Non-linear dynamics and chaos control of a physical pendulum with vibrating and rotating support

doi:10.1006/jsvi.2000.3353, available online at http://www.idealibrary.com on

NON-LINEAR DYNAMICS AND CHAOS CONTROL

OF A PHYSICAL PENDULUM WITH VIBRATING AND

ROTATING SUPPORT

Z.-M. GE, C.-H. YANG, H.-H. CHEN ANDS.-C. LEE

Department of Mechanical Engineering, National Chiao ¹ung ;niversity, Hsinchu, ¹aiwan, Republic of China

(Received 24 June 1999, and in ,nal form 12 July 2000)

The dynamic behavior of a physical pendulum system of which the support is subjected to both rotation and vertical vibration are studied in this paper. Both analytical and computational results are employed to obtain the characteristics of the system. By using Lyapunov's direct method the conditions of stability of the relative equilibrium position can be determined. Melnikov's method is applied to identify the existence of chaotic motion. The incremental harmonic balance method is used to "nd the stable and unstable periodic solutions for the strong non-linear system. By applying various numerical results such as phase portrait, PioncareH map, time history and power spectrum analysis, a variety of the periodic solutions and the phenomena of the chaotic motion can be presented. The e!ects of the changes of parameters in the system could be found in the bifurcation and parametric diagrams. Further, chaotic motion can be veri"ed by using Lyapunov exponent and Lyapunov dimension. The global analysis of basin boundary and fractal structure are observed by the modi"ed interpolated cell mapping method. Besides, non-feedback control, delayed feedback control, adaptive control, and variable structure control are used to control the chaos e!ectively.

 2001 Academic Press

1. INTRODUCTION

In the dynamics of a rigid body with a "xed point, the mechanics in question has three degrees of freedom. In engineering system, however, one often encounters rigid bodies attached to a base by a two-degree-of-freedom joint, consisting of a vertical axis and a horizontal one, which are mutually perpendicular. If a vertical rotation is given, the degree of freedom becomes one, yet the set of kinematically possible motions is still quite rich. The motion of such a physical pendulum with rotation and vibration of support will be considered in this paper, when there are no applied forces other than the gravity force [1, 2].

Most of the physical systems are non-linear in nature, and can be described by the non-linear equations of motion. Hence, the researches of non-linear systems are spreading quickly today. In the analysis of a non-linear dissipative system, "rst one usually tries to locate all the possible equilibrium positions and periodic solutions of the system, and to determine the stability of these solutions. Second, one observes how these solutions evolve as the system parameters are varied, which leads to di!erent forms of bifurcation. A further work is to "nd the basins of attraction in the state space for each attractor, and also to see how the basins of attraction change with the system parameters. Through extensive

analytical investigations, analogue and numerical simulations, as well as experimental observations, it has been shown that a pendulum exhibits a rich variety of non-linear bifurcational phenomena [3}8]. In particular, much work has been done on how point equilibria, periodic and chaotic attractors are created, changed or destroyed as system parameters are varied. The mechanisms include the well-known local bifurcation, together with subharmonic cascades, intermittencies, crises, etc. The geometry of the basins and their boundaries can often play an important role in the sensitivity of attractors to, say, noise-induced intermittency [8].

The analytical analyses for this non-linear dynamical system are obtained by Lyapunov's direct method [9}11], the incremental harmonic balance method [12}16], and Melnikov's method [17]. Only some special approaches can be used to reveal some important dynamical characteristics of the non-linear system. Lyapunov's direct method is applied to obtain the conditions of stability of the relative equilibrium position. Here an undamping-free oscillation system is employed. The incremental harmonic balance method (IHB) is used to "nd the stable and unstable periodic solutions for the strong non-linear system. Further, Melnikov's method is used to measure the distance between unstable and stable manifolds when the external disturbance is small.

