Synchronization of uncertain chaotic systems using active sliding mode control

Chaos, Solitons and Fractals 21 (2004) 999–1011

www.elsevier.com/locate/chaos

Synchronization of uncertain chaotic systems viabackstepping approach

Samuel Bowong a,*, F.M. Moukam Kakmeni b

a INRIA Lorraine, Project CONGE, I.S.G.M.P. Bat A, Ile de Saulcy, 57045 Metz Cedex 01, Franceb Laboratoire de M�ecanique, D�epartement de Physique, Facult�e des Sciences, Universit�e de Yaound�e I, B.P. 812 Yaound�e, Cameroon

Accepted 8 December 2003

Abstract

In this paper, adaptive synchronization of two uncertain chaotic systems is presented using adaptive backstepping

approach. The master system is any smooth nonlinear chaotic system, while the slave system is a nonlinear chaotic

system in the feedback form. Global stability and exponential synchronization between the master and slave systems

can be achieved. The proposed approach offers a systematic design procedure for adaptive synchronization of a large

class of continuous-time chaotic systems in the chaos research literature. Computer simulations are provided to verify

the operation of the designed synchronization scheme.

� 2004 Elsevier Ltd. All rights reserved.

1. Introduction

Motivated by potential applications in physics, electrical engineering, communication theory and many other fields,

the synchronization of chaotic systems has received an increasing interest [1–8]. It has been shown that two chaotic

systems exhibit identical oscillations, i.e., synchronization error tends to zero for all tP 0 [1]. But despite the amount of

theoretical and experimental results already obtained, chaos synchronization seems difficult task, over all if we think

that: (i) due to sensitive dependence of chaos on initial conditions, it is almost impossible to reduce the same starting

conditions, (ii) in matching exactly the master and slave systems, even infinitesimal parametric variations of any model

will eventually result in divergence of orbits starting nearby each other, and (iii) parametric differences between chaotic

systems (for instance, due to inaccuracy design or time variations) yield different attractors. To avoid these problems,

some strategies have been reported (see [3,4,7,9,11–14], and references therein). In particular, several authors have

reported adaptively estimation techniques. These techniques present an acceptable performance and allow synchro-

nization, although the parameters are not known [9,12,14] or they are time-varying [3,4,7,11,13]. But the only drawback

of these strategies is that the structure of parameters for a given model must be known. Although the structure of the

parameters can be known in some cases, it would be desirable to have a scheme to achieve synchronization even if slave

oscillator has little prior knowledge about the master system. This necessity of robustness can be required in some

systems (for instance, the multimode laser, animal gait or oscillatory neural systems).

In this paper, an algorithm for the adaptive chaos synchronization is developed. In particular, the synchronization

problem is interpreted as a stabilization one. The goal is to stabilize, at the origin, the discrepancy between the drive and

response systems. Discrepancy is defined as the dynamical differences between the drive and response systems and

include: (1) model mismatches, which means that the model of the drive system might not be the same as that of the

* Corresponding author. Address: B.P. 12639 Yaound�e, Cameroon. Tel.: +237-996-41-64/785-5788; fax: +237-231-9584.

E-mail addresses: [email protected] (S. Bowong), [email protected] (F.M. Moukam Kakmeni).

0960-0779/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.chaos.2003.12.084

mail to: [email protected]

1000 S. Bowong, F.M. Moukam Kakmeni / Chaos, Solitons and Fractals 21 (2004) 999–1011

response system; (2) unknown initial conditions, which implies that the time series of the master system cannot be equal

to that for the slave system; and (3) parametric uncertainty, which means that the slave system could be constructed

with inaccuracies [10].

In order to design the stabilizing feedback, the discrepancy system is transformed into a canonical form via geo-

metrical tools [15]. The main idea behind our proposal is, departing from the discrepancy system, to construct an

extended nonlinear system which should be dynamically equivalent to the canonical representation. In this way, the

discrepancy is lumped into a nonlinear function, which is rewritten into the extended nonlinear system as a state

variable [9,10]. The nonlinear feedback is designed from a simple algorithm based on backstepping approach. Back-

stepping design is a recursive procedure that combines the choice of a Lyapunov function with the design of a controller

[16]. By using the results reported by Teel and Praly [17], an observer can be constructed to get an estimated value of the

lumping nonlinear function via the augmented state variable.

Compared with the existing results in the literature, there are two advantages which make this approach attractive.

First, a systematic design procedure for adaptive synchronization is presented for a wide class of nonlinear systems with

many types of uncertainties, including most of the continuous-time chaotic and hyper-chaotic systems in the literature.

