Observer-based chaotic synchronization in the presence of unknown inputs

Observer-based chaotic synchronization in the presenceof unknown inputs

Moez Feki *, Bruno Robert

Universit�ee de Reims Champagne Ardenne-UFR Sciences Exactes, Moulin de la Housse, BP 1039, 51687 Reims cedex 2, France

Accepted 20 May 2002

Abstract

This paper deals with the problem of synchronization of chaotic dynamical systems. We consider a drive-response

type of synchronization via a scalar transmitted signal. Unlike most works we consider the presence of some unknown

inputs in the drive system and that no knowledge about their nature is available. A reduced-order observer-based

response system is designed to synchronize with the missing states. We show that under some assumptions the syn-

chronization is exponentially achieved. The efficiency of our method is confirmed by numerical simulations of two well-

known chaotic systems: Chua�s circuit and Lur�e system.� 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction

Over the past decades, synchronization of chaotic systems has been an intriguing concept and has received con-

siderable attention [1–3]. Indeed, the synchronization property of chaotic circuits has revealed potential applications to

secure communications [4–6]. In fact, since the seminal paper by Pecora and Carroll [1], the synchronization of chaotic

systems is based on the drive-response conception: a drive system drives via a scalar transmitted signal a custom de-

signed response system. Recently, the synchronization has been regarded as an observer design problem. In [7] Nijmeijer

presented different results on nonlinear observers design and how they can be adapted to chaotic synchronization. Liu

et al. [8] presented a global synchronization theorem for a class of chaotic systems with the observer feedback gain being

a function of free parameter. In [9,10] Morg€uul et al. and Liao showed results on local and global synchronization usingobserver design. In [7,8] authors assumed exact knowledge of their drive systems, whereas in [9,10] authors assumed

known bounds on parameter variations and disturbance.

In this paper we present a new result on chaotic synchronization. We assume that the drive system lies under the

effect of unknown inputs. We show that if the chaotic drive system has a special structure then without any premise on

the nature of the unknown inputs, a reduced-order response system (RORS) can be designed to estimate the unmea-

sured states. Our paper will be organized as follows: In Section 2 we present an observer-based full-order response

system (FORS) construction that can be applied to synchronize perturbation-free chaotic systems [11]. Section 3 gives

the design procedure of the RORS. In Section 4, the synchronization scheme is applied to two chaotic systems: modified

Chua�s circuit and Lur�e system, and simulation results are shown. Section 5 includes our concluding remarks.

*Corresponding author. Tel.: +33-3-2691-8579; fax: +33-3-2691-3106.

E-mail address: [email protected] (M. Feki).

0960-0779/03/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0960-0779 (02 )00164-9

Chaos, Solitons and Fractals 15 (2003) 831–840

www.elsevier.com/locate/chaos

mail to: [email protected]

2. Full-order response system design

We consider the chaotic drive system described by the following equations on Rn.

_xx ¼ Fxþ Guþ gðx; uÞ; u 2 Rm; ð1aÞ

y ¼ Hx; y 2 Rp; ð1bÞ

where x, u and y are respectively the state vector, the input and the output of the drive system. F, G and H are constant

matrices of appropriate dimensions and the nonlinear vector field g ¼ ðg1; g2; . . . ; gnÞT (T stands for the transpose) is

Lipschitz, with gð0; 0Þ ¼ 0.

We then consider the following response system

_̂xx̂xx ¼ F x̂xþ Guþ gðx̂x; uÞ þ fS�1HTðy � Hx̂xÞ; ð2aÞ

0 ¼ �hS � F TS � SF þ HTH ; ð2bÞ

where h is large enough and f P 1.

Remark 1. The matrix SðhÞ can be seen as the stationary solution of the differential equation

_SStðhÞ ¼ �hStðhÞ � F TStðhÞ � StðhÞF þ HTH ;

with initial condition S0 being positive definite. SðhÞ ¼ limt!1 StðhÞ, where StðhÞ 2 Sþ is the cone of symmetric positive

definite matrices.

