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The exponential synchronization of a class of fractional-order chaotic systems with discontinuous input
Accepted Manuscript Title: The exponential synchronization of a class of fractional-order chaotic systems with discontinuous input Author: Haipeng Su Runzi Luo Yanhui Zeng PII: DOI: Reference:
S0030-4026(16)31407-3 http://dx.doi.org/doi:10.1016/j.ijleo.2016.11.081 IJLEO 58496
To appear in: Received date: Accepted date:
22-9-2016 13-11-2016
Please cite this article as: Haipeng Su, Runzi Luo, Yanhui Zeng, The exponential synchronization of a class of fractional-order chaotic systems with discontinuous input, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2016.11.081 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
The exponential synchronization of a class of fractional-order chaotic systems with discontinuous input Haipeng Su Runzi Luo Yanhui Zeng Department of Mathematics, Nanchang University, 330031, P. R. China Abstract This paper investigates the exponential synchronization of a class of fractional-order chaotic systems. The response system is controlled by input which may be either discontinuous or continuous variable. Moreover, the input is assumed to be affected by external disturbance. Based on the Mittag-Leffler function, sufficient conditions for achieving exponential synchronization of fractional-order chaotic systems have been derived. Numerical examples are presented by taking the fractional-order chaotic economical system as an example to verify and demonstrate the effectivenessof the proposed schemes. Keywords: Exponential synchronization; Discontinuous input; Fractional-order chaotic system 1. Introduction Chaos synchronization is a common and widespread phenomenon in many science and engineering fields [1]. Since Pecora and Carroll [2] first introduced the master-slave concept for achieving the synchronization between two identical chaotic systems, chaos synchronization has received considerable attentions due to its potential applications in secure communication, biology, economics, signal generator design, and so on. Now, a variety types of chaos synchronization have been proposed, such as complete synchronization [2], phase synchronization [3], anti-synchronization [4], lag synchronization [5], generalized synchronization [6], projective synchronization [7], combination synchronization [8, 9], etc. Many different control methods for chaos synchronization have been developed, including the adaptive control method [10], impulsive approach [11], back-stepping technique [12], sliding control technique [13], sampled-data control scheme [14], and so on. Recently, research on synchronization of fractional-order chaotic has attracted increasing attention due to its potential application in secure communication and control processing. In [15] the function projective synchronization between two entirely different fractional-order chaotic systems with uncertain parameters was studied by using an adaptive controller. Based on fractional-order stability theory, Authors of [16] proposed a novel method to achieve robust modified projective synchronization of twouncertain fractional-order chaotic systems with external disturbance. Hyperchaos control and adaptive synchronization with uncertain parameter for fractional-order Mathieu-van der Pol systems was studied in [17]. Xue et al. [18] designed a nonlinear feedback controller to synchronize two identical fractional-order generalized augmented L u system. Based on sliding mode control, paper [19] investigated the synchronization of fractional order uncertain chaotic systems with input nonlinearity. Paper [20] considered the exponential synchronization of fractional-order chaotic systems via a non-fragile controller. Based on the Lyapunov stability theory and linear matrix inequalities approach a criterion for -exponential stability of an error system was proposed. It is easy to see that the aforementioned synchronization methods are valid only for the continuous controller. As it is well known that the transmission signals may be
interrupted by external disturbance in reality. Thus, the controllers which are always continuous are infeasible from the view of practical application. Therefore, it is necessary and important to investigate the chaos synchronization with the controller which can be either discontinuous or continuous variable. However, to the best of the author’s knowledge, the problem of exponential synchronization for fractional-order chaotic systems with discontinuous controller has not been fully investigated and remains open. Motivated by the above discussion, this paper will investigate the exponential synchronization of a class of fractional-order chaotic systems with or without uncertainties and disturbances. A novel input which may be either discontinuous or continuous variable is proposed. Based on the Mittag-Leffler function, sufficient conditions for achieving exponential synchronization of fractional-order chaotic systems have been obtained. The fractional-order chaotic economical system is taken as an example to demonstrate the effectiveness of the proposed schemes. The rest of the paper is organized as follows: In Section 2, some basic concepts and preliminary results of fractional-order derivative are presented. A brief description of a class of fractional-order chaotic systems is introduced in Section 3. Section 4investigates the exponential synchronization of a class of fractional-order chaotic systems, some novel criteria are proposed via the discontinuous controller. Section 5 includes several numerical examples to demonstrate the effectiveness of the proposedapproach. Some concluding remarks are drawn in Section 6. 2. Primaries of fractional-order derivative In this section, we introduce the definition of Caputo fractional-order derivative and present preliminary results needed in our proofs later. There are many ways to define fractional-order derivative [21]. Since the initial conditions for Caputo fractional-order derivatives take the same form as for integer-order differential. Thus, in this paper we use the Caputo fractional-order derivatives as a main tool to derive our results. The formula of the Caputo fractional-order derivative is defined as follows: Definition 1 (Caputo fractional-order derivative [21]). The Caputo fractional-order derivative of order is given as [21]: t 1 f ( n ) ( ) (n ) t0 (t )( n 1) d t t0 n 1 n t0 Dt f (t ) dn n f (t ) dt n where, n is a least integer no less than . The fractional-order here is limited as 0 1 . () is the Gamma function which is defined by the integral
