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Chaos, Solitons and Fractals 36 (2008) 1141–1156 www.elsevier.com/locate/chaos

A new model-free sliding observer to synchronization problem Rafael Martı´nez-Guerra *, Wen Yu, Enrique Cisneros-Saldan˜a Departamento de Control Automatico, CINVESTAV-IPN, A.P. 14-740, Av.IPN 2508, Me´xico, DF 07360, Mexico Accepted 20 July 2006

Abstract In this paper, a new observer is proposed for the synchronization problem, this new observer presents a simple structure that contains a sliding mode term which turns out to be robust against output noises as well as sustained disturbances, the slave system is a pure sliding-mode observer. As far as we know in the literature this class of observers have not been used in the synchronization problem. Comparisons with other two model-based observers, Thau observer and Bestle-Zeitz observer, are proposed. The performances of these observers are shown by using Lorenz system and Chua’s circuit. Ó 2006 Elsevier Ltd. All rights reserved.

1. Introduction In the last years, synchronization of chaotic systems problem has received a great deal of attention among scientist in many fields [1–3]. It is well known that the study of the synchronization problem for nonlinear systems has been very important for nonlinear science, in particular the applications to biology, medicine, cryptography, secure data transmission and so on. In general, the synchronization research has been focused onto two areas. The first one relates with the employ of state observers, where the main applications lies on the synchronization of nonlinear oscillators. The second one is the use of control laws, which allows to achieve the synchronization with different structure and order between nonlinear oscillators [4]. A particular interest is the connection between the observers for nonlinear systems and chaos synchronization, which is also known as master–slave configuration. Thus, chaos synchronization problem can be regarded as observer design procedure, where the coupling noise is viewed as output and the slave system is the observer [5]. The main approaches, which are related to the construction of asymptotic observers for nonlinear processes, use geometric differential methods. The idea is to find a state transformation to represent the system as a linear equation plus a nonlinear term, which is function of the system output. However, finding a nonlinear transformation that places a system of order n into the so-called observer canonical form requires the integration of n coupled partial differential equations. Furthermore, this approach needs accurate knowledge of the nonlinear dynamics of the system. Here, in this paper, we will present two observers based on geometric techniques: Thau observer [6] and Bestle-Zeitz observer [7], as well as a new observer is proposed for the synchronization problem, this new observer presents a simple structure that contains a sliding mode term which turns out to be robust against output noises as well as sustained disturbances, the *

Corresponding author. E-mail addresses: [email protected] (R. Martı´nez-Guerra), [email protected] (W. Yu), [email protected] (E. Cisneros-Saldan˜a). 0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.07.039

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slave system is a pure sliding-mode observer. This observer does not require an accurate model of the system since its structure just contains a proportional correction of the sign function of the measurement of the synchronization error. As far as we know from the literature, that class of observers have not been used in the synchronization problem. The main advantage of the observer is that the sliding contribution of measurement error provides robustness against perturbations to the system and noisy measurements with a high-gain like condition. The early works dealing with sliding mode observers which consider measurement noise were proposed by Utkin and Drakunov [8]. They discussed the state estimation using sliding mode technique. Anulova [9] treated an analysis of systems with sliding mode in presence of noises. Slotine et al. [10], successfully designed, so named, sliding-mode approach to construct observers which are highly robust with respect to noises in the input of the system. But, it turns out that the corresponding stability analysis can not be directly applied in the situations with the output noise (or, mixed uncertainty) presence. So, it is still a challenge to suggest a workable technique to analyze the stability of identification error generated by sliding-mode (discontinuous non linearity) type observers [1,11–14]. In this paper we propose a new model-free observer: sliding mode observer for the synchronization problem, and compare our observer with the other two model-based observers. The intention of choosing two examples as the Lorenz system and Chua’s circuit is to clarify the proposed methodology. However, it is worth to mention that this technique can be applied to many chaotic synchronization problems. The remainder of this paper is organized as follows: in Section 2 we give some definitions in a differential geometric setting. In Section 3 we introduce the new sliding mode observer structure to tackle the synchronization problem and compare it with the other two observers. Section 4 shows some numerical results. Finally, in Section 5 we will close the paper with some concluding remarks.

