Description

Automatica 36 (2000) 287}295
Brief Paper
Swinging up a pendulum by energy control
K.J. Astrom *, K. Furuta K s
Department of Automatic Control, Lund Institute of Technology, Box 118, S-22100 Lund, Sweden Department of Control and Systems Engineering, Tokyo Institute of Technology, Tokyo, Japan Received 13 November 1997; revised 10 August 1998; received in nal form 10 May 1999
Abstract Properties of simple strategies for swinging up an inverted pendulum are discussed. It is shown that the be

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Automatica 36 (2000) 287
}
295
Brief Paper
Swinging up a pendulum by energy control
K.J. A
s
stro
K
m
*
, K. Furuta
Department of Automatic Control, Lund Institute of Technology, Box 118, S-22100 Lund, Sweden
Department of Control and Systems Engineering, Tokyo Institute of Technology, Tokyo, Japan
Received 13 November 1997; revised 10 August 1998; received in
nal form 10 May 1999
Abstract
Properties of simple strategies for swinging up an inverted pendulum are discussed. It is shown that the behavior critically dependson the ratio of the maximum acceleration of the pivot to the acceleration of gravity. A comparison of energy-based strategies withminimum time strategy gives interesting insights into the robustness of minimum time solutions.
1999 Elsevier Science Ltd. Allrights reserved.
Keywords:
Inverted pendulum; Swing-up; Energy control; Minimum time control
1. Introduction
Inverted pendulums have been classic tools in thecontrollaboratories since the 1950s. They were srcinallyused to illustrate ideas in linear control such as stabiliz-ation of unstable systems, see e.g. Schaefer and Cannon(1967), Mori, Nishihara and Furuta (1976), Maletinsky,Senning and Wiederkehr (1981), and Meier, Farwig andUnbehauen (1990). Because of their nonlinear naturependulums have maintainedtheir usefulness and they arenow used to illustrate many of the ideas emerging in the
eld of nonlinear control. Typical examples are feedbackstabilization, variable structure control (Yamakita& Furuta, 1992), passivity based control (Fradkov,Guzenko, Hill & Pogromsky, 1995), back-stepping andforwarding (Krstic
H
, Kanellakopoulos & Kokotovic
H
,1994), nonlinear observers (Eker & A
s
stro
K
m, 1996), fric-tion compensation (Abelson, 1996), and nonlinear modelreduction. Pendulums have also been used to illustratetask oriented control such as swinging up and catching
This project was supported by the Swedish Research Council forEngineering Science under contract 95-759 and Nippon Steel Corpora-tion, Japan. This paper was presented at the 13th IFAC WorldCongress, which was held in San Francisco, U.S.A., July 1996. Thispaper was recommended for publication in revised form by AssociateEditor Henk Nijmeijer under the direction of Editor T. Basar.
*
Corresponding author. Tel.:
#
00-46-46-2228781; fax:
#
00-46-46-138118.
E-mail address:
kja
@
control.lth.se (K.J. A
s
stro
K
m)
the pendulum, see Furuta and Yamakita (1991), Furuta,Yamakita and Kobayashi (1992), Wiklund, Kristensonand A
s
stro
K
m (1993), Yamakita, Nonaka and Furuta(1993), Yamakita, Nonaka, Sugahara and Furuta (1994),Spong (1995), Spong and Praly (1995), Chung andHauser (1995), Yamakita, Iwashiro, Sugahara andFuruta (1995), Wei, Dayawansa and Levine (1995),Borto
!
(1996), Lin, Saberi, Gutmann and Shamash(1996), Fradkov and Pogromsky (1996), Fradkov,Makarov, Shiriaev and Tomchina (1997), Lozano andFantoni (1998). Pendulums are also excellently suited toillustrate hybrid systems (Guckenheimer, 1995; A
s
stro
K
m,1998) and control of chaotic systems (Shinbrot, Grebogi& Wisdom, 1992).In this paper we will investigate some properties of thesimple strategies for swinging up the pendulum based onenergy control. The position and the velocity of the pivotare not considered in the paper. The main results is thatthe global behavior of the swing up is completely charac-terized by the ratio
n
of the maximum acceleration of thepivot and the acceleration of gravity. For example, it isshownthat one swing is su
$
cient if
n
is larger than
. Theanalysis also gives insight into the robustness of min-imum time swing up in terms of energy overshoot.The ideas of energy control can be generalized in manydi
!
erent ways. Spong (1995) and Chung and Hauser(1995) have shown that it can be used to also control theposition of the pivot. An application to multiple pendu-lums is sketched in the end of the paper. The ideas havebeen applied to many di
!
erent laboratory experiments,
0005-1098/00/$-see front matter
1999 Elsevier Science Ltd. All rights reserved.PII: S0 0 05 -1 0 9 8 (9 9 ) 0 0 1 40 - 5
see e.g. Iwashiro, Furuta and A
s
stro
K
m (1996), Eker andA
s
stro
K
m (1996) and A
s
stro
K
m, Furuta, Iwashiro andHoshino (1995).
