inverted pendulum-bang bang (k'nın degerleri var)

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Robotica (1996) volume 14. pp397-405. 1996Cambridg e University Press

Swing-up control of inverted pendulum systemsAlan Bradshaw and Jindi ShaoEngineering Department, Lancaster University. Lancaster, LAI 4YR UK)(Received in Final Form: December 11. 1995)

SUMMARYIn Part I a technique for the swing-up control of singleinverted pendulum system ispresented. The requirementisto swing-up a carriage mounted pendulum system fromits natural pendent position to its inverted position. Itworks forallcarriage balan cing single inverted pend ulumsystems as the swing-up control algorithm does notrequire knowledge of the system parameters. Com-parison with previous swing-up controls shows that theproposed swing-up control is simpler, eaiser. moreefficient, and more robust.

In Part II the technique isextended to the case of theswing-up control of double inverted pendulum systems.Use is made of a novel selective partial-state feedbackcontrol law. The nonlinear, open-loop unstable,nonminimum-phase. and interactive MIMO pendulumsystem is actively linearised and decoupled about aneutrally stable equilibrium by the partial-s tate feedbackcontrol. This technique for swing-up control is not at allsensitive to uncertainties such as modelling error andsensor noise,and is both reliable and robust.

KEYWORDS: Inverted pendulum: Nonlinear unstable syst-ems: Swing-up control: Partial-state'feedback.P ART I: SINGLE INVERTE D PENDUL UMSYSTEM

1. IntroductionInverted pendulum systems' are well known in controlengineering for verification and practice of control theoryand robotics. 1 Mo dern inverted pend ulum systemsextend the inverted pendulum's working range fromaround the upright vertical to the full 360 angular range.This introduces an initialisation problem, that is. toswing-up the pendulum from its natural pendent positionto its inverted position.

Two approaches to swing-up control have beenpresented previously. The first uses a feed-forwardbang-bang control.4 This technique is very sensitive tomodelling error and noise due to the nature offeed-forward controland is not a reliable technique.Thesecond uses a feedback bang-bang co nt rol / This wassuccessfully demonstrated on a rotating-arm singleinverted pendulum system without rail length restriction.It is not suitable for carriage balancing single invertedpendulum (CBSIP) systems in which the rail length isfinite.

In this paper, a new approach to swing-up control isproposed. This approach depends on a sound under-standing of the physical behaviour of CBSIP systems. Itis based on three observations: (1) when disturbed, thependulum's natural behaviour is a nonlinear vibrationaround its pendent vertical position (stable equilibrium)with a decreasing amplitude and a varying (increasing)circular frequency: (2) by increasing the pendulum'skinetic energy, the amplitude of the pendulum'svibrationcan be increased:and (3) the kinetic energycanbe increased by moving the carriage co-ordinatelyfollowing the rhythm of the pendulum's vibration.

The major principle of the proposed swing-up controlis to follow the pendulum's natural vibration, andgradually turn it from damped vibration into resonancein a controlled manner. This will make the pendulumswing higher and higher. When it isclose to the invertedvertical,a full-state feedback controller will take overthecontrol and stabilise the whole system including theinverted pendulum.The proposed techniquehas two major advantages:(1)robustness, it works for allCBSIP systems regardlessofthe vaues of their parameters, and (2) simplicity, it issimple and easy to tune. Furthermore, the principle ofthe technique can be applied to double and triplependulum systems,and robotics.72. Inverted pendulum systemAs schematically shown in Figure 1. a CBSIP systemconsists of a carriage and a pendulum atop the carriage.The carriage has mass Mc. and translational frictioncoefficient kr along the rail.The pendulum hasmass Mp,length Lp, rotational friction coefficient ko, and gravitycentre at C.Control of thesystemis by means of forceuapplied horizontally to the carriage, the outputs are thecarriage's position .vc. and the pendulum's angle 0(0 0Lpii-

x,. =

+IMpLpsin (0)02+ 6cos(9)kt,9 - l. pgsin (20)

(4/W,+ M;>+ 3M,,sin2 (6)]LP-6M PLP cos{9)u + 6MPLP cos {8)k,xe- I.SM-LJsin (20)0 2- \2Mk,,B+6 MMpLpg sin 9)

[4M,+Mp+3/V/,, sin2(0)]/V/,,L :,1)


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398 Inverted pendulum systems

Fig. 1. Carriage balancing single inverted pendulum system.

