Non-fragile control and synchronization of a new ... - Semantic Scholar
Applied Mathematics and Computation 222 (2013) 712–721
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Non-fragile control and synchronization of a new fractional order chaotic system Mohammad Mostafa Asheghan a, Saleh S. Delshad b, Mohammad Taghi Hamidi Beheshti c,⇑, Mohammad Saleh Tavazoei d a
Communications, Universidad Carlos III de Madrid, Avenida de la Universidad 30, Leganes 28911, Madrid, Spain Control Engineering Group, Luleå University of Technology, Sweden c Control and Communication Networks Lab, Electrical Engineering Department, Tarbiat Modares University, Tehran, Iran d Electrical Engineering Department, Sharif University of Technology, Tehran, Iran b
a r t i c l e
i n f o
Keywords: Fractional order systems Chaos synchronization Robust control
a b s t r a c t In this paper, we address global non-fragile control and synchronization of a new fractional order chaotic system. First we inspect the chaotic behavior of the fractional order system under study and also find the lowest order (2.49) for the introduced dynamics to remain chaotic. Then, a necessary and sufficient condition which can be easily extended to other fractional-order systems is proposed in terms of Linear Matrix Inequality (LMI) to check whether the candidate state feedback controller with parameter uncertainty can guarantee zero convergence of error or not. In addition, the proposed method provides a global zero attraction of error that guarantees stability around all existing equilibrium points. Finally, numerical simulation are employed to verify the validity of the proposed algorithm. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Fractional calculus is a quite old issue that goes back to more than 300 years ago in history. Although it has a long history, the applications of fractional calculus to physics and engineering have just started [1]. Nowadays, there are many systems that fall into fractional order systems category such as viscoelastic systems, electrode–electrolyte polarization, electromagnetic waves quantum evolution of complex systems, quantitative finance and diffusion wave [1]. In recent years, study on the dynamics of fractional-order differential systems has attracted even more interests amongst researchers. There are plenty of references involved in investigation of the chaotic and hyper chaotic behavior of the fractional-order differential systems such as the fractional-order Chua circuit [2], the fractional-order Chen system [3], the fractional-order Lu system [4], the fractional-order Rossler system [5], the fractional-order Arneodo system [6] and the fractional-order Newton–Leipnik system [7]. Recently, a new four-dimensional fractional-order system has been proposed, which has the very interesting property of having no equilibrium point [8]. Over the last two decades, pioneering Pecora and Carroll [9], chaos synchronization has been widely explored and investigated in various fields such as physical biological, chemical and medical science and secure communications [10]. The problem of designing a system, whose behavior mimics that of another chaotic system, is called synchronization. There have been many prior studies that addressed chaos control and synchronization methods in integer-order systems [11–14]. For example, [11] designed a robust linear state feedback controller using Linear Matrix Inequalities (LMI) method, while [12]
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (M.T. Hamidi Beheshti). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.07.045
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presented a fuzzy sliding mode control based method for chaos synchronization. [13] utilizes back stepping method to synchronize two Chua chaotic circuits. Recently, synchronization of chaotic fractional-order systems has become more and more interesting to researchers due to its potential applications in secure communication and control processing [15]. For example, in [16] chaos synchronization of two fractional Lu systems has been studied. Also, synchronization of two chaotic fractional Chen systems and synchronization of two chaotic fractional Chua systems have been presented in [17,18], respectively. Some more attempts to synchronize fractional order systems can be found in [19–21]. As well as synchronization, control and stabilization of fractional order systems have received a great deal of interest among the researchers. As it is mentioned before, many real systems behave fractionally, so their control problem should be investigated in fractional order field. Additionally, mathematical fascination of fractional order equations is another reason for such an increasing attention of scientists to fractional dynamic filed. However, to our best knowledge, non-fragile control or synchronization of fractional order systems is not reported in the existing literature. In simple words, a controller for which a certain bounded variation on controller gain does not violate stability of the closed loop system is referred to as non-fragile. Some non-fragile approaches in field of ordinary systems can be found in [22–25]. When the non-fragility is not guaranteed, implementation of the controller in real world application becomes more expensive, due to necessity of using elements with ignorable tolerance in the nominal values. In [26], for instance, the controller matrix has one non-zero entry which is 10. Even though zero convergence of error has been theoretically shown by this value, however, there is no guarantee that if the controller is capable to endure any little perturbation or not. The rest of the paper is organized as follows. In Section 2, basic definitions in fractional calculus as well as some fundamental theorems about fractional systems are presented. In Section 3, a new fractional-order chaotic system and sensitivity of its dynamics to different values of the system coefficients are studied. Section 4 is devoted to non-fragile control and synchronization between two chaotic systems. Section 5 concludes the paper. 2. Primary definitions and stability theorem In this section, we will give preliminary knowledge for our main results. The differintegral operator, denoted by a Dqt , is a combined differentiation-integration operator commonly used in fractional calculus. This operator is a notation for taking both the fractional derivative and the fractional integral in a single expression and is defined by, q a Dt
8 >
:Rt a
q>0
;
1
ð1Þ
q¼0
ðdsÞ
q
q < x_ ¼ y ax þ byz y_ ¼ cy xz þ z > :_ z ¼ dxy Hz
ð7Þ
where x, y and z are state variables, and a, b, c, d and h are positive real constants. This system is found to be chaotic for the parameters a = 3, b = 2.7, c 2 ð4:45; 4:60Þ [ ð4:86; 4:94Þ [ ½5:17; 7Þ, d = 2 and h = 9, and has many interesting complex dynamical behaviors. In [32] it is shown that (7) has five equilibria. Setting the parameters as a = 3, b = 2.7, c = 4.7, d = 2 and h = 9 leads to the following equilibrium points
8 E1 ð0; 0; 0Þ > > > > > > < E2 ðþ5:1260; þ2:7094; þ2:3687Þ
E3 ðþ5:1260; 2:4045; þ2:7390Þ : > > > E > 4 ð4:1260; 2:0432; þ1:8734Þ > > : E5 ð4:1260; þ2:4471; 2:2438Þ
ð8Þ
The Jacobian matrix of (7), evaluated at (x⁄, y⁄, z⁄) has the following form.