A number of numerical results such as phase portraits, PoincareH maps, power spectrum analysis, bifurcation diagrams, and Lyapunov exponents [18], and the modi"ed interpolated cell mapping method [19] are used to study the dynamical behavior of the physical pendulum system. The phase portrait is a collection of trajectories that represent the solutions of these di!erential equations of motion in the phase space. A more informative representation of the periodic solutions is the PoincareH map method. The bifurcation is the special phenomenon in a non-linear system. As the parameters are changed in the non-linear dynamical system, the characteristics of the equilibrium points change as well. Chaotic motion is the motion that has a sensitive dependence on initial conditions in deterministic non-linear physical systems. The Lyapunov exponent test is a powerful method to measure the sensitivity of the dynamical system to changes in initial conditions. A new e!ective method, the modi"ed interpolated cell mapping [19, 20], is used to obtain the global analysis of dynamic behavior of the non-linear system. The di!erent initial conditions in di!erent basins of attraction may lead to the di!erent attractors. The fractal basin boundary is also obtained by this technique.

Various methods for control of chaos are presented. In practice, it is often desired that chaos can be avoided and that the system performance will be improved or changed in some way. Clearly, the ability to control chaos, that is to convert chaotic oscillations into desired regular ones with a periodic time dependence, would be bene"cial in working with a particular system. For this purpose, non-feedback control, delayed feedback control, adaptive control, and variable structure control are used to control chaos. As a result, the chaotic system can be controlled e!ectively.

2. EQUATION OF MOTION

A physical pendulum has a rotating and vibrating support with mutually perpendicular axes (Figure 1). The motion will be described in terms of two Cartestian co-ordinates, inertial system OXXX and a moving co-ordinate system oxxx rigidly attached to the physical pendulum. The origin of both co-ordinate systems is the point of intersection O of the joint axis; X- and x-axis are the "xed and moving axes of the joint respectively. All the kinematical possibility of the body relative to the inertial system OXXX can be described in terms of two angles: the anglea between the X- and x-axis, and the angle

Figure 1. A schematic diagram of the physical pendulum.

b between the x and XX plane. The angles a and b, which will be taken as generalized co-ordinates, may be treated as the angles of two successive rotations through which one can transfer the rigid body from its initial position (a"b"0, the moving co-ordinate system coincides with the "xed one) to the present position.

LetuG denote the projection of the angular velocity vector u of the body onto the xG (i"1, 2,3)-axis. The kinematic equations expressing the componentsuG by a, b, aR and bQ are u"bQ, u"aRsinb, u"aRcosb, (2.1) whereaR, bQ re generalized velocities. The kinetic energy of the motion of a rigid body with a "xed point is

¹" +-,2#J#+-,, (2.2)

where #J# is the inertia tensor of the body relative to the "xed point and +-,2" +u,u, u,.

Expanding the scalar product in equation (2.2) taking equation (2.1) into account, we obtain

KE"K(b)aR#JbQ!b(b)aRbQ, (2.3)

K (b)"Jsinb#Jcosb!2Jsinbcosb, b(b)"Jsinb#Jcosb,

where JGG (i"1, 2,3) are the axial moments of inertia and JGH"JHG (iOj, i, j"1, 2, 3) are the products of inertia of the body in Oxxx co-ordinate system.

The inertial system OXXX is now appointed to a vertical vibration with acceleration g csin ut where g is gravitational acceleration, 0(c(1 is constant, u is the given frequency of the vibration of support. The potential energy of the motion of the physical

pendulum in non-inertia system OXXX is

PE"mgh(1#c sin ut)cosb, (2.4)

where m is the mass of pendulum and h is the distance of center of mass to "xed point. By Lagrangian approach the Lagrangian has the expression

¸"K(b)aR#JbQ!b(b)aRbQ!mgh(1#ysinut)cosb. (2.5) When angular velocityaR"u(1#dsinuQt) is given to the vertical rotation, the degree of freedom of the system is reduced to one, whereu, 0(d(1, and uB are constants. The Lagrange equation corresponding to equation (2.5) is

JbG!(uuQdcosuQt)b(b)!Z(b)aR#CbQ!mgh(1#c sinut) sinb"0. (2.6) where

Z(b)"(J!J)sin2b!2Jcos2b and C is the coe$cient of damping.