The nonlinear equation of motion (1) can be linearisedabou t 0 = 0. and setting M, = 4M(. + A/,, the result canbe expressed in state-space formx = Ax +Bi t (2)where

A =

1- 4 * ,M,0

kr

06*,,

MXLP1

-12MA:,,

B =

0_4_M,0- 6

In equation (2). x = [.rt. ve 0 w ]r where v,. is the speedof the carriage and w is the an gular velocity of thependulum. The nonlinear equations (1) are used forcomputer simulation of the system, and the linearisedequation (2) is used for linear controller design. Becausethe linearised system shown in equation (2) is unstablebut con trollab le, a full-state feedback co ntrol of theformu = kxxr - kx (3)

with the constant control vector A= [kx kr ke ka] exists,which will stabilise the whole system and achieve thecarriage's position control. In equation (3), xr is thecommanded carriage position, which is set to zero in thisstudy. The control vector k can be calculated by theLQR (linear-quadratic regulator) design method 8 withthe performance criterion

4)= (x T Qx + uRu) dt

where Q is the state weighting matrix which must besymmetric and positive semi-definite, and R is the controlweighting coefficient which must be positive. In orderto compare the swing-up control results in this paper

with previous results, the system's parameter specifica-tions a re take n from Mori s system:4 Me = 0.48 kg.A:, = 3.83 [N s/m ], Mp = 0.16 kg. Lp = 0.5m . and ko =0.00218 [Nm s/R ad]. With Q = / and rt = 0.02. theoptimum control vector is found to bek =[-7.071 -15 .73 -59 .59 -12 .70] 5)

Only if the initial pendulum angle 0(0) is close to theinverted vertical, can the full-state feedback controlshown in equation (3) with the control vector (5) balancethe pendulum and position the carriage. Evidently, theinitialisation of the pendulum is necessary, which is tobring the inverted pendulum up from its natural pendentvertical position.3. Swing-up controlThe nonlinear system represented by equation (1) has astable equilib rium at jr, = [.rt. 0 180 0]T, and anunstable equilibrium at x2 = [.r,. 0 0 0]T. Both of theseequlibrium points can be obtained with an arbitrarycarriage position xc (therefore the carriage's positioncon trol is possible). With out the contro l (z< = 0), thesystem's natural behaviour can be observed by releasingit from the initial state x(0) = [0 0 0(0) 0]7 where0(0) * 0 and 0(0) ^ 180. With 0(0) = 10 the numericalsolution of equation (1) is computed and shown in Figure2. The natural behaviour of the system is dampedvibration of both the pendulum and the carriage. Twoimportant factors can be observed from the system'snatural behaviour: (1) due to the translational and therotational friction, the amplitudes of both the pendulumand the carriage decrease with the time: and (2) with thedecreasing amplitudes, the circular frequencies of thevibrations increase with the time, which is a nonlinearcharacteristic.

In order to swing-up the pendulum towards theinverted vertical, its kinetic energy level must beincreased. In this system, this can only be achieved byapplying force to the carriage, and this force must at least

Pendulum Angle (degree)

-0.063 4 5 6 7Time: 0-10 seconds

Fig. 2. Natural behaviour of carriage balancing single invertedpendulum system.


