Control and synchronization of chaotic systems

Soft Comput DOI 10.1007/s00500-012-0810-z

ORIGINAL PAPER

Control and synchronization of chaotic systems using a novel indirect model reference fuzzy controller Mojtaba Ahmadieh Khanesar • Mohammad Teshnehlab Okyay Kaynak

•

Ó Springer-Verlag 2012

Abstract This paper presents a robust indirect model reference fuzzy control scheme for control and synchronization of chaotic nonlinear systems subject to uncertainties and external disturbances. The chaotic system with disturbance is modeled as a Takagi–Sugeno fuzzy system. Using a Lyapunov function, stable adaptation laws for the estimation of the parameters of the Takagi–Sugeno fuzzy model are derived as well as what the control signal should be to compensate for the uncertainties. The synchronization of chaotic systems is also considered in the paper. It is shown that by the use of an appropriate reference signal, it is possible to make the reference model follow the master chaotic system. Then, using the proposed model reference fuzzy controller, it is possible to force the slave to act as the reference system. In this way, the chaotic master and the slave systems are synchronized. It is shown that not only can the initial values of the master and the slave be different, but also there can be parametric differences between them. The proposed control scheme is simulated on the control and the synchronization of Duffing oscillators and Genesio–Tesi systems. Keywords Chaos Fuzzy control Takagi–Sugeno fuzzy model Nonlinear system Duffing oscillators Genesio–Tesi systems

M. A. Khanesar (&) M. Teshnehlab K. N. Toosi University of Technology, Tehran, Iran e-mail: [email protected] M. Teshnehlab e-mail: [email protected] O. Kaynak Bogazici University, Istanbul, Turkey e-mail: [email protected]

1 Introduction A chaotic system has complex dynamical behaviors that possess some special features, such as excessive sensitivity to initial conditions, broad spectrums of Fourier transform and fractal properties of the motion in phase space (Chen and Dong 1998). Such a phenomena can frequently be observed in natural and engineering systems such as in physics (laser technology and plasma), biology, mechanical engineering and chemistry, and it has been widely studied in literature. In Ott et al. (1990), an analytical method was introduced by Ott, Grebogy and Yorke for stabilizing an unstable periodic orbit in a chaotic attractor for the first time. Since then, different papers have addressed the control and synchronization problem of chaotic systems. Some of the control approaches proposed in the literature are based on an exact model. However, if the chaos system is only partially known, for example when the differential equation describing its dynamics is known but some or all of its parameters are unknown, then such control methods fail to be a viable choice (Park et al. 2002). The use of adaptive controllers is a possible approach for the control and synchronization of chaotic systems when there are some uncertainties (Dadras and Momeni 2009; Feki 2003; Hua and Guan 2004). Tanaka et al. (1998) have proposed a unified approach for controlling chaos via a fuzzy control system design, based on linear matrix inequalities. However, they have not considered any uncertainties in the parameters of the chaotic system. Since then, different controllers based on intelligent control schemes have been proposed to control chaotic systems (Park et al. 2002; Kuo 2011; Yau et al. 2005, 2006; Yau and Chen 2007; Yau 2008; Bessa et al. 2009).

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M. A. Khanesar et al.

The problem of controlling nonlinear systems using fuzzy structures and classical control techniques is widely investigated in a number of previous studies. In some of these approaches, the fuzzy system is used as a powerful general function approximator, and different classical methods are used to estimate the parameters of the fuzzy control system. For example, Tao and Taur (2007) proposed a fuzzy sliding mode controller, and Chen et al. (2009) investigated a fuzzy identification-based backstepping controller. The model reference adaptive control system is an adaptive servo system, in which the desired performance is expressed in terms of a reference model (Astrom 2008). An adaptive controller is designed to ensure that the plant follows the behavior of the reference model. Model reference fuzzy controllers benefit from both the well-established stability proof of model reference adaptive control systems and its power to define the desired performance in terms of a reference model and the general function approximation property of fuzzy systems. They have previously been used to control and synchronize chaotic systems, widely referenced studies being Park et al. (2002) and Cho et al. (2002). However, the controller designed in Park et al. (2002) needs the solution of a number of linear matrix inequalities for the selection of the reference model. Additionally, the model reference fuzzy controllers proposed by Park et al. (2002), Cho et al. (2002) and Khanesar et al. (2011) cannot control nonlinear dynamical systems subject to external disturbances. In this paper, a novel indirect model reference adaptive fuzzy controller (IMRAFC) with disturbance rejection properties is proposed. As the name suggests, the proposed IMRAFC benefits from an identifier, and the parameters of the controller are calculated using the estimated parameters of the identifier. The stability of the proposed controller is analyzed and the adaptation laws for the identifier are derived using an appropriate Lyapunov function. The proposed IMRAFC is then used to control a class of chaotic systems subject to disturbances. In the paper, the synchronization of chaotic systems is also considered. It is shown that it is possible to take the reference signal in such a way that the reference model follows the behavior of the master chaotic system. It is then possible to use the proposed IMRAFC to make the slave follow the dynamics of the reference system. In this way, the two chaotic systems are synchronized. The proposed novel IMRAFC with disturbance rejection capabilities is simulated on the control and the synchronization of Duffing oscillator and Genesio–Tesi system. It is shown that the parameters of the master and the slave systems do not necessarily need to be the same. It is observed that both control and synchronization are achieved with satisfactory performance.

