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A Nussbaum gain adaptive synchronization of a new hyperchaotic system with input uncertainties and unknown parameters
Commun Nonlinear Sci Numer Simulat 14 (2009) 3439–3448
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A Nussbaum gain adaptive synchronization of a new hyperchaotic system with input uncertainties and unknown parameters Junwei Lei a,*, Xinyu Wang b, Yinhua Lei c a b c
Department of Automatic Control Engineering, Naval Aeronautical Engineering Academy, Yantai 264001, China Institute of Science and Technology for Opto-electronic Information, Yantai University, Yantai 264001, China Institute of Microelectronics, Peking University, Beijing 100871, China
a r t i c l e
i n f o
Article history: Received 9 August 2008 Received in revised form 24 September 2008 Accepted 1 December 2008 Available online 24 December 2008 PACS: 05.45.Vx 05.45.Gg 05.45.Xt Keywords: Input uncertainties Synchronization Chaos Global terminal Adaptive
a b s t r a c t The sign of unknown input coefficients is assumed to be known in most papers about the input uncertainties. In this paper, a Nussbaum gain method is adopted to cope with the situation that both the sign and the value of input are unknown. And the unknown parameters can be estimated under the situation of unknown sign of control. The synchronization is achieved for a class of hyperchaotic systems with unknown parameters and input uncertainties by adopting of the Nussbaum gain method and the global terminal adaptive method. And the conclusions are made as follows: First, the proposed method is effective in the situation that the sign of input is unknown. Second, the estimation of unknown parameters can be achieved only when the number of unknown parameters satisfied some condition and no uncertainty exist in the input of systems. Third, the unknown parameters cannot be estimated correctly with common adaptive method when there are input uncertainties in the system. But the Nussbaum gain method can get good result in the estimation of unknown parameters. At last, numerical simulations are done to show the effectiveness of the proposed method. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Chaos systems have complex dynamical behaviors that possess some special features such as being extremely sensitive to tiny variations of initial conditions, and having bounded trajectories with a positive leading Lyapunov exponent and so on [1–7]. In recent years, chaos synchronization has been attracted increasingly attentions due to their potential applications in the fields of secure communications. Synchronization of chaos systems with unknown parameters is investigated widely by researchers from various fields. In fact, input uncertainties usually exist in actual control systems, but the situations of chaos systems with input uncertainties are neglected in most papers [1–4]. We known, it is more difficult to estimate the unknown parameters exactly than to achieve synchronization. The estimation of unknown parameters often could not converge to its real value in many simulations [1–4]. It is a difficult problem that is often neglected by many researchers unconsciously or on purpose, and some researchers intended to find the answer for this question, such as [3,4]. But obviously the problem has not been solved successfully until now, even some wrong answers have been proposed in some papers [3,4]. In this paper, we conclude that the unknown parameters cannot be estimated in some situation. Also a sufficient condition of parameters estimation is proposed. This condition may be unnecessary, but it is * Corresponding author. E-mail address: [email protected] (J. Lei). 1007-5704/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2008.12.010
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still the best answer we can found for this question. Also, we pointed out that the unknown parameters cannot be estimated effectively in systems with input uncertainties. In the past decade, backstepping has become one of the most popular design methods for adaptive nonlinear control and synchronization because it can guarantee global stabilities for the broad class of strict-feedback systems [5–9]. The sign of input need to be known in many papers researched on the problem of input uncertainty. The Nussbaum gain theory is adopted to cope with this problem. The unknown parameters can be estimated effectively with the Nussbaum gain adaptive method. And the synchronization is achieved in finite time using a global terminal controller under the situation that the sign of input is unknown [9]. At last, we point out that the unknown parameters cannot be estimated with common adaptive method when there are input uncertainties in the system. The paper is organized as followed. In Section 2, the system model of our research is proposed. In Section 3, the global terminal control theory is introduced which can make the errors of system converge to 0 in finite time. In Section 4, a global terminal adaptive controller is designed for the chaos systems and synchronization is achieved. In Section 5, the numerical simulation is done to verify the conclusions of this paper. In Section 6, the conclusions are given and the future work is pointed out. 2. System description The new hyperchaotic system is investigated for its important use in the secret communication by many researchers recently, and its model can be written as:
x_ 1 ¼ a1 ðy1 x1 Þ y_ 1 ¼ b1 x1 k1 x1 z1 w1 z_ 1 ¼ c1 z1 þ h1 x21 _ w1 ¼ d1 x1
ð1Þ
where x1, y1, z1 are the states of the system, and a = 10, b = 40, c = 2.5, d = 10.6, k = 1, h = 4 are unknown parameters. We take the model (1) as the master system and assume the slave system have different unknown parameters as follows:
x_ 2 ¼ a2 ðy2 x2 Þ þ bu1 u1 y_ 2 ¼ b2 x2 k2 x2 z2 w2 þ bu2 u2 z_ 2 ¼ c2 z2 þ h2 x22 þ bu3 u3 _ 2 ¼ d2 x2 þ bu4 u4 w
ð2Þ
The master system and slave system are denoted by 1 and 2, respectively, where bui are input uncertainties. Our goal is to find a controller u ¼ ½ u1 u2 u3 to make the state of slave system track to the states of master system. The error system can be written as:
e_ 1 ¼ a2 ðy2 x2 Þ a1 ðy1 x1 Þ þ bu1 u1 e_ 2 ¼ ðb2 x2 k2 x2 z2 w2 Þ ðb1 x1 k1 x1 z1 w1 Þ þ bu2 u2 e_ 3 ¼ c2 z2 þ h2 x22 ðc1 z1 þ h1 x21 Þ þ bu3 u3 e_ 4 ¼ d2 x2 ðd1 x1 Þ þ bu4 u4
ð3Þ
where e1 = x2 x1,e2 = y2 y1,e3 = z2 z1,e4 = w2 w1. Assumption 1. The sign of bui is unknown. 3. Global terminal control To explain the concept of finite convergence time, the global terminal control theory is introduced as followed [2]. Definition 1 (Terminal function). For a kind of system having the form as defined in and the sliding mode control having the form as defined in, when S ? 0, for any initial system state e(t0), exists ts1 > 0 such thatts is bound and ts < ts1, where ts is defined as the time of the system need in order to converge the state from e(t0) to 0, then f(e) is defined as a terminal function. The system has the form as:
e_ ¼ hðeÞ þ uðtÞ
ð4Þ
We define the surface and control as:
S ¼ e_ þ f ðeÞ;
u ¼ gðS; eÞ
ð5Þ
J. Lei et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3439–3448
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Definition 2 (Global terminal function). For a kind of system having the form as defined in and the sliding mode control having the form as defined in (5), when S ? 0, exists ts1 > 0 for any initial system state e(t0), such that ts is bound and ts < ts1, where ts is defined as the time of the system need in order to converge the state from e(t0) to 0, then f(e) is defined as a global terminal function. The traditional terminal function is defined as f(e) = e1/3, and it is not a global terminal function, so it does not have the advantage of global terminal function. For system (4), we compute the convergence time as the following: When S ? 0, we get:
e_ ¼ f ðeÞ
ð6Þ
To ensure the system stable, make that:
ee_ ¼ ef ðeÞ < 0
ð7Þ
Define:
_ gðeÞ ¼ 1=f ðeÞ
ð8Þ
de=f ðeÞ ¼ dt ¼ dgðeÞ
ð9Þ
Then:
Do an integral computation on each side of the equation:
gðeðtf ÞÞ gðeðt0 ÞÞ ¼ t f t0
ð10Þ
Because the system is stable, so
eðtf Þ ¼ 0
ð11Þ
tf ¼ t 0 þ gð0Þ gðeðt0 ÞÞ
ð12Þ
Then:
The rule of constructing global terminal function can be found in Ref. [1]. For convenience, we take the following function as an example to compute the convergence time. 2=3 Consider the function f ðeÞ ¼ 32 e1=3 expðe2=3 Þ, We compute gðeÞ ¼ expðe2=3 Þ; tf ¼ t 0 þ gð0Þ gðeðt 0 ÞÞ ¼ 1 eðeð0Þ Þ < 1. So this function is not only a terminal function but also a global terminal function. 4. Nussbaum adaptive synchronization design The error system can be written as (13), and the control law can be designed as followed:
e_ 1 ¼ a1 f11 þ a2 f12 þ f13 þ bu1 u1 e_ 2 ¼ b1 f21 þ b2 f22 þ k1 f25 þ k2 f26 þ f23 þ bu2 u2
ð13Þ
e_ 3 ¼ c1 f31 þ c2 f32 þ h1 f35 þ h2 f36 þ f33 þ bu3 u3 e_ 4 ¼ d1 f41 þ c2 f42 þ f43 þ bu4 u4 where:
f11 ¼ ðy1 x1 Þ;
f 12 ¼ ðy2 x2 Þ;
f 22 ¼ x2 ;
f 25 ¼ x1 z1 ;
f31 ¼ z1 ;
f 32 ¼ z2 ;
f 35 ¼
x21 ;
f41 ¼ x1 ;
f 42 ¼ x2 ;
f 43 ¼ 0
f21 ¼ x1 ;
f 13 ¼ 0 f 26 ¼ x2 z2 ; f 36 ¼
Define:
f14 ¼ k11 e1 k12 je1ejþ1 e1 32 k13 d11 ed12 f24 ¼ k21 e2 k22 je2ejþ2 e2 32 k23 d21 ed22 f34 ¼ k31 e3 k32 je3ejþ3 e3 32 k33 d31 ed32 f44 ¼ k41 e4 k42 je4ejþ4 e4 32 k43 d41 ed42 where 2
di2 ¼ di1 ;
di1 ¼ e1=3 i ;
i ¼ 1; 2; 3; 4
x22 ;
f 23 ¼ w2 w1
f 33 ¼ 0
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Design
ubd i
ði ¼ 1; 2; 3Þ
^1 f11 a ^2 f12 f13 þ f14 ¼ ubd bu1 ud1 ¼ a 1 ^1 f21 b ^2 f22 k ^1 f25 k ^2 f26 f23 þ f24 ¼ ubd bu2 ud2 ¼ b 2 ^ f h ^ f f þ f ¼ ubd b ud ¼ ^c f ^c f h u3 3
1 31
2 32
1 25
2 26
33
34
ð14Þ
3
^1 f41 d ^2 f42 f43 þ f44 ¼ ubd bu4 ud3 ¼ d 3 where â1 is used to approximate a and define ã1 = a1 â1, and so are other variables. Because the control is unknown, and the sign of control is unknown, we adopt the Nussbaum function as follows: 2
NðkÞ ¼ k cos k
ð15Þ
Obviously, it has characters as follows:
lim sup
s!1
1 s
Z
s
NðkÞdk ¼ þ1;
lim inf
s!1
0
1 s
Z
s
NðkÞdk ¼ 1
ð16Þ
0
Design:
k_ bi ¼ ei ubd i ;
ui ¼ kni Ni ðkbi Þubd i ;
i ¼ 1; 2; 3; 4
ð17Þ
And define Lyapunov function as:
V1 ¼
4 1X e2 2 i¼1 i
And its time derivative described by:
V_ 1 ¼
4 X
e1 e_ 1
ð18Þ
i¼1 bd e1 e_ 1 ¼ e1 ½a1 f11 þ a2 f12 þ f13 þ ubd 1 þ ðb1 u1 u1 Þ
ð19Þ
Substituting (17)–(19), we get:
