Generalized synchronization of chaotic systems with different orders by fuzzy logic constant controller

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Generalized synchronization of chaotic systems with different orders

by fuzzy logic constant controller

Shih-Yu Li

⇑

, Zheng-Ming Ge

⇑⇑

Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, PR China

a r t i c l e

i n f o

Keywords:

Generalized synchronization Different orders

Fuzzy logic constant controller FLCC

a b s t r a c t

The fuzzy logic constant controller (FLCC) is introduced in this paper. Unlike traditional method, a sim-plest controller is proposed via fuzzy logic design and Lyapunov direct method. Controllers in traditional method by Lyapunov direct method are always complicated or the functions of errors. We propose a new idea to design constant numbers as controllers, while the constant numbers are decided by the upper bound and the lower bound of the error derivatives. Via fuzzy logic rules, the strength of controllers in our new approach can be adjusted according to the error derivatives. Consequently, the slave system becomes exactly and efﬁciently synchronized to the trajectory of master system through FLCC. Two examples, Lorenz system and four order Chen–Lee system, are presented to illustrate the effectiveness of the new controllers in chaos generalized synchronization.

1. Introduction

SincePecora and Carroll (1990)proposed the concept of chaotic synchronization, chaos synchronization has become a hot subject in the ﬁeld of nonlinear science due to its wide-scope potential application in various disciplines. The past two decades has wit-nessed signiﬁcant progress on chaotic synchronization in secure communication, life science and information engineering. Typical application of synchronization techniques are in the remote con-trol of nuclear systems and concon-trol of distributed power systems. Chaotic synchronization has been investigated extensively. Many kinds of synchronization phenomena and methods have been found in variety of chaotic systems, such as generalized synchroni-zation (Chen, 2009; Chen, Chang, Yan, & Liao, 2008), phase syn-chronization (Erjaee & Momani, 2008; Li, Chen, & Huang, 2008), lag synchronization (Chen, Chen, & Gu, 2007; Ge & Lin, 2007), inverse synchronization (Chang, Li, & Lin, 2009; Li, 2009), partially synchronization (Chen & Chen, 2009; Wu & Chen, 2009), projective synchronization (Chen, 2005; Hu, Yang, Xu, & Guo, 2008), Q–S synchronization (Hu & Xu, 2008;Wang & Chen, 2006), etc.

In recent years, some chaos synchronizations based on fuzzy systems have been proposed since the fuzzy set theory was initi-ated byZadeh (1988), such as fuzzy sliding mode controlling tech-nique (Bagheri& Moghaddam, 2009; Chen, Chen, & Chiang, 2009;

Hung & Chung, 2007; Hung, Lin, & Chung, 2007), LMI-based syn-chronization (Wang, Guan, & Wang, 2003) and extended backstep-ping sliding mode controlling technique (Li & Khajepour, 2005). The fuzzy logic control (FLC) scheme have been widely developed for almost 40 years and have been successfully applied to many applications (Li, Kuo, & Guo, 2007). Recently, Yau and Shieh (2008) proposed an amazing new idea in designing fuzzy logic controllers – constructing fuzzy rules subject to a common Lyapu-nov function such that the master–slave chaos systems satisfy stability in the Lyapunov sense. In Yau and Shieh (2008), there are two main controllers in their slave system. One is used in elim-ination of nonlinear terms and the other is built by fuzzy rules sub-ject to a common Lyapunov function. Therefore, the resulting controllers are nonlinear form. InYau and Shieh (2008), the regular form is necessary. In order to carry out the new method, the original system must to be transformed into their regular form.

In this paper, we propose a new strategy which is also con-structing fuzzy rules subject to a Lyapunov direct method. Error derivatives are used to be upper bound and lower bound. Through this new approach, a simplest controller, i.e. constant controller, can be obtained and the difﬁculty in realization of complicated controllers in chaos synchronization by Lyapunov direct method can be also coped. Unlike conventional approaches, the resulting control law has less maximum magnitude of the instantaneous control command and it can reduce the actuator saturation phenomenon in real physic system.