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2 Design of model reference fuzzy controller for nonlinear systems subject to disturbances 2.1 Takagi–Sugeno fuzzy model of a nonlinear chaotic system with disturbance The chaotic system considered in this paper is in the form of xn ¼ f ðxÞ þ gðxÞu þ dðx; tÞ; in which f ðxÞ and gðxÞ are unknown smooth functions and dðx; tÞ is the bounded external disturbance. This nonlinear chaotic system can be modeled using a Takagi–Sugeno (TS) fuzzy model. The main feature of a TS fuzzy model is to express a nonlinear dynamical system by a set of fuzzy rules. Each fuzzy rule is a linear dynamical system and the overall fuzzy model of the system is achieved by a fuzzy blending of each local model. The ith rule of the fuzzy model for the nonlinear system can be written as: Ri : If x1 is M1i ðx1 Þ and . . . and xn is Mni ðxn Þ Then x_ ¼ Ai xðtÞ þ Bi uðtÞ þ B1 dðx; tÞ

ð1Þ

for i ¼ 1; . . .; l where xðtÞ ¼ ½x1 ðtÞ; x2 ðtÞ; . . .; xn ðtÞ 2 Rn denotes the state vector; uðtÞ 2 R is the input; l is the number of rules; dðx; tÞ is the unknown bounded external disturbance which satisfies jdðx; tÞj\D; in which D is also an unknown upper bound of dðx; tÞ; and 2 3 2 3 0 1 0 0 0 . 6 7 6 ... 7 .. 7 ; Bi ¼ 6 7 ; Ai ¼ 6 40 405 0 0 1 5 n n1 2 1 ai ai nn ai ai bi n1 ð2Þ 2 3 0 6 ... 7 7 B1 ¼ 6 405 1 n1

ð3Þ

are the state matrices of the fuzzy system. The final output of the fuzzy model is inferred as follows: _ ¼ xðtÞ

l X

hi ðxðtÞÞfAi xðtÞ þ Bi uðtÞg þ B1 dðx; tÞ;

ð4Þ

i¼1

P in which hi ðxðtÞÞ ¼ wi ðxðtÞÞ= li¼1 wi ðxðtÞÞ is the firing Q strength of ith rule and wi ðxðtÞÞ ¼ nj¼1 Mji ðxj ðtÞÞ and Mji ðxðtÞÞ is the grade of membership function of xj ðtÞ in Mji : 2.2 Design of adaptation laws for the proposed controller The reference model for the system is taken as: x_ m ðtÞ ¼ Am xm ðtÞ þ Br rðtÞ; in which:

ð5Þ

Control and synchronization of chaotic systems

2

0

.. .

1

6 Am ¼ 6 4 0 anm

0 an1 m

0

0 a2m

0

3

7 7 ; 1 5 a1m nn

2

3 0 . 6 .. 7 7 Br ¼ 6 405 br

:

n1

ð6Þ An identifier is considered for the system as: x^_ ðtÞ ¼

l X

ð7Þ

in which Aî and Bî are the identified matrices for Ai and Bi ; so that: e_ ¼ x^_ x_ ¼

b_~i ¼ 2c2i hi PTn em u;

ð14Þ

in which Pn is the last column of the matrix P: The control signal is taken as: ( l X uðtÞ ¼ wi ðxðtÞÞjbî jf½ðam aî Þ=bî xðtÞg

l X

þ

l X

hi ðxðtÞÞfA~i xðtÞ þ B~i uðtÞg B1 dðx; tÞ:

(

e_ m ¼ x_ x_ m ¼ x_ x^_ þ x^_ x_ m ¼ e_ þ x^_ x_m ;

