Chaos, Solitons and Fractals 34 (2007) 1180–1187 www.elsevier.com/locate/chaos
Adaptive fuzzy approach to control unified chaotic systems Bing Chen
a,*
, Xiaoping Liu b, Shaocheng Tong
c
a Institute of Complexity Science, Qingdao University, Qingdao 266071, PR China Department of Electrical Engineering, Lakehead University, Thunder Bay, Ont., Canada P7B 5E1 c Department of Basic Mathematics, Liaoning Institute of Technology, Jinzhou 121000, PR China
b
Accepted 5 April 2006
Communicated by Prof. Gerando Iovane
Abstract An adaptive fuzzy control method is developed to control unified chaotic systems. Fuzzy logic systems are used to approximate nonlinear functions in the chaotic system and an adaptive technique is employed to construct controllers, which drive state variables into a small neighborhood of the origin. The simulation results are provided to demonstrate the effectiveness and feasibility of the proposed method. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, considerable attention has been paid to the control of chaos in nonlinear dynamical systems. Many authors have devoted their efforts to chaotic control, including stabilization of unstable equilibria. Techniques, such as OGY method, differential geometric approach, inverse optimal control, and adaptive control, have been developed to stabilize chaos systems (see [1–5], and the references therein). However, to the best of author’s knowledge, the problem of controlling chaos using adaptive fuzzy method has yet been investigated. It is shown that fuzzy logic systems can be used to approximate any nonlinear function defined on a compact set [6 and 7]. Based on this observation, some systematic fuzzy control methods have been developed to control nonlinear systems. In this letter, we discuss the problem of controlling a unified chaotic system [8] by an adaptive fuzzy control technique. The unified chaotic system is described by the following set of differential equations: 8 > < x_ 1 ¼ ð25a þ 10Þðx2 x1 Þ; x_ 2 ¼ ð28 35aÞx1 x1 x3 þ ð29a 1Þx2 ; ð1Þ > : aþ8 x_ 3 ¼ x1 x2 3 x3 ; where a 2 [0, 1] is an unknown parameter. The system described by (1) is called the Lorenz system for a 2 [0, 0.8), the Lu¨ system for a = 0.8, and the general Chen system for a 2 (0.8, 1] [4]. *
Corresponding author. E-mail addresses:
[email protected] (B. Chen),
[email protected] (X. Liu).
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.04.035
B. Chen et al. / Chaos, Solitons and Fractals 34 (2007) 1180–1187
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Recently, some authors have studied the problem of controlling the Lu¨ system and the unified chaotic system. In [9], the PC synchronization of the unified chaotic system has been investigated and the corresponding result has been applied to secure communication. In [10], the adaptive feedback synchronization has been investigated via three controllers. More recently, the parameter identification and backstepping control of uncertain Lu¨ system have been studied in [11 and 4]. It is noted that an existing adaptive technique can be applied to control uncertain unified chaotic systems, but it requires four update laws to estimate the unknown parameters since the uncertainty of a leads to system (1) containing four unknown parameters, 25a + 10, 28 35a, 29a 1 and aþ8 . In this letter, using an adaptive fuzzy control 3 technique, we propose a new controller design approach for the system (1) that requires only two adaptive laws.