The layout of the rest of the paper is as follows. In Section2, generalized synchronization by fuzzy logic constant controller (FLCC) scheme is presented. In Section3, simulation results are shown. In Section4conclusions are given.

⇑Corresponding author. Tel.: +886 3 5712121x55179; fax: +886 3 5720634. ⇑⇑Corresponding author. Tel.: +886 3 5712121x55119.

E-mail addresses:[email protected](S.-Y. Li),[email protected](Z.-M. Ge).

Contents lists available atScienceDirect

Expert Systems with Applications

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2. Generalized synchronization by FLCC scheme 2.1. Generalized synchronization scheme

There are two nonlinear dynamical systems, while the master system controls the slave system. The master system is given by

_x ¼ Ax þ f ðxÞ ð2-1Þ

where x = [x1, x2, . . . , xn]T2 Rn denotes a state vector, A is an n n

constant coefﬁcient matrix and f is a nonlinear vector function. The slave system is given by

_y ¼ By þ gðyÞ þ u ð2-2Þ

where y = [y1, y2, . . . , yn]T2 Rndenotes a state vector, B is an n n

constant coefﬁcient matrix, g is a nonlinear vector function, and u = [u1, u2, . . . , un]T2 Rnis a constant control input vector.

Our goal is to design appropriate fuzzy rules and corresponding constant controllers u so that the state vector of the chaotic system

(2-1) asymptotically approaches the state vector of the master system(2-2).

The generalized chaos synchronization can be accomplished in the sense that the limit of the error vector e(t) = [e1, e2, . . . , en]T

ap-proaches zero:

lim

t!1e ¼ 0 ð2-3Þ

where

e ¼ HðxÞ y ð2-4Þ

where H(x) is a given vector function of x. From Eq.(2-4)we have

_e ¼@HðxÞ

@x _x _y ð2-5Þ

_e ¼@HðxÞ

@x ½Ax þ f ðxÞ Ay f ðyÞ u ð2-6Þ

A Lyapnuov function V(e) is chosen as a positive deﬁnite function

VðeÞ ¼1 2e

T_e

ð2-7Þ

Its derivative along any solution of the differential equation system consisting of Eq.(2-6)is _ VðeÞ ¼ eT @HðxÞ @x ½Ax þ f ðxÞ Ay f ðyÞ u ð2-8Þ

If fuzzy constant controllers u can be appropriately chosen so that _

V ¼ CeTe, C is a diagonal negative deﬁnite matrix, and _V is a nega-tive deﬁnite function of e. By Lyapunov theorem of asymptotical stability:

lim

t!1e ¼ 0 ð2-9Þ

The generalized synchronization is obtained. The design process of FLCC is introduced in the following section.

2.2. Fuzzy logic constant controller design process

The basic conﬁguration of the fuzzy logic system is shown in

Fig. 1. It is composed of ﬁve function blocks (Shieh, 2003): 1. A rule base contains a number of fuzzy if-then rules.

2. A database deﬁnes the membership functions of the fuzzy sets used in fuzzy rules.

3. A decision-making unit performs the inference operations on the rules.

4. A fuzziﬁcation interface transforms the crisp inputs into degrees of match with linguistic value.

5. A defuzziﬁcation interface transforms the fuzzy results of the inference into a crisp output.

The fuzzy rules base consists of collection of fuzzy if-then rules expressed as the form if a is A then b is B, where a and b denote lin-guistic variables, A and B represent linlin-guistic values which are characterized by membership functions. All of the fuzzy rules can be used to construct the fuzzy associated memory.

