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Complete synchronization of commensurate fractional order chaotic systems using sliding mode control
Mechatronics xxx (2013) xxx–xxx
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Complete synchronization of commensurate fractional order chaotic systems using sliding mode control Abolhassan Razminia a, Dumitru Baleanu b,c,d,⇑ a
Electrical Engineering Department, School of Engineering, Persian Gulf University, Bushehr, Iran Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University, 06530 Ankara, Turkey c Institute of Space Sciences, Magurele–Bucharest, Romania d Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia b
a r t i c l e
i n f o
Article history: Received 4 December 2012 10 February 2013 Accepted 13 February 2013 Available online xxxx Keywords: Chaotic systems Fractional order dynamics Complete synchronization Sliding mode control
a b s t r a c t In this manuscript, we consider a new fractional order chaotic system which exhibits interesting behavior such as two, three, and four scrolls. Such systems can be found extensively in mechatronics and power electronic systems which exhibit self-sustained oscillations. Synchronization between two such systems is an interesting problem either theoretically or practically. Using a sliding mode control methodology, we synchronize a unidirectional coupling structure for the two chaotic systems. Numerical simulations are used to verify the theoretical analysis. Additionally, we report the robustness of the system in the presence of a noise in simulation. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Chaos theory and technology started to become well known as a promising research field with important impacts on an increasing number of novel, potentially attractive, time- and energy-critical engineering applications. As it is known the chaos control was developed by Ott et al. [1]. Since then, the effective methods such as adaptive method [2], back-stepping design [3], time-delay feedback control [4], active control [5], and nonlinear control [6] were devised to synchronize and control various chaotic systems. At present, most of methods for chaos control are especially designed for the physical systems, which are mainly used in the pure and applied science and engineering. This approach indicates the importance of the chaos control in the current technology. Examples of the chaos control in the mechatronics are abounding: piezoelectric vibration control [7], pendulum oscillation control [8], compass gait biped robot [9], self-sustained macroelectromechanical systems [10], and steer-by-wire vehicle systems [11]. Many mathematical definitions of chaos exist but roughly, it may be described as a type of dynamic behavior with the following characteristics [12]: extreme sensitivity to changes in initial conditions, random-like behavior, and deterministic motion. A regular chaotic system has one positive Lyapunov exponent. Systems with ⇑ Corresponding author at: Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University, 06530 Ankara, Turkey. Tel.: +90 3122331424. E-mail addresses: [email protected] (A. Razminia), [email protected] (D. Baleanu).
more than one positive Lyapunov exponent are called hyperchaotic and reveal more complicated dynamics that chaotic systems do. Generally, we have three problems in chaos literature: suppression, chaotization, and synchronization. In the following we describe briefly these basic topics: (1) The problems of stabilization of the unstable periodic solution (orbit) that is suppression of chaos [13], arise in suppression of noise and vibrations of various constructions, elimination of harmonics in the communication systems [14], electronic devices [15], and so on. These problems are distinguished by the fact that the controlled plant is strongly oscillatory. In other words, the eigenvalues of the matrix of the linearized system are close to the imaginary axis. The harmful vibrations can be either regular (quasiperiodic), or chaotic [16]. (2) The second class, chaotization, includes the control problems of excitation or generation of chaotic oscillations [17]. These problems are also called the chaotization or anticontrol. They arise where chaotic motion is the desired behavior of the system. The pseudorandom-number generators [17], sources of chaotic signals in communication and radar systems [12] are classical examples. Recent information suggests that chaotization of processes could produce an appreciable effect in the chemical and biological technologies [18], as well as in handling loose materials. These problems are characterized by the fact that the trajectory of the system phase vector is not predetermined, are unknown, or are of no consequence for attaining the objective.
0957-4158/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechatronics.2013.02.004
Please cite this article in press as: Razminia A, Baleanu D. Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics (2013), http://dx.doi.org/10.1016/j.mechatronics.2013.02.004
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(3) The third important class of the control objectives corresponds to the problem of synchronization or, more precisely, to achieve controllable synchronization as opposite to the auto-synchronization. Synchronization finds important applications in vibration technology [7], communications [14], biology and ecology [19], and many others. Several publications on control of synchronization of the chaotic processes and their application in the data transmission systems appeared during nineties [7]. On the other hand in the past few decades, much attention has been drawn for the study of fractional calculus [7]. As a branch of mathematical analysis, fractional calculus is a generalization of integration and differentiation to arbitrary non-integer orders. The applications of fractional calculus have been intensively investigated in many research fields, covering automatic control, informatics, materials, mechatronics, physics and so on. Especially, the study on the dynamics of fractionalorder differential systems has attracted increasing affection from many researchers [20–27]. In this manuscript we consider a new chaotic system which has been recently introduced in the scientific community of chaos. This system was presented and analyzed in [28]. After fractionalizing it, we will make a unidirectional coupling structure for two such systems that are run in different initial conditions. We call this structure as master–slave structure. As it was mentioned before, a small discrepancy between initial conditions makes the chaotic system behavior unpredictable in the long run. Thus designing a synchronization scheme via a control law may be useful in various applied fields. One of these fields is secure chaos-based communication systems that use the synchronization structure as its important parts. Using a sliding mode control the slave trajectories are forced to track the master trajectories asymptotically. Such type of synchronization is called as complete synchronization. The main objective of sliding mode control [29,30] is to switch the control law to force the states or pseudo-states of the system from the initial states onto some predefined sliding surface. The system on the sliding surface has desirable properties such as stability and disturbance attenuation capability. Sliding mode control is well known as an effective robust control strategy. Recently several works have been performed in the control of fractional order chaotic systems [30,31]. In this manuscript we propose an active sliding mode control that synchronizes two fractional order chaotic systems with various behaviors. The plan of the manuscript is as follows. Section 2 provides some preliminaries in fractional calculus. Section 3 briefly presents an introduction to the novel chaotic system. Section 4 studies the synchronization scheme using sliding mode techniques. The simulation results are presented in Section 5. Conclusions close the manuscript in Section 6. 2. Preliminaries In this section some mathematical backgrounds are presented. The definitions can be found in any standard textbook on fractional calculus; e.g. [21–23]. The fractional order integral operator of a Lebesgue integrable function x(t) is defined as follows: q a Dt xðtÞ
:¼
1 CðqÞ
Z
t
ðt sÞq1 xðsÞds;
q 2 Rþ ;
ð1Þ
a
R1 in which CðqÞ ¼ 0 ez zq1 dz; q > 0 is the Gamma function. The left fractional order derivative operator in the sense of Riemann–Liouville (LRL) is defined as:
m
a
RL
Dqt xðtÞ :¼ Dm a DðmqÞ xðtÞ ¼ t
1 d Cðm qÞ dtm
Z
t
ðt sÞmq1 xðsÞds;
a
m 1 < q < m 2 Zþ : ð2Þ Remark 1 [32]. For fractional derivative and integral RL operators we have:
q L a Dq t xðtÞ ¼ s XðsÞ; lim0 Dq t xðtÞ q!m
¼ 0 Dm t xðtÞ;
q > 0;
m 2 Zþ :
