AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 948782 10.1155/2013/948782 948782 Research Article Synchronization of Fractional-Order Chaotic Systems with Gaussian Fluctuation by Sliding Mode Control Xu Yong Wang Hua Jinhu 1 Department of Applied Mathematics Northwestern Polytechnical University Xi’an 710072 China nwpu.edu.cn 2013 24 11 2013 2013 04 09 2013 23 10 2013 2013 Copyright © 2013 Yong Xu and Hua Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Chaotic systems are always influenced by some uncertainties and external disturbances. This paper investigates the problem of practical synchronization of fractional-order chaotic systems with Gaussian fluctuation. A fractional integral (FI) sliding surface is proposed for synchronizing the uncertain fractional-order system, and then the sliding mode control technique is carried out to realize the synchronization of the given systems. One theorem about sliding mode controller is presented to prove that the proposed controller can make the system achieve synchronization. As a case study, the presented method is applied to the fractional-order Chen-Lü system, and simulation results show that the proposed control approach is capable to go against Gaussian noise well.

1. Introduction

Synchronization, which means “things occur at the same time or operate in unison,” has received a great deal of interest among scientists from various fields in the last few years, especially in fractional-order chaotic systems . It has been recognized that many systems in interdisciplinary fields can be elegantly described by fractional-order differential equations, such as viscoelastic materials , electrical circuits , population models , and financial systems . Meanwhile, most of precious studies have shown that some fractional-order systems exhibit chaotic behavior . In particular, fractional-order chaotic behavior has wide promising applications in information encryption, image processing, secure communication, and so forth . Therefore, synchronization of fractional chaotic systems starts to attract increasing attentions because it has a wide range of applications like the traditional (integer order) chaotic synchronization, which has been used in secure communication , complex dynamical network , and so on.

A basic configuration for chaos synchronization is the drive-response pattern, where the response of chaotic system must track the drive chaotic trajectory. Some approaches based on this configuration have been attained to achieve chaos synchronization in fractional-order chaotic systems, such as Pecora and Carroll (PC) control , active control , adaptive control [23, 24], sliding mode control (SMC) , and a scalar transmitted signal method , in which the sliding mode controller has some attractive advantages, including: (i) fast dynamic responses and good transient performance; (ii) external disturbance rejection; and (iii) insensitivity to parameter variations and model uncertainties [27, 28]. In addition, SMC method plays an important role in the application to practical problems. For example, in , Tavazoei and Haeri proposed a controller based on active sliding mode theory to synchronize fractional-order chaotic systems in master-slave structure. In , the problem of modified projective synchronization of fractional-order chaotic system was considered, and finite-time synchronization of nonautonomous fractional-order uncertain chaotic systems was investigated in .

All of the methods mentioned above have been used to synchronize the deterministic fractional-order chaotic systems. However, noise-induced synchronization in chaotic systems is a practical phenomenon due to the fact that noises are ubiquitous in natural and synthetic systems, and up till now, it has been studied by many investigators from different areas . What is more is that chaotic systems with fractional-order model influenced by random noise will be more challenging and difficult. In this investigation, our aim is to synchronize two fractional-order chaotic systems with uncertain environment. To achieve this goal, we propose fractional integral (FI) sliding mode surface which combines the property of fractional-order equation with sliding mode control method. Theoretical analysis about sliding mode controller is presented via series expansion and properties of Gaussian distribution to prove that the proposed controller can make the system synchronize well. Then a numerical example is given to verify the effectiveness of the mentioned method, and the good agreements are also found between the theoretical and the numerical results.

This paper is organized as follows. In Section 2, fractional-order chaotic systems with random factors and problem formulation are presented. In Section 3, we investigate the design method of sliding mode controller, and one theorem is obtained to prove the effectiveness of proposed controller. One example is presented to carry out the numerical simulations in Section 4. Finally, conclusions are presented to end this paper.

