Chaos synchronization and
parameter identification for gyroscope system
Z.-M. Ge
*, J.-K. Lee
Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, ROC
Abstract
Synchronization of chaos for a two-degree-of-freedom heavy symmetric gyroscope system are studied in this paper. Because of the nonlinear terms of the system, the system exhibits both regular and chaotic motions. By Lyapunovstability theory with control terms, by adaptive control and by random optimization method, the synchro-nization of two identical systems and tracking of the parameter of the systems are studied.
Ó 2004 Elsevier Inc. All rights reserved.
Keywords: Chaos; Synchronization; Parameter identification
1. Introduction
A lot of researches have shown that chaotic phenomena are observed in many physical systems that possess nonlinearity [1,2]. Chaotic motions also occur in many nonlinear control systems.
Most of physical systems in nature are nonlinear and can be described by the nonlinear equations of motion which in general cannot be linearized. So the studies of nonlinear systems spread quickly today. For the nonlinear system, the study of the types of system behavior, the effects to the behavior caused by different parameters and initial conditions, the behavior analysis of the system, consist of the major tasks. Besides, we are also interested in the understanding
*
Corresponding author.
E-mail address:[email protected](Z.-M. Ge).
0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.04.008
of the complicated phenomena arised from nonlinearity. The central charac-teristics are that a process like randomization happens in the deterministic system and small differences in the system parameters or initial conditions produce great ones in the final phenomena. The unpredictable and irregular motions of many nonlinear systems have been labeled ‘‘chaotic’’. By applying various numerical results, such as bifurcation, phase portraits, time history analysis, the behavior of the chaotic motion are presented. A large number of studies on the chaotic behavior have been reached up to now. First, the gov-erning equations of motion, the system model and differential equations of motion will be formulated.
Synchronization of chaos for a two-degree-of-freedom heavy symmetric gyroscope system are studied in this paper. By Lyapunovstability theory with control terms, by adaptive control and by random optimization method, the synchronization of two identical systems and tracking of the parameter of the systems are studied.
2. Equations of motion
The schematic diagram of a heavy symmetric gyroscope mounted on a vibrating base is shown in Fig. 1. The motion of this physical system can be described by Euler’s angles h, / and u. The vibration of the base can be
scribed as multiple harmonic motion Pnk¼1Aksin xkt. Let x¼ h, y ¼ _h and
z¼ _/, the state equations of the system are described by [3]: _x¼ y _y¼ ðb/bucos xÞðbub/cos xÞ I2 1sin 3x C I1yþ Mgl I1 sin x Mg I1 Pn k¼1Aksin xktsin x _z¼ 2 cos x sin x yzþ buy I1sin x 8 > > < > > : ; ð2:1Þ where I1, I3: the polar and equatorial moments of inertia of the symmetric
gyroscope, Mg: the gravity force, l: the distance between the center of gravity and O.
We set the parameters b/¼ 2, bu¼ 5, I1¼ 1, Mg ¼ 4, l ¼ 0:25, C ¼ 0:5,
x1¼ 1, x2¼ 2, x3¼ 3, xk¼ 0ðk > 3Þ, A1¼ A2¼ A3¼ ¼ Ak ¼ A.
3. Chaos synchronization for systems with unknown parameter
We investigate two third-order heavy symmetric gyroscope systems in this section. Both the systems have the same form and both the parameters are unknown. The drive system is described as Eq. (2.1). And the system (2.1) in which only first term is considered for the Fourier series and we set A1¼ A.
Eq. (2.1) becomes Eq. (3.1). The response system is described as Eq. (3.2). _x1¼ x2 _x2¼ ðb/bucos x1Þðbub/cos x1Þ I2 1sin 3x 1 C I1x2þ Mgl I1 sin x1 Mg I1 Asin x1tsin x1 _x3¼ 2 cos xsin x1 1 x2x3þ bux2 I1sin x1 8 > < > : ; ð3:1Þ _y1¼ y2 _y2¼ ðb/bucos y1Þðbub/cos y1Þ I2 1sin 3y 1 C I1y2þ Mgl I1 sin y1 Mg I1 Asin x1tsin y1 _y3¼ 2 cos ysin y11y2y3þ buy2 I1sin y1 8 > > < > > : : ð3:2Þ The true values of the ‘‘unknown’’ parameters are b/¼ 2, bu ¼ 5, I1¼ 1,
Mg¼ 4, l ¼ 0:25, C ¼ 0:5, x1¼ 1, A ¼ 12:1 in numerical simulation. The
initial conditions of the drive and the response systems are x1ð0Þ ¼ 0:5,
x2ð0Þ ¼ 1:2, x3ð0Þ ¼ 10 and y1ð0Þ ¼ y2ð0Þ ¼ y3ð0Þ ¼ 0:1, respectively. The
initial value of estimate for ‘‘unknown’’ parameter isC^
I1ð0Þ ¼ 0:1.