_ e1 e_ 1 ¼ ½e1 ða1 f11 þ a2 f12 þ f13 þ ubd 1 Þ þ ðb1 kn1 N 1 ðk1 Þ þ 1Þk1
ð20Þ
Design:
^_ 2 ¼ ka2 e1 f12 ^_ 1 ¼ ka1 e1 f11 ; a a Define:
1 2 1 2 1 2 ~ þ ~ e þ a a 2 1 2ka1 1 2ka2 2
ð21Þ
V_ 11 ¼ e1 f14 þ ðb1 kn1 N1 ðkb1 Þ þ 1Þk_ b1
ð22Þ
V 11 ¼ We get:
Consider that:
e1 f14 6 0 Do an integral computation we get:
V 11 ðtÞ V 11 ð0Þ
Z
t
e1 f14 ¼
0
Z
t
ðb1 kn1 N1 ðkb1 Þ þ 1Þk_ b1 dt
ð23Þ
0
Then
V 11 ðtÞ V 11 ð0Þ
Z
t
e1 f14 ¼
0
Z
k1 ðtÞ
b1 kn1 N1 ðkb1 Þdkb1 þ kb1 ðtÞ kb1 ð0Þ
k1 ð0Þ
Firstly, we assume kb1(t) ? +1 then
h
i R k ðtÞ ¼ k 1ðtÞ kb1b1ð0Þ b1 kn1 N 1 ðkb1 Þdkb1 1 þ kkb1 ð0Þ b1 ðtÞ b1 h i R k ðtÞ V 11 ðtÞ V 11 ð0Þ k 1ðtÞ 1 þ kkb1 ð0Þ < k 1ðtÞ k b1ð0Þ b1 kn1 N1 ðkb1 Þdkb1 ðtÞ ðV 11 ðtÞ V 11 ð0Þ b1
Rt 0
e1 f14 dtÞ k b1
1
b1 ðtÞ
b1
b1
ð24Þ
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Rs And it contradicts with lims!1 inf 1s 0 NðkÞdk ¼ 1. Secondly, we assume kb1(t) ? 1 then
V 11 ðtÞ V 11 ð0Þ
Z kb1 ðtÞ 1 kb1 ð0Þ 1 b1 kn1 N1 ðkb1 Þdkb1 1þ > kb1 ðtÞ kb1 ðtÞ kb1 ðtÞ kb1 ð0Þ
And it contradicts with lims!1 sup 1s gain as follows:
Rs 0
ð25Þ
NðkÞdk ¼ þ1. So kb1(t) is bound and e1 is bound. Also we can design other Nussbaum
_ e2 e_ 2 ¼ ½e2 ðb1 f21 þ b2 f22 þ k1 f25 þ k2 f26 þ f23 þ ubd 2 Þ þ ðb2 kn2 N 2 ðk2 Þ þ 1Þk2 bd _ e3 e_ 3 ¼ ½e3 ðc1 f31 þ c2 f32 þ h1 f35 þ h2 f36 þ f33 þ u Þ þ ðb3 kn3 N3 ðk3 Þ þ 1Þk3 3
_ e4 e_ 4 ¼ ½e4 ðd1 f41 þ c2 f42 þ f43 þ ubd 4 Þ þ ðb4 kn4 N 4 ðk4 Þ þ 1Þk4 Design
^_ 1 ¼ k e2 f21 ; b bb1 ^c_ 1 ¼ kc1 e3 f31 ; ^_ 1 ¼ k e4 f41 ; d d1
^_ 2 ¼ k e2 f22 ; b b2 ^c_ 2 ¼ kc2 e3 f32 ; ^_ 2 ¼ k e4 f42 d d2
^_ 1 ¼ k e2 f25 ; k k1 ^_ 1 ¼ k e3 f35 ; h h1
^_ 2 ¼ k e2 f26 ; k k2 ^_ 2 ¼ k e3 f36 ; h h2
It is easy to prove that kbi(t) and ei are bound. Because k_ bi ¼ ei ubd i , According to the Barbalat theorem, we getei ? 0. By a further analysis, we define new error variables gi as follows
~1 f11 þ a ~2 f12 ¼ g 1 e_ 1 ¼ a ~ ~2 f22 þ k ~1 f25 þ k ~2 f26 ¼ g e_ 2 ¼ b1 f21 þ b 2 ~1 f35 þ h ~2 f36 ¼ g e_ 3 ¼ ~c1 f31 þ ~c2 f32 þ h
ð26Þ
3
~ f þd ~ f ¼g e_ 4 ¼ d 1 41 2 42 4 When t ? 1, we havee˙i = 0, and then we get gi ? 0. If we assume some parameter is known, such as a2 = â2 then we get g1 = ã1, f11 = 0, and if f11 – 0, the parameter a1 can be estimated effectively, otherwise we can only get gi ? 0. In fact, it is easy to make the following conclusion: If only one unknown parameters exists in the one order differential equation, we can estimate the unknown parameter by constructing the adaptive law. If more than one unknown parameters exist in the one order differential equation, the unknown parameters cannot be estimated by common adaptive method. But we can define new error variables as gi, and gi will be converged to 0. If the uncertain input and one unknown parameter exist in the system, and the Nussbaum gain method is adopt, the unknown parameter can be estimated. Above all, only one unknown parameter can be estimated in one differential equation. 5. Numerical simulations We choose the parameters of the system and the control parameters as followed: a1 ¼ 10; b1 ¼ 40; c1 ¼ 2:5; d1 ¼ 10:6; ð2; 2; 2; 2Þ t < 2:5 , Choose k1 ¼ 1; h1 ¼ 4; a2 ¼ 10:5; b2 ¼ 40:5; c2 ¼ 2:6; d2 ¼ 11; k2 ¼ 1:1; h2 ¼ 4:4; ðb1 ; b2 ; b3 ; b4 Þ ¼ ð2; 2; 2; 2Þ t > 2:5
Fig. 1. Uncontrolled trajectories of system (1) and (2).
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the initial state of the master system as: x1(0) = y1(0) = z1(0) = w1(0) = 0.5, choose the initial state of the slave system as: x2(0) = y2(0) = z2(0) = w2(0) = 1, choose 0 to be the initial value of the estimation of unknown parameters. The simulation is done and Figs. 1–8 shows the results. The simulation result shows that the states of the slave system can drive to the states of the master system quickly and the control law is effective. The unknown parameters cannot be estimated accurately if more than one unknown parameters
Fig. 2. Synchronization of two systems.
Fig. 3. State x1 and state x2.
Fig. 4. State y1 and state y2.
J. Lei et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3439–3448
Fig. 5. State z1 and state z2.
Fig. 6. State w1 and state w2.
Fig. 7. Parameters ki.
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Fig. 8. Errors gi.
exist in the one order differential equation, but new error variable can be defined and it will be converged to 0 such as gi ? 0 in our paper. 6. Conclusions Two good results achieved by this paper as followed. (I) Input uncertainties is considered in the design of synchronization of two chaos systems, and an adaptive law is adopted to cope with the input uncertainties which are exist in most actual systems. (II) New variables are defined in systems with more than one unknown parameter in each one order differential equation. And in this situation the errors of estimation of unknown parameters may not converged to 0 but the new variables we defined will converged to 0. The conclusion is proposed that only one unknown parameters can be estimated in each one order differential equation. (III) The unknown parameter can be estimated in systems with uncertain input by introducing the Nussbaum gain technology, and it is difficult to get such good result by other common adaptive method. Why the unknown parameters cannot be estimated well? We will research on that in our future work, also we want to do some good work to improve the abilities of controller to estimate the unknown parameters. Addenda. The program of the simulation (written by m language of Matlab)