2. Preliminaries
Consider a single pendulum. Let its mass be
m
and letthemoment ofinertia with respectto the pivotpoint be
J
.Furthermore, let
l
be the distance from the pivot to thecenter of mass. The angle between the vertical and thependulum is
, where
is positive in the clockwise direc-tion. The acceleration of gravity is
g
and the accelerationof the pivot is
u
. The acceleration
u
is positive if it is in thedirection of the positive
x
-axis. The equation of motionfor the pendulum is
J
$
!
mgl
sin
#
mul
cos
0. (1)The system has two state variables, the angle
and therate of change of the angle
Q
. It is natural to let the statespace be a cylinder. In this state space the system has twoequilibria corresponding to
0,
Q
0, and
,
Q
0. If the state space is considered as
R
there arein
nitely many equilibria. There are many deeper di
!
er-ences between the choice of states.The model given by Eq. (1) is based on several assump-tions: friction has been neglected and it has been assumedthat the pendulum is a rigid body. It has also beenassumed that there is no limitation on the velocity of the pivot. The energy of the uncontrolled pendulum(
u
0) is
E
J
Q
#
mgl
(cos
!
1). (2)It is de
ned to be zero when the pendulum is in theupright position. The model given by Eq. (1) has fourparameters: the moment of inertia
J
, the mass
m
, thelength
l
, and the acceleration of gravity
g
. Introduce themaximum acceleration of the pivot
u
max
u
ng
. (3)Introduce the normalized variables
(
mgl
/
Jt
,
(
mgl
/
Jt
t
and
v
u
/
g
. The equation of motion(1) then becomesd
d
!
sin
#
v
cos
0,where
v
4
n
. The normalized total energy of the uncon-trolled system (
v
0) is
E
L
Emgl
12
d
d
#
cos
!
1. (4)The system is thus characterized by two parameters only,the natural frequency for small oscillations
(
mgl
/
J
and the normalized maximum acceleration of the pendu-lum
n
u
/
g
. The model given by Eq. (1) is locally
Fig. 1. Geometric illustration of a simple swing-up strategy. The srcinof the coordinate system is called O.
controllable when
O
/2, i.e. for all states except whenthe pendulum is horizontal.
2.1. A simple swing-up strategy
Before going into technicalitieswe will discuss a simplestrategy for swinging up the pendulum. Consider thesituation shown in Fig. 1 where the pendulum starts withzero velocity at the point A. Let the pivot accelerate withmaximum acceleration
ng
to the right. The gravity
ledseen by an observer
xed to the pivot has the directionOB where
arctan
n
, and the magnitude
g
(
1
#
n
.The pendulum then swings symmetrically around OB.The velocityis zero when it reaches the point C where theangle is
#
2
. The pendulum thus increases its swingangle by 2
for each reversal of the velocity. The simplestrategy we have described can be considered as a simpleway of pumping energy into the pendulum. In the nextsections we will elaborate on this simple idea.
3. Energy control
Many tasks can be accomplished by controlling theenergyof the pendulum instead of controllingits positionand velocity directly, see Wiklund et al. (1993). Forexample one way to swing the pendulum to the uprightposition is to give it an energy that corresponds to theupright position. This corresponds to the trajectory
E
J
(
Q
)
#
mgl
(cos
!
