Chaos synchronization in noisy environment using nonlinear filtering and sliding mode control

Chaos, Solitons and Fractals 36 (2008) 1295–1304

www.elsevier.com/locate/chaos

Chaos synchronization in noisy environment usingnonlinear filtering and sliding mode control

Mehdi Behzad *, Hassan Salarieh, Aria Alasty

Center of Excellence in Design, Robotics, and Automation (CEDRA), Department of Mechanical Engineering,

Sharif University of Technology, Postal Code 11365-9567, Azadi Avenue, Tehran, Iran

Accepted 28 July 2006

Communicated by Prof. Mohamed Saladen El Naschie

Abstract

This paper presents an algorithm for synchronizing two different chaotic systems, using a combination of theextended Kalman filter and the sliding mode controller. It is assumed that the drive chaotic system has a random exci-tation with a stochastically chaotic behavior. Two different cases are considered in this study. At first it is assumed thatall state variables of the drive system are available, i.e. complete state measurement, and a sliding mode controller isdesigned for synchronization. For the second case, it is assumed that the output of the drive system does not containthe whole state variables of the drive system, and it is also affected by some random noise. By combination of extendedKalman filter and the sliding mode control, a synchronizing control law is proposed. As a case study, the presentedalgorithm is applied to the Lur’e-Genesio chaotic systems as the drive-response dynamic systems. Simulation resultsshow the good performance of the algorithm in synchronizing the chaotic systems in presence of noisy environment.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

In the last few years, synchronization in chaotic dynamical systems has received a great deal of interest among sci-entists from various fields [1,2]. The results of chaos synchronization are utilized in biologic synchronization, chemicalreaction synchronization, secret communication and cryptography fields, nonlinear oscillation synchronization andsome other nonlinear fields. The first idea of synchronizing two identical chaotic systems with different initial conditionsis introduced by Pecora and Carrols [3–5], and the method is realized in electronic circuits. The methods for synchro-nization of the chaotic systems has been widely studied in recent years, and many different methods are applied theo-retically and experimentally to synchronize the chaotic systems [6–8]. A basic configuration for chaos synchronization isthe drive-response pattern, where the response chaotic system must track the drive chaotic trajectory. A number ofmethods based on this configuration have been proposed. In [9–14] synchronization in hyper-chaotic systems wereinvestigated and a generalized method for synchronization of chaotic systems was proposed [12,13]. Various active

0960-0779/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2006.07.058

* Corresponding author. Tel.: +98 21 6616 5509; fax: +98 21 6600 0021.E-mail addresses: [email protected] (M. Behzad), [email protected] (H. Salarieh), [email protected] (A. Alasty).

1296 M. Behzad et al. / Chaos, Solitons and Fractals 36 (2008) 1295–1304

and nonlinear control methods were used for chaos synchronization between two identical systems. For example, syn-chronization of coupled Lorenz system with active control [15], two Genesio systems via backstepping approach [16,17],two uncertain Liu systems with adptive control [18] were recently investigated. In [19] based on adaptive control,parameter identification and chaos synchronization for two identical chaotic systems were introduced. Also synchroni-zation between two different chaotic systems using different nonlinear control methods have been studied [20–23]. Para-metric adaptive control in chaos synchronization has been greatly used [24,25]. Besides, many techniques areinvestigated based on combination of observer and control systems [26–31] and applied for both chaos control andchaos synchronization. In [26] a sliding observer is designed for synchronization between two identical chaotic systems.Some criteria for discrete time generalized chaos synchronization are investigated in [27]. Observer based control ofchaos for a special class of chaotic systems whose nonlinearities depend on the output signal is presented in [28],and in [29] the structures of their attractors under synchronization are studied and some sufficient conditions for globalsynchronization are obtained. In [30] the problem of chaos synchronization is investigated using adaptive observerdesign and Lyapunov stability theory. Also some robust nonlinear methods such as variable structure [32,33], H1[34,35] and impulsive [36,37] robust controllers are used for chaos synchronization.

