Output synchronization of chaotic systems under nonvanishing perturbations

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Chaos, Solitons and Fractals 37 (2008) 1172–1186

www.elsevier.com/locate/chaos

Output synchronization of chaotic systems undernonvanishing perturbations

Didier Lopez-Mancilla a, Cesar Cruz-Hernandez b,*

a Departamento de Ciencias Exactas y Tecnologicas, Centro Universitario de los Lagos, Universidad de Guadalajara

(CULagos-UdeG), Enrique Dıaz de Leon s/n, 47460 Lagos de Moreno, Jal., Mexicob Electronics and Telecommunications Department, Scientific Research and Advanced Studies of Ensenada (CICESE), Km. 107,

Carretera Tijuana-Ensenada, 22860 Ensenada, B.C., Mexico

Accepted 13 October 2006

Communicated by Prof. M.S. El Naschie

Abstract

In this paper, an analysis for chaos synchronization under nonvanishing perturbations is presented. In particular, weuse model-matching approach from nonlinear control theory for output synchronization of identical and nonidenticalchaotic systems under nonvanishing perturbations in a master–slave configuration. We show that the proposedapproach is indeed suitable to synchronize a class of perturbed slaves with a chaotic master system; that is the synchro-nization error trajectories remain bounded if the perturbations satisfy some conditions. In order to illustrate this robust-ness synchronization property, we present two cases of study: (i) for identical systems, a pair of coupled Rosslersystems, the first like a master and the other like a perturbed slave, and (ii) for nonidentical systems, a Chua’s circuitdriving a Rossler/slave system with a perturbed control law, in both cases a quantitative analysis on the perturbation isincluded.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

Chaos synchronization has been in the last decades subject of great interest. Since the work of Pecora and Carroll [1],many researchers have proposed different approaches for chaos synchronization, see e.g., [2–9]. Such interest is becauseof chaos synchronization is useful in many cases of practical interest, like to design secure communication systems [10–16], and significantly when it occurs in living systems like could be the case for synchronization of the activity of groupsof neurons located in different brain areas [17–19] or, synchronization between heart and respiratory rates [20] or, syn-chronization of biological oscillators [21].

In all these cases it is important to be sure that the mechanisms that guide this synchrony are robust. Hence, a fun-damental problem in the convergence analysis of the synchronization of chaotic systems, is the robustness with respect

0960-0779/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2006.10.020

* Corresponding author. Address: CICESE, Electronics and Telecommunications Department, P.O. Box 434944, San Diego, CA92143-4944, USA. Tel.: +52 646 1750500; fax: +52 646 1750554.

E-mail addresses: [email protected] (D. Lopez-Mancilla), [email protected] (C. Cruz-Hernandez).

mailto:[email protected]

mailto:[email protected]

D. Lopez-Mancilla, C. Cruz-Hernandez / Chaos, Solitons and Fractals 37 (2008) 1172–1186 1173

to modelling error, aging of physical components, uncertainties, and perturbations that exist in any realistic problem.From a Lyapunov stability theory of nonlinear systems: The perturbation effect at the equilibrium point of the synchro-nization error dynamical system can be null or not. The first case, is called vanishing perturbation; this means that theequilibrium point of the perturbed synchronization error dynamical system remains the same as that of the equilibriumof the unperturbed one. The second case, is called nonvanishing perturbation, for which equilibrium points of the per-turbed and unperturbed synchronization error dynamical systems are not the same. However, the perturbed errordynamical system may not have an equilibrium at all, in which case, we cannot study the problem as the stability ofequilibria any longer. Thus, the best we can expect are ultimately bounded state trajectories for the synchronizationerror, if the perturbation satisfies some conditions.

In this paper, the stability of the synchronization error between two coupled chaotic systems through model-match-ing approach from modern control theory [22–24], and subjected to a class of nonvanishing perturbations, when theunperturbed error dynamical system has an uniformly asymptotically stable equilibrium point is discussed (see[25,26] for discrete-time systems). This work shows that the error trajectories remain bounded, if the perturbation sat-isfies some conditions. In order to illustrate this robustness synchronization property, two cases of study are included:for identical systems, a pair of chaotic Rossler systems is synchronized and, in this case, partial synchronization ofstates (output synchronization) of the Rossler systems will result in complete synchronization for all states. Moreover,considering nonidentical systems, the output synchronization of Chua’s circuit with a Rossler system, with a perturbedcontrol law is treated. In both cases, a quantitative analysis on the perturbation is included.

The paper is organized as follows. In Section 2, we present the problem statement under consideration, outputsynchronization of perturbed chaotic systems. In Section 3, we give a brief review of output synchronizationthrough model-matching approach. Section 4 establishes a description of the model-matching problem for per-turbed case of output synchronization, and the respective analysis is given. In Section 5, we apply this result intwo cases of study for perturbed output synchronization: identical chaotic systems (Rossler–Rossler) and noniden-tical chaotic systems (Chua’s circuit-Rossler), through model-matching approach and making a quantitative analysison the perturbation, some numerical simulations are presented. Finally, in Section 6, we give some concludingremarks.