2. System Description and Problem Formulation

Consider the following class of fractional-order chaotic system excited by Gaussian white noise, which is described by (1)Dαx=A1x+f1(x)+h(x,t)W(t),

where x=[x1,x2,,xn]TRn denotes the state vector, A1Rn×n is a constant matrix, f1:RnRn is nonlinear vector function, h is noise intensity function which is sufficient smooth and bounded, that is, |h(x,t)|H (H is a positive constant), and W(t)=[W1(t),W2(t),,Wn(t)]T is n-dimensional Brownian motion. Accordingly, W˙(t)=[W˙1,W˙2,,W˙n]T is a n-dimensional Gaussian white noise vector, in which every two noises are statistical independent. And α=[α1,α2,,αn]T. Dα denotes the Caputo derivative, which is defined as (2)Dαf(t)=1Γ(n-α)0tf(n)(τ)(t-τ)α-n+1dτ,  for  n-1<α<n

with Γ(z)=0e-ttz-1dt, the Euler’s Gamma function.

Let system (1) be the driving system; then response system with a controller u(t)=[u1(t),u2(t),,un(t)]T is given by (3)Dαy=A2y+f2(y)+u(t),

where y=[y1,y2,,yn]TRn is state vector, A2Rn×n is a coefficient matrix, and f2:RnRn is vector function. Let e=[e1,e2,,en]T=y-x be the error of systems (1) and (3). Then, from system (1) and (3), one has the error dynamics: (4)Dαe=A2y+f2(y)+u(t)-A1x-f1(x)-h(x,t)W˙Dαe=A2e+F(x,y)-h(x,t)W˙+u(t),

where F(x,y)=f2(y)-f1(x)+(A2-A1)x.

Thus, the control problem considered in this study is that, for chaotic driving system (1) and response system (3), they are to be synchronized by designing an appropriate controller u(t) satisfying (5)limte=limty-x=0,

where · is defined as e(t)=(E[e(t)Te(t)])1/2 and  E[·] is the expected value function.

3. Sliding Mode Controller Design and Analysis

In the following context, we will design sliding mode controller to establish synchronization between driving system (1) and response system (3).

3.1. Sliding Mode Controller Design Process

Now, the control input vector u(t) is defined to eliminate the nonlinear part of the error dynamics: (6)u(t)=H(t)-F(x,y),

where H(t)=Kw(t), K is a constant gain matrix, and w(t)Rn is the control input that satisfies (7)w(t)={w+(t)S(e)0,w-(t)S(e)<0,

in which S=S(e) is a switching surface to prescribe the desired sliding mode dynamics.

So the error system (4) is rewritten as (8)Dαe=A2e+H(t)-h(x,t)W˙.

Here, a new fractional integral (FI) switching surface is given as follows: (9)S=Dα-1e-0t(K+A2)e(τ)dτ,

where S=[s1,s2,,sn]T and K+A2 should be stable; namely, the eigenvalues λi(i=1,2,,n) of matrix K+A2 are negative (λi<0).

As we all know, when the system is controllable in the sliding mode, the switching surface should satisfy the following conditions: (10a)S(e)=0,

together with (10b)S˙(e)=0.

Substituting (8) and (9) into (10b), one obtains (11)S˙=Dαe-(K+A2)e(t)S˙=A2e+kw(t)-h(x,t)W˙-(K+A2)e(t)S˙=Kw(t)-h(x,t)W˙-Ke(t)S˙=0.

Therefore, the equivalent control law is obtained by (12)weq(t)=e(t)+K-1h(x,t)W˙.

In real-world applications, the Gaussian white noise W˙ is uncertain. Therefore, the equivalent control input is modified to (13)weq(t)=e(t).

To design the sliding mode controller, we consider the constant plus proportional rate reaching law [35, 36]; that, is (14)S˙=-rS-ρsgn(S),

where sgn(S)=[sgn(s1),sgn(s2),,sgn(sn)]T, r and ρ are all positive numbers, and sgn(x) represents sign function; that is (15)sgn(x)={1,x>0,0,x=0,-1,x<0.

So, we can get the controller (16)w(t)=K-1(Ke-rS-ρsgn(S)).

Further, according to the control law and the updated law, the controller is given by (17)u(t)=Ke-rS-ρsgn(S)-f2(y)+f1(x)-(A2-A1)x.

And the error system can be rewritten in the following differential form: (18)Dαe=(K+A2)e(t)-h(x,t)W˙-rS-ρsgn(S).