Theorem 1. If the system defined by (1a) and (1b) satisfies the following hypotheses

(H1) there exists a positive constant k such that

kgðx; uÞ � gðx̂x; uÞk6 kkx� x̂xk;

8ðx; x̂xÞ 2 Rn�n and 8u 2 Rm

(H2) the pair ðH ; F Þ is observable(H3) we can choose h > 0 and fP 1 such that

k <kminðhS þ ð2f � 1ÞHTHÞ

2kmaxðSÞ; ð3Þ

kminð�Þ and kmaxð�Þ denote the smallest and largest eigenvalues of the matrix ð�Þ.Then the response system defined by (2a) and (2b) globally asymptotically synchronizes with the drive system described

by (1a) and (1b) with an exponentially decaying error

kxðtÞ � x̂xðtÞk6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikmaxðSÞkminðSÞ

sexpð�l0tÞkxð0Þ � x̂xð0Þk;

where l0 ¼ ðkminðhS þ ð2f � 1ÞHTHÞ=2kmaxðSÞÞ � k.

Proof. Let us express Eq. (2b) in the form of a Lyapunov equation�� F T � h

2I�S þ S

�� h2I � F

�¼ �HTH

and define Fh ¼ �ðh=2ÞI � F . Then Fh is Hurwitz stable if

h > �2minfReðkÞjk 2 specðF Þg; ð4Þ

where specðF Þ is the spectrum of F. With this condition satisfied and since (H2) implies that ðH ; FhÞ is an observablepair, it follows that SðhÞ is positive definite.Let us define e ¼ x� x̂x and consider the error dynamics

_ee ¼ ðF � fS�1HTHÞeþ gðx; uÞ � gðx̂x; uÞ

832 M. Feki, B. Robert / Chaos, Solitons and Fractals 15 (2003) 831–840

and the Lyapunov function V ðeÞ ¼ eTSe, then we have

_VV ðeÞ ¼ 2eTSFe� 2feTHTHeþ 2eTSðgðx; uÞ � gðx̂x; uÞÞ:

Using Eq. (2b)

_VV ðeÞ ¼ �heTSe� ð2f � 1ÞeTHTHeþ 2eTSðgðx; uÞ � gðx̂x; uÞÞ

6 � eTðhS þ ð2f � 1ÞHTHÞeþ 2kkmaxðSÞkek2

6 � ðkminðhS þ ð2f � 1ÞHTHÞ � 2kkmaxðSÞÞkek2

6 � kminðhS þ ð2f � 1ÞHTHÞ2kmaxðSÞ

�� k

�2V ðeÞ

hence if (H3) is satisfied then we can have an exponentially decaying bound on the Lyapunov function.

V ðtÞ6 V ð0Þ expð�2l0tÞ

where l0 ¼ ðkminðhS þ ð2f � 1ÞHTHÞ=2kmaxðSÞÞ � k. Using the following inequality

kminðSÞkek26 eTSe6 kmaxðSÞkek2;

we have

keðtÞk26 V ðtÞkminðSÞ

6V ð0Þ expð�2l0tÞ

kminðSÞ6

kmaxðSÞkminðSÞ

expð�2l0tÞkeð0Þk2;

equivalently

keðtÞk6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikmaxðSÞkminðSÞ

sexpð�l0tÞkeð0Þk:

Therefore, the response system states converge exponentially fast to the drive system states. �

The response system suggested here has a general form compared to the response systems (observers) presented in

[7,8] with F being any matrix satisfying the observability assumption ðH2Þ. We can also note that this response systemcan be applied to nonautonomous chaotic systems such as the Duffing equation and the Van der Pol oscillator.

3. Reduced-order response system design

The foregoing design procedure is based on the exact knowledge of the nonlinear system (1a) and (1b). However, in

practice the existence of unknown perturbing signals is inevitable, consequently the obtained response system cannot

always synchronize with the drive system. To overcome this drawback, a reduced-order observer-based response system

is derived to ensure asymptotic synchronization. For the sake of simplicity, we will consider an autonomous example,

nevertheless, the generalization to a nonautonomous system can be easily extended. Our idea is based on finding a state

variable transformation which when applied will separate the unknown input and the state variables to be estimated.

We consider the following dynamical system:

_xx ¼ Axþ f ðxÞ þ Cn; ð5aÞ

y ¼ Cx; ð5bÞ

where x 2 Rn is the state vector, y 2 Rp is the output, n 2 Rm is an unknown input and C 2 Rn�m is its known injection

map. A and C are constant matrices of appropriate dimensions and f is a Lipschitzian vector field with f ð0Þ ¼ 0. We

assume that (5a) and (5b) is a drive system exhibiting a chaotic behaviour. We also suppose that the following hy-

potheses are pertaining to (5a) and (5b)

(H4) p > m(H5) rank CC ¼ m

M. Feki, B. Robert / Chaos, Solitons and Fractals 15 (2003) 831–840 833

Assumption (H4) implies that the number of unknown inputs is less than the number of the output variables. (H5)

is a technical assumption for mathematical completeness.