Second, it needs only one controller to realize synchronization between chaotic systems and it is easy to implemented.

The rest of the paper is organized as follows. The problem formulation is presented in Section 2. Adaptive back-

stepping is extended to the adaptive synchronization problem in Section 3. Here it is shown that the proposed design

algorithm yields robust adaptive feedback which leads to chaos synchronization in spite of the master and slave model

being nonidentical. In Section 4, two illustrative examples are presented. (1) The Chua oscillator is used to show that the

synchronization is attained in spite of parametric master/slave mismatches. (2) Two second-order driven oscillators are

presented to show that the synchronization can been achieved in spite of a strictly different model, which results in

different master/slave dynamics. Section 5 contains the conclusion.

2. Problem formulation

Consider the master system of nonlinear (chaotic) system as

_xm ¼ fmðxm; pm; tÞym ¼ x1m

�ð1Þ

where xm ¼ ½x1m; x2m; . . . ; xnm�T 2 Rn is the state vector; ym 2 R is the output, pm is the vector of unknown constant

parameters; and fm is an unknown smooth nonlinear function uniformly bounded in t.The slave system is in the form of feedback nonlinear (chaotic) system as

_xs ¼ fsðxs; t; psÞ þ Buys ¼ x1s

�ð2Þ

where xs ¼ ½x1s; x2s; . . . ; xns�T 2 Rn, ys 2 R and u 2 R are the state vector, output and control action, respectively; ps is the

vector of unknown constant parameters; and fs is an unknown smooth nonlinear function uniformly bounded in t andB a constant vector which defines the control channel.

The problem is to design an adaptive synchronization algorithm u to guarantee robust global stability and force the

states of the slave system (2) to exponentially synchronize with the states of the master system (1) despite modelling

errors, parameter variations, perturbing external forces and time lacks in the actuators, i.e., to achieve

limt!1

xs ¼ xm ð3Þ

Consider the variable e ¼ xs xm which is the measure of the nearness of the master system to the slave system.

Introducing e in Eq. (2), we obtain the following equation:

_e ¼ fsðeþ xm; t; psÞ fmðxm; t; pmÞ þ Buye ¼ e1

�ð4Þ

where ye is the system error output. The synchronization is achieved when e goes to zero as t increases or, practically, isless than a given precision.

Now, let us define a coordinates transformation z ¼ UðeÞ such that the synchronization error system (4) can be

globally transformed into the canonical form [15]

S. Bowong, F.M. Moukam Kakmeni / Chaos, Solitons and Fractals 21 (2004) 999–1011 1001

_zi ¼ ziþ1; i ¼ 1; 2; . . . ; q 1

_zq ¼ aðz; mÞ þ u

_m ¼ fðz; mÞy ¼ z1

8>>><>>>:

ð5Þ

where y is the system output, q is the relative degree of the synchronization error system (i.e., the lowest order time-

derivative such that the control command u is directly related to the output y) and m 2 Rnq is the unobservable state

vector (internal dynamics).

Often, the constant matrix B can be selected to make possible the transform from (4) to the assumed format (5).

The following must be pointed out. System (5) is very general, which contains most well known chaotic systems, and

some special models for chaotic systems like those proposed in [18]. For example, the Lorenz dynamical, the Rossler

system, and several types of Chua’s circuits can be transformed into the canonical form with a relative degree q < n. Onthe other hand, an nonautonomous second-order chaotic system such as Duffing oscillator and Van der Pol system can

be written as the canonical form with q ¼ n. In addition, if q ¼ n, the transformed system (5) is the so-called fully

linearizable nonlinear system, and if q < n, the system (5) is called a partially linearizable nonlinear system. Finally, if

the dynamical subsystem _m ¼ fð0; mÞ is asymptotically stable, we will say that the system (5) is minimum-phase [15].

However, several interesting chaotic systems are so-called minimum-phase.

Nevertheless, since the vector fields fmðxm; t; pmÞ and fsðxs; t; psÞ are uncertain, the coordinates transformation

z ¼ UðeÞ, bringing the error system (4) into the canonical form (5), is uncertain. In principle, since the coordinates

transformation is a diffeomorphism, one can suppose that the uncertain transformation exists and it is invertible.

However, since UðeÞ is uncertain, the nonlinear function aðz; mÞ is uncertain. The sources of such uncertainties could be

parameter mismatching, unknown initial conditions, or structural differences between models of master and slave

systems.