( z ) et t z 1dt 0
For the sake of convenience in writing, the Caputo fractional-order derivative operator will be replaced by D in the following. 0 Dt Lemma 1 [21] Let V (t ) be a continuous function on [t0 ) and satisfies t0
DtV (t ) V (t )
where 0 1 and is a constant, then V (t ) V (t0 ) E ( (t t0 ) ) where
E ( z ) k 0
zk ( k 1)
is the Mittag-Leffter function. Lemma 2 [22]. Let x(t ) R be a continuous and derivable function. Then, for any time instant t t0 D x 2 2 xD x (01] 3. System description Consider a class of fractional-order chaotic systems which can be described as: (1) D x Ax f (t x) where x ( x1 xn )T R n is the state vector of the system (1), A R nn is a constant parameter matrix. The function f (t x) R R n R n satisfies the Lipschitz condition: f (t x1 ) f (t x2 ) L x1 x2 (2) for all (t x1 ) and (t x2 ) in with a Lipschitz constant L . Remark 1. Model (1) covers most of typical fractional-order chaotic systems such as the fractional-order Lorenz system [23], the fractional-order L u system [24] and the fractional-order Genesio system [25]. 4. The synchronization scheme In this section, a discontinuous input control scheme is applied for reaching exponential synchronization of a class of fractional-order chaotic systems. In order to achieve synchronization, system (1) is taken as the drive system while the response system which is controlled by u is represented as follows: D y Ay f (t y ) u (3) where u is the controller to be designed later. Define the error state e y x (e1 e2 en )T where
e1 y1 x1 e y x 2 2 2 en yn xn Subtracting system (1) from system (3), we obtain the error dynamical system as follows: D e Ae f (t y ) f (t x) u (4) Before proceeding further, the following essential definition is introduced. Definition 2. The controlled system (3) is said to be globally exponentially synchronized with system (1) if there exist scalars ( 0) and r ( 0) such that
e(t ) e rt
t0
hold for any initial values. The task here is to design a discontinuous controller to achieve the exponential synchronization between the drive system (1) and the response system (3). In the present study, we choose the discontinuous controller as follows:
k e sign(e) u t [t2 m t2 m 1 ) t0 0 m 01 2 u m (5) 0 t [t2 m 1 t2 m 2 ) m 01 2 in which km and ( 0) are constants to be designed later, u is the disturbance of u . Remark 2. It is well known that noise disturbance is inevitable in practical situations, so in this paper we add the term u to denote the disturbance of input. To the best of our knowledge, there are few results on the disturbance input in the literature. Remark 3. From (5) it is easy to see that if u is continuous and t2 m 1 t2 m 2 km k , then the controller u is continuous variable. In other case the controller u is discontinuous variable. In order to show our main work, we need the following assumption: Assumption 1. The disturbance u is bounded with u i.e. u u Now, we are in a position to give our main results. Theorem 1. Suppose that the Assumption 1 is satisfied. Then the response chaotic system (3) can be globally exponentially synchronized with drive system (1) under the discontinuous controller (5) if there exist constants km and r ( r 0) such that the following inequalities hold: 1) u r (t t ) (6) 2) E ( m (t2 m 1 t2 m ) ) E ( (t2 m 2 t2 m 1 ) ) e 2 m2 2 m m 01 2 where m 2 L 2km , 2L and is the maximum eigenvalue of
A AT . Proof. Select the following Lyapunov candidate defined as: 1 V eT e 2 Using Lemma 2, the time derivative of V along the solution of (4) is 1 DV eT ( A AT )e eT ( f (t y ) f (t x)) eT u (t ) 2 1 eT e LeT e eT u (t ) ( 2 L)eT e eT u (t ) 2 2
(7)
For t [t2 m t2 m 1 ) , by plugging u into the above inequality, one gets
DV 12 ( 2 L )eT e eT ( km e sign(e) u ) 12 ( 2 L 2k m )eT e e u e Note that u we have 1 DV ( 2 L 2km )eT e mV 2 For t [t2 m 1 t2 m 2 ) , since in this case u 0 , thus from (7) we obtain 1 D V ( 2 L ) eT e V 2
(8)
(9)
When m 0 in inequality (8), by Lemma 1 for any t [0 t1 ) one gets V (t ) V (0) E ( m t )
This leads to V (t1 ) V (0) E ( mt1 )
(10)
In the same way for t [t1 t2 ) , we have V (t ) V (t1 ) E ( (t t1 ) ) V (0) E ( mt ) E ( (t t1 ) )
(11)
It is noted that for any t [0 ) there exists a positive integer m such that t [t2 m t2 m 1 ) or t [t2 m1 t2 m 2 ) . We discuss two cases according to t in different time intervals. Case 1: t [t2 m t2 m 1 ) . In this case, one finds that V (t ) V (0) E ( mt1 ) E ( (t2 t1 ) ) E ( m (t2 m 1 t2 m 2 ) ) E ( (t2 m t2 m 1 ) ) E ( m (t t2 m ) ) V (0)e rt2 e r ( t4 t2 ) e r ( t2 m t2 m2 ) E ( m (t t2 m ) ) V (0) E ( m (t t2 m ) )e r ( t t2 m ) e rt
(12)
It should be noted that the interval [t2 m t2 m 1 ) is bounded which means that E ( m (t t2 m ) )e r ( t t2 m ) is also bounded. Thus, there exists constant such that
V (0) E ( m (t t2 m ) )er (t t2 m ) 2 Therefore, inequality (12) yields 2
V (t )
2 2
e rt
In view of that V 12 eT e , we obtain rt
e e 2
(13)
Case 2: t [t2 m 1 t2 m 2 ) . In this case, we have V (t ) V (0) E ( m t1 ) E ( (t2 t1 ) ) E ( m (t2 m 1 t2 m ) ) E ( (t t2 m 1 ) ) V (0)e rt2 e r ( t4 t2 ) e r ( t2 m t2 m2 ) E ( m (t2 m 1 t2 m ) ) E ( (t t2 m 1 ) ) V (0)e rt2 m E ( m (t2 m 1 t2 m ) ) E ( (t t2 m 1 ) ) V (0)e r ( t t2 m ) E ( m (t2 m 1 t2 m ) ) E ( (t t2 m 1 ) )e rt
(14)
It should point out that the intervals [t2 m t2 m 1 ) and [t2 m 1 t2 m 2 ) are bounded which means that e r ( t t2 m ) E ( m (t2 m 1 t2 m ) ) E ( (t t2 m 1 ) ) is also bounded. Thus, there exists constant such that r ( t t2 m ) 2 V (0)e E ( m (t2 m1 t2 m ) ) E ( (t t2 m1 ) ) 2 Therefore, inequality (14) follows V (t )
2 2
e rt
By using V 12 eT e , we get rt
e e 2
(15)