2. Problem statement

Consider the master dynamics represented by the state equations

M :_xM ¼ fM ðxM Þ þ gM ðxM ÞuM ;

yM ¼ hM ðxM Þ;

�ð1Þ

where xM ðtÞ 2 Rn is the master state, uM ðtÞ 2 R is the master input, and yM ðtÞ 2 R is the master output. Assume thatfM(•) and gM(•) are smooth vector fields and that the master system M remains in chaotic motion for suitable inputsignals uM(t).

In addition, consider the family of systems referred as the class of perturbed slave systems, described by

P e :_x ¼ f ðxÞ þ Df ðxÞ þ ½gðxÞ þ DgðxÞ�u;y ¼ hðxÞ;

�ð2Þ

in which xðtÞ 2 Rn denotes the slave state, uðtÞ 2 R the control input, and yðtÞ 2 R the slave output. Also, f(•) and g(•)are smooth vector fields. The class of perturbed slaves Pe could result from modelling errors, aging of physical compo-nents, uncertainties, or constant disturbances that exist in practice. Thus, in Pe, Df(•) and Dg(•) represent perturbationterms, such that Pe is the perturbed case of the following unperturbed (or nominal) slave system

P :_x ¼ f ðxÞ þ gðxÞu;y ¼ hðxÞ:

�ð3Þ

We assume that x�M is an equilibrium point of the chaotic master system M, i.e. fM ðx�M Þ ¼ 0. Similarly, x� is an equi-librium point of the unperturbed slave system P.

Definition 1 (Output synchronization problem, OSP). The output of the unperturbed slave system P Eq. (3)synchronizes with the output of the chaotic master system M Eq. (1), if

limt!1jyðtÞ � yM ðtÞj ¼ 0; ð4Þ

1174 D. Lopez-Mancilla, C. Cruz-Hernandez / Chaos, Solitons and Fractals 37 (2008) 1172–1186

no matter which initial conditions xM(0) and x(0) have, and for suitable input signals u(t) and uM(t).

In previous definition, no matter if the chaotic master M and the unperturbed slave P are identical or nonidenticalsystems.

The class of perturbed slaves Pe cannot synchronize with chaotic master M in the sense that output synchronizationcondition (4) is not fulfilled. Therefore, we need to replace it by the following weaker condition:

limt!1jyðtÞ � yM ðtÞj 6 q ð5Þ

with a small constant q > 0. If condition (5) is fulfilled, then the class of perturbed slaves Pe and the chaotic master M

are approximately synchronized in their outputs, with level q.In this paper, we study approximate output synchronization between chaotic master M and the class of perturbed

slaves Pe. In the next sections, we will describe how to satisfy synchronization condition (5) from the perspective ofmodel-matching problem and Lyapunov-based stability from nonlinear control theory. The unperturbed case forOSP is reported in [7].

3. A brief review on output synchronization

From a modern control theory point of view, consider now the dynamical systems (1) and (3) like model M andunperturbed plant P, respectively. In particular, consider the following control objective: to design a control law u(t)for P, such that irrespectively of the initial conditions of P and M, the output y(t) of P converges asymptotically tothe output yM(t) produced by M under an arbitrary input uM(t). This problem is the so-called asymptotically model-

matching problem [22–24] and for discrete-time context, see [25–27]. It is well-known that, this problem is solved byreducing it to an equivalent problem of decoupling the output of a suitable auxiliary system from the input uM(t) ofmodel M. The auxiliary system is defined as follows:

E :_xE ¼ fEðxEÞ þ gðxEÞuþ gM xEð ÞuM ;

yE ¼ hEðxEÞ

�ð6Þ

with auxiliary state xE ¼ ðx; xM ÞT 2 R2n, auxiliary inputs u(t), and uM(t), and

fEðxEÞ ¼f ðxÞ

fM ðxM Þ

� �; gðxEÞ ¼

gðxÞ0

� �;

gM ðxEÞ ¼0

gM ðxM Þ

� �; hEðxEÞ ¼ hðxÞ � hM ðxM Þ;

which corresponds to a system having as ‘‘output’’ the difference between the outputs of P and M. In this equivalentproblem, the input signal uM(t) is considered as ‘‘perturbation’’ that acts on the auxiliary system E Eq. (6) and here, theobjective is to decouple it from the output yE(t). We are allowed to use perturbation ‘‘measurements’’ because uM(t) isthe input of model M, and thus the control law takes the form

u ¼ aðxEÞ þ cðxEÞuM þ bðxEÞv ð7Þ

with v(t) an additional input signal to achieve asymptotic stability in the closed-loop auxiliary system, which corre-sponds to the convergence rate of output synchronization between dynamical systems M and P.