3.2. Synchronization Analysis Theorem 1.

If the controller is selected as (17), with suitably selected r and ρ, then the synchronization of fractional-order chaotic systems between driving system (1) and response system (3) can be achieved (i.e., the synchronization error converges to zero in the mean square norm).

Proof.

Consider a Lyapunov function constructed by the mean square norm of S(t) and its differential form : (19)V=12S(t)2=12E[S2(t)],dV=12E[d(S2(t))].

According to the definition of derivative, it is found that (20)d(S2(t))=(S(t)+dS(t))2-S2(t)d(S2(t))=2S(t)dS(t)+dS(t)dS(t),

While, from (11) and (16), one gets (21)dS(t)=(-rS-ρsgn(S))dt-h(x,t)dW.

Substituting (21) into (20) results in (22)d(S2(t))=2S(t)((-rS-ρsgn(S))dt-h(x,t)dW)d(S2(t))=+h2(x,t)dt.

Taking expectations to (22) and using the properties of Brownian motion, we have (23)dV=E[S(t)((-rS-ρsgn(S))dt-h(x,t)dW)12dV=ED+12h2(x,t)dt]dV=E[S(t)((-rS-ρsgn(S))dt)+12h2(x,t)dt].

Therefore (24)V˙=E[S(t)(-rS-ρsgn(S))+12h2(x,t)]V˙-rE[S2]-ρE[|S|]+12H2.

Equation (24) implies that as long as suitable r and ρ which satisfy (1/2)H2rE[S2]+ρE[|S|] is selected, namely, V˙0, according to Barbalat’ Lemma , system (1) and system (3) can achieve synchronization under the controller law in (17).

This completes the proof.

4. Simulation

In this part, to confirm the validity of proposed method, we numerically examine the synchronization between fractional-order Chen system  and fractional-order Lü system . In the simulation, step-by-step method is performed to receive numerical solution, and the detailed descriptions of this algorithm are available in [41, 42].

Here, we assume that the Chen system drives the Lü system. Hence, the driving system (fractional-order Lü system) is described as (25)Dαx1=a1(x2-x1)+h1(x,t)W˙1,Dαx2=-x1x3+c1x2+h2(x,t)W˙2,Dαx3=x1x2-b1x3+h3(x,t)W˙3

which can also be written as (26)Dαx=A1x+f1(x)+h(t,x)W˙,

in which, (27)x=(x1x2x3),A1=[-a1a100c1000-b1],f1(x)=(0-x1x3x1x2),h(t,x)=(h1(t,x)h2(t,x)h3(t,x)),W˙=(W˙1W˙2W˙3).

It has been shown that the fractional-order Lü system can demonstrate chaotic behavior  when a1=35, b1=3, c1=28, and α=0.9.

The response system (fractional-order Chen system) is given as follows: (28)Dαy1=a2(y2-y1),Dαy2=(c2-a2)y1-y1y3+c2y2,Dαy3=y1y2-b2y3,

which can be written in the following form: (29)Dαy=A2y+f2(y),

where (30)y=(y1y2y3),A2=[-a2a20c2-a2c2000-b2],f2(y)=(0-y1y3y1y2),

in which system will exhibit chaotic behavior  when a2=35, b2=3, c2=28, and α=0.9. According to (3), the response system with controller can be described as follows: (31)Dαy=A2y+f2(y)+u(t),

where u(t)=[u1,u2,u3]T is the control vector.

Now, we apply the proposed sliding control approach to finish synchronization between fractional-order Lü system driven by Gaussian white noise and fractional-order Chen system. Here, we define the error states as (32)ei=yi-xi

and the sliding mode surface as (33)S=Dα-1e-0t(K+A2)e(τ)dτ,

The control law is given by (34)u(t)=Ke-rS-ρsgn(S)-f2(y)+f1(x)-(A2-A1)x.

