2017 IEEE 6th Data Driven Control and Learning Systems Conference May 26-27, 2017, Chongqing, China Modified Function Projective Synchronization of Fractional-order Hyperchaotic Systems Based on Active Sliding Mode Control Yuan Gao1,2,3, Hangfang Hu1, Ling Yu1, Haiying Yuan1,3, Xisheng Dai1,3 1.College of Electrical and Information Engineering, Guangxi University of science and technology, Liuzhou 545006,China E-mail: [email protected] 2. Guangxi Key Laboratory of Automobile Components and Vehicle Technology, Liuzhou 545006, China 3. Key Laboratory of Industrial Process Intelligent Control Technology of Department of Guangxi Education, Liuzhou 545006, China Abstract: Considering the time-varying scaling function matrix and system disturbances, a new sliding mode control strategy is proposed to realize modified function projective synchronization (MFPS) of two different fractional-order hyperchaotic systems, meanwhile improve the control robustness of synchronization system. From the MFPS error equations, combining a proper fractional-order exponential reaching raw, an active controller for MFPS is derived out via sliding mode control technology. By mean of the stability theorem, the asymptotic stability of synchronization error system is proved. Simulation results of the MFPS between fractional-order hyperchaoticLorenz system and Chen system demonstrate the validity of the presented method. Key Words: Fractional-order, Hyperchaotic system, Scaling function matrix, Modified function projective synchronization, Sliding mode control 1 (GPS) of the fractional-order hyperchaotic Chen system. Ref. [15] studied the MPS problem of two different fractional-order hyperchaotic systems with system parameter uncertainty. Ref. [16] presented a MFPS scheme for a class of partially linear fractional-order chaotic systems, in which the controlleris designed out by Routh-Hurwitz conditions. On the other hand, there are many uncertain factors and disturbances in the practical systems. Therefore, it is necessary to overcome the influence of disturbances on the chaos synchronization. The sliding mode control (SMC) is well known as a kind of robust control techniques [17], which has been widely used for controlling the disturbed nonlinear system. Nevertheless, most of the existing SMC methods for chaos (or hyperchaos) control or synchronization focus on the integral-order systems[18,19]. In recent years, there are a few results about the fractional-order chaotic systems[20, 21, 22] . Based on the principle of active control, Ref. [21] designed a new SMC controller with a fractional-order sliding mode surface to realize the PS of two fractional-order hyperchaotic systems. For a class of different structure synchronization problem of uncertain fractional-order chaotic systems, ref. [22] proposed an active feedback controller based on adaptive neural network, where the sliding mode controller is designed to compensate the influence of uncertain disturbances. In order to enhance the control robustness, according to the stability theorem of fractional-order system, ref. [15] and [23] studied active SMC method to obtain the MPS of two different fractional-order chaotic and hyperchaotic system, respectively. Inspired by the discussions above, in order to obtain more complex relationships of chaos synchronization, we dedicate to study the MFPS of two non-identical fractional-order hyperchaotic systems with disturbances, and propose a new active SMC method. In this paper, considering the time-varying scaling function matrix,we introduce a reaching raw of sliding mode surface, and then a proper sliding mode controller is derivedby mean of the stability theory of the fractional-order system. At last, the numerical results of the MFPS between the fractional-order Introduction Chaos synchronization has always been a hot issue because of great application prospect in the field of secure communication[1],since this interesting phenomenon was proved through a practical circuit experiment in 1990[2]. By now, complete synchronization[3], phase synchronization[4], generalized synchronization[5], projective synchronization (PS)[6], and other types of chaos synchronization have been studied[7]. PS is one of important chaos synchronization types, whose main characteristic is that the state variables of the drive system and response system are synchronized according to a certain proportion[6]. In recent years, inspired with the PS, people have further investigated a new scheme named MFPS[8]. The MFPS scheme is an extension of many existing synchronization schemes including modified projective synchronization(MPS)[9], function projective synchronization(FPS), et al.