As one can see RL differentiation of a constant is not zero; moreover its Laplace transform needs fractional derivatives of the function in initial time. The Caputo operator solves these problems. The left fractional order derivative operator in the sense of Caputo is defined as follows:
a
C
Dqt xðtÞ :¼ a RL DtðmqÞ Dm xðtÞ ¼
1 Cðm qÞ
Z
t
ðt sÞmq1 xðmÞ ðsÞds;
a
m 1 < q < m 2 Zþ : ð3Þ Remark 2 [32]. For fractional Caputo derivative operator we have the following properties: 0
0
0
C
Dqt c ¼ 0;
C
q RL q Dqt 0 Dq t xðtÞ ¼ 0 Dt 0 Dt xðtÞ ¼ xðtÞ;
RL
Dqt 0 RL Dq t xðtÞ ¼ xðtÞ;
0 < q < 1;
q 2 Rþ :
Usually a dynamical system with fractional order could be described by:
(
0
C
Dqt xðtÞ ¼ f ðxðtÞ; tÞ; m 1 < q < m 2 Zþ ;
x ðtÞjt ¼ 0 ¼ xk0 ; ðkÞ
k ¼ 1; 2; . . . ; m:
t>0
;
ð4Þ
where x 2 Rn , f : Rn R ! Rn , q ¼ ð q1 q2 qn ÞT are vector state, nonlinear vector field, and differentiation order vector. If q1 = q2 = = qn we call (4) commensurate fractional order dynamical system; otherwise we call it incommensurate one. For simplicity in notation, we use Dq for the Caputo derivative. Moreover, sum of the P orders of all involved derivatives in Eq. (4), i.e. ni¼1 qi is called the effective dimension of Eq. (4) [33]. The size of vector x in state space form (4), i.e. n, is called the inner dimension of system (4) [34]. Theorem 1 [34]. Consider the following linear fractional order system: 0
C
Dqt xðtÞ ¼ AxðtÞ;
xð0Þ ¼ x0 ;
ð5Þ T
with x e R , A e R , q ¼ ð q1 q2 qn Þ , 0 < qi 6 1 and qi ¼ ndii , gcdðni ; di Þ ¼ 1. Let M be the lowest common multiple of the denominators di’s. The zero solution of system (5) is globally asymptotically stable in the Lyapunov sense if all roots k’s of the equation: n
nn
DðkÞ ¼ detðdiagðkMqi Þ AÞ ¼ 0; Satisfy : j argðki Þj >
p 2M
ð6Þ
:
3. System description 3.1. A mechatronic prototype As a motivation it can be mentioned that chaos in power electronic circuits has been investigated in late 1980s [35]. It has been
Please cite this article in press as: Razminia A, Baleanu D. Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics (2013), http://dx.doi.org/10.1016/j.mechatronics.2013.02.004
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shown that simple power electronic circuits called DC–DC converters having switching nonlinearities have a rich chaotic behavior [36]. The chaotic behavior of industrial motor drives has been reported in [37]. It has also been attempted to investigate the chaotic behavior of brushless DC motor drives by ignoring the switching effect and approximately transforming into the Lorentz system [37]. In fact, the series motor is the most widely used DC motor for electric traction applications. Moreover, it is widely used in various mechatronic applications. Analytical investigation of chaotic behavior of DC series motor drive is an obscure topic in power electronics literature. The major reasons should be due to two factors: the complexity of analytical formulation when considering the nonlinearity of the motor itself along with nonlinearity caused by the switching effect; and memory effect of the configuration because of energy saving elements. For instance chaotic behavior of chopper fed DC series motor can generate various steady state behaviors. A schematic diagram of such system is shown in Fig. 1. A numerical study of this system has been presented in [38]. More discussion about advances in mechatronics can be found in [39]. Note that there are two complexities in this circuit: chaos and hereditary property which indicates the memory of the system. The former corresponds to nonlinearity in the system due to the switching operation and other nonlinear elements. However, the memory effect due to some physical phenomena may introduce the fractional order differential equations in describing the system. Some logical argument for using such tool is presented in the next subsection. 3.2. An interesting dynamical system Consider the three-dimensional autonomous system proposed in [28]. This system has very rich nonlinear dynamics, including chaos, period doubling bifurcations, and others. Moreover, this system can generate two-scroll chaotic attractor. However, by varying a single parameter, a new three-scroll chaotic attractor is detected in the novel three-dimensional smooth system. This three-scroll chaotic attractor evolves into a four-scroll chaotic attractor in some way. The chaotic system is described as [28]:
1 0 1 0 x_ y ax þ byz C B_C B @ y A ¼ @ cy xz þ z A; dxy Hz z_
ð7Þ
3
where [x(t), y(t), z(t)]T e R3 is the state vector, and {a, b, c, d, h} e R+ are some positive constants. It is well-known that the chaotic systems have strange limit sets with a non-integer Lyapunov dimension. Indeed chaotic behaviors are usual in self-sustained mechatronic devices. For instance, several Chua circuits have been developed with various limit sets [36]. This circuit is a useful prototype circuit that can generates several strange trajectories which are suitable for chaos research. Fractional order cases of such systems have not been studied thoroughly. The main reason is its mathematical complexity. Moreover, a prototype of a mechatronic system with chaotic behavior has been reported in the previous subsection. A proper tool for analyzing such systems is the fractional order modeling. Therefore a more accurate model for such systems is its fractional order dynamic which is now described. Our main motivation for considering the fractional order derivative instead of the classical integer order ones, is that for most physical systems which exhibit chaotic behavior, the invariant set is not an integer-order dimensional object. Therefore, the domain of the trajectories in the phase space is a strange field which called strange attractors whose Lyapunov dimensions are usually non-integer. The presented integer-order system in Eq. (7) has a chaotic behavior for a wide range of parameters. Such integer-order dynamical system may be a reduced model for a physical system. If such system is implemented using physical electronic devices, e.g., the DC series motor, the environment effects (aging of the elements, temperature, inaccurate values, and so on) on the elements may appear as a different behavior so that the response predicted by the model does not resemble by the actual system. Indeed for such chaotic systems, the super-sensitivity to tiny changes in the element values, cannot be considered clearly in the dynamical equations. The main reason is related to the fractal or holed basin of the invariant set of the system. In other words, some trace of the system trajectories may be seen in the observed coordinate (phase plane) and some of them lie in the unseen region which cannot be handled by the classical model (nonlinear ODEs). Thus a useful and proper tool for model of such phenomena is the fractional calculus. In particular, inserting the fractional orders operator instead of the integer ones, is a compensation for this strange behavior. The fractional order operators can modify the observed behavior by considering a kernel in the integral. This kernel can be treated as a weighting factor which generates a new response using existing vector fields. Regarding these discussions, the fractional order version of the system presented in [28] is as follows:
Fig. 1. The schematic diagram of DC series drive.
Please cite this article in press as: Razminia A, Baleanu D. Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics (2013), http://dx.doi.org/10.1016/j.mechatronics.2013.02.004
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0
1
0
1
y ax þ byz Dq x C B q C B @ D y A ¼ @ cy xz þ z A; q dxy Hz D z
ð8Þ
in which q is a real quantity. For finding the equilibrium points of the proposed system it is enough to equate the left hand side of (7) to zero. Thus, the equilibrium points are:
Q 1 : ð 0 0 0 Þ; pffiffiffi pffiffiffiffiffiffiffi p1þ ffiffiffi K Q 2 : dþ2d D hb 1þ dþ D pffiffiffi pffiffiffiffiffiffiffi p1þ ffiffiffi K Q 3 : dþ2d D hb 1 dþ D pffiffiffi pffiffiffiffiffiffiffi p1þ ffiffiffi K Q 4 : d2d D hb 1þ d D pffiffiffi pffiffiffiffiffiffiffi p1þ ffiffiffi K Q 5 : d2d D hb 1 d D
pffiffiffiffiffiffiffi 1þ 1þK 2b pffiffiffiffiffiffiffi 1 1þK ;; 2b pffiffiffiffiffiffiffi 1þ 1þC ; 2b pffiffiffiffiffiffiffi 1 1þC 2b
The Jacobian matrix for (7) evaluated in the equilibrium point Qi: (x⁄, y⁄, z⁄), i = 1, 2, . . ., 5, where x⁄, y⁄, and z⁄ are the coordination of the equilibrium, can be calculated as follows:
0
a
B J ¼ @ z dy
1 þ bz c
dx
by
1
C 1 x A :
ð10Þ
h
It has been shown that for the following quantities there exist various behaviors [28]:
ðiÞða; b; c; d; hÞ ¼ ð3; 2:7; 4:7; 2; 9Þ; ð9Þ
ðiiÞða; b; c; d; hÞ ¼ ð3; 2:7; 1:7; 2; 9Þ;
ð11Þ
ðiiiÞða; b; c; d; hÞ ¼ ð3; 2:7; 3:9; 2; 9Þ; where (i–iii) correspond to two, three, and four scrolls respectively. The numerical simulations for these chaotic attractors are depicted in Figs. 2–5.
