Observer-based chaotic synchronization in the presence of unknown inputs

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1 Chaos, Solitons and Fractals 5 (00) Observer-based chaotic synchronization in the presence of unknown inputs Moez Feki *, Bruno Robert Universite de Reims Champagne Ardenne-UFR Sciences Exactes, Moulin de la Housse, BP 09, 58 Reims cedex, France Accepted 0 May 00 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 synchronization is exponentially achieved. The efficiency of our method is confirmed by numerical simulations of two wellknown chaotic systems: ChuaÕs circuit and LurÕe system. Ó 00 Elsevier Science Ltd. All rights reserved.. Introduction Over the past decades, synchronization of chaotic systems has been an intriguing concept and has received considerable attention [ ]. Indeed, the synchronization property of chaotic circuits has revealed potential applications to secure communications [4 ]. In fact, since the seminal paper by Pecora and Carroll [], the synchronization of chaotic systems is based on the drive-response conception: a drive system drives via a scalar transmitted signal a custom designed response system. Recently, the synchronization has been regarded as an observer design problem. In [] 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 [9,0] Morg ul et al. and Liao showed results on local and global synchronization using observer design. In [,8] authors assumed exact knowledge of their drive systems, whereas in [9,0] 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 unmeasured states. Our paper will be organized as follows: In Section we present an observer-based full-order response system (FORS) construction that can be applied to synchronize perturbation-free chaotic systems []. Section gives the design procedure of the RORS. In Section 4, the synchronization scheme is applied to two chaotic systems: modified ChuaÕs circuit and LurÕe system, and simulation results are shown. Section 5 includes our concluding remarks. * Corresponding author. Tel.: ; fax: address: moez.feki@univ-reims.fr (M. Feki) /0/$ - see front matter Ó 00 Elsevier Science Ltd. All rights reserved. PII: S090-09(0)004-9

2 8 M. Feki, B. Robert / Chaos, Solitons and Fractals 5 (00) Full-order response system design We consider the chaotic drive system described by the following equations on R n. _x ¼ Fx þ Gu þ gðx; uþ; u R m ; ðaþ y ¼ Hx; y R p ; ðbþ 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 ¼ðg ; g ;...; g n Þ T (Tstands for the transpose) is Lipschitz, with gð0; 0Þ ¼0. We then consider the following response system _^x^x ¼ F ^x þ Gu þ gð^x; uþþfs H T ðy H ^xþ; ðaþ 0 ¼ hs F T S SF þ H T H; ðbþ where h is large enough and f P. Remark. The matrix SðhÞ can be seen as the stationary solution of the differential equation _S t ðhþ ¼ hs t ðhþ F T S t ðhþ S t ðhþf þ H T H; with initial condition S 0 being positive definite. SðhÞ ¼lim t! S t ðhþ, where S t ðhþ S þ is the cone of symmetric positive definite matrices. Theorem. If the system defined by (a) and (b) satisfies the following hypotheses (H) there exists a positive constant k such that kgðx; uþ gð^x; uþk kkx ^xk; 8ðx; ^xþ R nn and 8u R m (H) the pair ðh; F Þ is observable (H) we can choose h > 0 and f P such that k < k minðhs þðf ÞH T HÞ ; ðþ k max ðsþ k min ðþ and k max ðþ denote the smallest and largest eigenvalues of the matrix ðþ. Then the response system defined by (a) and (b) globally asymptotically synchronizes with the drive system described by (a) and (b) with an exponentially decaying error sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k max ðsþ kxðtþ ^xðtþk expð l k min ðsþ 0 tþkxð0þ ^xð0þk; where l 0 ¼ðk min ðhs þðf ÞH T HÞ=k max ðsþþ k. Proof. Let us express Eq. (b) in the form of a Lyapunov equation F T h I S þ S h I F ¼ H T H and define F h ¼ ðh=þi F. Then F h is Hurwitz stable if h > minfreðkþjk specðf Þg; ð4þ where specðf Þ is the spectrum of F. With this condition satisfied and since (H) implies that ðh; F h Þ is an observable pair, it follows that SðhÞ is positive definite. Let us define e ¼ x ^x and consider the error dynamics _e ¼ðF fs H T HÞe þ gðx; uþ gð^x; uþ