[10], where the drive system and the response system are synchronized up to a scaling function matrix instead of constant matrix, namely, state variables of two systems correlate each other in pairs by a scaling function. Due to the more unpredictability of the scaling function, the MFPS can further improve the anti-decryption of chaos secure communication, and attracts more attention on it. By now, some research results about the MFPS have been reported[11, 12, 13], however, most of these works focused on integral-order system. Chaos synchronization for the fractional-order system, due to its great potentials in the fields of automatic control andsecure communication, which has attracted increasing attention in the last few years. In [14], using the stability theorem of fractional-order system, Wu et al. provided a method for realizing generalized projective synchronization * This work was supported by the Natural Science Foundation of Guangxi Province of China (Grant No.2014GXNSFBA118284), the Scientific Research Fund of Guangxi Education Department (Grant No.KY2016YB244),the Opening Project of Guangxi Key Laboratory of AutomobileComponents and Vehicle Technology (Grant No. 2015KFZD03), the Project of Key laboratoryof Industrial Process Intelligent Control Techno- logyof Guangxi Higher Education Institutes (GrantNo. IPICT-2016-01 and IPICT-2016-05) 978-1-5090-5461-9/17/$31.00 ©2017 IEEE 445 DDCLS'17 hyperchaotic Chen system and Lorenz system verify the presented method. 2 DD x Consider the following fractional-order hyperchaotic drive system A1 x f1 ( x, t ) d x (t ) (1) DD x and the unidirectionally coupled response system Dq y A2 y f 2 ( y, t ) d y (t ) u (2) matrix Q satisfied f1 ( x, t ), f 2 ( y, t ) : R o R are the continuous nonlinear 3.2 (u1 , u2 ,}, un )T is a control input vector, d x (t ) (d x1 , d x 2 ,}, d xn ) R n and d y (t ) (d y1 , d y 2 , Dqe n t of where e(t ) (e1 (t ), e2 (t ),..., en (t )) R , and ei tof 3 3.1 (10) (11) n Dq e w(t ) d y (t ) H (t )d x (t ) A2e (13) In order to decrease the design complexity of sub-controller w (t ) , the sliding surface is directly defined as follows: (4) s(e) Ce yi (14) where C diag(c1 , c2 ,}, cn ) is a constant matrix. To design the sliding mode controller, an exponential reaching law is selected as (15) Dq s H sgn( s) Ps Active sliding mode control scheme of MFPS where P diag( p1 , p2 ,}, pn ) denotes a positive-definite matrix, sgn(.) represents the sign function, the gain factor H !0. From (13) and (14), one can have Stability theory for fractional-order system Lemma 1 For the following fractional-order system DD x (5) Ax nun where 0 D 1 , x R and A R . If and only if the Dq s C[ Kw(t ) d y (t ) H (t )d x (t ) A2e] n (16) Neglecting the disturbances in (16), and according to the fractional-order reaching law (15), we can derive the SMC input w (t ) as follows: following relationship is satisfied ,then system (5) is asymptotically stable [25]. A2 y f 2 ( y, t ) d y (t ) u H (t )( A1 x the control parameters, w (t ) R is a SMC input that need to be designed. Substituting (12) into (10), we have 0. arg(Oi ( A)) ! 0.5DS , i 1, 2,..., n Design of controller where K = diag(k1 , k2 ,}, kn ) is a constant gain matrix for hi (t ) xi (i 1, 2,}, n) . From (3), MFPS means that e(t) satisfies the relationship lim e . Considering the advantages of better stability and robustness of the SMC system,therefore, the controller can be written as u Kw(t ) F ( x , y ,t ) (12) 2,}, n) are constants, MFPS will be simplified into GPS. n (9) [27] H (t ) f1 ( x, t ) Dtq ( H (t ) x) where Н (t ) diag(h1 (t ), h2 (t ),..., hn (t )) is called the scaling function matrix, and diag(.) means diagonal matrix, scaling function factors hi (t ) z 0(i 1, 2, }, n) are differentiable bounded functions, ||.||denotes the matrix norm. Obviously, the MFPS scheme becomes FPS when. h1 (t ) h2 (t ) } hn (t ) Furthermore, while hi (t )(i 1, T xT QDD x d 0 F ( x , y ,t )= f 2 ( y, t ) H (t )( A1 A2 ) x (3) Definethe synchronization error vector as e (t ) = y H (t ) x Q ! 