Changing the time scaleq"uLt, equation (2.6) can be written in dimensionless form. De"ne

uL"mgh_J IfuQ"u then equation (2.6) becomes

bG!(ogd cos gq)bM(b)! ZM(b)aR#CMbQ!(1#csingq)sinb"0, (2.7) where o"u uL, g" u uL, bM (b)" b(b) J, ZM (b)" Z(b) J, CM " C JuL, q"uLt, aR"e(1#dsingq).

IfuQOu equation (2.6) becomes

bG!(ogQdcosgQq)bM(b)!ZM(b)aR#CMbQ!(1#csingq)sinb"0, (2.8) where

gQ"_uLuQ.

3. STABILITY ANALYSIS BY LYAPUNOV DIRECT METHOD

In this section, the stabilities of steady motion of the autonomous system are studied, whereaQ"u"const, c"0, C"0 and J is assumed to be zero. The kinetic energy for this system is

¹" (Jsinb#Jcosb)u#JbQ!(Jsinb#Jcosb)ubQ, (3.1) The potential energyP is

The Lagrangian of the system becomes

¸"¹!P" (Jsinb#Jcosb)u# JbQ !

(J sinb#Jcosb)ubQ!mgh cos b. (3.3) The Lagrange di!erential equation of motion is

d dt



Expanding sinb and cos b as power series, it can be written as bQ"b,

bQ" 1

J [(J!J)u#mgh]b#2, (3.7)

whereb"b, b"b, and the higher order terms are neither presented nor neglected. The stability of steady motionb"0 is studied. Since the Lagrangian of the system does not contain t explicitly, there exists the Jacobi integral [2]

¹!¹#P"const.,

where ¹ denotes the quadratic terms of bQ in kinetic energy and ¹ the terms free from bQ in kinetic energy.

The Jacobi integral is chosen as the Lyapunov function

<"¹!¹#P" [(J!J)u!mgh]b# JbQ#2. (3.8) If

(J!J) u!mgh'0, (3.9)

<M is positive de"nite for all higher order terms [9]. Since < is a "rst integral, d<

dt"0. (3.10)

By Lyapunov's stability theorem, the steady motion is stable [9]. For the study of instability, the Lyapunov function is chosen as

<"bb. (3.11)

The time derivative of < through equation (3.7) becomes

<Q "[mgh!(J!J)u]b#b#2 _(3.12) If

(J!J) u!mgh(0, (3.13)

<Q is positive de"nite for all higher order terms. By Lyapunov's instability theorem [9] equation (3.13) is the condition of instability.

4. INCREMENTAL HARMONIC BALANCE METHOD

The IHB method is a combination of the incremental method with the harmonic balance method. The steady state periodic solutions of equation (2.6) are obtained by the IHB method, which can deal very well with strong non-linearity and is convenient for computer implementation [12}16].

From equation (2.6), letq"ut, u"u, the dimensionless equation is given as ub!(uudcosq)bM(b)! ZM(b)aR#cubQ!v(1#csinq)sinb"0, (4.1) where b"db dq, c" C J, v"mghJ

The "rst step of the IHB method is a Newton}Raphson procedure. Letb, u, u and c be solutions; the neighboring state can be expressed by adding the corresponding increments to them as follows:

b"b#Db, u"u#Du, u"u#Du, c"c#Dc. (4.2) Substituting equation (4.2) into equation (4.1) and neglecting the small terms of higher order, the linearized incremental equation can be derived as

ub#cub#Dbg(b, q)"R#DuS#DuE#DcP, (4.3) where

g(b, q)"uubMd cosqJ#aRZM#(1#c sinqJ) cos b, R"!(ub#cub#g (b, q)),

g(b, q)"uubMd cosq#aRZM#(1#csinq) sin b, S"!(2ub#cb!ubMdcosq),

E"ubMdcosq#uaZM, P"!sinqsinb, b_{M "}J cosb!Jsinb

J ,

ZM "2((J!J) cos 2b#2Jsin2b) J

and R is a corrective vector which goes to zero when the solution is reached.