In the next section, the detailed design procedure of the feedback control law u is described with detailed expla-

nations.

3. Adaptive synchronization via backstepping

In order to design an adaptive synchronization algorithm to achieve objective (3), adaptive backstepping is em-

ployed. We present a robust control design for synchronizing uncertain chaotic systems in canonical form (5). The main

idea is to construct a dynamically equivalent system which is itself uncertain. Dynamic output feedback is applied to

perform chaos synchronization in spite of modelling errors, parameter variations, perturbing external forces and time

lack in the actuators.

3.1. Stability analysis

To describe the new design and analysis, the following assumptions are needed:

1. Only the system output y ¼ z1 is available for feedback.2. The nonlinear function aðz; mÞ is uncertain.3. The system (5) is minimum-phase.

Comments regarding the above assumptions are in order. The first assumption is realistic because in most cases only

one state is available for feedback from the coding (master) as well as the decoding (slave) circuit. For instance, in the

secure communication case only the transmitted signal x1m, and receiver signal, x1s are available for feedback from

measurements. Another example can be found in neuron synchronization where the master neuron transmits a scalar

signal. The slave neuron tracks the signal of the master neuron. Concerning the second assumption, we claim that it is a

general and practical situation because the term aðz; mÞ involves the differences in the master as well as slave systems.

The source of such uncertainties are: (i) unknown values of the model parameters or time-varying parameters; (ii) since

the master and slave systems are chaotic, their trajectories depend on the initial conditions which are often unknown

and (iii) structural differences between models of the master and slave systems. Regarding the third assumption,

minimum-phase implies that the zero dynamics fð0; mÞ converges to an attractor. In other words, the closed system is

internally stable [9]. From the control view-point this is a strong assumption. But this is reasonable for the boundness of

chaotic attractor in state space and the interaction of all the trajectories inside the attractor. So when we have taken


actions to achieve lim zi ! 0, i ¼ 1; . . . ; q 1, the part fðz; mÞ ! fð0; mÞ ! 0 asymptotically for the so-called minimum-

phase character.

Thus, let us define g ¼ Hðz; mÞ. Then, there exists a time-invariant manifold wðz; g; m; uÞ ¼ 0 such that the solution of

system (5) is a projection of the solution of the following dynamical system:

_zi ¼ ziþ1; 16 i6 q 1

_zq ¼ g þ u_g ¼ Nðz; g; m; uÞ_m ¼ fðz; mÞy ¼ z1

8>>>><>>>>:

ð6Þ

where

Nðz; g; m; uÞ ¼Xq1k¼1

zkþ1okHðz; mÞ þ ðg þ uÞoqHðz; mÞ þ omHðz; mÞfðz; mÞ

with okHðz; mÞ ¼ oHðz; mÞ=ok , and omHðz; mÞ ¼ oHðz; mÞ=om, i.e., system (6) is dynamically equivalent to system (5). It must

be pointed out that the manifold wðz; g; m; uÞ ¼ g Hðz; mÞ ¼ 0 is, by definition, time-invariant. In fact, it is straight-

forward to prove that the set

W ¼ wðz; g; m; uÞ(

¼ g Hðz; mÞ;

_g ¼Xq1k¼1

zkþ1okHðz; mÞ þ ðg þ uÞoqHðz; mÞ þ omHðz; mÞfðz; mÞ)

satisfies dw=dt ¼ 0 for all tP 0. Now, from the equality wðz; g; m; uÞ ¼ 0 and condition dw=dt ¼ 0, one can take the first

integral of system (6) to get g ¼ Hðz; mÞ. When the first integral is back-substituted into system (6), we obtain the

solution of system (5). Hence, the solution of system (5) is a projection of system (6) via the module p � ðz; g; mÞ ¼ ðz; mÞ.This is, system (6) is dynamically equivalent to system (5) if initial conditions, ðzð0Þ; gð0Þ; mð0ÞÞ, are contained in

wðz; g; u; mÞ, i.e, the augmented state g provides the dynamics of the uncertain function Hðz; mÞ which involves the

modelling errors, uncertain parameters and the unknown external disturbances.

For clarity and conciseness of presentation, we present the design procedure for q ¼ 5, although it can be easily

generalized to systems of any order. In this way, system (6) becomes

_zi ¼ ziþ1; 16 i6 4

_z5 ¼ g þ u_g ¼ Nðz; g; m; uÞ_m ¼ fðz; mÞy ¼ z1

8>>>><>>>>:

ð7Þ

with z ¼ ðz1; z2; z3; z4; z5ÞT.The global stability of the closed-loop system and the exponential synchronization of the error system (7) are

summarized in Theorem 1.