The objective of the model-matching problem is contained in the following definition.

Definition 2 (Model-matching problem, MMP). Given the unperturbed plant P Eq. (3) and the model M Eq. (1) aroundtheir respective equilibrium points x� and x�M , and a point x�E, the MMP consists in finding a state feedback control u(t)of the form (7) for the auxiliary system E Eq. (6), yielding a closed-loop auxiliary system in which the output yE(t) iscompletely independent of, or decoupled from, the disturbance uM(t), i.e. yielding yE(t) = y(t) � yM(t)! 0 as t!1.

The MMP is treated in terms of a relative degree associated with the outputs y(t) and yM(t).

Definition 3 (Relative degree [24]). The output y(t) of the unperturbed plant P Eq. (3), is said to have relative degree r

at a point x�, if

1. LgLkf hðxÞ ¼ 0 for all x in a neighborhood of x� and k < r � 1.

2. LgLr�1f hðx�Þ 6¼ 0.


A similar definition is given for the relative degree of the model M Eq. (1), rM near x�M .

Assuming that the outputs y(t) of P and yM(t) of M have a finite relative degree r and rM, respectively. Then, it iswell-known that, the MMP is locally solvable if, and only if [24],

r 6 rM : ð8Þ

Next, we show the unperturbed closed-loop auxiliary system – in terms of P and M – in a different coordinate frame.In this paper, we restrict our results to unperturbed plants P that satisfy,

(H1) The unperturbed slave/plant P Eq. (3) is fully linearizable, i.e. r = n.

From definition of relative degree r; hðxÞ; . . . ; Ln�1f hðxÞ are a set of independent functions from unperturbed plant P,

and can be chosen as new coordinates niðxÞ ¼ Li�1f hðxÞ and nMiðxMÞ ¼ Li�1

fMhM ðxMÞ from model M for i = 1, . . . ,n, around

x� and x�M , respectively. Let us now to consider the auxiliary system E Eq. (6) and the new coordinates [24]:

ðfðxEÞ; xM Þ ¼ /ðxEÞ ¼ /ðx; xM Þ;

where f(xE) = (f1(xE), . . . ,fn(xE))T, and

fiðxEÞ ¼ Li�1fE

hEðxEÞ ¼ niðxÞ � nMiðxM Þ; i ¼ 1; . . . ; n: ð9Þ

In these new coordinates, the unperturbed closed-loop auxiliary system, by using the following state feedback control:

u ¼ 1

LgLn�1f hðxÞ

v� Lnf hðxÞ þ Ln

fMhMðxM Þ þ LgM

Ln�1fM

hM ðxM ÞuM

� �ð10Þ

takes the form

_fi ¼ fiþ1; i ¼ 1; . . . ; n� 1; ð11Þ_fn ¼ v ¼ �c0f1 � � � � � cn�1fn;

_xM ¼ fM ðxMÞ þ gM ðxM ÞuM ;

yE ¼ f1:

We can identify two subsystems in the unperturbed closed-loop auxiliary system (11), namely:

1. The subsystem described by

_f ¼ A�f ð12Þ

with

A� ¼

0 1 0 � � � 0

0 0 1 � � � 0

..

. ... ..

. . .. ..

.

0 0 0 � � � 1

�c0 �c1 �c2 � � � �cn�1

0BBBBBB@

1CCCCCCA;

which represents the dynamics of yE(t), and2. The subsystem described by

_xM ¼ fM ðxMÞ þ gM ðxM ÞuM ;

which represents the dynamics of M.

The model M is stable by assumption and if we choose the control law u(t) Eq. (10), such that the eigenvalues ofmatrix A* have real part negative, then the unperturbed closed-loop auxiliary system (11) will be exponentially stable,and therefore the output synchronization condition (4) is satisfied for unperturbed case [7].


4. Perturbed output synchronization

Now, consider the class of perturbed slave systems (2) like perturbed plants Pe. Our objective is that the output of Pe

approximately synchronizes with the output of the chaotic master system M. Here, we discuss the stability of the syn-chronization error between the outputs of M and Pe, when the same control law (10) that solves the OSP for the unper-turbed case is applied to the class of perturbed slaves Pe. Our objective is achieved by using model-matching approach,in which we obtain a closed-loop auxiliary system subject to a class of nonvanishing perturbations; whereas the unper-turbed closed-loop auxiliary system has an asymptotically stable equilibrium point. It is desired that the proposed meth-odology guarantees that in presence of nonvanishing perturbations the output synchronization error be small, i.e. itremains bounded. In other words, the convergence analysis presented shows that the error trajectories remain bounded,if the nonvanishing perturbations satisfy some conditions. A similar convergence analysis is given in [28] with a differentmethodology for synchronization, using sliding modes control.

