Nonlinear Dynamics 10: 49-62, 1996.
@ 1996 Kluwer Academic Publishers. Printed in the Netherlands.
Global Stabilization of the Oscillating Eccentric Rotor
CHIH-JIAN WAN, DENNIS S. BERNSTEIN *, and VINCENT T. COPPOLA
**
Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI 48/09-2118, U.S.A.
(Received: 15 April 1994; accepted in revised fonn: 21 February 1995)
Abstract. The oscillating eccentric rotor has been widely studied to model resonance capture phenomena occurring
in dual-spin spacecraft and rotating machinery. This phenomenon arises during spin-up as a resonance condition is
encountered. We consider the related problem of rotor despin. Specifically, we detennine nonlinear feedback control
laws that not only despin the rotor but also bring its translational motion to rest. These globally asymptotically
stabilizing control laws are derived using partial feedback linearization and integrator backstepping schemes. For
the case in which the oscillating eccentric rotor is excited by a translational sinusoidal forcing function, the control
law is shown to attenuate the amplitude of the translational oscillation.
Key words: Passage through resonance, cascade system, integrator backstepping.
1. Introduction
.,
We are ultimately interested in controlling dual-spin spacecraft; however, in this paper, we
analyze a model that embodies similar dynamical behavior: the oscillating eccentric rotor.
The control problem involving dual-spin spacecraft is concerned with reducing nutation that
becomes excited during spin-up. The interaction between spin and nutation has been modelled
in [12] by means of the oscillating eccentric rotor where translation represents nutation. The
interaction between rotation and translation in the oscillating eccentric rotor is analogous to
the interaction between spin and nutation in dual-spin spacecraft.
The oscillating eccentric rotor consists of an unbalanced rotor attached to an elastic support.
This system has been used as a simplified model to study the resonance capture phenomenon
in dual-spin spacecraft during spin-up under small constant torque [4, 7, 12, 15]. The capture
phenomenon represents the failure of a rotating mechanical system to be spun up by a torquelimited motor to a desired terminal state due to its resonant interaction with another part of
the system [3, 12]. In [3, 12], the dynamics of the oscillating eccentric rotor were analyzed
using perturbation theory, while in [7] control laws for rotor spin-up under limited torque were
obtained and extended to dual-spin spacecraft.
In this paper, we consider the oscillating eccentric rotor but, rather than spinning up the
rotor, we seek feedback control laws that globally asymptotically stabilize the entire system,
both rotation and translation, to the rest state using only a torque actuator. Aside from potential
application to dual-spin spacecraft, such control laws for the oscillating eccentric rotor are
of independent interest. For example, the controlled oscillating eccentric rotor sewes as a
rotational actuator for suppressing translational vibration [2].
The .synthesis technique used to obtain globally asymptotically stabilizing control laws
involves partial feedback linearization [1, 10, 11] and integrator back stepping for cascade
systems [1, 6, 8, 13]. In particular, we apply partial feedback linearization to transform
*
..
Research supported in part by the Air Force Office of Scientific Research under Grant F49620-92-J-OI27.
Research supported in part by NSF Grant MSS-9309165.
50
C.-J. Wanet al.
Fig. 1. Oscillating eccentric rotor.
the dynamical equations of the oscillating eccentric rotor into a system in cascade form.
Then we derive stabilizing control laws for the system in cascade form using integrator
backstepping.Controllaws for the originalsystemare then obtainedby applyingthe reverse
transformation.
2. Dynamic Model of the Oscillating Eccentric Rotor
The oscillating eccentricrotor shown in Figure 1 consists of an unbalancedpoint mass of
mass m attachedto a rotary disk of inertia I which is riding on a cart constrainedto move
horizontally.The moving cart is attached to a wall by a spring with spring constant k. Let
M denote the total mass of the disk and the cart, and let e denote the distance between the
center of the disk and the unbalancedmass. For simplicity,we assume that the motion is
confinedto the horizontalplane so that there are no gravitationaleffects. Let Xc denote the
translationalpositionof the center of the disk from its equilibriumposition,and let 8 denote
the counter-clockwiserotationalangle of the unbalancedmass where 8 = 0 correspondsto a
90-degreerotationfromthe springaxis as shownin Figure 1.Let N denotethe controltorque
appliedto the disk and let F denote a translationaldisturbanceforce applied to the moving
cart. Note that I
= 0 is allowed, that is, the disk may be massless.