Theorem 1. The closed-loop adaptive system consisting of the master system (1), the slave system (2) and the backsteppingfeedback

u ¼ g 1

2

X5i¼1

ki

!z5

1

4k1k2

"þ k3ðk1 þ k2Þ þ k4

X3i¼1

ki

!þ k5

X4i¼1

ki

!#z4

1

8k1k2k3

"þ ðk4 þ k5Þ½k1k2

þ k3ðk1 þ k2Þ� þ k4k5X3i¼1

ki

!#z3

1

16½k1k2k3k4 þ k5½k1k2k3 þ k4�½k1k2 þ k3ðk1 þ k2Þ��z2

1

32k1k2k3k4z1 ð8Þ

where ki > 0, i ¼ 1; . . . ; 5 are constants which determine the convergence rate of the synchronization error system has aglobally exponentially stable equilibrium at ðz; g; mÞ ¼ ð0; 0; 0Þ, i.e., condition (3) is satisfied.

Proof. The backstepping design procedure is recursive. The backstepping controller design, in our case, consists of five

steps. The detail design procedures and stability analysis are presented below.


First step. In the z1-state of Eq. (7), one treats the second state variable z2 as a control input, and finds out the desiredz2 say z2 ¼ z2dðz1Þ that will force z1 to converge to zero at a prescribed speed. The desired z2d function is chosen as

z2d ¼ k12z1 ð9Þ

Define a Lyapunov function candidate Vz1 ¼ 12z21. From Eqs. (7) and (9), the time derivative of Vz1 is found to be

_Vz1 ¼ k1Vz1 . Consequently, z1 approaches zero exponentially fast in the following way:

jz1ðtÞj6 jz1ð0Þjeðk1=2Þt

Second step. In the z2-state of Eq. (7), one treats the third variable z3 as a control input, and finds out the desired

z3 ¼ z3dðz1; z2Þ that will force z2 to approach the previously designed function z2d at a prescribed speed. The desired z3dfunction is chosen as

z3dðz1; z2Þ ¼ 1

2ðk1 þ k2Þz2

1

4k1k2z1 ð10Þ

Define a second Lyapunov function candidate Vz2 ¼ 12z22. From Eqs. (7) and (10), one can easily verify that the time

derivative of Vz2 is given by _Vz2 ¼ k2Vz2 . Again, one has exponential convergence of z2 to zero


Third step. In the z3-state of Eq. (7), one treats the fourth variable z4 as a control input, and finds out the desired

z4 ¼ z4dðz1; z2; z3Þ that will force z3 to approach the previously designed function z3d at a prescribed speed. The desired

z4d function is chosen as

z4dðz1; z2; z4Þ ¼ 1

2

Xlim3

i¼1ki

�z3

1

4½k1k2 þ k3ðk1 þ k2Þ�z2

1

4k1k2k3z1 ð11Þ

Define a third Lyapunov function candidate Vz3 ¼ 12z23. From Eqs. (7) and (11), the time derivative of Vz3 leads to

_Vz3 ¼ k3Vz3 . Hence, it follows that


Fourth step. In the z4-state of Eq. (7), one treats the fifth variable z5 as a control input, and finds out the desired

z5 ¼ z5dðz1; z2; z3; z4Þ that will force z4 to approach the previously designed function z4d at a prescribed speed. The desiredz5d function is chosen as

z5dðz1; z2; z3; z4Þ ¼ 1

2

X4i¼1

ki

!z4

1

4k1k2


X3i¼1

ki

!#z3

1

8½k1k2k3 þ k3 þ k4½k1k2 þ k3ðk1 þ k2Þ�z2

1

16k1k2k3k4z1 ð12Þ

Define a fourth Lyapunov function candidate Vz4 ¼ 12z24. From Eqs. (7) and (12), the derivative of Vz4 is _Vz4 ¼ k4Vz4 .