clc,clear;startsimulation=0, dt=0.0002;tf=5; x1=0.5;y1=0.5;z1=0.5;w1=0.5; a1=10;b1=40;c1=2.5;d1=10.6;k1=1;h1=4;a2=10.5;b2=40.5;c2=2.6;d2=11;k2=1.1; h2=4.4; x2=1; y2=1; z2=1;w2=1;u1=0;u2=0;u3=0;u4=0; kv1=1;kv2=1;kv3=1;kv4=1; kn1=1;kn2=1;kn3=1;kn4=1; a1g=0;a2g=0;b1g=0;b2g=0;c1g=0;c2g=0;d1g=0;d2g=0;h1g=0;h2g=0;k1g=0;k2g=0; for i=1:tf/dt t=idt; bu1=21;bu2=2;bu3=2;bu4=2; if t > tf/2 bu1=-2;bu2=-2;bu3=-2;bu4=-2; end ex=x2-x1;ey=y2-y1;ez=z2-z1;ew=w2-w1; k11=5;k12=5;k13=0.2;esten1=0.5; k21=5;k22=5;k23=0.2;esten2=0.5;
J. Lei et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3439–3448
k31=5;k32=5;k33=0.2;esten3=0.5; k41=5;k42=5;k43=0.2;esten4=0.5; d11=sign(ex)abs(ex ˆ(1/3));d12=d11 ˆ 2; f14=-k11ex-k12ex/(abs(ex)+esten1)-3/2k13d11exp(d12); d21=sign(ey)abs(ey ˆ(1/3));d22=d21 ˆ 2; f24=-k21ey-k22ey/(abs(ey)+esten2)-3/2k23d21exp(d22); d31=sign(ez)abs(ez ˆ(1/3));d32=d931 ˆ 2; f34=-k31ez-k32ez/(abs(ez)+esten3)-3/2k33d31exp(d32); d41=sign(ew)abs(ewˆ (1/3));d42=d41 ˆ 2; f44=-k41ew-k42ew/(abs(ew)+esten4)-3/2k43d41exp(d42); f11=-(y1-x1);f12=y2-x2;f13=0; f21=-x1;f22=x2;f25=x1z1;f26=-x2z2;f23=w1-w2; f31=z1;f32=-z2;f35=-x1 ˆ 2;f36=x2 ˆ 2; f33=0; f41=x1;f42=-x2;f43=0; ka1=0.5;ka2=0.5;kb1=0.5;kb2=0.5;kc1=0.5;kc2=0.5;kd1=0.5;kd2=0.5;kh1=0.5; kh2=0.5;kk1=0.5;kk2=0.5; da1g=ka1exf11;a1g=a1g+da1gdt; da2g=ka2exf12;a2g=a2g+da2gdt; %a2g=a2; db1g=kb1eyf21;b1g=b1g+db1gdt; db2g=kb2eyf22;b2g=b2g+db2gdt; dk1g=kk1eyf25;k1g=k1g+dk1gdt; dk2g=kk2eyf26;k2g=k2g+dk2gdt; dc1g=kc1ezf31;c1g=c1g+dc1gdt; dc2g=kc2ezf32;c2g=c2g+dc2gdt; dh1g=kh1ezf35;h1g=h1g+dh1gdt; dh2g=kh2ezf36;h2g=h2g+dh2gdt; dd1g=kd1ewf41;d1g=d1g+dd1gdt; dd2g=kd2ewf42;d2g=d2g+dd2gdt; u1bd=-a1gf11-a2gf12-f13+f14; u2bd=-b1gf21-b2gf22-f23-k1gf25-k2gf26+f24; u3bd=-c1gf31-c2gf32-f33-h1gf35-h2gf36+f34; u4bd=-d1gf41-d2gf42-f43+f44; dkv1=-exu1bd;kv1=kv1+dkv1dt;n1kv1=kv1kv1cos(kv1); rou1g=-kn1n1kv1; dkv2=-eyu2bd;kv2=kv2+dkv2dt;n2kv2=kv2kv2cos(kv2); rou2g=-kn2n2kv2; dkv3=-ezu3bd;kv3=kv3+dkv3dt;n3kv3=kv3kv3cos(kv3); rou3g=-kn3n3kv3; dkv4=-ewu4bd;kv4=kv4+dkv4dt;n4kv4=kv4kv4cos(kv4); rou4g=-kn4n4kv4; % rou1g=1/b1;rou2g=1/b2;rou3g=1/b3; u1=rou1gu1bd;u2=rou2gu2bd;u3=rou3gu3bd;u4=rou4gu4bd; %u1=0;u2=0;u3=0;u4=0; dx1=a1(y1-x1); dy1=b1x1-k1x1z1-w1; dz1=-c1z1+h1x1 ˆ ;2; dw1=-d1x1; x1=x1+dx1dt; y1=y1+dy1dt; z1=z1+dz1dt; w1=w1+dw1dt; dx2=a2(y2-x2)+bu1u1; dy2=b2x2-k2x2z2-w2+bu2u2; dz2=-c2z2+h2x2 ˆ 2+bu3u3; dw2=-d2x2+bu4u4; x2=x2+dx2dt; y2=y2+dy2dt; z2=z2+dz2dt; w2=w2+dw2dt; tp(i)=t; x1p(i)=x1;y1p(i)=y1;z1p(i)=z1; w1p(i)=w1;w2p(i)=w2; x2p(i)=x2;y2p(i)=y2;z2p(i)=z2; k1p(i)=kv1;k2p(i)=kv2;k3p(i)=kv3;k4p(i)=kv4; g1p(i)=(a1-a1g)f11+(a2-a2g)f12; g2p(i)=(b1-b1g)f21+(b2-b2g)f22+(k1-k1g)f25+(k2-k2g)f26; g3p(i)=(c1-c1g)f31+(c2-c2g)f32+(h1-h1g)f35+(h2-h2g)f36; g4p(i)=(d1-d1g)f41+(d2-d2g)f42; a1gp(i)=a1g;a2gp(i)=a2g; c1gp(i)=c1g;c2gp(i)=c2g;k1gp(i)=k1g;k2gp(i)=k2g; b1gp(i)=b1g;b2gp(i)=b2g; d1gp(i)=d1g;d2gp(i)=d2g;h1gp(i)=h1g;h2gp(i)=h2g;
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end endsimulation=1, %figure(1);plot(x1p,y1p);xlabel(’t’);ylabel(’x1’); figure(18);plot3(x1p,y1p,z1p,’r’);xlabel(’x’);ylabel(’y’),zlabel(’z’); hold on;plot3(x2p,y2p,z2p,’b’); figure(3);plot3(x1p,y1p,z1p); figure(4);plot(tp,x1p,tp,x2p,tp,y1p,tp,y2p,tp,z1p,tp,z2p);xlabel(’t’); ylabel(’x’); figure(5);plot(tp,w1p,tp,w2p);xlabel(’t’);ylabel(’w’); figure(6);plot(tp,x1p,tp,x2p);xlabel(’t’);ylabel(’x’); figure(88);plot(tp,z1p,tp,z2p);xlabel(’t’);ylabel(’z’); figure(55);plot(tp,y1p,tp,y2p);xlabel(’t’);ylabel(’y’); figure(7);plot(tp,expp,’r’,tp,eyp,’b’,tp,ezp,’g’); figure(99);plot(tp,k1p);figure(98);plot(tp,k2p);figure(97);plot(tp,k3p); figure(100);plot(tp,k1p,tp,k2p,tp,k3p,tp,k4p); figure(16);plot(tp,alfa1gp,tp,alfa2gp,tp,beta1gp,tp,beta2gp,tp,gama1gp, tp,gama2gp);xlabel(’ t’);ylabel(’ex’); figure(16);plot(tp,alfa1gp,tp,beta1gp,tp,gama1gp);xlabel(’t’); ylabel(’ex’); figure(15);plot(tp,g1p,tp,g2p,tp,g3p,tp,g4p);xlabel(’t’);ylabel(’ey’); figure(11);plot(tp,a1gp);xlabel(’t’);ylabel(’ex’); figure(12);plot(tp,eyp);xlabel(’t’);ylabel(’ey’); figure(13);plot(tp,g4p);xlabel(’t’);ylabel(’ez’); figure(9);plot(tp,alfa1gp,tp,beta1gp,tp,gama1gp);xlabel(’t’);ylabel(’z’);