From chaotic master M Eq. (1) and class of perturbed slaves Pe Eq. (2), the perturbed auxiliary system results asfollows:

Ee :_xE ¼ fEðxEÞ þ DfEðxEÞ þ gðxEÞ þ DgðxEÞ½ �uþ gM ðxEÞuM ;

yE ¼ hEðxEÞ

�ð13Þ

with

DfEðxEÞ ¼Df ðxÞ

0

� �; DgðxEÞ ¼

DgðxÞ0

� �;

where fE(•), g(•), and gM(•) are defined as in Eq. (6). Df(•) and Dg(•) are perturbation terms due to parameter mis-matching, or structure differences on the unperturbed slave P Eq. (3).

If Assumption (H1) holds and the same state feedback control u(t) Eq. (10) is used, then the closed-loop perturbed

auxiliary system in new coordinates (9), takes the form

_fi ¼ fiþ1 þ LiDf hðxÞ; i ¼ 1; . . . ; n� 1;

_fn ¼ �c0f1 � � � � � cn�1fn þ LnDf hðxÞ þ

LDgLn�1f hðxÞ

LgLn�1f hðxÞ

� �c0f1 � � � � � cn�1fn � Lnf hðxÞ þ Ln

fMhM ðxM Þ þ LgM

Ln�1fM

hM ðxM ÞuM

� �; ð14Þ

_xM ¼ fMðxM Þ þ gM ðxM ÞuM ;

yE ¼ f1:

Now, we can identify a subsystem from (14), described by

_f ¼ A�fþWeðf; x; xM Þ ð15Þ

with perturbation term

Weðf; x; xM Þ ¼

LDf hðxÞL2

Df hðxÞ

..

.

Ln�1Df hðxÞ

LnDf hðxÞ þ wgðf; x; xM Þ

0BBBBBBBB@

1CCCCCCCCA;

where

wgðf; x; xM Þ ¼LDgLn�1

f hðxÞLgLn�1

f hðxÞ�c0f1 � � � � � cn�1fn � Ln

f hðxÞ þ LnfM

hM ðxM Þ þ LgMLn�1

fMhM ðxM ÞuM

� �:

Subsystem (15) differs from subsystem (12) by the perturbation term We(f,x,xM). Since a perturbation termexists in the closed-loop auxiliary system (15), it is important to assure that conditions exist so that, if exact outputsynchronization between Pe and M is not achieved, at least that, the synchronization error remains bounded. Fig. 1shows the block diagram for chaotic output synchronization under nonvanishing perturbations via model-matchingapproach.

Fig. 1. Block diagram for chaotic output synchronization under nonvanishing perturbations via model-matching approach.


4.1. Robust stability analysis on perturbed error synchronization

Consider subsystem (15) like a perturbed case of the unperturbed subsystem (12). The perturbed subsystem (15) isdue to applying the same linearizing process, where the linearizing is not fully due to the perturbation terms. So, for theclosed-loop unperturbed auxiliary system (11) the following assumption is made:

(H2) The origin of the unperturbed subsystem (12) is an exponentially stable equilibrium point.

Notice that, whenever Assumption (H2) is satisfied, then a converse Lyapunov theorem stability theory [29] guar-antees the existence of a Lyapunov function V ðfÞ ¼ fTPf for subsystem (12), where P ¼ P T > 0 is a solution of theLyapunov matrix equation A�TP þ PA� ¼ �Q, with Q = QT > 0, that satisfies:

�c1kfk26 V ðfÞ 6 �c2kfk2

; ð16ÞoVof

dfdt6 ��c3kfk2

; ð17Þ

oVof

�� 6 �c4kfk2; ð18Þ

where �c1 ¼ kminðP Þ;�c2 ¼ kmaxðP Þ;�c3 ¼ kminðQÞ;�c4 ¼ 2kmaxðP Þ, and k•k denotes the Euclidean norm. In addition, weintroduce the following assumptions on the bound of the perturbation term:

(H3) The function nM(xM) satisfies the bound

knM ðxM Þk 6 lxM ð19Þ

for all t > 0 and for all xM 2 Rn, and lxM a positive constant.

(H4) The perturbation term We(f,x,xM) satisfies the bound

kWeðf; x; xM Þk 6 lW þ d2kfk þ dknðxÞk ð20Þ

for all t > 0 and for all x; f 2 Rn and lW, d > 0.

Now, we are able to state the main result of this paper as follows.

Theorem 1. Consider the origin as an exponentially stable equilibrium point for unperturbed subsystem (12). Assume that

Assumptions (H3) and (H4) hold. Then, for all kfð0Þ; xð0Þ; xM ð0Þk <ffiffiffic1

c2

qrB, there exist constants �d;�lW;�lxM P 0 such that,

the solution f(t) for the perturbed subsystem (15) has an ultimate bound �b for all t > t0 in function of �d;�lW, and �lxM , and for

all perturbation term We(f,x,xM) that satisfies (H3) and (H4) with d 6 �d, lW 6�lW, and lxM 6

�lxM .