2. Adaptive fuzzy control with two controllers The controlled system can be expressed as 8 x_ 1 ¼ ð25a þ 10Þðx2 x1 Þ; > > > < x_ 2 ¼ ð28 35aÞx1 x1 x3 þ ð29a 1Þx2 þ u2 ; > > > : x_ 3 ¼ x1 x2 aþ8 x3 þ u3 ; 3
ð2Þ
where u2 and u3 are the controllers to be designed. The aim of this letter is to design controllers u2 and u3 to stabilize the system. Before developing the controller design procedure, we first introduce the following preliminaries. A fuzzy system consists of four main components: fuzzy rule base, fuzzy inference engine, fuzzifier and defuzzifier [7]. The fuzzy rule base is composed of a collection of IF–THEN inference rules Ri : IF x1 is F i1 and . . . and x3 is F i3 ; THEN y is Gi T
ði ¼ 1; 2; . . . ; NÞ;
3
where x = [x1, x3, x3] 2 R and y 2 R are the input and output of the fuzzy system, respectively, F ij and Gi are the fuzzy sets in R. The fuzzy inference engine maps fuzzy sets in Rn into a fuzzy set in R according to the IF–THEN rules in the fuzzy rule base and the compositional rule of inference. The fuzzifier maps a crisp point x = [x1, x3, x3]T 2 R3 into a fuzzy set Ax in R. The defuzzifier maps a fuzzy set in R to a crisp point in R. Singleton fuzzification, center-average defuzzifucation and product inference are used. The output of the fuzzy system can be formulated as PN Q3 j¼1 Uj i¼1 lF ji ðxi Þ i yðxÞ ¼ P hQ N 3 j¼1 i¼1 lF j ðxi Þ i
where Uj is the point at which the fuzzy Q3 membership function lGj ðUj Þ achieves its maximum value, and it is assumed l j ðxi Þ
i¼1 F
that lGj ðUj Þ ¼ 1. Define pj ðxÞ ¼ PN Q3 j¼1
fuzzy system can be expressed as
½
i
, P(x) = [p1(x), p2(x), . . . , pN(x)]T and U = [U1, . . . , UN]T. Then the above
l j ðxi Þ
i¼1 F
i
yðxÞ ¼ UT P ðxÞ:
ð3Þ
The main reason for using the fuzzy system in (3) as a basic building block of adaptive fuzzy controllers is that the fuzzy systems in the form of (3) have been proved to be universal approximators [6], i.e., for any given real continuous function f(x) on some compact set U, there exists a fuzzy system of (3) such that it can uniformly approximate f(x) over U with arbitrary accuracy. Therefore, the fuzzy system in (3) is suitable for modelling nonlinear systems. Lemma 1 [6]. Let f(x) be a continuous function defined on a compact set X. Then, for any given constant e > 0, there exists a fuzzy logic system of (3) such that sup jf ðxÞ UT P ðxÞj 6 e: x2X
In the following, we will develop an adaptive fuzzy controller design procedure. Consider the following Lyapunov function V ¼
3 3 X 1X 1 ~ hi ; x2i þ 2 i¼1 2k i i¼2
where ~hi ¼ /i hi ; /i is an unknown parameter, which will be specified later, and hi is the estimation of /i.
ð4Þ
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B. Chen et al. / Chaos, Solitons and Fractals 34 (2007) 1180–1187
Differentiating (4) along the trajectory of the system in (2) yields, V_ ¼ ð25a þ 10Þx21 þ
3 X
xi ðfi ðxÞ þ ui Þ
i¼2
3 X 1~_ hi hi ; k i i¼2
ð5Þ
x3 . Because a is unknown, the nonlinear funcwhere f2(x) = (38 10a)x1 x1x3 + (29a 1)x2, and f3 ðxÞ ¼ x1 x2 aþ8 3 tions fi(x) for i = 2, 3, are unknown. Thus by Lemma 1, for any given positive scalar e > 0 there are fuzzy logic systems in the form (3) such that for i = 2, 3 fi ðxÞ ¼ UTi P ðxÞ þ di ðxÞ;
kdi ðxÞk 6 e;
ð6Þ
where x = [x1, x2, x3]T and Ui for i = 2, 3 are unknown constant vectors. Then, it follows from substituting (6) into (5) that V_ ¼ ð25a þ 10Þx21 þ
3 X
xi ðUTi P ðxÞ þ di ðxÞ þ ui Þ
i¼2
In addition, xi ðUTi P ðxÞ þ di ðxÞ þ ui Þ ¼ xi
3 X 1~_ hi hi : k i i¼2
ð7Þ
kUi k T Ui P ðxÞ þ di ðxÞ þ ui kUi k
1 2 1 1 2 1 2 x / P T ðxÞP ðxÞ þ ri þ x þ li e þ xi ui 2ri i i 2 2li i 2 1 1 1 ¼ xi xi /i P T ðxÞP ðxÞ þ xi þ ui þ ðri þ li e2 Þ; 2ri 2li 2
6
ð8Þ