Based on Definition 2, from (13) and (15) it is easy to see that the origin of error system (4) is exponentially stable which means that the exponential synchronization between the drive system (1) and the response system (3) is achieved. This completes the proof of Theorem 1. In practical applications it is well known that some dynamical systems are inevitably disturbed by the noises from external circumstance. Furthermore, owing to the un-modeled dynamics and structural changes, these dynamical systems usually have some uncertainties. These uncertainties and noises will destroy the dynamical behaviors or even break the synchronization. Therefore, the synchronization between chaotic systems with uncertainties and disturbances are challenging jobs for researchers. Based on this consideration, in the following we consider the synchronization between systems (1) and (3) with uncertainties and disturbances. System (1) with uncertainties and disturbance is rewritten as: (16) D x ( A A1 ) x f (t x) f1 (t x ) d1 where x A and f (t x ) are defined as that in system (1). A1 f1 (t x) and d1 are parameter uncertainty, model uncertainty and external disturbance, respectively. Suppose system (16) is the drive system, in order to synchronize (16) the controlled response system can be represented as: (17) D y ( A A2 ) y f (t y ) f 2 (t y ) d 2 u where A2 f 2 (t y ) and d 2 are parameter uncertainty, model uncertainty and external disturbance, respectively. u is the controller to be designed later. Subtracting (16) from (17), the synchronization error system is achieved as follows: D e Ae ( A2 y A1 x ) f (t y ) f (t x ) ( f 2 (t y ) f1 (t x )) ( d 2 d1 ) u (18) Now, we introduce an Assumption which is useful in proving Theorem 2. Assumption 2. Suppose A1 A2 f 2 (t y ) f1 (t x)) and d1 d 2 are all bounded. Since x and y are two bounded variables, thus there exists a constant d ( 0) such that A2 y A1 x f 2 (t y ) f1 (t x) d 2 d1 d For the sake of achieving synchronization, we choose the discontinuous controller u as follows: k e sign(e) u t [t2 m t2 m1 ) t0 0 m 01 2 u m (19) t [t2 m1 t2 m 2 ) m 01 2 sign(e) u in which km and ( 0) are constants to be designed later, u is the disturbance of u . Remark 4. From (19) it is easy to see that in interval [t2 m 1 t2 m 2 ) we add controller sign(e) u Thus, controller (19) is different from (6). The reason we use term sign(e) in interval [t2 m 1 t2 m 2 ) is that sign(e) can eliminate the adverse effects caused by uncertainties and external disturbances. The following theorem will give sufficient conditions of exponential synchronization between systems (16) and (17) with parameter uncertainty, model uncertainty and external disturbance.
Theorem 2. Suppose that the Assumptions 1-2 are satisfied. Then the response chaotic system (17) can be globally exponentially synchronized with drive system (16) under the discontinuous controller (19) if there exist constants km and r ( r 0) such that the following inequalities hold: 1) u d r (t t ) (20) 2) E ( m (t2 m 1 t2 m ) ) E ( (t2 m 2 t2 m 1 ) ) e 2 m2 2 m m 01 2 where m 2 L 2km , 2L and is the maximum eigenvalue of
A AT . Proof. Take the following Lyapunov candidate: 1 V eT e 2 Its time derivative along the solution of (18) is 1 DV eT ( A AT )e eT ( A2 y A1 x f 2 (t y ) f1 (t x) d 2 d1 ) 2 eT ( f (t y ) f (t x)) eT u (t ) 1 eT e LeT e d e eT u (t ) ( 2 L )eT e d e eT u (t ) 2 2
(21)
For t [t2 m t2 m 1 ) , by substituting u (t ) into the above inequality, one gets
DV 12 ( 2 L )eT e d e eT ( k m e sign(e) u ) 12 ( 2 L 2k m )eT e d e e u e Keep in mind that u d we have 1 DV ( 2 L 2km )eT e mV 2 Similarly, for t [t2 m 1 t2 m 2 ) we obtain 1 D V ( 2 L ) eT e V 2
(22)
(23)
The rest of the proof is similar to that of Theorem 1 and is omitted here. Remark 5. From controllers (6) and (20) it is obvious that r is closely depended on m and the length of intervals [t2 m t2 m 1 ) [t2 m 1 t2 m 2 ) . Thus, for the given r we can choose proper km and intervals [t2 m t2 m 1 ) [t2 m 1 t2 m 2 ) such that (6) and (20) are satisfied which means that the speed of exponential synchronization can be determined by the controller freely. 5. Numerical simulations In this section, the fractional-order chaotic economical system [26] is taken as an example to verify and demonstrate the effectiveness of the proposed method. The fractional-order chaotic economical system to be investigated in present paper was proposed by Chen [26]. It has three state variables x1 x2 and x3 which stand
for the interest rate, the investment demand, and the price index, respectively. The dynamical equations of the fractional-order chaotic economical system is given as: a 0 1 x1 x1 x2 D x 0 b 0 x2 1 x12 Ax f ( x t ) (24) 1 0 c x 0 3 T where x ( x1 x2 x3 ) is the state variable of system (24), a b and c are three system parameters which are positive real constants. Fig. 1 illustrates the chaotic behavior of the financial system (24) for a 1 b 01 and c 1 .
In the simulation process, we take a 1 b 01 c 1 such that system (24) is chaotic. Furthermore, the order is fixed as 099 . Thus, we have 1 0 0 1 0 2 A 0 01 0 B AT A 0 02 0 1 0 0 0 1 2 The eigenvalues of B are 2 , 2 and 02 , respectively. Obviously, its maximum eigenvalue is 02 which implies that 02 . Note that y1 y2 x1 x2 (24) f (t y ) f (t x ) x12 y12 (25) 0
y2 x1 ( x1 y1 ) 0 0 0 y2 ( x1 y1 ) 0
0(26) y1 x1 (28) 0(27) y2 x2 (29) 0 y3 x3 x1 0 y1 x1 0 0 y2 x2 . 0 0 y3 x3
y2 x1 0 Therefore, we can take L max{ ( x1 y1 ) 0 0 } From Fig. 1 it is easy to see 0 0 0 that x1 2 x2 4 Thus, we can easily obtain L 73006 . Example I. The synchronization without uncertainties and disturbances Suppose system (24) is the drive system, in order to synchronize system (24) the response system with controller u is constructed as: a 0 1 y1 y1 y2 D y 0 b 0 y2 1 y12 u Ay f ( y t ) u (25) 1 0 c y 0 3
In order to achieve synchronization, we choose the discontinuous controller u
as follows: sint km e 2 sign(e) cost t [t2 m t2 m 1 ) t0 0 m 01 2 (26) u sintcost t [t2 m 1 t2 m 2 ) m 01 2 0 sint where cost denotes u and 2 . Thus, u 3 . Obviously, we have sintcost
u For convenience, we let km 20 t2 m 1 t2 m 002 t2 m 2 t2 m 1 001 m 01 2 then it is easy to check that 144012 . m 255988 0 and Thus, we can obtain E ( m (t2 m 1 t2 m ) ) E ( (t2 m 2 t2 m 1 ) ) 05867 11641 06830 e 126003 which means that r 126 Therefore, the conditions of Theorem 1 are all satisfied. According to Theorem 1, the exponential synchronization between the drive system (24) and the response system (25) will be achieved. Without loss of generality, in the simulation we choose the initial conditions as: x(0) (2 3 8)T y (0) (2 2 3)T . The simulation results shown in Figs. 2-4 which are respectively the time evolution of errors e1 e2 e3 between drive system (24) and the response system (25). FromFigs. 2-4 one can conclude that the synchronization errors converge quickly to zero and the exponential synchronization between system (24) and response system (25) is achieved.