ð10Þ

þ

e_ m ¼ Am em

) wi ðxðtÞÞjbî jfðbr =bî ÞrðtÞg

i¼1

(

l X

)1 wi ðxðtÞÞjbî j

ð17Þ

:

i¼1

This term corresponds to a fuzzy system whose ith rule is:

hi ðxðtÞÞf½0. . .0~ aTi T xðtÞ þ ½0. . .0b~i T uðtÞg

Ri : If x1 is N1i ðx1 Þ and . . . and xn is Nni ðxn Þ ð18Þ Then uðtÞ ¼ Ki xðtÞ þ Li rðtÞ; qffiffiffiffiffiffiffi n in which Nji ¼ jbî jMji is the grade of membership

i¼1

hi ðxðtÞÞfAî xðtÞ þ Bî uðtÞg

i¼1 l X

ð15Þ

i¼1 l X

so that:

l X

wi ðxðtÞÞjbî j

ð9Þ

in which a~i ¼ aî ai ; b~i ¼ bî bi and ai and aî are the last rows of Ai and Aî ; respectively. The equation for the tracking error is achieved as:

l X

)1

This control signal can be divided into two terms as: u ¼ uf þ ur where uf is the fuzzy part of the control signal and ur which is the robustness term of control signal. The former term uf is defined as: ( l X uf ðtÞ ¼ wi ðxðtÞÞjbî jf½ðam aî Þ=bî xðtÞg

i¼1

l X

ð8Þ

hi ðxðtÞÞf½0. . .0 a~Ti T xðtÞ þ ½0. . .0 b~i T uðtÞg

B1 dðx; tÞ;

þ

wi ðxðtÞÞjbî jfðbr =bî ÞrðtÞ qsgnðBT1 Pem Þ=jbî jg

i¼1

in which sgnð:Þ corresponds to the signum function as: 8 < 1; h [ 0 sgnðhÞ ¼ 0; h ¼ 0 ð16Þ : 1; h\0

Furthermore,

)

i¼1

i¼1

l X

ð13Þ

i¼1

hi ðxðtÞÞfAî xðtÞ þ Bî uðtÞg;

i¼1

e_ ¼

a~_ i ¼ 2c1i hi PTn em xT

hi ðxðtÞÞfAm xðtÞ þ Br rðtÞg þ B1 dðx; tÞ:

i¼1

ð11Þ In order to study the stability of the system, the following Lyapunov function is considered: Vðe; a~i ; b~i Þ ¼ eTm Pem þ

l X 1 T a~i a~i 2c 1i i¼1

l X 1 ~T ~ 1 þ bi bi þ ðq q Þ2 ; 2c c 2i i¼1

function of xj ðtÞ in Nji ; and: am aî br ; Li ¼ : Ki ¼ ^ bi bî

ð19Þ

The second part of the control signal ur (the robustness term) is defined as: ( )1 l l X X ur ðtÞ ¼ qsgnðBT1 Pem Þ wi ðxðtÞÞ wi ðxðtÞÞjbî j : i¼1

i¼1

ð12Þ

ð20Þ

in which P is a positive definite matrix and 0\c2i ; c1i ; c: The adaptation laws are considered as:

Using the control signal of (15) and the adaptation laws of (13) and (14), the time derivative of the Lyapunov function is achieved as:

123


V_ ¼ eTm ðATm P þ PAm Þem þ eTm PB1 dðx; tÞ qjBT1 Pem j 1 _ q Þ: þ qðq c

ð21Þ

Am is the state matrix of the reference model whose eigenvalues are all in the open left half plane, so that ATm P þ PAm ¼ Q; in which Q is a positive definite matrix. We then have: V_ kmin ðQÞkem k2 qjBT1 Pem j 1 _ q Þ þ DjBT1 Pem j þ qðq c

ð22Þ

model are uniformly bounded. Then the adaptation laws of (13) and (27) with the control signal of (15) guarantee the stability of the system and it follows that: 1. 2.

Proof In the case when jbî j [ b0 or jbî j ¼ b0 and PTn em u sgnðbi Þ [ 0; the simple adaptation laws of (13) and (27) are used. By using the similar procedure as that is used in R1 the previous section, we have 0 eTm ðsÞem ðsÞ ds\ Vð0ÞVð1Þ kmin ðQÞ : In the other condition, one obtains:

in which kmin ðQÞ corresponds to minimum eigenvalue of Q: Equation 22 can be written as:

2 _ V\k min ðQÞkem k

V_ kmin ðQÞkem k2 q jBT1 Pem j 1 _ q Þ: þ ðq qÞjBT1 Pem j þ DjBT1 Pem j þ qðq c

2 ð23Þ

ð24Þ

eTm ðsÞem ðsÞ ds

Vð0Þ Vð1Þ ; kmin ðQÞ

ð25Þ

ð26Þ

0

which means that em 2 L2 : Since xm 2 l2 ; we have x 2 l2 : In order to complete the stability of the system, the control signal must be bounded. To achieve a bounded control signal, bî must be prevented from being zero in the adaptation process. To do so, the adaptation process of bi is modified as follows: 8 < 2c2i hi PTn em u if jbî j [ b0 or _b~ ¼ jbî j ¼ b0 and PTn em u sgnðbi Þ [ 0 i : 0 otherwise; ð27Þ in which b0 is a selected lower bound for jbî j and it must be selected as b0 \jbi j; i ¼ 1; . . .; l: 2.3 Stability analysis of the proposed controller Theorem 1. Consider the nonlinear dynamical system with disturbance of (1) and the reference model (5) with control law of (15) and adaptation laws as (13) and (27). Assume that the reference signal and states of the reference