For synchronizing the two third-order heavy symmetric gyroscope systems, we add three controllers u1, u2, u3, on the first, second and third equation of
_y1¼ y2þ u1 _y2¼ ðb/bucos y1Þðbub/cos y1Þ I2 1sin 3y 1 C I1y2þ Mgl I1 sin y1 Mg I1 Asin x1tsin y1þ u2 _y3¼ 2 cos ysin y11y2y3þ buy2 I1sin y1þ u3 8 > > < > > : : ð3:3Þ First, subtracting Eq. (3.1) from (3.3), we can obtain the error dynamics as
_e1¼ e2þ u1; _e2¼ ðb/ bucos y1Þðbu b/cos y1Þ I2 1sin 3y 1 þðb/ bucos x1Þðbu b/cos x1Þ I2 1sin 3x 1 C I1 e2þ Mgl I1 ðsin y1 sin x1Þ 1 Mg I1
Asin x1tðsin y1 sin x1Þ þ u2;
_e3¼ 2 cos y1 sin y1 y2y3þ 2 cos x1 sin x1 x2x3þ buy2 I1sin y1 bux2 I1sin x1 þ u3; ð3:4Þ where e1¼ y1 x1, e2¼ y2 x2, e3¼ y3 x3.
Then, choosing a Lyapunovfunction of the form
V e1; e2; e3; e C I1 ! ¼1 2 e 2 1 0 @ þ e2 2þ e 2 3þ e C I1 !21 A; ð3:5Þ where eC I1¼ C I1 ^ C I1, and ^ C
I1 is estimate value of the unknown parameter
C
I1,
respectively [4].
Its derivative along the solution of Eq. (3.4) is _ V e1; e2; e3; e C I1 ! ¼ e1ðe2þ u1Þ þ e2 " ðb/ bucos y1Þðbu b/cos y1Þ I2 1sin 3y 1 þðb/ bucos x1Þðbu b/cos x1Þ I2 1sin 3 x1 C I1 e2þ Mgl I1 ðsin y1 sin x1Þ Mg I1 Asin x1t ðsin y1 sin x1Þ þ u2 # þ e3 2 cos y1 sin y1 y2y3 þ2 cos x1 sin x1 x2x3þ buy2 I1sin y1 bux2 I1sin x1 þ u3 þ Ce I1 ! C_^ I1 ! : ð3:6Þ
We select u1¼ e2 e1; u2¼ ðb/ bucos y1Þðbu b/cos y1Þ I2 1sin 3y 1 ðb/ bucos x1Þðbu b/cos x1Þ I2 1sin 3x 1 Mgl I1 ðsin y1 sin x1Þ þ Mg I1
Asin x1tðsin y1 sin x1Þ þ
^ C I1 1 ! e2; u3¼ 2 cos y1 sin y1 y2y3 2 cos x1 sin x1 x2x3 buy2 I1sin y1 þ bux2 I1sin x1 e3; _^ C I1 ¼ e2 2:
Then, Eq. (3.6) becomes _ Vðe1; e2; e3Þ ¼ e21 e 2 2 e 2 3<0: ð3:7Þ
This means that the synchronization of the two third-order heavy symmetric gyroscope systems can be achieved. The results are shown in Figs. 2–4.
0 50 100 150 200 250 -3 -2 -1 0 x1 0 50 100 150 200 250 -5 0 5 y1 0 50 100 150 200 250 0 2 4 6 e1 t t t (sec) (sec) (sec)
4. Parameter identification
4.1. Synchronization of uncertain chaotic systems via adaptive control
We investigate two third-order heavy symmetric gyroscope systems in this
section. Both the systems have the same form. But the parameter,C
I1ðtÞ of the
response system is time-varying and uncertain.