References [1] Elabbasy EM, Agiza HN, El-Dessoky MM. Adaptive synchronization of a hyperchaotic system with uncertain parameter. Chaos Solitons Fract 2006;30:1133–42. [2] Tang Fang, Wang Ling. An adaptive active control for the modified Chua’s circuit. Phys Lett A 2005;346:342–6. [3] Hu Manfeng, Xu Zhenyuan, Zhang Rong, Hu Aihua. Parameters identification and adaptive full state hybrid projective synchronization of chaotic (hyperchaotic) systems. Phys Lett A 2007;361:231–7. [4] Ge Zheng-Ming, Yang Cheng-Hsiung. Pragmatical generalized synchronization of chaotic systems with uncertain parameters by adaptive control. Physica D 2007;231:87–9. [5] Gao Tiegang, Chen Zengqiang, Yuan Zhuzhi, Yu Dongchuan. Adaptive synchronization of a new hyperchaotic system with uncertain parameters. Chaos Solitons Fract 2007;33:922–8. [6] Wang Xinyu, Lei Junwei, Pan Changpeng. Trigonometric RBF neural robust controller design for a class of nonlinear system with linear input unmodeled dynamics. Appl Math Comput 2007;185:989–1002. [7] Kim Seung-Hwan, Kim Yoon-Sik, Song Chanho. A robust adaptive nonlinear control approach to missile autopilot design. Control Eng Pract 2004;12:149–54. [8] Polycarpou MM, Ioannou PA. A robust adaptive nonlinear control design. Automatic 1996;32(3):423–42. [9] Wang Xinyu, Lei Junwei. Design of Global Identical Terminal Sliding Mode Controller for a Class of Chaotic System, 2006 IMACS Multiconference on Computational Engineering in Systems Applications (CESA0 2006). Tsinghua University Press; 2006.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
A Nussbaum gain adaptive synchronization of a new hyperchaotic system with input uncertainties and unknown parameters Junwei Lei a,*, Xinyu Wang b, Yinhua Lei c a b c
Department of Automatic Control Engineering, Naval Aeronautical Engineering Academy, Yantai 264001, China Institute of Science and Technology for Opto-electronic Information, Yantai University, Yantai 264001, China Institute of Microelectronics, Peking University, Beijing 100871, China
a r t i c l e
i n f o
Article history: Received 9 August 2008 Received in revised form 24 September 2008 Accepted 1 December 2008 Available online 24 December 2008 PACS: 05.45.Vx 05.45.Gg 05.45.Xt Keywords: Input uncertainties Synchronization Chaos Global terminal Adaptive
a b s t r a c t The sign of unknown input coefficients is assumed to be known in most papers about the input uncertainties. In this paper, a Nussbaum gain method is adopted to cope with the situation that both the sign and the value of input are unknown. And the unknown parameters can be estimated under the situation of unknown sign of control. The synchronization is achieved for a class of hyperchaotic systems with unknown parameters and input uncertainties by adopting of the Nussbaum gain method and the global terminal adaptive method. And the conclusions are made as follows: First, the proposed method is effective in the situation that the sign of input is unknown. Second, the estimation of unknown parameters can be achieved only when the number of unknown parameters satisfied some condition and no uncertainty exist in the input of systems. Third, the unknown parameters cannot be estimated correctly with common adaptive method when there are input uncertainties in the system. But the Nussbaum gain method can get good result in the estimation of unknown parameters. At last, numerical simulations are done to show the effectiveness of the proposed method. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Chaos systems have complex dynamical behaviors that possess some special features such as being extremely sensitive to tiny variations of initial conditions, and having bounded trajectories with a positive leading Lyapunov exponent and so on [1–7]. In recent years, chaos synchronization has been attracted increasingly attentions due to their potential applications in the fields of secure communications. Synchronization of chaos systems with unknown parameters is investigated widely by researchers from various fields. In fact, input uncertainties usually exist in actual control systems, but the situations of chaos systems with input uncertainties are neglected in most papers [1–4]. We known, it is more difficult to estimate the unknown parameters exactly than to achieve synchronization. The estimation of unknown parameters often could not converge to its real value in many simulations [1–4]. It is a difficult problem that is often neglected by many researchers unconsciously or on purpose, and some researchers intended to find the answer for this question, such as [3,4]. But obviously the problem has not been solved successfully until now, even some wrong answers have been proposed in some papers [3,4]. In this paper, we conclude that the unknown parameters cannot be estimated in some situation. Also a sufficient condition of parameters estimation is proposed. This condition may be unnecessary, but it is * Corresponding author. E-mail address: [email protected] (J. Lei). 1007-5704/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2008.12.010
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still the best answer we can found for this question. Also, we pointed out that the unknown parameters cannot be estimated effectively in systems with input uncertainties. In the past decade, backstepping has become one of the most popular design methods for adaptive nonlinear control and synchronization because it can guarantee global stabilities for the broad class of strict-feedback systems [5–9]. The sign of input need to be known in many papers researched on the problem of input uncertainty. The Nussbaum gain theory is adopted to cope with this problem. The unknown parameters can be estimated effectively with the Nussbaum gain adaptive method. And the synchronization is achieved in finite time using a global terminal controller