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Inverted pendulum systemsbe large enough to overcome the friction. The directionand the amplitude of the input force must always becontrolled so as to increase the pendulum's kineticenergy, despite the difficulties that (1) the force u appliesto the pendulum indirectly through the carriage. (2) thevibrations of the pendulum and the carriage arenonlinear, and (3) there is a phase-lag between these twovibrations. Failure to do so will at best lead to energyloss and make the initialisation unnecessarily longer andat worst lead to aforced vibration of the pendulum at aconstant amplitude and make the initialisation impos-sible. When the total energy of the pendulum equals PE,where

PE = M,,LPg (6 )that is. the pendulum's potential energy at the invertedvertical, the pendulum will be swung-up to its uprightposition. It is shown in this paper that the easiest andmost efficient way to achieve this is to apply the force atthe frequency of the pendulum's nonlinear vibration, soas to drive the pendulum into a resonance in a controlledmanner. This leads directly to the swing-up control law.as linear feedback of w according t o the equ atio n:

, = ks (7)Clearly, to start the swing-up control, a nonzero initialw(0) is required. This can be obtained either by releasingthe pendulum from a small angle away from the pendentvertical, or by applying a force to the carriage for a shorttime period before switching to the swing-up control. Tosimplify the discussion, the pendulum is released at[0(0) = 165 w(0) = 0j.

4. Tuning of swing-up controlThe tuning of the sole constant k, in equation (7) is easilyachieved by starting from a small value and graduallyincreasing it. either in a computer simulation or in a reallaboratory experiment. There exists a boundary kh of ks.which is determined by the friction and the parameters of

399the system. The relationship between ks an d kh is asfollows: (1) if k, kh then us is strong enough to swing-up thependulum towards the inverted vertical.

By substituting equation (7) into the nonlinearequation (1). kh could be obtained analytically, howeverthis is both difficult and unnecessary. The value of kh canbe found by numerical solution of equations (1) and (7)during the tuning of the swing-up control. For example,it is found that kh = 0.31 for the parameter values used inthis paper. The system's time responses with kx = 0A.0.31, and 0.4 are shown in Figure 3. Once ks>kh, thependulum will be swung-up towards the upward verticalby the control it, given by equation (7).The tuning of k, is judged by two factors, the requi red

rail space .vr, and the required swing-up time r, at whichthe pen dulum is close to the inverted vertical (no rmallywithin 10). Generally, increasing k, leads to decreasingts and increasing xn. For example, a swing-up time/., < 2 s can be achieved by choosing ks = 2 . The results ofthe swing-up control shown in Figure 4. in which thependulum is successfully swung-up towards the invertedvertical, are obtained with k, = 2. In Figure 4. the control., is limited by a maximum force m.,v = 10.8 N. whichwill be discussed shortly.5. Switching conditionsThere are three important factors which must beconsidered when switching between the swing-up controland the full-state feedback control. First, the closed-loopsystem with the full-state feedback control expressed inequation (3) has a restricted stable state region aroundthe origin, because the controller design is based on thelinearised system represented in equation (2). Second,the final states of the swing-up control must not force the

0.05

o .oi

Pendulum Angles (degree)

0.5 1.5 2 2.5 3 3.5Carriage Positions

4.5

4.5 5.5 2 2.5 3 3.5Time: 0 - 5 secondsFig. 3. The vibrations of the carriage balancing single invertedpendulum system.


0.2 0.4 1.4 1.6.6 0.8 1Time: 0-1 .6 secondsFig. 4. Swing-up c ontro l of the carriag e b alancin g singleinverted pendulum system.


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400 Inverted pendulum systemscarriage beyond the available rail space. Third, duringthe swing-up control the pendulum should not beallowed to go over the top. Once it crosses the upwardvertical, the pendulum will perform a nonlinear rotationaround its pivot with a nonzero angular velocity. In thissituation, even if the system is switched when thependulum is vertical, it is most likely that either thesystem is outside the stable region, or the full-statefeedback control runs out of rail space.

