Full text of DOI:10.2514/2.4783

JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS
–
Vol. 24, No. 4, July August 2001
Robust Minimum Power/Jerk Control of Maneuvering Structures
Timothy A. Hindle¤ and Tarunraj Singh†
State University of New York at Buffalo, Buffalo, New York 14260
The focus of this paper is on the development of weighted minimum power/jerk control proŽ les for the rest-
to-rest maneuver of a  exible structure. To account for modeling uncertainties, equations, which represent the
sensitivity of the system states to model parameters, are derived. The original state-space model of the  exible
structure is augmented with the sensitivity state equations with the constraint that the sensitivity state variables
are forced to be zero at the end of the maneuver. This requirement attenuates the residual vibration at the end
of the maneuver caused by errors in system parameters. A systematic procedure for the design of the controller
is developed by representing the linear-time-invariant system in its Jordan form. This decouples the modes of
the system permitting us to address smaller-order dynamical systems. The proposed technique is illustrated via a
benchmark  oating oscillator problem.
I. Introduction
HE controlof the benchmark two-mass/spring/damper system
undergoing a rest-to-rest maneuver is to be considered in this
topic of sensitivity equation formulation is considered in Sec. IV.
Section V gives the same numericalexample consideredin Sec. III,
with the addition of sensitivity equations. A comparison between
the robust and nonrobust solutions is also drawn in this section. Fi-
nally, the paper concludes with a summary of the results obtained
in Sec. VI.
T
paper. This problem, representativeof many  exible structures,has
one  exible mode and one rigid-body mode. A fairly comprehen-
sive treatment of this family of problems has been presented by
Junkins and Turner.1 In previous research on this topic, time opti-
mal control proŽ les have been derived by Singh et al.,2 Ben-Asher
II. Problem Formulation
The weighted minimum power/jerk cost function
et al.,3 Farrenkopf, and Hablani.5 Desensitizingthe controlproŽ les
4
to modeling errors has been addressed by Swigert,6 Liu and Wie,7
and Singh and Vadali.8 Closed-form solutions have been obtained
for the optimal controlof the rest-to-rest maneuverusing minimum
"
#
³
´
Z
2
T
u
1
2
d
2u2 C
( )
1
min
³
d¿
t
d
power and minimum jerk cost functions by Bhat and Miu.9;10 Min-
0
R
u2
imum power solutions are obtained by minimizing
d¿, while
R
is considered,1 subject to the constraint
2
u
.d
t
solutionsare obtained by minimizing
=d /
.
d¿
minimum jerk
u
The term in these cost functions is the control effort. Recently, it
has been of interest to develop optimal solutions using a weighted
cost function, such as the weighted fuel/time optimal control con-
sidered by Singh.11 Here, the closed-form solution for the opti-
mal control of the rest-to-restmaneuverusing a weightedminimum
power/jerk cost functionis of interest.In the weightedcost function
consideredhere, the user can select the relativeimportanceof mini-
( )
2
MxR C xP C K x D Pu
»
M
K
is the
where
is the mass matrix, » the damping matrix, and
P
u
x
stiffness matrix. is the control in uence vector, and and are
( )
T
the scalar control input and state vector, respectively. In Eq. 1
is the speciŽ ed Ž nal time, and ³ is the weighting parameter to be
varied. The equations of motion for this system can be given in
state-space form as
(
)
mum power or equivalently,minimum control effort to minimum
(
)
jerk or equivalently,the minimum rate of change of control effort .
