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Synchronization in reduced-order of chaotic systems via controlapproaches based on high-order sliding-mode observer

A. Rodríguez a, J. De León a,*, L. Fridman b

a Electrical Engineering Doctoral Program, Mechanical and Electrical Engineering Faculty, Autonomous University of Nuevo León,66450 San Nicolás de los Garza, N.L., Mexicob Department of Control, Division of Electrical Engineering, Engineering Faculty, National Autonomous University of Mexico, 04510 Mexico City, Mexico

a r t i c l e i n f o

Article history:Accepted 21 April 2009

Communicated by Prof. P. Grigolini

a b s t r a c t

The reduced-order synchronization problem of two chaotic systems (master–slave) withdifferent dimension and relative degree is considered. A control scheme based on a high-order sliding-mode observer-identifier and a feedback state controller is proposed, wherethe trajectories of slave can be synchronized with a canonical projection of the master.Thus, the reduced-order synchronization is achieved in spite of master/slave mismatches.Simulation results are provided in order to illustrate the performance of the proposed syn-chronization scheme.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

1.1. Antecedents

Synchronization of chaotic systems has attracted the attention of the researchers because many possible applications canbe foreseen from a proper understanding of the role of chaotic dynamics in interacting systems [26,2]. It is well known that achaotic system is a nonlinear deterministic system having a complex and unpredictable behavior such that the sensitivedependence on initial conditions and on system parameters variations are a prominent feature of chaotic behavior. However,though the trajectories from autonomous chaotic systems are very hard to predict, they respond enough mildly to synchro-nization attempts of their behavior with respect to other chaotic systems. Talking about classic dynamical systems, the syn-chronization phenomenon has been actively researched since the 17th century, starting with the Huygen’s work concerningtwo coupled pendulum clocks that become synchronized in anti-phase [15]. From then on, many others natural and man-made processes and systems have been discovered which exhibit the synchronization phenomenon, such as synchronizedlightning of fireflies, adjacent organ pipes, biological and physiological systems, synchrony of triode generators and otherelectronic devices, many rotating mechanical structures, and many classical cases of synchronization of periodic systems[3]. In such context, the meaning of synchronization is understood as an adjustment of rhythms of oscillating objects dueto weak interactions [22].

In spite of the classical concept of synchrony, varied types of synchronization are known from studies in this field withchaotic systems [20,25,4,16,9,21,23,5,10], which are dependent on signal parameters (frequency, phase, amplitude) to bematched and other technical concepts. This is very important because different kinds of synchrony can appear or be desiredin process for a common objective, as either identical or different systems, for instance, robot coordination and cooperationof manipulators or others electro-mechanical systems [7,24]. Then, some types of synchronization for chaotic systems can bereviewed. For example, the identical synchronization (IS) implies an actual equality (both amplitude and phase) of the

0960-0779/$ - see front matter � 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2009.04.055

* Corresponding author. Tel.: +52 818 329 4020x5773.E-mail addresses: angelrdz@gmail.com (A. Rodríguez), drjleon@gmail.com (J. De León), lfridman@servidor.unam.mx (L. Fridman).

Chaos, Solitons and Fractals 42 (2009) 3219–3233

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corresponding variables of two or more coupled identical systems [20]. The phase synchronization (PS) is displayed if thereexists certain relation between the phase of the system variables but the amplitudes remain chaotic and uncorrelated [21].The generalized synchronization (GS) is a generalization of the above concepts, in this sense it is said that two unidirectionallycoupled different systems are synchronized if a (static) functional relation exists between the variables of both systems [25].

Furthermore, there are different synchronization schemes dependent on the amount of signals to be synchronized in thesystems. In this way, complete or full synchronization is achieved for chaotic systems with same order if and only if all theirtrajectories are synchronized in some sense [9]. In another way, chaotic systems with same order are partially synchronizedif, at least, one of the trajectories is synchronized and only if, at least, one of the trajectories is not synchronized [9,23]. Now,considering two unidirectionally coupled systems, where the order of the slave system is less than the master one, synchro-nization of reduced-order occurs if all trajectories of a slave system are synchronized with projections of a master system[10,6,28].

It is worth mentioning that in unidirectional schemes, the master system independently evolves regarding to slave sys-tem, whereas the slave system is driven under coupling or control actions (called input variables).

Then, the reduced-order synchronization considered here is the problem of synchronizing a slave system with projectionsof a master system, under the assumption that master system has a relative degree equal or greater than slave system.

Since synchronization is an open field of research, and the subject in this work is related with the generalized synchro-nization in reduced-order, i.e., unidirectional synchronization of a slave system with projections of a master system, takinginto account that the dynamics describing the master and slave systems are completely different.

1.2. Motivation

Although the synchrony phenomenon occurs in nature between systems either with different model representation ororder; for instance, the cardiorespiratory system [27] and neural relationships [19]; the forced-synchronization between sys-tems with different model representation and different order can be difficult, when (in practical cases) these models haveparametric uncertainties, non-modeled dynamics, or external disturbances, such as occurs in laser systems [29], coordina-tion and cooperation of manipulator robots [7,24], and so on.

Moreover, the feedback control laws are usually expressed in terms of all state vector components and it is common thatcomplete state vector is not available in practical cases, then, in order to avoid this difficulty it is clearly necessary to recon-struct this information from the output. Indeed, even if all the state measurements are possible they are typically corruptedby noise. Additionally, the chaotic dynamic is extremely sensitive to their initial conditions.

Thus, to solve the chaotic synchronization problem with robustness, it should be taking into account the presence of thenoise and non-modeled dynamics. Besides the knowledge of all state vector components should be suitable in order toimplement some state feedback control law and have some information about the disturbances for compensating them.