Theequationsof motionforthesystemaregivenby [3,7, 12,15]
(M + m)xc + me(iJcos8- 02sin8) + kxc = F,
(1)
(I + me2)iJ+ mxcecos8= N.
(2)
Introducing the dimensionless variables [7]
Xd
u
M +m
=. V 1+
me2 Xc,
C= V(I + me2)(M + m) ,
=
.
T=
k(I
(M+m)
+ me2) N,
.
me
.~
V[""""k
t,
Fd
.
=
.IVM+m
k I + me2 F,
equations (1) and (2) are equivalent to the dimensionless equations
Xd + Xd
= c(02sin8 -
iJcos 8) + Fd,
(3)
Global Stabilization of the Oscillating Eccentric Rotor
jj = u
-
eXdCOSe,
51
(4)
where the dot and double dot now denote first and second derivatives with respect to T. By
defining x = (Xl,X2,X3,X4)T = (Xd,Xd,e, iJ)T,(3) and (4) can be writtenin first-orderform
as
(5)
x = I(x) + g(x)u + d(x)Fd,
where
X2
.
I(x)
A
(-Xl
+ ex~sinx3)/(1
-
e2 cos2 X3)
X4
ecos X3(Xl
-
ex~sinx3)/(1
-
e2 cos2 X3)
1
g(x)
A
d(x) ~ 1 _ e2 cos2 X3
Note that the denominator 1 - e2cos2X3is never zero, since,by definition,e < 1. It is easy
to check that if Fd = 0 and u = 0, then everyequilibriumstate I (x) = 0 of (5) i~of the form
Xl
X2
X4
0 with arbitrary X3, which corresponds to zero translational position and
velocity, zero angular velocity, and arbitrary rotation angle X3 = eof the unbalancedmass. In
this paper we are interested in the equilibrium state corresponding to X3 = O.In particular, we
are interested in deriving globally asymptotically stabilizing control laws for the equilibrium
=
state X
=
=
= O.
3. Cascade System Control Synthesis
Wenow considera single-inputgeneralnonlinearaffinecontrol systemof the form
x = I(x) + g(x)u,
where X E ]Rn, U E Ii, and we assume 1(0)
(6)
= O.In the case Fd = 0, the oscillating eccentric
rotor model (5) is a special case of (6) with n = 4. Our goal is to find a feedback control
u = u( x) that globally asymptotically stabilizes (6) to X O.This is accomplished by means
of a two-step procedure. First we transform (6) to a the simpler form
=
(7)
~=v,
(8)
using partial feedback linearization [1,10,11], where <p((,~) is ai, (E ]Rn-l and~, v E R
Having transformed (6) into the form (7) and (8), we then seek control laws to stabilize (7)
and (8). Control laws for the original system (6) can then be obtained by using the reverse
transformation.
As shown in [13], the cascade form of (7) and (8) is convenient for stabilization. Specifically, if (7) can be stabilized by viewing ~ as the control input, then the combined system (7),
52
C.-J.Wanet al.
(8) can be stabilizedby means of a suitablefunction v = v( (, ~). This procedure, which is
known as integratorbackstepping[6, 8], is based on the followingresult obtainedby Sontag
and Sussmann[13]and Kokotovicand Sussmann[9].