The drive system is described as Eq. (3.1). The response system is described by _y1¼ y2 _y2¼ ðb/bucos y1Þðbub/cos y1Þ I2 1sin 3y 1 C I1ðtÞy2þ Mgl I1 sin y1 Mg I1 Asin x1tsin y1 _y3¼ 2 cos ysin y1 1 y2y3þ buy2 I1sin y1 8 > > < > > : : ð4:1Þ For synchronizing two third-order heavy symmetric gyroscope systems, we add three controllers u1, u2, u3, on the first, second and third equation of
Eq. (4.1), respectively. 0 50 100 150 200 250 -10 0 10 x2 0 50 100 150 200 250 -50 0 50 y2 0 50 100 150 200 250 -50 0 50 e2 t t t (sec) (sec) (sec)
_y1¼ y2þ u1 _y2¼ ðb/bucos y1Þðbub/cos y1Þ I2 1sin 3y 1 C I1ðtÞy2þ Mgl I1 sin y1 Mg I1 Asin x1tsin y1þ u2 _y3¼ 2 cos ysin y11y2y3þ buy2 I1sin y1þ u3 8 > > < > > : : ð4:2Þ The drive and the response system can be written as [5,6]
_x¼ f ðxÞ þ F ðxÞ C I1 ; _y¼ f ðyÞ þ F ðyÞ C I1 ðtÞ þ U ; ð4:3Þ where fðxÞ ¼ x2 ðb/ bucos x1Þðbu b/cos x1Þ I2 1sin 3 x1 þMgl I1 sin x1 Mg I1 Asin x1tsin x1 2 cos x1 sin x1 x2x3þ bux2 I1sin x1 2 6 6 6 6 6 4 3 7 7 7 7 7 5 ; 0 50 100 150 200 250 5 10 15 20 x3 0 50 100 150 200 250 -100 0 100 200 y3 0 50 100 150 200 250 -100 0 100 200 e3 t t t (sec) (sec) (sec)
and FðxÞ ¼ 0 x2 0 2 4 3 5:
From Eq. (4.3), we can obtain the error dynamics _e¼ f ðyÞ f ðxÞ þ F ðyÞ C I1 ðtÞ F ðxÞ C I1 þ U : ð4:4Þ
Before solving our problem, we have some work to do first. Considering the special case when the drive and the response systems have the same parameters, which are time invariant. The drive and the response systems can be written as
_x¼ f ðxÞ þ F ðxÞ C I1 ; _y¼ f ðyÞ þ F ðyÞ C I1 þ U : ð4:5Þ
The error dynamics can be obtained _e¼ f ðyÞ f ðxÞ þ ðF ðyÞ F ðxÞÞ C
I
þ U : ð4:6Þ
Choosing a Lyapunovfunction of the form VðeÞ ¼1 2e Te¼1 2ðe 2 1þ e 2 2þ e 2 3Þ: ð4:7Þ
Its derivative along the solution of Eq. (4.6) is dV dt ¼ e T _e¼ eT fðyÞ f ðxÞ þ ðF ðyÞ F ðxÞÞ C I1 þ U : ð4:8Þ Select u1¼ e2 e1; u2¼ ðb/ bucos y1Þðbu b/cos y1Þ I2 1sin 3y 1 ðb/ bucos x1Þðbu b/cos x1Þ I2 1sin 3x 1 þ C I 1 e2 Mgl I1 ðsin y1 sin x1Þ þ Mg I1
Asin x1tðsin y1 sin x1Þ;
u3¼ 2 cos y1 sin y1 y2y3 2 cos x1 sin x1 x2x3 buy2 I1sin y1 þ bux2 I1sin x1 8 3e3: ð4:9Þ
Then, Eq. (4.8) can be rewritten as dV dt ¼ e 2 1 e 2 2 8 3e 2 360: ð4:10Þ
This means that the synchronization of the two systems is achieved.
Now, we use the results of this special case to solve our problem. Choosing a Lyapunovfunction for Eq. (4.4)
V1 e; C I1 ðtÞ C I1 ¼ V ðeÞ þ1 2 C I1 ðtÞ C I1 T C I1 ðtÞ C I1 : ð4:11Þ Let e C I1 ¼C I1 ðtÞ C I1 :
We can rewrite Eq. (4.11) as
V1 e; e C I1 ! ¼ V ðeÞ þ1 2 e C I1 !T e C I1 ! : ð4:12Þ
Its derivative along Eq. (4.4) satisfies dV1 dt ¼ dV dt þ _ C I1 ðtÞ !T e C I1 !
