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Stabilization of the Furuta pendulum around itshomoclinic orbitIsabelle Fantoni & Rogelio LozanoPublished online: 08 Nov 2010.

To cite this article: Isabelle Fantoni & Rogelio Lozano (2002): Stabilization of the Furuta pendulum around itshomoclinic orbit, International Journal of Control, 75:6, 390-398

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Stabilization of the Furuta pendulum around its homoclinic orbit

ISABELLE FANTONI{* and ROGELIO LOZANO{

We present an energy based control approach to control the Furuta pendulum. A controller is proposed for swinging thependulum and raise it to its uppermost unstable equilibrium position. The passivity properties of the system are used as aguideline in the control strategy. The stability analysis is carried out by using a Lyapunov technique.

1. Introduction

The inverted pendulum is a very popular experience

used for educational purposes in modern control theory.

The structure of the conventional inverted pendulum isthe rail-cart type which consists of a cart running on a

rail and a pendulum attached to the cart. The inverted

pendulum of this type has the movement limitation of its

cart as a restriction of the control system. On the other

hand, the Furuta pendulum has a di� erent structure. It

has a direct-drive motor as its actuator source and itspendulum attached to the rotating shaft of the motor.

This inverted pendulum on the rotating arm was ®rst

developed by K. Furuta at the Tokyo Institute of

Technology. The experiment is called the TITech pen-

dulum (see Futura et al. 1992, Yamakita et al. 1995,

Iwashiro et al. 1996).

Since the angular acceleration of the pole cannot becontrolled directly, the Furuta pendulum is an under-

actuated mechanical system. Therefore, the techniques

developed for fully-actuated mechanical robot manipu-

lators cannot be used to control the Furuta pendulum.

Furuta et al. (1992) proposed a robust swing-up con-

trol using a subspace projected from the whole statespace. Their controller uses a bang±bang pseudo-state

feedback control method.

Yamakita et al. (1995) considered di� erent methods

to swing up a double pendulum. One is based on an

energy approach and another one is based on a robust

control method.

Iwashiro et al. (1996) considered a golf shot with arotational (Furuta) pendulum using control methods

based on an energy approach.

Olfati-Saber (1999) proposed a semi-global stabiliz-

ation for the rotational inverted (or Furuta) pendulum

using ®xed point controllers as for the cart-pole system.

Then he introduced new cascade normal forms for

underactuated mechanical systems (Olfati-Saber 2000).The main bene®t of this transformation was to reducethe overall system to control a lower order non-linearsubsystem in the normal form. He illustrated his resultwith the example of the rotational pendulum. Contraryto the technique proposed here, the magnitude of thecontrol input in his scheme increases as the initial stateis further from the origin.

The stabilization algorithm proposed here is anadaptation of the technique presented in the work ofLozano et al. (2000) which deals with the inverted pen-dulum. We will consider the passivity properties of theFuruta pendulum and use an energy based approach toestablish the proposed control law. The control algor-ithm’s convergence analysis is based on Lyapunovtheory.

In } 2, we present the model of the Furuta pendulumobtained using Euler±Lagrange equations. We alsoestablish its passivity properties. The control law isdeveloped in } 3 and the stability analysis of theclosed-loop system are given in } 4. Simulations are pre-sented in } 5 and conclusions are ®nally given in } 6.

2. Modelling of the system

The Furuta pendulum is di� erent from the conven-tional cart-pole inverted pendulum. The Furuta pendu-lum requires less space and has fewer unmodelleddynamics owing to a power transmission mechanism,since the shaft around which the pendulum is rotatedis directly attached to the motor shaft. The coordinatesystem and notations are described in ®gure 1. We willassume that friction is negligible.

2.1. Energy of the system

The total energy of the system is the sum of thekinetic energy K and the potential energy P of the armand the pendulum.

2.1.1. The arm. The kinetic energy of the arm is givenby

K0 ˆ 12I0

_³³20 …1†

International Journal of Control ISSN 0020±7179 print/ISSN 1366±5820 online # 2002 Taylor & Francis Ltdhttp://www.tandf.co.uk/journalsDOI: 10.1080/0020717011011226

INT. J. CONTROL, 2002, VOL. 75, NO. 6, 390 ±398

Received 1 September 2000. Accepted 1 November 2001.Communicated by Professor H. Sira-Ramirez.