THEOREM 3.1. Suppose there exists a C1function
~ = k(()
with k(O)
(9)
= 0 such
that the origin (
= 0 of the system
(10)
( = <1>((,
k(())
is globally asymptotically stable. Furthermore,
define 'I/;: IRn-1 x IR -+ IRn-1 by
let YO(() be a Lyapunov function for (10), and
(11)
Then the feedback
v((,~)
= k(()
control law
globally asymptotically stabilizes the origin (
function for the closed loop system is
V((,~)
I
(12)
_ ~ + 8~~() <I>((,~)
_ 8~~() 'I/;((,~)
= 0, ~= 0 of(7),
(8). Moreover, a Lyapunov
= Vo(() + ~ [~- k(()f
(13)
,
with time derivative
V((,~) = 8~~() <1>((,
k(()) - [~- k(())f
(14)
Note that, accordingto I'Hopital's rule, 'I/;((,~) = 8<1>((,
k(())/8~, when~ = k(().
4. Stabilizing Control Laws for the Oscillating Eccentric Rotor
In this section, we synthesize control laws for the oscillating eccentric rotor that globally
despin the rotor and bring the translationalmotion of the cart to rest in the case Fd = O.
That is, we design control laws that globally asymptoticallystabilize (6) to the equilibrium
state x
=
0, where I(x) and g(x) are defined in Section 2. This is accomplished by first
usingpartial feedbacklinearizationto transform(6) into a systemhavingcascadeform. Then
we iterativelyapply Theorem 3.1 to synthesizecontrollaws for the system in cascade form.
Controllaws for the originalsystemsare then obtainedfrom the reverse tran~formation.
Since the linearizationof (6) about the origin is controllable,one can easily design linear
controllawsthatlocallyasymptoticallystabilize(6)to theorigin.Linearcontrollaws,however,
are guaranteedto asymptoticallystabilize(6) only for initial conditionssufficientlynear the
origin.Since we are interestedin the despinproblemfor which none of the statesare initially
small, we seek nonlinearcontrol laws that stabilize(6) for arbitraryinitial conditions.
Global Stabilizationof the OscillatingEccentricRotor
4.1.
53
PARTIAL FEEDBACK LINEARIZATION
To obtain globally asymptotically stabilizing control laws, we first check feedback linearizability of (6) [14]. It is straightforward to verify that [g(x), ad}g(x)] ~ ~2(X) ~
span{g(x),adf9(x),ad}g(x)},
where standard expressions for Lie brackets have been used
[5, 14]. Hence, L~dx) is not involutive [5]. Consequently, (6) is not feedback linearizable
[14].
Next we consider partial feedback linearization [1, 10]. Since ~1(X) = Li1(x) and
dim~l(x) = 2, where ~I(X) ~ span{g(x),adfg(x)},
and Li1(x) is the smallest involutive
distribution which contains ~I (x) [1], it follows that the dimension of the largest linearizable
subsystem is two [11]. To transform (6) into a cascade form with linear part having maximal
dimension, we apply the following procedure. Since ~ I(x) is involutive, it follows from the
Frobenius theorem [5, 14] that solutions to the partial differential equations
i
= 1,2,
(15)
exist, where
(16)
-1
o
One set of solutionsto these equationsis
ZI(X)
= XI + esinx3,
(17)
Z2(X)
= X2 + eX4cosx3.
(18)
The new state variables in the linearizedpart of the transformedsystem can be obtainedby
letting YI(x) be such that
(19)
and such that the vectors
:
( )
T
,
(~~ )
T
8z2
,
T
(8x )
(20)
'
are linearly independent [11]. The simplest solution to these constraints is Yl(Z)
Letting
=
X3.
(21)
one can rewrite equation (6) in terms of the new variables (Zl, Z2,Yl, Y2) as
.'
Z2
=
-:-Zl+ esinYl,
(22)
(23)
ill = Y2,
(24)
Y2 = v,
(25)
54
C.-I. Wan et al.
where
1
v=1
-
2 2.
