞX ðtÞ
i¼1
2 X
hi ðxs ðtÞÞðAi BF i ÞX s ðtÞ:
ð25Þ
i¼1
Theorem 2 (Linearization of T–S fuzzy system). The error system (25) is exactly linearized via the LMI-based fuzzy controller (24) as e_ ðtÞ ¼ HeðtÞ, where H ¼ Ai BF i ;
i ¼ 1; 2
if there exist the feedback gains Fi(i = 1, 2) such that A1 BF 1 ¼ A2 BF 2 :
ð26Þ
The proof is simple and hence omitted here. In this paper, we choose a stable matrix H as follows: 0 1 1 0 0 0 B 0 1 0 0 C B C H ¼B C: @ 0 0 1 0 A 0 0 0 1
20
30
15
20
10 10
ys(t)
xs(t)
5 0
-5 -
- 10
10
-15 -20 20 15 10 5 - - - -
(a)
- 20 0 x(t)
5
10
15
20
(b)
100
45
80
40
60
-20
-10
0 y(t)
10
20 w(t)
40
20
30
40
ws(t)
30
zs(t)
-30 -30
50
35
25 20 15 10 5 0 0
(c)
0
10
20
30 z(t)
40
50
(d)
20 0
- 20 - 40 - 60 - 80 -80 - 60 -40 -20
0
60
80
Fig. 3. Synchronization of the GLHS: (a) x(t) vs xs(t), (b) y(t) vs ys(t), (c) z(t) vs zs(t), (d) w(t) vs ws(t).
100
J. Wang et al. / Chaos, Solitons and Fractals 35 (2008) 922–930
929
Then we can get the following fuzzy controller to achieve synchronization for two coupled GLHS: uðtÞ ¼
2 X
hi ðxðtÞÞF i X ðtÞ þ
i¼1
2 X
hi ðxs ðtÞÞF i X s ðtÞ
ð27Þ
i¼1
with feedback gains 0
9 11
B 29 B F 1 ¼ B1 ðA1 H Þ ¼ B @ 1
0 1 11
B 29 B F 1 ¼ B1 ðA1 H Þ ¼ B @ 1
0
9
1
29
31 5=3
9 9
0
1
1
2C C C; 1A 1 1
31
2C C C: 1A
1
ð28Þ
1
1
29 5=3 1
1
ð29Þ
1
Fig. 3 shows the synchronization results, where the initial conditions of X(0) are slightly different from those of Xs(0). It can be seen from Fig. 3 that the fuzzy controller design to realize synchronization for GLHS is feasible, i.e., e(t) ! 0 as t ! +1.
5. Conclusions In this paper, we have represented the hyperchaos system with a Takagi–Sugeno fuzzy model. The controller for stability and synchronization of hyperchaos system based on T–S fuzzy model and PDC control design is given. Of course, the design procedure is conceptually simple and nature. Moreover, the sufficient conditions for hyperchaos stability and synchronization are reduced to linear matrix inequalities (LMI’s) and so are solved very efficiently in practice by convex programming techniques for LMI’s. Numerical simulations on GLHS are given to illustrate the effectiveness of the fuzzy model-based controller for hyperchaos stability and synchronization. To the best of our knowledge, this is the first report on fuzzy stability and synchronization of hyperchaos system. Finally, we note that this fuzzy modelbased controller design for stability and synchronization can be applied to other hyperchaos systems.
Acknowledgement This work was partially supported by the Natural Science Foundation of Guangdong Province, PR China (No. 50033047).
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