Example II. The synchronization with uncertainties and disturbances In order to show the robust of our synchronization scheme to uncertainties and external disturbances, we add some uncertainties and disturbances to system (24). Thus system (24) can be rewritten as D x1 x3 ( x2 (1 01)) x1 01x1 x2 01cos (t ) 2 (27) D x2 1 (01 001) x2 x1 02 x3 x2 02 sin(t ) D x x (1 02) x 03 x x 03cos 2 (t ) 3 1 3 1 3 where
terms
01x1 001x2 02 x3
are
parameter
uncertainties.
01x1 x2 02 x3 x2 03x1 x3 are model uncertainties. 01cos(t ) 02sin(t ) 03cos 2 (t ) are external disturbances. Suppose system (27) is the drive system, then the corresponding response system with uncertainties and disturbances is given as: D y1 y3 ( y2 (1 01)) y1 01 y12 y2 01cos 2 (t ) u1 2 2 (28) D y2 1 (01 001) y2 y1 02 y3 y2 02cos (t ) u2 D y y (1 02) y 03 y 2 y 03sin 2 (t ) u 3 1 3 1 3 3 where
terms
01y1 001y2 02 y3
01 y12 y2 02 y32 y2 03 y12 y3
01cos 2 (t ) 02cos(t ) 03sin 2 (t ) controllers to be designed later.
are
are
parameter model
are external disturbances. u1 u2
uncertainties. uncertainties. and u3
are
For the sake of achieving exponential synchronization, the controller u (u1 u2 u3 )T is chosen as: sint km e sign(e) cost t [t2 m t2 m 1 ) t0 0 m 01 2 sintcost u sint sign(e) cost t [t2 m 1 t2 m 2 ) m 01 2 sintcost
(29)
For simplicity, in the numerical simulation, we assume km 20 t2 m 1 t2 m 002 t2 m 2 t2 m 1 001 m 01 2 20 . Similar to example 1 one gets m 255988 0 and 144012 . Therefore, we derive E ( m (t2 m 1 t2 m ) ) E ( (t2 m 2 t2 m 1 ) ) 06830 e 126003 which implies that r 126 Based on Theorem 2, the exponential synchronization between the drive system (24) and the response system (25) will be achieved. To confirm the validity of our presented scheme, we give numerical simulation with the following choices of the initial conditions: x(0) (1 25 4)T y (0) (2 2 6)T . Numerical results of the time evolution of errors e1 e2 e3 between system (27) and observer (28) with controller (29) are shown in Figs. 5-7, respectively. From Figs. 5-7 it is easy to see that although there are uncertainties and disturbances both in the drive system and the response system, the synchronization errors converge quickly to zero which means that the exponential synchronization between system (27) and response system (28) is reached. Remark 6. In Theorems 1 and 2, it is not easy to calculate u d . However, we have
no need of computing them directly in numerical simulation. Since u and d are all bounded, we can select to be large enough such that conditions 1 in Theorems 1 and 2 are all met. Remark 7. The synchronization of system (24) have been discussed in papers [27, 28]. It is easy to see that the controllers presented in papers [27, 28] are continuous function of states variables. Furthermore, the synchronization schemes proposed in papers [27, 28] are valid only for the asymptotical synchronization not for the exponential synchronization. 6. Conclusions The exponential synchronization of a class of fractional-order chaotic systems has been investigated. The sufficient conditions are derived by implementing the drive-response synchronization theory together with the Mittag-Leffler function. Two examples are used by means of fractional-order chaotic economical system to show the effectiveness of the developed method. Acknowledgments This work was jointly supported by the National Natural Science Foundation of China under Grant No. 11361043 and the Natural Science Foundation of Jiangxi Province under Grant No. 20161BAB201008. References [1] A. Pikovsky, M. Roseblum, J. Kurths, Synchronization: A universal concept in nonlinear sciences, Cambridge University Press, New York, NY, 2003.
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Fig. 1. The chaotic trajectories of system (24) with Fig. 2. The time evolution of error Fig. 3. The time evolution of error Fig. 4. The time evolution of error Fig. 5. The time evolution of error Fig. 6. The time evolution of error Fig. 7. The time evolution of error
e1 e2 e3 e1 e2 e3
099
and
between systems (24) and (25). between systems (24) and (25). between systems (24) and (25). between systems (27) and (28). between systems (27) and (28). between systems (27) and (28).
x1 (0) 1 x2 (0) 1 x3 (0) 1 .