where /i = kUik2 for i = 2, 3 are unknown constants, ri and li are design parameters to be chosen by the designer. Therefore, an adaptive technique can be used to estimate the unknown parameters /i’s, i = 2, 3, and the adaptive fuzzy controllers are constructed as follows: 1 1 ui ¼ ki þ xi xi hi P T ðxÞP ðxÞ; i ¼ 2; 3; ð9Þ 2li 2ri where ki > 0 is the design parameter, and hi is the estimation of /i. By taking (8) and (9) into account, we obtain the following inequality: xi ðUTi P ðxÞ þ di ðxÞ þ ui Þ 6 ki x2i þ
1 2~ T 1 x hi P ðxÞP ðxÞ þ ðri þ li e2 Þ 2ri i 2
ð10Þ
with ~hi ¼ /i hi . Substituting (10) into (7) gives V_ 6 ð25a þ 10Þx21
3 X i¼2
ki x2i þ
3 3 X X 1 ki 2 T 1 hi þ xi P ðxÞP ðxÞ h_ i ~ ðri þ li e2 Þ: 2r k 2 i i i¼2 i¼2
ð11Þ
Further, choose adaptive laws h_ i in the following form: ki 2 T h_ i ¼ x P ðxÞP ðxÞ hi ; 2ri i
i ¼ 2; 3:
ð12Þ
Then, it follows from substituting (12) into (11) that V_ 6
3 X i¼1
ki x2i þ
3 3 X 1 ~ X 1 hi hi þ ðri þ li e2 Þ k 2 i i¼2 i¼2
ð13Þ
hi , we have that with k1 = (25a + 10). Since a 2 [0, 1], k1 = (25a + 10) P 10. In addition, for the term hi ~ 1 1 hi ~hi ¼ ð/i ~hi Þ~hi ¼ ~h2i /i ~hi 6 ~hi þ /2i : 2 2
ð14Þ
B. Chen et al. / Chaos, Solitons and Fractals 34 (2007) 1180–1187
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Thus, in light of (13) and (14), one can verify 3 3 X 1 ~2 X 1 1 hi þ ri þ li e2 þ /2i 2k i 2 ki i¼1 i¼2 i¼2 3 3 3 X X X 2ki 2 1 ~2 1 1 2 2 ri þ li e þ /i : ¼ x h þ 2k i i 2 ki 2 i i¼1 i¼2 i¼2
V_ 6
3 X
ki x2i
Now, define a0 = min{1, 2kiji = 2, 3} and d 0 ¼
P3
1 i¼2 2 ðr i
ð15Þ
þ li e2 þ k1i /2i Þ. Then, (15) becomes
V_ 6 a0 V þ d 0 ; which implies d 0 a0 t d 0 e þ ; V 6 V ð0Þ a0 a0
10 9 8 7 6 5 4 3 2 1 0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Fig. 1.1. The responses of the closed-loop system for a = 0.25.
10
8
6
4
2
0
–2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Fig. 1.2. The responses of the closed-loop system for a = 0.8.
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8
6
4
2
0
–2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Fig. 1.3. The responses of the closed-loop system for a = 1.
0
–50
–100
–150
–200
–250
–300 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Fig. 2.1. The action of u2 and u3 for a = 0.25.
for t P 0. According to the definition of V and the inequality above, the following inequality becomes true. 3 X
x2i 6
V ð0Þ
i¼1
d 0 a0 t 2d 0 e þ : a0 a0 2
Note that ki, ri, li and ki are all design parameters, and /i’s are constants. Thus for any given l > 0, the inequality da00 6 l2 can be obtained by appropriately choosing these design parameters. Furthermore, we have that limt!1 x2i 6 l2 ;
which implies that with the control laws (9) and adaptive laws (12), the state of system converges into a neighborhood of the origin, and the radius of this neighborhood can be made small enough by appropriately tuning the design parameters. Remark. In the system (1), there are four unknown parameters, (25a + 10), (28 35a), (29a 1) and aþ8 3 . Four update laws are required to estimate these four unknown parameters. However, our proposed approach requires only two adaptive laws to control this chaotic system.
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3. Computer simulations In this section, stabilizing control laws will be designed for system (2) by the approach proposed in Section 2. To this end, the fuzzy membership functions are chosen as lF 1i ðxi Þ ¼ exp½0:5ðxi þ 20Þ2 ;
lF 2i ðxi Þ ¼ exp½0:5ðxi þ 15Þ2 ;
lF 3i ðxi Þ ¼ exp½0:5ðxi þ 10Þ2 ;
lF 4i ðxi Þ ¼ exp½0:5ðxi þ 5Þ2 ;
lF 5i ðxi Þ ¼ exp½0:5ðxi Þ2 ;
lF 6i ðxi Þ ¼ exp½0:5ðxi 5Þ2 ;
lF 7i ðxi Þ ¼ exp½0:5ðxi 10Þ2 ;
lF 8i ðxi Þ ¼ exp½0:5ðxi 15Þ2 ;
lF 9i ðxi Þ ¼ exp½0:5ðxi 20Þ2 : l
Let pj ðxÞ ¼ P9
F
j ðx1 Þl j ðx2 Þl j ðx3 Þ F F 1 2 3
l
j¼1 F
. Then P(x) = [p1(x), p2(x), . . . , p9(x)]T. According to (9) and (12), the control laws u2,
j ðx1 Þl j ðx2 Þl j ðx3 Þ F F 1 2 3
u3, and adaptive laws h_ 2 and h_ 3 are constructed as follows:
50
0
–50
–100
–150
–200
–250
–300
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
4.5
5
Fig. 2.2. The action of u2 and u3 for a = 0.8.