3 and the Lyapunov function V ðeþ ¼e T Se, then we have _V ðeþ ¼e T SFe fe T H T He þ e T Sðgðx; uþ gð^x; uþþ: Using Eq. (b) _V ðeþ ¼ he T Se ðf Þe T H T He þ e T Sðgðx; uþ gð^x; uþþ e T ðhs þðf ÞH T HÞe þ kk max ðsþkek ðk min ðhs þðf ÞH T HÞ kk max ðsþþkek k minðhs þðf ÞH T HÞ k V ðeþ k max ðsþ hence if (H) is satisfied then we can have an exponentially decaying bound on the Lyapunov function. V ðtþ V ð0þ expð l 0 tþ M. Feki, B. Robert / Chaos, Solitons and Fractals 5 (00) where l 0 ¼ðk min ðhs þðf ÞH T HÞ=k max ðsþþ k. Using the following inequality k min ðsþkek e T Se k max ðsþkek ; we have keðtþk V ðtþ k min ðsþ V ð0þ expð l 0tÞ k maxðsþ k min ðsþ k min ðsþ expð l 0tÞkeð0Þk ; equivalently sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k max ðsþ keðtþk expð l k min ðsþ 0 tþkeð0þk: 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 [,8] with F being any matrix satisfying the observability assumption ðhþ. We can also note that this response system can be applied to nonautonomous chaotic systems such as the Duffing equation and the Van der Pol oscillator.. Reduced-order response system design The foregoing design procedure is based on the exact knowledge of the nonlinear system (a) and (b). 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: _x ¼ Ax þ f ðxþþcn; y ¼ Cx; ð5aþ ð5bþ where x R n is the state vector, y R p is the output, n R m is an unknown input and C R nm is its known injection map. A and C are constant matrices of appropriate dimensions and f is a Lipschitzian vector field with f ð0þ ¼0. We assume that (5a) and (5b) is a drive system exhibiting a chaotic behaviour. We also suppose that the following hypotheses are pertaining to (5a) and (5b) (H4) p > m (H5) rank CC ¼ m

4 84 M. Feki, B. Robert / Chaos, Solitons and Fractals 5 (00) 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 ½I p 0 Š ¼ I m I p 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 y ¼ I x m x y 0 I p m 0 5 ¼ x x x ) x ¼ 4 x x x 5 ¼ 4 Therefore, the drive system (5a) and (5b) can be expressed as follows: _x _y A f ðxþ C 4 _x 5 ¼ 4 _y 5 ¼ 4 A 5x þ 4 f ðxþ 5 þ 4 C 5n: ðþ _x _x A f ðxþ C It is easy to verify using (H4) and (H5) that rank C ¼ m: C Without loss of generality we assume that C is nonsingular. Thus the following transformation matrix is well defined: I 0 0 U ¼ 4 C C I 0 5: C C 0 I We note that U is chosen to annihilate the last two terms of C, thus premultiplying () by U yields to: _y A f ðxþ C 4 _y C C _y 5 ¼ A C C A 4 5x þ f ðxþ C C f 4 ðxþ 5 þ 4 0 5n; ðþ _x C C _y A C C A f ðxþ C C f ðxþ 0 hence the unknown input enters only through the first row. Let us define the following matrices and functions: ea i,a i C i C A i ¼ ; ; f~ i,f i C i C f i ¼ ; : The last two equations of () become _y C C _y ¼ ea x þ f ~ ðxþ; _x C C _y ¼ ea x þ f ~ ðxþ: If we further partition ea i in the following manner: we obtain ea i,½ ea i ea i ea i Š i ¼ ; ; _x C C _y ¼ ea y þ ea y þ ea x þ f ~ ðy ; y ; x Þ; ð8aþ _y C C _y ¼ ea y þ ea y þ ea x þ f ~ ðy ; y ; x Þ: ð8bþ In the sequel we will consider f ~ 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. y y x 5:

5 M. Feki, B. Robert / Chaos, Solitons and Fractals 5 (00) Whence, (8a) and (8b) can be expressed by the following equations _x ¼ ea x þ Bu þ gðx ; uþ; ð9aþ where: ~y ¼ ea x ; ð9bþ B ¼ C C ea ea ; u ¼ _y y T; y gðx ; uþ ¼ f ~ ðy ; y ; x Þ; ~y ¼ C C _y ea y þ _y ea y : and We have finally obtained a perturbation-free nonlinear dynamical system (9a) and (9b) in the form of (a) and (b) studied in the first section. Since x and x are already available by direct measurement then it is sufficient to design a RORS that (estimates) synchronizes with the third state variable x. From the previous section, a response system that will synchronize exponentially fast with (9a) and (9b) is given by: _^x^x ¼ ea ^x þ Bu þ gð^x ; uþþfs ea T ð~y ea ^x Þ; ð0aþ 0 ¼ hs ea T S S ea þ ea T e A : ð0bþ Substituting u and ~y in (0a), by their respective values yields to: _^x^x ¼ðeA fs ea T e A Þ^x þðc C fs ea T C C Þ_y þðea fs ea T e A Þy þðfs ea T Þ_y þðea fs ea T e A Þy þ gð^x ; uþ: ðþ We notice that the estimation of x depends on time derivative of the output y namely _y and _y which are not directly measured. Hence, the response system needs to be modified. Let us define a new state variable v ¼ ^x ðc C fs ea T C C Þy ðfs ea T Þy : ðþ Therefore, it comes out that in the new state space the response system () is expressed by _v ¼ Fv þ HY þ Gðv; uþ; ðþ where: F ¼ ea fs ea T e A ; " H ¼ ð ea fs ea T e # T A ÞþFðC C fs ea T C C Þ ðea fs ea T e A ÞþFðfS ea T Þ ; Y ¼ y ; y Gðv; uþ ¼gðv þðc C fs ea T C C Þy þðfs ea T Þy ; uþ: Finally, the estimation is given as a function of the new state variable v of the response system () and the output y of the drive system (5a) and (5b). ^x ¼ v þðc C fs ea T C C Þy þðfs ea T Þy : ð4þ Eventually, we can state the following theorem to summarize our result: Theorem. Consider the chaotic drive system described by (5a) and (5b), satisfying hypotheses (H4), (H5) and ~ f ¼ 0. We also assume that the reduced-order system (9a) and (9b) satisfies (H), (H) and (H), then the RORS described by () with the output equation given by (4) globally synchronizes with (5a) and (5b).

6 8 M. Feki, B. Robert / Chaos, Solitons and Fractals 5 (00) Examples In this section, we consider two well-known chaotic systems onto which we will apply and appraise the suggested method namely: a modified ChuaÕs circuit and LurÕe system. 4.. Modified Chua s circuit The modified ChuaÕs circuit shown in Fig. is different from ChuaÕs circuit only in that a RC-parallel circuit is added in series with the inductor. It has been shown in [] 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: 0 _x r r 0 0 x _x 4 _x 5 ¼ b 0 b 0 x x 5 þ _x a a x 4 y ¼ x x x 5 ¼ x x x anðx 4 Þ C A þ 4 c c c c 4 5n; ð5þ ; ðþ where, NðxÞ ¼bx þ 0:5ða bþðjx þ j jx jþ; ðþ and n is an added perturbation. Obviously (5) is in the form of (5a) and (5b). Let us consider the following numerical values: r ¼ :8 r ¼ 0:045 b ¼ 8 a ¼ 0 a ¼ :4 b ¼ 0:4 c ¼ 0:0 c ¼ 0:0 c ¼ 0:0 c 4 ¼ 0: Using the analysis of the previous section we have: rank CC ¼ ; C ¼ 0:0 is nonsingular; C ¼ 0:0; C ¼ 0:0 0: and U ¼ : Premultiplying (5) by U yields to: Fig.. A modified ChuaÕs circuit.

7 M. Feki, B. Robert / Chaos, Solitons and Fractals 5 (00) _y _y _y :8 0: :0 4 _x _y 5 ¼ :5 0: : :0 5 x þ þ n: _x 4 0 _y 45: 0: 0 0 0Nðx 4 Þ 0 It is obvious that f ~ ¼ 0 and that 0 gððx ; x 4 Þ; uþ ¼ :4x 4 ðjx 4 þ j jx 4 jþ is globally Lipschitz with k ¼ :4. Knowing that the pair ðea ; ea Þ is observable ea ¼ ea 0 0 ¼ ½ 8 0 Š then by choosing f ¼ 4 and h ¼ ; we obtain the following RORS: _v ¼ 89:88 0: 49:8 y v þ þ Gðv; uþ; 8: 0 4: 4: y where Gðv; uþ ¼gððv þ :y 4:94y ; v þ 0:4y 5:y Þ; uþ. The simulation results are obtained using fourth order Runge Kutta method of MATLAB. Fig. depicts the unknown perturbation signal. In Figs. and 4 we show the convergence of the RORS, whereas in Fig. 5 we show that the FORS described in Section 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 []. Hence two trajectories, which are close can be attracted each to a different manifold. 4.. Lur e system LurÕe system is a well-known chaotic system and it is described by the following state equations: _x _x 5 ¼ x 0 0 0: 4 x 5 0 A þ 4 0: 5n; ð8þ _x 0 :5 x Nðx Þ 0:0 y ¼ 0 0 x 4 x ¼ x ; ð9þ x x where n is an added perturbation signal and NðxÞ is a nonlinear function: Fig.. Perturbation signal nðtþ.