4 3
x3
2 1 0 −1 4 2
2 1
0
0
−2 x2
−1 −4
−2
x1
0 −0.5 −1
e1
−1.5 −2 −2.5 −3 −3.5 −4
0
0.1
0.2
0.3 t
0.4
0.5
5 4.5 4 3.5
e2
3 2.5 2 1.5 1 0.5 0
0
0.1
0.2
0.3 t
0.4
0.5
0 −0.5 −1 −1.5
e3
−2 −2.5 −3 −3.5 −4 −4.5 −5
0
0.1
0.2
0.3 t
0.4
0.5
3 2.5 2
e1
1.5 1 0.5 0 −0.5 −1
0
0.02
0.04
0.06
0.08 t
0.1
0.12
0.14
0.5 0 −0.5 −1
e2
−1.5 −2 −2.5 −3 −3.5 −4 −4.5
0
0.02
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S0030-4026(16)31407-3 http://dx.doi.org/doi:10.1016/j.ijleo.2016.11.081 IJLEO 58496
To appear in: Received date: Accepted date:
22-9-2016 13-11-2016
Please cite this article as: Haipeng Su, Runzi Luo, Yanhui Zeng, The exponential synchronization of a class of fractional-order chaotic systems with discontinuous input, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2016.11.081 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
The exponential synchronization of a class of fractional-order chaotic systems with discontinuous input Haipeng Su Runzi Luo Yanhui Zeng Department of Mathematics, Nanchang University, 330031, P. R. China Abstract This paper investigates the exponential synchronization of a class of fractional-order chaotic systems. The response system is controlled by input which may be either discontinuous or continuous variable. Moreover, the input is assumed to be affected by external disturbance. Based on the Mittag-Leffler function, sufficient conditions for achieving exponential synchronization of fractional-order chaotic systems have been derived. Numerical examples are presented by taking the fractional-order chaotic economical system as an example to verify and demonstrate the effectivenessof the proposed schemes. Keywords: Exponential synchronization; Discontinuous input; Fractional-order chaotic system 1. Introduction Chaos synchronization is a common and widespread phenomenon in many science and engineering fields [1]. Since Pecora and Carroll [2] first introduced the master-slave concept for achieving the synchronization between two identical chaotic systems, chaos synchronization has received considerable attentions due to its potential applications in secure communication, biology, economics, signal generator design, and so on. Now, a variety types of chaos synchronization have been proposed, such as complete synchronization [2], phase synchronization [3], anti-synchronization [4], lag synchronization [5], generalized synchronization [6], projective synchronization [7], combination synchronization [8, 9], etc. Many different control methods for chaos synchronization have been developed, including the adaptive control method [10], impulsive approach [11], back-stepping technique [12], sliding control technique [13], sampled-data control scheme [14], and so on. Recently, research on synchronization of fractional-order chaotic has attracted increasing attention due to its potential application in secure communication and control processing. In [15] the function projective synchronization between two entirely different fractional-order chaotic systems with uncertain parameters was studied by using an adaptive controller. Based on fractional-order stability theory, Authors of [16] proposed a novel method to achieve robust modified projective synchronization of twouncertain fractional-order chaotic systems with external disturbance. Hyperchaos control and adaptive synchronization with uncertain parameter for fractional-order Mathieu-van der Pol systems was studied in [17]. Xue et al. [18] designed a nonlinear feedback controller to synchronize two identical fractional-order generalized augmented L u system. Based on sliding mode control, paper [19] investigated the synchronization of fractional order uncertain chaotic systems with input nonlinearity. Paper [20] considered the exponential synchronization of fractional-order chaotic systems via a non-fragile controller. Based on the Lyapunov stability theory and linear matrix inequalities approach a criterion for -exponential stability of an error system was proposed. It is easy to see that the aforementioned synchronization methods are valid only for the continuous controller. As it is well known that the transmission signals may be
interrupted by external disturbance in reality. Thus, the controllers which are always continuous are infeasible from the view of practical application. Therefore, it is necessary and important to investigate the chaos synchronization with the controller which can be either discontinuous or continuous variable. However, to the best of the author’s knowledge, the problem of exponential synchronization for fractional-order chaotic systems with discontinuous controller has not been fully investigated and remains open. Motivated by the above discussion, this paper will investigate the exponential synchronization of a class of fractional-order chaotic systems with or without uncertainties and disturbances. A novel input which may be either discontinuous or continuous variable is proposed. Based on the Mittag-Leffler function, sufficient conditions for achieving exponential synchronization of fractional-order chaotic systems have been obtained. The fractional-order chaotic economical system is taken as an example to demonstrate the effectiveness of the proposed schemes. The rest of the paper is organized as follows: In Section 2, some basic concepts and preliminary results of fractional-order derivative are presented. A brief description of a class of fractional-order chaotic systems is introduced in Section 3. Section 4investigates the exponential synchronization of a class of fractional-order chaotic systems, some novel criteria are proposed via the discontinuous controller. Section 5 includes several numerical examples to demonstrate the effectiveness of the proposedapproach. Some concluding remarks are drawn in Section 6. 2. Primaries of fractional-order derivative In this section, we introduce the definition of Caputo fractional-order derivative and present preliminary results needed in our proofs later. There are many ways to define fractional-order derivative [21]. Since the initial conditions for Caputo fractional-order derivatives take the same form as for integer-order differential. Thus, in this paper we use the Caputo fractional-order derivatives as a main tool to derive our results. The formula of the Caputo fractional-order derivative is defined as follows: Definition 1 (Caputo fractional-order derivative [21]). The Caputo fractional-order derivative of order is given as [21]: t 1 f ( n ) ( ) (n ) t0 (t )( n 1) d t t0 n 1 n t0 Dt f (t ) dn n f (t ) dt n where, n is a least integer no less than . The fractional-order here is limited as 0 1 . () is the Gamma function which is defined by the integral