Hence, it comes that


Fifth step. Finally, in the z5-state of Eq. (7), one chooses the true control input u to drive z5 to approach the pre-

viously designed function z5d at a prescribed speed. The control input is chosen to be that in the control law (8). Define a

fifth Lyapunov function candidate Vz5 ¼ 12z25. From Eqs. (7) and (8), one can verify, although tediously, that the time

derivative of Vz5 is _Vz5 ¼ k5Vz5 . Hence


The convergence of gðtÞ and mðtÞ from zero follow from the fact that the closed-loop system is in cascade form [9] and

minimum-phase [15], respectively. Since one makes any constraints on the initial condition z1ð0Þ, z2ð0Þ, z3ð0Þ, z4ð0Þ, z5ð0Þ,gð0Þ and mð0Þ, this result is global. h

Nevertheless, the backstepping feedback controller (8) is not physically realizable because it requires measurements

of the states zðtÞ and the uncertain state gðtÞ. In this sense the following comments are in order: (i) the augmented state g


is not available for feedback. This fact is obvious because g represents, by definition, the mismatches between master

and slave systems; and (ii) it is desired that only one state z1 ¼ x1s x1m is available for feedback from on-line mea-

surements. Consequently, estimated values of the states ðz; gÞ are required for practical implementation. As it has been

established in [9,14], the problem of estimating ðz; gÞ can be addressed by using high-gain observer. To this end, the

following uncertainty estimator is proposed:

_zi ¼ ziþ1 þ hiCi6ðz1 z1Þ; 16 i6 4

_z5 ¼ g þ uþ h5C56ðz1 z1Þ

_g ¼ h6ðz1 z1Þ

8<: ð13Þ

where z and g are the estimated values of z and g, respectively, Ciq ¼ q!

i!ðqiÞ! and h > 0 is a high-gain parameter which can

be interpreted as the uncertainties estimation rate. Note that the uncertain term Nðz; g; m; uÞ has been neglected in the

construction of the observer (13).

Let ~e 2 R6 be an estimation error vector whose components are defined as follows: ~ei ¼ h5iðzi ziÞ, i ¼ 0; 1; . . . ; 5,and ~e6 ¼ g g. Then, the dynamics of the estimation error can be written as follows:

_~e ¼ hA~eþ Cðz; g; m; uÞ ð14Þ

where Cðz; g; m; uÞ ¼ ½0; . . . ; 0;Nðz; g; m; uÞ�T and the companion matrix is given by

A ¼

C16 1 0 . . . 0

C26 0 1 . . . 0

..

. ... . .

. . .. ..

.

C56 0 0 . . . 1

1 0 0 0 0

266666664

377777775

It is easy to prove that all the eigenvalues of the matrix A are located at 1, i.e., A is a Hurwitz matrix. Since the

trajectories zðtÞ are contained in a chaotic attractor, Cðz; g; m; uÞ is bounded. Consequently, for a large value of the high-gain estimation parameter h, the dynamics of the estimation error, ~e, converges asymptotically to zero, that is, ~e ! 0 as

t ! 1 which implies that ðz; gÞ ! ðz; gÞ.Thus, the backstepping feedback controller (8) with estimation uncertainty becomes

u ¼ g 1

2

Xlim5

i¼1ki

�z5

1

4k1k2


X3i¼1

ki

!þ k5

X4i¼1

ki

!#z4

1

8k1k2k3

"þ ðk4 þ k5Þ½k1k2 þ k3ðk1 þ k2Þ� þ k4k5

X3i¼1

ki

!#z3

1

16½k1k2k3k4 þ k5½k1k2k3 þ k4�½k1k2 þ k3ðk1 þ k2Þ��z2

1

32k1k2k3k4z1 ð15Þ

Notice that the backstepping controller (15) only uses estimates of Hðz; mÞ (by means g) and z which are provided by

estimator (13). Thus, the modified feedback control law is given by the dynamic compensator (13) and the backstepping

controller (15). It should be pointed out that the output backstepping controller (13), (15) yields practical stabilization,

i.e., the synchronization error system (7) converges to a ball B whose radius is of order of h1. That is, z ! B as t ! 1with rðBÞ ¼ Oðh1Þ.

4. Simulation studies

We present two examples in this section. The first example consists of two third-order systems whose model is similar

but parameter values are different. The aim is to show that the synchronization can be achieved in spite of parametric

variations and to illustrate that the chaotic minimum-phase assumption is satisfied. The second example consists of

two strictly different oscillators. Here, the goal is to show that the synchronization can be attained in spite of model

differences between master and slave systems. We choose two second-order driven oscillators to illustrate this case.

Thus, the drive system is given by the /6-Duffing equation whereas the response system is given by /6-Van der Pol

oscillator.