We will further assume that C has a specific structure

C ¼ Ip 0½ � ¼ Im 0 0

0 Ip�m 0

� �:

Note that this is not a restricting condition, indeed, if C is a full rank matrix then there always exist a transformation

matrix that transforms C into the above form.

Thus we have

y ¼ y1y2

� �¼ Im 0 0

0 Ip�m 0

� � x1x2x3

24 35 ¼ x1x2

� �) x ¼

x1x2x3

24 35 ¼y1y2x3

24 35:Therefore, the drive system (5a) and (5b) can be expressed as follows:

_xx1_xx2_xx3

24 35 ¼_yy1_yy2_xx3

24 35 ¼A1A2A3

24 35xþ f1ðxÞf2ðxÞf3ðxÞ

24 35þC1

C2

C3

24 35n: ð6Þ

It is easy to verify using (H4) and (H5) that

rankC1

C2

� �¼ m:

Without loss of generality we assume that C1 is nonsingular. Thus the following transformation matrix is well defined:

U ¼I 0 0

�C2C�11 I 0

�C3C�11 0 I

24 35:We note that U is chosen to annihilate the last two terms of C, thus premultiplying (6) by U yields to:

_yy1_yy2 � C2C

�11 _yy1

_xx3 � C3C�11 _yy1

264375 ¼

A1A2 � C2C

�11 A1

A3 � C3C�11 A1

264375xþ f1ðxÞ

f2ðxÞ � C2C�11 f1ðxÞ

f3ðxÞ � C3C�11 f1ðxÞ

264375þ

C1

0

0

24 35n; ð7Þ

hence the unknown input enters only through the first row.

Let us define the following matrices and functions:eAAi,Ai � CiC�11 A1 i ¼ 2; 3;

~ffi,fi � CiC�11 f1 i ¼ 2; 3:

The last two equations of (7) become

_yy2 � C2C�11 _yy1 ¼ eAA2xþ ~ff2ðxÞ;

_xx3 � C3C�11 _yy1 ¼ eAA3xþ ~ff3ðxÞ:

If we further partition eAAi in the following manner:

eAAi, ½ eAAi1eAAi2

eAAi3 � i ¼ 2; 3;

we obtain

_xx3 � C3C�11 _yy1 ¼ eAA31y1 þ eAA32y2 þ eAA33x3 þ ~ff3ðy1; y2; x3Þ; ð8aÞ

_yy2 � C2C�11 _yy1 ¼ eAA21y1 þ eAA22y2 þ eAA23x3 þ ~ff2ðy1; y2; x3Þ: ð8bÞ

In the sequel we will consider ~ff2 ¼ 0. This is not a very restrictive condition since we will see in the next section that it is

satisfied by many well-known chaotic systems.


Whence, (8a) and (8b) can be expressed by the following equations

_xx3 ¼ eAA33x3 þ Buþ gðx3; uÞ; ð9aÞ

~yy ¼ eAA23x3; ð9bÞ

where:

B ¼ C3C�11

eAA31 eAA32

� �; u ¼ _yy1 y1 y2

� �T; gðx3; uÞ ¼ ~ff3ðy1; y2; x3Þ; and

~yy ¼ �C2C�11 _yy1 � eAA21y1 þ _yy2 � eAA22y2:

We have finally obtained a perturbation-free nonlinear dynamical system (9a) and (9b) in the form of (1a) and (1b)

studied in the first section. Since x1 and x2 are already available by direct measurement then it is sufficient to design aRORS that (estimates) synchronizes with the third state variable x3.

From the previous section, a response system that will synchronize exponentially fast with (9a) and (9b) is given

by:

_̂xx̂xx3 ¼ eAA33x̂x3 þ Buþ gðx̂x3; uÞ þ fS�1eAAT23ð~yy � eAA23x̂x3Þ; ð10aÞ

0 ¼ �hS � eAAT33S � SeAA33 þ eAAT23eAA23: ð10bÞ

Substituting u and ~yy in (10a), by their respective values yields to:

_̂xx̂xx3 ¼ ðeAA33 � fS�1eAAT23eAA23Þx̂x3 þ ðC3C�11 � fS�1eAAT23C2C

�11 Þ _yy1 þ ðeAA31 � fS�1eAAT23eAA21Þy1

þ ðfS�1eAAT23Þ _yy2 þ ðeAA32 � fS�1eAAT23eAA22Þy2 þ gðx̂x3; uÞ: ð11Þ

We notice that the estimation of x3 depends on time derivative of the output y namely _yy1 and _yy2 which are not directlymeasured. Hence, the response system needs to be modified.