2
1 þ bz
a
6 J ¼ 4 z dy
3
7 1 x 5
c
by
dx
ð9Þ
h
The eigenvalues of (9) for any x⁄, y⁄ and z⁄ and the defined parameters a, b, c, d and h are as follows:
8 k11 > > > > > > < k21 k31 > > > > k41 > > : k51
¼ 3; k12 ¼ 4:7; k13 ¼ 9; ¼ 11:0247; k22 ¼ 1:8623 þ 6:6831i; k23 ¼ 1:8623 6:6831i; ¼ 11:7856; k32 ¼ 2:2428 þ 6:8580i; k33 ¼ 2:2428 6:8580i;
ð10Þ
¼ 10:7669; k42 ¼ 1:7335 þ 6:0024i; k43 ¼ 1:7335 6:0024i; ¼ 11:6813; k52 ¼ 2:1906 þ 6:1881i; k53 ¼ 2:1906 6:1881i;
where kij is jth eigenvalue of ith equilibrium point.Based on the descriptions provided in the previous section, we modify the derivative operator in (7) to be with respect to a fractional-order a. Thus, (7) is transformed into the following fractional system:
8 da x > a ¼ y ax þ byz > < dta d y ¼ cy xz þ z dt a > > : da z ¼ dxy Hz dta
ð11Þ
Necessary conditions to check the existence of chaos in fractional order systems with commensurate or incommensurate orders are given in [33,34] respectively. Based on [33], a necessary condition for fractional order system (11) to be chaotic is
a > max i
2
p
tan1
jImðki2 Þj Reðki2 Þ
a > maxð0; 0:827; 0:799; 0:821; 0:783Þ
ð12Þ
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715
According to inequality (12), the maximum order a to show the chaotic trajectories for commensurate system (11) and the same parameters as integer one is a 0:83. Figs. 1 and 2 show the numerical simulation results for a ¼ 0:83 and a ¼ 0:82 respectively. Numerical simulations prove that although for a ¼ 0:82 states of system converge to the equilibrium point E2 (initial states are set as [1,1,1]T), but just for 0.01 variation towards the higher value a ¼ 0:83, the system behaves completely different and in chaotic way. We have also calculated the value of maximum Lyapunov exponent for the system, which has been found 1.12, and certainly verifies the chaos existence in system (11) with a ¼ 0:83 (see Fig. 3.)
4. Non-fragile state feedback control Here, we investigate chaos control issue in system (11) with the state feedback scheme. Consider the following system:
8 a d x > a ¼ y ax þ byz þ u1 > > < dt da y
dt a > > > : da z dta
ð13Þ
¼ cy xz þ z þ u2
¼ dxy hz þ u3
Linearization of system (13) around the equilibrium point results in the following structure:
0
da x dta
1
0 1 0 1 x x B a C Bd yC B C B C ¼ A y þ b y B dta C @ A @ A @ A da z z z a
ð14Þ
dt
where
0
a
B A¼B @ z dy
1 þ bz c
dx
1 0 1 0 1 u1 x C B C B C C x þ 1 A; @ u2 A ¼ b@ y A
by
h
u3
ð15Þ
z
in which x⁄, y⁄ and z⁄ are state values of master system in equilibria, and b is the control gain matrix which should be designed. Based on Theorem 1, b must be chosen such that:
j argðeigðA þ bÞÞj > ap=2
ð16Þ
By linearization around each equilibrium point, we will have 5 matrices Ai which have the following values, if we set as a = 3, b = 2.7, c = 4.7, d = 2 and h = 9:
Fig. 1. Simulation results for fractional order system (11) with a ¼ 0:82.
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Fig. 2. Simulation results for fractional order system (11) with a ¼ 0:83.
Fig. 3. Disturbances in gain matrix (24). D1 (solid line) and D2 (dashed line).
0
3
B A1 ¼ @ 0 0 0
1
0
1
0
3
7:396
7:315
1
0
3
8:395
6:492
1
C B C B C 4:7 1 A A2 ¼ @ 2:369 4:7 4:216 A A3 ¼ @ 2:739 4:7 4:216 A 0 9 5:419 10:432 9 4:809 10:432 9 1 0 1 3 6:058 5:517 3 6:058 6:607 B C B C 4:7 5:126 A A5 ¼ @ 1:873 4:7 5:126 A A4 ¼ @ 1:873 4:086 8:252 9 4:8942 8:252 9
ð17Þ
For sake of controller design or stability analysis of a nonlinear fractional order system it should be linearized around the equilibria, whereas our mathematical tools for analyzing the nonlinear dynamics of a fractional derivative system is still very poor. For improvement of controller design, it is advised to regard all of the linear dynamics for multi-equilibrium systems. However, this is neglected in many previous literatures. For example in [26], two fractional order Chen systems were synchronized, when the Jacobian matrix is calculated only in the vicinity of point cþ ¼ ð7:9373; 7:9373; 21Þ. So the proposed controller of [26] or of many other similar letters cannot guarantee the desired performance around the other equilibria. Therefore, one of the contributions of this paper is designing a controller such that the ta stability will be demonstrated theoretically around each equilibrium point. Also, we consider the controller matrix b is involved with uncertainty. Based on generalized LMI framework, the following theorem is presented in [35].