The solution for the control proŽ le obtained for linear-time-
invariantsystems, like the system consideredin this paper, often as-
sumes known constantsystemparameters.With thisassumptionthe
simulatedsystem responsefor a rest-to-restmaneuverwill meet the
required endpointconditionswith zero residual vibration. In actual
physical systems it is impossible to know the exact values of the
system parameters. Thus, any solution using the control proŽ le ob-
tained assumingconstantsystem parameterswill have zero residual
vibrationonly when the actual system parametersexactlymatch the
design parameters used to obtain the control proŽ le. With this in
mind, it is the goal of the researchers to obtain a solution that is
µ
¶
µ
¶
I
0
0
( )
3
P D
w
C
u
w
¡M¡1 K ¡M¡1
M
¡1 P
»
|
{z
}
|
{z
}
A
B
( )
4
y D C C Du
w
( )
( )
Transformingthissystemof equations[Eqs. 3 and 4 ] into Jordan
canonical form gives
( )
zP D J
z
C
b
u
5
|{z}
|{z}
(
robust to errors in system parameters for example, damping ratio,
V w
V B
)
natural frequency . To do this, sensitivity equationsare derived and
y D C¤ z C Du
( )
6
added to the state-space equations before transforming them into
Jordan canonical form. It will be shown that with the addition of
these equations, which force the sensitivity state variables to zero
at the end of the maneuver, there is a reductionin residual vibration
caused by errors in system parameters.
|{z}
¡1
C V
J
A
V
where is the Jordan canonicalform of and is the transforma-
( )
tion matrix. The solution of Eq. 5 is given as
The paper begins with the problem formulation in Sec. II.
Section III gives a numerical example with results presented. The
Z
t
2
e¡Jt2 z t ¡ e¡Jt1 z t
D
e¡J¿ bu
.¿/ d¿
( )
7
. 2/
. 1/
t
1
Received 13 July 1999;revision received 15 December 1999;accepted for
To obtain the optimal control for this problem, calculus of vari-
ations will be used in order to perform the function optimization.
Using this method, for the chosen cost function [Eq. 1 ] to be min-
c
°
publication 25 October 2000. Copyright 2000 by the American Institute
of Aeronautics and Astronautics, Inc. All rights reserved.
¤Graduate Student, Mechanical and Aerospace Engineering.
†Associate Professor, Mechanical and Aerospace Engineering.
( )
imized, the following performance criteria must be minimized:
816

HINDLE AND SINGH
817
"
#
³
´
Z
2
n D
C
C
p
2
2
2
T
u
1
2
d
(
(
)
)
13
14
I D
2u2 C
³
d¿
(
)
(
)
(
)
for p  exible modes
for 1 rigid-body mode
# of B.C.s
t
d
0
¤
D
C
dimension.¸ /
2
(
)
# of B.C.s
µ
¶
Z
T
¤
T
¡JT z T ¡
e¡J¿ bu
.¿/ d¿
( )
8
C
e
.¸ /
. /
The cost function used here is the same as used by Junkins and
Turner.1 Junkinsand Turner convert the problem to a standard form
for Pontryagin’s Principle, whereas here the calculus of variations
is used for the function optimization. Also, Junkins and Turner de-
rive closed-form solutions for the case of rigid-body motion only,
whereasa systemwith rigid-bodyand  exiblemodesis addressedin
thispaper.The procedurefor obtainingthe controlusing the method
developed here will be demonstrated in the next section.
0
where ¸¤ in this equation represents the Lagrange multipliers.
( )
T
Equation 8 is derived by assuming the maneuver time to be
,
the initial time and initial conditionsto be zero, and by augmenting
the cost function with Eq. 7 . By taking the Ž rst variation of this
equation and setting it equal to zero, the that minimizes Eq. 1
can be obtained. The Ž rst variation is expressed as
( )
( )
u
µ
¶
³
´
Z
T
T
2u
t2
u
d
d
d
¤
( )
I D
2u ¡
¡
T e¡J¿
b
u
C
u
±
±
³
.¸ /
± d¿
9
III. Numerical Example 1
t
d
0
0
The benchmark two-mass/spring/damper problem will now be
considered. This system is a model of a  exible structure with one
rigid-bodymode and one  exible mode. Figure 1 shows the system
u
For this equation to be equal to zero for all ± , the quantitiesinside
the brackets must be equal to zero. This requirement results in a
differential equation in , which is
m
m
x
tobe considered,withthe two masses 0 and 1, thespringconstant
u
k
c
x
, anda viscousdamper .IntheŽ gure 0 and 1 arethedisplacement
of the Ž rst and secondmass, respectively.The inputforce is denoted
2u
t2
d
d
¤
u
y
2u D ¡
T e¡Jt b
as and the output as .