We use two signals, e(t) = [e1, e2, . . . , em, . . . , en]Tin Eq.(2-4)and

e(t) = [e˙1, e˙2, . . . , e˙m, . . . , e˙n]TEq.(2-5), as the antecedent part of the

proposed FLCC to design the control input u in Eq.(2-8)that will be used in the consequent part of the proposed FLCC as follows:

u ¼ ½u1;u2; . . .um; . . .un T

ð2-10Þ

where u is a constant column vector and the FLCC accomplishes the objective to stabilize the error dynamics(2-6). In this paper, we are not going to use the original fuzzy rule base, but using it in each er-ror dynamics separately. In order to obtain the simplest controllers, the ith if-then rule of the fuzzy rule base of the FLCC is of the follow-ing form:

Rule i : if emis Xi then _emis Yiand umi¼ constant ð2-11Þ where Xiis the input fuzzy sets of em, m = 1 n, Yiis the output

fuz-zy sets of e˙mand umiis the i-rd output of e˙mwhich is a constant

con-troller. For given input sign of the process variables em, then the

output sign of e˙mwould be decided and its degree of membership

l

Xi; i ¼ 1 3 called rule-antecedent weights are calculated. The

centriod defuzziﬁer evaluates the output of all rules as follows:

um¼ P3 i¼1

l

xi umi P3 i¼1

l

xi ð2-12Þ

The fuzzy rule base is listed inTable 1, in which the input variables in the antecedent part of the rules are emand the output variable, in

the consequent part are e˙mandumi.

The membership function is obtained via the method shown in

Fig. 2. After designing appropriate fuzzy logic constant controllers, Fig. 1. The conﬁguration of fuzzy logic controller.

Table 1 Rule-table of FLCC.

Rule Antecedent Consequent part 1 Consequent part 2

em e˙m umi

1 Positive (P) Negative (N) um1

2 Negative (N) Positive (P) um2

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a negative deﬁnite of _V in Eq.(2-9)can be obtained and the asymp-totically stability of Lyapunov theorem can be achieved.

3. Simulation results

3.1. Example 1: synchronization of master and slave Lorenz system The master Lorenz system (Lorenz, 1963) is:

dx1ðtÞ dt ¼ aðx2ðtÞ x1ðtÞÞ dx2ðtÞ dt ¼ cx1ðtÞ x1ðtÞx3ðtÞ x2ðtÞ dx3ðtÞ dt ¼ x1ðtÞx2ðtÞ bx3ðtÞ 8 > > < > > : ð3-1-1Þ

When initial condition (x10, x20, x30) = (0.1, 0.2, 0.3) and parameters

a = 10, b = 8/3 and c = 28, chaos of the Lorenz system appears. The chaotic behavior of Eq.(3-1-1)is shown inFig. 3.

The slave Lorenz system is: dy1ðtÞ dt ¼ aðy2ðtÞ y1ðtÞÞ þ u1 dy2ðtÞ dt ¼ cy1ðtÞ y1ðtÞy3ðtÞ y2ðtÞ þ u2 dy3ðtÞ dt ¼ y1ðtÞy2ðtÞ by3ðtÞ þ u3 8 > > < > > : ð3-1-2Þ

When initial condition (y10, y20, y30) = (0.5, 0.7, 1.5) and parameters

are the same as that of Eq.(3-1-1), chaos of the slave Lorenz system appears as well. u1, u2and u3are FLCC to synchronize the slave

Lor-enz system to master one, i.e.,

lim

t!1e ¼ 0 ð3-1-3Þ

where the error vector

½e ¼ e1ðtÞ e2ðtÞ e3ðtÞ 2 6 4 3 7 5 ¼ x1ðtÞ y1ðtÞ x2ðtÞ y2ðtÞ x3ðtÞ y3ðtÞ 2 6 4 3 7 5 ð3-1-4Þ

From Eq.(3-1-4), we have the following error dynamics:

_e1¼ aðx2 x1Þ ðaðy2 y1Þ þ u1Þ _e2¼ cx1 x1x3 x2 ððcy1 y1y3 y2Þ þ u2Þ _e3¼ x1x2 bx3 ððy1y2 by3Þ þ u3Þ 8 > < > : ð3-1-5Þ

Choosing Lyapunov function as:

V ¼1 2ðe 2 1þ e 2 2þ e 2 3Þ ð3-1-6Þ

Its time derivative is:

_

V ¼ e1_e1þ e2_e2þ e3_e3

¼ e1ðaðx2 x1Þ ðaðy2 y1Þ þ u1ÞÞ

þ e2ðcx1 x1x3 x2 ððcy1 y1y3 y2Þ þ u2ÞÞ

þ e3ðx1x2 bx3 ððy1y2 by3Þ þ u3ÞÞ ð3-1-7Þ In order to design FLCC, we divide Eq.(3-1-7)into three parts as fol-lows: Assume V ¼1

2 e21þ e22þ e23

¼ V1þ V2þ V3, then _V ¼ e1_e1þ

e2_e2þ e3_e3¼ _V1þ _V2þ _V3, where V1¼1₂e21; V2¼1₂e22and V3¼1₂e23.

Part 1 : _V1¼ e1_e1¼ e1ðaðx2 x1Þ ðaðy2 y1Þ þ u1ÞÞ

Part 2 : _V2¼ e2_e2¼ e2ðcx1 x1x3 x2 ððcy1 y1y3 y2Þ þ u2Þ

Part 3 : _V3¼ e3_e3¼ e3ðx1x2 bx3 ððy1y2 by3Þ þ u3ÞÞ

Part 1: FLCC in Part 1 can be obtained via the fuzzy rules inTable 1 as follows and the maxima value and minima value of e˙1

(without any controller) can be observed in time history of error Fig. 2. Membership function.

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derivatives drawn inFig. 4. We choose f1to be the upper bound

value and g1to be the lower bound value of e˙1(without any

con-troller), they are satisﬁed with f1< e˙1 (without any

control-ler) < g1and f1, g1are all constants.

Rule 1: if e1is P, then e˙1is N and we take u11= f1

Rule 2: if e1is N, then e˙1is P and we take u12= g1

Rule 3: if e1is Z, then e˙1is Z and we take u13= 0 = e1

where f1= g1= constant = 400 and we choose u13= 0 = e1when

e1approaches to zero. We take Rule 1 3 in Part 1, _V1¼ e1_e1,

for explaining:

Rule 1 : if e1is P; then _e1is N and we take u11¼ f1:

_

V1¼ e1_e1¼ e1ðaðx2 x1Þ aðy2 y1Þ f1Þ

where e1> 0 and (a(x2 x1) a(y2 y1) f1) = (e˙1(without

control-ler) f1) < 0. Therefore, V_1¼ e1_e1¼ e1ðaðx2 x1Þ aðy2 y1Þ

f1Þ < 0 and is going to approach asymptotically stable. Rule 2 : if e1is N; then _e1is P and we take u12¼ g1

_

V1¼ e1_e1¼ e1ðaðx2 x1Þ aðy2 y1Þ g1Þ

where e1< 0 and (a(x2 x1) a(y2 y1) g1) = (e˙1(without

control-ler) g1) > 0. Therefore, V_1¼ e1_e1¼ e1ðaðx2 x1Þ aðy2 y1Þ

g1Þ < 0 and is going to approach asymptotically stable. Rule 3 : if e1is Z; then _e1is Z and we take u13¼ 0 ¼ e1

_

V1¼ e1_e1¼ e1ðaðx2 x1Þ aðy2 y1Þ e1Þ

where e1= 0 and we donot need any controller now. Therefore,

_

V1¼ e1_e1¼ 0 and achieve asymptotically stable. As a results, FLCC

in Part 1 can be obtained from Rule 1, 2 and 3:

u1¼

l

P u11þ

l

N u12þ

l

Z u13

l

_Pþ

l

_Nþ

l

_Z ð3-1-8Þ Part 2: FLCC in Part 2 can be obtained via the fuzzy rules inTable 1as follows and the maxima value and minima value of e˙2(without any controller) can be observed in time history

of error derivatives drawn inFig. 4. We choose f2to be the upper

bound value and g2to be the lower bound value of e˙2(without

any controller), they are satisﬁed with f2< e˙2(without any

con-troller) < g2and f2, g2are all constants.