2. Canonical form of nonlinear system Consider the vector-valued functions f : Rn ! Rn and g : Rn ! Rn , where f, g are vector fields in C1. The Lie bracket is defined by ½f ; g,

of og g f ox ox

where of and og are the Jacobian matrices of f and g respectively. Using an alternative notation, it is possible to represent ox ox the Lie bracket as ½f ; g ¼ ðad 1 f ; gÞ It is also defined ðad k f ; gÞ ¼ ½f ; ðad k1 f ; gÞ;

16k6n

where by definition ðad 0 f ; gÞ ¼ g Let now consider a C1 function h : Rn ! R. Let hÆ, Æi denote the standard dot product on Rn . Let dh denote the gradient of h (dh = $Th) with respect to x. Then the Lie derivative of h with respect to f is defined by Lf h ¼ Lf ðhÞ ¼ hdh; f i ¼ rT h  f The following notation will be employed: L0f h ¼ h L1f h ¼ Lf h .. . Lkf h ¼ Lf ðLk1 f hÞ The Lie derivative of dh with respect to the vector field f is defined by !T oðdhÞT of f þ ðdhÞ Lf ðdhÞ ¼ ox ox One may easily verify that these Lie derivatives obey the following so-called Leibnitz formula:

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L½f ;g h ¼ hdh; ½f ; gi ¼ Lg Lf h  Lf Lg h Furthermore, the following relation is valid: dLf h ¼ Lf ðdhÞ We consider a nonlinear system which is described as follows: ( n_ ¼ f ðnÞ þ gðnÞu y ¼ hðnÞ

ð1Þ

where n 2 Rn is the state of the plant, y 2 R is a measurable output. f, g and h are smooth functions. If the system has uniform relative degree n, i.e. Lg hðnÞ ¼    ¼ Lg Ln2 f hðnÞ ¼ 0;

Lg Ln1 f hðnÞ 6¼ 0

So there exists a mapping g ¼ T ðnÞ

ð2Þ

which can transform the system (1) into the following canonical form: g_ i ¼ giþ1 ; i ¼ 1; . . . ; n  1 g_ n ¼ Uðg; uÞ

ð3Þ

y ¼ g1 where U(g, u) is a continuous nonlinear function. Here, we consider the output y = g1 + d, with d an additive bounded noise.

3. Sliding mode observer to synchronization problem Synchronization of chaotic systems can be classified into two types called mutual synchronization and master–slave synchronization according to coupling configuration. The former is a system with bidirectional coupling, and the latter is that with unidirectional coupling. In this paper we consider master–slave configuration. It is also known as driveresponse. First consider a system described as follows:  x_ ¼ f ðxÞ ð4Þ R: y ¼ hðxÞ where x 2 Rn and y 2 R. Then consider any kind of observer for the system (4) with state ^x. System (4) will be the master and the observer will be the slave. Definition 1. The slave system synchronizes with the master system if jxðtÞ  ^xðtÞj ! 0 t!1

for almost all (with respect to Lebesgue measure) combinations of initial states of the master and slave systems. In this paper, we propose a new model-free observer, sliding mode observer, to design the slave system. It has the following form: 8 i _ > < ^gi ¼ ^giþ1 þ mi s signðy  ^y Þ; i ¼ 1; . . . ; n  1 _^gn ¼ mn sn signðy  ^y Þ ð5Þ > : ^y ¼ g^1 where 1 > s > 0, the constants mi are chosen such that the polynomial mncn + mn1cn1 +    + m1 = 0 has all its roots in the left half-plane (LHP). The function signðy  ^y Þ is defined as follows: 8 if ðy  ^y Þ > 0 > : 0 if ðy  ^y Þ ¼ 0

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First we consider a simple case, two-dimension. The slave is g^_ 1 ¼ ^g2 þ ms1 signðy  ^y Þ; ^g_ 2 ¼ m2 s2 signðy  ^y Þ

m>0

ð6Þ

where ^g1 , g^2 are the states in the slave system and ^y the estimate of the output y, m1 = m, m2 = m2 and s are small positive parameters. Since the output of the master system is y = g1 + d. Let us define the synchronization error as e1 ¼ g1  ^g1 1 e2 ¼ ðg2  ^ g2 Þ m

ð7Þ

The recovered noise at slave is ^d ¼ y  ^y ¼ e1 þ d From (6) and (7) the synchronization dynamical error can be formed as e_ ¼ Al e  K signðCe þ dÞ þ Df with  Al ¼

 l m ; 0 l

K ¼ ms1



 1 ; ms1

le1

Df ¼

l > 0;