¼ ðgrad V ðeÞ; _eÞ þ C_ I1 ðtÞ !T e C I1 !
¼ grad VðeÞ; f ðyÞ f ðxÞ þ F ðyÞ C I1 ðtÞ F ðxÞ C I1 þ U þ C_ I1 ðtÞ !T e C I1 !
¼ grad VðeÞ; f ðyÞ f ðxÞ þ F ðyÞ C I1 ðtÞ F ðxÞ C I1 ðtÞ þ U þ grad VðeÞ; F ðxÞ Ce I1 !! þ C_ I1 ðtÞ !T e C I1 ! : ð4:13Þ
Choosing u1¼ e2 e1; u2¼ ðb/ bucos y1Þðbu b/cos y1Þ I2 1sin 3 y1 ðb/ bucos x1Þðbu b/cos x1Þ I2 1sin 3 x1 þ C I ðtÞ 1 e2 Mgl I1 ðsin y1 sin x1Þ þMg I1
Asin x1tðsin y1 sin x1Þ;
u3¼ 2 cos y1 sin y1 y2y3 2 cos x1 sin x1 x2x3 buy2 I1sin y1 þ bux2 I1sin x1 8 3e3; _ C I1 ðtÞ ¼ FTðxÞðgrad V ðeÞÞT ¼ x2e2:
Eq. (4.13) can be rewritten as
_ V1 e;Ce I1 ¼
grad VðeÞ; f ðyÞ f ðxÞ þ F ðyÞ C I1 ðtÞ F ðxÞ C I1 þ U ¼ _V ðeÞ ¼ e2 1 e 2 2 8 3e 2 3<0: ð4:14Þ
In this section, the parameter C
I1ðtÞ is unknown and the values of the other
parameters are given in Section 3. Let A¼ 12:1 and the concerned functions
becomes as
fðxÞ ¼
x2
ð2 5 cos x1Þð5 2 cos x1Þ sin3x1
þ sin x1 4A sin t sin x1
2 cos x1 sin x1 x2x3þ 5x2 sin x1 2 6 6 6 6 4 3 7 7 7 7 5; fðyÞ ¼ y2 ð2 5 cos y1Þð5 2 cos y1Þ sin3y1
þ sin y1 4A sin t sin y1
2 cos y1 sin y1 y2y3þ 5y2 sin y1 2 6 6 6 6 6 4 3 7 7 7 7 7 5 :
The controllers becomes u1¼ e2 e1; u2¼ ð2 5 cos y1Þð5 2 cos y1Þ sin3y1 ð2 5 cos x1Þð5 2 cos x1Þ sin3x1 þ C I1 ðtÞ 1
e2 ðsin y1 sin x1Þ þ 4A sin tðsin y1 sin x1Þ;
u3¼ 2 cos y1 sin y1 y2y3 2 cos x1 sin x1 x2x3 5y2 sin y1 þ 5x2 sin x1 8 3e3:
In numerical simulation, the initial conditions of the drive and the response
systems are x1ð0Þ ¼ 0:5, x2ð0Þ ¼ 1:2, x3ð0Þ ¼ 10 and y1ð0Þ ¼ 1:5,
y2ð0Þ ¼ 2:4, y3ð0Þ ¼ 6, respectively. We find that the synchronization can be
achieved, and the results are shown in Figs. 5–7. The identification of
0 5 10 15 20 25 30 35 40 45 50 -3 -2 -1 0 x1 0 5 10 15 20 25 30 35 40 45 50 -4 -2 0 2 y1 0 5 10 15 20 25 30 35 40 45 50 0 1 2 e1 t t t (sec) (sec) (sec)
parameter is also achieved. It also means that the value ofC
I1ðtÞ arrives the value
ofC
I1¼ 0:5. The result is shown in Fig. 8.
4.2. Parameters identification by random optimization
We investigate two identical third-order heavy symmetric gyroscope systems in this section. Both systems have the same parameters, but the parameter a of the response system is unknown. Our work is to identify the unknown parameter.