* Author for correspondence. e-mail: Isabelle.Fantoni@hds.utc.fr

{ Heudiasyc, UTC, UMR CNRS 6599, BP 20529, 60205CompieÁ gne, France.

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Its potential energy is null, since no gravitational forcesact on the horizontal arm.

2.1.2. The pendulum. The kinetic energy of the pendu-lum is given by

K1 ˆ 12J1

_³³21 ‡ 1

2m1

d

dt…L0 sin ³0 ‡ l1 sin ³1 cos ³0†

» ¼µ 2

‡ d

dt…L0 cos ³0 ¡ l1 sin ³1 sin ³0†

» ¼2

‡ d

dt…l1 cos ³1†

» ¼2#

…2†

where the ®rst term corresponds to the kinetic energydue to the angular velocity of the pendulum while thelast three terms are due to the tangential velocity, theradial velocity and the vertical velocity of the pendulumrespectively. After some simple computations K1

reduces to

K1 ˆ 12J1

_³³21 ‡ 1

2m1L2

0_³³20 ‡ 1

2m1l

21

_³³21

‡ 12m1l

21 sin2 ³1

_³³20 ‡ m1L0l1 cos ³1

_³³0_³³1

…3†

Its potential energy is given by

P1 ˆ m1gl1…cos ³1 ¡ 1† …4†

2.2. Euler±Lagrange dynamics equations

The equations of motion can be obtained using anEuler±Lagrange formulation

d

dt

@L

@ _³³i

³ ´¡ @L

@³i

ˆ Fi …5†

where L ˆ K ¡ P, K ˆ K0 ‡ K1 and P ˆ P1. We have

@L

@ _³³0

³ ´ˆ ‰I0 ‡ m1…L2

0 ‡ l21 sin2 ³1†Š _³³0 ‡ m1l1L0 cos ³1_³³1

@L

@³0

³ ´ˆ 0

@L

@ _³³1

³ ´ˆ m1l1L0 cos ³1

_³³0 ‡ ‰J1 ‡ m1l21 Š _³³1

@L

@³1

³ ´ˆ m1l

21 sin ³1 cos ³1

_³³20 ¡ m1l1L0 sin ³1

_³³1_³³0

‡ m1gl1 sin ³1

and thus the system is given by

½ ˆ ‰I0 ‡ m1…L20 ‡ l21 sin2 ³1†Š �³³0 ‡ m1l1L0 cos ³1

�³³1

‡ m1l21 sin…2³1† _³³0

_³³1 ¡ m1l1L0 sin ³1_³³21 …6†

0 ˆ m1l1L0 cos ³1�³³0 ‡ ‰J1 ‡ m1l

21 Š �³³1

¡ m1l21 sin ³1 cos ³1

_³³20 ¡ m1gl1 sin ³1 …7†

In compact form, the system can be written

D…q† �qq ‡ C…q; _qq† _qq ‡ g…q† ˆ F …8†

where

q ˆ³0

³1

" #

D…q† ˆI0 ‡ m1…L2

0 ‡ l21 sin2 ³1† m1l1L0 cos³1

m1l1L0 cos³1 J1 ‡ m1l21

2

4

3

5 …9†

C…q; _qq† ˆ12m1l21 sin…2³1† _³³1 ¡m1l1L0 sin ³1

_³³1 ‡ 12m1l

21 sin…2³1† _³³0

¡12m1l

21 sin…2³1† _³³0 0

2

4

3

5

…10†

g…q† ˆ0

¡m1gl1 sin ³1

" #and F ˆ

½

0

" #…11†

Note that D…q† is symmetric and also

d11 ˆ I0 ‡ m1…L20 ‡ l21 sin2 ³1† …12†

¶ I0 ‡ m1L20 > 0 …13†

and

det…D…q†† ˆ …I0 ‡ m1…L20 ‡ l21 sin2 ³1††…J1 ‡ m1l21†

¡ m21l21L2

0 cos2 ³1 …14†

ˆ …I0 ‡ m1l21 sin2 ³1†…J1 ‡ m1l21† ‡ J1m1L

20

‡ m21l21L2

0 sin2 ³1 > 0 …15†

Stabilization of the Furuta pendulum 391

Figure 1. The Furuta pendulum system. I0, inertia of thearm; L0 , total length of the arm; m1, mass of the pen-dulum; l1, distance to the centre of gravity of the pen-dulum; J1, inertia of the pendulum around its centre ofgravity; ³0, rotational angle of the arm; ³1, rotationalangle of the pendulum; and ½ , input torque applied onthe arm.