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Theorem 2 [35]. Fractional system DVx(t) = Ax(t) of order 0 < v < 1 is t a asymptotically stable if and only if there exist positive definite matrices X 1 ¼ X 1 2 C nn and X 2 ¼ X 2 2 C nn such that
rX 1 A0 þ rAX 1 þ rX 2 A0 þ r AX 2 < 0;
ð18Þ
jð1v Þp
2 . where r ¼ e The following theorem is proposed for the deterministic matrix A. But in the light of convexity nature of LMI problems, we upgraded it for sake of our purpose. Consider the uncertain matrix b such that b 2 bI , where bI is set of all possible values for matrix b, according to the uncertainties of the parameters. The following theorem gives a necessary and sufficient condition for check of any candidate interval matrix b 2 bI as a non-fragile controller.
Theorem 3. Uncertain matrix b is a non-fragile stabilizer for systems (14) if and only if two positive definite matrices X1 and X2 can be found such that the following LMIs are satisfied:
rX 1 ðAi þ b Þ0 þ rðAi þ b ÞX 1 þ rX 2 ðAi þ b Þ0 þ r ðAi þ b ÞX 2 < 0; jð1v Þp 2
where r ¼ e
i ¼ 1; 2; 3; 4; 5:
ð19Þ
I
, for all vertex matrices b 2 b .
Proof. Based on Theorem 2, if the above LMIs are satisfied simultaneously, the stability of dynamic system (14) around each is guaranteed. Also, due to the convexity of LMI problems, satisfying the equilibria with any vertex matrix of b, (i.e. b and b) following inequalities means that the closed loop system is stable for all interval matrices b. The inverse proof is straightforward and omitted here. h Remark 1. The structure of the proposed controller is static state feedback, which is classified as one of the most popular methods, due to simplicity of implementation in real world. In some papers that utilize active mode control and have a synchronizer signal including nonlinear fractional order terms, some interesting results are reported. In [36], for instance, achieving synchronization between an ordinary and a fractional-order system is reported with a nonlinear fractional order synchronizer signal. Remark 2. Since the condition (19) is necessary and sufficient, we may exclude any subset of equilibrium points to guarantee the stability around the other ones. For instance, if one is interested in zero convergence of states, just needs to keep LMI (19) in its current form for A1 (which is associated with equilibrium point (0, 0, 0)) and then converts ‘‘‘‘ for i = 2–5 in (19). This guarantees that any matrix b 2 bI satisfying these LMI’s simultaneously provides states attraction only to the origin, and not for any other equilibria.
5. Numerical simulations In this section, we present some computer simulation results that illustrate the achievement of non-fragile control and then synchronization of system (11) in scenarios considered in Section 4. Consider the following system:
8 d0:9 x 1 > ¼ y1 3x1 þ 2:7y1 z1 þ u1 > > dt0:9 < d
0:9
y1
dt 0:9 > > > : d0:9 z1 dt0:9
0
¼ 4:7y1 x1 z1 þ z1 þ u2 ;
¼ 2x1 y1 9z1 þ u3
1 0 1 u1 x B C B C @ u2 A ¼ b@ y A z u3
ð20Þ
A simple choice of b would be as
2 6 b¼4
6 þ D1
0
0
6 þ D2
0
0
0
3
7 0 5;
D1 2 ½1; 1;
D2 2 ½1; 1
ð21Þ
0
where D1 and D2 denote the uncertainty associated with b. Based on Theorem 3, uncertain matrix b could be utilized as a stabilizer if and only if symmetric positive definite matrices X1 and X2 could be found such that following LMIs are satisfied:
rX 1 ðAi þ bj Þ0 þ rðAi þ bj ÞX 1 þ rX 2 ðAi þ bj Þ0 þ r ðAi þ bj ÞX 2 < 0;
i ¼ 1; 2; 3; 4; 5; j ¼ 1; 2; 3; 4: 2
3 2
5 0 0 7 0 p where r ¼ ej20 , Ais are as given in (17) and bj s are vertex matrices 4 0 5 0 5, 4 0 5 0 0 0 0 0 2 3 7 0 0 4 0 7 0 5. Using Yalmip toolbox of MATLAB program, the following matrices were found: 0 0 0
ð22Þ 3 2
0 5 0 5, 4 0 0 0
0 7 0
3
0 0 5 and 0
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Fig. 4. Simulation results for system (20).