where 2
D ¼ d þ 4chd;
K¼
pffiffiffiffi 2ab d þ 2ch þ D ; h
pffiffiffiffi 2ab d þ 2ch D : C¼ h
4. Synchronization scheme Consider the master–slave synchronization scheme of two autonomous different fractional order chaotic systems:
Fig. 2. Chaotic attractor of system (8) with (a, b, c, d, h) = (3, 2.7, 4.7, 2, 9), (2-scroll) with initial conditions (x0, y0, z0) = (6, 1, 3) and order q = 0.9.
Fig. 3. Phase portrait of the system (8) with order q = 0.9 and (a, b, c, d, h) = (3, 2.7, 4.7, 2, 9).
Please cite this article in press as: Razminia A, Baleanu D. Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics (2013), http://dx.doi.org/10.1016/j.mechatronics.2013.02.004
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Fig. 4. Chaotic attractor of system (8) with (a, b, c, d, h) = (3, 2.7, 3.9, 2, 9), (4-scroll) with initial conditions (x0, y0, z0) = (6, 1, 3) and orders q = 0.9.
Fig. 5. Phase portrait of the system (8) with order q = 0.9 and (a, b, c, d, h) = (3, 2.7, 3.9, 2, 9).
Dq xðtÞ ¼ f ðxÞ;
Master : Slave :
ð12Þ
q
D yðtÞ ¼ gðyÞ þ u;
where x, y e Rn represent the states of the drive and the response systems, respectively. Moreover, f: Rn ? Rn, g: Rn ? Rn are the vector fields of the drive and response systems, respectively. The aim is to choose a suitable control function u = (u1, u2, . . ., un)T such that the states of the drive and response systems are synchronized asymptotically, i.e. lim t?1ky(t) x(t)k = 0. For the system (8), we construct the following master–slave structure in which the subscripts m and s stand for master and slave systems, respectively:
0
1 0 1 ym axm þ bym zm Dq xm B q C B C Master system : @ D ym A ¼ @ cym xm zm þ zm A; ; dxm ym Hzm D q zm ðxm ð0Þ; ym ð0Þ; zm ð0ÞÞ ¼ ðxm0 ; ym0 ; zm0 Þ
where u1, u2 and u3 are nonlinear controllers to be designed such that the two chaotic system can be synchronized. For achieving this goal, let define the error variables as follows:
e1 ðtÞ ¼ xs ðtÞ xm ðtÞ; e2 ðtÞ ¼ ys ðtÞ ym ðtÞ;
ð15Þ
ee ðtÞ ¼ zs ðtÞ zm ðtÞ: Substituting Eqs. (13) and (14) in (15) we have:
Dq e1 ¼ e2 ae1 þ bys e3 þ bzm e2 þ u1 ; Dq e2 ¼ ce2 þ e3 xs e3 zm e1 þ u2 ;
ð16Þ
Dq e2 ¼ he3 þ dxs e2 þ dym e1 þ u3 : Now we select the control laws as:
ð13Þ
u1 ¼ bzm e2 bys e3 þ v 1 ; u2 ¼ xs e3 þ zm e1 þ v 2 ;
ð17Þ
u3 ¼ dym e1 dxs e2 þ v 3 : and
0
D q xs
1
0
ys axs þ bys zs
1
0
u1
As one knows in active sliding mode control, vi(t)’s (i = 1, 2, 3) are designed based on a sliding mode control law:
1
B C B C B C Slave system : @ Dq ys A ¼ @ cys xs zs þ zs A þ @ u2 A; q u3 dxs ys Hzs D zs ðxs ð0Þ; ys ð0Þ; zs ð0ÞÞ ¼ ðxs0 ; ys0 ; zs0 Þ;
ð14Þ
v i ðtÞ ¼ ki wðtÞ;
ð18Þ
where ki’s (i = 1, 2, 3) are constant gains and w(t) is the actuating signal with the following rule:
Please cite this article in press as: Razminia A, Baleanu D. Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics (2013), http://dx.doi.org/10.1016/j.mechatronics.2013.02.004
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A. Razminia, D. Baleanu / Mechatronics xxx (2013) xxx–xxx
wðtÞ ¼
wþ ðtÞ sðeÞ P 0; ; w ðtÞ sðeÞ < 0
ð19Þ
in which s = s(e) is a switching surface that prescribes the desired dynamics. Thus the error dynamics (16) can be rewritten as:
Dq e1 ¼ e2 ae1 þ k1 w; Dq e2 ¼ ce2 þ e3 þ k2 w;
ð20Þ
k1 ðk1 l1 þ k2 l2 þ k3 l3 Þ e1 Dq e1 ¼ al1 a k1 ðk1 l þ k2 l2 þ k3 l3 Þ 1 ðk l þ k l þ k l Þ l1 þ cl2 1 1 2 2 3 3 e2 þ ðhl3 l2 Þe3 ; k1 k2 ðk1 l1 þ k2 l2 þ k3 l3 Þ ðk1 l1 þ k2 l2 þ k3 l3 Þ al1 e1 l1 þ cl2 þ c e2 k2 ðk1 l1 þ k2 l2 þ k3 l3 Þ e3 ; þ hl3 l2 þ k2
Dq e2 ¼
Dq e2 ¼ he3 þ k3 w: A formal sliding surface can be selected as:
sðeÞ ¼ l1 e1 þ l2 e2 þ l3 e3 ;
ð21Þ
where li’s are some constants. The equivalent control is found by the fact that s_ ðeÞ ¼ 0 is a necessary condition for the pseudo-state trajectories to stay on the switching surface s(e) = 0. Hence, when in sliding mode, the controlled system satisfies the following conditions:
sðeÞ ¼ 0;
s_ ðeÞ ¼ 0:
ð22Þ
Using the chain rule one can write:
ð23Þ
ðk1 l1 D1q þ k2 l2 D1q þ k3 l3 D1q ÞwðtÞ ð24Þ
As the second step let design the sliding mode control law. To do this the reaching law is selected as:
ð27Þ
Dq s ¼ l1 Dq e1 þ l2 Dq e2 þ l3 Dq e3
ð28Þ
Dq s ¼l1 Dq e1 þ l2 Dq e2 þ l3 Dq e3 ¼ l1 ðe2 ae1 þ k1 wÞ þ l2 ðce2 þ e3 þ k2 wÞ
which results in:
þ l3 ðhe3 þ k3 wÞ ¼ cs ¼ cl1 e1 cl2 e2 cl3 e3 :
1 ðal1 e1 ðl1 þ cl2 Þe2 ðk1 l1 þ k2 l2 þ k3 l3 Þ þ ðhl3 l2 Þe3 Þ:
al1 e1 ðl1 þ cl2 Þe2 ðk1 l1 þ k2 l2 þ k3 l3 Þ ðk1 l1 þ k2 l2 þ k3 l3 Þ þ hl3 l2 h e3 : k3
in which the obtained error dynamics in Eq. (26) should be inserted. Now using (27) and (28) the control signal w can be computed as in the previous step. Indeed we obtain:
A direct result is:
wðtÞ ¼
The positive gain c is determined such that the sliding condition is satisfied and the sliding mode motion occurs. Based on Eq. (21) one can easily deduce that:
¼ l1 D1q ðe2 ae1 þ k1 wÞ þ l2 D1q ðce2 þ e3 þ k2 wÞ
¼ l1 D1q ðe2 ae1 Þ l2 D1q ðce2 þ e3 Þ þ l3 D1q ðhe3 Þ;
k3
Dq s ¼ cs:
@sðeÞ e_ @e 1q ¼ l1 D ðDq e1 Þ þ l2 D1q ðDq e2 Þ þ l3 D1q ðDq e3 Þ
s_ ðeÞ ¼
þ l3 D1q ðhe3 þ k3 wÞ ¼ 0:
Dq e2 ¼
ð26Þ
ð29Þ
Therefore we conclude that:
ð25Þ
Notice that this control term is realizable if and only if the term k1l1 + k2l2 + k3l3 is not zero. Thus, the error dynamics can be written as follows:
w¼
c ðk1 l1 þ k2 l2 þ k3 l3 Þ
½l1 þ aðk1 l1 þ k2 l2 þ k3 l3 Þl1 e1
þ½l2 þ ðl1 cl2 Þðk1 l1 þ k2 l2 þ k3 l3 Þe2 þ½l3 þ ðl2 þ hl3 Þðk1 l1 þ k2 l2 þ k3 l3 Þe3 :
ð30Þ
Fig. 6. Time responses for synchronization using sliding mode control with initial conditions (xm0, ym0, zm0) = (6, 1, 3) and (xs0, ys0, zs0) = (2, 4, 1).