under the situation that the sign of input is unknown [9]. At last, we point out that the unknown parameters cannot be estimated with common adaptive method when there are input uncertainties in the system. The paper is organized as followed. In Section 2, the system model of our research is proposed. In Section 3, the global terminal control theory is introduced which can make the errors of system converge to 0 in finite time. In Section 4, a global terminal adaptive controller is designed for the chaos systems and synchronization is achieved. In Section 5, the numerical simulation is done to verify the conclusions of this paper. In Section 6, the conclusions are given and the future work is pointed out. 2. System description The new hyperchaotic system is investigated for its important use in the secret communication by many researchers recently, and its model can be written as:
x_ 1 ¼ a1 ðy1 x1 Þ y_ 1 ¼ b1 x1 k1 x1 z1 w1 z_ 1 ¼ c1 z1 þ h1 x21 _ w1 ¼ d1 x1
ð1Þ
where x1, y1, z1 are the states of the system, and a = 10, b = 40, c = 2.5, d = 10.6, k = 1, h = 4 are unknown parameters. We take the model (1) as the master system and assume the slave system have different unknown parameters as follows:
x_ 2 ¼ a2 ðy2 x2 Þ þ bu1 u1 y_ 2 ¼ b2 x2 k2 x2 z2 w2 þ bu2 u2 z_ 2 ¼ c2 z2 þ h2 x22 þ bu3 u3 _ 2 ¼ d2 x2 þ bu4 u4 w
ð2Þ
The master system and slave system are denoted by 1 and 2, respectively, where bui are input uncertainties. Our goal is to find a controller u ¼ ½ u1 u2 u3 to make the state of slave system track to the states of master system. The error system can be written as:
e_ 1 ¼ a2 ðy2 x2 Þ a1 ðy1 x1 Þ þ bu1 u1 e_ 2 ¼ ðb2 x2 k2 x2 z2 w2 Þ ðb1 x1 k1 x1 z1 w1 Þ þ bu2 u2 e_ 3 ¼ c2 z2 þ h2 x22 ðc1 z1 þ h1 x21 Þ þ bu3 u3 e_ 4 ¼ d2 x2 ðd1 x1 Þ þ bu4 u4
ð3Þ
where e1 = x2 x1,e2 = y2 y1,e3 = z2 z1,e4 = w2 w1. Assumption 1. The sign of bui is unknown. 3. Global terminal control To explain the concept of finite convergence time, the global terminal control theory is introduced as followed [2]. Definition 1 (Terminal function). For a kind of system having the form as defined in and the sliding mode control having the form as defined in, when S ? 0, for any initial system state e(t0), exists ts1 > 0 such thatts is bound and ts < ts1, where ts is defined as the time of the system need in order to converge the state from e(t0) to 0, then f(e) is defined as a terminal function. The system has the form as:
e_ ¼ hðeÞ þ uðtÞ
ð4Þ
We define the surface and control as:
S ¼ e_ þ f ðeÞ;
u ¼ gðS; eÞ
ð5Þ
J. Lei et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3439–3448
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Definition 2 (Global terminal function). For a kind of system having the form as defined in and the sliding mode control having the form as defined in (5), when S ? 0, exists ts1 > 0 for any initial system state e(t0), such that ts is bound and ts < ts1, where ts is defined as the time of the system need in order to converge the state from e(t0) to 0, then f(e) is defined as a global terminal function. The traditional terminal function is defined as f(e) = e1/3, and it is not a global terminal function, so it does not have the advantage of global terminal function. For system (4), we compute the convergence time as the following: When S ? 0, we get:
e_ ¼ f ðeÞ
ð6Þ
To ensure the system stable, make that:
ee_ ¼ ef ðeÞ < 0
ð7Þ
Define:
_ gðeÞ ¼ 1=f ðeÞ
ð8Þ
de=f ðeÞ ¼ dt ¼ dgðeÞ
ð9Þ
Then:
Do an integral computation on each side of the equation:
gðeðtf ÞÞ gðeðt0 ÞÞ ¼ t f t0
ð10Þ
Because the system is stable, so
eðtf Þ ¼ 0
ð11Þ
tf ¼ t 0 þ gð0Þ gðeðt0 ÞÞ
ð12Þ
Then:
The rule of constructing global terminal function can be found in Ref. [1]. For convenience, we take the following function as an example to compute the convergence time. 2=3 Consider the function f ðeÞ ¼ 32 e1=3 expðe2=3 Þ, We compute gðeÞ ¼ expðe2=3 Þ; tf ¼ t 0 þ gð0Þ gðeðt 0 ÞÞ ¼ 1 eðeð0Þ Þ < 1. So this function is not only a terminal function but also a global terminal function. 4. Nussbaum adaptive synchronization design The error system can be written as (13), and the control law can be designed as followed:
e_ 1 ¼ a1 f11 þ a2 f12 þ f13 þ bu1 u1 e_ 2 ¼ b1 f21 þ b2 f22 þ k1 f25 þ k2 f26 þ f23 þ bu2 u2
ð13Þ
e_ 3 ¼ c1 f31 þ c2 f32 þ h1 f35 þ h2 f36 þ f33 þ bu3 u3 e_ 4 ¼ d1 f41 þ c2 f42 þ f43 þ bu4 u4 where:
f11 ¼ ðy1 x1 Þ;
f 12 ¼ ðy2 x2 Þ;
f 22 ¼ x2 ;
f 25 ¼ x1 z1 ;
f31 ¼ z1 ;
f 32 ¼ z2 ;
f 35 ¼
x21 ;
f41 ¼ x1 ;
f 42 ¼ x2 ;
f 43 ¼ 0
f21 ¼ x1 ;
f 13 ¼ 0 f 26 ¼ x2 z2 ; f 36 ¼
Define:
f14 ¼ k11 e1 k12 je1ejþ1 e1 32 k13 d11 ed12 f24 ¼ k21 e2 k22 je2ejþ2 e2 32 k23 d21 ed22 f34 ¼ k31 e3 k32 je3ejþ3 e3 32 k33 d31 ed32 f44 ¼ k41 e4 k42 je4ejþ4 e4 32 k43 d41 ed42 where 2
di2 ¼ di1 ;
di1 ¼ e1=3 i ;
i ¼ 1; 2; 3; 4
x22 ;
f 23 ¼ w2 w1
f 33 ¼ 0
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Design
ubd i
ði ¼ 1; 2; 3Þ
^1 f11 a ^2 f12 f13 þ f14 ¼ ubd bu1 ud1 ¼ a 1 ^1 f21 b ^2 f22 k ^1 f25 k ^2 f26 f23 þ f24 ¼ ubd bu2 ud2 ¼ b 2 ^ f h ^ f f þ f ¼ ubd b ud ¼ ^c f ^c f h u3 3
1 31
2 32
1 25
2 26
33
34
ð14Þ
3