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Therefore D…q† is positive de®nite for all q. From (9) and(10) it follows that

_DD…q† ¡ 2C…q; _qq†

ˆ0 m1l1…¡l1 sin…2³1† _³³0 ‡ L0 sin ³1

_³³1†

m1l1…l1 sin…2³1† _³³0 ¡ L0 sin ³1_³³1† 0

" #

…16†which is a skew-symmetric matrix. This constitutes animportant property which will be used in establishingthe passivity of the Furuta pendulum

zT… _DD…q† ¡ 2C…q; _qq††z ˆ 0 8z …17†

The potential energy of the system is de®ned asP ˆ m1gl1…cos ³1 ¡ 1†. Note that P is related to …g…q†† as

g…q† ˆ @P

@qˆ

0

¡m1gl1 sin ³1

" #…18†

2.3. Passivity properties of the Furuta pendulum

The total energy of the system is given by

E ˆ K…q; _qq† ‡ P…q†

ˆ 12_qqTD…q† _qq ‡ m1gl1…cos ³1 ¡ 1†

…19†

Therefore, from (8)±(11) and (16)±(18) we obtain

_EE ˆ _qqTD…q† �qq ‡ 12_qqT _DD…q† _qq ‡ _qqTg…q†

ˆ _qqT…¡C…q; _qq† _qq ¡ g…q† ‡ F† ‡ 12_qqT _DD…q† _qq ‡ _qqTg…q†

ˆ _qqTF ˆ _³³0½

…20†

Integrating both sides of the above equation we obtain…t

0

_³³0½dt ˆ E…t† ¡ E…0† ¶ ¡2m1gl1 ¡ E…0† …21†

Therefore the system having ½ as input and _³³0 as outputis passive. Note that for ½ ˆ 0 and ³0 2 ‰0; 2º‰ the system(8) has a subset of two equilibrium set of points. …³0, _³³0,

³1, _³³1† ˆ …¤; 0; 0; 0† is an unstable equilibrium set ofpoints and …³0; _³³0; ³1; _³³1† ˆ …¤; 0; º; 0† is a stable equilib-rium set of points. The total energy E…q; _qq† is equal to 0for the unstable equilibrium set of points and to¡2m1gl1 for the stable equilibrium set of points. Thecontrol objective is to stabilize the system around itsunstable equilibrium point …³0; _³³0; ³1; _³³1† ˆ …0; 0; 0; 0†,i.e. to bring the pendulum to its upper position andthe arm angle to zero simultaneously.

3. Stabilization algorithm

Let us ®rst note that in view of (19), (9) and (10), if_³³0 ˆ 0 and E…q; _qq† ˆ 0 then

12…J1 ‡ m1l21† _³³2

1 ˆ m1gl1…1 ¡ cos ³1† …22†

The above equation de®nes a particular trajectory whichcorresponds to a homoclinic orbit. Note that _³³1 ˆ 0 onlywhen ³1 ˆ 0. This means that the pendulum angularposition moves clockwise or counter-clockwise until itreaches the equilibrium point …³1; _³³1† ˆ …0; 0†. Thus ourobjective can be reached if the system can be brought tothe orbit (22) for _³³0 ˆ 0, ³0 ˆ 0 and E ˆ 0. Bringing thesystem to this homoclinic orbit solves the problem of`swinging up’ the pendulum. In order to balance thependulum at the upper equilibrium position the controlmust eventually be switched to a controller which guar-antees (local) asymptotic stability of this equilibrium(Spong 1994). By guaranteeing convergence to theabove homoclinic orbit, we guarantee that the trajectorywill enter the basin of attraction of any (local) balancingcontroller. We do not consider here the design of thebalancing controller in this paper.