( z ) et t z 1dt 0
For the sake of convenience in writing, the Caputo fractional-order derivative operator will be replaced by D in the following. 0 Dt Lemma 1 [21] Let V (t ) be a continuous function on [t0 ) and satisfies t0
DtV (t ) V (t )
where 0 1 and is a constant, then V (t ) V (t0 ) E ( (t t0 ) ) where
E ( z ) k 0
zk ( k 1)
is the Mittag-Leffter function. Lemma 2 [22]. Let x(t ) R be a continuous and derivable function. Then, for any time instant t t0 D x 2 2 xD x (01] 3. System description Consider a class of fractional-order chaotic systems which can be described as: (1) D x Ax f (t x) where x ( x1 xn )T R n is the state vector of the system (1), A R nn is a constant parameter matrix. The function f (t x) R R n R n satisfies the Lipschitz condition: f (t x1 ) f (t x2 ) L x1 x2 (2) for all (t x1 ) and (t x2 ) in with a Lipschitz constant L . Remark 1. Model (1) covers most of typical fractional-order chaotic systems such as the fractional-order Lorenz system [23], the fractional-order L u system [24] and the fractional-order Genesio system [25]. 4. The synchronization scheme In this section, a discontinuous input control scheme is applied for reaching exponential synchronization of a class of fractional-order chaotic systems. In order to achieve synchronization, system (1) is taken as the drive system while the response system which is controlled by u is represented as follows: D y Ay f (t y ) u (3) where u is the controller to be designed later. Define the error state e y x (e1 e2 en )T where
e1 y1 x1 e y x 2 2 2 en yn xn Subtracting system (1) from system (3), we obtain the error dynamical system as follows: D e Ae f (t y ) f (t x) u (4) Before proceeding further, the following essential definition is introduced. Definition 2. The controlled system (3) is said to be globally exponentially synchronized with system (1) if there exist scalars ( 0) and r ( 0) such that
e(t ) e rt
t0
hold for any initial values. The task here is to design a discontinuous controller to achieve the exponential synchronization between the drive system (1) and the response system (3). In the present study, we choose the discontinuous controller as follows:
k e sign(e) u t [t2 m t2 m 1 ) t0 0 m 01 2 u m (5) 0 t [t2 m 1 t2 m 2 ) m 01 2 in which km and ( 0) are constants to be designed later, u is the disturbance of u . Remark 2. It is well known that noise disturbance is inevitable in practical situations, so in this paper we add the term u to denote the disturbance of input. To the best of our knowledge, there are few results on the disturbance input in the literature. Remark 3. From (5) it is easy to see that if u is continuous and t2 m 1 t2 m 2 km k , then the controller u is continuous variable. In other case the controller u is discontinuous variable. In order to show our main work, we need the following assumption: Assumption 1. The disturbance u is bounded with u i.e. u u Now, we are in a position to give our main results. Theorem 1. Suppose that the Assumption 1 is satisfied. Then the response chaotic system (3) can be globally exponentially synchronized with drive system (1) under the discontinuous controller (5) if there exist constants km and r ( r 0) such that the following inequalities hold: 1) u r (t t ) (6) 2) E ( m (t2 m 1 t2 m ) ) E ( (t2 m 2 t2 m 1 ) ) e 2 m2 2 m m 01 2 where m 2 L 2km , 2L and is the maximum eigenvalue of
A AT . Proof. Select the following Lyapunov candidate defined as: 1 V eT e 2 Using Lemma 2, the time derivative of V along the solution of (4) is 1 DV eT ( A AT )e eT ( f (t y ) f (t x)) eT u (t ) 2 1 eT e LeT e eT u (t ) ( 2 L)eT e eT u (t ) 2 2
(7)
For t [t2 m t2 m 1 ) , by plugging u into the above inequality, one gets
DV 12 ( 2 L )eT e eT ( km e sign(e) u ) 12 ( 2 L 2k m )eT e e u e Note that u we have 1 DV ( 2 L 2km )eT e mV 2 For t [t2 m 1 t2 m 2 ) , since in this case u 0 , thus from (7) we obtain 1 D V ( 2 L ) eT e V 2
(8)
(9)
When m 0 in inequality (8), by Lemma 1 for any t [0 t1 ) one gets V (t ) V (0) E ( m t )
This leads to V (t1 ) V (0) E ( mt1 )
(10)
In the same way for t [t1 t2 ) , we have V (t ) V (t1 ) E ( (t t1 ) ) V (0) E ( mt ) E ( (t t1 ) )
(11)
It is noted that for any t [0 ) there exists a positive integer m such that t [t2 m t2 m 1 ) or t [t2 m1 t2 m 2 ) . We discuss two cases according to t in different time intervals. Case 1: t [t2 m t2 m 1 ) . In this case, one finds that V (t ) V (0) E ( mt1 ) E ( (t2 t1 ) ) E ( m (t2 m 1 t2 m 2 ) ) E ( (t2 m t2 m 1 ) ) E ( m (t t2 m ) ) V (0)e rt2 e r ( t4 t2 ) e r ( t2 m t2 m2 ) E ( m (t t2 m ) ) V (0) E ( m (t t2 m ) )e r ( t t2 m ) e rt
(12)
It should be noted that the interval [t2 m t2 m 1 ) is bounded which means that E ( m (t t2 m ) )e r ( t t2 m ) is also bounded. Thus, there exists constant such that
V (0) E ( m (t t2 m ) )er (t t2 m ) 2 Therefore, inequality (12) yields 2
V (t )
2 2
e rt
In view of that V 12 eT e , we obtain rt
e e 2
(13)
Case 2: t [t2 m 1 t2 m 2 ) . In this case, we have V (t ) V (0) E ( m t1 ) E ( (t2 t1 ) ) E ( m (t2 m 1 t2 m ) ) E ( (t t2 m 1 ) ) V (0)e rt2 e r ( t4 t2 ) e r ( t2 m t2 m2 ) E ( m (t2 m 1 t2 m ) ) E ( (t t2 m 1 ) ) V (0)e rt2 m E ( m (t2 m 1 t2 m ) ) E ( (t t2 m 1 ) ) V (0)e r ( t t2 m ) E ( m (t2 m 1 t2 m ) ) E ( (t t2 m 1 ) )e rt
(14)
It should point out that the intervals [t2 m t2 m 1 ) and [t2 m 1 t2 m 2 ) are bounded which means that e r ( t t2 m ) E ( m (t2 m 1 t2 m ) ) E ( (t t2 m 1 ) ) is also bounded. Thus, there exists constant such that r ( t t2 m ) 2 V (0)e E ( m (t2 m1 t2 m ) ) E ( (t t2 m1 ) ) 2 Therefore, inequality (14) follows V (t )
2 2
e rt
By using V 12 eT e , we get rt
e e 2
(15)