Rule 1: if e1is P, then e˙1is N and we take u11= f

Rule 1: if e2is P, then e˙2is N and u21= f2

Rule 2: if e2is N, then e˙2is P and u22= g2

Rule 3: if e2is Z, then e˙2is Z and u23= 0 = e

e2approaches to zero. The process of FLCC designing is the same

as Part 1, as a results, FLCC in Part 2 can be obtained from Rule 1, 2 and 3 and are going to take _V2¼ e2_e2<0:

u2¼

l

P u21þ

l

N u22þ

l

Z u23

l

Pþ

l

Nþ

l

Z

ð3-1-9Þ

(without any controller) can be observed in time history of error derivatives drawn inFig. 4. We choose f3to be the upper bound

value and g3to be the lower bound value of e˙3 (without any

controller), they are satisﬁed with f3< e˙3(without any

Rule 3: if e3is Z, then e˙3is Z and u33= 0 = e3

e3 approaches to zero. The process of FLCC designing is the

same as Part 1, as a results, FLCC in Part 3 can be obtained from Rule 1, 2 and 3 and are going to take _V3¼ e3_e3<0:

u3¼

l

P u31þ

l

N u32þ

l

Z u33

l

Pþ

l

Nþ

l

Z

ð3-1-10Þ

FLCC are proposed in Part 1, 2 and 3 and are going to take _

V1¼ e1_e1<0; _V2¼ e2_e2<0 and _V3¼ e3_e3<0. Hence, we have

_

V ¼ _V1þ _V2þ _V3<0. It is clear that all of the rules in our FLC

can lead the Lyapunov function to approach asymptotically stable and the simulation results are shown inFigs. 5 and 6.

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3.2. Example 2: generalized synchronization of different order chaotic system- Lorenz and New Chen–Lee system

Chen & Lee (2004)gave a new chaotic system, which is now called the Chen–Lee system (Tam & Tou, 2008). The system is scribed by the following nonlinear differential equations and is de-noted as system (1): dz1ðtÞ dt ¼ z2ðtÞz3ðtÞ þ a1z1ðtÞ dz2ðtÞ dt ¼ z1ðtÞz3ðtÞ þ b1z2ðtÞ dz3ðtÞ dt ¼ 1 3z1ðtÞz2ðtÞ þ cz3ðtÞ 8 > > < > > : ð3-2-1Þ

where z1, z2and z3are state variables, and a1, b1and c1are three

system parameters. When (a1, b1, c1) = (5, 10, 3.8), system

(3-2-1) is a chaotic attractor. The positive Lyapunov exponent of this attractor is k1= 0.88, while the other ones are k2= 0 and

k3= 13.57, respectively. It is clear that the Chen–Lee system is a

regular chaotic system. For more-detailed dynamics of the Chen– Lee system, seeChen & Lee (2004).

It is known that in order to obtain hyper-chaos, there are two important requisites: (1) the minimal dimension of the phase space that embeds a hyper-chaotic attractor should be at least four, which requires a minimum of four couple ﬁrst-order autonomous ordinary differential equations; and (2) the number of terms in Fig. 5. Time histories of errors for Example 1- the FLCC is coming into after 30s.

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the couple equations giving rise to instability should be at least two, of which at least one should be a nonlinear function. In (Chen et al., 2009), Chen and Lee introduce a nonlinear feedback control-ler to the third equation of system(3-2-1), the following dynamic system can be obtained:

dz1ðtÞ dt ¼ z2ðtÞz3ðtÞ þ a1z1ðtÞ dz2ðtÞ dt ¼ z1ðtÞz3ðtÞ þ b1z2ðtÞ dz3ðtÞ dt ¼ 1 3z1ðtÞz2ðtÞ þ c1z3ðtÞ þ15z4ðtÞ dz4ðtÞ dt ¼ d1z1ðtÞ þ 1 2z2ðtÞz3ðtÞ þ 1 20z4ðtÞ 8 > > > > > > > < > > > > > > > : ð3-2-2Þ

