Anticontrol and synchronization of chaos for an autonomous rotational machine system with a hexagonal centrifugal governor

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JOURNAL OF SOUND AND VIBRATION

Journal of Sound and Vibration 282 (2005) 635–648

Anticontrol and synchronization of chaos for an

autonomous rotational machine system with a hexagonal

centrifugal governor

Zheng-Ming Ge

, Ching-I Lee

Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsuej Road, Hsinchu 30050, Taiwan, Republic of China

Received 2 June 2003; received in revised form 13 January 2004; accepted 2 March 2004 Available online 24 November 2004

Abstract

Anticontrol and synchronization of chaos for an autonomous rotational machine system with a hexagonal centrifugal governor are studied in the paper. Two different procedures for the design of the controller are pro-posed to anticontrol the governor system effectively. Finally, ﬁve methods are studied for chaos synchronization. r2004 Elsevier Ltd. All rights reserved.

1. Introduction

Anticontrol and synchronization of chaos have received great attention for many research activities in recent years [1–7]. Sometimes, chaos is not only useful but actually important. For example, chaos is desirable in many applications of liquid mixing while the required energy is minimized. For this purpose, making a non-chaotic dynamical system chaotic is called ‘‘anticontrol of chaos’’. Besides, secure communication and information processing are proposed as potential applications of synchronization of chaotic systems.

In previous researches, most of them were concentrated to several well-known systems, such as Lorenz system, Ro¨ssler system and Chua’s circuits system, etc. In this paper, an autonomous hexagonal centrifugal governor system is studied. It plays an important role in many rotational

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machines such as diesel engine, steam engine and so on. Two different procedures, linear and nonlinear controllers with certain feedback gain are proposed to anticontrol, i.e. to chaotify, the governor system. Four methods, linear feedback, nonlinear feedback, adaptive feedback, backstepping design and parameter evaluation from time sequences approaches are also discussed for synchronization of two coupled chaotic system.

2. Equations of motion

The rotational machine with centrifugal governor is depicted inFig. 1. Somebasic assumptions for thesystem are

(1) neglecting the mass of the rods and the sleeve;

(2) viscous damping in rod bearing of the ﬂy-ball is presented by damping constant c. From Fig. 1, the kinetic and potential energies of the system are written as follows:

T ¼ 2 n₂1m ðr þ l sin fÞh 2Z2þl2f_2io¼mZ2ðr þ l sin fÞ2þml2f_2, V ¼ 2kl2ð1 cos fÞ2þ2mglð1 cos fÞ,

where l, m, r and f represent the length of the rod, the mass of ﬂy ball, the distance between the rotational axis and the suspension joint, and the angle between the rotational axis and the rod. It is easy to obtain the Lagrangian

L ¼ T V ¼ mZ2ðr þ l sin fÞ2þml2f_22kl2ð1 cos fÞ22mglð1 cos fÞ.

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Then using Lagrange equation, the equation of motion for the governor can be derived as follows:

2 mlh 2f mrlZ€ 2 cos f ð2k þ mZ2Þl2 sin f cos f þ ð2kl þ mgÞl sin fi¼ c _f, (2.1) where c is the damping coefﬁcient.

For the rotational machine the net torque is the difference between the torque Q produced by theengineand theload torqueQL, which is available for angular acceleration. That is,

Jdo

dt ¼Q QL, (2.2)

where J is themoment of inertia of themachine. As theanglef varies, the position of control valve which admits the fuel is also varied. The dynamical equation (2.2) can be written in the form[8]

J _o ¼ g cos f b, (2.3)

where g40 is a proportionality constant and b is an equivalent torque of the load. Eq. (2.3) is the second differential equation of motion for the system.

Usually, the governor is geared directly to the output shaft such that its speed of rotation is proportional to the engine speed, i.e. Z ¼ no: The operation of the fly-ball governor can be briefly described as follows. At first, set the speed of engine at o0: If the speed of engine drops down, the

centrifugal force acting on the ﬂy-ball would decrease, thus the control valve of fuel will open wider. When more fuel is supplied, the speed of the engine increases until equilibrium is again reached. Similarly, if the speed rises up, the fuel supply is reduced and the speed decreases until o0

is recovered.