1)
0,which passes through the unstable equilibrium at theupright position. A di
!
erent strategy is used to catchthe pendulum as it approaches the equilibrium. Such
288
K.J. A
s
stro
(
m, K. Furuta
/
Automatica 36 (2000) 287
}
295
a strategy can also catch the pendulum even if there is anerror in the energy control so that the constant energystrategy does not pass through the desired equilibrium,see A
s
stro
K
m (1999).The energy
E
of the uncontrolled pendulum is given byEq. (2). To perform energy control it is necessary tounderstand how the energy is in
#
uenced by the acceler-ation of the pivot. Computing the derivative of
E
withrespect to time we
ndd
E
d
t
J
Q
$
!
mgl
Q
sin
!
mul
Q
cos
, (5)where Eq. (1) has been used to obtain the last equality.Eq. (5) implies that it is easy to control the energy. Thesystem is simply an integrator with varying gain. Con-trollability is lost when the coe
$
cient of
u
in the right-hand side of (5) vanishes. This occurs for
Q
0 or
$
/2, i.e., when the pendulum is horizontal or whenit reverses its velocity. Control action is most e
!
ectivewhen the angle
is 0 or
and the velocity is large. Toincrease energy the acceleration of the pivot
u
should bepositive when the quantity
Q
cos
is negative. A controlstrategy is easily obtained by the Lyapunov method.With the Lyapunov function
<
(
E
!
E
)
/2, and thecontrol law
u
k
(
E
!
E
)
Q
cos
, (6)we
nd thatd
<
d
t
!
mlk
((
E
!
E
)
Q
cos
)
.The Lyapunov function decreases as long as
Q
O
0 andcos
O
0. Since the pendulum cannot maintain a station-ary position with
$
/2 strategy (6) drives the en-ergy towards its desired value
E
. There are many othercontrol laws that accomplishes this. To change the en-ergy as fast as possible the magnitude of the controlsignalshouldbe as largeas possible.Thisis achievedwiththe control law
u
ng
sign((
E
!
E
)
Q
cos
), (7)which drives the function
<
E
!
E
to zero and
E
towards
E
. Control law (7) may result in chattering.This is avoided with the control law
u
sat
LE
(
k
(
E
!
E
)sign(
Q
cos
)), (8)where sat
LE
denotes a linear function which saturates at
ng
. Strategy (8) behaves like linear controller (6) for smallerrors and like strategy (7) for large errors. Notice thatthe function sign is not de
ned when its argument is zero.If the value is de
ned as zero the control signal will bezero when the pendulum is at rest or when it is horizon-tal. If the pendulum starts at rest in the downwardposition strategies (7), (6) and (8) all give
u
0 and thependulum will remain in the downward position.The parameter
n
is crucial because it gives the max-imum control signal and thus the maximum rate of energy change, compare with Eq. (5). Parameter
n
drasti-cally in
#
uences the behavior of the swing up as will beshown later. Parameter
k
determines the region wherethe strategy behaves linearly. For large values of
k
strategy (8) is arbitrarily close to the strategy that givesthe maximum increase or decrease of the energy. Inpracticalexperiments,the parameteris determined by thenoise levels on the measured signals.
4. Swing-up behaviors
We will now discuss strategies for bringing the pendu-lum to rest in the upright position. The analysis will becarried out for the strategy given by Eq. (7). The signfunction in Eq. (7) is de
ned to be
#
1 when the argu-ment is zero. The energyof the pendulum given by Eq. (2)is de
ned so that it is zero in the stable upright positionand
!
2
mgl
in the downward position. With these con-ventions the acceleration is always positive when thependulum starts at rest in the downward position.Energy control with
E
0 gives the pendulum thedesired energy. The motion approaches the manifoldwhere the energy is zero. This manifold contains thedesired equilibrium. With energy control the equilibriumis an unstable saddle. It is necessary to use anotherstrategy to catch and stabilize the pendulum in the up-right position. In Malmborg, Bernhardsson and A
s
stro
K
m(1996) it is shown how to design suitable hybridstrategies. Before considering the details we will make ataxonomy of the di
!
erent strategies. We will do this bycharacterizing the gross behavior of the pendulum andthe control signal during swing-up. The number of swings the pendulum makes before reaching the uprightposition is used as the primary classi
er and the numberof switches of the control signal as a secondary classi
er.It turns out that the gross behavior is entirely determinedby the maximum acceleration of the pivot
ng
. The behav-ior during swing up is simple for large values of
ng
andbecomes more complicated with decreasing values of
ng
.