1
e 2 cos 2 X3 (eXICOSX3-e X4cosX3smX3)
+ 1-
e 2 cos 2 X3 U
is the transformed control input. Note that (22), (23) are of the form (7) with
(=
Z
= (ZI,Z2l
(26)
~=
YI,
and
4J«('~) = 4J(z,Yd =
.
(27)
eXI cos X3 + e2X~ COSX3 sin X3.
(28)
[ -Zl
Z2.
+ esmYI
]
Solvingfor u in terms of v yields
u
= (1 -
e2 cos2 X3)V
-
Having transformed (6) into the form (22)-(25), we now seek a control law v that stabilizes
(22)-(25) to the origin YI = Y2 = Zl = Z2 = 0, whichcorrespondsto x = 0 in (6).
4.2.
INTEGRATOR BACKSTEPPING
In this subsectionwe synthesizea control law v for (22)-(25) by iteratively applying Theorem 3.1. First, in Step 1, we view YI as the control input to (23) and seek a control law
YI = kl (z) that stabilizes(22) and (23). Then, in Step 2, we use Theorem 3.1 with Y2as the
controlinput for (24) to obtaina controllaw Y2 = k2(z, YI) that stabilizes (22)-(24). Finally,
in Step 3, we use Theorem3.1 again to derive a controllaw v that stabilizes(22)-(25).
Step 1. Considerthe problem of stabilizing(22), (23) with YIas the control input. To do
this, let
(29)
YI = kl(z) ~ -COtan-I Z2,
where 0 < CO< 2, and consider the Lyapunov candidate
Vo(z)
=~
(30)
(zf + zi),
wherePo > O.Then it followsthat
Vo(z) = -ePOZ2sin(co tan-I Z2),
(31)
which is nonpositive.Next, consider the set e = {z E JR2: Vo(z) = O}.It is easy to see
thate = {z E JR2: Z2 = O}.Consider Z2(t) = 0 for t ~ tl. From (23) and (29), we have
Zl(t) = Z2(t) = 0, t ~ tl. Hence, the largest invariant set in e is {O}. It thus follows from
the LaSalle-Krasovskii theorem [14] that the states of the closed-loop system (22), (23) with
Yl defined in (29) approach the origin as t -+ 00. Finally, since Vo(z) -+ 00 as IIzlI -+ 00, it
follows that the control law (29) globally asymptotically stabilizes (22), (23).
Step 2. We now use Theorem 3.1 to obtain a control law for (22)-(24) with Y2as the control
input. That is, we seek a feedback control law Y2 = ~(z, Yl) such that (22}.{24) are globally
asymptotically stable. From Step 1 and Theorem 3.1 we have k«() = kl(z) = -co tan-I Z2.
A direct calculation shows that (10) is now
4>«,k«))
= 4>(z, k, (z)) =
[ -ZI _ e sin%o tan-I Z2) ]
,
(32)
Global Stabilization of the Oscillating Eccentric Rotor
55
while V;«(,~)definedby (11) is V;«(,~) = V;(z, YI) and
0
.
-v;(z,yd=
.
[ c[smYI + sm(cotan
_
_'
(33)
I Z2)J!(YI + cotan 1Z2) ]
With the Lyapunov candidate
PI
Po 2
2
-1
2
(
)
(
)
Vi Z,YI ="2 zl + Z2 +"2 (YI+ COtan Z2) ,
where PI
> 0, Theorem 3.1 yields the control law Y2 = k2(z,Yd, where
I
~
.
(34)
Zl
k2(z, Yd = -CI (YI+ COtan- Z2)+ CO
cPOZ2
-
PI ( YI + COtan
- c sin2 YI
1 + z2
.
-1
\
Z2
.
-I
[sm YI + sm( COtan
(35)
Z2,)]
and CI > o. The time derivativeof Vi(z, YI)along the closed-looptrajectory(22)-(24) with
the controllaw Y2definedby (35) is
Vi (z, YI)
= -cPOZ2 sin( COtan-I
Z2) - PI CI(YI + COtan-I Z2)2.