( )
10
¡
³
.¸ /
The differentialequation that governs this system is given by
u
The control in this equation can be determined by solving for the
homogeneousand particular parts of this equation and then adding
(
)
MxR C xP C K x D Pu
»
15
u
thesesolutionstogetherto obtainthe totalsolutionfor . The bound-
u
ary conditions for are obtained from the necessary conditions for
where
( )
optimality. From Eq. 9 , the conditions for optimality require that
u
the second bracketed term must be equal to zero. Because ± is
µ
¶
µ
¶
m0
k
¡k
0
u
t
arbitrary, this implies that d =d at the initial and Ž nal time must
be equalto zero.There is no requirementthatthe control be forced
(
(
)
)
M D
K D
;
16
17
u
m1
¡k
k
0
to zero at the initial and Ž nal time, as is done in the minimum jerk
solution obtained in Ref. 10. In that paper the control is forced to
zero at the initial and Ž nal times in order to make the controlpracti-
cal to input on a real physical system, although the cost function is
not minimized.Thus, the minimum jerk solutionobtainedin Ref. 10
is suboptimal.Here, the optimal solution of the weighted minimum
power/jerk solution will be considered, and it will be shown that
as ³ goes to zero the solution converges to the optimal minimum
jerk solution; conversely, the optimal minimum power solution is
obtained as ³ goes to inŽ nity.
µ
¶
µ ¶
µ ¶
c
¡c
x0
1
D
x D
P D
»
;
;
¡c
c
x1
0
(
)
The linear ordinary coupled differentialequation from Eq. 15 can
be written in the state-space form as
(
(
)
)
P D A C Bu
w
w
18
19
A general closed-form solution for the weighted minimum
(
)
power/jerk control can be found by solving Eq. 10 with the nec-
essary condition for optimality that the derivative of the control at
initial and Ž nal time is set equal to zero. The form of the solution
will depend on the size of the system as well as the number of
modes present. The general closed-formsolution for a system with
y D C C Du
w
where
p
2 3
2
3
one rigid-body mode and  exible modes, for ³ > 0, is given as
x0
0
0
0
1
0
0
1
³ t
u t D
. /
C
t C
e
C
e¡³ t
¸
4
¸
¸
¸
6 7
6
6
4
7
7
5
1
2
3
x1
0
6 7
(
)
D
A D
w
;
20
4 5
p
xP0
¡k m
k m
=
¡c m
c m
=
=
=
X
0
0
0
0
£
¤
¡a t
i
¡a t
i
( )
11
C
e
b t C
e
b t
cos. /
i
¸
sin.
/
¸
.3 C 2i/
i
.4 C 2i/
xP1
k m
¡k m
c m
¡c m
=
=
=
=
1
1
1
1
i D 1
a
i
b
i
where is the real part of the th complex conjugate pole and is
i
i
the imaginary part of the th complex conjugatepole of the system.
(
)
D
For ³ 0 the solution of Eq. 10 is
u t D
. /
C
t C t2 C t3
¸
¸
¸
¸
4
1
2
3
p
X
£
¤
¡a t
¡a t
i
i
( )
12
C
e
b t C
e
b t
cos. /
i
¸
sin.
/
¸
.3 C 2i/
i
.4 C 2i/
i D 1
which corresponds to the minimum jerk solution. The parameters
(
)
(
)
( )
¸
i
in Eq. 11 are found by simultaneouslysolving Eq. 7 and the
( )
( )
boundaryconditionsfrom Eq. 9 . The number of parameters ¸i in
n
this solution is , which in a generalcase depends on the size of the
system. For the system considered here with one rigid-body mode,
n D
C
p p
2 , where the scalar is the number of  exible modes of
4
n
the system.The valueobtainedfor can be brokendown as follows:
Fig. 1 Two-mass/spring/damper system.