Based on Definition 2, from (13) and (15) it is easy to see that the origin of error system (4) is exponentially stable which means that the exponential synchronization between the drive system (1) and the response system (3) is achieved. This completes the proof of Theorem 1. In practical applications it is well known that some dynamical systems are inevitably disturbed by the noises from external circumstance. Furthermore, owing to the un-modeled dynamics and structural changes, these dynamical systems usually have some uncertainties. These uncertainties and noises will destroy the dynamical behaviors or even break the synchronization. Therefore, the synchronization between chaotic systems with uncertainties and disturbances are challenging jobs for researchers. Based on this consideration, in the following we consider the synchronization between systems (1) and (3) with uncertainties and disturbances. System (1) with uncertainties and disturbance is rewritten as: (16) D x ( A A1 ) x f (t x) f1 (t x ) d1 where x A and f (t x ) are defined as that in system (1). A1 f1 (t x) and d1 are parameter uncertainty, model uncertainty and external disturbance, respectively. Suppose system (16) is the drive system, in order to synchronize (16) the controlled response system can be represented as: (17) D y ( A A2 ) y f (t y ) f 2 (t y ) d 2 u where A2 f 2 (t y ) and d 2 are parameter uncertainty, model uncertainty and external disturbance, respectively. u is the controller to be designed later. Subtracting (16) from (17), the synchronization error system is achieved as follows: D e Ae ( A2 y A1 x ) f (t y ) f (t x ) ( f 2 (t y ) f1 (t x )) ( d 2 d1 ) u (18) Now, we introduce an Assumption which is useful in proving Theorem 2. Assumption 2. Suppose A1 A2 f 2 (t y ) f1 (t x)) and d1 d 2 are all bounded. Since x and y are two bounded variables, thus there exists a constant d ( 0) such that A2 y A1 x f 2 (t y ) f1 (t x) d 2 d1 d For the sake of achieving synchronization, we choose the discontinuous controller u as follows: k e sign(e) u t [t2 m t2 m1 ) t0 0 m 01 2 u m (19) t [t2 m1 t2 m 2 ) m 01 2 sign(e) u in which km and ( 0) are constants to be designed later, u is the disturbance of u . Remark 4. From (19) it is easy to see that in interval [t2 m 1 t2 m 2 ) we add controller sign(e) u Thus, controller (19) is different from (6). The reason we use term sign(e) in interval [t2 m 1 t2 m 2 ) is that sign(e) can eliminate the adverse effects caused by uncertainties and external disturbances. The following theorem will give sufficient conditions of exponential synchronization between systems (16) and (17) with parameter uncertainty, model uncertainty and external disturbance.
Theorem 2. Suppose that the Assumptions 1-2 are satisfied. Then the response chaotic system (17) can be globally exponentially synchronized with drive system (16) under the discontinuous controller (19) if there exist constants km and r ( r 0) such that the following inequalities hold: 1) u d r (t t ) (20) 2) E ( m (t2 m 1 t2 m ) ) E ( (t2 m 2 t2 m 1 ) ) e 2 m2 2 m m 01 2 where m 2 L 2km , 2L and is the maximum eigenvalue of
A AT . Proof. Take the following Lyapunov candidate: 1 V eT e 2 Its time derivative along the solution of (18) is 1 DV eT ( A AT )e eT ( A2 y A1 x f 2 (t y ) f1 (t x) d 2 d1 ) 2 eT ( f (t y ) f (t x)) eT u (t ) 1 eT e LeT e d e eT u (t ) ( 2 L )eT e d e eT u (t ) 2 2
(21)
For t [t2 m t2 m 1 ) , by substituting u (t ) into the above inequality, one gets
DV 12 ( 2 L )eT e d e eT ( k m e sign(e) u ) 12 ( 2 L 2k m )eT e d e e u e Keep in mind that u d we have 1 DV ( 2 L 2km )eT e mV 2 Similarly, for t [t2 m 1 t2 m 2 ) we obtain 1 D V ( 2 L ) eT e V 2
(22)
(23)
The rest of the proof is similar to that of Theorem 1 and is omitted here. Remark 5. From controllers (6) and (20) it is obvious that r is closely depended on m and the length of intervals [t2 m t2 m 1 ) [t2 m 1 t2 m 2 ) . Thus, for the given r we can choose proper km and intervals [t2 m t2 m 1 ) [t2 m 1 t2 m 2 ) such that (6) and (20) are satisfied which means that the speed of exponential synchronization can be determined by the controller freely. 5. Numerical simulations In this section, the fractional-order chaotic economical system [26] is taken as an example to verify and demonstrate the effectiveness of the proposed method. The fractional-order chaotic economical system to be investigated in present paper was proposed by Chen [26]. It has three state variables x1 x2 and x3 which stand
for the interest rate, the investment demand, and the price index, respectively. The dynamical equations of the fractional-order chaotic economical system is given as: a 0 1 x1 x1 x2 D x 0 b 0 x2 1 x12 Ax f ( x t ) (24) 1 0 c x 0 3 T where x ( x1 x2 x3 ) is the state variable of system (24), a b and c are three system parameters which are positive real constants. Fig. 1 illustrates the chaotic behavior of the financial system (24) for a 1 b 01 and c 1 .
In the simulation process, we take a 1 b 01 c 1 such that system (24) is chaotic. Furthermore, the order is fixed as 099 . Thus, we have 1 0 0 1 0 2 A 0 01 0 B AT A 0 02 0 1 0 0 0 1 2 The eigenvalues of B are 2 , 2 and 02 , respectively. Obviously, its maximum eigenvalue is 02 which implies that 02 . Note that y1 y2 x1 x2 (24) f (t y ) f (t x ) x12 y12 (25) 0
y2 x1 ( x1 y1 ) 0 0 0 y2 ( x1 y1 ) 0
0(26) y1 x1 (28) 0(27) y2 x2 (29) 0 y3 x3 x1 0 y1 x1 0 0 y2 x2 . 0 0 y3 x3
y2 x1 0 Therefore, we can take L max{ ( x1 y1 ) 0 0 } From Fig. 1 it is easy to see 0 0 0 that x1 2 x2 4 Thus, we can easily obtain L 73006 . Example I. The synchronization without uncertainties and disturbances Suppose system (24) is the drive system, in order to synchronize system (24) the response system with controller u is constructed as: a 0 1 y1 y1 y2 D y 0 b 0 y2 1 y12 u Ay f ( y t ) u (25) 1 0 c y 0 3
In order to achieve synchronization, we choose the discontinuous controller u
as follows: sint km e 2 sign(e) cost t [t2 m t2 m 1 ) t0 0 m 01 2 (26) u sintcost t [t2 m 1 t2 m 2 ) m 01 2 0 sint where cost denotes u and 2 . Thus, u 3 . Obviously, we have sintcost
u For convenience, we let km 20 t2 m 1 t2 m 002 t2 m 2 t2 m 1 001 m 01 2 then it is easy to check that 144012 . m 255988 0 and Thus, we can obtain E ( m (t2 m 1 t2 m ) ) E ( (t2 m 2 t2 m 1 ) ) 05867 11641 06830 e 126003 which means that r 126 Therefore, the conditions of Theorem 1 are all satisfied. According to Theorem 1, the exponential synchronization between the drive system (24) and the response system (25) will be achieved. Without loss of generality, in the simulation we choose the initial conditions as: x(0) (2 3 8)T y (0) (2 2 3)T . The simulation results shown in Figs. 2-4 which are respectively the time evolution of errors e1 e2 e3 between drive system (24) and the response system (25). FromFigs. 2-4 one can conclude that the synchronization errors converge quickly to zero and the exponential synchronization between system (24) and response system (25) is achieved.