^1 f41 d ^2 f42 f43 þ f44 ¼ ubd bu4 ud3 ¼ d 3 where â1 is used to approximate a and define ã1 = a1 â1, and so are other variables. Because the control is unknown, and the sign of control is unknown, we adopt the Nussbaum function as follows: 2
NðkÞ ¼ k cos k
ð15Þ
Obviously, it has characters as follows:
lim sup
s!1
1 s
Z
s
NðkÞdk ¼ þ1;
lim inf
s!1
0
1 s
Z
s
NðkÞdk ¼ 1
ð16Þ
0
Design:
k_ bi ¼ ei ubd i ;
ui ¼ kni Ni ðkbi Þubd i ;
i ¼ 1; 2; 3; 4
ð17Þ
And define Lyapunov function as:
V1 ¼
4 1X e2 2 i¼1 i
And its time derivative described by:
V_ 1 ¼
4 X
e1 e_ 1
ð18Þ
i¼1 bd e1 e_ 1 ¼ e1 ½a1 f11 þ a2 f12 þ f13 þ ubd 1 þ ðb1 u1 u1 Þ
ð19Þ
Substituting (17)–(19), we get:
_ e1 e_ 1 ¼ ½e1 ða1 f11 þ a2 f12 þ f13 þ ubd 1 Þ þ ðb1 kn1 N 1 ðk1 Þ þ 1Þk1
ð20Þ
Design:
^_ 2 ¼ ka2 e1 f12 ^_ 1 ¼ ka1 e1 f11 ; a a Define:
1 2 1 2 1 2 ~ þ ~ e þ a a 2 1 2ka1 1 2ka2 2
ð21Þ
V_ 11 ¼ e1 f14 þ ðb1 kn1 N1 ðkb1 Þ þ 1Þk_ b1
ð22Þ
V 11 ¼ We get:
Consider that:
e1 f14 6 0 Do an integral computation we get:
V 11 ðtÞ V 11 ð0Þ
Z
t
e1 f14 ¼
0
Z
t
ðb1 kn1 N1 ðkb1 Þ þ 1Þk_ b1 dt
ð23Þ
0
Then
V 11 ðtÞ V 11 ð0Þ
Z
t
e1 f14 ¼
0
Z
k1 ðtÞ
b1 kn1 N1 ðkb1 Þdkb1 þ kb1 ðtÞ kb1 ð0Þ
k1 ð0Þ
Firstly, we assume kb1(t) ? +1 then
h
i R k ðtÞ ¼ k 1ðtÞ kb1b1ð0Þ b1 kn1 N 1 ðkb1 Þdkb1 1 þ kkb1 ð0Þ b1 ðtÞ b1 h i R k ðtÞ V 11 ðtÞ V 11 ð0Þ k 1ðtÞ 1 þ kkb1 ð0Þ < k 1ðtÞ k b1ð0Þ b1 kn1 N1 ðkb1 Þdkb1 ðtÞ ðV 11 ðtÞ V 11 ð0Þ b1
Rt 0
e1 f14 dtÞ k b1
1
b1 ðtÞ
b1
b1
ð24Þ
J. Lei et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3439–3448
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Rs And it contradicts with lims!1 inf 1s 0 NðkÞdk ¼ 1. Secondly, we assume kb1(t) ? 1 then
V 11 ðtÞ V 11 ð0Þ
Z kb1 ðtÞ 1 kb1 ð0Þ 1 b1 kn1 N1 ðkb1 Þdkb1 1þ > kb1 ðtÞ kb1 ðtÞ kb1 ðtÞ kb1 ð0Þ
And it contradicts with lims!1 sup 1s gain as follows:
Rs 0
ð25Þ
NðkÞdk ¼ þ1. So kb1(t) is bound and e1 is bound. Also we can design other Nussbaum
_ e2 e_ 2 ¼ ½e2 ðb1 f21 þ b2 f22 þ k1 f25 þ k2 f26 þ f23 þ ubd 2 Þ þ ðb2 kn2 N 2 ðk2 Þ þ 1Þk2 bd _ e3 e_ 3 ¼ ½e3 ðc1 f31 þ c2 f32 þ h1 f35 þ h2 f36 þ f33 þ u Þ þ ðb3 kn3 N3 ðk3 Þ þ 1Þk3 3
_ e4 e_ 4 ¼ ½e4 ðd1 f41 þ c2 f42 þ f43 þ ubd 4 Þ þ ðb4 kn4 N 4 ðk4 Þ þ 1Þk4 Design
^_ 1 ¼ k e2 f21 ; b bb1 ^c_ 1 ¼ kc1 e3 f31 ; ^_ 1 ¼ k e4 f41 ; d d1
^_ 2 ¼ k e2 f22 ; b b2 ^c_ 2 ¼ kc2 e3 f32 ; ^_ 2 ¼ k e4 f42 d d2
^_ 1 ¼ k e2 f25 ; k k1 ^_ 1 ¼ k e3 f35 ; h h1
^_ 2 ¼ k e2 f26 ; k k2 ^_ 2 ¼ k e3 f36 ; h h2
It is easy to prove that kbi(t) and ei are bound. Because k_ bi ¼ ei ubd i , According to the Barbalat theorem, we getei ? 0. By a further analysis, we define new error variables gi as follows
~1 f11 þ a ~2 f12 ¼ g 1 e_ 1 ¼ a ~ ~2 f22 þ k ~1 f25 þ k ~2 f26 ¼ g e_ 2 ¼ b1 f21 þ b 2 ~1 f35 þ h ~2 f36 ¼ g e_ 3 ¼ ~c1 f31 þ ~c2 f32 þ h
ð26Þ
3
~ f þd ~ f ¼g e_ 4 ¼ d 1 41 2 42 4 When t ? 1, we havee˙i = 0, and then we get gi ? 0. If we assume some parameter is known, such as a2 = â2 then we get g1 = ã1, f11 = 0, and if f11 – 0, the parameter a1 can be estimated effectively, otherwise we can only get gi ? 0. In fact, it is easy to make the following conclusion: If only one unknown parameters exists in the one order differential equation, we can estimate the unknown parameter by constructing the adaptive law. If more than one unknown parameters exist in the one order differential equation, the unknown parameters cannot be estimated by common adaptive method. But we can define new error variables as gi, and gi will be converged to 0. If the uncertain input and one unknown parameter exist in the system, and the Nussbaum gain method is adopt, the unknown parameter can be estimated. Above all, only one unknown parameter can be estimated in one differential equation. 5. Numerical simulations We choose the parameters of the system and the control parameters as followed: a1 ¼ 10; b1 ¼ 40; c1 ¼ 2:5; d1 ¼ 10:6; ð2; 2; 2; 2Þ t < 2:5 , Choose k1 ¼ 1; h1 ¼ 4; a2 ¼ 10:5; b2 ¼ 40:5; c2 ¼ 2:6; d2 ¼ 11; k2 ¼ 1:1; h2 ¼ 4:4; ðb1 ; b2 ; b3 ; b4 Þ ¼ ð2; 2; 2; 2Þ t > 2:5
Fig. 1. Uncontrolled trajectories of system (1) and (2).
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the initial state of the master system as: x1(0) = y1(0) = z1(0) = w1(0) = 0.5, choose the initial state of the slave system as: x2(0) = y2(0) = z2(0) = w2(0) = 1, choose 0 to be the initial value of the estimation of unknown parameters. The simulation is done and Figs. 1–8 shows the results. The simulation result shows that the states of the slave system can drive to the states of the master system quickly and the control law is effective. The unknown parameters cannot be estimated accurately if more than one unknown parameters
Fig. 2. Synchronization of two systems.
Fig. 3. State x1 and state x2.
Fig. 4. State y1 and state y2.
J. Lei et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3439–3448
Fig. 5. State z1 and state z2.
Fig. 6. State w1 and state w2.
Fig. 7. Parameters ki.
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Fig. 8. Errors gi.