4.1. Synchronization in spite of parametric variation

Chua circuit has been chosen to illustrate the proposed synchronization scheme. This oscillator is widely studied as a

carrier for synchronization schemes. The drive system can be written in dimensionless form as follows:

_x1m ¼ p1m½x2m x1m f ðx1mÞ�_x2m ¼ x1m x2m þ x3m_x3m ¼ p2mx2m

8<: ð16Þ

where f ðx1mÞ ¼ p3mx1m þ 1=2ðp4m p3mÞðjx1m þ 1j jx1m 1jÞ. Suppose that the same configuration is used as a re-

sponse system. However, assume that there are differences between the electronic devices (for instance, the nonlinear

resistance). That is, the parameter values of the response system are different than the drive system. In this way the

response system becomes

_x1s ¼ p1s½x2s x1s f ðx1sÞ� þ u_x2s ¼ x1s x2s þ x3s_x3s ¼ p2sx2s

8<: ð17Þ

where f ðx1sÞ ¼ p3sx1s þ 1=2ðp4s p3sÞðjx1s þ 1j jx1s 1jÞ, with ps 6¼ pm and u the control input which has to be chosen.From the differences ei ¼ xis xim, i ¼ 1; 2; 3, the uncertain system (4) can be obtained as follows:

_e1 ¼ p1sðe2 e1Þ p1sf ðe1 þ x1sÞ þ ðp1m p1sÞðx2m x1mÞ þ p1mf ðx1mÞ þ u_e2 ¼ e1 e2 þ e3_e3 ¼ p2se2 þ ðp2m p2sÞx2m

8<: ð18Þ

Now defining the response output by ys ¼ x1s and the drive output by ym ¼ x1m, one has that ye ¼ e1. This implies thatthe smallest integer is q ¼ 1. In this way the coordinates transformation is globally defined and becomes z1 ¼ e1, m1 ¼ e2and m2 ¼ e3. Then the uncertain system can be rewritten as

_z1 ¼ Dg1 þ u_m1 ¼ Dg2_m2 ¼ Dg3y ¼ z1

8>>><>>>:

ð19Þ

where y is the output of the uncertain system and Dgi, i ¼ 1; 2; 3 are unknown functions. In order to illustrate that

system (19) satisfies the minimum-phase assumption, one can show that Dg2 ¼ z1 m1 þ m2 and Dg3 ¼ p2sm1 þ d, whered ¼ ðp2m p2sÞx2m converge to zero when z1 ¼ 0. As z1 ! 0 (zero dynamics), one has that

_m ¼ Em þ F

where F ¼ ½0; d�T and

E ¼ 1 1

p2s 0

� �

which is Hurwitz because p2s > 0. Hence, since d is bounded, the zero dynamics subsystem _m ¼ Em þ F is asymptotically

stable. That is, the discrepancy between systems (16) and (17) is a minimum-phase system. Since Assumptions 1 to 3 are

satisfied, the augmented state can be defined as g ¼ Dg1. Then system (6) can be constructed and we get the extended

state observer (13) as the following form:

_z1 ¼ z2 þ uþ 2hðz1 z1Þ_g ¼ h2ðz1 z1Þ:

�ð20Þ

So the backstepping controller (15) can be described by

u ¼ g k12z1: ð21Þ

The parameter values of the master system are p1m ¼ 10, p2m ¼ 14:87, p3m ¼ 1:27 and p4m ¼ 0:68, whereas theparameter values of the slave system are p1s ¼ 9, p2s ¼ 15, p3s ¼ 1:5 and p4s ¼ 0:8. Note that the minimum and

maximum errors on the parameter values are 6% and 15%, respectively. The initial conditions were arbitrarily chosen as


follows: xmð0Þ ¼ ð0:1; 0; 0Þ, x1sð0Þ ¼ ð0:2; 0; 0Þ and ðz1ð0Þ; gð0ÞÞ ¼ ð1; 0Þ. The control gain k1 ¼ 8 and the high-gain

estimation parameter value h ¼ 20 were chosen.

Fig. 1 shows the synchronization of the master and slave systems. The feedback was tuned on at t ¼ 50s. Note thatthe dynamic output feedback (21) yields complete synchronization, i.e., the trajectories of the discrepancy system

converge exactly to the origin.

Fig. 2 shows the synchronization error e1 ¼ x1s x1m for three different values of the control gain k1. One can see

that, as predicted by the analysis in Theorem 1, the synchronization error system state convergence rate is determined

by the control gain k1. The larger the control gain, the faster the convergence rate is.