Proof. We use V ðfÞ ¼ fTPf as a Lyapunov function candidate for the perturbed subsystem (15). The derivative of V(f)along the trajectories of this subsystem is given by

_V ðfÞ ¼ fT A�TP þ PA�

fþ 2fTPWe

if (H3) and (H4) are satisfied, and using properties (16)–(18), we have


_V ðfÞ 6� �c3kfk2 þ �c4kfkðlW þ d2kfk þ dknðxÞkÞ6� �c3 � �c4d

2

kfk2 þ �c4lWkfk þ �c4dkfkknðxÞk:

From (9), we can write kfk = kn(x) � nM(xM)kP kn(x)k � knM(xM)k. Then we have,

knðxÞk 6 kfk þ knM ðxM Þk 6 kfk þ lxM ;

thus

_V ðfÞ 6 � �c3 � �c4d2 � �c4d

kfk2 þ �c4ðlW þ dlxM Þkfk:

If d is enough small to satisfy the bound

d 6 �d < � 1

2þ 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4�c3 þ �c4

�c4

s> 0; ð21Þ

then

_V ðfÞ 6 �~akfk2 þ �c4ðlW þ dlxM Þkfk;

where ~a ¼ �c3 � �c4d2 � �c4d > 0. Moreover,

_V ðfÞ 6 �ð1� hÞ~akfk2 � h~akfk2 þ �c4ðlW þ dlxM Þkfk

for some constant h such that 0 < h < 1. If lW and lxM satisfy the bounds:

lW 6�lW <

ffiffiffiffi�c1

�c2

rh~a�c4

rB � dlxM > 0; ð22Þ

lxM 6�lxM <

ffiffiffiffi�c1

�c2

rh~a�c4d

rB; ð23Þ

then

_V ðfÞ 6 �ð1� hÞ~akfk2 8kfkP l ¼ �c4

h~aðlW þ dlxM Þ;

and an ultimate bound is given by

�b ¼ �c4

h~a

ffiffiffiffi�c2

�c1

rðlW þ dlxM Þ: � ð24Þ

Note that the ultimate bound �b is proportional to the upper bound on the perturbation lW. This result can be viewedas a robustness property of the unperturbed synchronization error system, having asymptotically stable equilibria at theorigin. In other words, this result shows that arbitrarily small nonvanishing perturbations will not result in large outputsynchronization error.

5. Output synchronization of perturbed slave systems

5.1. Rossler–Rossler synchronization

In order to illustrate the previous results, let us to consider the output synchronization of Rossler system given by

_x1 ¼ �x2 � x3;

_x2 ¼ x1 þ ax2;

_x3 ¼ aþ x3 x1 � lð Þ:

With the parameter values a = 0.2 and l = 7, the Rossler system exhibits chaotic dynamics [30]. Assume that theparameters of Pe are slightly different from those of master M and is given by


P e :

_x1

_x2

_x3

0B@1CA ¼ �x2 � x3

x1 þ eax2ea þ x3 x1 � elð Þ

0B@1CAþ 0

0

1

0B@1CAu;

y ¼ x2:

8>>><>>>: ð25Þ

If ~a ¼ aþ Da and ~l ¼ lþ Dl, where Da and Dl are small differences for parameters a and l, respectively, then Pe canbe represented in the form (2) as follows:

P e :

_x1

_x2

_x3

0B@1CA ¼ �x2 � x3

x1 þ ax2

aþ x3ðx1 � lÞ

0B@1CAþ 0

Dax2

Da� Dlx3

0B@1CAþ 0

0

1

0B@1CAu;

y ¼ x2:

8>>><>>>: ð26Þ

Note that, the class of perturbed slaves Pe has relative degree re = 3. Let us to propose a master M for Pe, by usinganother Rossler system in the form (1) with relative degree rM = 3,

M :

_xM1

_xM2

_xM3

0B@1CA ¼ �xM2 � xM3

xM1 þ axM2

aþ xM3ðxM1 � lÞ

0B@1CAþ 0

0

1

0B@1CAuM ;

yM ¼ xM2:

8>>><>>>: ð27Þ

Suppose that it is desired that the output y(t) = x2(t) of (26) follows the chaotic output trajectory yM(t) = xM2(t) of (27),i.e. that output synchronization condition is satisfied. The perturbed auxiliary system from (26) and (27) can be writtenin the form (13):

Ee :

_x1

_x2

_x3

_xM3

_xM2

_xM3

0BBBBBBBB@

1CCCCCCCCA¼

�x2 � x3

x1 þ ax2

aþ x3ðx1 � lÞ�xM2 � xM3

xM1 þ axM2

aþ xM3ðxM1 � lÞ

0BBBBBBBB@

1CCCCCCCCAþ

0

Dax2

Da� Dlx3

0

0

0

0BBBBBBBB@

1CCCCCCCCAþ

0

0

1

0

0

0

0BBBBBBBB@

1CCCCCCCCAuþ

0

0

0

0

0

1

0BBBBBBBB@

1CCCCCCCCAuM ;

yE ¼ x2 � xM2;