From these reasons, this paper is motivated by the next:

� The synthesis of synchronization for chaotic systems should be focused from a robust viewpoint [1,8].� The reconstruction of unavailable state components and disturbances variables allows to design feedback control laws in

order to eliminate the above obstacles.� Additionally, it is well known that sliding-mode techniques are robust about parametric uncertainties and some distur-

bances. Particularly, hierarchical sliding-mode observers based on super-twisting are widely used due to their attractivefeatures:

(i) insensitivity (more than robustness) with respect to unknown inputs;(ii) the conditions required for the design of the observers are sufficient and necessary for the case when the unknowninputs are bounded;(iii) identification for unknown input is attained;(iv) convergence in finite time to the exact system state, even in presence of unknown inputs.

1.3. Contributions

A control scheme is proposed to solve the synchronization problem for a class of nonlinear chaotic systems. This controlscheme, constituted by a feedback linearizing controller based on a high-order sliding-mode observer, ensures asymptoticstability and robustness under presence of uncertain disturbances and non-modeled dynamics. Since the high-order slid-ing-mode observer converges in finite-time, then does not require the separation principle to be proved.

Thus, the advantages of the proposed synchronization scheme can be summarized as follows:

� Synchronization is achieved between two chaotic systems with different relative degree.� The synchronization is achieved in spite of unknown disturbances and non-modeled dynamics, only partial knowledged is

required (the relative degree and disturbance characteristic index).� The proposed scheme allows synchronization considering only the outputs of both systems and an observer is used to

reconstruct the non-measurable state vector.� No separation principle is required to analyze the closed-loop system stability, due to finite-time observer convergence.

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1.4. Structure of the paper

The paper is organized as follows. First, in Section 2 the problem statement is presented. In Section 3, we introduce themethodology based on a state feedback linearization control law using a high-order sliding-mode observer designed in orderto estimate the unmeasurable state under unknown dynamics or uncertainties of slave and master systems. Next, in Section4, the proposed control scheme is implemented in two strictly different chaotic systems in order to illustrate the synchro-nization scheme. Finally, some conclusions are given.

2. Problem statement

Now, the generalized synchronization problem between two different chaotic systems is analyzed, assuming that the or-der of the master system is greater than the order of the slave system. In what follows this problem will be called generalizedsynchronization in reduced-order.

Consider the following master system

RM :_xM ¼ fMðxMÞ þ gMðxMÞuM;

yM ¼ hMðxMÞ;

�ð1Þ

where xM ¼ ½x1;M; x2;M; . . . ; xnM ;M �T 2 RnM is the state vector, uM 2 R is an input, and yM 2 R is the output of the master system,

fM and gM are smooth vector fields, and (1) has relative degree rM (see Appendix). Consider a slave system described by

RS :_xS ¼ fSðxSÞ þ gSðxSÞuS;

yS ¼ hSðxSÞ;

�ð2Þ

where xS ¼ ½x1;S; x2;S; . . . ; xnS ;S�T 2 RnS is the state vector, uS 2 R is the control input, and yS the output, fS and gS are smooth

vector fields, with relative degree rS.We assume that the order nS of the slave system is less than the order nM of the master system (nS < nM). It is clear that

nS ¼ nM represents a particular case of this analysis.Now, the synchronization problem considered can be established as follows.The reduced-order generalized synchronization objective: Given two chaotic system with different order, find a control to

force the states of the slave system (2) to be synchronized with some projections of state vector of the master system (1).In order to attain this objective, let us define the synchronization error

e :¼ xS �UðxMÞ;

where U is a functional relation. Then, generalized synchronization is defined as follows:

Definition 1 ([25,16,30]). A slave system (2) exhibits generalized synchronization with the master system (1), if there existsa functional relation U, such that

limt!1

e ¼ limt!1½xS �UðxMÞ� ¼ 0 ð3Þ

for all t P 0 and any initial condition eðt0Þ ¼ xSðt0Þ �UðxMðt0ÞÞ.

Definition 1 is meaningful because (3) implies xS ! UðxMÞ for all t P 0 and any initial difference eðt0Þ ¼ xSðt0Þ �UðxMðt0ÞÞ,which is understood as the synchronization of system (2) to a projection of state vector of the system (1). Then, the dynam-ical synchronization error is given by

Re :_e ¼ feðeÞ þ geðeÞuS þ qðe; nÞ;ye ¼ heðeÞ;

�ð4Þ

where e 2 RnS , ye is a function of the synchronization error which is available for feedback, fe and ge are smooth vector fieldsand qðe; nÞ is a smooth vector field containing terms arisen from difference between master and slave models, which dependon the synchronization error e, the states xS and xM and control uM , where n :¼ nðxM;uMÞ. The components of vector fieldqðe; nÞ are considered as perturbations into the system. In addition, the vector field satisfies qðe;0Þ ¼ 0, 8e 2 RnS . It is clearthat for n ¼ 0, we obtain the nominal system, i.e., dynamical system without perturbations.

Furthermore, we assume that the system (4) has relative degree rS; i.e., the control uS explicitly appears first time in therSth total time derivative of yS. Consequently, the relative degree can be interpreted as the dimension of system (4) that canbe linearized by a change of coordinates and a state feedback uS. Furthermore, taking into account the presence of the per-turbation terms associated to n, we introduce the following definition, which is important for synchronizing both systems.

Definition 2 (Disturbance Characteristic Index [18]). The disturbance characteristic index m is defined for system (4) as theinteger such that:

LqLife heðeÞ ¼ 0; 0 6 i 6 m� 2; 8e 2 Rns ; 8n 2 X

LqLm�1fe heðeÞ – 0; for some n 2 X; for some e 2 Rns :

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We set m ¼ 1 if

LqLife heðeÞ ¼ 0; i P 0; 8e 2 Rns ; 8n 2 X:

That is, the disturbance characteristic index is the number m of differentiations of the output yS which are required to showexplicitly the term n. The function n can be considered as a unknown input.

Then, the importance of the relationship between m and rS is clarified by the following result.

Lemma 1 [18]. Assume that relative degree for the nominal system is rS 6 nS. Then, there exist nS � rS functions e?i ðeÞ,1 6 i 6 nS � rS, such that:

(i) the functions heðeÞ; . . . ; LrS�1fe

heðeÞ; e?1 ðeÞ; . . . ; e?nS�rSðeÞ form a local diffeomorphism about the origin;

(ii) hde?i ; qi ¼ 0, 1 6 i 6 nS � rS.