818
HINDLE AND SINGH
2
3
(
)
From Eq. 24 the unknown ¸ vector is determined using
0
0
2
3
6
6
4
7
7
£
¤
e¡J.4¼/
z
.4¼/
( )
21
B D
C D 0
1
0
0
D D
;
;
0
5
6
7
5
m
1=
¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢
D S¡1
0
(
(
)
)
¸
27
28
4
0
0
0
When the state-spaceequationsare convertedinto Jordancanonical
Using the numerical values for this example, ¸ is given as
(
)
form, Eq. 18 can be rewritten as
2
3
0:106437
2 3
2
3 2 3
2
3
6
7
zP1
z1
b11
b21
b31
b41
0
0
0
0
1
0
0
0
0
0
0
0
¡
0:016940
6
6
6
6
7
7
7
7
6 7
6
6
4
7 6 7
6
6
4
7
7
5
zP2
z2
E¡
0:15
6
6 7
7 6 7
D
¸
(
)
D
C
u
22
4 5
5 4 5
zP
p1
0
z3
0
¡
¡
0:043842
zP34
p2
z4
6
4
7
5
0:026902
|
{z
} | {z } | {z }
0
J
V w
V B D b
(
)
These ¸ values are substituted into Eq. 11 , and the solution is
obtained. The control for the rest-to-rest maneuver given in this
example is then
J
A
V
where is the Jordan canonical form of the matrix,
transformation matrix, and
is the
p1
p2
are the complex conjugate
and
poles of the  exible mode.
u t D
¡
t C
E¡ et
6/
. / 0:106437 0:01694
.0:15
The rest-to-rest maneuver of the undamped two-mass/spring/
damper benchmark problem is considered here. For simplicity, the
following parameters are used:
e¡t
¡
t
( )
29
¡
.0:043842/
0:026902 sin. /
(
)
It can be noted from Eq. 26 that there exist large variations in
the magnitude of the elements of the matrix, which might lead to
numericalinstabilitiesbecausethe matrix requiresinversion.This
is as a result of the
inordinately large for large maneuver times. This can be remedied
by normalizationof the maneuver time to one.
Figure 2 is a plot of the control proŽ le [Eq. 29 ] and the position
1
2
S
(
)
m D m D
k D
c D
t D
1;
;
0;
0 initial time
0
1
1
S
³ t
e
term in the controlproŽ le, which can become
(
)
(
)
23
t D
D
4¼ Ž nal time ;
³
1
2
m
and
m
The input is on 0, and the output is the position of 1. The initial
(
)
m
m
are both zero, and the Ž nal positions are
1
positions of
0
(
)
m
usingthe valueschosenin Eq. 23 . The
1
(
)
m
of both masses 0 and
(
)
chosen arbitrarily to be one. For this system the natural frequency
rest-to-restmaneuver is completed without any residual vibration.
The next point of interest is the effect varying the weighting pa-
rameter³ has onthevaluesof ¸. First,thegeneralformof the control
of the elastic mode is equal to 1. Using the parameterized closed-
(
)
u
form solution of [Eq. 11 ], the solution is found by rewriting
( )
Eq. 7 as
(
)
[Eq. 11 ] for this example is given as
2
3
e¡J.4¼ /
¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢
z
.4¼/
³ t
u t D
. /
C
t C
e
C
e¡³ t
C
e¡at
bt C e¡at
bt
30
¸
¸
¸
¸
¸
sin.
/
¸
6
cos.
