Indirect adaptive control of discrete chaotic systems

Chaos, Solitons and Fractals 34 (2007) 1188–1201 www.elsevier.com/locate/chaos

Indirect adaptive control of discrete chaotic systems Hassan Salarieh b

a,1

, Mohammad Shahrokhi

b,*

a Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran

Accepted 31 March 2006

Communicated by Professor Mohamed Saladin El Naschie

Abstract In this paper an indirect adaptive control algorithm is proposed to stabilize the fixed points of discrete chaotic systems. It is assumed that the functionality of the chaotic dynamics is known but the system parameters are unknown. This assumption is usually applicable to many chaotic systems, such as the Henon map, logistic and many other nonlinear maps. Using the recursive-least squares technique, the system parameters are identified and based on the feedback linearization method an adaptive controller is designed for stabilizing the fixed points, or unstable periodic orbits of the chaotic maps. The stability of the proposed scheme has been shown and the effectiveness of the control algorithm has been demonstrated through computer simulations. Ó 2006 Elsevier Ltd. All rights reserved.

1. Introduction Chaotic phenomena have been observed in numerous fields of science such as physics, chemistry, biology and ecology. Chaotic behavior has been detected in some fields such as laser technology, plasma, mechanical engineering and chemistry. An interesting subject in chaos theory is to eliminate the chaotic behavior by means of control systems. However, in some occasions, it is desired to excite the chaotic mode of the system, e.g. to obtain a special vibrating motion. The first documents in control of chaotic systems return to 1990s. Ott et al. [1] proposed an innovative scheme to control chaos for which a small parameter perturbation converts the chaotic attractor to an attractor with large number of attracting periodic motions (see [2] for extensions of this method). Pyragas, based on delayed feedback control, proposed another control scheme [3]. In recent years different nonlinear control techniques are used for chaos control, e.g. feedback linearization [4,5], variable structure controllers [6–8], fuzzy methods and neural networks [9–13]. Besides non-adaptive controllers, different methods for chaos control via adaptive control techniques are developed [14]. In adaptive version, the uncertain chaotic systems are controlled. Adaptive controllers are used for both chaos synchronizing [15–17] and stabilizing the chaotic systems [18–22]. Chaotic systems may be appeared in discrete or continuous form [23], hence control algorithms are developed based on continuous [18–20] and discrete [24–28] dynamic *

1

Corresponding author. Tel: +98 21 6616 5419. E-mail addresses: [email protected] (H. Salarieh), [email protected] (M. Shahrokhi). Tel.: +98 21 6616 5586.

0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.03.115

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models. In Ref. [24] an algorithm for discrete time adaptation of one parameter is proposed using the one dimensional Poincare map for system modeling. In [27] a dynamic feedback with a perturbed parameter is used as an adaptive technique for control of chaos. Also some experimental studies for chaos control of low dimensional discrete systems are performed. As an example in [28], the OGY method is applied to a chaotic system. In this paper, control of a discrete chaotic system with known functionalities and unknown parameters is considered. This is the case for many discrete chaotic systems, for example the Logistic and the Henon maps have quadratic forms with some unknown parameters. Using the recursive least squares and the feedback linearization method, an indirect adaptive controller is designed to stabilize the fixed points, or unstable periodic orbits of a chaotic system. The proposed method can be used for stabilizing the fixed points of any order, and the formulation is developed for stabilizing a d-cycle fixed point where d may be arbitrary large. The asymptotic stability of the proposed control scheme is proved analytically and its effectiveness has been shown through simulation. 2. Problem statement Consider a discrete time chaotic system as: xj ½k þ 1 ¼ /TP j ðx½kÞhP j þ ð/TQj ðx½kÞhQj Þuj ½k;

j ¼ 1; . . . ; N

ð1Þ

where /P j ðÞ and /Qj ðÞ, j=1, . . . , N are known smooth functions, and hP j and hQj , j = 1, . . . , N are unknown constant coefficients. If the above system is combined d times with itself, a delayed discrete time system is obtained as: xj ½k þ d ¼ /Tfj ðx½k; u½k; . . . ; u½k þ d  2Þhfj þ ð/Tgj;1 ðx½k; u½k; . . . ; u½k þ d  2Þhgj;1 Þ  uj ½k þ d  1 þ    þ ð/Tgj;d ðx½kÞhgj;d Þuj ½k