Please cite this article in press as: Razminia A, Baleanu D. Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics (2013), http://dx.doi.org/10.1016/j.mechatronics.2013.02.004
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Now it is easy to see that using this w the error dynamics reported in Eq. (20) is asymptotically stable. Indeed it is straightforward to show that one of the eigenvalues of the error dynamic is c. The other two eigenvalues depends on the coefficients of kis and lis which can be selected such that they lie in the stability region. In the next section we examine numerically the proposed method. 5. Simulation results In this section, to demonstrate the effectiveness of the proposed methods, we will present the numerical results for synchronizing chaotic systems (13) and (14) under the control laws provided in the previous section. We selected the initial conditions (xm0, ym0, zm0) = (6, 1, 3) for the master system and (xs0, ys0, zs0) = (2, 4, 1) for the slave system. Note that we have chosen the initial conditions very different intentionally. Driving the simulations we turn on the control at the instant t = 1.5. As it can be seen from Fig. 6, after an initial transient, synchronization is achieved completely and the errors converge to zero asymptotically. An important point that must be noted is that the above results have been derived based on the general values of parameters. Therefore the proposed method is valuable for synchronizing the chaotic systems where they exhibit 2-scrolls, 3-scrolls or 4-scrolls. 6. Conclusions This work discusses a sliding mode control scheme for synchronizing two fractional order chaotic systems. Indeed the integer order chaotic system is a novel that recently has been introduced in the research societies. The main feature of this system is its interesting behavior. Indeed the system presents various chaotic attractors such as 2, 3, or 4 scrolls. However, since most electronic configurations with self-sustained oscillations have strange limit set which can be regarded as a fractal object, the more proper model for such systems is the fractional order one. Based on the sliding mode control technique, we developed a procedure that guarantees the asymptotical stability of the error dynamics which means the synchronization is achieved completely. Our proposed method is valuable for each case: both master and slave systems are 2scrolls, or 3-scrolls, or 4-scrolls. Numerical simulations have been used for clarify the effectiveness of the proposed control laws. Acknowledgement Authors would like to thank to the referees for their interesting comments and remarks. References [1] Ott E, Grebogi C, Yorke JA. Controlling chaos. Phys Rev Lett 1990;65:1196–9. [2] Shi XRg, Wang ZL. Adaptive added-order anti-synchronization of chaotic systems with fully unknown parameters. Appl Math Comput 2009;215:1711–7. [3] Chen F, Zhang W. Stabilization of parameters perturbation chaotic system via adaptive backstepping technique. Appl Math Comput 2008;200:101–9. [4] Park JH, Ji DH, Won SC, Lee SM. Hinf synchronization of time-delayed chaotic systems. Appl Math Comput 2008;204:170–7. [5] Razminia A, Majd VJ, Baleanu D. Chaotic incommensurate fractional order Rossler system: active control synchronization. Adv Differ Equ 2011;15:1–12. [6] Zhu C. Feedback control methods for stabilizing unstable equilibrium points in a new chaotic system. Nonlinear Anal-Theor 2009;71:2441–6. [7] Li S, Qiu j, Ji H, Zhu K, Li J. Piezoelectric vibration control for all-clamped panel using DOB-based optimal control. Mechatronics 2011;21:1213–21.
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[8] Fradkov A, Andrievsky B, Boykov K. Control of the coupled double pendulums system. Mechatronics 2005;15:1289–303. [9] Gritli H, Khraief N, Belghith S. Chaos control in passive walking dynamics of a compass-gait model. Commun Nonlinear Sci Numer Simulat 2013;18:2048–65. [10] Kwuimy CAK, Nana B, Woafo P. Experimental bifurcations and chaos in a modified self-sustained macro electromechanical system. J Sound Vib 2010;329:3137–48. [11] Chang SC. Synchronization in a steer-by-wire vehicle dynamic system. Int J Eng Sci 2007;45:628–43. [12] Zhang H, Liu D, Wang Z. Controlling chaos, suppression, synchronization and chaotification. Springer; 2009. [13] Pai NS, Yau HT. Suppression of chaotic behavior in horizontal platform systems based on an adaptive sliding mode control scheme. Commun Nonlinear Sci Numer Simulat 2011;16:133–43. [14] Yang T. A survey of chaotic secure communication systems. Int J Comput Cogn 2004;2:81–130. [15] Hung YC, Liao TL, Yan JJ. Adaptive variable structure control for chaos suppression of unified chaotic systems. Appl Math Comput 2009;209:391–8. [16] Ge ZM, Yang CH. Chaos synchronization and chaotization of complex chaotic systems in series form by optimal control. Chaos, Solitons Fract 2009;42:994–1002. [17] Fradkov AL, Evans RJ. Control of chaos: methods and applications in engineering. Ann Rev Control 2005:33–56. [18] Yu H, Zhong S, Agrawal RP. Mathematics analysis and chaos in an ecological model with an impulsive control strategy. Commun Nonlinear Sci Numer Simulat; in press, accessed 24.04.10. [19] Chen DY, Zhao WL, Ma XY, Zhang RF. No-chattering sliding mode control chaos in Hindmarsh–Rose neurons with uncertain parameters. Comput Math Appl 2011;61:3161–71. [20] Hilfer R. Applications of fractional calculus in physics. Hackensack (NJ): World Scientific; 2001. [21] Miller KS, Ross B. An introduction to the fractional calculus and fractional differential equations. New York (NY, USA): A Wiley-Interscience Publication, JohnWiley & Sons; 1993. [22] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. Amsterdam: Elsevier Science; 2006. [23] Baleanu D, Diethelm K, Scalas E, Trujillo JJ. Fractional calculus models and numerical methods (series on complexity, nonlinearity and chaos). World Scientific; 2012. [24] Ionescu C, De Keyser R. Relations between fractional order model parameters and lung pathology in chronic obstructive pulmonary disease. IEEE Trans Biomed Eng 2009;56(4):978–87. [25] Ionescu C, Segers P, DeKeyser R. Mechanical properties of the respiratory system derived from morphologic insight. IEEE Trans Biomed Eng 2009;56(4):949–59. [26] Baleanu D, Mustafa OG, Agarwal RP. An existence result for a superlinear fractional differential equation. Appl Math Lett 2010;23:1129–32. [27] Baleanu D, Machado JAT, Luo A. Fractional dynamics and control. Springer; 2012. [28] Dadras S, Momeni HR. A novel three-dimensional autonomous chaotic system generating two, three and four-scroll attractors. Phys Lett A 2009;373:3637–42. [29] Yin C, Zhong S, Chen W. Design of sliding mode controller for a class of fractional-order chaotic systems. Commun Nonlinear Sci Numer Simulat 2012;17:356–66. [30] Hosseinnia SH, Ghaderi R, Ranjbar AN, Mahmoudian M, Momani S. Sliding mode synchronization of an uncertain fractional order chaotic system. Comput Math Appl 2010;59:1637–43. [31] Tavazoei MS, Haeri M. Synchronization of chaotic fractional-order systems via active sliding mode controller. Physics A 2008;387:57–70. [32] Li C, Deng W. Remarks on fractional derivatives. Appl Math Comput 2007;187:777–84. [33] Wang Y, Li C. Does the fractional Brusselator with efficient dimension less than 1 have a limit cycle? Phys Lett A 2007;363:414–9. [34] Matignon D. Stability results for fractional differential equations with applications to control processing. In: Computational engineering in systems and application multi-conference, IMACS, IEEE-SMC Proceedings, vol. 2. Lille, France, July 1996. p. 963–8. [35] Deane JHB, Hamill DC. Instability, sub harmonics and chaos in power electronic system. IEEE Trans Power Electron 1990;5:260–8. [36] Chakrabarty K, Poddar G, Banerjee S. Bifurcation behavior of the buck converter. IEEE Trans Power Electron 1999;11:439–47. [37] Hemati N. Strange attractor in brushless DC motor. IEEE Trans Circ Syst – I: Fund Theory Appl 1994;41:40–5. [38] Chakrabarty K, Kar U, Kundu S. Chaotic behavior in a DC series drive. Roorkee: Indian Institute of Technology; 2010. [39] Rosario JM, Dumur D, Machado JAT. Control of a 6-DOF parallel manipulator through a mechatronic approach. J Vib Control 2007;13:1431–46.