2
4:008
0:688
6 X 1 ¼ X 2 ¼ 4 0:688
1:931
0:440
3
7 0:114 5
0:440 0:114
ð23Þ
3:509
As result of existence of these matrices, we conclude that interval matrix b stabilizes dynamic system (20) with any b 2 bI . Numerical results shown in Fig. 4 are computed by the following controller:
2
6 þ D1
0
0
6 þ D2
0
0
6 b¼4
0
3
7 05
ð24Þ
0
Remark 3. It is notable to remind that by today, a method to find a state feedback matrix for stability of fractional-order system has not been proposed yet. All available methods dealing with state feedback scheme in fractional order systems, including Theorems 1 and 2 can only analyze given gain matrices. We remind that for the ordinary version of system (6), matrix b such that u ¼ bx be a stabilizer for the ordinary system
(
x_ ¼ Ax þ Bu y ¼ Cx
;
xð0Þ ¼ x0 ;
could be easily found by solving the LMI AX þ XAT þ BW þ W T BT < 0; with b ¼ WX 1 [37]. In the sequel, according to Theorem 3, synchronization between two fractional order systems (11) is investigated. With applying the familiar method for constructing the drive–response configuration by PC method [19], we build a drive–response configuration described as following differential equations. The master system is given by
8 0:9 d x1 > > 0:9 ¼ y1 3x1 þ 2:7y1 z1 > > < dt0:9 d y1 ¼ 4:7y1 x1 z1 þ z1 : dt 0:9 > > > 0:9 > : d z1 ¼ 2x y 9z 1 1 1 dt0:9
ð25Þ
The response system is described by
8 0:9 d x2 > > 0:9 ¼ y2 3x2 þ 2:7y2 z1 þ u1 > > < dt0:9 d y2 ¼ 4:7y2 x2 z1 þ z1 þ u2 : dt 0:9 > > > 0:9 > : d z2 ¼ 2x y 9z þ u 2 2 1 3 dt0:9
ð26Þ
M.M. Asheghan et al. / Applied Mathematics and Computation 222 (2013) 712–721
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Fig. 5. Synchronization errors of state x (solid line), state y (dotted line) and state z (dashed line).
Fig. 6. Disturbances in gain matrix (28). D1 (solid line) and D2 (dashed line).
Define synchronization errors as e1 ¼ x2 x1 , e2 ¼ y2 y1 and e3 ¼ z2 z1 . Subtracting the equations of system (25) from the equations of system (26) yields the following error system 0:9
d e dt
0:9
0
a
B ¼ A:e þ u; A ¼ @ z1 dy1
1 c dx1
0
1
C 0 A;
u ¼ be
ð27Þ
0
where A ¼ Df , f ¼ ½f1 ; f2 ; f3 is linearization of f around x1 and y1 . In spite of the previous section in which we linearized the system just around its equilibria points, here we should hold synchronization in vicinity of any point on trajectory. To achieve this aim, we utilize convexity property of LMI problems. Indeed, if we show that some inequalities are satisfies for some vertex matrices, then we can conclude that they are also satisfied for any matrices located in convex space described by those boundary matrices. As a result of being a chaotic system, all states of master system (25) are bounded. According to Fig. 1, one can easily find the upper bound x1 , y1 and z1 as 15. So matrix A in (27) is always in a convex area with the following vertex matrices:
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0
1
a
B A1 ¼ @ 15 0
c
0
1
B A5 ¼ @ 15
c
0
a
C B 0 A; A2 ¼ @ 15
15d 15d 0 a
1
0
1
0
c
0
a
1 c
1
0
a
C B 0 A; A3 ¼ @ 15
15d 15d 0
C B 0 A; A6 ¼ @ 15
15d 15d 0
1
0
1
0
15d a
C B 0 A; A7 ¼ @ 15
15d 15d 0
1 c
0
1
0
a
C B 0 A; A4 ¼ @ 15
1
0
1
C 0 A;
c
15d 0 15d 1 0 1 0 a C B c 0 A; A8 ¼ @ 15
15d 0 1 1 0 C c 0 A: 15d 15d 0
15d 15d 0
A simple choice of b would be as
2
0 0 6 b ¼ 4 0 12 þ D1 0
0
3
0 0
7 5;
D1 ; D2 2 ½2; 2
ð28Þ
12 þ D2
where D1 and D2 denote the uncertainty associated with b. Based on Theorem 3, uncertain matrix b could be utilized as a synchronizer if an only if a couple of symmetric positive definite matrices X1 and X2 could be found such that following LMIs are satisfied:
rX 1 ðAi þ bj Þ0 þ rðAi þ bj ÞX 1 þ rX 2 ðAi þ bj Þ0 þ r ðAi þ bj Þ < 0 i ¼ 1; . . . ; 8 j ¼ 1; 2; 3; 4: 2
3
2
3
2
3
ð29Þ 2
0 0 0 0 0 0 0 0 0 0 p 0 5, b2 ¼ 4 0 12 0 5, b3 ¼ 4 0 10 0 5, b4 ¼ 4 0 where r ¼ ej20 , b1 ¼ 4 0 10 0 0 10 0 0 10 0 0 12 0 given in (28). Using Yalmip toolbox of MATLAB program, the following matrices were found:
2
0:021
6 X 1 ¼ X 2 ¼ 4 0:071 0
0:071
0
0:451
0
0
15:25
0 12 0
3
0 0 5, and Ais are as 12
3 7 5:
ð30Þ
As result of existence of these matrices, we conclude that interval matrix b stabilizes error dynamic system (27) with any value in (28). Numerical simulations shown in Fig. 5 are computed by the controller (28), where D1 and D2 represent disturbances shown in Fig. 6. 6. Conclusion Chaotic behavior of a new fractional order chaotic system is investigated in this work. Chaos control and synchronization between two identical fractional order systems are then presented. Also, we extended our work to non-fragile control field, when based on a rigorous theorem presented in [35], any interval matrix could be checked as a non-fragile controller candidate, using LMI method. The proposed method is necessary and sufficient. Because of independency of proposed criterion to a special case, it can be easily extended to other fractional order chaotic systems. Numerical simulations show the efficiency of the proposed method to control and synchronize chaotic fractional-order systems while the control parameters are perturbed. Non-fragile synchronization of two incommensurate fractional order systems is another issue that can be investigated in our future work. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/ j.amc.2013.07.045. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Non-fragile control and synchronization of a new fractional order chaotic system Mohammad Mostafa Asheghan a, Saleh S. Delshad b, Mohammad Taghi Hamidi Beheshti c,⇑, Mohammad Saleh Tavazoei d a
Communications, Universidad Carlos III de Madrid, Avenida de la Universidad 30, Leganes 28911, Madrid, Spain Control Engineering Group, Luleå University of Technology, Sweden c Control and Communication Networks Lab, Electrical Engineering Department, Tarbiat Modares University, Tehran, Iran d Electrical Engineering Department, Sharif University of Technology, Tehran, Iran b
a r t i c l e
i n f o
Keywords: Fractional order systems Chaos synchronization Robust control