In local coordinates e ¼ ½e; e?�,

e ¼ ½heðeÞ; . . . ; LrS�1fe

heðeÞ; e?1 ðeÞ; . . . ; e?nS�rSðeÞ� ¼ /ðeÞ

system (4) is expressed as follows:

For the case m > rS,

_ei ¼ eiþ1; 1 6 i 6 rS � 1_erS ¼ LrS

fehe þ uSLge L

rS�1fe

he;

_e?j ¼ Lfe e?j þ Lqe?j ; 1 6 j 6 nS � rS

ye ¼ e1:

8>>><>>>: ð5Þ

For the case m ¼ rS,

_ei ¼ eiþ1; 1 6 i 6 rS � 1_erS ¼ LrS

fehe þ uSLge L

rS�1fe

he þ LqLrS�1fe

he;

_e?j ¼ Lfe e?j þ Lqe?j ; 1 6 j 6 nS � rS

ye ¼ e1:

8>>><>>>: ð6Þ

For the case m < rS,

_ei ¼ eiþ1; 1 6 i 6 m� 1_ei ¼ eiþ1 þ LqLi�1

fe he; m 6 i 6 rS � 1

_erS ¼ LrSfe

he þ uSLge LrS�1fe

he þ LqLrS�1fe

he;

_e?j ¼ Lfe e?j þ Lqe?j ; 1 6 j 6 nS � rS

ye ¼ e1:

8>>>>>><>>>>>>:ð7Þ

There exists a global change of coordinates transforming system (4) into (5)–(7), if the vector fields

~f e ¼ fe �Lr

fe he

Lge Lr�1fe he

ge; ~ge ¼1

Lge Lr�1fe he

ge

are complete and the global relative degree is well defined.

From now on, it is assumed the next:

Assumption 1. The error system (4) is assumed to be of minimum phase (see Appendix).

Remark 1. It is clear that the case m < rS is not considered because no control can be designed in order to reject disturbanceand guarantee the synchronization error tends to zero. The case m > rS is trivial since the disturbance does not affect the con-trol objective, then synchronization can be achieved. Therefore, the interesting case is when m ¼ rS, which is considered inthis paper.

Taking into account the above remark, the following assumption is introduced.

Assumption 2. For synchronization error system (4), disturbance characteristic index is assumed to be m ¼ rS.

Thus, the synchronization error system (4) can be written as (where e ¼ /ðeÞ):

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Re :

_e ¼ Aeeþ Be bSðeÞuS þ nðe; uMÞ½ �_e? ¼ gðe; e?Þye ¼ Cee

8><>: ð8Þ

with e 2 RrS , e? 2 RnS�rS , uS 2 R, n 2 R; and

Ae ¼

0 1 0 � � � 00 0 1 � � � 0... ..

. ... . .

. ...

0 0 0 � � � 10 0 0 � � � 0

26666664

37777775; Be ¼

00...

01

26666664

37777775; Ce ¼ 1 0 � � � 0 0½ �;

where e :¼ xS �UðxMÞ, e ¼ ½e; e?�, and e ¼ /ðeÞ 2 RnS is the synchronization error vector, uS 2 R is the forcing input, ye 2 R isthe output of error system (8), the function nðe;uMÞ is considered as perturbation term. The subsystem _e? ¼ gðe; e?Þ repre-sents the internal dynamic whose zero dynamics _e? ¼ gð0; e?Þ, is assumed to be asymptotically stable, i.e., the system is min-imum phase.

It is evident for obtaining the above representation is necessary to know the functional relation U, which usually is un-known. Then, the following result summarized the conditions which allow to find a functional relation U and to transformthe systems (1) and (2) into a normal form.

Lemma 2 [18]. Assume that rS 6 nS for system (2). Then there exist nS � rS functions �z?i;SðxSÞ, 1 6 i 6 nS � rS, such that:

(i) the functions hSðxSÞ; . . . ; LrS�1fS

hSðxSÞ;�z?1;SðxSÞ; . . . ;�z?nS�rS ;SðxSÞ form a local diffeomorphism /SðxSÞ ¼ ½�/SðxSÞ; �/?S ðxSÞ�T about the

origin;(ii) hd�z?i;S; gSi ¼ 0, 1 6 i 6 nS � rS.

In local coordinates ½�zS;�z?S � ¼ ½hSðxSÞ; . . . ; LrS�1fS

hSðxSÞ;�z?1;SðxSÞ; . . . ;�z?nS�rS ;SðxSÞ�, i.e., zS ¼ ½�zS;�z?S �

T ¼ ½�/SðxSÞ; �/?S ðxSÞ�T ¼ /SðxSÞ,system (2) is expressed in normal form

_�zS ¼ AS�zS þ CSð�zS;�z?S ;uSÞ_�z?S ¼ uSð�zS;�z?S ÞyS ¼ CS�zS

8><>: ð9Þ

with

AS ¼

0 1 0 � � � 00 0 1 � � � 0... ..

. ... . .

. ...

0 0 0 � � � 10 0 0 � � � 0

26666664

37777775; CSðzS; uSÞ ¼

00...

0aSðzSÞ þ bSðzSÞuS

26666664

37777775; CS ¼ 1 0 � � � 0 0½ �;

where aSðzSÞ ¼ LrSfS

hS and bSðzSÞ ¼ LgSLrS�1

fShS.

Additionally, if the global relative degree rS is well defined with rS 6 nS and

(iii) the vector fields

~f S ¼ fS �LrS

fShS

LgSLrS�1

fShS

gS; ~gS ¼1

LgSLrS�1

fShS

gS

are complete, then exists a global diffeomorphism transforming (2) into the normal form (9).

Thus, we have the following result:

Lemma 3 [18]. If system (2) has relative degree rS 6 nS, then it is locally partially state feedback linearizable with index rS.

Assumption 3. The slave system has relative degree rS ¼ nS, where nS is the order of slave system, i.e., all components of theslave system will be synchronized with some projections of the master system.