Please cite this article in press as: Razminia A, Baleanu D. Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics (2013), http://dx.doi.org/10.1016/j.mechatronics.2013.02.004
Contents lists available at SciVerse ScienceDirect
Mechatronics journal homepage: www.elsevier.com/locate/mechatronics
Complete synchronization of commensurate fractional order chaotic systems using sliding mode control Abolhassan Razminia a, Dumitru Baleanu b,c,d,⇑ a
Electrical Engineering Department, School of Engineering, Persian Gulf University, Bushehr, Iran Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University, 06530 Ankara, Turkey c Institute of Space Sciences, Magurele–Bucharest, Romania d Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia b
a r t i c l e
i n f o
Article history: Received 4 December 2012 10 February 2013 Accepted 13 February 2013 Available online xxxx Keywords: Chaotic systems Fractional order dynamics Complete synchronization Sliding mode control
a b s t r a c t In this manuscript, we consider a new fractional order chaotic system which exhibits interesting behavior such as two, three, and four scrolls. Such systems can be found extensively in mechatronics and power electronic systems which exhibit self-sustained oscillations. Synchronization between two such systems is an interesting problem either theoretically or practically. Using a sliding mode control methodology, we synchronize a unidirectional coupling structure for the two chaotic systems. Numerical simulations are used to verify the theoretical analysis. Additionally, we report the robustness of the system in the presence of a noise in simulation. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Chaos theory and technology started to become well known as a promising research field with important impacts on an increasing number of novel, potentially attractive, time- and energy-critical engineering applications. As it is known the chaos control was developed by Ott et al. [1]. Since then, the effective methods such as adaptive method [2], back-stepping design [3], time-delay feedback control [4], active control [5], and nonlinear control [6] were devised to synchronize and control various chaotic systems. At present, most of methods for chaos control are especially designed for the physical systems, which are mainly used in the pure and applied science and engineering. This approach indicates the importance of the chaos control in the current technology. Examples of the chaos control in the mechatronics are abounding: piezoelectric vibration control [7], pendulum oscillation control [8], compass gait biped robot [9], self-sustained macroelectromechanical systems [10], and steer-by-wire vehicle systems [11]. Many mathematical definitions of chaos exist but roughly, it may be described as a type of dynamic behavior with the following characteristics [12]: extreme sensitivity to changes in initial conditions, random-like behavior, and deterministic motion. A regular chaotic system has one positive Lyapunov exponent. Systems with ⇑ Corresponding author at: Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University, 06530 Ankara, Turkey. Tel.: +90 3122331424. E-mail addresses: [email protected] (A. Razminia), [email protected] (D. Baleanu).
more than one positive Lyapunov exponent are called hyperchaotic and reveal more complicated dynamics that chaotic systems do. Generally, we have three problems in chaos literature: suppression, chaotization, and synchronization. In the following we describe briefly these basic topics: (1) The problems of stabilization of the unstable periodic solution (orbit) that is suppression of chaos [13], arise in suppression of noise and vibrations of various constructions, elimination of harmonics in the communication systems [14], electronic devices [15], and so on. These problems are distinguished by the fact that the controlled plant is strongly oscillatory. In other words, the eigenvalues of the matrix of the linearized system are close to the imaginary axis. The harmful vibrations can be either regular (quasiperiodic), or chaotic [16]. (2) The second class, chaotization, includes the control problems of excitation or generation of chaotic oscillations [17]. These problems are also called the chaotization or anticontrol. They arise where chaotic motion is the desired behavior of the system. The pseudorandom-number generators [17], sources of chaotic signals in communication and radar systems [12] are classical examples. Recent information suggests that chaotization of processes could produce an appreciable effect in the chemical and biological technologies [18], as well as in handling loose materials. These problems are characterized by the fact that the trajectory of the system phase vector is not predetermined, are unknown, or are of no consequence for attaining the objective.