The second step of the IHB method is the Galerkin procedure. For steady state response, an approximate periodic solution may be assumed as

b" , H





corresponding to a solution of period 2n*the torque period, where q is the order of subharmonic. Galerkin's method is used withDaH and DbH as generalized co-ordinates:



 +ub#cub#Dbg(b,q),d(Db) dq "





(R#DuS#DuE#DcP)d(Db)dq. (4.5)

An incremental system of 2N linear equations in terms ofDaH and DbH is obtained from equation (4.5):

CDa"R#DuS#DuE#DcP, (4.6)

where R is the corrective vector, S is the unbalance torque vector due to unit change ofDu, and P is the exciting torque increment vector.

The increments can be solved from the following equation:

CDa"R. (4.7)

The procedure is repeated until the magnitude of the corrected vector R is acceptably small and the solution is obtained.

With the system parameter varied, the system results obtained by the IHB method are compared with the results obtained by numerical integration in the phase planes. There was good agreement between IHB and numerical results calculated by forth order Runge}Kutta method. The phase plane for the non-linear system, equation (2.7), is depicted in Figure 2(a)}2(d) for d"0.2, d"0,c"0,1 and c"0 where the symbols&&*'' and &&*'' indicate the results obtained by IHB and numerical integration respectively.

5. MELNIKOV'S METHOD

Melnikov [17] developed a global analysis technique on the occurrence of a heteroclinic (or homoclinic) bifurcation. Such a bifurcation is said to have occurred if a heteroclinic (homoclinic) set is either created or destroyed as a parameters is varied. The Melnikov function is a measure of the distance between stable and unstable manifolds when that distance is small. Both the damping and the amplitude of the external torque of the non-linear systems, equation (2.7), are assumed to be small. Assume d"0 and to express equation (2.7) in the dimensionless form, let

uL"mgh_J, q"uLt and rewrite the equation in the following form:

bQ"b, bQ"sinb# ZM( bM)o#e[csingqsinb!CMb], (5.1) where o"u uL, g" u n, ZM (b)" Z(b) J, CM " C JuL, q"uLt.

Figure 2. Comparison between the IHB and numerical integration methods: (a) d"0)2, (b) d"0, (c)c"0)1, (d)c"0.

Whene"0, the system of equations (5.1) is an unperturbed system and can be reduced to bQ"b,

bQ"sinb#12







H"1

2b#cosb# 1

4o







M$

G (q)"



$

G(q)[!cNb$G(q)#c sin bsing(q#q)] dq (5.4) for the homoclinic orbitsC$

G . Suppose that M$

G (q) has a simple zero, i.e., there exists a point q"q such that M$

G (qN)"0, *M $ G

Figure 3. Comparison between the Melnikov and R}K numerical method values

Then =Q(pMG) and =S(pMG) intersect transversely and there exist transverse homoclinic orbits. Sinceb$

G(q) is an odd function of q, equation (5.4) becomes M$ G (q)"!cN







G(q) sin bsingq, dqsingq "!cN A$ G #cB$ G singq, (5.6) where A$ G "





G . From equation (5.3) we have db

dq"G



2o







c cN'

G (q) has a simple zero and consequently chaotic dynamics may occur in system (5.1). Criterion (5.9) provides a remarkably good lower bound for the regions of chaos in the (g, c/c) space. Comparisons of the Melnikov critical value and the R}K simulation value are shown in Figure 3.

6. PHASE PORTRAITS, POINCARED MAP AND POWER SPECTRUM ANALYSIS The evolution of a set of trajectories emanating from various initial condition is presented in the phase plane. When the solution becomes stable, the asymptotic behaviors of the phase

Figure 4. PoincareH maps and phase portraits for different values of_{c for dO0: (a) u"uQ, c"2)0, (b) u"uQ,} c"2)01, (c) u'uQ, c"1)5, (d) u'uQ, c"3)0.

trajectories are particularly interesting and the transient behaviors in the system are neglected. The phase portraits of the physical pendulum system, equations (2.7) and (2.8), are plotted in Figure 4(a)}4(d).