123

ð28Þ

Using the priori knowledge that b0 \jbi j it follows that This means that the modified adaptation law of (27) makes V_ more negative so that in all cases we have R1 T e ðsÞem ðsÞ ds\ Vð0ÞVð1Þ : Since bî is non-zero, the 0

This implies that V 2 l1 ; so that em ; a~i ; b~i 2 l1 ; i ¼ 1; . . .; l: Furthermore, Z1

hi PTn em ub~i q jBT1 Pem j þ DjBT1 Pem j:

PTn em ub~i \0:

Considering the design parameter q as D q ; it is concluded that: 2 _ V\k min ðQÞkem k :

l X i¼1

By assuming q_ ¼ cjBT1 Pem j; one obtains: 2 T T _ V\k min ðQÞkem k q jB1 Pem j þ DjB1 Pem j:

em ! 0 as t ! 1 _ a^_ i ; bî ! 0 as t ! 1:

m

kmin ðQÞ

control signal is also bounded so that e_m 2 l1 : As T previously achieved em 2 l1 l2 and em 2 l1 and using the Barbalat’s lemma, limt!1 em ðtÞ ! 0: Considering the adaptation laws of (13) and (27), since limt!1 em ðtÞ ! 0 _ and x; u 2 l1 ; it is clear that a^_ i ; bî ! 0 as t ! 1: 3 Synchronization of uncertain chaotic systems This section deals with the synchronization of chaotic systems by the use of the IMRAFC discussed in the previous section. First a TS fuzzy model is found which exactly models the master chaotic system (exact modeling of nonlinear dynamical systems using a TS fuzzy model has previously been considered in Tanaka et al. (1998) and Tanaka (2001). Then, the reference signal for the reference model is chosen so that the states of the reference model follow those of the chaotic system. The equations that describe the error dynamics are derived and it is shown that the states of the reference model track the states of the chaotic system and the error converges to zero. Then the proposed IMRAFC is used to make the slave chaotic system follow the reference model and hence the master system. In this way, the master and the slave chaotic systems are synchronized. Consider the TS fuzzy model which exactly represents the master chaotic system with disturbance as: l X x_ R ðtÞ ¼ vi ðzR ðtÞÞARi xR ðtÞ þ B1 dðx; tÞ; ð29Þ i¼1

in which vi is the firing of the ith fuzzy rule; zR is any smooth function of xR which enables the exact modeling

Control and synchronization of chaotic systems

P and li¼1 vi ðzR ðtÞÞ ¼ 1; l is the number of the rules; dðx; tÞ is unknown bounded disturbance. It is assumed that: jdðx; tÞj\D and D is the unknown upper bound of dðx; tÞ: ARi is considered in the form of: 2 3 0 1 0 0 .. 6 7 6 7 . ARi ¼ 6 ð30Þ 7: 4 0 0 0 1 5 aiRn aiRðn1Þ aiR2 aiR1 One obtains: e_ R ðtÞ ¼ x_ m ðtÞ x_ R ðtÞ ¼ Am eR ðtÞ þ

l X

vi ðxR ðtÞÞðAm ARi ÞxR ðtÞ

ð31Þ

i¼1

þ BR rðtÞ B1 dðx; tÞ: Furthermore,

l X

In this section, the proposed control scheme is used to control the states of modified Duffing system. Consider the following Duffing system, which may exhibit chaotic behavior (Layeghi et al. 2008):

vi ðxR ðtÞÞð½0. . .0aTm T ½0. . .0aTRi T ÞxR ðtÞ

i¼1

þ ½0. . .0br T rðtÞ B1 dðx; tÞ: ð32Þ For the stability analysis of the system, a Lyapunov function is chosen as: eTR PeR

þ ðq1

q1 Þ2 :

ð33Þ

Defining the reference signal as: rðtÞ ¼

l X

vi ðxR ðtÞÞ

i¼1

aRi am 1 xR ðtÞ sgnðBT1 PeR Þ br br

ð34Þ

and the adaptation law for q1 as: q_ 1 ¼ jBT1 PeR ðtÞj;

ð35Þ

it is obtained that: _ ¼ kmin ðQÞkeR k2 þ DjBT1 PeR ðtÞj q1 jBT1 PeR ðtÞj: VðtÞ ð36Þ _ kmin ðQÞkeR k2 : It follows If D q1 ; we have: VðtÞ R1 that V; eR ðtÞ 2 l1 : In addition, since 0 keR ðsÞk2 ds\ Vð0ÞVð1Þ kmin ðQÞ ; eR

In this section, the proposed IMRAFC is simulated on the control and the synchronization of chaotic systems. The chaotic systems which are selected here are subjected to uncertainties in the form of external disturbances and uncertainty in the parameters. Since the proposed method benefits from a stable adaptive parameter estimation law and disturbance rejection capabilities, it is expected that the controller will result in a successful response despite the presence of disturbances and the variations in the parameters of the system.