0957-4158/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechatronics.2013.02.004
Please cite this article in press as: Razminia A, Baleanu D. Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics (2013), http://dx.doi.org/10.1016/j.mechatronics.2013.02.004
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(3) The third important class of the control objectives corresponds to the problem of synchronization or, more precisely, to achieve controllable synchronization as opposite to the auto-synchronization. Synchronization finds important applications in vibration technology [7], communications [14], biology and ecology [19], and many others. Several publications on control of synchronization of the chaotic processes and their application in the data transmission systems appeared during nineties [7]. On the other hand in the past few decades, much attention has been drawn for the study of fractional calculus [7]. As a branch of mathematical analysis, fractional calculus is a generalization of integration and differentiation to arbitrary non-integer orders. The applications of fractional calculus have been intensively investigated in many research fields, covering automatic control, informatics, materials, mechatronics, physics and so on. Especially, the study on the dynamics of fractionalorder differential systems has attracted increasing affection from many researchers [20–27]. In this manuscript we consider a new chaotic system which has been recently introduced in the scientific community of chaos. This system was presented and analyzed in [28]. After fractionalizing it, we will make a unidirectional coupling structure for two such systems that are run in different initial conditions. We call this structure as master–slave structure. As it was mentioned before, a small discrepancy between initial conditions makes the chaotic system behavior unpredictable in the long run. Thus designing a synchronization scheme via a control law may be useful in various applied fields. One of these fields is secure chaos-based communication systems that use the synchronization structure as its important parts. Using a sliding mode control the slave trajectories are forced to track the master trajectories asymptotically. Such type of synchronization is called as complete synchronization. The main objective of sliding mode control [29,30] is to switch the control law to force the states or pseudo-states of the system from the initial states onto some predefined sliding surface. The system on the sliding surface has desirable properties such as stability and disturbance attenuation capability. Sliding mode control is well known as an effective robust control strategy. Recently several works have been performed in the control of fractional order chaotic systems [30,31]. In this manuscript we propose an active sliding mode control that synchronizes two fractional order chaotic systems with various behaviors. The plan of the manuscript is as follows. Section 2 provides some preliminaries in fractional calculus. Section 3 briefly presents an introduction to the novel chaotic system. Section 4 studies the synchronization scheme using sliding mode techniques. The simulation results are presented in Section 5. Conclusions close the manuscript in Section 6. 2. Preliminaries In this section some mathematical backgrounds are presented. The definitions can be found in any standard textbook on fractional calculus; e.g. [21–23]. The fractional order integral operator of a Lebesgue integrable function x(t) is defined as follows: q a Dt xðtÞ
:¼
1 CðqÞ
Z
t
ðt sÞq1 xðsÞds;
q 2 Rþ ;
ð1Þ
a
R1 in which CðqÞ ¼ 0 ez zq1 dz; q > 0 is the Gamma function. The left fractional order derivative operator in the sense of Riemann–Liouville (LRL) is defined as:
m
a
RL
Dqt xðtÞ :¼ Dm a DðmqÞ xðtÞ ¼ t
1 d Cðm qÞ dtm
Z
t
ðt sÞmq1 xðsÞds;
a
m 1 < q < m 2 Zþ : ð2Þ Remark 1 [32]. For fractional derivative and integral RL operators we have:
q L a Dq t xðtÞ ¼ s XðsÞ; lim0 Dq t xðtÞ q!m
¼ 0 Dm t xðtÞ;
q > 0;
m 2 Zþ :
As one can see RL differentiation of a constant is not zero; moreover its Laplace transform needs fractional derivatives of the function in initial time. The Caputo operator solves these problems. The left fractional order derivative operator in the sense of Caputo is defined as follows:
a
C
Dqt xðtÞ :¼ a RL DtðmqÞ Dm xðtÞ ¼
1 Cðm qÞ
Z
t
ðt sÞmq1 xðmÞ ðsÞds;
a
m 1 < q < m 2 Zþ : ð3Þ Remark 2 [32]. For fractional Caputo derivative operator we have the following properties: 0
0
0
C
Dqt c ¼ 0;
C
q RL q Dqt 0 Dq t xðtÞ ¼ 0 Dt 0 Dt xðtÞ ¼ xðtÞ;
RL
Dqt 0 RL Dq t xðtÞ ¼ xðtÞ;
0 < q < 1;
q 2 Rþ :
Usually a dynamical system with fractional order could be described by:
(
0
C
Dqt xðtÞ ¼ f ðxðtÞ; tÞ; m 1 < q < m 2 Zþ ;
x ðtÞjt ¼ 0 ¼ xk0 ; ðkÞ
k ¼ 1; 2; . . . ; m:
t>0
;
ð4Þ
where x 2 Rn , f : Rn R ! Rn , q ¼ ð q1 q2 qn ÞT are vector state, nonlinear vector field, and differentiation order vector. If q1 = q2 = = qn we call (4) commensurate fractional order dynamical system; otherwise we call it incommensurate one. For simplicity in notation, we use Dq for the Caputo derivative. Moreover, sum of the P orders of all involved derivatives in Eq. (4), i.e. ni¼1 qi is called the effective dimension of Eq. (4) [33]. The size of vector x in state space form (4), i.e. n, is called the inner dimension of system (4) [34]. Theorem 1 [34]. Consider the following linear fractional order system: 0
C
Dqt xðtÞ ¼ AxðtÞ;
xð0Þ ¼ x0 ;
ð5Þ T
with x e R , A e R , q ¼ ð q1 q2 qn Þ , 0 < qi 6 1 and qi ¼ ndii , gcdðni ; di Þ ¼ 1. Let M be the lowest common multiple of the denominators di’s. The zero solution of system (5) is globally asymptotically stable in the Lyapunov sense if all roots k’s of the equation: n
nn
DðkÞ ¼ detðdiagðkMqi Þ AÞ ¼ 0; Satisfy : j argðki Þj >
p 2M
ð6Þ
:
3. System description 3.1. A mechatronic prototype As a motivation it can be mentioned that chaos in power electronic circuits has been investigated in late 1980s [35]. It has been
Please cite this article in press as: Razminia A, Baleanu D. Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics (2013), http://dx.doi.org/10.1016/j.mechatronics.2013.02.004
A. Razminia, D. Baleanu / Mechatronics xxx (2013) xxx–xxx
shown that simple power electronic circuits called DC–DC converters having switching nonlinearities have a rich chaotic behavior [36]. The chaotic behavior of industrial motor drives has been reported in [37]. It has also been attempted to investigate the chaotic behavior of brushless DC motor drives by ignoring the switching effect and approximately transforming into the Lorentz system [37]. In fact, the series motor is the most widely used DC motor for electric traction applications. Moreover, it is widely used in various mechatronic applications. Analytical investigation of chaotic behavior of DC series motor drive is an obscure topic in power electronics literature. The major reasons should be due to two factors: the complexity of analytical formulation when considering the nonlinearity of the motor itself along with nonlinearity caused by the switching effect; and memory effect of the configuration because of energy saving elements. For instance chaotic behavior of chopper fed DC series motor can generate various steady state behaviors. A schematic diagram of such system is shown in Fig. 1. A numerical study of this system has been presented in [38]. More discussion about advances in mechatronics can be found in [39]. Note that there are two complexities in this circuit: chaos and hereditary property which indicates the memory of the system. The former corresponds to nonlinearity in the system due to the switching operation and other nonlinear elements. However, the memory effect due to some physical phenomena may introduce the fractional order differential equations in describing the system. Some logical argument for using such tool is presented in the next subsection. 3.2. An interesting dynamical system Consider the three-dimensional autonomous system proposed in [28]. This system has very rich nonlinear dynamics, including chaos, period doubling bifurcations, and others. Moreover, this system can generate two-scroll chaotic attractor. However, by varying a single parameter, a new three-scroll chaotic attractor is detected in the novel three-dimensional smooth system. This three-scroll chaotic attractor evolves into a four-scroll chaotic attractor in some way. The chaotic system is described as [28]:
1 0 1 0 x_ y ax þ byz C B_C B @ y A ¼ @ cy xz þ z A; dxy Hz z_
ð7Þ
3
where [x(t), y(t), z(t)]T e R3 is the state vector, and {a, b, c, d, h} e R+ are some positive constants. It is well-known that the chaotic systems have strange limit sets with a non-integer Lyapunov dimension. Indeed chaotic behaviors are usual in self-sustained mechatronic devices. For instance, several Chua circuits have been developed with various limit sets [36]. This circuit is a useful prototype circuit that can generates several strange trajectories which are suitable for chaos research. Fractional order cases of such systems have not been studied thoroughly. The main reason is its mathematical complexity. Moreover, a prototype of a mechatronic system with chaotic behavior has been reported in the previous subsection. A proper tool for analyzing such systems is the fractional order modeling. Therefore a more accurate model for such systems is its fractional order dynamic which is now described. Our main motivation for considering the fractional order derivative instead of the classical integer order ones, is that for most physical systems which exhibit chaotic behavior, the invariant set is not an integer-order dimensional object. Therefore, the domain of the trajectories in the phase space is a strange field which called strange attractors whose Lyapunov dimensions are usually non-integer. The presented integer-order system in Eq. (7) has a chaotic behavior for a wide range of parameters. Such integer-order dynamical system may be a reduced model for a physical system. If such system is implemented using physical electronic devices, e.g., the DC series motor, the environment effects (aging of the elements, temperature, inaccurate values, and so on) on the elements may appear as a different behavior so that the response predicted by the model does not resemble by the actual system. Indeed for such chaotic systems, the super-sensitivity to tiny changes in the element values, cannot be considered clearly in the dynamical equations. The main reason is related to the fractal or holed basin of the invariant set of the system. In other words, some trace of the system trajectories may be seen in the observed coordinate (phase plane) and some of them lie in the unseen region which cannot be handled by the classical model (nonlinear ODEs). Thus a useful and proper tool for model of such phenomena is the fractional calculus. In particular, inserting the fractional orders operator instead of the integer ones, is a compensation for this strange behavior. The fractional order operators can modify the observed behavior by considering a kernel in the integral. This kernel can be treated as a weighting factor which generates a new response using existing vector fields. Regarding these discussions, the fractional order version of the system presented in [28] is as follows:
Fig. 1. The schematic diagram of DC series drive.