The technique introduced by PoincareH deals with the question of the three-dimensional phase space (b, b, t) whenever t is a multiple of ¹"2n/u or 2n/uQ. Here ¹ is the period of the external torque. It is constructed by viewing the phase space diagram stroboscopically in such a way that the motion is observed periodically. By using the fourth order Runge}Kutta numerical integration method, the solution of the physical pendulum system obtained by PoincareH maps are shown in Figure 4(a)}4(d), which can be compared with the phase portraits.

Any functionb(q) may be represented as a superposition of di!erent periodic components. The determination of their relative strength is called spectral analysis. Due to the character of the functionb(q), there are two di!erent methods to express b(q). If it is periodic, the spectrum may be a linear combination of oscillations whose frequencies are integer multiple of a basic frequency. The linear combination is called a Fourier series. If it is not periodic, then the spectrum must be in terms of oscillations with a continuum of frequencies. Such a representation of the spectrum is called Fourier integral of b(q). The power spectrums of the non-linear dynamical system, equations (2.7) and (2.8) are shown in Figure 5(a)}5(d) respectively. Apparently, the spectrum of the periodic motion only consists of discrete frequencies. The noise-like spectrum is the characteristic of a chaotic dynamical system.

Figure 5. Power spectrum for di!erent values of_{c for dO0: (a) u"uQ, c"2)0, (b) u"uQ, c"2)01, (c) u'uQ,} c"1)5, (d) u'uQ, c"3)0.

7. BIFURCATION DIAGRAM AND LYAPUNOV EXPONENT

The information about the dynamics of a non-linear system for speci"c values of the parameters is provided. The dynamics may be viewed more completely over a range of parameter values. As the parameter is changed, the periodic solutions are created or destroyed, or their stability may be changed. The phenomenon of sudden change in the motion as a parameter is varied is called bifurcation, and the parameter value at which it occurs is called bifurcation point.

The bifurcation diagrams of the non-linear system of equations (2.7) and (2.8) are depicted in Figure 6(a) and 6(b). They are calculated by the fourth order Runge}Kutta numerical integration. At each d orc, the points of PoincareH map in the transient state of motion are discarded.

For Figure 6(a),u"uQ, dO0, the pitch fork bifurcations are obvious. But for Figure 6(b),u'uQ, dO0 the pattern is changed, the pitch fork bifurcations disappear and the regions of chaos are increased to a large extent.

The Lyapunov exponent may be used to measure the sensitive dependence upon initial conditions. It is an index for chaotic behavior. Di!erent solutions of dynamical system, such as "xed points, periodic motions, quasiperiodic motion, and chaotic motion can be distinguished by it. If two trajectories start close to one another in phase space, they will move exponentially away from each other for short periods of time on the average. Thus, if d is a measure of the initial distance between the two starting points, the distance is

Figure 6. Bifurcation diagram ofc versus angular velocity for: (a) u"uQ, dO0, (b) u'uQ, dO0 and Lyapunov exponents diagram, (c)u"uQ, dO0, (d) u'uQ, dO0.

d(q)"d2HO. The symbol j is called Lyapunov exponent. The divergence of chaotic orbits can only be locally exponential, because if the system is bounded, d(q) cannot grow to in"nity. A measure of this divergence of orbits is that the exponential growth at many points along a trajectory has to be averaged. The Lyapunov exponent can be expressed as

j" 1 q,!q , Ilog d(qI) d(qI!1). (7.1)

The signs of the Lyapunov exponents provide a qualitative picture of a system dynamics. The criterion is

j'0 (chaotic),

j)0 (regular motion).

The Lyapunov exponents of the non-linear dynamical systems, equations (2.7) and (2.8), are plotted in Figure 6(c) and 6(d).

The bifurcation diagram provides a summary of the essential dynamics and is therefore a useful way to observe non-linear dynamical behavior. To investigate bifurcation further, the phase portraits, PoincareH maps, and power spectra are used. The periodic and chaotic motions could be distinguished by the bifurcation diagram, but the quasiperiodic motion and chaotic motion may be confused. However, they can be distinguished by the Lyapunov exponent method.