j ¼ 1; . . . ; N

ð2Þ

where /fj ðÞ and /gj;i ðÞ, i = 1, . . . , d, j = 1, . . . , N are known smooth functions, and hfj and hgj;i , i = 1, . . . , d, j = 1, . . . , N are unknown constant coefficients. Eq. (2) is a general form of a d-self-combined discrete system (1) and is used for stabilizing a d-cycle unstable fixed point of a discrete time chaotic system. Assume that the nonlinear coefficient of uj[k + d  1] is invertible, all of the nonlinear functions are sufficiently smooth and the state variables are available. The main goal is to stabilize the given fixed points of system (2). Note that using some numerical techniques, the fixed points of a chaotic system whose states are accessible can be calculated without using its dynamic equation [11,29,30], hence it is assumed that the fixed points of the system are obtained by some numerical algorithms without using the system parameters. 3. Identification method To identify the unknown parameters of system (2), it is written in the following form: 2 3 hfj 6h 7 6 gj;1 7 6 7 T T T xj ½k þ d ¼ ½/fj ðÞ /gj;1 ðÞun ½k þ d  1    /gj;d ðÞuj ½k6 . 7 6 .. 7 4 5 hgj;d

ð3Þ

Now define, gj ½k ¼ xj ½k þ d h T T Uj ½k ¼ /fj ðÞ /gj;1 ðÞuj ½k þ d  1 h iT T T T Hj ¼ hfj hgj;1    hgj;d



/Tgj;d ðÞuj ½k

iT

ð4Þ ð5Þ ð6Þ

Using the above definitions, Eq. (4), can be written as: gj ½k ¼ UTj ½kHj b j ½k is the estimate of Hj and defined as below: Now assume that H h iT b j ½k ¼ ^hTf ½k ^hTg ½k    ^hTg ½k H j j;1 j;d

ð7Þ

ð8Þ

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The error vector can be written as: b j ½k  1 ej ½k ¼ gj ½k  UTj ½k H

ð9Þ

b j ½k, the least squares technique is used. Consider the following objective To obtain the estimated parameters, H functions: Jk ¼

k X b j ½kÞ2 ðgj ½n  UTj ½n H

ð10Þ

n¼1

b and set it to zero we have: By differentiating Eq. (10) with respect to H 0 2 T 311 2 3 gj ½1 Uj ½1 B 6 7C 6 7 b j ½kB½ Uj ½1    Uj ½k 6 .. 7C ½ Uj ½1    Uj ½k 6 .. 7 H @ 4 . 5A 4 . 5 gj ½k UTj ½k

ð11Þ

Let 2

0 B P j ½k ¼ B @½ Uj ½1

6 Uj ½k 6 4



UTj ½1 .. .

311 7C 7C 5A

ð12Þ

UTj ½k Substituting Eq. (12) into Eq. (11) yields: b j ½k ¼ P j ½k H

k X

Uj ½ngj ½n ¼ P j ½k

n¼1

k1 X

! Uj ½ngj ½n þ Uj ½kgj ½k

n¼1

b ¼ P j ½kP 1 j ½k  1 H j ½k  1 þ P j ½kUj ½kgj ½k

ð13Þ

Using Eq. (12) one can write: T 1 P 1 j ½k  1 ¼ P j ½k  Uj ½kUj ½k

ð14Þ

Substituting Eq. (14) into Eq. (13) results in: b j ½k  1 þ P j ½kUj ½kej ½k b j ½k ¼ H H

ð15Þ

After some matrix manipulations, Eqs. (14) and (15) can be rewritten as: P j ½k ¼ P j ½k  1 