4.1.1 Control of Duffing oscillator

¼ Am eR ðtÞ

VðeR Þ ¼

4 Simulation results

4.1 Control and synchronization of Duffing oscillator

e_ R ðtÞ ¼ x_ m ðtÞ x_ R ðtÞ

þ

the master system and produces the desired trajectory for the slave system.

2 l1 and since

e_ R ðtÞ ¼ Am eR ðtÞ þ B1 sgnðBT1 PeR ðtÞÞ

ð37Þ

e_ R ðtÞ 2 l1 and using Barbalat’s lemma, limt!1 eR ðtÞ ¼ 0: This analysis shows that if the reference signal is selected as (34), the states of the reference model track the states of

x_ 1 ¼ x2 x_ 2 ¼ ax1 þ bx31 þ dx2 þ f0 cosðxtÞ þ ð3 þ cosðx1 ÞÞu; ð38Þ in which u is the external input signal and f0 cosðxtÞ is the external bounded disturbance dðx; tÞ: Figure 1 shows the chaotic behavior of the Duffing system when running autonomously ðu ¼ 0Þ in phase space with its parameters selected as a ¼ 1; b ¼ 1; d ¼ 0:15; f0 ¼ 0:3 and x ¼ 1: The initial values selected for the system are xð0Þ ¼ ½1 0:1T : A TS fuzzy system is used to model the Duffing oscillator. The rules of the fuzzy system are considered as: _ ¼ A1 xðtÞ þ B1 uðtÞ: If x1 is negative then xðtÞ _ ¼ A2 xðtÞ þ B2 uðtÞ: If x1 is zero then xðtÞ _ ¼ A3 xðtÞ þ B3 uðtÞ: If x1 is positive then xðtÞ The membership functions selected for the fuzzy system are as in Fig. 2. The initial value of the state matrices of the system are selected as: 0 1 0 1 A1 ¼ A3 ¼ ; A2 ¼ 2 0:15 1 0:15 ð39Þ 0 0 B1 ¼ B3 ¼ ; B2 ¼ : 3:54 4 The reference model matrices for the system are considered as:

123


assumed that the parameters b; d vary in the interval of [1.5, -0.5] and [-0.1, -0.2], respectively. The sinusoidal response of the system in the presence of these uncertainties is shown in Fig. 4. The trajectories of the Duffing oscillator and of the reference model in phase space are shown in Fig. 4a. The control signal is shown in Fig. 4b. The time response of the states of Duffing oscillator x1 ; x2 and the reference signal for them are shown in Fig. 4c and d, respectively. As can be seen from the figures, the control signal and the adaptation law cope with the uncertainties in the parameters and responses are quite satisfactory.

1.5

1

x2

0.5

0

−0.5

−1

4.1.2 Synchronization of Duffing oscillators with different parameters

−1.5 −2

−1

0

1

2

x1

As mentioned earlier, for the synchronization of two chaotic systems by the proposed method, the reference signal is chosen in such a way that the reference model is synchronized with the master chaotic system and then the proposed IMRAFC is used to force the states of slave system follow the states of the master system. Since the synchronization of the model reference system with the master and the model following of the slave with respect to the reference model are independent of each other, it is possible to have different master and slave parameters. Therefore, consider a master– slave Duffing–Holmes systems as follows:

Fig. 1 Chaotic behavior of the Duffing system in phase space 1

0.8

About −1

0.6

About 0

About 1

x_ 1 ¼ x2 x_ 2 ¼ ax1 þ bx31 þ dx2 þ f0 cosðxtÞ

0.4

ð41Þ

y_ 1 ¼ y2 y_ 2 ¼ a0 y1 þ b0 y31 þ d0 y2 þ f00 cosðx0 tÞ

0.2

ð42Þ

þ Mf ðyÞ þ ð3 þ cosðx1 ÞÞu; 0 −2

−1

0

1

2

Fig. 2 The membership functions considered for the Duffing oscillator

Am ¼

0 1 ; 10 7

Br ¼

0 : 1

ð40Þ

The eigenvalues of Am are f2; 5g and the reference system is stable. Figure 3 shows the result of applying the proposed controller to the Duffing oscillator. The trajectories of the Duffing oscillator and of the reference model in phase space are shown in Fig. 3a. The control signal is shown in Fig. 3b. The time response of the states of Duffing oscillator x1 ; x2 and the reference signal for them are shown in Fig. 3c and d, respectively. As can be seen from the figure, the Duffing oscillator tracks the sinusoidal reference input with very high performance. In addition, in order to show the performance of the system in the presence of uncertain parameters, it is