In the numerical simulations, the initial conditions are set as (x1(0),x2(0),x3(0))=(7,-4,4), (y1(0),y2(0),y3(0))=(1,3,-1). The noise intensity matrices are presumably given in the form of (h1(t,x),h2(t,x),h3(t,x))=(0.3,0.4,sint). In fact, controller parameters can be chosen as r=5, ρ=0.5, and K=[34-3507-290002]. The time step size is h=0.0005. Then the simulation results are summarized in Figures 14. The state trajectories of the system (26) and system (31) under the sliding mode control method are shown in Figure 1(a) (signals x1;y1), Figure 1(b) (signals x2;y2), and Figure 1(c) (signals x3;y3), respectively. Note that the driving system is shown by solid line whereas response system is shown by dashed line. As one can see, the designed controller is effectively capable achieving the synchronization of fractional-order Chen chaotic system; that is, the state variables (y1,y2,y3) follow the trail of (x1,x2,x3) well. Then the synchronization errors between the uncertain fractional-order chaotic Lü system and fractional-order chaotic Chen system are depicted in Figure 2(a) (signal e1), and Figure 2(b) (signal e2), Figure 2(c) (signal e3). As it is expected, the synchronization errors (32) close to zero. Further, the expectation and variance of error vectors e1, e2 and e3 converge to zero, as displayed in Figures 3 and 4, which all indicate that the chaos synchronization between uncertain fractional-order chaotic Lü and Chen systems are indeed realized.

(a) The state trajectories of the system (26) and system (31) with the sliding mode control method. (Signals x1;y1). (b) The state trajectories of the system (26) and system (31) with the sliding mode control method. (Signals x2;y2). (c) The state trajectories of the system (26) and system (31) with the sliding mode control method. (Signals x3;y3).

(a) The time evolution of synchronization error e1 of the drive system (26) and response system (31). (b) The time evolution of synchronization error e2 of the drive system (26) and response system (31). (c) The time evolution of synchronization error e3 of the drive system (26) and response system (31).

Mean value of synchronization errors e1,e2,e3 between system (26) and system (31) using sliding mode method.

Variance of the state estimation errors e1,e2,e3 of the system (26) and system (31) using the sliding mode method.

From the simulation results, it can be concluded that the obtained theoretic results are efficient and feasible for synchronizing fractional uncertain dynamical systems, and the proposed controller guarantees the convergence of the error system.

5. Conclusions

In this paper, we focus on the problem of synchronization between fractional-order chaotic systems with Gaussian fluctuation by the method of fractional-order sliding mode control. A fractional sliding surface is introduced, and the sliding mode controller is proposed for synchronization. Furthermore, convergence property has been analyzed for the error dynamics after adding proposed controllers. It has been shown that the fractional-order chaotic systems under uncertain environment can achieve synchronization by proper choice of control parameters (r and ρ). Finally, to further illustrate the effectiveness of the proposed controllers, one applies the presented algorithm to the fractional-order Chen and fractional-order Lü systems through numerical simulations. From the simulation results, it is obvious that a satisfying control performance can be achieved by using the proposed method.

Acknowledgments

This work was supported by the NSF of China (Grant nos. 11372247 and 11102157), Program for NCET, the Shaanxi Project for Young New Star in Science & Technology, NPU Foundation for Fundamental Research, and SRF for ROCS, SEM.