4.2. Synchronization in spite of strictly different model

The goal of this example is to illustrate that the synchronization can be attained in spite of different model for the

drive and response systems, which is the extreme case of drive/response mismatch and external perturbations by

oscillatory signals which can be interpreted as noise. Thus, we choose the /6-Duffing oscillator as the drive system and

the /6-Van der Pol oscillator as response system. The equation of the drive system is given as follows:

Fig. 1

system

_xm ¼ fmðxm; tÞ ð22Þ

where fmðxm; tÞ ¼ ½x2m;r1x2m r2x1m r3x31m r4x51m þ r5 cosxt�T. The equation of the response system is given by

_xs ¼ fsðxs; tÞ þ Bu ð23Þ

. Complete exact synchronization of the drive and response systems via the dynamics output feedback (21). States of the drive

(––) are tracked by states of the response system (– – –).

0 0.5 1 1.5 2 2.5 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

---- k1=8

- - - k1=10

.... k1=12

Time(s)

e 1=

x 1s-x

1m

Fig. 2. The synchronization error e1 ¼ x1s x1m for three different values of the control gain k1.


where fsðxs; tÞ ¼ ½x2s; lð1 x21sÞx2s x20x1s dx31s kx51s þ f0 cosXt�T, B ¼ ½0; 1�T and u the control input which has to be

chosen. If ym ¼ x1m and ys ¼ x1s are the outputs of the drive and response systems, respectively, and by defining

ei ¼ xim xis, i ¼ 1; 2, one gets the following uncertain system:

_e ¼ fsðeþ xm; tÞ fmðxm; tÞ þ Buye ¼ e1

�ð24Þ

Thus the coordinates transformation is given by z1 ¼ e1 and z2 ¼ e2 in such a way that, system (24) is transformed into

_z1 ¼ z2_z2 ¼ aðz; xm; tÞ þ uy ¼ z1

8<: ð25Þ

where y is the output of the uncertain system and aðz; xm; tÞ ¼ fsðzþ xm; tÞ fmðxm; tÞ denotes the uncertainties of thesetwo systems (parameter mismatching and external perturbations). Note that system (25) is fully linearizable, i.e., there is

no unobservable states m in the uncertain system because the relative degree q ¼ n. Now, defining g ¼ aðz; xm; tÞ, thedynamical system (6) can be constructed. So the extended state observer (13) can be described in the following form:

_z1 ¼ z2 þ 3hðz1 z1Þ_z2 ¼ g þ uþ 3h2ðz1 z1Þ_g ¼ h3ðz1 z1Þ

8<: ð26Þ

Hence, the backstepping controller (15) can be described by

u ¼ g 1

2ðk1 þ k2Þz2

1

4k1k2z1 ð27Þ

The parameter values were taken as r1 ¼ 1, r2 ¼ 1, r3 ¼ 3, r4 ¼ 1:5, r5 ¼ 1:5 and x ¼ 1:221 in which case the /6-

Duffing oscillator has a chaotic behavior. For the parameter values l ¼ 0:4, x0 ¼ 0:46, d ¼ 1, k ¼ 0:1, f0 ¼ 4:5 and

X ¼ 0:86, the /6-Van der Pol oscillator displays chaotic behavior. The initial conditions were chosen as follows:

xmð0Þ ¼ ð0; 0Þ, xsð0Þ ¼ ð0:1; 0Þ and ðz1ð0Þ; z2ð0Þ; gð0ÞÞ ¼ ð1; 0; 0Þ. The controller is acted at time t ¼ 40 s.

Fig. 3. Synchronization of a /6-Duffing oscillator (––) and a /6-Van der Pol oscillation (– – –): (a) position trajectory and (b) velocity

trajectory.


Fig. 3(a) and (b) shows the synchronization of the master and slave system position and velocity with k1 ¼ k2 ¼ 5

and h ¼ 30. Note that the backstepping controller (27) leads to the synchronization of the chaotic oscillators.

After the synchronization of transmitter (drive) and receiver (response), one would like to known if a message signal

can be recovered in spite of model differences between transmitter and receiver and external perturbations by oscillatory

signals which can be interpreted as noise. We can use many ways to transmit information [7,19]. Here we use the

demodulation scheme reported in [7] to discuss it simply. The transmitted signal is a sum of the information and the

output of the chaotic transmitter. In addition, the transmitter signal is also injected into the transmitter and, simul-

taneously, transmitted to the receiver. By the proposed synchronization technique, a chaotic receiver is then derived to

recover the information signal at the receiving end of communication. Fig. 4 illustrates the typical frame of the

communication. We add information signal to the right-hand side of the second equation of the /6-Duffing system.