123

in which the parameters of a; b and d are selected as in the previous section. But the parameters of the slave are selected as: a0 ¼ 0:5; b0 ¼ 1; d0 ¼ 0:1; f00 ¼ 0:2 and x0 ¼ 1:5: First, the exact TS fuzzy model for the slave system is constructed. The rules of exact TS fuzzy model for the master system can be written as: _ ¼ A1 xðtÞ þ B1 uðtÞ: If z is about(0) then xðtÞ _ ¼ A2 xðtÞ þ B2 uðtÞ; If z is about(1) then xðtÞ in which z ¼ bx21 þ a and the membership functions for the label about(0) is l1 ðzÞ ¼ 1 z and the membership function for about(1) is l2 ðzÞ ¼ z and: 0 1 0 1 A1 ¼ ; A2 ¼ ; 0 0:075 1 0:075 ð43Þ 0 B1 ¼ B2 ¼ 1 It can be shown that this TS fuzzy model can exactly represent the master Duffing oscillator in the interval of

Control and synchronization of chaotic systems 0.5 0.5

x2

0

0

−0.5

−1

Trajectory of the system Reference Trajectory of the system

−1.5

−0.5 0

0.5

x

1

0

5

10

15

20

time (sec)

1

(a)

(b)

1.2

0.6

State of system x1

1

0.4

Reference state of system x

1m

0.8

0.2

0.6

0

0.4

−0.2

0.2

−0.4

0

−0.6

−0.2

−0.8

State of system x

2


2m

−0.4

0

5

10

15

20

−1

0

5

10

time (sec)

time (sec)

(c)

(d)

15

20

Fig. 3 a The trajectory of the Duffing oscillator and the reference model, b control signal, c time response of the states of Duffing oscillator x1 and the reference signal for x1 and d time response of the state of Duffing oscillator x2 and the reference signal for x2

x1 2 ½1; 1 except the cosine term. The effect of this term can be considered as a disturbance and the reference model is forced to follow the master system using an additional term in control signal to ensure robustness. Taking the reference signal as in (34), the reference model follows the states of the master, and the control law of (15) is used to force the slave to behave as the reference system and the master. The design procedure of IMRAFC for the slave is similar to procedure described in the previous section. However, it should be mentioned that in order to exhibit less chattering phenomena, the saturation function is used instead of the signum function in the control signal of (15). Figure 5 shows the simulation results of the synchronization of the chaotic systems. The slave system is run for 5 s without input and at the fifth second the control signal is turned on. The initial values

for the master, the slave and the reference model are selected as x1 ¼ 0:1; x2 ¼ 0:1; y1 ¼ 0:1; y2 ¼ 0:1 and xm1 ¼ 0:8; xm2 ¼ 0; respectively. As can be seen from Fig. 5, the performance of the proposed controller in the presence of disturbance and different parameters for master and slave is satisfactory. 4.2 Control and synchronization of Genesio–Tesi chaotic system The Genesio–Tesi system, proposed by Genesio and Tesi (1992), is one of the paradigms of chaos since it captures many features of chaotic systems. It includes a simple square part and three simple ordinary differential equations that depend on three positive real parameters (Park 2007). The dynamic equation of the system is as follows:

123

M. A. Khanesar et al. 0.5 0.5

x

2

0

0

−0.5

−1 Trajectory of the system Reference Trajectory of the system

−1.5

−0.5 0

0.5

0

1

5

10

x1

time (sec)

(a)

(b)

1.2

15

20

0.6

State of system x1 1

Reference state of system x1m

0.4

0.8

0.2

0.6

0

0.4

−0.2

0.2

−0.4

0

−0.6

−0.2

−0.8

−0.4

−1

State of system x

2


2m

0

5

10

15

20

10

time (sec)

(c)

(d)

15

20

oscillator x1 and the reference signal for x1 in the presence of uncertainties and d time response of the state of Duffing oscillator x2 and the reference signal for x2 in the presence of uncertainties

0\jDf ðx; tÞj\F; 0\jdðtÞj\D: ð44Þ

where x1 ; x2 and x3 are state variables, and a; b and c are the positive real constants satisfying ab\c: The uncertain Genesio–Tesi system can be written as (Dadras and Momeni 2009): 8 < x_ 1 ¼ x2 x_ 2 ¼ x3 : x_ 3 ¼ cx1 bx2 ax3 þ mx21 þ Df ðx; tÞ þ dðtÞ þ uðtÞ; ð45Þ in which x ¼ ½x1 ; x2 ; x3 T is the state vector, Df ðx; tÞ is time varying unknown bounded uncertainty and dðtÞ is the bounded external disturbance of the system such that:

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5

time (sec)

Fig. 4 a The trajectories of the Duffing oscillator and the reference model in the presence of uncertainties, b control signal in the presence of uncertainties, c time response of the state of Duffing

8 < x_ 1 ¼ x2 x_ ¼ x3 : 2 x_ 3 ¼ cx1 bx2 ax3 þ mx2

0

It is assumed that Df ðx; tÞ ¼ 0:5 sinðpx1 Þ sinð2px2 Þ sinð3px3 Þ and dðtÞ ¼ 0:2 cosðtÞ: Figure 6 illustrates the chaotic behavior of the uncertain system when u ¼ 0 and xð0Þ ¼ ½1:1; 0:1; 0:1T :

4.2.1 Control of Genesio–Tesi system The rules of the fuzzy system considered for the identification of Genesio–Tesi system are as: If If If If

x1 ðtÞ x1 ðtÞ x1 ðtÞ x1 ðtÞ

is is is is

_ ¼ A1 xðtÞ þ B1 uðtÞ: negative then xðtÞ _ ¼ A2 xðtÞ þ B2 uðtÞ: zero then xðtÞ _ ¼ A3 xðtÞ þ B3 uðtÞ: positive then xðtÞ _ ¼ A4 xðtÞ þ B4 uðtÞ: positive big then xðtÞ

Control and synchronization of chaotic systems 10

2

8

1 6 4

x

2

0

−1

0

State of slave system y1

control in action

−2

−2

State of master system x1

−4

Reference state of system x1m −3

2

0

10

20

30

40

−6

50

−2

0

2

time (sec)

4

6

4

6

x1

(a)

(a) 15

0.5 10

0

x3

5

−0.5

State of slave system y2

−1

Control in action

0

−5

State of master system x

2

−10


2m

−1.5 0

10

20

30

40

50

−15

time (sec)

−2

0

2

x1

(b)

(b)

1.5 15

maximum of u =8.87 1

10

5

x

3

0.5

0

0

−5

−10

−0.5

0

10

20

30

40

50

time (sec)

(c)

−15 −6

−4

−2

0

2

4

6

8

10

x

2

Fig. 5 a Time responses of the (x1 ) controlled chaotic Duffing state. b Time responses of the (x2 ) controlled chaotic Duffing state. c The control signal note that control u(t) is activated at t ¼ 5s

(c) Fig. 6 The chaotic behavior of the uncertain Genesio-Tesi system in phase space. a x2 versus x1 ; b x3 versus x1 and c x3 versus x2

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M. A. Khanesar et al. 1.4

1

0.8

State of system x

Positive Big

Negative

1

Zero

1

0.8

Positive

0.6


1.2

0.6

0.4

0.4

0.2

0.2 0

0

1

2

3

4

5

time (sec) 0 −4

−2

0

2

4

(a)

6 1.5

Fig. 7 The membership functions considered for the Genesio–Tesi system

State of system x2 Reference state of system x

2m

1

0.5

The initial values considered for system matrices are as: 2 3 2 3 0 1 0 0 1 0 6 7 6 7 A1 ¼ 4 0 0 1 5 ; A2 ¼ 4 0 0 1 5; 10 2:9 1:2 3 0 1 0 6 7 A3 ¼ 4 0 0 1 5; 2

6 2:9 1:2

−0.5

−1

2 2:9 1:2 3 2 3 0 1 0 0 6 7 6 7 A4 ¼ 4 0 0 1 5; B1 ¼ B2 ¼ B3 ¼ B4 ¼ 4 0 5: 2

2 2:9 1:2

0

1

−1.5 0

1

2

3

4

5

time (sec)

(b) 2

ð46Þ The membership functions considered for the identification of the Genesio–Tesi system are shown in Fig. 7. The reference matrices for the system are considered as: 2 3 2 3 0 1 0 0 Am ¼ 4 0 0 1 5; Br ¼ 4 0 5: ð47Þ 100 80 17 1

1 0 −1 −2 −3 −4

Figure 8 shows the regulation response of the states of the Genesio–Tesi system. The initial values for the chaotic system are considered as: xð0Þ ¼ ½1:1; 0:1; 0:1T : As can be seen from the figure, the states of the Genesio–Tesi system track the states of the reference model and all of the states converge to zero in finite time. The tracking of a sinusoidal reference signal with the same initial values is also shown in Fig. 9. As can be seen from the figure, sinusoidal tracking of the proposed IMRAFC is also satisfactory.