Zhang R. Yang S. Robust synchronization of two different fractional-order chaotic systems with unknown parameters using adaptive sliding mode approach Nonlinear Dynamics 2013 71 269 278 Wang X. Zhang X. Ma C. Modified projective synchronization of fractional-order chaotic systems via active sliding mode control Nonlinear Dynamics 2012 69 511 517 2-s2.0-82755190929 10.1007/s11071-011-0282-1 Zhou P. Ding R. Cao Y. Multi drive-one response synchronization for fractional-order chaotic systems Nonlinear Dynamics 2012 70 1263 1271 Li C. P. Deng W. H. Xu D. Chaos synchronization of the Chua system with a fractional order Physica A 2006 360 2 171 185 2-s2.0-27744554781 10.1016/j.physa.2005.06.078 Wilkie K. P. Drapaca C. S. Sivaloganathan S. A nonlinear viscoelastic fractional derivative model of infant hydrocephalus Applied Mathematics and Computation 2011 217 21 8693 8704 2-s2.0-79956150959 10.1016/j.amc.2011.03.115 ZBL1215.92036 Luo Y. Chen Y. Pi Y. Experimental study of fractional order proportional derivative controller synthesis for fractional order systems Mechatronics 2011 21 1 204 214 2-s2.0-79551504874 10.1016/j.mechatronics.2010.10.004 Rivero M. Trujillo J. J. Vázquez L. Pilar Velasco M. Fractional dynamics of populations Applied Mathematics and Computation 2011 218 3 1089 1095 2-s2.0-80052273583 10.1016/j.amc.2011.03.017 ZBL1226.92060 Laskin N. Fractional market dynamics Physica A 2000 287 3-4 482 492 2-s2.0-0034517069 10.1016/S0378-4371(00)00387-3 Ahmad W. M. Sprott J. C. Chaos in fractional-order autonomous nonlinear systems Chaos, Solitons and Fractals 2003 16 2 339 351 2-s2.0-0037332868 10.1016/S0960-0779(02)00438-1 ZBL1033.37019 Hartley T. T. Lorenzo C. F. Qammer H. K. Chaos in a fractional order Chua's system IEEE Transactions on Circuits and Systems I 1995 42 8 485 490 2-s2.0-0029359880 10.1109/81.404062 Li C. Chen G. Chaos and hyperchaos in the fractional-order Rössler equations Physica A 2004 341 1–4 55 61 2-s2.0-3342927052 10.1016/j.physa.2004.04.113 Li C. Peng G. Chaos in Chen's system with a fractional order Chaos, Solitons and Fractals 2004 22 2 443 450 2-s2.0-1842832060 10.1016/j.chaos.2004.02.013 ZBL1060.37026 Sheu L.-J. Chen H.-K. Chen J.-H. Tam L.-M. Chen W.-C. Lin K.-T. Kang Y. Chaos in the Newton-Leipnik system with fractional order Chaos, Solitons and Fractals 2008 36 1 98 103 2-s2.0-35348886158 10.1016/j.chaos.2006.06.013 ZBL1152.37319 Zhou P. Ding R. Modified function projective synchronization between different dimension fractional-order chaotic systems Abstract and Applied Analysis 2012 2012 12 862989 10.1155/2012/862989 Kiani-B A. Fallahi K. Pariz N. Leung H. A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter Communications in Nonlinear Science and Numerical Simulation 2009 14 3 863 879 2-s2.0-50249121925 10.1016/j.cnsns.2007.11.011 ZBL1221.94049 Sheu L. J. A speech encryption using fractional chaotic systems Nonlinear Dynamics 2011 65 1-2 103 108 2-s2.0-79959528048 10.1007/s11071-010-9877-1 ZBL1251.94013 Singh N. Sinha A. Optical image encryption using fractional Fourier transform and chaos Optics and Lasers in Engineering 2008 46 2 117 123 2-s2.0-36048937251 10.1016/j.optlaseng.2007.09.001 Zhou N. Wang Y. Gong L. He H. Wu J. Novel single-channel color image encryption algorithm based on chaos and fractional Fourier transform Optics Communications 2011 284 12 2789 2796 2-s2.0-79955045282 10.1016/j.optcom.2011.02.066 Lu J. Wu X. J. Synchronization of a unified chaotic system and the application in secure communication Physics Letters A 2002 305 6 365 370 2-s2.0-0037121821 10.1016/S0375-9601(02)01497-4 ZBL1005.37012 J. Chen G. A time-varying complex dynamical network model and its controlled synchronization criteria IEEE Transactions on Automatic Control 2005 50 6 841 846 2-s2.0-21344466279 10.1109/TAC.2005.849233 Li C. Deng W. Chaos synchronization of fractional-order differential systems International Journal of Modern Physics B 2006 20 7 791 803 2-s2.0-33645166246 