Then, from the above assumption it follows that zS ¼ �zS and

_�zS ¼ AS�zS þ CSð�zS;uSÞ;yS ¼ CS�zS:

(ð10Þ

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Now, from Lemma 2, the master system (1) can be transformed, by means of the diffeomorphism zM ¼ ½�zM;�z?M�T

¼ ½�/MðxMÞ; �/?MðxMÞ�T ¼ /MðxMÞ, into a normal form:

_�zM ¼ AM�zM þ CMð�zM ;�z?M;uMÞ ð11aÞ_�z?M ¼ uMð�zM;�z?MÞ ð11bÞyM ¼ CM�zM ð11cÞ

with

AM ¼

0 1 0 � � � 00 0 1 � � � 0... ..

. ... . .

. ...

0 0 0 � � � 10 0 0 � � � 0

26666664

37777775; CMð�zM;�z?M;uMÞ ¼

00...

0aMð�zM ;�z?MÞ þ bMð�zM ;�z?MÞuM

26666664

37777775;CM ¼ 1 0 � � � 0 0½ �;

where zM ¼ ½�zM ;�z?M �T 2 RnM , with �zM 2 RrM , �z?M 2 RnM�rM , aMð�zM;�z?MÞ and bMð�zM ;�z?MÞ are functions of zM .

The new synchronization error is given by e ¼ �zS � /projð�zMÞ, where /proj is the projection map from RrM into RrS . This syn-chronization error can be written in original coordinates as follows: e ¼ �/SðxSÞ � /proj � �/MðxMÞ. Then, to obtain the functionalrelation U (see Fig. 1), it is easy to see that,

e ¼ �/SðxSÞ � /proj � �/MðxMÞ;�/�1

S ðeÞ ¼ xS � �/�1S � /proj � �/MðxMÞ;

�/�1S ðeÞ ¼ xS �UðxMÞ;

�/�1S ðeÞ ¼ e:

ð12Þ

Then, it follows that Uð�Þ ¼ �/�1S � /proj � �/Mð�Þ and

e ¼ �/SðeÞ: ð13Þ

Finally, using this transformation and the Assumption 1, the system (8) is minimum phase, we obtain that

_e ¼ Aeeþ Be bSðeÞuS þ nðe;uMÞ½ �: ð14Þ

Remark 2. The exact synchronization of (10) with (11a), (11c) can be solved by stabilizing the synchronization error system(8) at the origin.

Notice that the synchronization of e ¼ �zS � /projð�zMÞ is affected by unknown dynamics and/or parametric uncertaintiespresent in function nðe;uMÞ that can be arisen from both models, and due to that the complete states xM and xS could notbe available for feedback control laws. Then, in the sequel the term nðe;uMÞ will be considered as external disturbances intothe system.

Fig. 1. Blocks diagram illustrating the relationship between original variables and transformed variables.

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Now, in order to synchronize the systems (1) and (2), i.e., the synchronization error e! 0, or equivalently e! 0, it is nec-essary to design a controller to stabilize system (8) under presence of external perturbations n and/or parametric uncertain-ties. Then, a state feedback controller is designed. However, this controller requires all components of the state vectoravailable, which limits its implementation. Therefore, in order to overcome this difficulty, an high-order sliding-mode obser-ver is proposed for solving mentioned problems about state estimation and identification.

3. Synchronization approach

Since synchronization between chaotic systems (10) and (11a) can be interpreted as a stabilizing problem for error sys-tem (8). In this section, some well-known results for stabilization of dynamical systems at the origin will be introduced.

3.1. State feedback linearization control

Assuming that all components of the state vector are available, then the state feedback linearization technique can beapplied in order to globally stabilize the nonlinear dynamical system. Then, the nonlinear system (8) is transformed intoa linear and non-interacting system in the Brunovsky form by means of the control

uS ¼�nðe; uMÞ þ vS

bSðeÞ; ð15Þ

where the controller vS is an auxiliary control selected as follows:

vS ¼XnS

i¼1

kiei; ð16Þ

where the gain values ki, i ¼ f1;2; . . . ;nSg, are chosen such that snS þ knS snS�1 þ � � � þ k2sþ k1 ¼ 0 be a Hurwitz polynomial.However, it is required to know all components of e and n in order to implement the controller (15) and (16).

3.2. High-order sliding-mode observer design

In order to estimate all the components of e and identification of unknown input n, the design of an observer to recon-struct the no measurable variables (preferably finite-time convergent and exact) is presented.

Take into account that the error system (8) can be written in extended form as:

_e1 ¼ e2;

_e2 ¼ e3;

..

. ...

_enS�1 ¼ enS ;

_enS ¼ nðe; �xM; �x?M; uM; tÞ þ uS;

_n ¼ vðtÞ;ye ¼ Cee ¼ e1;

8>>>>>>>>>>>><>>>>>>>>>>>>:ð17Þ

where e 2 RnS , ye 2 R is the measured output, n 2 R is an unknown smooth function containing nonlinear and disturbanceterms, uS 2 R is the control input, vðtÞ is an unknown bounded continuous signal, and Ce is a matrix of appropriatedimension.

Therefore, the selection of output ye is important due to the relationship between relative degree with respect to the un-known input nðtÞ:

Definition 3 [14]. System (17) is called (strongly) observable if for any initial state eð0Þ and nðtÞ � 0 (any input nðtÞ),yeðtÞ � 0 (8t P 0) implies x � 0 (8t P 0).

The observability is related with relative degree in the following form:

Theorem 1 [12]. The system (17) is strongly observable if and only if the output of the system (17) has relative degree nS withrespect to the unknown input nðtÞ.

Then, from [12], it follows that system (17) satisfies:

Assumption 4. The system (17) has relative degree nS with respect to the unknown input nðtÞ.

This assumption means that the system is strongly observable. The above assumption has the next two variants:

Assumption 5. The unknown input nðtÞ is a bounded Lebesgue-measurable function, jnðtÞj 6 nþ.