8. MODIFIED INTERPOLATED CELL MAPPING METHOD

It is well known that di!erent initial conditions may lead to di!erent attractors when the governing di!erential equations are non-linear. Hence, how to determine which solution will occur for a given initial conditions is the major task. The attractors and corresponding basins of attraction of this system could be found by using the modi"ed interpolated cell mapping method (MICM) which improved from the interpolated cell mapping method [21]. For a two-dimensional system, the interesting rectangular region of dimensions n;n is divided into N;N cells with size h;h. The mapping through MIMC can then be expressed as

PM (xG#mh, yH#gh)"(1!m)(1!g)P(xG, yH)#m(1!g)P(xG>, yH)

#(1!m)gP(xG, yH>)#mgP(xG>, yH>), (8.1) where

0)m, g)1.

P is the approximate mapping through MICM, P(xG, yH) is the actual reference mapping of the point (xG, yH) by numerical integration and (xG,yH) is the co-ordinate of the center of the (xG, yH)th cell.

The wonderful phenomena are called fractal, and the boundary is called fractal basin boundary. In order to observe it, the structure of the fractal basin boundary is enlarged in Figure 7(a)}7(d). Hence, small uncertainties in initial conditions or other system parameters may lead to uncertainties in the consequence of the state of the non-linear system. Thus predictability is not always possible.

9. NON-FEEDBACK CONTROL METHODS

These methods modify the underlying chaotic dynamical system weakly so that stable solutions appear [21], which are arranged as follows

(1) Controlling of chaos by addition of constant torque. Interestingly, one can even add just a constant torque to control or quench the chaotic attractor to a desired periodic one in a typical non-linear system. It ensures e!ective controlling in a very simple way. Examining the e!ect of the constant torque, the added torque is assumed to be present in equation (2.7). Consider the e!ect of the constant torque by increasing it from zero upwards; the chaotic behavior is then modi"ed. In Figure 8(a), using constant torque control, chaotic behaviors become P-8T motion for constant torque 0.8.

(2) Controlling chaos by the second periodic torque: One can also control system dynamic by the addition of the external second period force in the chaotic state. Equation (2.7) with the second periodic torquecsinXq can be written as

bQ"b,

bQ"(ogd cos gq) bM (b)# ZM(b)aR!CMbQ

#(1#c sin gq) sin bG#csinXq. (9.1) Whenc3[1, 5] and XOg, the detailed structure of Lyapunov exponent versus c is shown in Figure 8(b).

Figure .7 (a) Five attractors coexisting (b)}(d) enlarged fractal diagram of diagram (a).

10. DELAYED FEEDBACK CONTROL

In this section the application of delayed feedback control [22] is introduced. The di!erence between the delayed output signalb(q!qB) and the output signal b(q) is used as a control signal:

F(q)"K [b (q!qB)!b(q)], (10.1)

where K is the weight of control signal andqB is the delay time. Adjusting K and qB, we can convert the chaotic motion to periodic motion or even quasi-periodic motion. Figure 8(c), where K"0.02 and qB"8n presents a period-4T periodic motion. Figure 8(d), where K"0.04 presents two quasi-periodic mapping diagrams.

11. ADAPTIVE CONTROL

Adaptive control algorithm was recently suggested [23, 24] for multi-parameter and higher-dimensional non-linear systems. This control mechanism is remarkably e!ective in

Figure 8. (a) PoincareH maps and phase portraits of non-feedback control of constant torque, (b) Lyapunov exponent for non-feedback control of periodic force, (c)}(d) delayed feedback control.

returning a system to its original dynamics after a sudden perturbation in the system parameter changes the dynamical behavior. This error signal governs the change of the parameter of the system, which readjusts so as to reduce the error to zero. For a general N-dimensional dynamical system

bQ,db

dq"F (b, q, k), (11.1)

where b,(b, b,2, b,) are variables and k,(k, k,2, k+) are parameters, which determine the nature of the dynamics, the prescription for e!ecting adaptive control is through the additional dynamics

kR"f (b!bQ), (11.2)

wherebQ is the desired steady state value and f indicates the sti!ness of control.