10.1142/S0217979206033620 ZBL1101.37025 Bhalekar S. Daftardar-Gejji V. Synchronization of different fractional order chaotic systems using active control Communications in Nonlinear Science and Numerical Simulation 2010 15 11 3536 3546 2-s2.0-77952237940 10.1016/j.cnsns.2009.12.016 ZBL1222.94031 Zhang R. Yang S. Adaptive synchronization of fractional-order chaotic systems via a single driving variable Nonlinear Dynamics 2011 66 4 831 837 2-s2.0-82255185667 10.1007/s11071-011-9944-2 ZBL1242.93097 Odibat Z. M. Adaptive feedback control and synchronization of non-identical chaotic fractional order systems Nonlinear Dynamics 2010 60 4 479 487 2-s2.0-77955654277 10.1007/s11071-009-9609-6 ZBL1194.93105 Chen D.-Y. Liu Y.-X. Ma X.-Y. Zhang R.-F. Control of a class of fractional-order chaotic systems via sliding mode Nonlinear Dynamics 2012 67 1 893 901 2-s2.0-82255193992 10.1007/s11071-011-0002-x ZBL1242.93027 Lu J. G. Synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal Chaos, Solitons and Fractals 2006 27 2 519 525 2-s2.0-22844431613 10.1016/j.chaos.2005.04.032 ZBL1086.94007 Boiko I. Fridman L. Iriarte R. Pisano A. Usai E. Parameter tuning of second-order sliding mode controllers for linear plants with dynamic actuators Automatica 2006 42 5 833 839 2-s2.0-33645165555 10.1016/j.automatica.2006.01.009 ZBL1137.93335 Slotine J. J. Li W. Applied Nonlinear Control 1991 Upper Saddle River, NJ, USA Prentice Hall Tavazoei M. S. Haeri M. Synchronization of chaotic fractional-order systems via active sliding mode controller Physica A 2008 387 1 57 70 2-s2.0-35748971445 10.1016/j.physa.2007.08.039 Aghababa M. Finite-time chaos control and synchronization of fractional-order nonautonomous chaotic (hyperchaotic) systems using fractional nonsingular terminal sliding mode technique Nonlinear Dynamics 2012 69 247 261 Pikovsky A. S. Comment on ‘chaos, noise, and synchronization’ Physical Review Letters 1994 73 21 2931 2-s2.0-0001196018 10.1103/PhysRevLett.73.2931 Nagai K. H. Kori H. Noise-induced synchronization of a large population of globally coupled nonidentical oscillators Physical Review E 2010 81 6 2-s2.0-77954225339 10.1103/PhysRevE.81.065202 065202 Herzel H. Freund J. Chaos, noise, and synchronization reconsidered Physical Review E 1995 52 3 3238 3241 2-s2.0-0000922144 10.1103/PhysRevE.52.3238 Lai C.-H. Zhou C. Synchronization of chaotic maps by symmetric common noise Europhysics Letters 1998 43 4 376 380 2-s2.0-0032529059 10.1209/epl/i1998-00368-1 Lin J.-S. Yan J.-J. Liao T.-L. Chaotic synchronization via adaptive sliding mode observers subject to input nonlinearity Chaos, Solitons and Fractals 2005 24 1 371 381 2-s2.0-9644300919 10.1016/j.chaos.2004.09.042 ZBL1094.93512 Yau H.-T. Design of adaptive sliding mode controller for chaos synchronization with uncertainties Chaos, Solitons and Fractals 2004 22 2 341 347 2-s2.0-1842842989 10.1016/j.chaos.2004.02.004 ZBL1060.93536 Salarieh H. Alasty A. Chaos synchronization of nonlinear gyros in presence of stochastic excitation via sliding mode control Journal of Sound and Vibration 2008 313 3–5 760 771 2-s2.0-41549136143 10.1016/j.jsv.2007.11.058 Tao G. A simple alternative to the Barbǎlat lemma IEEE Transactions on Automatic Control 1997 42 5 698 2-s2.0-0031143646 10.1109/9.580878 ZBL0881.93070 J. Chen G. Cheng D. Celikovsky S. Bridge the gap between the Lorenz system and the Chen system International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 2002 12 12 2917 2926 2-s2.0-0036999538 10.1142/S021812740200631X ZBL1043.37026 J. Chen G. A new chaotic attractor coined International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 2002 12 3 659 661 2-s2.0-0036011505 10.1142/S0218127402004620 ZBL1063.34510 Petráš I. Fractional-Order Nonlinear Systems Modeling, Analysis and Simulation 2011 Berlin, Germany Springer Škovránek T. Podlubny I. Petráš I. Modeling of the national economies in state-space: a fractional calculus approach Economic Modelling 2012 29 1322 1327