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−5

State of system x3

−6 −7 0

Reference state of system x3m 1

2

3

4

5

time (sec)

(c) Fig. 8 The regulation response of an uncertain Genesio-Tesi system. a The time response of x1 ; b the time response of x2 and c the time response of x3

Control and synchronization of chaotic systems State of slave system x

State of system x

1

1

1



10

1


8

0.5

Control in action

6 4

0 2 0

−0.5

−2

0

5

10

15

20 0

time (sec)

10

20

30

40

50

time (sec)

(a)

(a) 2

State of system x

2


1.5

State of system x2

15


2

Reference state of system x2m 1

10

Control in action 0.5 5

0 −0.5

0

−1 −5

−1.5

0

5

10

15

0

20

10

20

30

time (sec)

time (sec)

(b)

(b)

3

30

2

25

50

State of system x3 State of master system x

3


20

1

40

Control in action 15

0

10

−1 5

−2 0

−3 −5

−4

−10

State of system x

3

−5


3m

−6

0

5

10

15

20

time (sec)

(c) Fig. 9 The sinusoidal response of an uncertain Genesio-Tesi system. a The time response of x1 ; b the time response of x2 and c the time response of x3

−15

0

10

20

30

40

50

time (sec)

(c) Fig. 10 The synchronization of uncertain Genesio-Tesi systems. a The time response of the state of slave y1 versus the state of master x1 ; b the time response of the state of slave y2 versus the state of master x2 and c the time response of the state of slave y3 versus the state of master x3

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4.2.2 Synchronization of two Genesio–Tesi systems Two master slave Genesio–Tesi systems with uncertainty is considered as: 8 < x_ 1 ¼ x2 x_ 2 ¼ x3 : x_ 3 ¼ c0 x1 b0 x2 a0 x3 þ m0 x21 þ Df ðx; tÞ þ dðtÞ ð48Þ 8 < y_ 1 ¼ y2 y_ ¼ y3 ; : 2 y_ 3 ¼ cy1 by2 ay3 þ my21 þ Df ðy; tÞ þ dðtÞ þ uðtÞ ð49Þ in which the parameters of slave is as in control part, but the parameters of master are selected as: a0 ¼ 1; b0 ¼ 2:8; c0 ¼ 5 and m0 ¼ 1:1: The exact TS fuzzy model for the master system is achieved as: If x1 is N, then x_ 1 ¼ A1 x þ B1 u: If x1 is P, then x_ 1 ¼ A2 x þ B2 u; in which: 2 0 6 A1 ¼ 4 0

3 0 7 1 5; 1:6 2:8 1 2 3 0 6 7 B1 ¼ B2 ¼ 4 0 5 1 0

2

0 6 A2 ¼ 4 0 8:3

1 0 2:8

3 0 7 1 5; 1 ð50Þ

1 where the membership function for N and P are as: lN ¼ and lP ¼ 1 x19þ3 : It can be shown that this TS fuzzy system can exactly approximate the chaotic system of (48) in the interval of x1 2 ½3 6 except the uncertainty of the system. The uncertainty is compensated by the robustness term of (34) to make the reference model act as the master system. Then by the use of the controller which is designed in the previous section, it is possible to control the slave system to behave as the reference model and in this way the master and the slave systems are synchronized. Figure 10 shows the result of synchronization. Here, the slave system is run for 20 s without any input and the controller is turned on at the 20th second. As can be seen from the figure, the reference model, the master and the slave systems are synchronized using the proposed IMRAFC scheme. x1 þ3 9

5 Conclusions In this paper, a novel IMRAFC for a class of nonlinear chaotic systems subject to disturbances is proposed. A robust model reference fuzzy controller and the adaptation laws for the identification system are derived using an

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appropriate Lyapunov function. It is shown that the adaptation law and the proposed control signal guarantee the boundedness of all the signals in the closed-loop system and ensure that the states of the chaotic system track those of the reference model asymptotically for any bounded reference input signal. The proposed adaptive fuzzy controller is then used to synchronize uncertain chaotic systems. It is shown that it is possible to select the reference signal of the system in such a way that the states of the reference model track the states of the master chaotic system. Then, using the proposed IMRAFC, it is possible to force the states of the slave chaotic system to track the states of the reference model and hence become synchronized with the master chaotic system. It is also shown that not only can the initial values of the master and the slave systems be different but also there can be parametric differences between them. The proposed adaptive fuzzy control scheme is used for the stabilization and synchronization of Duffing oscillator and Genesio–Tesi system to verify the validity and the effectiveness of the proposed control scheme in the presence of uncertainties and disturbances. From the simulation results, it is concluded that the suggested scheme can effectively solve the control and synchronization problems of chaotic systems which suffer from disturbances and uncertainties.

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