To prevent the pendulum from crossing the upwardvertical, its total energy must be less than PE as given byequation (6) (Clearly a larger Mp or Lp leads to a largerPE, and hence a larger force or a longer switching time.).This can be achieved by saturating the control to amaximum uma x. If the pendulum starts rotating, therotation can be controlled back to a vibration byresetting uma x.During swing-up control, the best switching moment iswhen the pendulum is close to the upward vertical and atits maximum angle of vibration, then, the system's statesare nearest to the equilibrium condition. Although .r (. isat its maximum position, it does not affect the system'sstability. vc is close to zero (there is a phase lag betweenthe vibrations of the carriage and the pendulum). \9 \is atthe minim um (0 = 360 also stand s for the invertedvertical), and w = 0. This case can be observ ed in Figure5 which shows the states of the swing-up control withks = 2 and uma x =10.8 N.

Moreover, when close to the upward vertical, thefull-state feedback control uf can quickly stabilise thependulum (small 9 and w = 0) a nd then drive thecarriage towa rds the middle of the rail (.r, = 0), so thatthe recovery does not require much extra rail length.Thus, the switching conditions ate \9\^9S and w = 0,where the boundary 9S is determin ed mainly by theconstants of the full-state feedback control. Generally,9Scan be set to 10 to ensure sm ooth sw itching. Ho wev er,the results shown in Figure 5 indicate that the system'sbest switching time is ts= 1.58 s, at which time 9 =17.5.Even with this larger angle (9> 10), the system can be

stabilised easily and smoothly, by switching on thefull-state feedback contro l. The com plete swing-upcontrol of the system is computed and shown in Figure 6.The flow chart of a computer algorithm thatautomatically tunes and switches the swing-up control isshown in Figure 7. In the algorithm, H is the step sizewith which to increase ks from zero, and h is the step sizewith which to decrease m ax from the real maximumforce of the system.Refer to Figure 7, when the switch conditions are met.the full-state feedbac k co ntrol is switched on to com plet ethe swing-up control. If the pendulum starts rotating[9 = 0), uma s will be decreased by h and the experimentor simulation will start again. On the other hand, if theswitching conditions are not met by t= ts, ks will beincreased by H and the experiment or simulation willstart again.

6. Discussion of swing-up controlThe results of the swing-up control shown in Figure 4 arealmost identical to those in Mori's nonlinear optimalswing-up control.4 Mori's approach uses a bang-bangtype feed-forward control, this is very sensitive tochanges of the parameter values, the experimentconditions, and noise. Also the recovery can be difficult,as the system might not be at the desired state at the endof the preset control sequence. On the other hand, theapproach in this paper uses a linear feedback controlequa tion (7). which does not require detailed knowledgeof the system. As the system is switched at a state nearthe equilibrium, the whole control is smooth and reliable.Therefore, the proposed swing-up control approach issimple and robust.

Based on a rotating-arm balancing inverted pendulumsystem. Furuta's approach used a feedback control 36 an dwas therefore more robust than Mori's approach.However, as it was still of the bang-bang type, the resultwas not as smooth as the result of the approachpresented in this paper. Furthermore, Furuta 's approachwas not as simple as that presented in this paper.Because the switching of the bang-bang control was


-120 0.2 0.4 0.6 0.8 1 1.2 f.4 1.6Time: 0 - 2 seconds 1.8Fig. 5. Com plete respon ses of the carriage balanc ing singleinverted pendulum system.

4 5 6 7Time: 0 -10 seconds 10Fig. 6. C om plete initialisation process.


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Inverted pendulum systems 401

x 0) -fO Initialisation0 165 0],ix, ti, h, 11, fc/i

* , = 0,

k,=k, H

t = 0

Wait till / - r , Experiment orSimulation

Switching to the full-state feedback control

Fig. 7. The Algorithm of the swing-up control.

based on the phase plane of (dw). knowledge of thesystem param eter values w as requried in Furuta sapproach. Finally unlike CBSP systems, rail length is notan issue in the rotating-arm type of inverted pendulumsystem.PART II: DOUB LE INVER TED PEND ULUMSYSTEM/ . IntroductionExcept for the case of swing-up control. 46 nearly allprevious theoretical and'experimental studies of invertedpendulum systems have been based on linear analyses.All reported double pendulum systems, such a s .y u l havebeen single-input systems. The only two MIMO(multi-input multi-output) pendulum systems were bothtr iple pendulum systems. 1 2 Although these two systemswere able to achieve nonlinear pendulum positioncontrol, they were both limited to small perturbationsfrom the vertical equilibrium position, and so wereessentially examples of linear control systems.