U m

þ le2

! ;

C ¼ ð1 0Þ

ð8Þ

l is a regularizing parameter, Df is an uncertainty term (or unmodeled dynamic term). The following assumptions are used for our theoretical result: A1. There exist nonnegative constants L0f, L1f such that for any e the following generalized quasi-Lipschitz conditions holds: kDf k 6 L0f þ ðL1f þ kAl kÞkek

ð9Þ kdk2K

T

þ

¼ d Kd 6 d < 1 where K is a symmetric definite positive A2. Information noise is assumed to be bounded as matrix. A3. There exist a positive definite matrix Q0 ¼ QT0 > 0 such that the following matrix Riccati equations: PAl þ ATl P þ PRP þ Q ¼ 0

ð10Þ

where R ¼ K1 f þ 2kKf kL1f I;

0 < Kf ¼ KTf

ð11Þ

Q ¼ Q0 þ 2ðL1f þ kAl kÞ2 I has a positive definite solution P = PT > 0. Since P > 0, there exists k > 0 such that K = kP1CT.

Remark 1. Notice that the dynamic U(g1, g2) in (3) is Lipschitz respect to g1 and g2, so the assumption A1 is satisfied for chaotic systems. The measurement is corrupted by a noise d which is bounded, it is assumption A2. To calculate the solution to the Riccati equation (10), the following parameters have been selected: Kf ¼ kf I;

L1f ¼ l;

R ¼ ðk1 f þ 2kf lÞI;

Q0 ¼ q0 I;

Q ¼ ðq0 þ 8l2 ÞI

with kf = 20, l = 0.0001, q0 = l2, we obtain   3:16099 0:22096 P ¼ 103 >0 0:22096 3:16099 which satisfies assumption A3. The main result is the following: Theorem 1. The sliding-mode observer (6) can realize synchronization and converges to the following residual set:

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ðkÞg De ¼ fejkekP 6 l

1145

ð12Þ

where P is a solution of the Riccati equation (10) 0 12 qðkÞ B C ðkÞ ¼ @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l A 2 ðkap Þ þ qðkÞaQ þ kap

ð13Þ

where qðkÞ ¼ 2kKf kL20f þ 4k

qffiffiffiffiffiffiffiffiffiffiffi dþ nK1 f

kap ¼ kðkmin ðP 1=2 C T CP 1=2 ÞÞ aQ ¼ kmin ðP 1=2 QT QP 1=2 Þ > 0 where n is the dimension of the chaotic system. Proof. We select the Lyapunov function candidate V(e) as V ðeÞ ¼ kek2P ¼ eT Pe;

0 < P ¼ PT

ð14Þ

and using the matrix inequality X T Y þ Y T X 6 X T Kf X þ Y T K1 f Y nxm

valid for any X ; Y 2 R

, 0 < Kf ¼

ð15Þ

KTf ,

it follows:

V_ ðeÞ ¼ 2eT P e_ ¼ 2eT P ðAl e  K signðCe þ dÞ þ Df Þ 6 2eT PAl e  2keT C T signðCe þ dÞ þ 2eT P Df 6 eT ðPAl þ ATl P Þe  2keT C T signðCe þ dÞ þ eT P K1 Pe þ DT f Kf Df 6 eT ðPAl þ ATl P þ PRP þ QÞe  eT Qe þ ðL20f þ ðL1f þ kAl kÞ2 kek2 Þ2kKf k  2keT C T signðCe þ dÞ ¼ eT ðPAl þ ATl P þ PRP þ QÞe  eT Qe þ 2kKf kL20f  2kðCeÞT signðCe þ dÞ

ð16Þ

by using xT sign½x þ z P

n X

pffiffiffi jxi j  2 nkzi k

i¼1

Then V_ ðeÞ 6 e Qe þ T

2kKf kL20f

 2k

n X

! n X pffiffiffi jðCeÞi j  2 nkdk 6 eT Qe  2k jðCeÞi j þ qðkÞ

i¼1

where qðkÞ ¼ 2kKf kL20f þ 4k

i¼1

pffiffiffiffiffiffiffiffiffiffiffi nK1 dþ

Thus V_ ðeÞ 6 kekQ  2kaP kekP þ qðkÞ

ð17Þ

where n X

!2 jðCeÞi j

P

Xn i¼1

ðjðCeÞi jÞ2 ¼ kCek2 ¼ kCP 1=2 P 1=2 ek2 P ap eT Qe

ð18Þ

i¼1

with aP ¼ kmin ðP 1=2 C T CP 1=2 Þ P 0 So that, from (17) we obtain pffiffiffiffiffiffiffiffiffiffi V_ ðeÞ ¼ aQ V ðeÞ  # V ðeÞ þ b