This algorithm is remarkably e!ective and rapid, and is of utility in a large variety of systems, ranging from biological units to control engineering. The e$cacy of this idea in application to discrete maps with a signal control parameter has been proved. The recovery time, de"ned as the time taken to reach the desired state within "nite precision after a shock was found to be inversely proportional to the sti!ness of control. Adaptive controlling can change chaos motion into periodic motion. The result is shown in Figure 9(a) and 9(b).

Figure 9. PoincareH maps and phase portraits of (a)}(b) adaptive control, (c)}(d) variable structure control.

12. VARIABLE STRUCTURE CONTROL

In this section, we present the basic principles of the variables structure control theory [25, 26] and their applications in robust control for a class for a class of non-linear oscillators with chaotic attractor. Let us consider the following non-linear oscillators:

bQG"bG>, i"1,2, n!1,

bQL"f (b, q)#u(q) (12.1)

whereb"[b bQ 2bL\]2 is the state vector. The non-linear function f (b, q) is unknown, but estimated as fK (b, q). The estimation error f (b, q) is assumed to be bounded by some known function F (b, q)

" f (b, q)!fK (b, q) "(F (b, q). (12.2) Assume that the aim of the control system is to track a given desired state vector bB"[bB bQB2bL\B ]2 in spite of model uncertainties. Let e"bB!b be the tracking error in the variableb. Furthermore, let us de"ne a time-varying surface S (q) in the state-space RL by the scalar equation

s (b, q)"0, (12.4)

s (b, q)"



L\

wherej is a strictly positive constant, will be called the sliding surface. For instance, if n"2,

s"eR #je. (12.6)

It is apparent also that system dynamics while on the sliding surface is determined not from di!erential equation (12.1) but from equation (12.4) and therefore it is invariant to the parameter variations and structure uncertainty. Initially, however, the state vectorb and desired vector bB do not generally coincide. Let us consider the Lyapunov function candidate

<(b)" s. (12.7)

The derivative of < (b) along the trajectories of the system is given by <Q (b)"(bGB!f!u#jeR)s.

The best approximation uL of a continuous control law that would achieve sR"0 is then

uL "!bGB#fK!jeR. (12.8)

The u_{L can be interpreted as our best estimate of the equivalent control. We add to} u_{L a discontinuous term across the surface s"0:}

uL "uL![F (b, q)#g] sgn(s), with g'0. (12.9) Finally, let us consider our dynamic system*a physical pendulum driven by two periodic external forces. The equations considered are

bQ"b,

bQ"(ogdcosgq)bM(b)# ZM(b)aR!CMbQ

#(1#c sin gq) sin b#u. (12.10)

The result is shown in Figure 9(c) and 9(d).

13. CONCLUSION

The dynamical system of the physical pendulum with damping subjected to two exciting torques exhibits a rich variety of non-linear behaviors as unequivocal parameters are varied. Due to the e!ect of non-linearity, regular or chaotic motions may appear. In this paper, analytical, computational methods and controlling of chaos have been employed to study the dynamical behaviors of the non-linear system.

The stability conditions for the physical pendulum system have been found by using the Lyapunov direct method. The IHB method uses fast-Fourier transform (FFT) and must be used by trying various initial conditions. Further, the existence of chaotic motion has been identi"ed by Melnikov's method.

The computational analyses have been performed. The bifurcation of the parameter-dependent system has been studied numerically. The time evolutions of non-linear dynamical system response have been described using the phase portraits via the PoincareH map technique. The occurrence and nature of chaotic attractors have been veri"ed by evaluating Lyapunov expenont and power spectra. Finally, global analysis of the basin boundary and fractal structure have been observed by the MICM method.

We have demonstrated that a simple control strategy can be e!ectively used to suppress chaos in a non-linear dynamical system. It is our hope that similar control strategies can be successfully implemented for more situations. By using a number of analytical or

computational methods, the non-linear behaviors of the physical pendulum like the di!erent types of periodic solutions, the e!ects on the solutions caused by di!erent parameters and initial condition, the stability analysis of solutions have been studied. In spite of the fact that these methods are di!erent, the results obtained match each

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