where d is a constant, determining the dynamic behaviors of the system (3-2-2) and a1, b1, and c1 are three system

parameters. Thus, controller z4 causes chaotic system (3-2-1) to

become a four-dimensional system, which has four Lyapunov exponents. This may lead to a hyper-chaotic system. When (a1, b1, c1) = (5, 10, 3.8) and we choose d = 1.3, system (3-2-2)

is a hyper-chaotic attractor. The projection of phase portraits of system (3-2-2) with hyper-chaotic behaviors is shown in

Fig. 7.

Eq.(3-1-2)is chosen as slave system to be synchronized with the master system (3-2-2). Our goal is [e] = [e1(t), e2(t), e3(t)] =

[z1(t)y1(t), z3(t)y2(t), z4(t)y3(t)]. As a result, we get the

follow-ing error dynamics:

_e1¼ z2z3þ a1z1 ðaðy2 y1Þ þ u1Þ _e2¼13z1z2þ c1z3 ððcy1 y1y3 y2Þ þ u2Þ _e3¼ d1z1þ1₂z2z3þ₂₀1z4 ððy1y2 by3Þ þ u3Þ 8 > > < > > : ð3-2-3Þ

Choosing Lyapunov function as:

V ¼1 2ðe 2 1þ e 2 2þ e 2 3Þ ð3-2-4Þ

Its time derivative is:

_ V ¼ e1_e1þ e2_e2þ e3_e3 ¼ e1ðz2z3þ a1z1 ðaðy2 y1Þ þ u1ÞÞ þ e2 1 3z1z2þ c1z3 ððcy1 y1y3 y2Þ þ u2Þ þ e3 d1z1þ 1 2z2z3þ 1 20z4 ððy1y2 by3Þ þ u3Þ ð3-2-5Þ

We divide Eq.(3-2-5)into three parts as follows: Assume V ¼1 2ðe21þ e22þ e23Þ ¼ V1þ V2þ V3, then V ¼ e_ 1_e1þ e2_e2þ e3_e3¼ _V1þ _V2þ _V3, where V1¼12e21; V2¼12e22and V3¼12e23. Part 1 : _V1¼ e1_e1¼ e1ðz2z3þ a1z1 ðaðy2 y1Þ þ u1ÞÞ Part 2 : _V2¼ e2_e2¼ e2 13z1z2þ c1z3 ððcy1 y1y3 y2Þ þ u2Þ Part 3 : _V3¼ e3_e3¼ e3 d1z1þ12z2z3þ201z4 ððy1y2 by3Þ þ u3Þ

value and g4to be the lower bound value of e˙1 (without any

controller), they are satisﬁed with f4< e˙1(without any

Rule 1: if e1is P, then e˙1is N and we take u11= f4

Rule 2: if e1is N, then e˙1is P and we take u12= g4

Rule 3: if e1is Z, then e˙1is Z and we take u13= 0 = e1

where f4= g4= constant = 2000 and we choose u13= 0 = e1

when e1 approaches to zero. We take Rule 1 3 in Part 1,

_

V1¼ e1_e1, for explaining:

Rule 1 : if e1is P; then _e1is N and we take u11¼ f4:

_

V1¼ e1_e1¼ e1ðx2x3þ a1x1 aðy2 y1Þ f4Þ

where e1> 0 and (z2z3+ a1z1 a(y2 y1) f4) = (e˙1(without

con-troller) f4) < 0. Therefore, V_1¼ e1_e1¼ e1ðz2z3þ a1z1 aðy2

y1Þ f4Þ < 0 and is going to approach asymptotically stable.