/
)
1
2
3
4
5
6
4
7
5
(
(
)
24
D S
¸
0
where
0
(
)
a D
b D
0;
1
31
S
where is given as
With these valuesof ¸ clearly deŽ ned, the effect of varying ³ on the
parameters can be determined and is illustrated in Figs. 3 5. The
parameter ¸6 is not shown because it is zero or negligible for all ³
values for this undamped system under consideration.The smallest
2
6
6
6
6
6
4
3
7
7
7
7
7
5
Z
–
t
2
e¡J¿ bu
.¿/ d¿
6
6
6
7
7
7
t
1
¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢
D
value of ³ plotted for all cases is ³ 0:01, which is chosen to be
( )
25
S D
u
Jacobian
d
w.r.t. ¸
a small number greater than zero because the weighted solution is
t
. 2/
D
D
not valid for ³ 0. Recall that in the case of ³ 0, the minimum
t
d
(
)
jerk solution is obtained [Eq. 12 ].
u
d
d
It willbe shownherethatas ³ goesto zero the weightedminimum
power/jerk control proŽ le converges to the minimum jerk control
proŽ le. This will be shown for the example system consideredhere.
The convergence of the weighted minimum power/jerk control to
the minimum power control as ³ tends toward inŽ nity will also be
t
. 1/
t
S
The matrix for thisundampedsystemunderconsiderationis given
analyticallyas
2
3
e4¼³ ¡ e4¼³
C
e¡4¼³ C e¡4¼³
¡
64
3
4¼³
1 4¼³
1
2
3
¡
¡
¡
8 ¼
¼
4¼
0
6
7
2
2
³
³
6
6
6
7
7
7
¡ C e4¼³
¡ C e¡4¼³
1
1
2
¡
6
4¼
0
8¼
0
0 7
6
7
³
³
6
6
6
6
7
7
7
7
¡ C e4¼³
¡e¡4¼³
C
¡
p
p
1
1
¡
¡
¡
4
1¼
p
p
2
1¼ 2¼
(
)
S D
26
C
¡
¡
³
1
³
1
6
7
6
6
6
6
7
7
7
7
¡ C e4¼³
¡e¡4¼³
C
¡
p
p
1¼ 2¼
1
1
¡
¡
¡
0
4
1¼
p
p
2
¡
³
¡
C
³
1
³
1
6
7
6
4
7
5
e4¼³
¡ e¡4¼³
0
0
1
1
³
1
1
0
0
¡
³
³

HINDLE AND SINGH
819
Fig. 2 Undamped system minimum power/jerk rest-to-rest maneuver.
Fig. 3
¸
and ¸2 vs weighting parameter ³ for undamped system minimum power/jerk control.
1
demonstratedfor this two-mass/spring/damper example. Before the
convergence of the solution in the limits of ³ can be shown, the
minimum power and minimum jerk solutions must be stated. The
minimumpowercontrolproŽ le for thissystem,which is determined
using the closed-form minimum power solution given by Bhat and
Miu,10 is
The minimum jerk solution is obtained using the same procedure
as in Ref. 10, except that the derivative of the control is forced
to zero at initial and Ž nal time. Bhat and Miu10 require the con-
trol to be zero at initial and Ž nal times, resulting in a suboptimal
solution. These solutions are shown here for demonstration of the
convergenceof the weighted minimum power/jerk to the minimum
power and minimum jerk solutions in the limits of the weighting
parameter ³. Figures 6 and 7 show the convergenceof the weighted
minimum power/jerk solution in the limits of ³. A ³ value of 10 is
chosen to demonstrate the convergence of the weighted minimum
power/jerk solution to the minimum power solution as ³ goes to
inŽ nity. Figure 6 plots the solutionobtained for this case along with
(
)
u t D
¡
t ¡
t
. / 0:08961 0:01426
0:02852 sin. /
32
while the minimum jerk solution is
u t D
C
t ¡
t2
. / 0:05017 0:02026
0:00674
(
)
the minimum power solution given by Eq. 32 . A ³ value of 0.005
(
)
C
t3 ¡
t
0:00036
0:02026 sin. /
33
is chosento demonstratethe convergenceof the weightedminimum

820
HINDLE AND SINGH
Fig. 4
¸
and ¸4 vs weighting parameter ³ for undamped system minimum power/jerk control.
3
Fig. 5 ¸5 vs weighting parameter ³ for undamped system minimum power/jerk control.
power/jerk solution to the minimum jerk solution as ³ goes to zero.
where for this case
Figure 7 plotsthe solutionobtainedfor thiscase along with the min-
(
)
imum jerk solution of Eq. 33 . In both Figs. 6 and 7 there appears
to be only one curve because the two curves in each plot lie on top
of each other.