0 , and the following relationship is According to the principle of active control, F ( x, y, t ) is introduced as following form Definition1 For arbitrary initial values x (0) and y (0) , system(1) and system(2) are said to be MFPS if there exists control vector u such that 0 . If there exists a f1 ( x, t ) d x (t )) D q ( H (t ) x ) disturbance of drive system and response system, respectively. lim y H (t ) x (8) From (1), (2) and (4), we can obtain the following dynamical model of the MFPS error }, d yn ) R represent the bounded and differentiable T nun ,then system (8) is asymptotically stable n T A( x) x T ) ( x) y ( y1 , y2 ,}, yn )T Rn denote system state variables, A1 and A2 represent constant matrix, functions, u F ( x) where 0 D 1 , x R and A R where D indicates the fractional derivative of Caputo definition[24], fractional order number q (0,1) , x ( x1 , n n n q x2 ,}, xn )T R n and (7) where 0 D 0 , x R and F R If all the eigenvalues O of the Jacobian matrix J wF ( x) / wx calculated at the equilibrium points satisfy | arg(O ) |! DS / 2 , then system (7) is locally asymptotically stable. Lemma 3 Consider an autonomous system described as MFPS of fractional-order hyperchaotic system Dq x F ( x) n (6) w(t )= (CK )1[H sgn( s) Ps CA2e] (17) wherethe switching part plays a role of suppressing disturbances. Substituting (17) into (12), thus, the controller can be expressed as follows Lemma 2 For a commensurate fractional-order nonlinear dynamical system[26] 446 DDCLS'17 u K (CK )1[H sgn( s) Ps CA2e ] f 2 ( y, t ) Dtq y1 b1 ( y2 y1 ) y4 d y1 u1 ° q ° Dt y2 b2 y1 y1 y3 y2 d y 2 u2 ® q ° Dt y3 y1 y2 b3 y3 d y 3 u3 °Dq y y y b y d u 1 3 4 4 4 y4 ¯ t 4 H (t )( A1 A2 ) x H (t ) f1 ( x, t ) D q ( H (t ) x ) (18) 3.3 Stability analysis Selecting a candidate function is selected as follows \ If q 0.95, b1 10, b2 28, b3 8 / 3, b4 1 , the Lorenz system displays the hyperchaotic behavior[29]. eT D q e (19) ,substituting Eqs.(10),(18) into (19), and using commutative law of multiplication between diagonal matrixes, we have \ From (23) and (24), one can see that x and y e T { A2 e K (CK ) 1 C [H sgn(e ) ( P A2 )e ] u2 , u3 , u4 )T is control input vector, corresponding system d y (t ) H (t )d x (t )} matrixes and nonlinear functionvectors are described as follows T T e T [ H (t )d x (t )] n H ¦ ei e T Pe e T d y (t ) e T [ H (t )d x (t )] i 1 n n i 1 i 1 d H ¦ ei r e (a b)¦ ei (20) where max{| h1 (t )d x1 (t ) |, | h2 (t )d x 2 (t ) |,},| hn (t )d xn (t ) |} , b max{| d y1 (t ) , d y 2 (t ) , }, d y n (t ) |} , r min{ p1 , 20,12) Let us presume the following conditions satisfied , then we get \ d (H a b)¦ ei r e d 0 d y (t ) , y(0) (12, 9,15,3) T , the disturbances >0.1sin(t ),0.5sin(t ),0.2sin(t ),0.3sin(t )@ scaling function matrix H (t ) (22) T , the diag(2sin(t ),sin(t ) 4, cos(3t ),cos(t ) 0.2) , the parameters and gain matrixes of the controller r 1.2 , K = diag(1,1,1,1) , P diag(50, 70,42,35) and C diag(5,5,5,5) , step size is set as i 1 From the Lemma 3, the error system is asymptotically stable. Simulation results 0.002. For indicating the reliability of the controller, we use the predictor-corrector algorithm to obtain numerical solution of the fractional-order systems[30], and implement the control operation after t 4s . From figure 1 and 2, we see that synchronization errors ei (i 1,2,3,4) converge to zeros quickly after control, and In this section,an example of the MFPS is provided to verify the effectiveness of our method, in which fractional-order hyperchaotic Chen system and Lorenz system are chosen asthe drive system and response system, respectively. The fractional-order Chen system with external disturbances is expressed as Dtq x1 a1 ( x2 x1 ) x4 d x1 ° q ° Dt x2 a2 x1 x1 x3 a3 x2 d x 2 ® q ° Dt x3 x1 x2 a4 x3 d x 3 ° D q x x x a x d x4 ¯ t 4 2 3 5 4 T d x (t ) [0.8sin(0.2t ),0.6sin(0.5t ),0.4cos(2t ),0.5cos(0.6t )]T , (21) n (8, 5, In the simulation, initial values of system x(0) p2 ,}, pn } . H t ab ( y1 , y2 , y3 , y4 )T are system state vectors, u (u1 , 1 · § 0 · § 35 35 0 ¨ ¸ ¨ ¸ x x 7 12 0 0 ¸ , f1 ( x ,t ) ¨ 1 3 ¸ A1 = ¨ ¨ x1 x2 ¸ ¨ 0 0 3 0 ¸ ¨ ¸ ¨ ¸ 0 0 0.5 ¹ © 0 © x2 x3 ¹ 0 1· § 0 · § 10 10 ¨ ¸ ¨ ¸ y1 y3 ¸ 28 1 0 0¸ ¨ ¨ , f 2 ( y ,t ) A2 = ¨ y1 y2 ¸ ¨ 0 0 8 / 3 0 ¸ ¨ ¸ ¨ ¸ 0 0 1¹ © 0 © y1 y3 ¹ H e sgn(e ) re e e d y (t ) T 4 ( x1 , x2 , x3 , x4 )T eT Dqe e T [H sgn(e ) Pe d y (t ) H (t )d x (t )] a (24) the relationship yi hi (t ) xi (i 1,2,3, 4) can be satisfied, which means the MFPS between system (23) and system (24) is obtained. (23) when q 0.95, a1 35, a2 7, a3 12, a4 3, a5 0.5 , the Chen system shows hyperchaos[28]. The fractional-order Lorenz system with disturbances is describedas follows 447 DDCLS'17 Fig. 2: The state trajectories of the drive and response systems 5 Conclusions In this paper, a new active SMC method for realizing MFPS of two different fractional-order hyperchaotic systems with external disturbances is presented. 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