(36)
It is straightforward to check that the largest invariant set in Hz, YI) : Vi (z, Yd = O} is
{z = 0,YI = O}.Hence, the control Y2 in (35) globally asymptotically stabilizes (22)-(24) to
z = 0, YI= O.
Step3. Nowwe applyTheorem3.1 to obtaina stabilizingcontrollaw for the entire system
(22)-(25). Note that (22)-(24) and (25) are again in the cascade form of (7) and '(8) with
~ = Y2,(= (ZI,z2,ydT and
,p«(,€)= ,p«(,1I2)= [ -ZI ~sinY1 ] .
Furthermore, in Step 2 we have already established a 01 function Y2 = k2(z, Yd with
k2(0,0) = 0 that globally asymptotically stabilizes (22)-(24) and a Lyapunov function
Vi(z, YI)definedin (34).HenceweapplyTheorem3.1againto obtainthe stabilizingcontrol
law
v(z, y) = -C2(Y2- k2(Z,Yd) + KI (Z2)Z2+ K2(Z,YI)(-ZI + c sinYI)
+ K3(z, YdY2
-
PI (YI + COtan-I Z2),
P2
for (22)-(25), where Y = (YI, Y2l,
(37)
C2 > 0, k2(z, YI) is defined in (35), and KI (Z2), K2(z, YI),
K3(z, Yd are defined by
KI (Z2) £
(
K2 Z,YI
)
~,
1+ z2
.
.
~
cpo
= - PI (YI +
COtan -1 Z2) [ smYI + sm(cotan
---
COCI
1 + z~
(38)
.
zI-csinYI
2COZ2
(1 + z~)2
+-
-1
cos(cotan-I
Z2) + COZ2
Z2)
1 + z22
cPOCOZ2[sinYI+sin(COtan-Iz2)]
PI
(1 + Z~)(YI+ COtan-I Z2)2 '
]
(39)
56
C.-f. Wanet al.
and
K 3(Z, Y1)
f>
=
-C1 -
c:eocos Y1
1 + Z~
c:po
- -
Z2 COSY1
P1 Y1+ eotan-1 z2
+ c:po Z2[S~Y1 ~
P1
s~(~~~~ Z2)],
(40)
The Lyapunov function that guarantees global asymptotic stability of the closed-loop system
(22)-(25) with v defined in (37) is
1t2(z,y)
= P; (zr + z~) + ~
(Y1 + eotan-1 Z2)2+
~
[Y2- k2(z, ydf,
(41)
where P2 > O.Note that V2(z, y) is proper and positive definite. The time derivativeof1t2(z, y)
is
V2(Z,y)
=
-C:POZ2sin(eotan-1
Z2)
- P1C1(Y1+ eotan-1 Z2)2 - P2C2[Y2- k2(Z,Y1)]2,
(42)
which is negative semi-definite. It can be shown that along the closed-loop trajectories the
largest invariant set in ((z,y) : V2(Z,y) = O} is {z = Y = O}. Since V2(Z,y) is proper, it
follows that v given by (37) globally asymptotically stabilizes the system (22)-(25). Note that
k2(z, Y1), K 2(z, Y1) and K 3(z, Y1) are continuous for all their arguments.
. Finally, the feedback control law u that globally asymptotically stabilizes the original system (6) to the origin is obtained by substituting (37) into (28).
REMARK 1. In Step 1, we chose k1(z) in (29) in terms of the inverse tangent function.
Other choices of k1(z) may be made in accordance with Theorem 3.1. It can be shown that
if k1(z) is a C1 bounded function with k1(0) 0 and Z2k1(z) < 0, z E JR.2,then Y1= k1(z)
globally asymptotically stabilizes (22), (23).
=
REMARK 2. The six parameters Po, P1,P2, eo, C1and C2which appear in the control law are
allowedto vary over specifiedrangesPo > 0, P1 > 0, P2 > 0, 0 < eo < 2, C1 > 0 and C2 > O.