8>>>>>>>>>><>>>>>>>>>>:ð28Þ

where the second term of the right-hand side corresponds to DfE(•) and Dgð�Þ ¼ 0 We consider Da = 0.04sin(2pt/T)and Dl = 1.4sin(2pt/T), such that the parameters for Pe are ~a ¼ 0:2þ 0:04 sinð2pt=T Þ and ~l ¼ 7þ 1:4 sinð2pt=T Þ.This variation is permitted in Pe such that the parameter values oscillate between ±20%, acting as a perturbationfor the slave P. Considering that these small parameter variations or parameter mismatch are not available for mea-surement, or are simply not considered in the design of the control law u(t), we can design it from the unperturbedcase, i.e. Da = Dl = 0 (it means that DfE(•) = 0). Then, the control law of the form (10) for unperturbed slave P, isgiven by

u ¼ �vþ ða2 � 1Þðx1 � xM1Þ þ aða2 � 2Þðx2 � xM2Þ þ aðx3 � xM3Þ þ x3ðx1 � lÞ � xM3ðxM1 � lÞ þ uM

with uM(t) = 0.3sin(t). The perturbed closed-loop auxiliary system (28), takes the form (15), with

Weðf; x; xM Þ ¼�Dax2

�Dax1 þ ð2aþ DaÞDax2

ð2aþ DaÞDax1 þ ð2þ 3a2 þ 3aDaþ D2aÞDax2 þ ðDaþ DlÞx3 � Da

0B@1CA:

Although exponential convergence is guaranteed for unperturbed case, for robust stability analysis of perturbed syn-chronization will be useful the existence of a Lyapunov function for unperturbed subsystem (12). Thus, let us to proposethe following candidate Lyapunov function

V ðfÞ ¼ 7f21 þ 10f1f2 þ f1f3 þ

19

2f2

2 þ 6f2f3 þ 2f23; ð29Þ

positive definite. Its derivative along the trajectories of the unperturbed subsystem (12) is_V ðfÞ ¼ �2f2

1 � 2f22 � 2f2

3 < 0, choosing the eigenvalues of A* placed in �4. The Lyapunov function (29) satisfies(16)–(18) with �c1 ¼ 0:57542, �c2 ¼ 17:007, �c3 ¼ 4, and �c4 ¼ 34:014. Moreover, it has been considered that the maxi-mum value of perturbation e is jejmax = 1.4. Then, choosing the bound d ¼ 0:002 < �d ¼ 0:1063, from (21), and the


value h = 0.99 leads the bound lW = 2.21 · 10�2 and lxM ¼ 36:8374, by using (22) and (23), with lW ¼ 0:7�lW, andlxM ¼ 0:7�lxM . Finally, in accordance with Theorem 1, if we take a ball of radius rB = 5, then an ultimate boundfor f is �b ¼ kfðtÞkmax ¼ 4:55. In the sequel, a fourth-order Runge–Kutta integration algorithm with step time of size0.0001, was used to obtain numerical results.

Fig. 2 illustrates the complete synchronization of Rossler–Rossler systems under nonvanishing perturbatios: (a) Thestate slave x1(t) following xM1(t), (b) the output of the perturbed slave x2(t) following the output of the chaotic masterxM2(t), (c) the state slave x3(t) following xM3(t), and (d) the control law u(t) taking action after 40 s. The initial condi-tions f(0) = (5,5,5) are considered. While, Fig. 3 shows the state behaviors f1(t), f2(t), and f3(t), once approximate out-put synchronization was obtained, where jf1(t)j = 0.2841, jf2(t)j = 0.4189, and jf3(t)j = 1.0685, and thereforekf(t)k = 1.1814.

The behavior of the synchronization between the states x(t) and xM(t), and a comparison between unperturbed andperturbed case, is shown in Fig. 4: (a), (b), and (c), phase portrait of synchronization: x1 vs xM1, x2 vs xM2, and x3 vsxM3 for unperturbed case; (d), (e), and (f) phase portrait of synchronization: x1 vs xM1, x2 vs xM2, and x3 vs xM3 forperturbed case.

Note that the effect of perturbation e on state f(t) is negligible. From this result, it is easily verified that the norm ofstate f(t) is completely within the considered ultimate bound �b.