Then, the /6-Duffing system can be described by

_x1m ¼ x2m_x2m ¼ r1x2m r2x1m r3x31m r4x51m þ r5 cosxt þ sðtÞyT ¼ ym þ sðtÞ

8<: ð28Þ

where sðtÞ is the information signal and masked by the system’s output ym ¼ x1m, and yT ¼ x1m þ s is the chaoticallytransmitted signal, which drives the receiver [7]. Here, the information signal was chosen to be a periodic function

sðtÞ ¼ 2 sin t. The frequency was chosen such that the dynamics behavior of the drive system remains chaotic. The

receiver is the /6-Van der Pol system given by

Fig. 4. The typical frame of the communication transmission.

-3-6

-4

-2

0

2

4

6

x 2m

Fig. 5

dynam


_x1s ¼ x2sx2s ¼ lð1 x21sÞx2s x2

0x1s dx31s kx51s þ f0 cosXtys ¼ x1s

8<: ð29Þ

where ys ¼ x1s is the receiver’s output. According to Fig. 4, the recovered signal is achieved by sRðtÞ ¼ yTðtÞ ysðtÞ and,by using the synchronization scheme developed in Section 3, we have

limt!1

sRðtÞ ¼ limt!1

ðx1m þ s x1sÞ ¼ limt!1

ðe1 þ sðtÞÞ ¼ sðtÞ ð30Þ

Fig. 5 presents the behavior of the communication system described above for k1 ¼ k2 ¼ 5 and h ¼ 30. Fig. 5(a) presents

the ðx1m; x2mÞ-phase plane which shows the chaotic nature of the transmitter dynamics. The waveform of the transmitted

signal yT is shown in Fig. 5(b). The information signal sðtÞ (––) and the recovered signal sRðtÞ (– – –) is depicted in Fig.

5(c).

Suppose now that the message sðtÞ is an amplitude-modulated message of the form sðtÞ ¼ Smð1þ g sinX0tÞ sinðXtÞwhere Sm ¼ 2 and X ¼ 1 are the amplitude and frequency of the message, while g ¼ 1 is a parameter modulating at

frequency X0 ¼ 0:2 the amplitude of the message. In this case, results similar to those in the first case are obtained (see

Fig. 6).

-2 -1 0 1 2 3x1m

0 10 20 30 40 50 60 70 80-4

-3

-2

-1

0

1

2

3

4

5

Time(s)

y T

0 10 20 30 40 50 60 70 80-4

-3

-2

-1

0

1

2

3

4

5

6

Time(s)

s,s R

(a) (b)

(c)

. Behavior of the communication system for a single tone message sðtÞ ¼ 2 sin t: (a) phase portrait ðx1m; x2mÞ of the transmitterics; (b) transmitted signal yTðtÞ ¼ x1m þ sðtÞ and (c) information signal sðtÞ (––) and recovered signal sRðtÞ (– – –).

-3 -2 -1 0 1 2 3-6

-4

-2

0

2

4

6

x1m

x 2m

0 10 20 30 40 50 60 70 80-6

-4

-2

0

2

4

6

8

Time(s)

y T

0 10 20 30 40 50 60 70 80-6

-4

-2

0

2

4

6

8

Time(s)

s,s R

(c)

(a) (b)

Fig. 6. Behavior of the communication system for an incoming amplitude modulated message sðtÞ ¼ 2ð1þ sin 0:2tÞ sin t: (a) phaseportrait ðx1m; x2mÞ of the transmitter dynamics; (b) transmitted signal yT ðtÞ ¼ x1m þ sðtÞ and (c) information signal sðtÞ (––) and

recovered signal sRðtÞ (– – –).


5. Conclusion

An approach for adaptive synchronization of uncertain chaotic systems using backstepping approach has been

presented in this paper. The design algorithm yields continuous feedback which allows handling realistic situations such

as unmeasured states and uncertainties.

The resulting synchronization scheme is obtained by posing a control problem in the synchronization error

dynamics. The control problem consists in the stabilization at the origin of the synchronization error system. The

proposed scheme comprises a backstepping-like controller and a state/uncertainty estimator. A central feature of our

approach is that uncertainties of the underlying vector field are lumped in an extended state whose dynamics is

reconstructed form measurements of the system output. Consequently, strong properties of global stability and

exponential synchronization can been achieved in a finite number of steps. The proposed approach offers a systematic

design procedure for the adaptive synchronization of most of chaotic systems in the chaos research literature.

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Synchronization of uncertain chaotic systems using active sliding mode control