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Assumption 6. The unknown input nðtÞ is a bounded function, jnðtÞj 6 nþ, with successive derivatives up to the order kbounded by the same constant vþ. The kth derivative is a Lipschitzian function with the Lipshitz constant not exceedingvþ. Thus, nðkþ1ÞðtÞ exists almost everywhere and is a bounded Lebesgue-measurable function, jnðkþ1ÞðtÞj 6 vþ.

Then, in order to be able to identify the unknown input nðtÞ, it is necessary to assume that the Assumptions 4 and 6 hold.Furthermore, for the extended error system (17) it is supposed that the function n is bounded and smooth. Then, a high-ordersliding-mode observer [13] for the extended state ðe; nÞT 2 RnSþ1 of (17) is built is given by

_e1 ¼ �k1M1=ðnSþ1Þje1 � e1jnS=ðnSþ1Þsignðe1 � e1Þ þ e2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}v1

;

_e2 ¼ �k2M1=nS je2 � v1jðnS�1Þ=nS signðe2 � v1Þ þ e3|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}v2

;

..

.

_enS ¼ �knS M1=2jenS � vnS�1j1=2signðenS � vnS�1Þ þ n|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}vnS

þuS;

_n ¼ �knSþ1Msignðn� vns Þ;

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

ð18Þ

where M is a constant chosen sufficiently large and the constants ki are chosen as in [17]. Then, the following result can beestablished:

Theorem 2. Consider the extended system (17) where nðtÞ is a unknown function to be identified. Then system (18) is an observerfor the system (17) where the estimation error ð~e; ~nÞT ¼ ðe; nÞT � ðe; nÞT converges to zero in finite-time, i.e., the estimates convergeto the actual states ei, i ¼ 1;2; . . . ;nS; and the unknown function nðtÞ.

Proof. This theorem can be proved as follows. Defining the estimation errors ~ei :¼ ei � ei (i ¼ 1;2; . . . ;n), and ~n :¼ n� n then

_~e1 ¼ �k1M1=ðnSþ1Þj~e1jnS=ðnSþ1Þ signð~e1Þ þ ~e2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}~v1 :¼v1�e2

;

_~e2 ¼ �k2M1=nS j~e2 � ~v1jðnS�1Þ=nS signð~e2 � ~v1Þ þ ~e3|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}~v2 :¼v2�e3

;

..

.

_~enS ¼ �knS M1=2j~enS � ~vns�1j1=2signð~enS � ~vnS�1Þ þ ~n|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}~vnS :¼vns 1�n

;

_~n ¼ �knSþ1Msignð~n� ~vnS Þ � vðtÞ;

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

ð19Þ

where ei are the estimation of ei, and n is the identified value of n. The parameter M is chosen sufficiently large, in particularM > vþ. The constants ki are chosen recursively sufficiently large as in [17].

To prove finite-time convergence of the estimation error (18), we need just to rewrite (20) into the form of differentialinclusion

_~e1 ¼ �k1M1=ðnSþ1Þj~e1jnS=ðnSþ1Þ signð~e1Þ þ ~e2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}~v1 :¼v1�e2

;

_~e2 ¼ �k2M1=nS j~e2 � ~v1jðnS�1Þ=nS signð~e2 � ~v1Þ þ ~e3|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}~v2 :¼v2�e3

;

..

.

_~ens ¼ �knS M1=2j~enS � ~vnS�1j1=2signð~enS � ~vnS�1Þ þ ~n|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}~vnS :¼vns�n

;

_~n 2 �knSþ1Msignð~n� ~vnS Þ � ½�vþ;þvþ�:

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

ð20Þ

The inclusion is understood in Filippov sense [11]. Then, using similar arguments as in proof of Lemma 8 in [17], the finitetime convergence of the estimation error can be proved. h

Then, the reduced-order synchronization problem between to systems of different order can be establish as follows:Consider system (2) with known relative degree rS, which will be synchronized with the projections of the system (1), are

transformed using diffeomorphisms zS ¼ /SðxSÞ and zM ¼ /MðxMÞ, into the canonical form (10) and (11a), respectively. Then,

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the reduced-order synchronization problem is equivalent to analyze the stability of the error dynamics (17) in closed-loopwith control (15) using the estimates given by the observer (18), i.e.,

System

_e1 ¼ e2

_e2 ¼ e3

..

. ...

_enS�1 ¼ enS

_enS ¼ nþ uS

_n ¼ vðtÞ

8>>>>>>>>><>>>>>>>>>:ð21Þ

Controller

uS ¼ �bn �XnS

i¼1

kibei ð22Þ

Observer

_e1 ¼ �k1M1=ðnSþ1Þje1 � e1jnS=ðnSþ1Þsignðe1 � e1Þ þ e2

_e2 ¼ �k2M1=nS je2 � v1jðnS�1Þ=nS signðe2 � v1Þ þ e3

..

.

_enS ¼ �knS M1=2jenS � vnS�1j1=2signðenS � vnS�1Þ þ nþ uS

_n ¼ �knSþ1Msignðn� vnS Þ

8>>>>>>>><>>>>>>>>:ð23Þ

The following theorem allows us to known if the stability of the closed-loop system is preserved when the state is replacedby its estimates in the control law.

Theorem 3. Consider the system (21), for which the linearized feedback control (22) is designed. Suppose that observer (23) witha finite-time of convergence is designed. Then, the augmented closed-loop system is asymptotically stable.

Proof. The proof is straightforward. Since the observer is finite-time convergent, i.e., after a time t > T , the trajectories of theobserver converges to those of the system. Then, the control can be applied to the system, and since the control law stabilizethe system at the origin. Then, this means that the trajectories of the slave system track the trajectories of the master system.In this way, the reduced-order synchronization problem is solved. h

4. Implementation

To illustrate the implementation of the proposed scheme, let us consider the synthesis of the synchronization between afourth-order system (master system) and a second-order driven oscillator (slave system). More precisely, the synchronizationof Duffing attractor with a projection in a canonical plane of a master, as is presented in [6].