By changing thetimescalet ¼ Ont; Eqs. (2.1) and (2.3) can be written in nondimensional form

_ f ¼ j,

_

j ¼ do2cos f þ ðe þ po2Þsin f cos f sin f bj, _ o ¼ q cos f F , ð2:4Þ where q ¼ g JOn ; F ¼ b JOn ; d ¼ n 2_mr 2kl þ mg; e ¼ 2kl 2kl þ mg, p ¼ n 2_ml 2kl þ mg; b ¼ c 2ml2On ; On¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2kl þ mg ml r ,

and the dot presented the derivative with respect to t, j is df=dt: Hence, the dynamics of the system of rotational machine with centrifugal governor is described by a three-dimensional autonomous system. Denoting f ¼ x; _f ¼ y; o ¼ z; Eq. (2.4) is rewritten in the form

_ x ¼ y, _y ¼ dz2cos x þ1 2 e þ pz 2

sin 2x sin x by,

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3. Anticontrol of chaos

Anticontrol of chaos is making a nonchaotic dynamical system chaotic. This implies that the regular behaviors will be destroyed and replaced by chaotic behavior. In the real world, chaotic behavior is important. Examples include liquid mixing, human heartbeat regulations, resonance prevention in mechanical systems and secure communication [3]. In this section, Eqs. (2.5) are modiﬁed by the addition of a linear and a nonlinear feedbacks to chaotify the system, respectively.

3.1. Adding a linear feedback

The state equations of the centrifugal governor system with a linear feedback controller are represented as

_

x ¼ y þ a1x,

_y ¼ dz2cos x þ1₂e þ pz2sin 2x sin x by þ a2y,

_z ¼ q cos x F þ a3z. ð3:1Þ

Here a1, a2, a3are feedback gains and the values of parameters d, e, p, q, F, b aregiven as 0.008, 0.8, 0.04, 3, 2, 0.4, respectively.

By numerical integration method, the phase portrait of the system, Eq. (3.1), is plotted inFig. 2

for a1=a2=a3=0. Clearly, the motion is periodic. But Eq. (3.1) exhibits both strange attractors and limit cycles for certain choices of a1, a2 and a3. For example, when a1=0.2, a2=0.1, a3=0.1, one can observe a chaotic attractor as depicted in Fig. 3. By simulation results, for certain interval of parameters, the maximum Lypunov exponent of the system is positive, i.e., the system exhibits the strange attractor, we defined that chaotic region. Inspired by the consideration of chaotification, we found regions of specific feedback gains for which this system is chaotic as shown in Fig. 4.

3.2. Adding a nonlinear feedback

For our purpose, the nonlinear feedback controller, xjxj; is added to the right-hand side of the ﬁrst equation of Eq. (2.5). Then the system equations are represented as

_

x ¼ y þ xjxj,

_y ¼ dz2cos x þ1₂e þ pz2sin 2x sin x by,

_z ¼ q cos x F . ð3:2Þ

System (3.2) is obtained for which certain value of e (for example, 0:008pp0:032) has strange attractor by numerical solution. As illustrated inFig. 5, chaotic motion is observed from system (3.2) with ¼ 0:01:

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Fig. 2. Phase portrait of uncontrolled system (a1=a2=a3=0).

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4. Chaos synchronization

A characteristic property of chaotic dynamics is the sensitive dependence on initial condition. Different initial conditions may cause entirely different trajectories for the system. However, Pecora and Carroll[4]showed that synchronization can be achieved for the chaotic systems. This interest phenomenon plays a signiﬁcant role in the chaotic dynamics of communication signals and may be applied to the real-time recovery of signals that have been masked in a strange attractor and thus to encode communication. Other applications of synchronization of chaos also have expectative potential [2]. A natural way to develop synchronization for chaotic systems is through system decomposition. Chaotic system (3.1) is decomposed into two subsystems as follows: Drive system: _ x1¼y1þ0:2x1, _y₁¼dz2₁cos x1þ1₂ e þ pz2₁

sin 2x1sin x1by10:1y1,

_z1¼q cos x1F 0:1z1. ð4:1Þ Response system: _ x2¼y2þ0:2x2, _y₂¼dz2₂cos x2þ1₂ e þ pz22

_z2¼q cos x2F 0:1z2. ð4:2Þ

In the following, linear feedback, nonlinear feedback, adaptive feedback and backstepping design approaches are discussed.