All six parameters appear explicitly in the control law and time derivative of the Lyapunov
function and thus affect the convergence rate, the transient response and the required control
effort.
In the next section we choose values for these parameters and perform numerical simulations.
5. Simulation Results
For simulationwe consider the dimensionlessequations (5) and assume c: = 0.1. We first
consider the stabilization problem in which there is no external force applied to the cart, that
is, Fd = O.In particular, we numerically show that the feedback control law u in (28) with v
defined in (37) stabilizes (6) to the origin. 1\\'0 sets of control parameters are chosen for the
control law, specifically,
Controller 1: Po = P1 = P2 = eo = C1= C2= 1;
Controller 2: Po = 10,P1 = 0.1, eo = 1.9,P2 = C1= C2= 1.
Global Stabilization of the Oscillating Eccentric Rotor
57
..
..
:1 \:
:' ':
" I:
j
x0
"
\
I
~
I
~
:I
:1
:, I
,/ \:
:/ I:
;, i
j
.
" ,
:\ (
;\ ':
,J
,,
..
.
"
Doned!line: U~co~trolled m6tion
Das,h I!ne: Co~tr~lIer 1
. .
. .
~.
-1
o
..
..
..
5
10
15
20
Soli\i I!ne: COn!r~lIer 2
. .
, .
..
25
TIME
30
35
40
45
50
Fig. 2. Time history of cart translation Xt of the uncontrolled and controlled motions with x(O)
= (1,0,0,of.
150
100
.......
50
,-50
Dotted
.100
5
10
15
20
line:
Uncontrolled
Dash
line: Controller
1
Solid
line: Controller
2
25
TIME
30
35
motion
40
45
50
Fig. 3. Time history of eccentric mass rotation X3 of the uncontrolled and controlled motions with
x(O)
= (l,O,O,O)T.
In the first example, we consider the case in which the cart has an initial displacement
while the disk has zero initial angular velocity,that is, x(O) = (l,O,O,O)T.Figures 2 and 3
illustratethe cart translationXl(r) and eccentricmass rotationX3(r) of the uncoq,trolledand
controlled motions.Note that the rotor is free to spin for the uncontrolledsystem.Figure 4
showsthe control torque u for Controllers 1 and 2 with respect to time. From Figures 2 and
3 it can be seen that the translationalmotion of the uncontrolledsystem is oscillatorywith
fixedamplitudeand frequencywhile the rotor angle is increasing.For the controlledsystem,
both Controllers1 and 2 drive the translationaland rotational oscillationsto rest. Note that
58
C.-I. Wan et al.
12
10
8
Dash line: Controller 1
Solid line: Controller 2
6
4
:> 2
0
.2
.4
.6
-8
0
5
10
15
20
25
TIME
30
35
40
Fig. 4. Control torque u for Controller I and 2 with x(O)
45
50
= (1,0,0,O)T.
0.8
Dotted
0.6
line:
Uncontrolled
motion
Solid line: Controller 2
0.4
0.2
-0.2
-0.4
-0.6
5
10
15
20
25
TIME
30
35
40
45
50
Fig. 5. Tune history of cart translation XI of the uncontrolled and controlled motions with x(O)
= (0,0,0, l)T.
Controller2 drives the motion to rest fasterthan Controller1. Sincethis was found to be the
case for all initialconditionstested, Controller1 will not be consideredfurther.
Next we considerthe case in which the cart is at rest while the desk has angular velocity
equal to the resonance frequency of the translational motion, that is, x( 0 f = (0,0, 0, 1)T.
Figures 5, 6 and 7 show the cart translation Xl(T), eccentric mass rotation X3(T), and angular
velocity X4(T) of the uncontrolled and controlled motions. It can be seen in Figure 7 that the
angular velocity X4(T) remains close to 1 for the uncontrolled motion, which shows that the
eccentric mass lies within the resonance capture zone. However, as shown in Figures 5, 6 and
Global Stabilizationof the OscillatingEccentricRotor
59
200
...
