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Implementation of Partial Synchronization of Different Chaotic Systems by Field Programmable Gate Array

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Implementation of Partial Synchronization of Different Chaotic Systems by Field Programmable Gate Array View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by DSpace@IZTECH Institutional Repository Implementation of Partial Synchronization of Different Chaotic Systems by Field Programmable Gate Array Can Eroglu˘ F. Acar Savacı Department of Electrical and Department of Electrical and Electronics Engineering Electronics Engineering Izmir Institute of Technology Izmir Institute of Technology Izmir, Turkey 35430 Izmir, Turkey 35430 Email: [email protected] Email: [email protected] Abstract—In this study, the synchronization of the master- system is implemented by the help of ISE and FUSE programs. slave systems has been achieved and implemented on Field Finally, in Section V, conclusions are presented. Programmable Gate Array (FPGA). In this paper, the master system and the slave system have been chosen as Lorenz and Rossler systems, respectively. The feedback control rule has been derived by feedback linearization method. By feedback II. SYNCHRONIZATION PROBLEM linearization, the coordinate transformation has been achieved then the control command for synchronization has been obtained. In this section, the partial synchronization problem in the In order to implement designed synchronized system, Matlab Simulink design of the system has been translated to Xilinx System sense of exact synchronization and practical synchronization Generator design to generate Very-High-Speed Integrated Circuits will be considered. The definitions given below describes Hardware Description Language (VHDL) code which is used to exact synchronization, practical synchronization and partial produce bitstream file. By Xilinx Integrated Software Environment synchronization, respectively. (ISE) program, VHDL code is converted to bitstream file which has been embedded into FPGA by Field Upgradeable Systems Environment (FUSE). Finally, the designed synchronized system Definition 2.1: It is said that two chaotic systems are ex- has been observed on the HP 54540C oscilloscope. actly synchronized if the synchronization error, ei = xi − yi , exponentially converges to the origin. This implies that at a I. INTRODUCTION finite time xi = yi [8]. Definition 2.2: It is said that two chaotic systems are prac- In the nonlinear science, control of chaotic systems have tically synchronized if the trajectories of the synchronization been studied recently owing to its particular importance in ap- error ei = xi − yi converges to a neighborhood around the plications such as communications, physics etc. [1], [2]. There origin. This implies that for all time t ≤ t∗ the trajectories are several kinds of synchronizations such as generalized of the slave system are close to the master trajectories, i.e., synchronization [3], complete synchronization [4], [5], partial xi ≈ yi [8]. synchronization [6], phase synchronization [7] and almost synchronization [8]. The pioneering work [5], has increased Definition 2.3: It is said that two chaotic systems are the interest in synchronization after having recently found partially synchronized if, at least, one of the states of the many applications particularly in telecommunications [9], in synchronization error system is either practically or exactly mechanical systems [10] and in control theory [11]. Partial synchronized and only if, at least, one of the states of the synchronization problem has been studied in populations of synchronization error system is neither practically nor exactly pulse-coupled oscillators [12]. synchronized [8]. This paper is organized as follows: In Section II the partial The difference between master and slave system is called as synchronization problem is defined and the feedback lineariza- an error system which can be constructed using the definition tion method is explained. In Section III, as a case study given below. Lorenz system is chosen as the master system and Rossler Definition 2.4: Let x˙ = FM (x) and y˙ = FS(y)+g(y)u(y) n system is chosen as the slave system then by the suitable be two chaotic systems in a manifold M ⊂ R . FM , FS feedback, Rossler system is synchronized to Lorenz system. smooth vector fields with scalar output functions sM = h(x), n n In Section IV, the proposed control system is simulated by sS = h(y) and x, y ∈ R and g(y) ∈ R is a smooth input Matlab Simulink then the simulated design is converted to vector where subscripts M and S stands for the master and Xilinx System Generator design and the designed synchronized slave, respectively [4]. 