ð19Þ

ð20Þ

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where aQ ¼ kmin ðP 1=2 QT QP 1=2 Þ > 0 # ¼ 2kap ; b ¼ qðkÞ

ð21Þ

If the assumptions A1–A3 are satisfied then   l  !0 1 V þ

ð22Þ

where the function [Æ]+ is defined as  z if z P 0 ½zþ ¼ 0 if z < 0

ð23Þ

The proof of this result is given in appendix. The theorem actually states that the weighted estimation error V(e) = eTPe ðkÞ asymptotically, that is, it is ultimately bounded. converges to the zone l T ðkÞ P e Pe P eT1 Pe1 l  Remark 2. Because 0

12

B C qffiffiffiffiffiffiffiffiffiffiffi B C 2kKf kL20f þ 1 B C d þ 4 nK B C f k B C ðkÞ ¼ Bvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l pffiffiffiffiffiffiffiffi 3 ffi 2 C u Bu C 4 nK1 dþ Bu 2 C f 2kKf kL20f 5aQ þ ap A @tap þ 4 2 þ k k

ð24Þ

Since P is bounded, we canselect sarbitrary small (the gain of the slave (8) becomes bigger) in order to make k very big 1 ), so the first term of (24) goes to zero. K1 (since K ¼ kP 1 C T ¼ ms1 f in A3 is any positive matrix, we can ms1 choose it any small such that the second term of (24) goes to zero. Remark 3. Although we have restricted ourselves to the case of second-order chaotic system, the observer construction and convergence analysis can be extended to n-dimensional case. The master system is g_ j ¼ gjþ1 ;

j ¼ 1; . . . ; n  1

g_ n ¼ H ðgÞ y ¼ g1 the sliding-mode observer-based slave is constructed as g^_ j ¼ ^gjþ1 þ mj sj signðy  ^y Þ; ^g_ n ¼ mn sn signðy  ^y Þ

j ¼ 1; . . . ; n  1

ð25Þ

where the constants mj are chosen such that the polynomial mnln + mn1ln1 +    + m1 = 0 has all its roots in the open left-hand side of the complex plane. As the second-order case, it can be proved that the synchronization error converges to any accuracy by selecting sufficiently small values of observer’s gain.

4. Model-based observers to synchronization problem Now we compare our model-free observer with the other model-based observers for the synchronization problem. The role of Lie derivatives in the nonlinear state observation problem will be considered. 4.1. Bestle-Zeitz observer [7] for the synchronization problem We consider the class of time-invariant nonlinear systems described by (4). It will be assumed that f is a C1 vector field in Rn and h is a C1 function. It is desirable to find a one-to-one onto C1 nonlinear transformation T : Rn ! Rn , where

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x ¼ T ðzÞ

ð26Þ

such that system (4) may be transformed into the canonical form defined below 8 2 3 2  3 f0 ðzÞ > 0  0 > > > 61 7 6 f  ðzÞ 7 > > 7 6 1 7  > < z_ ¼ 6 6 7 6 7 z  .. 7 .. 6 6 .. 7,f ðzÞ 4 . . 5 4 5 . > > >  > 1 0 > fn1 ðzÞ > > : y ¼ ½ 0    0 1 z

ð27Þ

Taking the derivative of (26) with respect to time yields x_ ¼ where

oT  f ðzÞ oz

ð28Þ

  oT oT oT ¼  oz oz1 ozn

Making some algebraic manipulations and using the Lie derivatives notation it is possible to obtain the following:   oT oT k1 ðk ¼ 1; . . . ; nÞ ¼ ad f ; ozk oz1 Thus it is now possible to express all the columns of oT in terms of a single starting vector oz h    i oT 1 n1 oT oT oT ad    ad f ; f ; ¼ ad 0 f ; oz oz oz 1 1 1 oz

oT oz1

as follows [7]:

Now the following equation must be employed to obtain an expression for the starting vector y ¼ hðxÞ ¼ zn

ð29Þ oT . oz1

ð30Þ

Taking the partial derivative of (30) with respect to z yields ohðxÞ oT ¼ ½0 ox oz



0 1

Note that the first component of (31) may be written as

 oh oT oT oT ¼ dh; ¼0 ¼ L0f ðdhÞ ox oz1 oz1 oz1

ð31Þ

ð32Þ

Similarly, Leibnitz’s formula may be used to simplify the second element of (31). And by repeated application of Leibnitz’s formula and use of (29), the following matrix involving the starting vector may be obtained 2 0 3 2 3 Lf ðdhÞðxÞ 0 6 1 7 6 .. 7 6 Lf ðdhÞðxÞ 7 oT 6.7 6 7 ¼6 7 6 7 7 .. 6 7 oz1 6 405 . 4 5 1 Ln1 ðdhÞðxÞ f

The matrix 2

L0f ðdhÞðxÞ

3

7 6 1 6 Lf ðdhÞðxÞ 7 7 6 OðxÞ ¼ 6 7 .. 7 6 . 5 4 n1 Lf ðdhÞðxÞ oT is so-called the observability matrix of the system defined by (4). Thus the starting vector oz is equal to the last column 1 of O1 . The observer in the new coordinate system is given by

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8 2 2  3 3 f0 ðzÞ > 0  0 > > > 61 6 f  ðzÞ 7 7 > > 6 1 7 7 > < ^z_ ¼ 6 6 7 7 .. .. 7^z  6 6 6 .. 7  Kð^y  yÞ 4 . . 4 5 . 5 > > >  > 1 0 > fn1 ðzÞ > > : ^y ¼ ½ 0    0 1 ^z where ^y and K ¼ ½ k 0



ð33Þ

k n1 T . Define the error as

ez ¼ ^z  z Hence the error obeys the homogeneous differential equation 3 2 0    0 k 0 61 k 1 7 7 6 7ez e_ z ¼ 6 .. .. 7 6 5 4 . . 1

ð34Þ

0 k n1

The characteristic polynomial of (34) is given by pðsÞ ¼ k 0 þ k 1 s þ    þ k n1 sn1 þ sn

ð35Þ

Thus we may easily assign the spectrum of (35) via an appropriate selection of K. 4.2. Thau observer [6] for the synchronization problem Consider the nonlinear system described by  x_ ¼ Ax þ f ðxÞ þ Bu y ¼ Cx

ð36Þ

where f ðÞ : Rn ! Rn is continuous; A 2 Rnn , B 2 Rnm and C 2 Rpn . The nonlinear function f(Æ) may contain linear terms in x. It is assumed that the pair (A, C) is completely observable. Therefore it is possible to find K 2 Rnp such that the eigenvalues of A0 = A  KC are in open LHP. Let ^x, the estimate of the true state. Then ^x satisfies the equation ( ^x_ ¼ A0^x þ f ð^xÞ þ Ky þ Bu ð37Þ ^y ¼ C^x Let e denotes e ¼ ^x  x Thus e satisfies the differential equation e_ ¼ A0 e þ f ð^x  f ðxÞ ¼ A0 e þ f ðx þ eÞ  f ðxÞ

ð38Þ

Since the spectrum of A0 is contained in the LHP, for any given positive-defined Q 2 Rnn there exists a unique positivedefined P 2 Rnn such that AT0 P þ PA ¼ 2Q Next consider the following Lyapunov-function candidate V ðeÞ ¼ eT Pe The derivative of V(e) evaluated along the solution of the error differential equation (38) is given by V_ ðeÞ ¼ e_ T Pe þ eT ¼ 2eT Qe þ 2eT P ½f ðx þ eÞ  f ðxÞ It is necessary to impose the additional constraint that the function f(Æ) is locally Lipschitz about the origin; that is, there exists a positive constant L such that kf ðx1 Þ  f ðx2 Þk 6 Lkx1  x2 k for all x1, x2 in some open region R containing the origin. Therefore if e is contained in R then the following inequalities are valid:

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V_ ðeÞ 6 2eT Qe þ 2LkPekkek 6 ð2a þ 2LkP kÞkek where a is the minimum eigenvalue of Q and kPk is the maximum eigenvalue of P. Hence if a >L kP k

ð39Þ

then e = 0 is an asymptotically stable equilibrium point of (38). Remark 4. Model-based observers (Thau observer and Bestle-Zeitz observer) require complete information of the master system. Bestle-Zeitz observer needs a complex transformation, the synchronization error can converge to zero. The gain of Thau observer is not difficult to be obtained, the synchronization error converges to a bounded zone. The model-free observer proposed in this paper does not require any information of the master system, it is robust to the bounded noise, but there exists chattering compared with the model-based observers. We will show these results in the following examples.