a b s t r a c t In this paper, we address global non-fragile control and synchronization of a new fractional order chaotic system. First we inspect the chaotic behavior of the fractional order system under study and also find the lowest order (2.49) for the introduced dynamics to remain chaotic. Then, a necessary and sufficient condition which can be easily extended to other fractional-order systems is proposed in terms of Linear Matrix Inequality (LMI) to check whether the candidate state feedback controller with parameter uncertainty can guarantee zero convergence of error or not. In addition, the proposed method provides a global zero attraction of error that guarantees stability around all existing equilibrium points. Finally, numerical simulation are employed to verify the validity of the proposed algorithm. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Fractional calculus is a quite old issue that goes back to more than 300 years ago in history. Although it has a long history, the applications of fractional calculus to physics and engineering have just started [1]. Nowadays, there are many systems that fall into fractional order systems category such as viscoelastic systems, electrode–electrolyte polarization, electromagnetic waves quantum evolution of complex systems, quantitative finance and diffusion wave [1]. In recent years, study on the dynamics of fractional-order differential systems has attracted even more interests amongst researchers. There are plenty of references involved in investigation of the chaotic and hyper chaotic behavior of the fractional-order differential systems such as the fractional-order Chua circuit [2], the fractional-order Chen system [3], the fractional-order Lu system [4], the fractional-order Rossler system [5], the fractional-order Arneodo system [6] and the fractional-order Newton–Leipnik system [7]. Recently, a new four-dimensional fractional-order system has been proposed, which has the very interesting property of having no equilibrium point [8]. Over the last two decades, pioneering Pecora and Carroll [9], chaos synchronization has been widely explored and investigated in various fields such as physical biological, chemical and medical science and secure communications [10]. The problem of designing a system, whose behavior mimics that of another chaotic system, is called synchronization. There have been many prior studies that addressed chaos control and synchronization methods in integer-order systems [11–14]. For example, [11] designed a robust linear state feedback controller using Linear Matrix Inequalities (LMI) method, while [12]
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (M.T. Hamidi Beheshti). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.07.045
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713
presented a fuzzy sliding mode control based method for chaos synchronization. [13] utilizes back stepping method to synchronize two Chua chaotic circuits. Recently, synchronization of chaotic fractional-order systems has become more and more interesting to researchers due to its potential applications in secure communication and control processing [15]. For example, in [16] chaos synchronization of two fractional Lu systems has been studied. Also, synchronization of two chaotic fractional Chen systems and synchronization of two chaotic fractional Chua systems have been presented in [17,18], respectively. Some more attempts to synchronize fractional order systems can be found in [19–21]. As well as synchronization, control and stabilization of fractional order systems have received a great deal of interest among the researchers. As it is mentioned before, many real systems behave fractionally, so their control problem should be investigated in fractional order field. Additionally, mathematical fascination of fractional order equations is another reason for such an increasing attention of scientists to fractional dynamic filed. However, to our best knowledge, non-fragile control or synchronization of fractional order systems is not reported in the existing literature. In simple words, a controller for which a certain bounded variation on controller gain does not violate stability of the closed loop system is referred to as non-fragile. Some non-fragile approaches in field of ordinary systems can be found in [22–25]. When the non-fragility is not guaranteed, implementation of the controller in real world application becomes more expensive, due to necessity of using elements with ignorable tolerance in the nominal values. In [26], for instance, the controller matrix has one non-zero entry which is 10. Even though zero convergence of error has been theoretically shown by this value, however, there is no guarantee that if the controller is capable to endure any little perturbation or not. The rest of the paper is organized as follows. In Section 2, basic definitions in fractional calculus as well as some fundamental theorems about fractional systems are presented. In Section 3, a new fractional-order chaotic system and sensitivity of its dynamics to different values of the system coefficients are studied. Section 4 is devoted to non-fragile control and synchronization between two chaotic systems. Section 5 concludes the paper. 2. Primary definitions and stability theorem In this section, we will give preliminary knowledge for our main results. The differintegral operator, denoted by a Dqt , is a combined differentiation-integration operator commonly used in fractional calculus. This operator is a notation for taking both the fractional derivative and the fractional integral in a single expression and is defined by, q a Dt
8 >
:Rt a
q>0
;
1
ð1Þ
q¼0
ðdsÞ
q
q < x_ ¼ y ax þ byz y_ ¼ cy xz þ z > :_ z ¼ dxy Hz
ð7Þ
where x, y and z are state variables, and a, b, c, d and h are positive real constants. This system is found to be chaotic for the parameters a = 3, b = 2.7, c 2 ð4:45; 4:60Þ [ ð4:86; 4:94Þ [ ½5:17; 7Þ, d = 2 and h = 9, and has many interesting complex dynamical behaviors. In [32] it is shown that (7) has five equilibria. Setting the parameters as a = 3, b = 2.7, c = 4.7, d = 2 and h = 9 leads to the following equilibrium points
8 E1 ð0; 0; 0Þ > > > > > > < E2 ðþ5:1260; þ2:7094; þ2:3687Þ
E3 ðþ5:1260; 2:4045; þ2:7390Þ : > > > E > 4 ð4:1260; 2:0432; þ1:8734Þ > > : E5 ð4:1260; þ2:4471; 2:2438Þ
ð8Þ
The Jacobian matrix of (7), evaluated at (x⁄, y⁄, z⁄) has the following form.