The master model is described by the following set of two nonlinearly coupled differential equations:

€yþ a _yþ yþ by3 � f wþ w2

2

� �¼ uMðtÞ;

€wþ c _wþ d2ðw� y�wyÞ ¼ 0;

8<: ð24Þ

where w and y are the coordinates of both oscillators, a, b, c, d, and f are constant coefficients. The external excitation is givenby uMðtÞ. It is worth mentioning that system (24) describes a simplified version of loudspeakers and microphones.

Now, consider a slave model, represented by the Duffing equation with two external forces, as follows:

€qþ q _qþx0qþ cq3 ¼ K1 cosðx1t þ h1Þ þ K2 cosðx2t þ h2Þ þ uS; ð25Þ

where q, x0, and c are systems’ variables and Ki, xi and hi, i ¼ 1;2 are the amplitudes, frequencies and phases of the externalforces, respectively.

Taking the following change of coordinates: x1;M ¼ w, x2;M ¼ _w, x3;M ¼ y, and x4;M ¼ _y, the master system (24) can be writ-ten in the following form:

_x1;M ¼ x2;M ;

_x2;M ¼ �cx2;M � d2ðx1;M � x3;M � x1;Mx3;MÞ;_x3;M ¼ x4;M ;

_x4;M ¼ �ax4;M � x3;M � bx33;M þ f ðx1;M þ x2

1;M=2Þ þ uMðtÞ:

8>>>><>>>>: ð26Þ

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We can see from the above model, with output yM ¼ x1;M , has relative degree rM ¼ 4, then the system is of the form (11a) and(11b) with /MðxMÞ ¼ I4�4. The parameter were chosen as follows: a ¼ 0:03, b ¼ 1, c ¼ 0:3, d ¼ 0:985, f ¼ 0:1. Furthermore,the external excitation is taken as a single sinusoid function uMðtÞ ¼ E cosðxetÞ, with E ¼ 0:35, and xe ¼ 1. Initial conditionswere arbitrarily located at origin.

The Duffing equation (25) is represented in the controlled form (10), with zS ¼ xS and /SðxSÞ ¼ I2�2, as follows:

_x1;S ¼ x2;S;

_x2;S ¼ �qx2;S þx0x1;S � cx31;S þ K1 cosðx1t þ h1Þ þ K2 cosðx2t þ h2Þ þ uS;

(ð27Þ

where x1;S ¼ q, x2;S ¼ _q, and uS is the coupling force (controller). The relative degree of the slave system with respect to theoutput yS ¼ x1;S is rS ¼ 2.

Simulation results are obtained by choosing the following values of the parameters: q ¼ 1, x0 ¼ 10, c ¼ 100, K1 ¼ K2 ¼ 1,x1 ¼ 2, x2 ¼ 4, and h1 ¼ h2 ¼ 0, by setting the initial conditions as xSð0Þ ¼ ð0:1; 0ÞT .

Let us define the synchronization error as e ¼ ðe1; e2ÞT , where e1 :¼ x1;S � x1;M , and e2 :¼ _e1 ¼ _x1;S � _x1;M ¼ x2;S � x2;M ./projðzMÞ ¼ ðz1;M; z2;MÞT ¼ ðx1;M; x2;MÞT , zS ¼ ðz1;S; z2;SÞT ¼ ðx1;S; x2;SÞT . Thus, taking the first and second time derivative of the syn-chronization error, we have

_e2 ¼ _x2;S � _x2;M ¼ �qe2 þx0e1 � cðe1 þ x1;MÞ3 þ ðc � qÞx2;M þ ðd2 þx0Þx1;M

� d2ð1þ x1;MÞx3;M þ K1 cosðx1t þ h1Þ þ K2 cosðx2t þ h2Þ þ uS:

Then, the dynamics of synchronization error is given by

_e1 ¼ e2;

_e2 ¼ nðe; xM; tÞ þ uS;

�ð28Þ

where the function nð�Þ, arisen from difference between master and slave models, is given by

nðe; xM ; tÞ ¼ �qe2 þx0e1 � cðe1 þ x1;MÞ3 þ ðc � qÞx2;M þ ðd2 þx0Þx1;M � d2ð1þ x1;MÞx3;M

þ K1 cosðx1t þ h1Þ þ K2 cosðx2t þ h2Þ:

In the following, a control law uS will be designed in order to synchronize the systems, which is equivalent to stabilize thesystem (28) at the origin.

4.1. State feedback linearization control law

The synchronization of (27) to (26) can be viewed as a control problem of stabilizing the system (28) at the origin. Then,the system (28) can be written as follows:

_e ¼_e1

_e2

� �¼

e2

n

� �þ

01

� �uS: ð29Þ

Besides, considering heðeÞ ¼ e1 as the output of the above system. Then, it follows that the control law, which stabilizes (28),is given by

uS ¼ �nðe; xM; tÞ � k1e1 � k2e2 ð30Þ

with k1 and k2 choosing such that s2 þ k2sþ k1 ¼ 0 be a Hurwitz polynomial.

4.2. States and unknown terms estimation

The controller designed above requires the knowledge of full state vector e ¼ ðe1; e2Þ and the unknown function nðe; xM ; tÞfor stabilizing system (28). However, for implementation purposes, we have assumed that only e1 ¼ x1;S � x1;M is available.Then, to implement the stabilizing controller (30), it is necessary estimate the components of the state vector e of states andthe unknown function n, by using the following third-order sliding-mode observer

_e1 ¼ �k1M1=3je1 � e1j2=3signðe1 � e1Þ þ e2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}v1

;

_e2 ¼ �k2M1=2je2 � v1j1=2signðe2 � v1Þ þ n|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}v2

þuS;

_n ¼ �k3Msignðn� v2Þ:

8>>>>>>><>>>>>>>:ð31Þ

To illustrate the performance of the observer, digital simulations are performed using the following parameters of the slid-ing-mode observer: M ¼ 1950, k1 ¼ 3:0, k2 ¼ 1:5, k3 ¼ 1:1, and initial conditions e1ð0Þ ¼ 4, e2ð0Þ ¼ �2, nð0Þ ¼ 1.