978-1-4244-3896-9/09/$25.00 ©2009 IEEE 559 [F, [F, [..., [F, G], ..., ]]] for d =1, ..., n where [F, G] is called the Lie bracket of F and G [4]. ˙ = ( ) x FM x , (1) Corollary 2.8: Two chaotic systems with the same order are e˙ = FM (x) − FS(x, e) − g(x, e)u(x, e), (2) completely synchronizable if and only if the dynamical error se = h(x, e), (3) system is feedback linearizable at (x, 0) [4]. Feedback linearization problem: The system in (4) is called feedback linearizable if there exist a smooth reversible change where se is the output of the synchronization error system and of coordinates z =Φ(x, e) and smooth transformation of the extended synchronization error system can be written in affine feedback [13], [14]. form as: u = λ(x, e)+μ(x, e)v, (5) X˙ = F(X)+G(X)u(X), (4) where v ∈ Rm is the new control if the closed-loop is linear and then the resulting variables z and v satisfy linear T T where X =[x, e] , F(X)=[FM (x), FM (x) − FS(y)] and dynamical system in the form of G(X)=[0, −g(x, e)]T . In order to achieve synchronization, synchronization error z˙ = Az + bv ρ−1 T system in Eq. 2 should be stabilized around the equilibrium z =Φ(x, e)=[h(x, e),LF h(x, e), ..., LF h(x, e)] point e∗ =0. The definitions and theorems given in the 1 = (− ρ ( )+ ) following sequel will be used to find the proper invertible u ρ−1 LF h x, e v (6) LGLF h(x, e) transformation which will be used to derive the control command for synchronization. ρ −L h(x, e) λ(x, e)= F ρ−1 ( ) Definition 2.5: System (4) is said to have relative degree ρ, LGLF h x, e n 1 ρ ≤ n at point x0 ∈ R with respect to the output μ(x, e)= ρ−1 LGLF h(x, e) se = h(x) ∗ ν = Ki(zi − zi ) if for any x ∈ Ω where Ω is some neighborhood of x0,the following conditions are valid where Ki with i =1, ..., ρ are the control gains and chosen in k such a way that the closed-loop subsystem z˙ converges to the (i)LGLF h(x)=0, k =0, 1, ...ρ − 2, ∀x in a neighborhood ∗ origin and zi s are the coordinates of the stabilization point. In of x0 and k<ρ− 1, ∗ order to achieve complete synchronization zi s are set to zero. ρ−1 ( ) =0 (ii)LGLF h x0 . III. SYNCHRONIZATION OF CHAOTIC SYSTEMS WITH n DIFFERENT MODEL ∂φ Recall that Lψφ(x)= ψ(x) stands for the Lie Lorenz system [15] is chosen as the master system and its ∂xi i=1 state equations can be written as follows : derivative of the vector function φ along the vector field ψ. Relative degree ρ is exactly equal to the number of times x˙1 = α(x2 − x1) one has to differentiate the output in order to have the input x˙2 = βx1 − x2 − x1x3 (7) explicitly appearing in the equation which describes the (r) x˙3 = x1x2 − γx3 evolution of se (t) in the neighborhood of x0 [2], [13]. Theorem 2.6: System (4) is feedback linearizable in the where α =10, β =28,and γ =8/3. Ω ∈ Rn neighborhood of a point x0 if and only if there exists a Rossler system [16] is chosen as the slave system and is ( ) Ω smooth scalar function h x defined in such that the relative described as: degree ρ of (3) and (4) is equal to n [2]. Theorem 2.7: Consider the system (4). Suppose that there ˙ = −( + )+ ( ) exist 2n − ρ functions Φi(x, e) such that LGΦi(x, e)=0, y1 y2 y3 g1 y u i = ρ+1, ..., 2n. This system is feedback linearizable at (x, 0) y˙2 = y1 + θy2 + g2(y)u (8) if and only if there exists a function h(x, e) such that k−1 y˙3 = δ + y3(y1 − φ)+g3(y)u (i) <∂h,adF G > (x, e)=0for k =1, ..., ρ − 1; ρ>1 and (x, e) in a neighborhood Ω of (x, 0), where g1(y), g2(y) and g3(y) are control i (ii) <∂h,adF G > (x, 0) =0for i = ρ, ..., n at (x, 0), inputs and θ =0.2, δ =0.2, φ =5.7. where ρ = d stands for the dimension of the tangent space and The extended synchronization error system for coupled the accessibility distribution function C(x, e) can be expressed Lorenz-Rossler system can be written by calculating the error d−1 d−1 as Cd = span{adF G} where adF =[F, G] and adF = system as: 560 C3(x, 0) can be written as: ⎧⎡ ⎤⎫ ⎪ 00 0⎪ e˙1 = α(x2 − x1)+(x2 − e2)+(x3 − e3) − g1(x, e)u ⎪ ⎪ ⎪⎢ 00 0⎥⎪ ˙ = − − − ( − ) − ( − ) − ( ) ⎨⎪⎢ ⎥⎬⎪ e2 βx1 x2 x1x3 x1 e1 θ x2 e2 g2 x, e u ⎢ 00 0⎥ ( 0) = ⎢ ⎥ e˙3 = x1x2 − γx3 − δ − (x3 − e3)[(x1 − e1) − φ] − g3(x, e)u C3 x, span ⎪⎢ 0 −1 ⎥⎪ (11) ⎪⎢ θ ⎥⎪ ⎪⎣−1 θ 1 − θ2⎦⎪ After having found error sytem, the extended synchroniza- ⎩⎪ ⎭⎪ 00 x3 tion error system can be written as: In order to derive the control command, we need to deter- x˙1 = α(x2 − x1) mine the dimension of the tangent space d which is generated x˙2 = βx1 − x2 − x1x3 by the corresponding distribution so the conditions of Theorem x˙3 = x1x2 − γx3 2.7 must be satisfied.
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