The failure to extend inverted pendulum systemstudies into large amplitude, nonlinear regions is mainlydue to the fact that a nonlinear unstable pendulumsystem is extremely difficult to balance and control.Furthermore, at interactive MIMO pendulum systemalso requires decoupling. With the uncertainties ofmodelling error and noise, decoupling is difficult, even inlinear unstable systems.In nonlinear MIM O doub le pendulum systems, achallenging control object is to swing-up the double

pendulum from its natural pendent position to theinverted position. As these systems are nonlinear,open-loop unstable, nonminimum-phase. and interactive,it is impossible to achieve swing-up control by using alinear full-state feedback control law. As a result,swing-up control of double pendulum system has notbeen studied previously in the literature.For nonlinear, open-loop unstable, and interactingMIMO systems, full-state feedback is not always the bestsolution. In fact, by a suitable partitioning of thestate-space, it is possible to achieve several controlobjectives. Indeed, a new controller design strategybased on a selective partial-state feedback controlapproach is presented in this paper. A special feature ofthe approach is that it can actively decouple and lineariseinteracting nonlinear unstable MIMO systems. Further-more, as the partial-state feedback control itself is linear,linear control theory can be used for system analysis andcontroller design.

Based on both the partial-state feedback control andthe single pendulum swing-up control presented in Part I,the swing-up control of a double pendulum system isachieved successfully in this study. The theoreticalstudies and simulations in this paper show that the oncecomplicated control problem, which is associated withnonlinearity and interaction, is solved in a simple andeasy manner. Unlike conventional linearisation anddecoupling which are associated with fixed operatingpoints, the double pendulum is actively linearised anddeco upled ab out its neutrally stable equilibrium by thepartial-state feedback control. This makes the doublependulum swing-up control very reliable and robust.2. S elective partial-state feedback controlTo date, the most common procedure for stabilising andcontrolling a nonlinear unstable MIMO system is: (1)linearise the system at its operating point which must bean equilibrium state of the system. (2) decouple,stabilise, and control the linearised system by usingfull-state feedback, and (3) use the resulting linearcontroller to control the original nonlinear systemaround the operating point. As linear control theory hasbeen well developed, the above procedure often workssuccessfully.There are three major disadvantages of the aboveprocedure which are now discussed,(i) The resulting linear controller can only stabilise thenonlinear system around the operating point. Tocover a wide range of the state-space, the system hasto be linearised at several operating points. Also thelinear controller design has to be carried out at eachoperating point. When the system works over largeregions of the state-space, the number of requiredoperating points increase sharply,(ii) The decoupling control is based on the linearisedmodel of the nonlinear system, and is therefore verysensitive to the system's modelling error and noise.

Moreover, unwanted steady-state errors are intro-duced w hen the system is stabilised betw een theoperating points.


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402 Inverted pendulum systems(iii) The idea of stab ility and full-state feedb ackhave greatly restricted the search for better control,as control engineers are used to think in theseterms. For example, the first double pendulumsystem was built more than 30 years ago in 1963.1*itscontrol system was designed using a linearisedmodel and full-state feedback. Moreover, there hasbeen no significant progress in the control of doublependulum systems since.