The drive system is described by _x1¼ x2 _x2¼ ðb/bucos x1Þðbub/cos x1Þ I2 1sin 3x 1 C I1x2þ Mgl I1 sin x1 Mg I1 Asin x1tsin x1 _x3¼ 2 cos xsin x1 1 x2x3þ bux2 I1sin x1 8 > > < > > : : ð4:15Þ 0 5 10 15 20 25 30 35 40 45 50 -10 0 10 x2 0 5 10 15 20 25 30 35 40 45 50 -5 0 5 10 y2 0 5 10 15 20 25 30 35 40 45 50 -2 0 2 4 e2 t t t (sec) (sec) (sec) Fig. 6. Time history of x2, y2and the error between them.
The response system is described by _y1¼ y2 _y2¼ ðb/bucos y1Þðbub/cos y1Þ I2 1sin 3y 1 ay2þ Mgl I1 sin y1 Mg I1 Asin x1tsin y1 _y3¼ 2 cos ysin y1 1 y2y3þ buy2 I1sin y1 8 > < > : : ð4:16Þ To synchronize two identical third-order heavy symmetric gyroscope sys-tems, we add one coupling term, kðx1 y1Þ on the first equation of (4.16).
_y1¼ y2þ kðx1 y1Þ _y2¼ ðb/bucos y1Þðbub/cos y1Þ I2 1sin 3y 1 ay2þ Mgl I1 sin y1 Mg I1 Asin x1tsin y1 _y3¼ 2 cos ysin y11y2y3þ buy2 I1sin y1 8 > > < > > : : ð4:17Þ Define the difference by
U¼
Z T 0:9T
jx1 y1j2dt; ð4:18Þ
where T is the simulation time.
0 5 10 15 20 25 30 35 40 45 50 5 10 15 20 x3 0 5 10 15 20 25 30 35 40 45 50 0 10 20 y3 0 5 10 15 20 25 30 35 40 45 50 -4 -2 0 e3 t t t (sec) (sec) (sec)
) ( 1 t I C 0 5 10 15 20 25 30 35 40 45 50 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 t (sec)
Fig. 8. Time history ofC I1ðtÞ. -0.5 0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 α
The difference U can be considered as a function of a and k. If k is suffi-ciently large and a is close toC
I1, the difference U would tend to zero. In other
words, with sufficiently large value of k, if U is small, a would be close toC I1. The
result is shown in Fig. 9.
To identify the unknown parameter of the response system, we use the random optimization method [7]. The algorithm is as follows:
First, choose a sufficiently large value of k. In our case, we choose k¼ 200. By estimating initial value of a, we can calculate the difference U .
The parameter a is randomly modified as
am¼ a þ r; ð4:19Þ
where r is a random number which obeys the Gaussian distribution with
variance r¼ 0:0025.
Substituting the modified parameter am into Eq. (4.17), we can obtain y10.
The difference between two systems is U0¼ Z T 0:9T jx1 y10j 2 dt: ð4:20Þ
If the difference U0is smaller than U , the parameter is changed from a to a
m.
On the other hand, if the difference U0 is larger than U , the parameter is
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 104 0.42 0.44 0.46 0.48 0.5 0.52 step number α
unchanged and kept to be a. The processes are repeated until the difference U tends to zero.
In numerical simulation, we assume that only one parameter, a is unknown. Parameter identification can be achieved. The result is shown in Fig. 10. 5. Conclusions
The main studies in this paper is the study of synchronization of two sys-tems. In this paper, both analytical and computational methods have been used to study the dynamical behaviors of the nonlinear system.
We investigate two third-order heavy symmetric gyroscope systems in Sec-tion 3. Both the systems have the same form and both the parameters are unknown. The true values of the ‘‘unknown’’ parameters are selected. We choose three suitable controllers, and add them into the slave when t¼ 50. By using Lyapunovstability theory, the synchronization of the two third-order heavy symmetric gyroscope systems can be achieved successfully.
In Section 4, we investigate two third-order heavy symmetric gyroscope systems, while the parameter,C
I1ðtÞ of the response system is time varying and
uncertain. There are two purposes in this Section, one is to synchronize the two identical systems, by the Lyapunovfunction and controllers which are different
from that of Section 3. The another is to track the parameter C
I1 of the drive
system. In numerical simulation, the unknown parameterC
I1ðtÞ and a tracks the
known parameterC
I1successfully.
Acknowledgement
This research was supported by the National Science Council, Republic of China, under Grant Number NSC 91-2212-E-009-025.
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