5.2. Chua’s circuit-Rossler synchronization

Now, consider the output synchronization of Chua’s circuit [31] with a Rossler system, the first like a master M,whose normalized equations can be written in the form (1) as follows:

30 40 50 60–20

–10

–20

–10

–20

–10

0

10

20

time (sec)

ampl

itude

30 40 50 60

0

10

20

time (sec)

ampl

itude

30 40 50 60

0

10

20

30

40

50

time (sec)

ampl

itude

40 60 80 100

0

10

20

time (sec)

u(t)

a b

dc

Fig. 2. Complete synchronization of Rossler–Rossler under nonvanishing perturbations using model-matching approach: (a) The statex1 (dashed line) following xM1 (continuous line), (b) the output of the perturbed slave x2 (dashed line) following the output of themaster xM2 (continuous line), (c) the state x3 (dashed line) following xM3 (continuous line), and (d) the control law u(t).

Fig. 3. Behavior of the perturbed state f(t) with maximum magnitude kf(t)k = 1.1814, completely within the ultimate bound �b ¼ 4:55.

–15 –10 –5 0 5 10 15

–10

0

10

x1

x1

xM

1

–15 –10 –5 0 5 10

–10

0

10

x2

xM

2

0 10 20 30

0

10

20

30

x3

xM

3

0 10 20 30

0

10

20

30

x3

xM

3

–15 –10 –5 0 5 10

–10

0

10

x2

xM

2

–15 –10 –5 0 5 10 15

–10

0

10

xM

1

a d

eb

c f

Fig. 4. Phase portrait of complete synchronization of Rossler–Rossler. Unperturbed case: (a) x1 vs xM1, (b) x2 vs xM2, and (c) x3 vsxM3. Perturbed case: (d) x1 vs xM1, (e) x2 vs xM2, and (f) x3 vs xM3.


M :

_xM1

_xM2

_xM3

0B@1CA ¼ p½xM2 � xM1 � f ðxM1Þ�

xM1 � xM2 þ xM3

�qxM2

0B@1CAþ 1

0

0

0B@1CAuM ;

y ¼ x3

8>>><>>>:
with nonlinear function
f ðxM1Þ ¼ bxM1 þ1

2a� b� �

ðjxM1 þ 1j � jxM1 � 1jÞ;


and the other like a class of perturbed slaves Pe, given by

Fig. 6.

Fig. 5.(e), an

P e :

_x1

_x2

_x3

0B@1CA ¼ �x2 � x3

x1 þ eax2

~aþ x3ðx1 � ~lÞ

0B@1CAþ 0

0

1

0B@1CA~u;

y ¼ x2:

8>>><>>>:

Behavior of the perturbed state f(t) with maximum magnitude kf(t)k = 0.8076, completely within the ultimate bound �b ¼ 1:82.

0 50 100 150– 4

0

4

time (sec)

x M1

0 50 100 150–1

0

1

time (sec)

x M2

0 50 100 150–8

0

8

time (sec)

x M3

0 50 100 150–15

0

15

time (sec)

x 1

0 50 100 150–8

0

8

time (sec)

x 2

0 50 100 150– 35

0

35

time (sec)

x 3

a

b

c f

e

d

Output synchronization of Chua’s circuit with a Rossler system: (a), (b), and (c), xM1(t), xM2(t), and xM3(t), respectively, (d),d (f), x1(t), x2(t), and x3(t), respectively. The control law u(t) starts at t = 50s.


Both with the same relative degree rM = r = 3. Chua’s circuit exhibits chaotic dynamics with parameter valuesa = �1.27, b ¼ �0:68, p ¼ 10, and q ¼ 14:87 [31].

Consider the same parameter mismatch given in previous example. Suppose that these small variations are now pres-ent in the implementation of the control law, which uses similar parameters to those of M. Then, the control law is givenby

~u ¼ � vþ ~q ~p þ 1ð ÞxM1 � ~p � ~qþ 1ð ÞxM2 þ xM3 þ ~pf ðxM1Þ � uM½ �f �ða2 � 1Þx1

�aða2 � 2Þx2 þ aþ xM3ðxM1 � lþ aÞ � uM

�;

where ~p ¼ p þ Dp and ~q ¼ qþ Dq. We consider now Dp = sin(2pt/T) and Dq = 1.478sin(2pt/T), such that the param-eters for control ~uðtÞ are ~p ¼ 10þ sinð2pt=T Þ and ~q ¼ 14:78þ 1:478 sinð2pt=T Þ. Such variation affects the nominalparameter values oscillating between ±10%, acting again as perturbations on nominal control u(t). Then, consideringthis perturbed control ~uðtÞ, it is possible to express the auxiliary system in the form (13), with

DfEðxEÞ þ DgðxEÞu ¼ gðxEÞDu

separating the perturbation part from the control and leaving only the nominal control, that is u(t) = unom(t) + Du(t).Then, the perturbed auxiliary system becomes

Ee :