A system is defined as unstable if some of its outputincrease without a boundary or outside a presetboundary when time tends to infinity. However, manyreal controls are only required temporarily, such asthe initialisation of an inverted pendulum. By usingthe concept of BIBS (bounded-input bounded-state)stability. the behaviou r of those unstable outp uts isoften well understood and predictable. For a nonlinearunstable MIMO system with some equilibrium positions,instead of linearising and decoupling the nonlinearsystem around some operating points, partial-statefeedback could be used directly to decouple, linearise,and stabilise the system at the equilibrium. A nonlinear,unstable, and interactive MIMO system with linearsub-state xt and nonlinear sub-state x,, can be expressedin the partitioned state-space form

x 1 ffix x u u )xj Lfll(Xi.x l,,u,.ul,)\where u, is the control ofx, an d u, , is the control of x,,. Ifequation ( ) has a neutrally stable equilibrium [xu. x,,e]T,then it is possible to find a partial-sta te feedback

un = rH-K nxH (9)to stabilise the subsystem of x, , at xm. In equation (9).r,


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Inverted pendulum systems 403pendulums can be positioned at an arbitrary angle,however the other pendulum must meet equation (13) tokeep both of the pendulums balanced: and (3) a nonzerosteady-state torque Teis required to keep the pendulumsat the required angles. Equation (13) defines a neutrallystable equilibrium, as the carriage position and one ofthe pendulum angles can be arbitrary. It can be seenfrom Figure 8 that the maximum moment of the higherpendulum about O occurs at j3 = - 90 . Furth er, it can beseen from equation (13) that, in order to balance thependulums, the maximum lower pendulum angle is

(14)which is defined by the pendulums' lengths and masses.It follows from equations (12) and (13). in order toachieve the full 360 angle pendulum control, that

0.5Mp] + Mp2)L rl1 (15)

For demonstration purposes, the following valuesare assigned to the system parameters: Mt = 0.5 kg.k,.= 3.8 [N s/m ]. M,,, = 0 .1 kg. L,,, = 0.25 m. A:,,, =0.002 [Nms/Rad]. M,r = 0.1 ke. Lr =0.25 m. A> =0.002 [Nms/Rad]. and g = 9.8 [mis1].4. Swing-up controlThe natural behaviour of the nonlinear double pendulumsystem can be computed and observed from equation(12) by setting U = T = 0 and x (0 ) = [0 0 10 0 10 O]7 ,as shown in Figure 9. The carriage and the pendulums allperform damped nonlinear vibrations. At the beginningthe two pendulums vibrate differently, therefore theinteraction between the pendulums makes the system'smotion complicated. However, the pendulums tend tovibrate at the same circular frequency, and the vibrationsare towa rds the stable equ ilib rium (0 = 180 /3 = 180).The system's responses are much smoother when thependulums vibrate at the same frequency.

Clearly, in order to swing-up the two pendulums, the

higher pendulum has to follow the lower pendulum as itswings up towards the inverted vertical. Therefore, theangle (/3 - 8) between the pendulums has to becontrolled. Because (1) the pendulum system's state-variables can be partitioned asA\ , = [9 6 (16)

and (2) there is a neutrally stable equilibrium describedby equation (13). the partial-state feedback expressed inequation (9) exists in the double pendulum system with# = Trs.K,, = [-K,, -Ky Kp KY], and u, ,= Tr.'where

By substituting equation (17) into equation (12). thenonlinear system's steady-state responses under thepartial-state feedback are given byP, sin (0) + A s n (ft) = 0 (IS)

Comparison of equations (13) and (18) shows that Trcan balance the pendulums at the equilibrium angles 0,.and ft.. Therefore, the angle between the pendulums iscontrolled, and the whole nonlinear system is bothlinearised and decoupled actively. For example, bysetting K0 =10. Kv = 5. and 9, = 10c. it follows fromequation (IS) that ft. = -31.40 and Tr= -7.162. If thenonlinear system described by equation (12) is releasedfrom .v(0) = [ 0 0 0 0 10c 0] ' . ' with U =0 and T= 7 ,, thesystem's time responses can be computed and are shownin Figure 10. Altho ugh the pend ulum s vibrate towardsthe stable equ ilibri um (0,. + 180 ft. + 180). the angle(13 - 9) between the pendulums is balanced at(ft. - 9C)= -41.40 . With the control Tr. the doublependulum can be safely regarded as an equivalent singlependulum, which can be described by the lowerpendulum's state xtl = [9 9]T.Obviously, for the swing-up control of the doublependulum system, in equation (17) 9C= 0. and thereforeit follows from equation (18) that ft. = Tr=0. M oreover,from the single pendulum swing-up control presented in

Carriage Position

3 4 5 6 7Pendulum Angles (degree)

4 5 6 7Time: 0- 10 seconds 10Fig. 9. Natural behaviour of the pendulum system.