'.
150
"
motiOn;:
../~otted Iin~~UncontrO!j~d
.
! Solidline:fC;ontroller:2: . .
100
W
w
a:
C)
w
...
50
e.
M
x
-50
-100
,
..
-150
.200
0
5
10
15
20
25
TIME
30
35
40
45
50
Fig. 6. Time history of eccentric mass rotation X3 of the uncontrolled and controlled motions with
x(o) = (0,0,0, l)T.
1.2
...
0.8
'.
.....
0.6
0.4
....
x
0.2
o
-0.2
Dotted
-0.4
line:
Uncontrolled
motion
Solid line: Controller 2
5
10
15
20
25
TIME
30
35
40
45
50
Fig. 7. Time history of eccentric mass angular velocity X4 of the uncontrolled and controlled motions with
x(O) = (0,0,0,If.
.
7, the states of the controlled motion approach the origin very rapidly, while the control torque
u (not shown)decreaserapidly with the maximumtorque lul = 3.2 needed at T = O.
Next, we consider the case in which there are external disturbances applied to the cart and
show that the control law, although designed without regard to disturbances, atteimates the
forced translational oscillation. We thus consider (5) with sinusoidal forcing function
(43)
60
C.-I. Wanet al.
5
4
..
..
3
_1L
-2
..
Donedline:Uncontrolledmotion
-3,
-4
-5
0
Solid line: Controller2
10
20
30
40
50
TIME
60
70
80
90
100
Fig. 8. Tune history of cart translation Xl of the uncontrolled and controlled motions for resonant forcing WF
at amplitude AF
= 0.1
with x(O)
= (0,0, -11"/2,O)T.
=1
80
60
40
20
ffi"
a: 0
CJ
w
Q. -20
M
x
-40
-60
Doned line: Uncontrolled
-80
-100
o
motion
Solid line: Controller 2
10
20
30
40
50
TIME
60
70
80
90
100
Fig. 9. Time history of eccentric mass rotation X3 of the uncontrolled and controlled motions for resonant forcing
WF
1 at amplitude AF = 0.1 with x(O) = (0,0,-1I"/2,0)T.
=
where AF, WF are the amplitude and frequency of the external forcing. We also assume that
the unbalanced mass is initially aligned with the spring and that both the s:art and the disk
are at rest, that is, x(O) = (0,0 - 7rj2,0)T. Figures 8 and 9 show the translationxl(r) and
rotations X3(r) of the uncontrolled and controlled motions for resonant forcing WF = 1 at
amplitude AF = 0.1. It can be seen from Figures 8 and 9 that the controller attenuates the
amplitude of the translational oscillation by preventing the linear amplitude growth of the
uncontrolled motion. Figure 10 shows the time history of the control torque.
Global Stabilization of the Oscillating Eccentric Rotor
61
2
1.5
I
I
0.5
II
I I'
M
:>
..
0
-0.5
-1
\I
I
10
I
20
I
30
\
I
I
40
Fig. 10. Control torque u forresonant forcing WF
\
\
50
TIME
60
\
I
70
I
80
90
100
= I at amplitude AF = 0.1 withx(O) = (0,0,-7r /2, O)T.
40
30
20
10
'€ 0
.- --- -.-
-- -.-
--.:..;.:..;
-10
-201
Dotted line: Uncontrolled motion
Solid line: Controller 2 (AF=0.1)
.30r
Dash line: Controller 2 (AF=I)
10°
10'
FREQUENCY
Fig. 11. Amplitude-dependelt: frequency response of the translational oscillation of the uncontrolled and controlled
motions under sinusoidal forcing with amplitudes AF = 0.1 and AF = 1.