exist in the one order differential equation, but new error variable can be defined and it will be converged to 0 such as gi ? 0 in our paper. 6. Conclusions Two good results achieved by this paper as followed. (I) Input uncertainties is considered in the design of synchronization of two chaos systems, and an adaptive law is adopted to cope with the input uncertainties which are exist in most actual systems. (II) New variables are defined in systems with more than one unknown parameter in each one order differential equation. And in this situation the errors of estimation of unknown parameters may not converged to 0 but the new variables we defined will converged to 0. The conclusion is proposed that only one unknown parameters can be estimated in each one order differential equation. (III) The unknown parameter can be estimated in systems with uncertain input by introducing the Nussbaum gain technology, and it is difficult to get such good result by other common adaptive method. Why the unknown parameters cannot be estimated well? We will research on that in our future work, also we want to do some good work to improve the abilities of controller to estimate the unknown parameters. Addenda. The program of the simulation (written by m language of Matlab)
clc,clear;startsimulation=0, dt=0.0002;tf=5; x1=0.5;y1=0.5;z1=0.5;w1=0.5; a1=10;b1=40;c1=2.5;d1=10.6;k1=1;h1=4;a2=10.5;b2=40.5;c2=2.6;d2=11;k2=1.1; h2=4.4; x2=1; y2=1; z2=1;w2=1;u1=0;u2=0;u3=0;u4=0; kv1=1;kv2=1;kv3=1;kv4=1; kn1=1;kn2=1;kn3=1;kn4=1; a1g=0;a2g=0;b1g=0;b2g=0;c1g=0;c2g=0;d1g=0;d2g=0;h1g=0;h2g=0;k1g=0;k2g=0; for i=1:tf/dt t=idt; bu1=21;bu2=2;bu3=2;bu4=2; if t > tf/2 bu1=-2;bu2=-2;bu3=-2;bu4=-2; end ex=x2-x1;ey=y2-y1;ez=z2-z1;ew=w2-w1; k11=5;k12=5;k13=0.2;esten1=0.5; k21=5;k22=5;k23=0.2;esten2=0.5;
J. Lei et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3439–3448
k31=5;k32=5;k33=0.2;esten3=0.5; k41=5;k42=5;k43=0.2;esten4=0.5; d11=sign(ex)abs(ex ˆ(1/3));d12=d11 ˆ 2; f14=-k11ex-k12ex/(abs(ex)+esten1)-3/2k13d11exp(d12); d21=sign(ey)abs(ey ˆ(1/3));d22=d21 ˆ 2; f24=-k21ey-k22ey/(abs(ey)+esten2)-3/2k23d21exp(d22); d31=sign(ez)abs(ez ˆ(1/3));d32=d931 ˆ 2; f34=-k31ez-k32ez/(abs(ez)+esten3)-3/2k33d31exp(d32); d41=sign(ew)abs(ewˆ (1/3));d42=d41 ˆ 2; f44=-k41ew-k42ew/(abs(ew)+esten4)-3/2k43d41exp(d42); f11=-(y1-x1);f12=y2-x2;f13=0; f21=-x1;f22=x2;f25=x1z1;f26=-x2z2;f23=w1-w2; f31=z1;f32=-z2;f35=-x1 ˆ 2;f36=x2 ˆ 2; f33=0; f41=x1;f42=-x2;f43=0; ka1=0.5;ka2=0.5;kb1=0.5;kb2=0.5;kc1=0.5;kc2=0.5;kd1=0.5;kd2=0.5;kh1=0.5; kh2=0.5;kk1=0.5;kk2=0.5; da1g=ka1exf11;a1g=a1g+da1gdt; da2g=ka2exf12;a2g=a2g+da2gdt; %a2g=a2; db1g=kb1eyf21;b1g=b1g+db1gdt; db2g=kb2eyf22;b2g=b2g+db2gdt; dk1g=kk1eyf25;k1g=k1g+dk1gdt; dk2g=kk2eyf26;k2g=k2g+dk2gdt; dc1g=kc1ezf31;c1g=c1g+dc1gdt; dc2g=kc2ezf32;c2g=c2g+dc2gdt; dh1g=kh1ezf35;h1g=h1g+dh1gdt; dh2g=kh2ezf36;h2g=h2g+dh2gdt; dd1g=kd1ewf41;d1g=d1g+dd1gdt; dd2g=kd2ewf42;d2g=d2g+dd2gdt; u1bd=-a1gf11-a2gf12-f13+f14; u2bd=-b1gf21-b2gf22-f23-k1gf25-k2gf26+f24; u3bd=-c1gf31-c2gf32-f33-h1gf35-h2gf36+f34; u4bd=-d1gf41-d2gf42-f43+f44; dkv1=-exu1bd;kv1=kv1+dkv1dt;n1kv1=kv1kv1cos(kv1); rou1g=-kn1n1kv1; dkv2=-eyu2bd;kv2=kv2+dkv2dt;n2kv2=kv2kv2cos(kv2); rou2g=-kn2n2kv2; dkv3=-ezu3bd;kv3=kv3+dkv3dt;n3kv3=kv3kv3cos(kv3); rou3g=-kn3n3kv3; dkv4=-ewu4bd;kv4=kv4+dkv4dt;n4kv4=kv4kv4cos(kv4); rou4g=-kn4n4kv4; % rou1g=1/b1;rou2g=1/b2;rou3g=1/b3; u1=rou1gu1bd;u2=rou2gu2bd;u3=rou3gu3bd;u4=rou4gu4bd; %u1=0;u2=0;u3=0;u4=0; dx1=a1(y1-x1); dy1=b1x1-k1x1z1-w1; dz1=-c1z1+h1x1 ˆ ;2; dw1=-d1x1; x1=x1+dx1dt; y1=y1+dy1dt; z1=z1+dz1dt; w1=w1+dw1dt; dx2=a2(y2-x2)+bu1u1; dy2=b2x2-k2x2z2-w2+bu2u2; dz2=-c2z2+h2x2 ˆ 2+bu3u3; dw2=-d2x2+bu4u4; x2=x2+dx2dt; y2=y2+dy2dt; z2=z2+dz2dt; w2=w2+dw2dt; tp(i)=t; x1p(i)=x1;y1p(i)=y1;z1p(i)=z1; w1p(i)=w1;w2p(i)=w2; x2p(i)=x2;y2p(i)=y2;z2p(i)=z2; k1p(i)=kv1;k2p(i)=kv2;k3p(i)=kv3;k4p(i)=kv4; g1p(i)=(a1-a1g)f11+(a2-a2g)f12; g2p(i)=(b1-b1g)f21+(b2-b2g)f22+(k1-k1g)f25+(k2-k2g)f26; g3p(i)=(c1-c1g)f31+(c2-c2g)f32+(h1-h1g)f35+(h2-h2g)f36; g4p(i)=(d1-d1g)f41+(d2-d2g)f42; a1gp(i)=a1g;a2gp(i)=a2g; c1gp(i)=c1g;c2gp(i)=c2g;k1gp(i)=k1g;k2gp(i)=k2g; b1gp(i)=b1g;b2gp(i)=b2g; d1gp(i)=d1g;d2gp(i)=d2g;h1gp(i)=h1g;h2gp(i)=h2g;
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J. Lei et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 3439–3448
end endsimulation=1, %figure(1);plot(x1p,y1p);xlabel(’t’);ylabel(’x1’); figure(18);plot3(x1p,y1p,z1p,’r’);xlabel(’x’);ylabel(’y’),zlabel(’z’); hold on;plot3(x2p,y2p,z2p,’b’); figure(3);plot3(x1p,y1p,z1p); figure(4);plot(tp,x1p,tp,x2p,tp,y1p,tp,y2p,tp,z1p,tp,z2p);xlabel(’t’); ylabel(’x’); figure(5);plot(tp,w1p,tp,w2p);xlabel(’t’);ylabel(’w’); figure(6);plot(tp,x1p,tp,x2p);xlabel(’t’);ylabel(’x’); figure(88);plot(tp,z1p,tp,z2p);xlabel(’t’);ylabel(’z’); figure(55);plot(tp,y1p,tp,y2p);xlabel(’t’);ylabel(’y’); figure(7);plot(tp,expp,’r’,tp,eyp,’b’,tp,ezp,’g’); figure(99);plot(tp,k1p);figure(98);plot(tp,k2p);figure(97);plot(tp,k3p); figure(100);plot(tp,k1p,tp,k2p,tp,k3p,tp,k4p); figure(16);plot(tp,alfa1gp,tp,alfa2gp,tp,beta1gp,tp,beta2gp,tp,gama1gp, tp,gama2gp);xlabel(’ t’);ylabel(’ex’); figure(16);plot(tp,alfa1gp,tp,beta1gp,tp,gama1gp);xlabel(’t’); ylabel(’ex’); figure(15);plot(tp,g1p,tp,g2p,tp,g3p,tp,g4p);xlabel(’t’);ylabel(’ey’); figure(11);plot(tp,a1gp);xlabel(’t’);ylabel(’ex’); figure(12);plot(tp,eyp);xlabel(’t’);ylabel(’ey’); figure(13);plot(tp,g4p);xlabel(’t’);ylabel(’ez’); figure(9);plot(tp,alfa1gp,tp,beta1gp,tp,gama1gp);xlabel(’t’);ylabel(’z’);
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