Example II. The synchronization with uncertainties and disturbances In order to show the robust of our synchronization scheme to uncertainties and external disturbances, we add some uncertainties and disturbances to system (24). Thus system (24) can be rewritten as D x1 x3 ( x2 (1 01)) x1 01x1 x2 01cos (t ) 2 (27) D x2 1 (01 001) x2 x1 02 x3 x2 02 sin(t ) D x x (1 02) x 03 x x 03cos 2 (t ) 3 1 3 1 3 where
terms
01x1 001x2 02 x3
are
parameter
uncertainties.
01x1 x2 02 x3 x2 03x1 x3 are model uncertainties. 01cos(t ) 02sin(t ) 03cos 2 (t ) are external disturbances. Suppose system (27) is the drive system, then the corresponding response system with uncertainties and disturbances is given as: D y1 y3 ( y2 (1 01)) y1 01 y12 y2 01cos 2 (t ) u1 2 2 (28) D y2 1 (01 001) y2 y1 02 y3 y2 02cos (t ) u2 D y y (1 02) y 03 y 2 y 03sin 2 (t ) u 3 1 3 1 3 3 where
terms
01y1 001y2 02 y3
01 y12 y2 02 y32 y2 03 y12 y3
01cos 2 (t ) 02cos(t ) 03sin 2 (t ) controllers to be designed later.
are
are
parameter model
are external disturbances. u1 u2
uncertainties. uncertainties. and u3
are
For the sake of achieving exponential synchronization, the controller u (u1 u2 u3 )T is chosen as: sint km e sign(e) cost t [t2 m t2 m 1 ) t0 0 m 01 2 sintcost u sint sign(e) cost t [t2 m 1 t2 m 2 ) m 01 2 sintcost
(29)
For simplicity, in the numerical simulation, we assume km 20 t2 m 1 t2 m 002 t2 m 2 t2 m 1 001 m 01 2 20 . Similar to example 1 one gets m 255988 0 and 144012 . Therefore, we derive E ( m (t2 m 1 t2 m ) ) E ( (t2 m 2 t2 m 1 ) ) 06830 e 126003 which implies that r 126 Based on Theorem 2, the exponential synchronization between the drive system (24) and the response system (25) will be achieved. To confirm the validity of our presented scheme, we give numerical simulation with the following choices of the initial conditions: x(0) (1 25 4)T y (0) (2 2 6)T . Numerical results of the time evolution of errors e1 e2 e3 between system (27) and observer (28) with controller (29) are shown in Figs. 5-7, respectively. From Figs. 5-7 it is easy to see that although there are uncertainties and disturbances both in the drive system and the response system, the synchronization errors converge quickly to zero which means that the exponential synchronization between system (27) and response system (28) is reached. Remark 6. In Theorems 1 and 2, it is not easy to calculate u d . However, we have
no need of computing them directly in numerical simulation. Since u and d are all bounded, we can select to be large enough such that conditions 1 in Theorems 1 and 2 are all met. Remark 7. The synchronization of system (24) have been discussed in papers [27, 28]. It is easy to see that the controllers presented in papers [27, 28] are continuous function of states variables. Furthermore, the synchronization schemes proposed in papers [27, 28] are valid only for the asymptotical synchronization not for the exponential synchronization. 6. Conclusions The exponential synchronization of a class of fractional-order chaotic systems has been investigated. The sufficient conditions are derived by implementing the drive-response synchronization theory together with the Mittag-Leffler function. Two examples are used by means of fractional-order chaotic economical system to show the effectiveness of the developed method. Acknowledgments This work was jointly supported by the National Natural Science Foundation of China under Grant No. 11361043 and the Natural Science Foundation of Jiangxi Province under Grant No. 20161BAB201008. References [1] A. Pikovsky, M. Roseblum, J. Kurths, Synchronization: A universal concept in nonlinear sciences, Cambridge University Press, New York, NY, 2003.
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Fig. 1. The chaotic trajectories of system (24) with Fig. 2. The time evolution of error Fig. 3. The time evolution of error Fig. 4. The time evolution of error Fig. 5. The time evolution of error Fig. 6. The time evolution of error Fig. 7. The time evolution of error
e1 e2 e3 e1 e2 e3
099
and
between systems (24) and (25). between systems (24) and (25). between systems (24) and (25). between systems (27) and (28). between systems (27) and (28). between systems (27) and (28).
x1 (0) 1 x2 (0) 1 x3 (0) 1 .
4 3
x3
2 1 0 −1 4 2
2 1
0
0
−2 x2
−1 −4
−2
x1
0 −0.5 −1
e1
−1.5 −2 −2.5 −3 −3.5 −4
0
0.1
0.2
0.3 t
0.4
0.5
5 4.5 4 3.5
e2
3 2.5 2 1.5 1 0.5 0
0
0.1
0.2
0.3 t
0.4
0.5
0 −0.5 −1 −1.5
e3
−2 −2.5 −3 −3.5 −4 −4.5 −5
0
0.1
0.2
0.3 t
0.4
0.5
3 2.5 2
e1
1.5 1 0.5 0 −0.5 −1
0
0.02
0.04
0.06
0.08 t
0.1
0.12
0.14
0.5 0 −0.5 −1
e2
−1.5 −2 −2.5 −3 −3.5 −4 −4.5
0
0.02
0.04
0.06
0.08 t
0.1
0.12
0.14
3 2.5 2
e3
1.5 1 0.5 0 −0.5 −1
0
0.02
0.04
0.06
0.08 t
0.1
0.12
0.14