_x1

_x2

_x3

_xM1

_xM2

_xM3

0BBBBBBBB@

1CCCCCCCCA¼

�x2 � x3

x1 þ ax2

aþ x3ðx1 � lÞp½xM2 � xM1 � f ðxM1Þ�

xM1 � xM2 þ xM3

�qxM2

0BBBBBBBB@

1CCCCCCCCAþ

0

0

1

0

0

0

0BBBBBBBB@

1CCCCCCCCAunom þ

0

0

0

1

0

0

0BBBBBBBB@

1CCCCCCCCAuM þ

0

0

1

0

0

0

0BBBBBBBB@

1CCCCCCCCADu;

yE ¼ x2 � xM3:

8>>>>>>>>>><>>>>>>>>>>:ð30Þ

The perturbed closed-loop auxiliary system (30), yields the subsystem (15) with

– 5 0 5–5

0

5

xM3

x 2

–5 0 5–5

0

5

xM3

x 2

a b

Fig. 7. Output synchronization of Chua’s circuit-Rossler: x2 vs xM3. (a) Unperturbed case, (b) perturbed case.

5

0

5

1

0.5

0

0.5

1

4

3

2

1

0

1

2

3

4

xM3

xM2

x M1

20

0

20

5

0

5

30

20

10

0

10

20

30

x1

x2

x 3

20

0

20

5

0

5

30

20

10

0

10

20

30

x1

x2

x 3

Chua’s circuit/master

NominalRössler/slave

PerturbedRössler/slave

a b c

Fig. 8. Chaotic attractors: (a) Chua’s circuit/master, (b) unperturbed Rossler/slave driven by Chua’s circuit, and (c) perturbed Rossler/slave driven by Chua’s circuit.


Weðf; x; xM Þ ¼0

0

ðqþ DqÞ½DpxM1 � ðDp � DqÞxM2 � Dpf ðxM1Þ� þ DqUMðxM ; uM Þ

0B@1CA;

and

UM ðxM ; uM Þ ¼ ½ðp þ 1ÞxM1 � ðp � qþ 1ÞxM2 þ xM3 þ pf ðxM1Þ � uM �

such that the same analysis like in Theorem 1 applies. The same Lyapunov function (29) satisfies (16)–(18) for this case,with �c1 ¼ 0:57542, �c2 ¼ 17:007, �c3 ¼ 4, and �c4 ¼ 34:014. A ball of radius rB = 2 is enough to satisfy Assumption (H4)for Chua’s circuit, such that choosing a small bound d ¼ 0:002 < �d ¼ 0:1063, from (21), and the value h = 0.99 resultsthe bound lW = 8.8 · 10�3 and lxM ¼ 14:7350 , by (22) and (23) with lW ¼ 0:7�lW, and lxM ¼ 0:7�lxM . Then, an ultimatebound for f(t) is �b ¼ kfðtÞkmax ¼ 1:82.

Fig. 5 depicts the time evolution of the states (a), (b), and (c) xM1(t), xM2(t), and xM3(t), respectively, (d), (e), and (f)x1(t), x2(t), and x3(t), respectively. Notice that the state x2(t) of Pe follows to xM3(t) of M once the control law u(t)begins, after 50 seconds. The rest of the states of Pe does not synchronize with any other state of M, but they remainbounded. The initial conditions f(0) = (0.9,0.9,0.9) are considered. Fig. 6 shows the behavior of the states f1(t), f2(t),and f3(t), once approximate output synchronization was obtained, where j f1(t)j = 0.2904, jf2(t)j = 0.4597, andjf3(t)j = 0.5972, and then, kf(t) k = 0.8076. Phase portrait of synchronization: x2 vs xM3 for unperturbed and perturbedcases are illustrated in Fig. 7: (a) Unperturbed case, (b) perturbed case. A comparison between Chua’s chaotic attractorand the resulting attractor after output synchronization of Rossler driving by Chua’s circuit is shown in Fig 8. Noticethe similar dynamics of Rossler/slave with Chua’s circuit/master.

6. Conclusions

In this paper, a model-matching approach for synchronizing chaotic (identical or nonidentical) systems under non-vanishing perturbations was presented. Based on model-matching approach and Lyapunov-based stability from non-linear control theory, a control law for an unperturbed chaotic slave/plant was designed such that a class of perturbed


chaotic slaves/plants can be synchronized with a chaotic master/model dynamics. Rossler system and Chua’s circuitwere used as illustrative examples to verify and visualize the synchronization strategy. Under the proposed controlmethod, the (output) synchronization error converges in a finite short time to an ultimate (calculated) bound. Bothanalysis and numerical simulation reveal that the proposed model-matching approach has great potential for synchro-nizing chaotic systems under nonvanishing perturbations.

In a forthcoming work we will be concentrated with an application of robust output synchronization of chaotic sys-tems to transmit encrypted information, as reported for unperturbed case in [7].

Acknowledgement

This work was supported by the CONACYT, Mexico under Research Grants No. J49593-Y and No. P50051-Y.

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Output synchronization of chaotic systems under nonvanishing perturbations

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