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Rule 2 : if e1is N; then _e1is P and we take u12¼ g4

_

V1¼ e1_e1¼ e1ðx2x3þ a1x1 aðy2 y1Þ g4Þ

where e1< 0and (x2x3+ a1x1 a(y2 y1) g4) = (e˙1(without

con-troller) g4) > 0. Therefore, V_1¼ e1_e1¼ e1ðx2x3þ a1x1 aðy2

y1Þ g4Þ < 0 and is going to approach asymptotically stable. Rule 3 : if e1is Z; then _e1is Z and we take u13¼ 0 ¼ e1

_

V1¼ e1_e1¼ e1ðx2x3þ a1x1 aðy2 y1Þ e1Þ

where e1= 0 and we donot need any controller now. Therefore,

_

V1¼ e1_e1¼ 0 and achieve asymptotically stable. As a results, FLCC

in Part 1 can be obtained from Rule 1, 2 and 3:

u1¼

l

P u11þ

l

N u12þ

l

Z u13

l

Pþ

l

Nþ

l

Z

ð3-2-6Þ

Fig. 8. Time histories of error derivatives for master and slave chaotic systems without controllers.

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when e2approaches to zero. The process of FLCC designing is

the same as Part 1, as a results, FLCC in Part 2 can be obtained from Rule 1, 2 and 3 and are going to take _V2¼ e2_e2<0:

u2¼

l

_P u21þ

l

N u22þ

l

Z u23

l

Pþ

l

Nþ

l

Z

ð3-2-7Þ

when e3approaches to zero. The process of FLCC designing is

the same as Part 1, as a results, FLCC in Part 3 can be obtained from Rule 1, 2 and 3 and are going to take _V3¼ e3_e3<0:

u3¼

l

P u31þ

l

N u32þ

l

Z u33

l

Pþ

l

Nþ

l

Z

ð3-2-8Þ

FLCC are proposed in Eq.(3-2-6), (3-2-7) and (3-2-8)and are going to take _V1¼ e1_e1<0; _V2¼ e2_e2<0 and _V3¼ e3_e3<0 separately.

Hence, we have _V ¼ _V1þ _V2þ _V3<0. It is clear that all of the rules

in our FLC can lead the Lyapunov function to approach asymptot-ically stable and the simulation results are shown inFigs. 9 and 10.

4. Conclusions

In this paper, a simplest controller - fuzzy logic constant con-troller (FLCC) is introduced. Based on Lyapunov direct method and the upper bound and lower bound of the error derivatives, we construct the fuzzy rules and the simplest corresponding con-stant controllers. Complicated and nonlinear controllers would no longer appear and are replaced with simple and constant con-trollers through our new strategy. Simulation results in

synchroni-zation show that FLCC is effective enough and give very satisfactory results. Through this new approach, not only all cases in chaos synchronization or control can be achieved, but also the implement or experimental application of chaos synchronization could be attained much more easily.

Acknowledgment

This research was supported by the National Science Council, Republic of China, under Grant No. NSC 96-2221-E-009-145-MY3.

References

Bagheri, A., & Moghaddam, J. J. (2009). Decoupled adaptive neuro-fuzzy (DANF) sliding mode control system for a . chaotic problem. Expert Systems with Applications, 36(3), 6062.

Chang, S. M., Li, M. C., & Lin, W. W. (2009). Asymptotic synchronization of modiﬁed logistic hyper-chaotic systems and its applications. Nonlinear Analysis: Real World Applications, 10, 869.

Chen, H. K. (2005). Synchronization of two different chaotic systems: a new system and each of the dynamical systems Lorenz, Chen and Lü. Chaos, Solitons & Fractals, 25, 1049.

Chen, C. S. (2009). Quadratic optimal neural fuzzy control for synchronization of uncertain chaotic systems. Expert Systems with Applications, 36(9), 11827. Chen, H. C., Chang, J. F., Yan, J. J., & Liao, T. L. (2008). EP-based PID control design for

chaotic synchronization with application in secure communication. Expert Systems with Applications, 34(2), 1169.