The passivity property of the system suggests us touse the total energy E in (19) in the controller design.Since we wish to bring to zero ³0, _³³0 and E, we proposethe Lyapunov function candidate

V…q; _qq† ˆ kE

2E…q; _qq†2 ‡ k!

2_³³20 ‡ k³

2³2

0 …23†

where kE , k! and k³ are strictly positive constants to bede®ned later. Note that V…q; _qq† is a positive semi-de®nitefunction. Di� erentiating V and using (20) we obtain

_VV ˆ kEE _EE ‡ k!_³³0

�³³0 ‡ k³³0_³³0

ˆ kEE _³³0½ ‡ k!_³³0

�³³0 ‡ k³³0_³³0

ˆ _³³0…kEE½ ‡ k!�³³0 ‡ k³³0†

…24†

Let us now compute �³³0 from (8). The inverse of D…q† canbe obtained from (9) and (15) and is given by

D¡1…q† ˆ ‰det…D…q††Š¡1

£J1 ‡ m1l

21 ¡m1l1L0 cos ³1

¡m1l1L0 cos ³1 I0 ‡ m1…L20 ‡ l21 sin2 ³1†

" #

…25†with

det…D…q†† ˆ …I0 ‡ m1l21 sin2 ³1†…J1 ‡ m1l

21†

‡ J1m1L20 ‡ m2

1l21L20 sin2 ³1

Therefore from (8)±(11) we have

�³³0

�³³1

" #ˆ ‰det…D…q††Š¡1

…J1 ‡ m1l21†½

¡…m1l1L0 cos ³1†½

Á !

¡ D¡1…q† C…q; _qq†_³³0

_³³1

" #‡ g…q†

Á !

392 I. Fantoni and R. Lozano

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�³³0 can thus be written as

�³³0 ˆ 1

det…D…q††‰…J1 ‡ m1l21 †½ ¡ …J1 ‡ m1l21†m1l21 sin…2³1† _³³0

_³³1

¡ 12m

21l

31L0 cos ³1 sin …2³1† _³³2

0 ‡ …J1 ‡ m1l21†m1l1L0 sin ³1_³³21

¡ m21l21L0g cos ³1 sin ³1Š

…26†

De®ning

F…³0; _³³0; ³1; _³³1† ˆ ‰¡…J1 ‡ m1l21†m1l21 sin…2³1† _³³0

_³³1

¡ 12m2

1l31L0 cos ³1 sin…2³1† _³³20

‡ …J1 ‡ m1l21†m1l1L0 sin ³1

_³³21

¡ m21l

21L0g cos ³1 sin ³1Š

…27†

we get

�³³0 ˆ 1

det…D…q††‰…J1 ‡ m1l

21†½ ‡ F…q; _qq†Š …28†

Introducing the above in (24) we have

_VV ˆ _³³0 ½ kEE ‡ k!…J1 ‡ m1l21†det…D…q††

Á !‡ k!F…q; _qq†

det…D…q††‡ k³³0

" #

…29†

We propose a control law such that

½ kEE ‡ k!…J1 ‡ m1l21†

det…D…q††

Á !‡ k!F…q; _qq†

det…D…q††‡ k³³0 ˆ ¡k¯

_³³0

…30†

which will lead to

_VV ˆ ¡k¯_³³20 …31†

Note that other functions f … _³³0† such that _³³0 f … _³³0† > 0are also possible, in the RHS of (30). The control lawin (30) will have no singularities provided that

kEE ‡ k!…J1 ‡ m1l21†

det…D…q††

Á !6ˆ 0 …32†

Note from (19) that E ¶ ¡2m1gl1. Thus (32) alwaysholds if the following inequality is satis®ed

k!…J1 ‡ m1l21†max³1

…det…D…q††† > kE…2m1gl1† …33†

This gives the lower bound for k!=kE

k!

kE

> 2m1gl1…I0 ‡ m1l21 ‡ m1L20† …34†

Note that when using the control law (30), the pendulumcan still get stuck at the (lower) stable equilibrium point,…³0; _³³0; ³1; _³³1† ˆ …0; 0; º; 0† for which ½ ˆ 0. In order toavoid this singular point, which occurs when

E ˆ ¡2m1gl1 (see (19)), it su� ces that the followingholds

jEj < 2m1gl1 …35†

Since V is a non-increasing function (see (31)), (35) willhold if the initial conditions are such that

V…0† < 2kEm21g

2l21 …36†

The above de®nes the region of attraction (see Lozano etal. 2000 for more details).