Most of the recent works are applied for two identical chaotic systems. In practice it is difficult to find two exactlyidentical chaotic systems. Hence, the synchronization of two different chaotic systems plays a significant role in practicalapplications, and this problem will be more challenging and difficult if the parameters of two chaotic systems have someuncertainty or the environment apply some random noise to the dynamic systems or measured output. Actually in prac-tice, it hardly occurs that both the drive and response systems have exactly the same configuration or the output signalsare not influenced by some stochastic noises.

In this paper the sliding mode controller is used to synchronize the behavior of two different chaotic systems. Thedrive system used in this study and the output signal are affected by some independently random noise. Moreover theresponse chaotic system of this research has some uncertainties in its dynamic equation. For chaos synchronization, forthe first step it is assumed that all of the state variables of the drive system are available, and a sliding mode controllerwith the complete state measurement is designed. In many cases, complete state measurements may not be possible. Soin the next step, for such cases, a nonlinear estimator or observer along with the extended Kalman filter is designed toestimate all of the drive state variables. The sliding mode control obtained for complete state measurement is modifiedfor incomplete state measurement and then the stability analysis for the controlled system is investigated analytically.The method presented in this paper is applied to a Lur’e like and the Genesio chaotic systems as the drive and theresponse systems respectively. The simulation results show the effectiveness of the proposed method in chaos synchro-nization of two different uncertain chaotic systems with incomplete noisy measurement.

2. Chaos synchronization with complete state measurement

A system with the following equation is considered as the drive system:

xðnÞ ¼ f ðx; tÞ þ cðtÞv ð1Þ

where x ¼ ðx; _x; . . . ; xðn�1ÞÞ ¼ ðx1; x2; . . . ; xnÞ 2 Rn is the state vector, f : Rn �Rþ ! R is a nonlinear, and sufficientlysmooth function, v is a bounded continuous random process, jvj < N, e.g. a saturated Wienner process. c(t) is a realbounded and continuous function. It is assumed that the chaotic behavior of the system in Eq. (1), i.e. x, is uniformlybounded with respect to v, that is:

9M > 0; 9Mi > 0; jcðtÞvj < M ) jxiðtÞj < Mi ð2Þ

The response system that must be controlled for synchronization is given by:

yðnÞ ¼ gðy; tÞ þ bðy; tÞu ð3Þ

where y ¼ ðy; _y; . . . ; yðn�1ÞÞ ¼ ðy1; y2; . . . ; ynÞ 2 Rn is the state vector, u 2 R is the controller variable of the system, andg; b : Rn �Rþ ! R are sufficiently smooth functions. The functions g and b have some uncertainties and their nom-inal values are shown by g and b. It is also assumed that the function b is a positive definite function which has astrictly positive lower bound bm, and upper bound bM. The difference between g and g is also bounded by a knownfunction G, so:

jgðy; tÞ � gðy; tÞj < Gðy; tÞ ð4Þ0 < bmðy; tÞ 6 bðy; tÞ; bðy; tÞ 6 bM ðy; tÞ ð5Þ

M. Behzad et al. / Chaos, Solitons and Fractals 36 (2008) 1295–1304 1297

The synchronization problem is to design a controller u which synchronizes the state of both the drive and the responsesystems, in such a way that the response trajectories follow the drive trajectories. At first, it is assumed that all the statesof the drive system can be measured, i.e. complete state measurement is available. Now by subtracting (1) from (3) one gets,

eðnÞ ¼ gðy; tÞ � f ðx; tÞ þ bðy; tÞu� cðtÞv ð6Þ

In which e = y � x. The aim of synchronization is to make

limt!1keðtÞk ¼ 0 ð7Þ

and this objective is achieved by designing a sliding mode control. To design a sliding mode control for chaos synchro-nization, let a sliding surface as