2
1 þ bz
a
6 J ¼ 4 z dy
3
7 1 x 5
c
by
dx
ð9Þ
h
The eigenvalues of (9) for any x⁄, y⁄ and z⁄ and the defined parameters a, b, c, d and h are as follows:
8 k11 > > > > > > < k21 k31 > > > > k41 > > : k51
¼ 3; k12 ¼ 4:7; k13 ¼ 9; ¼ 11:0247; k22 ¼ 1:8623 þ 6:6831i; k23 ¼ 1:8623 6:6831i; ¼ 11:7856; k32 ¼ 2:2428 þ 6:8580i; k33 ¼ 2:2428 6:8580i;
ð10Þ
¼ 10:7669; k42 ¼ 1:7335 þ 6:0024i; k43 ¼ 1:7335 6:0024i; ¼ 11:6813; k52 ¼ 2:1906 þ 6:1881i; k53 ¼ 2:1906 6:1881i;
where kij is jth eigenvalue of ith equilibrium point.Based on the descriptions provided in the previous section, we modify the derivative operator in (7) to be with respect to a fractional-order a. Thus, (7) is transformed into the following fractional system:
8 da x > a ¼ y ax þ byz > < dta d y ¼ cy xz þ z dt a > > : da z ¼ dxy Hz dta
ð11Þ
Necessary conditions to check the existence of chaos in fractional order systems with commensurate or incommensurate orders are given in [33,34] respectively. Based on [33], a necessary condition for fractional order system (11) to be chaotic is
a > max i
2
p
tan1
jImðki2 Þj Reðki2 Þ
a > maxð0; 0:827; 0:799; 0:821; 0:783Þ
ð12Þ
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715
According to inequality (12), the maximum order a to show the chaotic trajectories for commensurate system (11) and the same parameters as integer one is a 0:83. Figs. 1 and 2 show the numerical simulation results for a ¼ 0:83 and a ¼ 0:82 respectively. Numerical simulations prove that although for a ¼ 0:82 states of system converge to the equilibrium point E2 (initial states are set as [1,1,1]T), but just for 0.01 variation towards the higher value a ¼ 0:83, the system behaves completely different and in chaotic way. We have also calculated the value of maximum Lyapunov exponent for the system, which has been found 1.12, and certainly verifies the chaos existence in system (11) with a ¼ 0:83 (see Fig. 3.)
4. Non-fragile state feedback control Here, we investigate chaos control issue in system (11) with the state feedback scheme. Consider the following system:
8 a d x > a ¼ y ax þ byz þ u1 > > < dt da y
dt a > > > : da z dta
ð13Þ
¼ cy xz þ z þ u2
¼ dxy hz þ u3
Linearization of system (13) around the equilibrium point results in the following structure:
0
da x dta
1
0 1 0 1 x x B a C Bd yC B C B C ¼ A y þ b y B dta C @ A @ A @ A da z z z a
ð14Þ
dt
where
0
a
B A¼B @ z dy
1 þ bz c
dx
1 0 1 0 1 u1 x C B C B C C x þ 1 A; @ u2 A ¼ b@ y A
by
h
u3
ð15Þ
z
in which x⁄, y⁄ and z⁄ are state values of master system in equilibria, and b is the control gain matrix which should be designed. Based on Theorem 1, b must be chosen such that:
j argðeigðA þ bÞÞj > ap=2
ð16Þ
By linearization around each equilibrium point, we will have 5 matrices Ai which have the following values, if we set as a = 3, b = 2.7, c = 4.7, d = 2 and h = 9:
Fig. 1. Simulation results for fractional order system (11) with a ¼ 0:82.
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Fig. 2. Simulation results for fractional order system (11) with a ¼ 0:83.
Fig. 3. Disturbances in gain matrix (24). D1 (solid line) and D2 (dashed line).
0
3
B A1 ¼ @ 0 0 0
1
0
1
0
3
7:396
7:315
1
0
3
8:395
6:492
1
C B C B C 4:7 1 A A2 ¼ @ 2:369 4:7 4:216 A A3 ¼ @ 2:739 4:7 4:216 A 0 9 5:419 10:432 9 4:809 10:432 9 1 0 1 3 6:058 5:517 3 6:058 6:607 B C B C 4:7 5:126 A A5 ¼ @ 1:873 4:7 5:126 A A4 ¼ @ 1:873 4:086 8:252 9 4:8942 8:252 9
ð17Þ
For sake of controller design or stability analysis of a nonlinear fractional order system it should be linearized around the equilibria, whereas our mathematical tools for analyzing the nonlinear dynamics of a fractional derivative system is still very poor. For improvement of controller design, it is advised to regard all of the linear dynamics for multi-equilibrium systems. However, this is neglected in many previous literatures. For example in [26], two fractional order Chen systems were synchronized, when the Jacobian matrix is calculated only in the vicinity of point cþ ¼ ð7:9373; 7:9373; 21Þ. So the proposed controller of [26] or of many other similar letters cannot guarantee the desired performance around the other equilibria. Therefore, one of the contributions of this paper is designing a controller such that the ta stability will be demonstrated theoretically around each equilibrium point. Also, we consider the controller matrix b is involved with uncertainty. Based on generalized LMI framework, the following theorem is presented in [35].