4.1. Single-swing double-switch beha
v
ior
There are situations where the pendulum swings insuch a way that the angle increases or decreases mono-tonically. This is called the single-swing behavior. If theavailable acceleration is su
$
ciently large, the pendulumcan be swung up simply by using the maximum acceler-ation until the desired energyis obtainedand then settingacceleration to zero. With this strategy the control signalswitches from zero to its largest value and then back tozero again. This motivates the name of the strategy.To
nd the strategy we will consider a coordinatesystem
xed to the pivot of the pendulum and regard the
K.J. A
s
stro
(
m, K. Furuta
/
Automatica 36 (2000) 287
}
295
289
Fig. 2. Simulation of a single-swing double-switch strategy. The para-meters are
n
2.1,
1 and
k
100.
force due to the acceleration of the pivot as an externalforce. In this coordinate system the center of mass of thependulum moves along a circular path with radius
l
. Itfollows from Eq. (8) that the desired energy must bereached before the pendulum is horizontal.The energy supplied to a mass when it is moved from
a
to
b
by a force
F
is
=
?@
@?
F
d
x
. (9)To swing up the pendulum with only two switches of thecontrol signal the pendulum must have obtained therequired energy before the pendulum is horizontal. Ina coordinate system
xed to the pivot the center of massof the pendulum has moved the distance
l
when it be-comes horizontal. The horizontal force is
mng
and itsenergy has thus been increased by
mngl
. The energyrequired to swing up the pendulum is 2
mgl
and we thus
nd that the maximum acceleration must be at least 2
g
for single-swing double-switch behavior. If the acceler-ation is larger than 2
g
the acceleration will be switchedo
!
when the pendulum angle has changed by
H
. Thecenter of the mass has moved the distance
l
sin
H
and theenergy supplied to the pendulum is
nmgl
sin
H
. Equatingthis with 2
mgl
gives sin
H
2/
n
.
4.1.1. Example 1
*
simulation of SSDS beha
v
ior
The single-swing double-switch strategy is illustratedin Fig. 2 which shows the angle, the normalized energy,and the control signal. The simulation is made using thenormalized model with
1 and the control law givenby Eq. (8) with
n
2.1 and
k
100. With this value of
k
the behavior is very close to a pure switching strategy.Notice that it is required to have
n
5
2 to have thesingle-swing double switch behavior for pure switching.Slightly larger values of
n
are required with the controllaw (8). For the simulations in Fig. 2 we used
n
2.1. Forapure switchingstrategy (7) the control signalis switchedto zero when the pendulum is 17.8
3
below the horizontalline.
4.2. Single-swing triple-switch beha
v
ior
To obtain the single-swing double-switch behavior thependulum must be given su
$
cient energy before itreaches the horizontal position. In the previous sectionwefound that the condition is
n
'
2. It is possible to havesingle-swing behavior for smaller values of
n
but thecontrol signal must then switch three times because theacceleration must be reversed when the pendulum ishorizontal. Since the pendulum must reach the horizon-tal in one swing we must still require that
n
'
1. To
ndout how much larger it has to be we will consider thesituation illustrated in Fig. 3. The pendulum starts at restat position A. The pivot is then accelerated
ng
in thedirection of the positive
x
-axis. An observer
xed to thepivot sees a gravitational
eld in the direction OB withthe strength
w
g
(
1
#
n
and the pendulum swingsclockwise. When the pendulum moves from A to D itloses the potential energy
mwa
, which is converted tokinetic energy. To supply energy as fast as possible to thependulum it follows from Eq. (5) that acceleration shouldbe reversed when the pendulum reaches the point D. Anobserver in a coordinate frame
xed to the pivot thensees a gravitational
eld with strength
w
in the directionOC. The kinetic energy is continuous at the switch butthe potential energy is discontinuous. The pendulum willswing towards the upright position if its kinetic energy isso large that it reaches the point E. The kinetic energy atF must thus be at least
mbw
. The condition for this is
a
5
b
. (10)It follows from Fig. 3 that
a
sin
!
cos
and
b
1
!
sin
. Condition (10) then becomes2sin
5
1
#
cos
. (11)Introducing
n
tan
and using equality in Eq. (11)gives2
n
1
#
(
1
#
n
.This equation has the solution
n
. To have a single-swing triple-switch behavior the acceleration of the pivotmustthus be at least 4
g
/3. If
n
the pivot accelerates tothe right until the pendulum reaches the horizontal. Thepivot is then accelerated to the left until the desiredenergy is obtained. This happens when the pendulum is30
3
from the vertical and the acceleration is then set tozero.If
n
is greaterthan
the accelerationofthe pivot can beset to zero before the pendulum reaches the point E. Let
H
be the angle of the pendulum when the acceleration of
290
K.J. A
s
stro
(
m, K. Furuta
/
Automatica 36 (2000) 287
}
295

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