5. Two synchronization problems 5.1. Lorenz system The Lorenz system is a nonlinear system with the following dynamics: 8 x_ 1 ¼ rðx2  x1 Þ > > > < x_ ¼ qx þ x  x x 2 1 2 1 3 RL : > _ x ¼ x x  bx 3 1 2 3 > > : y ¼ x1

ð40Þ

and it is well known that with r = 10, q = 28 and b ¼ 83, Lorenz system exhibits chaos. 5.1.1. Sliding mode observer (SMO) to synchronization of Lorenz system The SMO can not be applied directly to Lorenz system, thus applying the transformation (2), Lorenz system becomes in the following canonical form: 8 z_ 1 ¼ z2 > > > < z_ 2 ¼ z3 RLC : > z_ 3 ¼ f ðzÞ > > : y ¼ z1 1 and f1 ðzÞ ¼ z2 þz ; f 2 ðzÞ ¼ where f(z) = r{q(z2  z1) + f1(z) + z1f2(z)  z2f2(z)  z1[z1f1(z)  bf2(z)]  z3} r z2 ðrþ1Þþz3 q1 . z1 Now the SMO can be applied to Lorenz system and its dynamics can be described as follows: 8_ ^z1 ¼ ^z2 þ k 1 signðy  ^y Þ > > > < ^z_ ¼ ^z þ k signðy  ^y Þ 2 3 2 ð41Þ > ^z_ 3 ¼ k 3 signðy  ^y Þ > > : ^y ¼ z^1

We choose k1 = 100, k2 = 70, k3 = 100, the synchronization results are shown in Figs. 1 and 2. The synchronization states in Fig. 2 shows that the sliding mode observer can be successfully applied for the synchronization problem, although there exists a little chattering in synchronization error (Fig. 1). 5.1.2. Bestle-Zeitz observer to synchronization of Lorenz system By applying the above described method, we have obtained the Jacobian of the desired transformation for the Lorenz system: 2 3 0 0 1 7 1 1 oT 6  r1  x1 rxb1 þ x2xx 0 2 7 r ¼6 1 4 5 oz b x2 x1 1  rx1 rx1 þ x2 mðxÞ 1

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1150

50

Synchronization status 40 30 20 10 0 Time -1 0

0

5

10

15

Fig. 1. Synchronization error of Lorenz system via SMO.

30

Synchronization status z2 , zˆ2

20 10 0 -10 -20 -30

Time 0

5

10

15

Fig. 2. Synchronization states of Lorenz system via SMO.

where mðxÞ ¼

    x1 b x2  x1 qx1  x2  x1 x3 b x2  x1 2 þ þ   2  x b þ rðx2  x1 Þ 1 rx1 rx21 r x21 x21 x31

Now we only need to integrate considering T(0) = 0 to obtain the desired transformation, then, applying this transformation to the coordinate system yields the observer canonical form of the Lorenz system: 3 f0 ðzÞ 7 7 6 6 z_ ¼ 4 1 0 0 5z  4 f1 ðzÞ 5 > f2 ðzÞ 0 1 0 > > : y ¼ ½ 0 0 1 z 8 > > >
> > < ^x_ ¼ 6 q 1 0 7^x þ 6 ^x ^x 7  Kð^y  yÞ 4 5 4 1 35 > ^x1^x2 0 0 b > > : ^y ¼ ½ 1 0 0 ^x We select K ¼ ½ 13 13 13 T , the synchronization results are shown in Fig. 4. The synchronization error in Fig. 4 is bigger than the Bestle-Zeitz observer in Fig. 3. Although it is from a model-based observer, Thau observer needs less information than Bestle-Zeitz observer. 5.2. Chua’s circuit Some chaotic systems do not have the normal form as (3), we cannot apply sliding-mode observer directly on the slave, for example Chua’s circuit 8 _ C 1 n1 ¼ Gðn2  n1 Þ  gðn1 Þ þ u > > > < C n_ ¼ Gðn  n Þ þ n 2 2 1 2 3 ð42Þ _ > L n ¼ n > 3 2 > : y ¼ n3 where gðn1 Þ ¼ m0 g1 þ 12 ðm1  m0 Þ½jn1 þ Bp j þ jn1  Bp j, n1, n2, n3 denote the voltages across C1, C2 and L. It is known that with C 1 ¼ 19, C2 = 1, L ¼ 17, G = 0.7, m0 = 0.5, m1 = 1.5, Bp = 1 the circuit displays double scroll. If we make the transformation g = T(n) as g1 ¼ n3 g2 ¼ Ln2 LG L g3 ¼ ðn2  n1 Þ  n3 C2 C2

