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Synchronization in a Class of Fractional-Order Chaotic Systems

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Synchronization in a Class of Fractional-Order Chaotic Systems Advances in Science, Technology and Engineering Systems Journal Vol. 6, No. 3, 146-154 (2021) ASTES Journal www.astesj.com ISSN: 2415-6698 Special Issue on Multidisciplinary Sciences and Engineering Synchronization in a Class of Fractional-order Chaotic Systems via Feed- back Controllers: a Comparative Study Juan Luis Mata-Machuca* Instituto Politecnico Nacional, UPIITA, Department of Advanced Technologies, Mexico City, 07340, Mexico ARTICLEINFOABSTRACT Article history: In this paper a synchronization methodology of two fractional-order chaotic oscillators under Received: 03 December, 2020 the framework of identical synchronization and master-slave configuration is introduced. The Accepted: 04 May, 2021 proposed methodology is based on a fractional-order feedback control design under the frame Online: 20 May, 2021 of control theory, the feedback controllers provide synchronization convergence. A comparative study between a proportional control, a nonlinear fractional-order proportional-integral control Keywords: and an active control is presented. The above is showed via an analysis of the dynamic of Fractional-order chaotic the called synchronization error. Numerical experiments using the mathematical model of the system fractional-order unified chaotic system and its equivalent electronic circuit corroborate the Proportional control satisfactory results of the proposed schemes. Nonlinear fractional-order PI control Active control Master-slave synchronization Equivalent electronic circuit 1 Introduction that two seemingly random and unpredictable chaotic behaviors can converge in a single trajectory, new expectations arose around chaos theory, trying to solve its control, understanding and prediction. A significant development of chaos occurred after the publication by Lorenz [1] of a work related with the existence of non-periodic The synchronization of chaotic systems consists of coupling solutions in a model formed by nonlinear differential equations. two or more chaotic systems, after a transitory time, they exhibit Lorenz’s contribution provided the interest in studying deterministic identical chaotic oscillations. Synchronization can be solved from systems that generate dynamic trajectories strongly influenced by the point of view of control theory, designing a slave system which sensitivity to initial conditions. However, it was until 1975 when follows the variables of the master system. In the open literature the term chaos was introduced [2], since then chaotic systems have has been presented a lot of important contributions for the design of been studied intensively, and this is one important area of current control schemes for synchronization purposes in chaotic oscillators research in physics, mathematics, and engineering. The analysis of [9]–[11], observer based controllers [12, 13], adaptive algorithms nonlinear complex dynamic behaviors has led to important techno- [14, 15], optimal controllers [16, 17] and so on [18]–[20]. logical application. In particular the analysis of chaotic oscillators From the above, control theory has been looking for advanced has had important advances in process engineering, cryptography techniques in order to reach a high performance in the operation of of images, secure data transmission, life science, and information processes, in particular with dynamic systems with complex behav- processing, between others [3]–[5]. ior, such as the nonlinear fractional-order systems [21]–[23]. The There are several definitions about chaotic systems. However, in fractional calculus for the study of dynamic systems with chaotic general terms, a chaotic system is a deterministic dynamic system behavior has emerged as an alternative framework, since it has been that exhibits irregular behavior, similar to random behavior [6, 7]. proved that it is possible that a class of systems exhibits chaos de- A very important aspect in relation to chaos theory is the synchro- spite its order is fractional [24]–[27]. Generally, nonlinear systems nization of chaotic systems. Since Pecora and Carrol [8] showed display complex dynamic behavior as steady state multiplicity, in- *Corresponding Author: Juan Luis Mata-Machuca, Email: [email protected] www.astesj.com 146 https://dx.doi.org/10.25046/aj060317 J.L. Mata-Machuca / Advances in Science, Technology and Engineering Systems Journal Vol. 6, No. 3, 146-154 (2021) stabilities, complex oscillations and so on under different initial 2.1 Proportional control conditions, external disturbances and time-varying parameters, lead- Let us propose the controlled slave system of fractional order α as ing to chaotic dynamic behaviors. However, besides of the scientific follows, interest on the study and analysis of nonlinear system with