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4.1. Linear feedback synchronization

In this approach, the error between the output of the identical drive and response is used as the control signal. For the unidirectional case, where only ﬁrst equation of response system (4.2) is combined with a linear feedback, while the equations of drive remain the same[4]:

_

x2¼y2þ0:2x2þk1ðx1x2Þ,

_y₂¼dz2₂cos x2þ1₂ e þ pz22

sin 2x2sin x2by20:1y2

_z2¼q cos x2F 0:1z2, ð4:3Þ

where k1 is the constant feedback gain. With k1=0.2, thesynchronization errors, ex ¼x2

x1; ey ¼y2y1; and ez¼z2z1; areshown in Fig. 6. In this case, k1=0.15 is a critical value, below which no complete synchronization occurs.

4.2. Nonlinear feedback synchronization

The chaotic response system (4.2) by adding nonlinear coupling term are written as _

x2¼y2þ0:2x2þk2sinðx1x2Þ,

_z2¼q cos x2F 0:1z2. ð4:4Þ

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With k2=0.2, thesynchronization errors areshown inFig. 7. In this case, k2=0.15 is a critical value, below which no completely synchronization occurs.

4.3. Adaptive feedback synchronization

Some adaptive control strategies can direct a chaotic trajectory to stable orbits but not unstable ones. However, it is possible to combine the feedback method for chaos synchronization[2].

For response system (4.2), the linear feedback (4.3) is replaced by _

x2¼y2þ0:2x2þk3ðx1x2Þ,

_z2¼q cos x2F 0:1z2,

_q ¼ k₄ðy₁y₂Þ. ð4:5Þ

where the system parameter q is used as an adjustable function for adaptation, and k4is a constant adaptive control gain to be determined in the design. Using this method, the response can be synchronized by the chaotic drive, as shown inFig. 8.

4.4. Backstepping design

Backstepping design is a recursive procedure that combines the choice of a Lyapunov function for selecting a proper controller in chaos synchronization [9]. The drive system is expressed as

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Eq. (4.1) and a controller u is added to the right-hand side of the second equation of response system (4.2). Let the state errors between the response system and drive system be

ex¼x2x1; ey¼y2y1; ez¼z2z1; (4.6)

then error system can be derived as _ex ¼eyþ0:2ex, _ey¼d cos x2ðe2zþ2z1ezÞ þ z2 1 2 ðe 2 xþ2x1exÞ þ ðe 1Þex þp 2 sin 2x2ðe 2 zþ2z1ezÞ þ2z21ex ðb þ 0:1Þeyþu þ h:o:t:, _ez ¼ q 2ex0:1ez. ð4:7Þ

If system (4.7) did not have u, it would have an equilibrium point (0,0,0). The problem of synchronization between drive and response system can be transformed into how to ﬁnd a control law u for stabilizing the error variables of system (4.7) at the origin.

First we consider the stability of system as follows: _ez¼

q

2ex0:1ez, (4.8)

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where ex is regarded as a controller and it makes system (4.8) asymptotically stable. Choose Lyapunov function V1ðezÞ ¼e2z=2: Thederivativeof V1is

_

V1¼ 0:1e2zþ

q

2ezex. (4.9)

Assumecontroller ex¼a1ðezÞ and a1ðezÞ ¼0; then

_

V1¼ 0:1e2zo0 (4.10)

makes system (4.8) asymptotically stable. Function a1ðezÞ is an estimative function when ex is considered as a controller. The error between ex and a1ðezÞ is

w2¼exa1ðezÞ. (4.11)

Study ðez; w2Þ system

_e_z¼ q

2w20:1ez, _

w2¼eyþ0:2w2. ð4:12Þ

Consider ey¼a2ðez; w2Þas a controller in system (4.12). Choose Lyapunov function V2ðez; w2Þ ¼

V1ðezÞ þw22=2: Thederivativeof V2 is _

V2¼ 0:1e2zþ0:2w22þ

q

2w2ezþw2ey. (4.13)

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If a2ðez; w2Þ ¼ q₂ezw2; then

_

V2¼ 0:1e2z0:8w22o0 (4.14)

makes system (4.12) asymptotically stable. Deﬁne the error variable w3 as

w3¼eya2ðez; w2Þ. (4.15)

Study full dimension ðez; w2; w3Þ system

_ez ¼ q 2w20:1ez, _ w2¼w3þa2þ0:2w2, _ w3¼d cos x2ðe2zþ2z1ezÞ þ z2 1 2 ðw 2 2þ2x1w2Þ þ ðe 1Þw2 ðb þ 0:1Þðw3þa2Þ þp 2 sin 2x2ðe 2 zþ2z1ezÞ þ2z 2 1w2 da2 dt þu þ h:o:t:, ð4:16Þ where da2 dt ¼ q 2 q 2w20:1ez h i ðw3þa2þ0:2w2Þ.