ð43Þ

the Chua’s circuit becomes the normal form g_ 1 ¼ g2 g_ 2 ¼ g3 g_ 3 ¼ f ðg1 ; g2 ; g3 Þ þ gu y ¼ g1

ð44Þ

2

1 1 g1  CG2 g3  C11 gðg1  GL g1  CG2 g3 Þ, g ¼ C1GC2 . Now sliding-mode where f ðg1 ; g2 ; g3 Þ ¼  CG2 g3  C12 L g1  CG1 C2 ½2g2  GL observer-based slave (25) can be applied. The Chua’s circuit (42) can be transformed in (44) from (43). So we design the sliding-mode slave as (41).

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1.2

Synchronization error

1 0.8 0.6 0.4 0.2 0 -0.2

Time 0

5

10

15

Fig. 5. Synchronization error of Chua’s circuit via SMO.

1 Synchronization error

0.5

0

-0.5

Time 0

5

10

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Fig. 6. Synchronization error of Chua’s circuit via Bestle-Zeitz observer.

1.5 Synchronization error 1 0.5 0 -0.5 -1 Time -1.5

0

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Fig. 7. Synchronization error of Chua’s circuit via Thau observer.

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Synchronization status η2 ,ηˆ 2

2 1 0 -1 -2 -3

Time 0

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Fig. 8. Synchronization states of Chua’s circuit via Thau observer.

We choose k1 = 300, k2 = 100, k3 = 2, the synchronization results of SMO are shown in Fig. 5. We can see that the synchronization error of Chua’s circuit is a little bigger than the Lorenz system. But it is satisfied for the synchronization. We choose K ¼ ½ 13 13 13 T , the synchronization results via Bestle-Zeitz observer are shown in Fig. 6. We choose K ¼ ½ 3 3 3 T , the synchronization results via Thau observer are shown in Figs. 7 and 8. We can see that in both systems the best performance is given by the model-based Bestle-Zeitz Observer, but, even if the model-based Thau observer also shows better performance than the model-free Sliding Mode Observer, i.e., both Bestle-Zeitz and Thau require complete information about the system, but Sliding Mode Observer does not.

6. Concluding remarks In this paper, we have proposed a new observer for the synchronization problem, this new observer presents a simple structure that contains a sliding mode term which turns out to be robust against output noises as well as sustained disturbances, the slave system is a pure sliding-mode observer. As far as we know in the literature this class of observers have not been used in the synchronization problem. The performance of the observer was shown using numerical simulation.

Appendix A Consider the Lyapunov function V(e) verifying the equality pffiffiffiffi V_ ¼ aV  # V þ b The equilibrium point V* of this equation, satisfying pffiffiffiffiffiffi aV   # V  þ b ¼ 0 is as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðb=aÞ2 V ¼ ð#=2aÞ2 þ b=a  #=2a ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð#=2aÞ2 þ b=a þ #=2a

Proof. Defining D :¼ (V  V*)2, we derive

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h i pffiffiffiffi D_ ¼ 2ðV  V  ÞV_ 6 2ðV  V  Þ aV  # V þ b h pffiffiffiffi pffiffiffiffiffiffi i h i pffiffiffiffiffiffi pffiffiffiffi ¼ 2ðV  V  Þ aV  # V þ b þ ðaV  þ # V   bÞ 2ðV  V  Þ aðV  V  Þ  # V  V  pffiffiffiffi pffiffiffiffiffiffi pffiffiffiffi pffiffiffiffiffiffi 2 ¼ 2aðV  V  Þ2  2# V þ V  V  V