exotic dynamic behaviors, the applications for engineering purposes have α D zi = fi(z1; z2;:::; zn) + ui (3) been growing importantly [28]–[31]. ui = ki(xi − zi) (4) There are many engineering applications based on dynamical T n systems with fractional operators, for instance: fractional modelling with zi(0) = zi0, i = 1; 2;:::; n, z = [z1; z2;:::; zn] 2 Ω ⊂ R is of the human arm dynamics [32], modelling and identification of the state vector of the slave system, and ki is the proportional gain. viscoelastic mechanical systems [33], modelling of electrical sys- Let us define the synchronization error as, tems [34], analysis and control of economics and financial systems [35, 36], vibration and acoustics phenomena [37], modeling of "i = xi − zi: (5) friction in electric machines [38], problems in electrochemistry, bio- The synchronization performance is obtained by means of the physics and bioengineering [39]–[41], methods for signal and image analysis of the dynamics of the synchronization error. processing [42, 43], applications in automatic control, mechatronics First, we take the time derivative of order α of (5), and robotics [44]–[46], among others [47]–[51]. However, there are α α α a few practical applications in this field to prove the feasibility of D "i = D xi − D zi (6) the physical realization of the proposed techniques [52]–[54]. Some works deal with the master-slave synchronization of fractional-order Now, replacing the fractional dynamics of equations (1) and (3) systems via unidirectional linear coupling [55]–[58], where the pro- into (6), α posed methods are applied in the synchronization of fractional-order D "i + ki"i = fi(x) − fi(z) (7) Rossler,¨ Chua, Chen, unified systems and so on. Then, the solution of (7) is given by, The main contribution of this work is to achieve the synchro- α nization of a class of fractional-order systems and its verification via "i(t) = "i0Eα(−kit )+ Z t the equivalent electronic circuit by means of a proportional control, α−1 α (8) + (t − τ) Eα,α(ki(t − τ) ) fi(x) − fi(z) dτ a nonlinear fractional-order proportional-integral control and an 0 active control. The synchronization implies that all trajectories of where "i0 = xi0 − zi0. the slave system tend to follow all trajectories of the master system. In (8), the Mittag-Leffler function is defined as [? ], In this sense, the main problem is to find a control input such that the synchronization is possible. The suitable conditions for the syn- X1 q j E (q) = ; q 2 C; Re(α) > 0 (9) chronization are obtained by solving the dynamical equation of the α Γ(α j + 1) synchronization error. Furthermore, to verify the effectiveness of the j=0 suggested approach we have implemented an electronic circuit using 1 X qi Matlab-Simulink’s Simscape library to obtain the synchronization E (q) = ; q; α 2 C; Re(α) > 0 (10) α,α Γ(α j + α) regime between two fractional-order unified chaotic systems for the j=0 same order. To guarantee the convergence of the synchronization error in (8), Assumption1 and2 are introduced, Assumption 1 j f (x) − f (z)j ≤ L < 1 2 Fractional-order feedback controllers i i i Assumption 2 The proportional gain is positive. The master system of fractional order α can be represented by, Applying Assumption1 in (8) we have, α α D xi = fi(x1; x2;:::; xn); (1) j"i(t)j ≤ j"i0Eα(−kit )j+ Z t j − α−1 − − α j T n +Li (t τ) Eα,α( ki(t τ) ) dτ with xi(0) = xi0, i = 1; 2;:::; n, x = [x1; x2;:::; xn] 2 Ω ⊂ R is 0 the state vector, and f : Ω ! Rn is a nonlinear continuous function. Finally, by Assumption2, In (1), the fractional derivative of xi is defined as [? ], Li lim j"i(t)j ≤ (11) t!1 ki α α α d xi(t) D xi = t0 Dt xi(t) = α From the previous analysis, the following proposition is estab- dt lished. 1 Z t dm x (τ) = i (t − τ)m−α−1 dτ; − m Proposition 1 If Assumptions1 and2 are fulfilled, then the syn- Γ(m α) t0 dτ (2) chronization of the master system (1) and the controlled slave system m + d xi(τ) where: α 2 R , m − 1 ≤ α < m, m 2 N, dτm is the m-th time (3)-(4) is achieved, where, the trajectories of "i belongs to a closed derivative of xi and Γ is the gamma function. ball with radius proportional to Li=ki. www.astesj.com 147 J.L. Mata-Machuca / Advances in Science, Technology and Engineering Systems Journal Vol. 6, No. 3, 146-154 (2021) 2.2 Nonlinear fractional-order PI control In (22), g : Ω ! Rn is a nonlinear continuous function and the matrix A 2 Rn×n is defined by, Let us consider (1) as the master system. In this case, the slave system is described by, 2 3 2 3 6 a11 a12 ··· a1n 7 6 A1 7 6 7 6 7 α 6 a21 a22 ··· a2n 7 6 A2 7 D zi = fi(z1; z2;:::; zn) + ui; i = 1; 2;:::; n (12) A = 6 7 = 6 7 (23) 6 : : : 7 6 : 7 u = f (x) − f (z) + k " + K e (13) 6 : : : 7 6 : 7 i i i i i Ii i 46 57 46 57 α an1 an2 ::: ann An D ei = "i (14) h i where, "i = xi − zi, ki is the proportional gain and KIi is the integral where, ai j are constant coefficients and Ai = ai1 ai2 ··· ain gain.
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