P j ½k  1Uj ½kUTj ½kP j ½k  1

b j ½k ¼ H b j ½k  1 þ H

1 þ UTj ½kP j ½k  1Uj ½k P j ½k  1Uj ½kej ½k 1 þ UTj ½kP j ½k  1Uj ½k

ð16Þ

The above recursive equations can be solved using a positive definite initial matrix for Pj[0] and an arbitrary initial vecb j ð0Þ. tor for H The identification method given by Eq. (16) implies that: b j ½k  1 ! UT ½kHj UTj ½k H j

as k ! 1

ð17Þ

To show the property (17) [31], note that due to Eq. (12) P[k] is a positive definite matrix. Define: b j ½k  Hj dj ½k ¼ H

ð18Þ

Consider the following Lyapunov function: V j ½k ¼ dTj ½kP 1 j ½kdj ½k

ð19Þ

Using Eq. (16) we have: dj ½k ¼ P j ½kP 1 j ½k  1dj ½k  1

ð20Þ

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1191

DVj[k] can be obtained as follows: V j ½k  V j ½k  1 T 1 ¼ dTj ½kP 1 j ½kdj ½k  dj ½k  1P j ½k  1dj ½k  1 T 1 ¼ dTj ½kP 1 j ½k  1dj ½k  1  dj ½k  1P j ½k  1dj ½k  1

b T ½k  1ÞP 1 ½k  1ð H b j ½k  1  Hj Þ b T ½k  H ¼ ðH j j j b j ½k  1Þ2 ðgj ½k  UTj ½k H

¼

1 þ UTj ½kP j ½k  1Uj ½k

¼

1þ

e2j ½k T Uj ½kP j ½k 

ð21Þ

1Uj ½k

Eq. (21) shows that Vj[k] is a negative decreasing sequence, and consequently: n X k¼1

1þ

e2j ½k T Uj ½kP j ½k 

1Uj ½k

1þ

e2j ½k T Uj ½kP j ½k 

1Uj ½k

¼ V j ½0  V j ½n < 1

ð22Þ

Hence, lim

k!1

¼0

or

lim ej ½k ¼ 0

ð23Þ

k!1

And consequently: b j ½k  1 ! gj ½k ¼ UT ½kHj ; UTj ½k H j

as k ! 1

ð24Þ

4. Indirect adaptive control of chaos Assume that xFj ¼ ðxFj ½0; xFj ½2; . . . ; xFj ½d  1Þ is the fixed point of Eq. (2) when u[k] = 0, i.e. xFj ½0 ¼ xFj ½d ¼ /fj ðxF ½0; 0; . . . ; 0Þhfj

ð25Þ

xFj ½k þ d ¼ xFj ½k The main goal is stabilizing the fixed point xFj . Let, h qj ½k ¼ /Tfj ðx½k; u½k; . . . ; u½k þ d  2Þhfj  ð/Tgj;2 ðx½k; u½k; . . . ; u½k þ d  3Þhgj;2 Þuj ½k þ d  2      ð/Tgj;d ðx½kÞhgj;d Þuj ½k þ xFj ½k þ d 

d X

kj ðxj ½k þ d  j  xFj ½k þ d  jÞ

 ð26Þ

j¼1

h ^j ½k ¼ /Tfj ðx½k; u½k; . . . ; u½k þ d  2Þ^hfj ½k  1 q  ð/Tgj;2 ðx½k; u½k; . . . ; u½k þ d  3Þ^hgj;2 ½k  1Þuj ½k þ d  2      ð/Tgj;d ðx½kÞ^hgj;d ½k  1Þuj ½k þ xFj ½k þ d 

d X

kj ðxj ½k þ d  j  xFj ½k þ d  jÞ

 ð27Þ

j¼1

where ki are chosen such that all roots of the following polynomial lie inside the unit circle. zd þ k1 zd1 þ    þ kd ¼ 0 Assume that

/Tgj;1 ðx½k; u½k; . . . ; u½k

uj ½k þ d  1 ¼

ð28Þ þ d  2Þhgj;1 ½k  1 6¼ 0 and consider a controller as given below:

^ j ½k q T /gj;1 ðx½k; u½k; . . . ; u½k þ

d  2Þ^hgj;1 ½k  1

ð29Þ

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To show that the above controller can stabilize the fixed point of Eq. (2), note that from Eq. (24) we have: lim /Tfj ðÞ^hfj ½k  1 þ

k!1

d X

/Tgj;i ðÞ^hgj;i ½k  1uj ½k þ d  i ¼ /Tfj ðÞhfj þ

i¼1

d X

/Tgj;i ðÞhgj;i uj ½k þ d  i

ð30Þ

i¼1

One can write the above equation in the following form: ej ½k ¼ /Tfj ðÞð^hfj ½k  1  hfj Þ þ

d X

/Tgj;i ðÞð^hgj;i ½k  1  hgj;i Þuj ½k þ d  i

i¼1

lim ej ½k ¼ 0

ð31Þ

k!1

Using Eqs. (26), (27) and (31) one can obtain: ^j ½k  qj ½k ¼ ej ½k þ /Tgj;1 ðÞð^hgj;1 ½k  1  hgj;1 Þuj ½k þ d  1 q

ð32Þ

Using Eqs. (2), (26), (27) and (29) the controlled system can be written as: xj ½k þ d ¼ qj ½k þ xFj ½k þ d 

d X

ki ðxj ½k þ d  i  xFj ½k þ d  iÞ

i¼1

/Tg ðx½k; u½k; . . . ; u½k þ d  2Þhgj;1 ^j ½k q þ T j;1 / ðx½k; u½k; . . . ; u½k þ d  2Þh^g ½k  1 gj;1

ð33Þ

j;1

Using Eq. (32) in Eq. (33), we have: qj ½k þ ej ½k þ xFj ½k þ d  xj ½k þ d ¼ ^

d X

ki ðxj ½k þ d  i  xFj ½k þ d  iÞ

i¼1

/Tg ðx½k; u½k; . . . ; u½k þ d  2Þ^hgj;1 ½k ^j ½k q þ Tj;1 / ðx½k; u½k; . . . ; u½k þ d  2Þ^hg ½k gj;1

ð34Þ

j;1

After some manipulations we get: xj ½k þ d ¼ xFj ½k þ d 

d X

ki ðxj ½k þ d  i  xFj ½k þ d  iÞ þ ej ½k

ð35Þ

i¼1

or, ðxj ½k þ d  xFj ½k þ dÞ þ

d X

ki ðxj ½k þ d  i  xFj ½k þ d  iÞ ¼ ej ½k

ð36Þ

i¼1

Define y 1 ½k ¼ xj ½k  xFj ½k;

y 2 ½k ¼ xj ½k þ 1  xFj ½k þ 1; . . . ; y d ½k ¼ xj ½k þ d  1  xFj ½k þ d  1

Eq. (36) can be re-written as: 2 3 2 y 1 ½k þ 1 0 6 7 6 y 2 ½k þ 1 7 6 6 6 7 6 0 6 7¼6 .. 6 7 4 0 . 4 5 k1 k2 y d ½k þ 1

I ðd1Þðd1Þ 

32 y ½k 3 2 3 0 1 7 6.7 76 y ½k 6 7 76 2 7 6 .. 7 76 7 þ6 76 .. 7 6 7ej ½k 54 . 7 4 5 0 5 kd 1 y d ½k

ð37Þ

ð38Þ

Let 2

3

0

6 6 0 G¼6 6 0 4 k1

7 7 7; 7 5

I ðd1Þðd1Þ k2



kd

3 0 6 .. 7 6 . 7 7 H ¼6 7 6 4 0 5 2

1

ð39Þ

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1193

Eq. (38) can be written as: Y ½k þ 1 ¼ GY ½k þ H ej ½k

ð40Þ

Taking z-transform from both sides of Eq. (40) we get: Y ðzÞ ¼ ðzI  GÞ1 zY ð0Þ þ ðzI  GÞ1 H ej ðzÞ

ð41Þ

Taking inverse yields: Y ðkÞ ¼ Gk Y ½0 þ z1 ½ðzI  GÞ1 H ej ðzÞ

ð42Þ

Note that lim GkY[0] = 0 as k ! 1 (because all eigen-values of G lie inside the unit circle), besides lim ej[k] = 0 as k ! 1 and G is a stable matrix, therefore: lim z1 ½ðzI  GÞ1 H ej ðzÞ ¼ 0