Pendulum Angles (degree)

40 L

1 : 3 4 5 6 7Angle between the Pendulums (degree)

10

p-e4 5 6 7Time: 0-10 secondsFig. 10. Angle control between the pendulums.


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404 Inverted pendulum systemsPart I (Section 3), it can be seen that the control Us toswing-up the double pendulum is

(19)where Ks is the controller constant, and max is themaximum force. Obviously, Uxis a positive partial-statefeedback. The swing-up control tuning process describedin Part I (Section 4) gives the result that Ks = 4.\ an dmux =10.6 N. Substitution of eq uatio ns (17) and (19)into equation (12) gives the swing-up control of thedouble pendulum system which has been computed andis shown in Figure 11.

As shown in Figure 11, both of the pendul ums areswung-up from (around) the pendent vertical to near theinverted vertical. The best switching time is at ts= 2.07 swith 0( 0= 11 .0 3 and ^ ( O = 10.78. Therefore, theoriginally nonlinear swing-up control of the doublependulum system has been successfully achieved by usingtwo simple linear partial-state feedback control lawssimultaneously.5. Stabilisation of the systemFollowing the discussion in Section 2, after the swing-upcontrol has been accomplished, the whole system can bestabilised by using equation (17) and another partial-state feedback Uh

Uh = Ur- Kxxc - KJC - Kg6 - Kj (20)where Kx, K v, Kg, an d Kw are the controller constants,U r is the reference input. The purpose of Ur is to effectthe desired carriage and pendulum positions in thesteady state. It can be deduced by consideration of thesteady state behaviour of the system obtained whenU = Uh in equation (12). Thus

U,= Kxxr + Ke6r (21)where xr is the commanded carriage position and 6ris thecommanded lower pendulum angle. In equation (21)Kg6r is for remov ing the interaction from the do uble

pendulum to the carriage. Comparing equations (20) and(21) with equation (10). it can be seen that K, = [Kx Kv),KM =[KB K J . r,= Kxxr, an d r,n=Kedr As the non-linear double pend ulum system is dynamically co nvertedinto an equivalent linear single pendulum system by Tp,the controller constants of equation (20) can be obtainedby following the design procedure of the single pendulumsystem in Section 2 (Part I). The e quiv alent singlependulum has the parameters Mp =0.2 kg, L, , = 0.5 m,and k,, = 0.002 [Nms/Rad]. Setting Q = I and R = 0.01 inthe LQR design,8 the controller constants are obtainedas# , = [ -10 -19 .96] KM =[-78.74 -17.20J (22)

Using the p artial-state, feedback equatio ns (17) and(20) together, asymptotically stable nonlinear positioncontrol of the double pendulum system can be achieved.By switching the control from equations (17) and (19) toequation s (17) a n d ' (20) at /, = 2.07 s, the doublependulum system is initialised and then stabilised, asshown in Figure 12.Supported by the robust partial-state feedback controlapproach, the swing-up control can also be achieved withan angle between the pendulums. With 0 r = l O ,/3 , =-3 1 .4 0 , 7 ; =- 7 . 16 2 , ATV = 4.1, and /m a x = 1 0 . 3 N ,the partial-state feedback equations (17) and (19) canswing-up the pendulums towards 6r and jSr from0(0) = /3(0) = 180. The best sw itching time is found atr, = 1.606 s, with jc t. (O = - 0 . 8 m, 0 ( 0 = 24.69, and/3(O = -14.8 7. Afterwards, with xr =Q an d Ur=-13.74, equations (17) and (20) can stabilise the wholesystem at the required positions: xr = 0, 0r= 10, and