SðtÞ ¼ d

dtþ k

� �n�1

eðtÞ ¼Xn�1

m¼0

Cn�1m eðmÞkn�1�m; Cn�1

m ¼ ðn� 1Þ!m!ðn� 1� mÞ! ð8Þ

where k > 0 is chosen arbitrarily. Our aim is to design a controller that enables the system reaching to the sliding surfaceas smoothly as possible in a finite time. It is obvious that in the sliding surface the system trajectories approach to theorigin, because in the sliding surface the dynamics of the system is degenerated to

SðtÞ ¼Xn�1

m¼0

Cn�1m eðmÞkn�1�m ¼ 0 ð9Þ

So in the sliding surface one has a linear system such that its eigen-values are equal to k.A Lyapunov function V ¼ 1

2SðtÞ2, in which V is a positive definite function can now be defined and one can write,

_V ¼ SðtÞ _SðtÞ ¼ SðtÞXn�1

m¼0

Cn�1m eðmþ1Þkn�1�m

" #ð10Þ

A controller u, such that the trajectories of the system approach the sliding surface in a finite time is proposed as:

u ¼ 1

bmf � g �

Xn�2

m¼0

Cn�1m eðmþ1Þkn�1�m � KsignðSðtÞÞ

" #ð11Þ

where K will be determined in such a way that the sliding surface be attracting in a finite time. Substituting (11) in (10),one has

_V ¼ SðtÞ bbm� 1

� �ðf � gÞ þ g � g � cðtÞvþ 1� b

bm

� � Xn�2

m¼0

Cn�1m eðmþ1Þkn�1�m

" #� b

bmKsignðSðtÞÞ

( )ð12Þ

By setting

K P Gðy; tÞ þ bM

bm� 1

� �jf ðx; tÞ � gðy; tÞj þ

Xn�2

m¼0

Cn�1m eðmþ1Þkn�1�m

����������

!þ jcðtÞjN þ h

( ); h > 0 ð13Þ

where h is an arbitrary positive number and it is easily seen that:

_V 6 �hjSðtÞj ð14Þ

This implies that the sliding surface be attracting in a finite time, and the trajectories of the system approach the slidingsurface globally. So the controlling law in (11) can be easily used for chaos synchronization between the drive-responsesystems (1) and (3).

3. Estimation and incomplete state measurement

In practice all of the state variables of the drive system are not available and a measurement equation is added to thederive system described by Eq. (1):

xðnÞ ¼ f ðx; tÞ þ cðtÞv ð15ÞzðmÞ ¼ hðx; tÞ þ dðtÞw ð16Þ

1298 M. Behzad et al. / Chaos, Solitons and Fractals 36 (2008) 1295–1304

Eq. (16) is the measurement dynamics, and ðz; _z; . . . ; zðmÞÞ is the measurement vector, w is a continuous bounded randomprocess which is independent of v, h : Rn �Rþ ! R is sufficiently smooth function, and d(t) is a continuous boundedfunction. As it is seen the measurement dynamics is also affected by noise. In this case a nonlinear filter is designed toestimate x(t) with respect to all measured signals obtained from ðzðsÞ; _zðsÞ; . . . ; zðmÞðsÞÞ for s < t. To design such a filter,at first the drive system equations must be rewritten in the following form:

_x ¼ F 1ðx; tÞ þ CðtÞv ð17Þ_z ¼ H 1ðx; tÞ þ DðtÞw ð18Þ

where,

F 1ðx; tÞ ¼

x2

..

.

xn

f ðx; tÞ

266664377775; H 1ðx; tÞ ¼

z2

..

.

zm

hðx; tÞ

266664377775; CðtÞ ¼

0

..

.

0

cðtÞ

266664377775; DðtÞ ¼

0

..

.