ð43Þ

k!1

Consequently we have: lim Y ½k ¼ 0 ) xj ½k ! xFj ½k; . . . ; xj ½k þ d  1 ! xFj ½k þ d  1

k!1

ð44Þ

From Eq. (44) the stability of the proposed controller is established. Remark 1. In practice, control law (29) works well but in theory there is the remote possibility of division by zeros in calculating u(t). This can easily be avoided. For example u(t) can be calculated as follows: 8 ^j ½k q > > < l ½k ; jlj ½kj 6¼ 0 j uj ½k þ d  1 ¼ ð45Þ > ^j ½k > :q ; jlj ½kj ¼ 0 me where lj ½k ¼ /Tgj;1 ðx½k; u½k; . . . ; u½k þ d  2Þ^hgj;1 ½k  1

ð46Þ

and me > 0 is a small positive real number. Remark 2. For handling the time varying systems, least squares algorithm with variable forgetting factor can be used. The corresponding updating rule is given below (the Fortescue method [32]): " # P j ½k  1Uj ½kUTj ½kP ½k  1 1 P j ½k  1  P j ½k ¼ ð47Þ mj ½k mj ½k þ UTj ½kP j ½k  1Uj ½k ^ j ½k ¼ H ^ j ½k  1 þ H where,

P j ½k  1Uj ½kej ½k mj ½k þ UTj ½kP j ½k  1Uj ½k

(

e2j ½k mj ½k ¼ max 1  ; kmin 1 þ e2j ½k

) ð48Þ

and 0 < kmin < 1 usually set to 0.95.

5. Simulation results In this section through simulation, the performance of the proposed adaptive controller is evaluated. Example 1. Consider the logistic map given below: x½k þ 1 ¼ lx½kð1  x½kÞ þ u½k

ð49Þ

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For l P 3.567 and u[k] = 0 the behavior of the system is chaotic. In this example stabilization of the 2-cycle fixed point of the following logistic map is considered. In this case the governing equation is: x½k þ 2 ¼ l2 x½k  ðl2 þ l3 Þx2 ½k þ 2l3 x3 ½k  l3 x4 ½k þ lu½k  2l2 x½ku½k þ 2l2 x2 ½ku½k  lu2 ½k þ u½k þ 1 ð50Þ and its fixed points for l = 3.6 are xF[1] = 0.8696, xF[2] = 0.4081 and for l = 3.9 are xF[1] = 0.8974, xF[2] = 0.3590. Eq. (50) can be written in the following form: 2 3 a1 2 3 6 7 b1 6 a2 7 7 6     7 7 þ 1 x½k x2 ½k 6 x½k þ 2 ¼ x½k x2 ½k x3 ½k x4 ½k u2 ½k 6 a b ð51Þ 4 3 2 5u½k þ cu½k þ 1 6 7 6 7 b3 4 a4 5 a5

Fig. 1. Controlled response of the logistic map (51), for stabilizing the 2-cycle fixed point, when the parameter l changes from l = 3.6 to l = 3.9 at k = 150.

Fig. 2. Parameter estimates for the logistic map (51), when the parameter l changes from l = 3.6 to l = 3.9 at k = 150.

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It is assumed that the system parameter l changes from l = 3.6 to l = 3.9 at k = 150. The results are shown in Figs. 1 and 2. As can be seen the 2-cycle fixed point of the system is stabilized and the state error tends to zero, although the parameter estimates are not converged to their actual values. To evaluate the performance of the controller in the presence of noise, 5% white noise is added to the states. The results are shown in Figs. 3 and 4. As can be seen the control objective is achieved successfully.

Fig. 3. Controlled response of the logistic map (51), for stabilizing the 2-cycle fixed point, when the system states are corrupted with noise.

Fig. 4. Parameter estimates for the logistic map (51), when the system states are corrupted with noise.