Finally, the control law can be written as

½ ˆ ¡k!F…q; _qq† ¡ det…D…q††…k¯_³³0 ‡ k³³0†

det…D…q††kEE ‡ k!…J1 ‡ m1l21†…37†

with kE and k! satisfying (34).

4. Stability analysis

The stability analysis will be based on LaSalle’sinvariance theorem (see for instance Khalil 1996, p.117). In order to apply LaSalle’s theorem we requireto de®ne a compact (closed and bounded) set O withthe property that every solution of system (8) whichstarts in « remains in « for all future time. SinceV…q; _qq† in (23) is a non-increasing function, (see (31)),then ³0, _³³0 and _³³1 are bounded. Since cos ³0; sin ³0; cos ³1

and sin ³1 are bounded functions, we can de®ne a stateof z of the closed-loop system composed of

³0; _³³0; cos ³1; sin ³1 and _³³1. Therefore, the solution ofthe closed loop system _zz ˆ F…z† remains inside a com-pact set O that is de®ned by the initial state values. Let Gbe the set of all points in O such that _VV…z† ˆ 0. Let M bethe largest invariant set in G. LaSalle’s theorem insuresthat every solution starting in O approaches M ast ! 1. Let us now compute the largest invariant setM in G.

In the set G (see (31)), _VV ˆ 0 and _³³0 ˆ 0 whichimplies that ³0 and V are constant. From (23) it followsthat E is also constant. From (24), (29) and (30) it fol-lows that the control law has been chosen such that

¡k¯_³³0 ˆ kEE½ ‡ k!

�³³0 ‡ k³³0 …38†

From the above equation we conclude that E½ is con-stant in G. Since E is also constant, we either have (a)E ˆ 0 or (b) E 6ˆ 0.

. Case a: If E ˆ 0 then, from (38), ³0 ˆ 0. Notethat ½ in (37) is bounded in view of (32)±(36).Recall that E ˆ 0 means that the trajectories arein the homoclinic orbit (22). In this case we con-clude that ³0; _³³0 and E converge to zero. Notefrom (27) and (37) that ½ does not necessarily con-verge to zero.

. Case b: If E 6ˆ 0 and since E½ is constant, then ½is also constant. However a force input ½ constant

Stabilization of the Furuta pendulum 393

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and di� erent from zero would lead to a contra-diction. We will give below a mathematical proofof the fact that if E 6ˆ 0 then ½ ˆ 0 in G.

Proof: We will prove that when _³³0 ˆ 0, E is constantand E 6ˆ 0, and ½ is constant, then ½ should be zero.From (6) and (7) we get

m1l1L0 cos ³1�³³1 ¡ m1l1L0 sin ³1

_³³21 ˆ ½ …39†

‰J1 ‡ m1l21 Š �³³1 ¡ m1gl1 sin ³1 ˆ 0 …40†

Moreover, the energy E (19) is constant and given by

E ˆ 12…J1 ‡ m1l21† _³³2

1 ‡ m1gl1…cos ³1 ¡ 1† 7 E1 …41†

Introducing (40) in (39), we obtain

sin ³1…a cos ³1 ¡ _³³21† ˆ ½

b…42†

with a ˆ m1gl1=…J1 ‡ m1l21† and b ˆ m1l1L0. The expres-

sion (41) gives us

_³³21 ˆ E2 ‡ c…1 ¡ cos ³1† …43†

with E2 ˆ 2E1=…J1 ‡ m1l21† and c ˆ 2m1gl1=…J1 ‡ m1l21†.