50 0 –50 –100 –150 –200 –250 –300 –350
0
0.5
1
1.5
2
2.5
3
3.5
4
Fig. 2.3. The action of u2 and u3 for a = 1.
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1 1 xi ui ¼ ki þ xi hi P T ðxÞP ðxÞ; 2li 2ri ki 2 T x P ðxÞP ðxÞ hi ; i ¼ 2; 3; h_ i ¼ 2ri i
i ¼ 2; 3;
ð16Þ ð17Þ
with k2 = k3 = 27.5, l2 = l3 = 20, r2 = r3 = 0.1, k2 = k3 = 0.05. The simulations are carried out for three cases: a = 0.25, a = 0.8 and a = 1. It is evident that for different a values, the system (2) denotes three different chaotic systems, namely, Lorenz system, Lu¨ system and Chen system, respectively. The control laws (16) and the adaptive laws (17) are used to control these different chaotic systems simultaneously. In all the simulations, the initial conditions are chosen as [x1(0), x2(0), x3(0)] = [10, 10, 10] and [h2, h3] = [0, 0]. Figs. 1.1–1.3 show the responses of the closed-loop system. Figs. 2.1–2.3 display the action of u2 and u3; Figs. 3.1–3.3 show h~2 , and h~3 . From the simulation results, it can be clearly seen that the proposed controllers are effective to simultaneously control three different chaotic systems. The steady-state errors are zero and there are no overshoots. The settling time is 2 s for the case of a = 0.25, about 0.8 s for a = 0.8, and about 0.3 s for a = 1.
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
4
4.5
5
Fig. 3.1. h~2 , and h~3 for a = 0.25.
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
0.5
1
1.5
2
2.5
3
3.5
Fig. 3.2. h~2 , and h~3 for a = 0.8.
B. Chen et al. / Chaos, Solitons and Fractals 34 (2007) 1180–1187
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0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Fig. 3.3. h~2 , and h~3 for a = 1.
4. Conclusion The problem of controlling unified chaotic systems has been discussed. A new systematic procedure is proposed to design adaptive fuzzy controllers for unified chaotic systems. Our proposed method requires less number of update laws and guarantees that the state variables of the closed-loop systems go into a neighborhood of the origin whose radius can be made arbitrarily small by tuning the design parameters. The simulation results are provided to demonstrate the effectiveness of the proposed approach.
Acknowledgements This work is partially supported by the Natural Sciences and Engineering Research Council of Canada, and The national Key Basic Research and Development Programme of China (2002CB312200), and the Taishan Scholar Programs Foundation of Shandong Province.
References [1] Ott E, Grebogi C, Yorke JK. Controlling chaos. Phys Rev Lett 1999;64:1196–9199. [2] Fuh CC, Tung PC. Controlling chaos using differential geometric method. Phys Rev Lett 1995;75:2952–5. [3] Wang Y, Guan Z, Wen X. Adaptive synchronization for Chen chaotic system with fully unknown parameters. Chaos, Solitons & Fractals 2004;19:899–903. [4] Wu X, Lu J. Adaptive control of uncertain Lu¨ system. Chaos, Solitons & Fractals 2004;22:375–81. [5] Feki M. An adaptive feedback control of linearizable chaotic systems. Chaos, Solitons & Fractals 2003;15:883–90. [6] Wang LX, Mendel JM. Fuzzy basis functions, universal approximation, and orthogonal least squares learning. IEEE Trans Neural Networks 1992;3(5):807–14. [7] Wang LX. Adaptive fuzzy systems and control: design and stability. Englewood Cliffs, NJ: Prentice-Hall; 1994. ˇ elikoveskyˇ S. Bridge the gap between the Lorenz system and the Chen system. Int J Bifurc Chaos [8] Lu¨ J, Chen G, Chen D, C 2002;12:2917–26. [9] Lu J, Wu X, Lu J. Synchronization of a unified chaotic system and application in secure communication. Phys Lett A 2002;305:365–70. [10] Chen S, Lu¨ J. Synchronization of an uncertain unified chaotic system via adaptive control. Chaos, Solitons & Fractals 2002;14:643–7. [11] Wu X, Lu J. Parameter identification and backstepping control of uncertain Lu¨ system. Chaos, Solitons & Fractals 2003;18:721–9.