Please cite this article in press as: Razminia A, Baleanu D. Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics (2013), http://dx.doi.org/10.1016/j.mechatronics.2013.02.004
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A. Razminia, D. Baleanu / Mechatronics xxx (2013) xxx–xxx
0
1
0
1
y ax þ byz Dq x C B q C B @ D y A ¼ @ cy xz þ z A; q dxy Hz D z
ð8Þ
in which q is a real quantity. For finding the equilibrium points of the proposed system it is enough to equate the left hand side of (7) to zero. Thus, the equilibrium points are:
Q 1 : ð 0 0 0 Þ; pffiffiffi pffiffiffiffiffiffiffi p1þ ffiffiffi K Q 2 : dþ2d D hb 1þ dþ D pffiffiffi pffiffiffiffiffiffiffi p1þ ffiffiffi K Q 3 : dþ2d D hb 1 dþ D pffiffiffi pffiffiffiffiffiffiffi p1þ ffiffiffi K Q 4 : d2d D hb 1þ d D pffiffiffi pffiffiffiffiffiffiffi p1þ ffiffiffi K Q 5 : d2d D hb 1 d D
pffiffiffiffiffiffiffi 1þ 1þK 2b pffiffiffiffiffiffiffi 1 1þK ;; 2b pffiffiffiffiffiffiffi 1þ 1þC ; 2b pffiffiffiffiffiffiffi 1 1þC 2b
The Jacobian matrix for (7) evaluated in the equilibrium point Qi: (x⁄, y⁄, z⁄), i = 1, 2, . . ., 5, where x⁄, y⁄, and z⁄ are the coordination of the equilibrium, can be calculated as follows:
0
a
B J ¼ @ z dy
1 þ bz c
dx
by
1
C 1 x A :
ð10Þ
h
It has been shown that for the following quantities there exist various behaviors [28]:
ðiÞða; b; c; d; hÞ ¼ ð3; 2:7; 4:7; 2; 9Þ; ð9Þ
ðiiÞða; b; c; d; hÞ ¼ ð3; 2:7; 1:7; 2; 9Þ;
ð11Þ
ðiiiÞða; b; c; d; hÞ ¼ ð3; 2:7; 3:9; 2; 9Þ; where (i–iii) correspond to two, three, and four scrolls respectively. The numerical simulations for these chaotic attractors are depicted in Figs. 2–5.
where 2
D ¼ d þ 4chd;
K¼
pffiffiffiffi 2ab d þ 2ch þ D ; h
pffiffiffiffi 2ab d þ 2ch D : C¼ h
4. Synchronization scheme Consider the master–slave synchronization scheme of two autonomous different fractional order chaotic systems:
Fig. 2. Chaotic attractor of system (8) with (a, b, c, d, h) = (3, 2.7, 4.7, 2, 9), (2-scroll) with initial conditions (x0, y0, z0) = (6, 1, 3) and order q = 0.9.
Fig. 3. Phase portrait of the system (8) with order q = 0.9 and (a, b, c, d, h) = (3, 2.7, 4.7, 2, 9).
Please cite this article in press as: Razminia A, Baleanu D. Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics (2013), http://dx.doi.org/10.1016/j.mechatronics.2013.02.004
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Fig. 4. Chaotic attractor of system (8) with (a, b, c, d, h) = (3, 2.7, 3.9, 2, 9), (4-scroll) with initial conditions (x0, y0, z0) = (6, 1, 3) and orders q = 0.9.
Fig. 5. Phase portrait of the system (8) with order q = 0.9 and (a, b, c, d, h) = (3, 2.7, 3.9, 2, 9).
Dq xðtÞ ¼ f ðxÞ;
Master : Slave :
ð12Þ
q
D yðtÞ ¼ gðyÞ þ u;
where x, y e Rn represent the states of the drive and the response systems, respectively. Moreover, f: Rn ? Rn, g: Rn ? Rn are the vector fields of the drive and response systems, respectively. The aim is to choose a suitable control function u = (u1, u2, . . ., un)T such that the states of the drive and response systems are synchronized asymptotically, i.e. lim t?1ky(t) x(t)k = 0. For the system (8), we construct the following master–slave structure in which the subscripts m and s stand for master and slave systems, respectively:
0
1 0 1 ym axm þ bym zm Dq xm B q C B C Master system : @ D ym A ¼ @ cym xm zm þ zm A; ; dxm ym Hzm D q zm ðxm ð0Þ; ym ð0Þ; zm ð0ÞÞ ¼ ðxm0 ; ym0 ; zm0 Þ
where u1, u2 and u3 are nonlinear controllers to be designed such that the two chaotic system can be synchronized. For achieving this goal, let define the error variables as follows:
e1 ðtÞ ¼ xs ðtÞ xm ðtÞ; e2 ðtÞ ¼ ys ðtÞ ym ðtÞ;
ð15Þ
ee ðtÞ ¼ zs ðtÞ zm ðtÞ: Substituting Eqs. (13) and (14) in (15) we have:
Dq e1 ¼ e2 ae1 þ bys e3 þ bzm e2 þ u1 ; Dq e2 ¼ ce2 þ e3 xs e3 zm e1 þ u2 ;
ð16Þ
Dq e2 ¼ he3 þ dxs e2 þ dym e1 þ u3 : Now we select the control laws as:
ð13Þ
u1 ¼ bzm e2 bys e3 þ v 1 ; u2 ¼ xs e3 þ zm e1 þ v 2 ;
ð17Þ
u3 ¼ dym e1 dxs e2 þ v 3 : and
0
D q xs
1
0
ys axs þ bys zs
1
0
u1
As one knows in active sliding mode control, vi(t)’s (i = 1, 2, 3) are designed based on a sliding mode control law:
1
B C B C B C Slave system : @ Dq ys A ¼ @ cys xs zs þ zs A þ @ u2 A; q u3 dxs ys Hzs D zs ðxs ð0Þ; ys ð0Þ; zs ð0ÞÞ ¼ ðxs0 ; ys0 ; zs0 Þ;
ð14Þ
v i ðtÞ ¼ ki wðtÞ;
ð18Þ
where ki’s (i = 1, 2, 3) are constant gains and w(t) is the actuating signal with the following rule:
Please cite this article in press as: Razminia A, Baleanu D. Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics (2013), http://dx.doi.org/10.1016/j.mechatronics.2013.02.004
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A. Razminia, D. Baleanu / Mechatronics xxx (2013) xxx–xxx
wðtÞ ¼
wþ ðtÞ sðeÞ P 0; ; w ðtÞ sðeÞ < 0
ð19Þ
in which s = s(e) is a switching surface that prescribes the desired dynamics. Thus the error dynamics (16) can be rewritten as:
Dq e1 ¼ e2 ae1 þ k1 w; Dq e2 ¼ ce2 þ e3 þ k2 w;
ð20Þ
k1 ðk1 l1 þ k2 l2 þ k3 l3 Þ e1 Dq e1 ¼ al1 a k1 ðk1 l þ k2 l2 þ k3 l3 Þ 1 ðk l þ k l þ k l Þ l1 þ cl2 1 1 2 2 3 3 e2 þ ðhl3 l2 Þe3 ; k1 k2 ðk1 l1 þ k2 l2 þ k3 l3 Þ ðk1 l1 þ k2 l2 þ k3 l3 Þ al1 e1 l1 þ cl2 þ c e2 k2 ðk1 l1 þ k2 l2 þ k3 l3 Þ e3 ; þ hl3 l2 þ k2
Dq e2 ¼
Dq e2 ¼ he3 þ k3 w: A formal sliding surface can be selected as:
sðeÞ ¼ l1 e1 þ l2 e2 þ l3 e3 ;
ð21Þ
where li’s are some constants. The equivalent control is found by the fact that s_ ðeÞ ¼ 0 is a necessary condition for the pseudo-state trajectories to stay on the switching surface s(e) = 0. Hence, when in sliding mode, the controlled system satisfies the following conditions:
sðeÞ ¼ 0;