/3 r~ -31.40. The complete swing-up control is shown inFigu re 13. Th e large angle /3 r- 6,= -41 .40 is achievedand kept throughout the whole initialisation process.PART III: CONCLUSIONA practical and robust swing-up control approach ispresente d in Part I. The robust approach gives a smoothperformance, and is able to bring a pendulum up from its

Carriage Position

I 1.5Pendulum Angles (degree)

Carriage Position

1 1.5Time: 0 - 2.5 seconds

2 3 4 5Pendulum Angles (degree)

2 3 4Time: 0-7 secondsFig. 11. Swinging-up the pendulums. Fig. 12. Repositioning the double pendulum system.


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Inverted pendulum systemsCarriage Position

3 5 6 7Pendulum Angles (degree)

10

3 5 6 7T i m e :0 - 1 0 seconds 10Fig. 13. Pendulum control with a large angle.natural pendent position and balance it in the invertedposition. It works for all CBSIP systems regardless of theparame ter values. The principle employed in theapproach can be used to swing-up a double or triplependulum. Furthermore, the principle can also beapplied to new kinds of artificial intelligent machines.7

As presented in Part II, the selective partial-statefeedback control approach has been proved extremelysuccessful in the nonlinear, unstable, and interactivedouble pendulum system. It can also be applied to triplependulum systems for both swing-up control andnonlinear pendulum position control. Indeed, partial-state feedback control can be applied with advantage to awide range of nonlinear, unstable systems. All that isrequired is that the system equations can be partitionedas in equation (8). Many machines and processes aregoverned by such equations. In particular, many

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dynamical models used in the study of legged robots 21are governed by partitionable, nonlinear systems ofequations.References

1. J. Shao, Dynam ics & Nonlinear Control of UnstableInverted Pendulum Systems PhD Thesis (EngineeringDepar tment of Lancaster University, Britain, 1995).2. D.J.Todd, Walking Machines-an Introduction to LeggedRobots (Kogan Page Ltd, London, 1985).3. M.H. Raibert, Legged Robots that Balance (The MITPress.London, 1986).4. S.Mori,H.Nishiharaand K.Furuta, Contro l of UnstableMechanical System, Control of Pendulum Int. J. Control23 , No. 5, 673-692 (1976).5. K. Furuta, M. Yamakita and S. Kobayashi, Swing UpControl of Inverted Pendulum IECON 9/ (1991) pp .2193-2198.6. K. Furuta, M. Yamakita, S. Kobayashi and M. Nishimura A New Inverted Pendulum A pparatu s for EducationIF AC Advances in Control Education, Massachusetts,USA (1991) pp.133-138.

7. A.Bradshaw, D.W. Sewardand F.N. Nagy Robo Sapiens:a Personal Assistant Robot Proc. of Joint Hungarian-British Mechatronks Conf., Budapest, (Sept. 1994) pp.159-164.8. W.L.Brogan Modern Control Theory (Prentice-Hall,NewJersey, 1991).9. D.T. Higdon and R.H. Cannon On the Control ofUnstable Multiple-Output Mechanical Systems ASMEWinter Annual Meeting, Philadelphia (November, 1963)paper No. 63-WA-148.10. K. Furuta, H.Kajiwara and K.Kosuge Digital Controlofa Double Inverted Pendulum on an Inclined Rail Int. J.Control 32, No.5. 907-924 (1980).11. K. Furuta, T. Ochiai and N. Ono Attitude Control of aTriple Inverted Pendulum Int. J. Control 39, No. 6,1351-1365(1984) .12. H.M.Z. Farwig and H. Unbenauen Discrete ComputerControl of a Triple Inverted Pendu lum Optimal ControlApplications andMethods 11, 157-171 (1990).

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inverted pendulum-bang bang (k'nın degerleri var)