0

dðtÞ

266664377775 ð19Þ

The filter has the well-known form of:

_xe ¼ F 1ðxe; tÞxe � KðH 1ðxe; tÞ � _zÞ ð20Þ

where K is the filter gain and must be designed to obtain an appropriate estimation for x(t), and xe (t) is the estimate ofx(t). Indeed the nonlinear filter is obtained by feeding back the measurement error to a non-excited drive system. Toobtain K, the error dynamics between the filter and the drive systems is derived by setting g = x � xe:

_g ¼ F 1ðgþ xe; tÞ � F 1ðxe; tÞ þ CðtÞv� K½H 1ðxe; tÞ � Hðgþ xe; tÞ � DðtÞwðtÞ� ð21Þ

Now define:

F ¼ oF 1ðx; tÞox

����x¼xe

; H ¼ oH 1ðxÞox

����x¼xe

ð22Þ

Using the Taylorian linearization, Eq. (21) is rewritten in the following form:

_g ¼ ½F � KH �gþ KDðtÞwþ CðtÞvþ h:o:t ð23Þ

where h.o.t denotes the higher order terms in Taylor series. If (F,H) pair is observable one can design K(t) in such a waythat the linearized part of Eq. (23) is assymptotically stable, hence the error of estimation, g = x � xe, is locallybounded. There are several methods to calculate K, and one of them is the Kalman–Bucy filtering method [38,39], wherean optimal filter is obtained when the system dynamics is linear and the excitation signals are white noise, i.e. the deriv-ative of the Wienner process [39]. In this case matrix K(t) is obtained as:

K ¼ QH T ðDDT Þ�1 ð24Þ

where Q satisfies the following dynamics:

_Q ¼ FQþ SQT � QH T ðDDT Þ�1HQþ CCT ð25Þ

In above equations, if the matrix DDT is not invertible, one must use its pseudo-inverse instead of (DDT)�1. If (F,H)pair is observable, i.e. there exists K(t) such that the phase space origin in equation _g ¼ ðF � KHÞg be assymptoticallystable, the filter gain from Eqs. (24) and (25) will also stabilize the phase space origin, because in linear systems theKalman–Bucy filter is an optimal filter. Therefore if the initial condition of nonlinear filter, xe(0), is sufficiently nearto its actual value, x(0), then there exists Me > 0 such that jgj < Me, in Eq. (23). This is resulted from the local stabilityof the filter and hence from its local boundedness. In this paper it is assumed that the pair (F,H) be observable and theoptimum Kalman–Bucy gain is used for estimator system (20).

The initial condition for Eqs. (20) and (25) are the mean value of x and its covariance matrix at the first instance, i.e.t = 0.

4. Chaos synchronization with incomplete state measurement

Using the estimate value for x one can apply the sliding mode control by using xe instead of x in Section 2. Nowdefine:

M. Behzad et al. / Chaos, Solitons and Fractals 36 (2008) 1295–1304 1299

e ¼ y � xe; ei ¼ yi � xei ! eðiÞ ¼ yðiÞ � xðiÞe ð26Þ

and a new function, like the sliding surface (8) as,

bSðtÞ ¼Xn�1

m¼0

Cn�1m eðmÞkn�1�m ð27Þ

The difference between the sliding surface (8) and (27) may be written as:

SðtÞ ¼ bSðtÞ þ dðtÞ ð28Þ

where d(t) is dependent on x � xe and its derivatives:

dðtÞ ¼Xn�1

m¼0

Cm�1m ðxe � xÞðmÞkn�1�m ð29Þ

Using the error and the sliding surface defined in (26) and (27) the controller action in Eq. (11) changes as follows:

u ¼ 1

bmf ðxe; tÞ � gðy; tÞ �

Xn�2

m¼0

Cn�1m emþ1k

n�1�m � KsignðbSðtÞÞ" #ð30Þ

Consequently the derivative of the Lyapunov function along the trajectories of the error dynamics are:

_V ¼ SðtÞ bbm� 1

� �ðf ðxe; tÞ � gðy; tÞÞ þ f ðxe; tÞ � f ðx; tÞ þ gðy; tÞ � gðy; tÞ � cðtÞv

�þ 1� b

bm

� � Xn�2

m¼0

Cn�1m emþ1k

n�1�m

" #þXn�2

m¼0

Cn�1m ðxe � xÞðmþ1Þkn�1�m � b

bmKsignðbSðtÞÞ) ð31Þ

Our aim is to obtain a suitable bound for K such that _V becomes negative definite. Note that f(x, t) is a smooth functionand its responses satisfy the condition in Eq. (2), so the partial derivatives of f(x, t) in the region defined by condition (2)are bounded. Now using the mean value theorem and regarding to compactness of the region in condition (2), one canconclude, there exists an L > 0 such that,

jf ðx; tÞ � f ðxe; tÞj < Lkx� xek ð32Þ

Using inequality (32), and considering that both x and xe satisfy the condition in Eq. (2), and also assuming that thecovariance of estimator be finite, a suitable value for K is obtained from:�

K P Gðy; tÞ þ bM

bm� 1

� �jgðy; tÞ � f ðxe; tÞj þ

Xn�1

m¼1

Cn�1m�1eðmÞkn�1�m

����������

!þ LMe þ

Xn�1

m¼1

Cn�1m�1Mek

n�1�m

����������þ jcðtÞjN þ h

( )ð33Þ

Now substituting the condition (33) in (31) and manipulating some calculations, one obtains:

_V 6 �hjbSðtÞj � KdðtÞsignðbSðtÞÞ ð34Þ

Eq. (34) may be rewritten as:

_V 6 �hjSðtÞ � dðtÞj � KdðtÞsignðSðtÞ � dðtÞÞ 6 �hjSðtÞj þ ðhþ KÞjdðtÞj ð35Þ

So the negative definiteness of _V is not completely achieved, and it is perturbed by the estimation error. This produces aboundary layer for sliding surface, hence a steady state error for synchronization. Because, if _V P 0, the above inequal-ity results in:

jSðtÞj 6 hþ KhjdðtÞj ð36Þ

Note that S(t) = 0 makes a linear system that is asymptotically stable, so inequality (36) dictates a bounded steady stateerror for e = y � x. To show this, Eq. (36) is rewritten in the form of:

m¼0

Cn�1m eðmÞkn�1�m ¼ eðtÞ; jeðtÞj 6 hþ K

hjdðtÞj ð37Þ

So the steady state error of linear system (37) is bounded by:

1300 M. Behzad et al. / Chaos, Solitons and Fractals 36 (2008) 1295–1304

jeð1Þj 6 1

kn�1keðtÞk1 ð38Þ

where e(1) denotes the steady state error and k.k1 denotes the supremum norm. The results obtained may be summa-rized through the following theorem.

Theorem. Consider the drive system with its measurement dynamics in Eqs. (15) and (16). Suppose that the pair (F,H)

defined in Eq. (22) be observable, then the filtering and synchronizing laws, presented in Eqs. (20), (24), (25), (30) and (33)

can locally synchronize the behavior of the response system (6) with the drive system with the steady state error bound

defined in (38).

5. Simulation and results

The synchronization algorithm is explained in this section by using an example. Lur’e dynamic system has beenselected as the drive chaotic system. Dynamic equations for this system can be written as

_x1 ¼ x2

_x2 ¼ x3

_x3 ¼ a1x1 þ a2x2 þ a3x3 þ 12uðx1Þ þ c sin t:vðtÞð39Þ

where

uðx1Þ ¼kx1 jx1j < 1=k

signðx1Þ otherwise

�ð40Þ

For c = 0, a1 = �7.4, a2 = �4.1, a3 = �1, and k = 3.6 the behavior of the system is chaotic [35]. In this examplec(t) = c sin t, and v(t) is a saturated Wienner process with N = 10. In this example the parameters in Eq. (2) areM = 10, Mi = 15, i = 1,2,3. The Gensio chaotic system is chosen as the response system. The dynamic equation ofthe Genesio system is

_y1 ¼ y2

_y2 ¼ y3

_y3 ¼ b1y1 þ b2y2 þ b3y3 þ y21 þ ð0:5þ jy1jÞu

ð41Þ

Fig. 1. Chaotic attractor of Lur’e system, a1 = �7.4, a2 = �4.1, a3 = �1, k = 3.6 and c = 0.