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Example 2. For the second example, the Henon map is considered, x1 ½k þ 1 ¼ 1  ax21 ½k þ x2 ½k þ x2 ½k þ u1 ½k x2 ½k þ 1 ¼ bx1 ½k þ u2 ½k

ð52Þ

where for a = 1.4, b = 0.3 and u1[k] = u2[k] = 0, the behavior of the system is chaotic. First the 1-cycle fixed point is regarded for stabilization. Eq. (52) can be written in the following form: x1 ½k þ 1 ¼ /Tf 1 ðx1 ½k; x2 ½kÞhf 1 þ /Tg1 ðx1 ½k; x2 ½kÞhg1 u1 ½k x2 ½k þ 1 ¼ /Tf 2 ðx1 ½k; x2 ½kÞhf 2 þ /Tg2 ðx1 ½k; x2 ½kÞhg2 u2 ½k

ð53Þ

where,  /f1 ðx1 ½k; x2 ½kÞ ¼ 1 x2 ½k

x21 ½k

T

;

/f2 ðx1 ½k; x2 ½kÞ ¼ x1 ½k

/g1 ðx1 ½k; x2 ½kÞ ¼ /g2 ðx1 ½k; x2 ½kÞ ¼ 1

ð54Þ

and h hf1 ¼ h1f1

h2f1

h3f1

iT

;

hf2 ¼ h1f2 ;

hgi ¼ h1gi ;

i ¼ 1; 2

ð55Þ

Again the 1-cycle fixed point is obtained, using numerical methods, ðxFi ¼ 0:6314, xF2 ¼ 0:1894Þ. Figs. 5 and 6 show the results of applying the proposed adaptive controller to the Henon map. It is observed that the 1-cycle fixed point of the system is stabilized and all of the system parameter estimates are converged. It must be noted that if the exact functionality is not known, a more general form with additional parameters can be considered. For example system (53) can be modeled as follows:  T ð56Þ /fi ðx1 ½k; x2 ½kÞ ¼ /gi ðx1 ½k; x2 ½kÞ ¼ 1 x1 ½k x2 ½k x21 ½k x1 ½kx2 ½k x22 ½k and h iT hfi ¼ h1fi h2fi h3fi h4fi h5fi h6fi h iT hgi ¼ h1gi h2gi h3gi h4gi h5gi h6gi

Fig. 5. Controlled response of the Henon map (53), for stabilizing the 1-cycle fixed point.

ð57Þ

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1197

Fig. 6. Parameter estimates of the Henon map (53).

As it is illustrated through Figs. 7–9, the 1-cycle fixed point is stabilized and the system parameters are converged to fixed values. It shows that in cases where our knowledge about the system dynamics is not complete, the proposed method can be applied successfully using the over-parameterized model. Example 3. Finally, the proposed scheme is used for stabilizing the 2-cycle fixed point of the Henon map. The self-combined Henon map is given below: x1 ½k þ 2 ¼ 1  a þ bx1 ½k  2ax2 ½k þ 2a2 x21 ½k  ax22 ½k þ 2a2 x21 ½kx2 ½k  a3 x41 ½k  2au1 ½k þ u2 ½k  au21 ½k  2ax2 ½ku1 ½k þ 2a2 x21 ½ku1 ½k þ u1 ½k þ 1

ð58Þ

x2 ½k þ 2 ¼ b þ bx2 ½k  abx21 ½k þ bu1 ½k þ u2 ½k þ 1 Eq. (58) can be written in the following form: x1 ½k þ 2 ¼ /Tf1 ðx1 ½k; x2 ½k; u1 ½k; u2 ½kÞhf1 þ /Tg1;1 ðx1 ½k; x2 ½k; u1 ½k; u2 ½kÞ  hg1;1 u1 ½k þ 1 þ /Tg1;2 ðx½k; y½kÞhg1;2 u1 ½k

ð59Þ

x2 ½k þ 2 ¼ /Tf2 ðx1 ½k; x2 ½k; u1 ½k; u2 ½kÞ  hf2 þ /Tg2;1 ðx1 ½k; x2 ½k; u1 ½k; u2 ½kÞhg2;1 u2 ½k þ 1 þ /Tg2;2 ðx½k; y½kÞhg2;2 u2 ½k where, /f1 ðx1 ½k; x2 ½k; u1 ½k; u2 ½kÞ ¼ ½ 1 x1 ½k

x2 ½k

/f2 ðx1 ½k; x2 ½k; u1 ½k; u2 ½kÞ ¼ ½ 1 x2 ½k

x21 ½k 

x21 ½k

x22 ½k

x21 ½kx2 ½k

x41 ½k

u21 ½k u2 ½k 

/g1;1 ðx1 ½k; x2 ½k; u1 ½k; u2 ½kÞ ¼ /g2;1 ðx1 ½k; x2 ½k; u1 ½k; u2 ½kÞ ¼ 1 /g1;2 ðx1 ½k; x2 ½k; u1 ½k; u2 ½kÞ ¼ ½ 1 x2 ½k

x21 ½k ;