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Theorem 2 [35]. Fractional system DVx(t) = Ax(t) of order 0 < v < 1 is t a asymptotically stable if and only if there exist positive definite matrices X 1 ¼ X 1 2 C nn and X 2 ¼ X 2 2 C nn such that
rX 1 A0 þ rAX 1 þ rX 2 A0 þ r AX 2 < 0;
ð18Þ
jð1v Þp
2 . where r ¼ e The following theorem is proposed for the deterministic matrix A. But in the light of convexity nature of LMI problems, we upgraded it for sake of our purpose. Consider the uncertain matrix b such that b 2 bI , where bI is set of all possible values for matrix b, according to the uncertainties of the parameters. The following theorem gives a necessary and sufficient condition for check of any candidate interval matrix b 2 bI as a non-fragile controller.
Theorem 3. Uncertain matrix b is a non-fragile stabilizer for systems (14) if and only if two positive definite matrices X1 and X2 can be found such that the following LMIs are satisfied:
rX 1 ðAi þ b Þ0 þ rðAi þ b ÞX 1 þ rX 2 ðAi þ b Þ0 þ r ðAi þ b ÞX 2 < 0; jð1v Þp 2
where r ¼ e
i ¼ 1; 2; 3; 4; 5:
ð19Þ
I
, for all vertex matrices b 2 b .
Proof. Based on Theorem 2, if the above LMIs are satisfied simultaneously, the stability of dynamic system (14) around each is guaranteed. Also, due to the convexity of LMI problems, satisfying the equilibria with any vertex matrix of b, (i.e. b and b) following inequalities means that the closed loop system is stable for all interval matrices b. The inverse proof is straightforward and omitted here. h Remark 1. The structure of the proposed controller is static state feedback, which is classified as one of the most popular methods, due to simplicity of implementation in real world. In some papers that utilize active mode control and have a synchronizer signal including nonlinear fractional order terms, some interesting results are reported. In [36], for instance, achieving synchronization between an ordinary and a fractional-order system is reported with a nonlinear fractional order synchronizer signal. Remark 2. Since the condition (19) is necessary and sufficient, we may exclude any subset of equilibrium points to guarantee the stability around the other ones. For instance, if one is interested in zero convergence of states, just needs to keep LMI (19) in its current form for A1 (which is associated with equilibrium point (0, 0, 0)) and then converts ‘‘‘‘ for i = 2–5 in (19). This guarantees that any matrix b 2 bI satisfying these LMI’s simultaneously provides states attraction only to the origin, and not for any other equilibria.
5. Numerical simulations In this section, we present some computer simulation results that illustrate the achievement of non-fragile control and then synchronization of system (11) in scenarios considered in Section 4. Consider the following system:
8 d0:9 x 1 > ¼ y1 3x1 þ 2:7y1 z1 þ u1 > > dt0:9 < d
0:9
y1
dt 0:9 > > > : d0:9 z1 dt0:9
0
¼ 4:7y1 x1 z1 þ z1 þ u2 ;
¼ 2x1 y1 9z1 þ u3
1 0 1 u1 x B C B C @ u2 A ¼ b@ y A z u3
ð20Þ
A simple choice of b would be as
2 6 b¼4
6 þ D1
0
0
6 þ D2
0
0
0
3
7 0 5;
D1 2 ½1; 1;
D2 2 ½1; 1
ð21Þ
0
where D1 and D2 denote the uncertainty associated with b. Based on Theorem 3, uncertain matrix b could be utilized as a stabilizer if and only if symmetric positive definite matrices X1 and X2 could be found such that following LMIs are satisfied:
rX 1 ðAi þ bj Þ0 þ rðAi þ bj ÞX 1 þ rX 2 ðAi þ bj Þ0 þ r ðAi þ bj ÞX 2 < 0;
i ¼ 1; 2; 3; 4; 5; j ¼ 1; 2; 3; 4: 2
3 2
5 0 0 7 0 p where r ¼ ej20 , Ais are as given in (17) and bj s are vertex matrices 4 0 5 0 5, 4 0 5 0 0 0 0 0 2 3 7 0 0 4 0 7 0 5. Using Yalmip toolbox of MATLAB program, the following matrices were found: 0 0 0
ð22Þ 3 2
0 5 0 5, 4 0 0 0
0 7 0
3
0 0 5 and 0
718
M.M. Asheghan et al. / Applied Mathematics and Computation 222 (2013) 712–721
Fig. 4. Simulation results for system (20).