Combining (43) and (42) yields

sin ³1……a ‡ c† cos ³1 ‡ d† ˆ ½

b…44†

with d ˆ ¡…E2 ‡ c†. Taking the time derivative of (44),we obtain

_³³1……a ‡ c†…cos2 ³1 ¡ sin2 ³1† ‡ d cos ³1† ˆ 0 …45†

If _³³1 ˆ 0, then �³³1 ˆ 0 and from (40) we conclude thatsin ³1 ˆ 0. If _³³1 6ˆ 0, then (45) becomes

…a ‡ c†…cos2 ³1 ¡ sin2 ³1† ‡ d cos ³1 ˆ 0 …46†

Di� erentiating (46), it follows

¡ _³³1 sin ³1‰4…a ‡ c† cos ³1 ‡ d Š ˆ 0

If cos ³1 ˆ ¡d=4…a ‡ c† then ³1 is constant which implies_³³1 ˆ 0, and so sin ³1 ˆ 0 (see (40)). In each possible casewe conclude that sin ³1 ˆ 0. Then, _³³1 ˆ 0. From (39) itfollows that ½ ˆ 0. &

We therefore conclude that ½ ˆ 0 in G. From (38) itthen follows that ³0 ˆ 0 in G. It only remains to beproved that E ˆ 0 when ³0 ˆ 0, _³³0 ˆ 0 and ½ ˆ 0.

Since sin ³1 ˆ 0 it follows that ³0 ˆ 0 (mod º). Since³0 ˆ º (mod 2º) has been excluded by imposing con-dition (36). Therefore, ³0 ˆ 0 …mod 2º†, _³³0 ˆ 0, _³³1 ˆ 0imply that E ˆ 0. This contradicts the assumptionE 6ˆ 0 in case (b) and thus the only possible case isE ˆ 0.

The main result can be summarized in the followingtheorem.

394 I. Fantoni and R. Lozano

Figure 2. Simulation results.

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Stabilization of the Furuta pendulum 395

Figure 3. Simulation results.

Figure 4. Simulation results with a saturated controller between ¡3 and 3.

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396 I. Fantoni and R. Lozano

Figure 5. Simulation results with a saturated controller between ¡3 and 3.

Figure 6. Simulation results with a saturated controller between ¡0:25 and 0.25.

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Theorem 1: Consider the Furuta pendulum system (6)and (7) and the controller in (37) with strictly positiveconstants kE ; k!; k³ and k¯ satisfying (34). Provided thatthe state initial conditions satisfy inequality (36), thenthe solution of the closed-loop system converges to theinvariant set M given by the homoclinic orbit (22) with…³0; _³³0† ˆ …0; 0†. Note that ½ does not necessarily con-verge to zero.

Remark 1: The above result is local in the sense thatthe system initial state should belong to the domain ofattraction de®ned in (36). However, the same resultwill hold for arbitrary initial conditions except for aparticular manifold bringing the system to the stableequilibrium position with ³1 ˆ º.

5. Simulation results

Figures 2 and 3 show the performance of the pro-posed control law. The initial position is

³0 ˆ ¡ º

2_³³0 ˆ 0

³1 ˆ 2:5º

3_³³1 ˆ 0

and the parameters are I0 ˆ 1:75 £ 10¡2, L0 ˆ 0:215,m1 ˆ 5:38 £ 10¡2, l1 ˆ 0:113 and J1 ˆ 1:98 £ 10¡4.

The gains have been chosen such as kE ˆ 480,

k³ ˆ 1, k! ˆ 1 and k¯ ˆ 1.

On ®gure 3, the control input signal is large at

the start of the experiment (about ¡8). We ran simu-

lations including a saturation in the controller to protectthe actuator. First, ®gures 4 and 5 show the performance

of the controller with a saturation between ¡3 and 3.

The simulations are very similar to the one without

saturation. Furthermore, we ran simulations with a

saturation between ¡0:25 and 0.25 (see ®gures 6 and7). The results show that the controller performs well

in both cases. The convergence is only slower.

6. Conclusions

We have proposed a control strategy to `swing

up’ the Furuta pendulum. The control design is

based on the passivity properties of this rotational

inverted pendulum. Convergence of the trajectoriesof the system to a homoclinic orbit has been proved

by using LaSalle’s invariance theorem. We have pre-

sented simulations showing the performance of the pro-

posed strategy.

Stabilization of the Furuta pendulum 397

Figure 7. Simulation results with a saturated controller between ¡0:25 and 0.25.

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398 I. Fantoni and R. Lozano

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