Let us define a new state variable

v ¼ x̂x3 � ðC3C�11 � fS�1eAAT23C2C

�11 Þy1 � ðfS�1eAAT23Þy2: ð12Þ

Therefore, it comes out that in the new state space the response system (11) is expressed by

_vv ¼ Fv þHYþGðv; uÞ; ð13Þ

where:

F ¼ eAA33 � fS�1eAAT23eAA23;

H ¼ ðeAA31 � fS�1eAAT23eAA21Þ þ FðC3C�11 � fS�1eAAT23C2C

�11 Þ

ðeAA32 � fS�1eAAT23eAA22Þ þ FðfS�1eAAT23Þ

" #T;

Y ¼ y1y2

� �;

Gðv; uÞ ¼ gðv þ ðC3C�11 � fS�1eAAT23C2C

�11 Þy1 þ ðfS�1eAAT23Þy2; uÞ:

Finally, the estimation is given as a function of the new state variable v of the response system (13) and the output y of

the drive system (5a) and (5b).

x̂x3 ¼ v þ ðC3C�11 � fS�1eAAT23C2C

�11 Þy1 þ ðfS�1eAAT23Þy2: ð14Þ

Eventually, we can state the following theorem to summarize our result:

Theorem 2. Consider the chaotic drive system described by (5a) and (5b), satisfying hypotheses (H4), (H5) and ~ff2 ¼ 0.

We also assume that the reduced-order system (9a) and (9b) satisfies (H1), (H2) and (H3), then the RORS described by(13) with the output equation given by (14) globally synchronizes with (5a) and (5b).


4. Examples

In this section, we consider two well-known chaotic systems onto which we will apply and appraise the suggested

method namely: a modified Chua�s circuit and Lur�e system.

4.1. Modified Chua’s circuit

The modified Chua�s circuit shown in Fig. 1 is different from Chua�s circuit only in that a RC-parallel circuit is addedin series with the inductor. It has been shown in [12] that this circuit exhibits chaotic behaviour. Writing the state

equations of the circuit and using appropriate normalization of variables we obtain the following state model:

_xx1_xx2_xx3_xx4

26643775 ¼

�r1 �r2 0 0b 0 �b 0

0 1 �1 1

0 0 a �a

26643775

x1x2x3x4

26643775þ

00

0

aNðx4Þ

0BB@1CCAþ

c1c2c3c4

26643775n; ð15Þ

y ¼ 1 0 0 0

0 1 0 0

� � x1x2x3x4

26643775 ¼ x1

x2

� �; ð16Þ

where,

NðxÞ ¼ bxþ 0:5ða� bÞðjxþ 1j � jx� 1jÞ; ð17Þ

and n is an added perturbation. Obviously (15) is in the form of (5a) and (5b). Let us consider the following numerical

values:

r1 ¼ 6:8 r2 ¼ 0:045 b ¼ 18 a ¼ 10 a ¼ �1:34 b ¼ �0:74

c1 ¼ 0:03 c2 ¼ 0:02 c3 ¼ 0:01 c4 ¼ 0:2

Using the analysis of the previous section we have:

rank CC ¼ 1; C1 ¼ 0:03 is nonsingular; C2 ¼ 0:02; C3 ¼0:010:2

� �and

U ¼

1 0 0 0

� 23

1 0 0

� 13

0 1 0

� 203

0 0 1

2666437775:

Premultiplying (15) by U yields to:

Fig. 1. A modified Chua�s circuit.