Finally, Figure 11 shows the amplitude-dependent frequency response of the translational
oscillation of the uncontrolled and controlled motions under sinusoidal forcing with amplitudes
AF = 0.1 and AF = 1. Controller2 is used for the controlledmotion. The vertical axis of
Figure 11 is the ratio of the steady-state amplitude of the translational oscillation to the
forcing amplitude. The forcing frequency ranges from 0.1 to 10 rad/sec. It is observed that
the controller is most effective near and above resonance and is more effective for smaller
disturbance amplitude except for frequencies near twice the resonance.
62
C.-l. Wanet aI.
6. Conclusions
Wehave deriveda globallyasymptoticallystabilizingcontrollaw for the oscillatingeccentric
rotorthat usestorquecontrolto simultaneouslybringboth translationaland rotationalmotions
to rest. Six independentparametersin the control law allow adjustmentof the closed-loop
response. In addition, numerical simulationsshow that the control law, although designed
withoutregard to disturbances,can attenuatesinusoidaldisturbancesover a broad frequency
range.
In a recent paper [2], we explored an application of this work, a rotational actua,torfor
suppressingtranslationalmotion.Two control laws were compared:(i) the control based on
integrator backsteppinggiven herein and (ii) an active implementationof a passive control
law. The passive control law emulates a vibration absorber that damps out both rotational
and translationalenergy.Notethat a simplerotationalrate dampercannotgloballyasymptotically stabilize the translationalmotion. The two approacheshave different advantages and
disadvantagesthat are currentlybeinginvestigated;interestedreaders shouldconsult [2].
References
Bacciotti, A., Local Stabilizability of Nonlinear Control Systems, Series on Advances in Mathematics for
Applied Sciences, Vol. 8, World Scientific, Singapore, 1992.
Bupp, R., Wan, C.-l, Coppola, V. T., and Bernstein, D. S., 'Design of a rotational actuator for global
stabilization of translational motion', in Active Control of Vibration and Noise, ASME Winter Meeting,
DE-Vol. 75, 1994, pp. 449-456.
Ewan-Iwanowski, R. M., Resonance Oscillations in Mechanical Systems, Elsevier, Amsterdam, 1976.
Hall, C. D. and Rand, R. H., 'Spinup dynamics of axial dual-spin spacecraft', Journal of Guidance, Control
and Dynamics 17(1), 1994,30-37.
Isidori, A., Nonlinear Control Systems, Second Edition, Springer-Verlag, New York, 1989.
Kanellakopoulos, I., Kokotovic, P. V., and Morse, A. S., 'A toolkit for nonlinear feedback design', Systems
and Control Letters 18, 1992, 83-92.
Kinsey, R. J., Mingori, D. L., and Rand, R. H., 'Nonlinear controller to reduce resonance effects during despin
of a dual-spin spacecraft through precession phase lock', in Proceedings of the Conference on Decision and
Control, Tucson, AZ, December, 1992,3025-3030.
Kokotovic, P. V., 'The joy of feedback: nonlinear and adaptive', IEEE Control Systems Magazine, June,
1992,7-17.
Kokotovic, P. V. and Sussmann, H. J., 'A positive real condition for global stabilization of nonlinear systems',
Systems and Control Letters 13, 1989, 125-133.
Marino, R., 'On the largest feedback linearizable subsystem', Systems and Control Letters 6, 1984, 345-352.
Marino, R., 'High gain stabilization and partial feedback linearization', in Proceedings of the Conference on
Decision and Control, Athens, Greece, December, 1986,209-213.
Rand, R. H., Kinsey, R. J., and Mingori, D. L., 'Dynamics of spinup through resonance', International
Journal of Non-Linear Mechanics 27, 1992,489-502.
Sontag, E. D. and Sussmann, H. J., 'Further comments on the stabilizability of the angular velocity of a rigid
body', Systems and Control Letters 12, 1989,213-217.
Vidyasagar, M., Nonlinear Systems Analysis, Second Edition, Prentice Hall, Englewood Cliffs, NJ, 1993.
Yee, R. K., 'Spinup dynamics of a rotating system with limiting torque', Master's Thesis, UCLA, 1981.