Chen, C. S., & Chen, H. H. (2009). Robust adaptive neural-fuzzy-network control for the synchronization of uncertain chaotic systems. Nonlinear Analysis: Real World Applications, 10, 1466.

Chen, P. C., Chen, C. W., & Chiang, W. L. (2009). GA-based modiﬁed adaptive fuzzy sliding mode controller for nonlinear systems. Expert Systems with Applications, 36(3), 5872.

Chen, Y., Chen, X., & Gu, S. (2007). Lag synchronization of structurally nonequivalent chaotic systems with time delays. Nonlinear Analysis: Theory, Methods & Applications, 66, 1929.

Chen, H. K., & Lee, C. I. (2004). Anti-control of chaos in rigid body motion. Chaos, Solitons & Fractals, 21, 957.

Erjaee, G. H., & Momani, S. (2008). Phase synchronization in fractional differential chaotic systems. Physics Letters A, 372, 2350.

Ge, Z. M., & Lin, G. H. (2007). The complete, lag and anticipated synchronization of a BLDCM chaotic system. Chaos, Solitons & Fractals, 34, 740.

Hung, L. C., & Chung, H. Y. (2007). Decoupled control using neural network-based sliding-mode controller for nonlinear systems. Expert Systems with Applications, 32(4), 1168.

Hung, L. C., Lin, H. P., & Chung, H. Y. (2007). Design of self-tuning fuzzy sliding mode control for TORA system. Expert Systems with Applications, 32(1), 201. Fig. 10. Time histories of states for Example 2-the FLCC is coming into after 30 s.

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Hu, M., & Xu, Z. (2008). A general scheme for Q-S synchronization of chaotic systems. Nonlinear Analysis: Theory, Methods & Applications, 69, 1091. Hu, M., Yang, Y., Xu, Z., & Guo, L. (2008). Hybrid projective synchronization in a

chaotic complex nonlinear system. Mathematics and Computers in Simulation, 79(3), 449.

Li, G. H. (2009). Inverse lag synchronization in chaotic systems. Chaos, Solitons & Fractals, 40(3), 1076.

Li, C., Chen, Q., & Huang, T. (2008). Coexistence of anti-phase and complete synchronization in coupled chen system via a single variable. Chaos, Solitons & Fractals, 38, 461.

Li, G., & Khajepour, A. (2005). Robust control of a hydraulically driven ﬂexible arm using backstepping technique. Journal of Sound and Vibration, 280, 759.

Li, T. H. S., Kuo, C. L., & Guo, N. R. (2007). Design of an EP-based fuzzy sliding-mode control for a magnetic ball suspension system. Chaos, Solitons & Fractals, 33, 1523.

Lorenz, E. N. (1963). Deterministic non-periodic ﬂows. J. Atoms, 20, 130.

Pecora, L. M., & Carroll, T. L. (1990). Synchronization in chaotic systems. Physical Review Letters, 64(8), 821.

Shieh, C. S. (2003). Nonlinear rule-based controller for missile terminal guidance. IEE Proceedings–Control Theory and Applications, 150(1), 45.

Tam, L. M., & Tou, W. M. S. (2008). Parametric study of the fractional-order Chen– Lee system. Chaos, Solitons & Fractals, 37, 817.

Wang, Q., & Chen, Y. (2006). Generalized Q–S (lag, anticipated and complete) synchronization in modiﬁed Chua’s circuit and Hindmarsh–Rose systems. Applied Mathematics and Computation, 181, 48.

Wang, Y. W., Guan, Z. H., & Wang, H. O. (2003). LMI-based fuzzy stability and synchronization of Chen’s system. Physics Letters A, 320, 154.

Wu, W., & Chen, T. (2009). Partial synchronization in linearly and symmetrically coupled ordinary differential systems. Physica D: Nonlinear Phenomena, 238(4), 355.

Yau, H. T., & Shieh, C. S. (2008). Chaos synchronization using fuzzy logic controller. Nonlinear Analysis: Real World Applications, 9, 1800.