M. Behzad et al. / Chaos, Solitons and Fractals 36 (2008) 1295–1304 1301

where u is the controlling signal, which is used for synchronization purpose. For u = 0, b1 = �5.6, b2 = �2.74,b3 = �1.1 the dynamic behavior of the Genesio system is chaotic [16]. The chaotic attractors of the Lur’e and the Gene-sio system in the phase space are shown in Figs. 1 and 2.

Fig. 2. The chaotic attractor of Genesio system, b1 = �5.6, b2 = �2.74, b3 = �1.1.

0 2 4 6 8 10-5

0

5

x 1, y 1

0 2 4 6 8 10

-5

0

5

x 2, y2

0 2 4 6 8 10-20

0

20

x 3, y3

0 2 4 6 8 10-20

-10

0

10

Time

u

Lur’eGenesio

Fig. 3. Synchronization results between the Lur’e and the genesio dynamic system. xi’s are the Lur’e dynamic states and yi’s are theGenesio dynamic states.

0 2 4 6 8 10-5

0

5

x 1, xe1

0 2 4 6 8 10

-5

0

5

Time

x 1, xe1

0 2 4 6 8 10

-10

0

10

Time

x 3, xe3

Fig. 4. State estimation of the Lur’e dynamic system using the EKF method.

1302 M. Behzad et al. / Chaos, Solitons and Fractals 36 (2008) 1295–1304

It is assumed that the measuring signal is x1(t) but due to noise effects the measured output have some noise, so themeasuring equation is

_zðtÞ ¼ x2ðtÞ þ wðtÞ ð42Þ

Besides, the nominal values for bi’s are shown by b0is and their values are b1 ¼ �4:6, b2 ¼ �3:24 and b3 ¼ �2:1. Basedon Eqs. (39) and (41), functions f(x, t), F1(x, t), c(t), d(t), h(x, t), g(y, t), b(x, t), gðy; tÞ, bðx; tÞ and bm used in Sections 3 and4 are as

f ðx; tÞ ¼ a1x1 þ a2x2 þ a3x3 þ 12uðx1Þ; F 1ðx; tÞ ¼ x2 x3 f ðx; tÞ½ �T; hðx; tÞ ¼ x2; cðtÞ ¼ 0:2 sin t;

dðtÞ ¼ 0:2; gðy; tÞ ¼ b1y1 þ b2y2 þ b3y3 þ y21; gðy; tÞ ¼ b1y1 þ b2y2 þ b3y3 þ y2

1;

bðy; tÞ ¼ 0:5þ jy1j; bðy; tÞ ¼ 1þ jy1j; bM ðy; tÞ ¼ 1þ 2jy1j; bm ¼ 1

Using the method presented in Sections 3 and 4, the synchronization process between the Lur’e and the Genesio systemsas the drive and the response system is performed. The results of synchronization are illustrated in Figs. 3 and 4. Fig. 3illustrates the controller performance. It can be seen that the tracking process and hence the synchronization of thedrive-response systems is properly achieved. Fig. 4 shows that the filtering process acts properly and the states ofthe system are estimated after about 1 s. In Fig. 4, it is shown that despite the noise and other random uncertaintiesthe estimated signal approach the actual signal.

6. Conclusion

In this paper a new method for synchronizing two different chaotic systems is introduced. The drive chaotic system isexcited by a bounded random process, and the response system has some uncertainties in its dynamic equation. Twodifferent cases are considered; complete state measurement and incomplete state measurement where the measurementdynamics is affected by some random process too. Using the sliding mode control a synchronizing algorithm is obtainedfor complete state measurement case, and then this control method is coupled with a nonlinear filter and modified forapplying to the incomplete measurement case. The stability analysis in synchronization is investigated analytically andthe effectiveness of the method is examined through simulations. The results of applying the presented algorithm to the

M. Behzad et al. / Chaos, Solitons and Fractals 36 (2008) 1295–1304 1303

Lur’e-Gensio dynamic systems with incomplete state measurement show the robustness and proper performance of themethod in synchronization and estimation in noisy environments.

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