1
(
(
)
)
a D ¡
35
36
p4
¯
b D
15 4
Following the same procedure used to determine the con-
trol in Fig. 2, the control for the damped benchmark two-mass/
spring/damper problem can be obtained. Using the values given in
Eq. 23 for this system, this time with
the control for the rest-to-rest maneuver is found to be
(
)
Figure 8 is a plot of the control proŽ le [Eq. 34 ] and the position
(
)
c D
c D
0:25 instead of
0,
(
)
m0
m
of both masses
and
for the system using the values given
for this damped system. As with the undampedcase, the rest-to-rest
maneuver is completed without any residual vibration.
This section has demonstrated the validity of the weighted min-
imum power/jerk control determination as applied to a rest-to-rest
maneuver of a  exible structure, with and without the inclusion of
damping. The next section extends this idea to develop a robust
solution when there are errors present in system parameters.
1
u t D
¡
t C
E¡ et
6/
. / 0:108260 0:019430
.0:40
£¡p ¯ ¢ ¤
¡
¡
e¡t
¡
e.t=4/
t
15 4
.0:023246/
0:003277
sin
p
£¡
¯ ¢ ¤
e.t=4/
t
(
)
34
0:002571
cos
15 4

HINDLE AND SINGH
821
Fig. 6 Minimum power control vs weighted minimum power/jerk control when ³ = 10.
Fig. 7 Minimum jerk control vs weighted minimum power/jerk control when ³ = 0.005.
k
desensitize the system with respect to the stiffness , the damping
IV. Robust Solution — Sensitivity Equations
(
c
, or the natural frequency taking into considerationboth errors in
The goal of this paper is to formulate a control that minimizes
a weighted minimum power/jerk cost function while being robust
to errors in system parameters. The motivation comes from the fact
that in real physical systems the values which are assumed to be
)
mass and stiffness . Here, the sensitivitywith respectto the stiffness
(
)
is of interest thereby the natural frequency . It will be shown that
the control proŽ le obtained with the addition of these sensitivity
k
equations reduces residual vibration when errors in the value of
(
)
constant e.g., stiffness or natural frequency are not known exactly
or havebeenestimated.When thisis the case,for the rest-to-restma-
neuver considered here the control proŽ le determinedusing known
constant system parameters will bring the system to rest only when
the actual system parameters are exactly the same as those assumed
to obtainthe controlproŽ le. When the actual system parametersare
notthesame,residualvibrationwillexistattheendofthe rest-to-rest
maneuver.The goalis to minimizethisresidualvibration.To do this,
sensitivity equations are derived, which represent the sensitivity of
the system to model parameters. This procedure can be applied to
(
)
stiffness are present. The equations of motion for the benchmark
two-mass/spring/damper problem considered in Sec. III are
xR D ¡k m x C k m x C ¡c m xP C c m xP C u m
.
=
/
. =
/
.
=
/
. =
/
=
0
0
0
0
1
0
0
0
1
0
(
(
)
)
37
xR D k m x C ¡k m x C c m xP C ¡c m xP
. = . =
/
.
=
/
/
.
=
/
38
1
1
0
1
1
1
0
1
1
k
By taking the derivativeof these two equationswith respect to ,
the following equationsare obtained:

822
HINDLE AND SINGH
Fig. 8 Damped system minimum power/jerk rest-to-rest maneuver.