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From Fig. 2, we can see that the estimation errors ~e1 :¼ e1 � e1, ~e2 :¼ e2 � e2, and ~e3 :¼ e3 � e3, converge to the origin infinite-time.

The values of overshoot and settling time depend on the gain tuning of sliding-mode observer SMO (31). Moreover, thegains could be tuned taking into account performance and design specifications.

In order to illustrate the performance of SMO observer (31), in this section we present another observer-based controllerfor evaluating and comparing the performances of the proposed scheme. In this study two characteristics are considered toevaluate these observers: overshoots and settling time.

Consider an adaptive observer AO presented in [6], which is used for estimating the unknown function n and the non-measurable components of the error vector e.

The adaptive observer AO for system 28 is given by:

_e1 ¼ e2 þ 3pðe1 � e1Þ;_e2 ¼ nþ 3p2ðe1 � e1Þ þ uS;

_n ¼ p3ðe1 � e1Þ;

8>><>>: ð32Þ

where e1 and e2 are the estimated states and n the estimated unknown function. p is a parameter used to tune the adaptiveobserver AO.

The estimation errors are plotted in Fig. 3, using observers (31) and (32), and taking into account the approximated set-ting time (tS 1). We can see when adaptive observer AO (32) is applied, the overshoots appearing into the responses aregreater than those obtained using sliding-mode observer (31). Furthermore, the convergence of sliding-mode observerSMO is in finite-time, whereas the convergence of the adaptive observer AO is asymptotical, which constitutes an importantdifference. These responses have been obtained using the following gains for the sliding-mode observer: M ¼ 50, k1 ¼ 3:0,k2 ¼ 1:5, k3 ¼ 1:1, and whereas the gain of the adaptive observer AO was p ¼ 12.

Fig. 4 shows the estimation errors responses, taking into account that both responses have similar overshoots. The gainsused for tuning the sliding-mode observer were chosen as: M ¼ 50, k1 ¼ 3:0, k2 ¼ 1:5, k3 ¼ 1:1, whereas for adaptive obser-ver was p ¼ 4. The settling time for SMO was tS 1, whereas the settling time obtained using adaptive observer AO (32) isgreater than that obtained by applying SMO (31).

On the other hand, non-measurable components of the states and unknown function will be replaced in the control law(30), as follows

uS ¼ �n� k1e1 � k2e2: ð33Þ

In this way, reduced-order synchronization between (27) and system (26) is achieved by means of (31)–(33), in spite ofuncertain differences between both systems. In Figs. 5 and 6, the synchronization of x1;S with x1;M and x2;S with x2;M are illus-trated on time-domain. The synchronization is achieved after of apply the control action.

Fig. 7 shows the control signal uS applied to slave system (27) in order to achieve the synchronization with master system(26).

−4

−2

0a)

0

10

20

30b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−100

0

100

c)

time

Fig. 2. Estimation errors: (a) ~e1 ¼ e1 � e1, (b) ~e2 ¼ e2 � e2, (c) ~e3 ¼ n� n.

A. Rodríguez et al. / Chaos, Solitons and Fractals 42 (2009) 3219–3233 3229

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Finally, from these results, we can see SMO converges faster than AO. In addition, there is a noteworthy difference be-tween SMO and AO schemes. Since SMO converges in finite-time, this implies that the separation principle is not required,whereas the AO converges asymptotically. This implies that a stability analysis is required in order to guarantee the stabilityof the closed-loop system. In this way, there exists a compromise between the controller and observer design, which can notbe designed independently. Usually, in these cases, the observer must be tuned such that its converges must be sufficientlyfast in order to the controller performs correctly, then it constitutes an important limitation.

5. Conclusions

In this work, an approach for synchronizing chaotic systems with different dimension has been presented. We have pre-sented the matching of a class of master–slave chaotic systems in canonical forms. In this sense, results have shown thatreduced-order synchronization can be attained by nonlinear systems. Master–slave synchronization is allowed by a slid-ing-mode-based state feedback, which yields a bounded coupling force. A simple algorithm based on an geometric control

−4

−2

0a)

SlidingAdaptive

0

10

20

30

b) SlidingAdaptive

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−50

0

50

100

time

c)SlidingAdaptive

Fig. 3. Comparison – approximated settling time tS of estimation errors: (a) ~e1, (b) ~e2, (c) ~e3.

−4

−2

0a)

SlidingAdaptive

0

5

10

b)

SlidingAdaptive

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−20

−10

0

10

time

c)

SlidingAdaptive

Fig. 4. Comparison – approximated overshoot peaks of estimation errors: (a) ~e1, (b) ~e2, (c) ~e3.

3230 A. Rodríguez et al. / Chaos, Solitons and Fractals 42 (2009) 3219–3233

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strategy for chaos suppression has been designed. However, this state feedback strategy requires the state information to beavailable and only the process output is measured. Under such circumstances, a state reconstruction has been analyzed. Theconvergence of the high-order sliding-mode observer for estimating the components of the state and unknown function hasbeen shown. Simulation results have been obtained to illustrate the performance of the proposed scheme and additionally, acomparative study has been done in order to show the advantages of this scheme.

Acknowledgments

Authors thank Jorge Dávila, for his helpful support. A. Rodrıguez thanks to Mexican CONACyT (Consejo Nacional de Cien-cia y Tecnología), for scholarship Grant 45665. The work of J. de León was supported by Mexican SEP-CONACyT under Grant

0 10 20 30 40 50−3

−2

−1

0

1

2

3

time

Control off Control on

x1,Sx1,M

Fig. 5. Synchronization of x1;S to x1;M .

0 10 20 30 40 50−6

−5

−4

−3

−2

−1

0

1

2

time

Control off Control on

x2,Sx2,M

Fig. 6. Synchronization of x2;S to x2;M .

A. Rodríguez et al. / Chaos, Solitons and Fractals 42 (2009) 3219–3233 3231

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59987. The work of L. Fridman was supported by the Mexican CONACyT under Grant 56819, and by PAPIIT (Programa deApoyo a Proyectos de Investigación e Inovación Tecnológica) UNAM under Grant 111208.