s_ ðeÞ ¼ 0:
ð22Þ
Using the chain rule one can write:
ð23Þ
ðk1 l1 D1q þ k2 l2 D1q þ k3 l3 D1q ÞwðtÞ ð24Þ
As the second step let design the sliding mode control law. To do this the reaching law is selected as:
ð27Þ
Dq s ¼ l1 Dq e1 þ l2 Dq e2 þ l3 Dq e3
ð28Þ
Dq s ¼l1 Dq e1 þ l2 Dq e2 þ l3 Dq e3 ¼ l1 ðe2 ae1 þ k1 wÞ þ l2 ðce2 þ e3 þ k2 wÞ
which results in:
þ l3 ðhe3 þ k3 wÞ ¼ cs ¼ cl1 e1 cl2 e2 cl3 e3 :
1 ðal1 e1 ðl1 þ cl2 Þe2 ðk1 l1 þ k2 l2 þ k3 l3 Þ þ ðhl3 l2 Þe3 Þ:
al1 e1 ðl1 þ cl2 Þe2 ðk1 l1 þ k2 l2 þ k3 l3 Þ ðk1 l1 þ k2 l2 þ k3 l3 Þ þ hl3 l2 h e3 : k3
in which the obtained error dynamics in Eq. (26) should be inserted. Now using (27) and (28) the control signal w can be computed as in the previous step. Indeed we obtain:
A direct result is:
wðtÞ ¼
The positive gain c is determined such that the sliding condition is satisfied and the sliding mode motion occurs. Based on Eq. (21) one can easily deduce that:
¼ l1 D1q ðe2 ae1 þ k1 wÞ þ l2 D1q ðce2 þ e3 þ k2 wÞ
¼ l1 D1q ðe2 ae1 Þ l2 D1q ðce2 þ e3 Þ þ l3 D1q ðhe3 Þ;
k3
Dq s ¼ cs:
@sðeÞ e_ @e 1q ¼ l1 D ðDq e1 Þ þ l2 D1q ðDq e2 Þ þ l3 D1q ðDq e3 Þ
s_ ðeÞ ¼
þ l3 D1q ðhe3 þ k3 wÞ ¼ 0:
Dq e2 ¼
ð26Þ
ð29Þ
Therefore we conclude that:
ð25Þ
Notice that this control term is realizable if and only if the term k1l1 + k2l2 + k3l3 is not zero. Thus, the error dynamics can be written as follows:
w¼
c ðk1 l1 þ k2 l2 þ k3 l3 Þ
½l1 þ aðk1 l1 þ k2 l2 þ k3 l3 Þl1 e1
þ½l2 þ ðl1 cl2 Þðk1 l1 þ k2 l2 þ k3 l3 Þe2 þ½l3 þ ðl2 þ hl3 Þðk1 l1 þ k2 l2 þ k3 l3 Þe3 :
ð30Þ
Fig. 6. Time responses for synchronization using sliding mode control with initial conditions (xm0, ym0, zm0) = (6, 1, 3) and (xs0, ys0, zs0) = (2, 4, 1).
Please cite this article in press as: Razminia A, Baleanu D. Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics (2013), http://dx.doi.org/10.1016/j.mechatronics.2013.02.004
A. Razminia, D. Baleanu / Mechatronics xxx (2013) xxx–xxx
Now it is easy to see that using this w the error dynamics reported in Eq. (20) is asymptotically stable. Indeed it is straightforward to show that one of the eigenvalues of the error dynamic is c. The other two eigenvalues depends on the coefficients of kis and lis which can be selected such that they lie in the stability region. In the next section we examine numerically the proposed method. 5. Simulation results In this section, to demonstrate the effectiveness of the proposed methods, we will present the numerical results for synchronizing chaotic systems (13) and (14) under the control laws provided in the previous section. We selected the initial conditions (xm0, ym0, zm0) = (6, 1, 3) for the master system and (xs0, ys0, zs0) = (2, 4, 1) for the slave system. Note that we have chosen the initial conditions very different intentionally. Driving the simulations we turn on the control at the instant t = 1.5. As it can be seen from Fig. 6, after an initial transient, synchronization is achieved completely and the errors converge to zero asymptotically. An important point that must be noted is that the above results have been derived based on the general values of parameters. Therefore the proposed method is valuable for synchronizing the chaotic systems where they exhibit 2-scrolls, 3-scrolls or 4-scrolls. 6. Conclusions This work discusses a sliding mode control scheme for synchronizing two fractional order chaotic systems. Indeed the integer order chaotic system is a novel that recently has been introduced in the research societies. The main feature of this system is its interesting behavior. Indeed the system presents various chaotic attractors such as 2, 3, or 4 scrolls. However, since most electronic configurations with self-sustained oscillations have strange limit set which can be regarded as a fractal object, the more proper model for such systems is the fractional order one. Based on the sliding mode control technique, we developed a procedure that guarantees the asymptotical stability of the error dynamics which means the synchronization is achieved completely. Our proposed method is valuable for each case: both master and slave systems are 2scrolls, or 3-scrolls, or 4-scrolls. Numerical simulations have been used for clarify the effectiveness of the proposed control laws. Acknowledgement Authors would like to thank to the referees for their interesting comments and remarks. References [1] Ott E, Grebogi C, Yorke JA. Controlling chaos. Phys Rev Lett 1990;65:1196–9. [2] Shi XRg, Wang ZL. Adaptive added-order anti-synchronization of chaotic systems with fully unknown parameters. Appl Math Comput 2009;215:1711–7. [3] Chen F, Zhang W. Stabilization of parameters perturbation chaotic system via adaptive backstepping technique. Appl Math Comput 2008;200:101–9. [4] Park JH, Ji DH, Won SC, Lee SM. Hinf synchronization of time-delayed chaotic systems. Appl Math Comput 2008;204:170–7. [5] Razminia A, Majd VJ, Baleanu D. Chaotic incommensurate fractional order Rossler system: active control synchronization. Adv Differ Equ 2011;15:1–12. [6] Zhu C. Feedback control methods for stabilizing unstable equilibrium points in a new chaotic system. Nonlinear Anal-Theor 2009;71:2441–6. [7] Li S, Qiu j, Ji H, Zhu K, Li J. Piezoelectric vibration control for all-clamped panel using DOB-based optimal control. Mechatronics 2011;21:1213–21.
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Please cite this article in press as: Razminia A, Baleanu D. Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics (2013), http://dx.doi.org/10.1016/j.mechatronics.2013.02.004