/g2;2 ðx1 ½k; x2 ½ku1 ½k; u2 ½kÞ ¼ 1

ð60Þ

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Fig. 7. Controlled response of the Henon map (53), for stabilizing the 1-cycle fixed point, using over-parameterized model.

Fig. 8. hjfi parameter estimates of the Henon map (53), using over-parameterized model.

and hf1 ¼ b h1f1

h2f1



h9f1 cT ; hf2 ¼ b h1f2

hg1;1 ¼ h1g1;1 ; hg2;1 ¼ h1g2;1 h iT 1 2 3 hg1;2 ¼ hg1;2 hg1;2 hg1;2 ; hg2;2 ¼ h1g2;2

h2f2

h3f2 cT ð61Þ

H. Salarieh, M. Shahrokhi / Chaos, Solitons and Fractals 34 (2007) 1188–1201

1199

Fig. 9. hjgi parameter estimates of the Henon map (53), using over-parameterized model.

The 2-cycle fixed points of the system are xF1 ½1 ¼ 0:9758, xF2 ½1 ¼ 0:4758, xF1 ½2 ¼ 0:1427, xF2 ½2 ¼ 0:2927. Fig. 10 shows the result of the system behavior using the proposed controller. As before the parameter estimates are converged (not shown). For this case if the exact functionalities of the system are not known, the over-parameterized model can be used. For example the functionalities of a self-combined Henon map is considered as below:

Fig. 10. Controlled response of the Henon map (59), for stabilizing the 2-cycle fixed point.

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H. Salarieh, M. Shahrokhi / Chaos, Solitons and Fractals 34 (2007) 1188–1201

Fig. 11. Controlled response of the Henon map (59), for stabilizing the 2-cycle fixed point, using over-parameterized model.

 /fi ðx1 ½k; x2 ½k; u1 ½k; u2 ½kÞ ¼ 1 x1 ½k

x2 ½k x21 ½k x1 ½kx2 ½k x22 ½k x31 ½k 2 x1 ½kx2 ½k x32 ½k x41 ½k x31 ½kx2 ½k x21 ½kx22 ½k  x42 ½k u21 ½k u1 ½ku2 ½k u2 ½k u22 ½k

/gi;1 ðx1 ½k; x2 ½k; u1 ½k; u2 ½kÞ ¼ ½ 1 x1 ½k x2 ½k u1 ½k u2 ½k   /gi;2 ðx1 ½k; x2 ½k; u1 ½k; u2 ½kÞ ¼ 1 x1 ½k x2 ½k x21 ½k x1 ½kx2 ½k

x22 ½k

x21 ½kx2 ½k x1 ½kx32 ½k ð62Þ



The unknown parameters of the system are chosen as: h iT hfi ¼ h1fi h2fi    h19 fi h iT 1 2 5 hgi;1 ¼ hgi;1 hgi;1    hgi;1 h iT 1 2 6 hgi;2 ¼ hgi;2 hgi;2    hgi;2

ð63Þ

The proposed algorithm can be used for estimating the unknown parameters and stabilizing the 2-cycle fixed point as shown in Fig. 11.

6. Conclusions In this paper an indirect adaptive controller is designed to stabilize the d-cycle fixed points of a large class of discrete chaotic systems with unknown parameters. It is assumed that the functionality of the chaotic system is known but the system parameters are unknown. Using the least squares technique, an identifier is designed to estimate the unknown parameters of the system. Combining the recursive least squares algorithm with a nonlinear feedback controller that cancels the nonlinear terms of the system, an indirect adaptive controller is designed and applied to the chaotic systems. The boundedness and asymptotic stability of the proposed scheme have been shown, and the effectiveness of the technique has been demonstrated through computer simulations.

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