2
4:008
0:688
6 X 1 ¼ X 2 ¼ 4 0:688
1:931
0:440
3
7 0:114 5
0:440 0:114
ð23Þ
3:509
As result of existence of these matrices, we conclude that interval matrix b stabilizes dynamic system (20) with any b 2 bI . Numerical results shown in Fig. 4 are computed by the following controller:
2
6 þ D1
0
0
6 þ D2
0
0
6 b¼4
0
3
7 05
ð24Þ
0
Remark 3. It is notable to remind that by today, a method to find a state feedback matrix for stability of fractional-order system has not been proposed yet. All available methods dealing with state feedback scheme in fractional order systems, including Theorems 1 and 2 can only analyze given gain matrices. We remind that for the ordinary version of system (6), matrix b such that u ¼ bx be a stabilizer for the ordinary system
(
x_ ¼ Ax þ Bu y ¼ Cx
;
xð0Þ ¼ x0 ;
could be easily found by solving the LMI AX þ XAT þ BW þ W T BT < 0; with b ¼ WX 1 [37]. In the sequel, according to Theorem 3, synchronization between two fractional order systems (11) is investigated. With applying the familiar method for constructing the drive–response configuration by PC method [19], we build a drive–response configuration described as following differential equations. The master system is given by
8 0:9 d x1 > > 0:9 ¼ y1 3x1 þ 2:7y1 z1 > > < dt0:9 d y1 ¼ 4:7y1 x1 z1 þ z1 : dt 0:9 > > > 0:9 > : d z1 ¼ 2x y 9z 1 1 1 dt0:9
ð25Þ
The response system is described by
8 0:9 d x2 > > 0:9 ¼ y2 3x2 þ 2:7y2 z1 þ u1 > > < dt0:9 d y2 ¼ 4:7y2 x2 z1 þ z1 þ u2 : dt 0:9 > > > 0:9 > : d z2 ¼ 2x y 9z þ u 2 2 1 3 dt0:9
ð26Þ
M.M. Asheghan et al. / Applied Mathematics and Computation 222 (2013) 712–721
719
Fig. 5. Synchronization errors of state x (solid line), state y (dotted line) and state z (dashed line).
Fig. 6. Disturbances in gain matrix (28). D1 (solid line) and D2 (dashed line).
Define synchronization errors as e1 ¼ x2 x1 , e2 ¼ y2 y1 and e3 ¼ z2 z1 . Subtracting the equations of system (25) from the equations of system (26) yields the following error system 0:9
d e dt
0:9
0
a
B ¼ A:e þ u; A ¼ @ z1 dy1
1 c dx1
0
1
C 0 A;
u ¼ be
ð27Þ
0
where A ¼ Df , f ¼ ½f1 ; f2 ; f3 is linearization of f around x1 and y1 . In spite of the previous section in which we linearized the system just around its equilibria points, here we should hold synchronization in vicinity of any point on trajectory. To achieve this aim, we utilize convexity property of LMI problems. Indeed, if we show that some inequalities are satisfies for some vertex matrices, then we can conclude that they are also satisfied for any matrices located in convex space described by those boundary matrices. As a result of being a chaotic system, all states of master system (25) are bounded. According to Fig. 1, one can easily find the upper bound x1 , y1 and z1 as 15. So matrix A in (27) is always in a convex area with the following vertex matrices:
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M.M. Asheghan et al. / Applied Mathematics and Computation 222 (2013) 712–721
0
1
a
B A1 ¼ @ 15 0
c
0
1
B A5 ¼ @ 15
c
0
a
C B 0 A; A2 ¼ @ 15
15d 15d 0 a
1
0
1
0
c
0
a
1 c
1
0
a
C B 0 A; A3 ¼ @ 15
15d 15d 0
C B 0 A; A6 ¼ @ 15
15d 15d 0
1
0
1
0
15d a
C B 0 A; A7 ¼ @ 15
15d 15d 0
1 c
0
1
0
a
C B 0 A; A4 ¼ @ 15
1
0
1
C 0 A;
c
15d 0 15d 1 0 1 0 a C B c 0 A; A8 ¼ @ 15
15d 0 1 1 0 C c 0 A: 15d 15d 0
15d 15d 0
A simple choice of b would be as
2
0 0 6 b ¼ 4 0 12 þ D1 0
0
3
0 0
7 5;
D1 ; D2 2 ½2; 2
ð28Þ
12 þ D2
where D1 and D2 denote the uncertainty associated with b. Based on Theorem 3, uncertain matrix b could be utilized as a synchronizer if an only if a couple of symmetric positive definite matrices X1 and X2 could be found such that following LMIs are satisfied:
rX 1 ðAi þ bj Þ0 þ rðAi þ bj ÞX 1 þ rX 2 ðAi þ bj Þ0 þ r ðAi þ bj Þ < 0 i ¼ 1; . . . ; 8 j ¼ 1; 2; 3; 4: 2
3
2
3
2
3
ð29Þ 2
0 0 0 0 0 0 0 0 0 0 p 0 5, b2 ¼ 4 0 12 0 5, b3 ¼ 4 0 10 0 5, b4 ¼ 4 0 where r ¼ ej20 , b1 ¼ 4 0 10 0 0 10 0 0 10 0 0 12 0 given in (28). Using Yalmip toolbox of MATLAB program, the following matrices were found:
2
0:021
6 X 1 ¼ X 2 ¼ 4 0:071 0
0:071
0
0:451
0
0
15:25
0 12 0
3
0 0 5, and Ais are as 12
3 7 5:
ð30Þ
As result of existence of these matrices, we conclude that interval matrix b stabilizes error dynamic system (27) with any value in (28). Numerical simulations shown in Fig. 5 are computed by the controller (28), where D1 and D2 represent disturbances shown in Fig. 6. 6. Conclusion Chaotic behavior of a new fractional order chaotic system is investigated in this work. Chaos control and synchronization between two identical fractional order systems are then presented. Also, we extended our work to non-fragile control field, when based on a rigorous theorem presented in [35], any interval matrix could be checked as a non-fragile controller candidate, using LMI method. The proposed method is necessary and sufficient. Because of independency of proposed criterion to a special case, it can be easily extended to other fractional order chaotic systems. Numerical simulations show the efficiency of the proposed method to control and synchronize chaotic fractional-order systems while the control parameters are perturbed. Non-fragile synchronization of two incommensurate fractional order systems is another issue that can be investigated in our future work. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/ j.amc.2013.07.045. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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