ChooseLyapunov function V3ðez; w2; w3Þ ¼V2ðez; w2Þ þw23=2: Thederivativeof V3is _ V3¼ 0:1e2z0:8w22þw3 2z1ezðq cos x2þ p 2sin 2x2Þ þew2 ðb þ 0:1Þðw3þa2Þ þ dz2 1 2 ðw 2 2þ2x1w2Þ þpz 2 1w2 da2 dt þu þh:o:t: ð4:17Þ Let u ¼ ðb þ 0:1Þa22z1ez q cos x2þ p 2 sin 2x2 ew2 dz 2 1 2 w 2 2þ2x1w2 pz2₁w2þ da2 dt then _ V3¼ 0:1e2z0:8w22 ðb þ 0:1Þw23þh:o:t.

is quadric negative deﬁnite. Whereas x1, z1and x2arebounded, wecan concludethat theequilibrium point (0,0,0) of error system (4.7) is locally asymptotically stable. For proper initial errors between drive and response systems, the initial errors will converge to zero and synchroniza-tion between two chaotic subsystems will be achieved. The numerical results with initial condition (x1ð0Þ ¼ 1:42; y1ð0Þ ¼ 2:1; z1ð0Þ ¼ 5:55; x2ð0Þ ¼ 1:4; y2ð0Þ ¼ 2; z2ð0Þ ¼ 5:5Þ

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4.5. Parameter evaluation from time sequences

In this section, we study another method to estimate the parameter of chaotic system by a random optimization method for chaos synchronization[10]. For system (4.3), q is theunknown parameter and k1is a coupling constant. The difference of the two time sequences is calculated as

U ¼ Z T

0

jx2x1j2 dt,

theintegral timeT is larger than a typical period of the chaotic oscillation in the governor system and the parameter q is randomly modiﬁed as

q0¼q þ z,

where z is a random number. We can obtain a time sequence x0

2ðtÞ by numerical simulation of Eq.

(4.3) with the modiﬁed parameter q0_{: Then the difference of the two time sequences is calculated as}

U0¼ Z T

0

jx0₂x1j2dt.

If the difference U0_{is smaller than U, the parameter is changed from q to q}0_{: On theother hand,}

if the difference U0 is larger than U, the parameter is unchanged and kept to be q. Theobtained parameter value is expected to be the desired parameter until the difference U becomes zero, i.e., complete chaos synchronization is achieved.

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Fig. 10 displays theminimum valueof U with respect to the coupling constant k1. The numerical simulation shows that the complete chaos synchronization x2ðtÞ ¼ x1ðtÞ occurs for

k140.15. This result is the same as that in Section 4.1. The difference U as a function of the parameter q for k1=0.2 is shown inFig. 11. The complete chaos synchronization is attained when thevalueof U takes a minimum value 0 at q=3. Time evolution of the parameter q by therandom optimization process is shown in Fig. 12.

Fig. 10. Theminimum valueof U with respect to the coupling constant k1.

Fig. 11. The difference U versus the parameter q for k1=0.2.

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5. Conclusions

Theproblems of anticontrol and synchronization of chaos for an autonomous rotational machine system with a hexagonal centrifugal governor has been discussed in this paper. For anticontrol of chaos, two different procedures to design the controller have been presented. The periodic motion of the system disappeared and was replaced by chaotic motion effectively by adding a linear and a nonlinear feedback term, respectively.

Synchronization of two chaotic oscillators is also studied in this paper. For two identical chaotic systems, increase of coupling strength leads to the occurrence of complete synchroniza-tion. Chaos synchronization of the autonomous governor system has been presented by adding linear feedback term, adding sinusoidal term and adaptive feedback methods. Chaos synchronization is also attained by a recursive procedure, backstepping design that combines the choice of a Lyapunov function for selecting a proper controller. Finally, the parameter of chaotic system is estimated from time sequences for chaos synchronization is studied. In this paper, the theoretical study of the governor system for anticontrol and synchronization of chaos had been proposed. The knowledge of anticontrol tells us the various conditions for steadily running chaotic process, prevent it from going into some catastrophic event that is known to occur whenever the chaotic orbit wanders into some particular regions of state space. As for synchronization, our study affords a more complex model for secure communication than the Lorenz system and Ro¨ssler system to obtain better security.

Acknowledgment

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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