Appendix

Now, we introduced the following. Consider the following nonlinear system

R :_x ¼ f ðxÞ þ gðxÞu;y ¼ hðxÞ;

�ð34Þ

where x ¼ ½x1; x2; . . . ; xn�T 2 Rn is the state vector, u 2 R is an input, and y 2 R is an output variable of the system, f and g aresmooth vector fields.

A vector field is said to be complete if the solutions to the differential equation _x ¼ f ðxÞ may be defined for all t 2 R [18].

Definition 4 ((Global Relative Degree) [18]). The global relative degree r of (34) is defined as the integer such that

LgLif hðxÞ ¼ 0; 8x 2 Rn; 0 6 i 6 r � 2

LgLr�1f hðxÞ – 0; 8x 2 Rn:

If

LgLif hðxÞ ¼ 0; 8x 2 Rn; 8i P 0

we say that r ¼ 1.

Now, we introduce the following definition.

Definition 5 ((Minimum Phase) [18]). Systems (5)–(7), with rS 6 nS are said to be minimum phase if the origin �e? ¼ 0 is anasymptotically stable equilibrium point for the zero dynamics. A system which is not minimum phase is said to be non-minimum phase.

References

[1] Andrievskii B, Fradkov A. Control of chaos: methods and applications. I. Methods. Autom Remote Control 2003;64(5):673–713.[2] Andrievskii B, Fradkov A. Control of chaos: methods and applications. II. Applications. Autom Remote Control 2004;65(4):505–33.[3] Blekhman I. Synchronization in science and technology. New York: ASME; 1988.[4] Blekhman I, Landa S, Rosenblum M. Synchronization and chaotization in interacting dynamical systems. ASME Appl Mech Rev 1995;48:733–52.[5] Boccaletti S, Kurths J, Osipov G, Valladares D, Zhou C. The synchronization of chaotic systems. Phys Rep 2002;366:1–101.[6] Bowong S. Stability analysis for the synchronization of chaotic systems with different order: application to secure communications. Phys Lett A

2004;326:102–13.[7] Brunt M. Coordination of redundant systems. Ph.D., Technical University of Delft, Delft, The Netherlands; 1998.

0 10 20 30 40 50−1000

−800

−600

−400

−200

0

200

400

600

800Control off Control onu

S(t)

time

Fig. 7. Control signal uS .

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[8] Femat R, Ramírez JA, Fernández-Anaya G. Adaptive synchronization of high-order chaotic systems: a feedback with low order parametrization. PhysicaD 2000;130:231–46.

[9] Femat R, Solís-Perales G. On the chaos synchronization phenomena. Phys Lett A 1999;262:50–60.[10] Femat R, Solís-Perales G. Synchronization of chaotic systems with different order. Phys Rev E 2002;65:036226.[11] Filippov A. Differential equations with discontinuous right-hand side. Dordrecht, The Netherlands: Kluwer; 1988.[12] Fridman L, Levant A, Dávila J. Observation of linear systems with unknown inputs via high-order sliding-modes. Int J Syst Sci 2007;38(10):773–91.[13] Fridman L, Shtessel Y, Edwards C, Yan X. Higher-order sliding-mode observer for state estimation and input reconstruction in nonlinear systems. Int J

Robust Nonlinear Control 2008;18(4–5):399–413.[14] Hautus M. Strong detectability and observers. Linear Algebra Appl 1983;50:353–68.[15] Huygens C. Horoloquim oscilatorium, Paris, France; 1673.[16] Kocarev L, Partlitz U. Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems. Phys Rev Lett

1996;76(11):1816–9.[17] Levant A. High-order sliding modes: differentiation and output feedback control. Int J Control 2003;76(9/10):924–41.[18] Marino R, Tomei P. Nonlinear control design: geometric, adaptive, and robust. Englewood Cliffs, NJ: Prentice Hall; 1995.[19] Neiman A, Pei X, Russell D, Wojtenek W, Wilkens L, Moss F, et al. Synchronization of noisy electrosensitive cells in the paddlefish. Phys Rev Lett

1999;82:660–3.[20] Pecora L, Carroll T. Synchronization in chaotic systems. Phys Rev Lett 1990;64(8):821–4.[21] Pikovsky A, Rosenblum M, Kurths J. Phase synchronization in regular and chaotic systems. Int J Bifurcat Chaos 2000;10(10):2291–305.[22] Pikovsky A, Rosenblum M, Kurths J. Synchronization: a universal concept in nonlinear sciences. Cambridge, UK: Cambridge University Press; 2001.[23] Pogromsky A, Santoboni G, Nijmeijer H. Partial synchronization: from symmetry towards stability. Physica D 2002;172:65–87.[24] Rodríguez-Angeles A, Nijmeijer H. Coordination of two robot manipulators based on position measurements only. Int J Control 2001;74(13):1311–23.[25] Rulkov N, Sushchik M, Tsimring L, Abarbanel H. Generalized synchronization of chaos in directionally coupled chaotic system. Phys Lett E

1995;51:980–94.[26] Sarasola C, Torrealdea F, d’Anjou A, Moujahid A, Grana M. Nonzero error synchronization of chaotic systems via dynamic coupling. Physica D

2003;177:39–49.[27] Schafer C, Rosenblum M, Kurths J, Abel H. Heartbeat synchronized with ventilation. Nature 1998;392:239.[28] Solís-Perales G, Bowong S, Femat R. Synchronization of dynamical systems with different order and topology. In: First IFAC conference on analysis and

control of chaotic systems, Reims, France. International Federation of Automatic Control; 2006. p. 175–80.[29] Sukow D, Blackburn K, Spain A, Babcock K, Bennett J, Gavrielides A. Experimental synchronization of chaos in diode lasers with polarization-rotated

feedback and injection. Opt Lett 2004;29(20):2393–5.[30] Yang S, Duan C. Generalized synchronization in chaotic systems. Chaos, Solitons and Fractals 1998;9(10):1703–7.

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