_yy1_yy2 � 2

3_yy1

_xx3 � 13_yy1

_xx4 � 203_yy1

266664377775 ¼

�6:8 �0:045 0 0

22:53 0:03 �18 0

2:27 1:02 �1 1

45:33 0:3 10 �10

26643775xþ

0

0

0

10Nðx4Þ

26643775þ

0:030

0

0

26643775n:

It is obvious that ~ff2 ¼ 0 and that

gððx3; x4Þ; uÞ ¼0

�7:4x4 � 3ðjx4 þ 1j � jx4 � 1jÞ

� �is globally Lipschitz with k ¼ 13:4. Knowing that the pair ðeAA23; eAA33Þ is observable

eAA33 ¼ �1 1

10 �10

� � eAA23 ¼ �18 0½ �

then by choosing f ¼ 4 and h ¼ 22; 11 we obtain the following RORS:

_vv ¼ �89:88 1

�83:77 �10

� �v þ �202:1 439:8

�242:2 466:2

� �y1y2

� �þ Gðv; uÞ;

where Gðv; uÞ ¼ gððv1 þ 3:63y1 � 4:94y2; v2 þ 10:14y1 � 5:2y2Þ; uÞ.The simulation results are obtained using fourth order Runge–Kutta method of MATLAB 6. Fig. 2 depicts the

unknown perturbation signal. In Figs. 3 and 4 we show the convergence of the RORS, whereas in Fig. 5 we show that

the FORS described in Section 2 fails to synchronize. This failure, due to perturbation, occurs when the trajectory of the

drive system passes simultaneously in the vicinity of two different stable manifolds [13]. Hence two trajectories, which

are close can be attracted each to a different manifold.

4.2. Lur’e system

Lur�e system is a well-known chaotic system and it is described by the following state equations:

_xx1_xx2_xx3

24 35 ¼0 1 0

0 0 1

0 �1:25 �1

24 35 x1x2x3

24 35þ0

0

Nðx1Þ

0@ 1Aþ0:120:10:06

24 35n; ð18Þ

y ¼ 1 0 0

0 1 0

� � x1x2x3

24 35 ¼ x1x2

� �; ð19Þ

where n is an added perturbation signal and NðxÞ is a nonlinear function:

Fig. 2. Perturbation signal nðtÞ.


Fig. 3. Convergence of the RORS: x3ðtÞ and x̂x3ðtÞ.

Fig. 4. Convergence of the RORS: x4ðtÞ and x̂x4ðtÞ.


Fig. 5. Wrong estimation of the FORS.

Fig. 6. (Lur�e system) Correct synchronization of the RORS. Failure of the FORS.


NðxÞ ¼1:8x jxj6 1�3:6xþ 5:4signðxÞ 1 < jxj6 3�5:4signðxÞ 3 < jxj

8<:In this example x1 and x2 are the driving signals, and a RORS is to be designed to synchronize with x3. We can verify that

rank CC ¼ 1; C1 ¼ 0:12 is nonsingular; C2 ¼ 0:1; C3 ¼ 0:06

and

U ¼1 0 0

� 1012

1 0

� 12

0 1

264375:

Premultiplying (15) by U yields to:

_yy1_yy2 � 10

12_yy1

_xx3 � 12_yy1

264375 ¼

0 1 0

0 �0:83 �10 �1:75 �2

24 35xþ 0

0

NðxÞ

24 35þ0; 120

0

24 35n:

It is obvious that ~ff2 ¼ 0, ðeAA23; eAA33Þ is observable and that gðx3; uÞ ¼ Nðy1Þ is globally Lipschitz with k ¼ 3:6. Thereforeby choosing f ¼ 4 and h ¼ 3 we obtain a convergence rate l0 ¼ 1:4 and the following RORS:

_vv ¼ �6v þ ½ 17 �22:42 � y1y2

� �þ Gðv; uÞ;

where Gðv; uÞ ¼ gðv � 2:83y1 þ 4y2; uÞ. Using a fourth-order Runge–Kutta algorithm of MATLAB 6 we have simulated

the above example and the results are delineated inFig. 6. It can easily be seen that theRORSquickly synchronizeswith the

drive system, however the FORS which was designed without considering the perturbation fails to track the drive system.

5. Conclusion

In this paper we have proposed a new design procedure for chaotic system synchronization. Under some structural

assumptions of the drive system, and assuming the presence of unknown perturbation signal, an observer-based RORS

was derived to synchronize with the drive system. Driven by the transmitted signal, the RORS correctly reproduces the

remaining states of the drive system. To illustrate the efficiency of our method, two well-known chaotic systems were

considered: a modified Chua�s circuit and Lur�e system. Our method can also be applied to several other chaotic systemssuch as R€oossler�s hyperchaotic system. It is also interesting to exploit the robustness of our method to design a com-munication scheme using chaotic encryption.

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Observer-based chaotic synchronization in the presence of unknown inputs

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