8 88 M. Feki, B. Robert / Chaos, Solitons and Fractals 5 (00) Fig.. Convergence of the RORS: x ðtþ and ^x ðtþ. Fig. 4. Convergence of the RORS: x 4 ðtþ and ^x 4 ðtþ.

9 M. Feki, B. Robert / Chaos, Solitons and Fractals 5 (00) Fig. 5. Wrong estimation of the FORS. Fig.. (LurÕe system) Correct synchronization of the RORS. Failure of the FORS.

10 840 M. Feki, B. Robert / Chaos, Solitons and Fractals 5 (00) < :8x jxj NðxÞ ¼ :x þ 5:4signðxÞ < jxj : 5:4signðxÞ < jxj In this example x and x are the driving signals, and a RORS is to be designed to synchronize with x. We can verify that rank CC ¼ ; C ¼ 0: is nonsingular; C ¼ 0:; C ¼ 0:0 and 0 0 U ¼ : 0 Premultiplying (5) by U yields to: _y ; 4 _y 0 _y 5 ¼ 4 0 0:8 5x þ þ 4 0 5n: _x _y 0 :5 NðxÞ 0 It is obvious that f ~ ¼ 0, ðea ; ea Þ is observable and that gðx ; uþ ¼Nðy Þ is globally Lipschitz with k ¼ :. Therefore by choosing f ¼ 4 and h ¼ we obtain a convergence rate l 0 ¼ :4 and the following RORS: _v ¼ v þ½ :4 Š y y þ Gðv; uþ; where Gðv; uþ ¼gðv :8y þ 4y ; uþ. Using a fourth-order Runge Kutta algorithm of MATLAB we have simulated the above example and the results are delineated in Fig.. It can easily be seen that the RORS quickly synchronizes with 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 ChuaÕs circuit and LurÕe system. Our method can also be applied to several other chaotic systems such as R osslerõs hyperchaotic system. It is also interesting to exploit the robustness of our method to design a communication scheme using chaotic encryption. References [] Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 990;4(8):8 4. [] Ogorzalek M. Taming chaos part I: Synchronization. IEEE Trans Circ Syst-I 99;40(0):9 9. [] Morg ul O, Feki M. On the synchronization of chaotic systems by using occasional coupling. Phys Rev E 99;55(5): [4] Wu CW, Chua LO. A simple way to synchronize chaotic systems with applications to secure communication systems. Int J Bifurcation Chaos 99;():9. [5] Cuomo KM, Oppenheim AV, Strogatz SH. Synchronization of lorenzed-based chaotic circuits with applications to communications. IEEE Trans Circ Syst-II 99;40(0):. [] Morg ul O, Feki M. A chaotic masking scheme by using synchronized chaotic systems. Phys Lett A 999;5():9. [] Nijmeijer H, Mareels IMY. An observer looks at synchronization. IEEE Trans Circ Syst-I 99;44(0): [8] Liu F, Ren Y, Shan X, Qiu Z. A linear feedback synchronization theorem for a class of chaotic systems. Chaos, Solitons & Fractals 00;(4): 0. [9] Morg ul O, Solak E. On the synchronization of chaotic systems by using state observers. Int J Bifurcation Chaos 99;():0. [0] Liao T-L, Tsai S-H. Adaptive synchronization of chaotic systems and its application to secure communications. Chaos, Solitons & Fractals 000;(9):8 9. [] Moez Feki. Synthese de commandes et dõobservateurs pour les systemes non-lineaires: Application aux systemes hydrauliques. PhD thesis, Universite de Metz, Metz-France, Juin 00. [] Yuan-Zhao Y. Synchronization of chaos in a modified ChuaÕs circuit using continuous control. Int J Bifurcation Chaos 99;():0. [] Kennedy MP. Three steps to chaos part II: A ChuaÕs circuit primer. IEEE Trans Circ Syst-I 99;40(0):5 4.