³
³
³
³
´
³
´
where
xR
k
x
x
c
xP
d
d
d
d
k
d
0
0
1
0
C
C
¡
C
¡
C
¡
C
C
k
m0
k
k
d
m0
d
d
2
3
´
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
1
0
0
¡c
xP
x0
x1
d
1
6
6
6
6
6
6
6
4
7
7
7
7
7
7
7
5
(
(
)
)
C
D
³
0
39
40
m0
k
m0
m0
d
´
´
A D
xR
k
x
x
¡c
xP
d
d
d
d
k
d
1
0
1
0
¡k m k m
¡c m c m
=
=
=
=
¡
k
m1
k
k
d
m1
d
d
k m ¡k m
c m ¡c m
=
=
=
=
´
¡
m
m
¡ k m
2 =
¡ c m
2 =
1=
1=
0
0
c
xP
x0
x1
d
1
¡
D
0
(
)
)
47
m1
k
m1
m1
d
To simplify formulationwhile still demonstratingthe beneŽ t of this
method, the values of the two masses are assumed to be equal as
m D m
2
3
0
0
0
in the preceding example .
case, but it makes for simple understanding of the procedure. If
this assumption is not made, instead of having the single sensitiv-
ity equation given in Eq. 44 , Eqs. 39 and 40 would represent
the sensitivity equations with which the original state-space model
would be augmented.Using the equal mass assumption,the follow-
1/. This does not have to be the
6
6
6
6
6
7
7
7
7
0
£
¤
(
B D
C D 0
1
0
0
0
0
D D
;
;
0
48
7
m
1=
(
)
(
)
(
)
6
7
5
4
0
0
(
)
(
)
ing equation is obtained using Eqs. 39 and 40 :
2 3
x
x
d
d
0
1
x0
(
(
)
)
D ¡
41
42
k
k
d
d
6 7
x1
6 7
(
)
Substituting into Eq. 39 gives
6 7
x2
6 7
(
)
³
´
³
´
D
w
49
6 7
xP0
xR
c
xP
k
x
d
x0
m
x1
m
d
d
0
0
0
6 7
C
C
C
¡
D
2
2
0
4 5
xP
k
m
k
m
k
d
d
d
xP12
DeŽ ning a new state variable by
x
d
0
(
(
)
)
D x2
43
44
Once the sensitivity equations are added and the system equations
are placedinstate-spaceform,the sameprocedureusedto determine
the control from Sec. II can be used. It can be shown for this
case that the same control proŽ le will be obtained if the sensitivity
k
d
(
)
gives the sensitivity equation, from Eq. 42 , to be
(
)
xR C c m xP C k m x C x m ¡ x m D
2. = 2. =
/
/
=
=
0
2
2
2
0
1
c
istakenwithrespecttothedamping . Thisimpliesthatdesensitizing
the system with respect to stiffness simultaneouslydesensitizesthe
system with respect to damping. By forcing the sensitivitystates to
zero at Ž nal time, the residual vibration is reduced. The example
considered in Sec. III will be considered in the next section, this
time with the additionof the derivedsensitivityequations.A general
closed-form solution for the robust weighted minimum power/jerk
Now, this equation will be added to the dynamical equationsof the
system and written in state-space form. The new system equations
(
)
m D m
are assuming
0
1
(
(
)
)
P D A C Bu
w
w
45
46
( )
controlcanbefoundbysolvingEq. 10 withthenecessarycondition
y D C C Du
w

HINDLE AND SINGH
823
Fig. 9 Undamped system nonrobust and robust control without error in k.
for optimality that the derivative of the control at initial and Ž nal
and the control is given as
u t D
(
)
J
time is set equalto zero.Also, the matrixgiven in Eq. 10 must be
augmented with the sensitivity equations to analytically obtain the
¡
t C
E¡ et ¡
6/
e¡t
.0:042876/
. / 0:106792 0:016996
.0:14
(
)
robustminimum power/jerk closed-formsolution given in Eq. 50 .
The form of the solution will depend on the size of the system
as well as the number of modes present. The general closed-form
solution for the benchmark problem having one rigid-body mode
(
)
¡
t ¡
t C
t
t
0:026739sin. / 0:005400cos. / 0:000859 cos. /
52
(
)
(
m
out-
1
Figure 9 is a plot of the control input and position of
)
(
)
put for both the nonrobust solution [Eq. 29 ] and robust solution
p
and  exible modes with the addition of the derived sensitivity
(
)
k
[Eq. 52 ] when there are no errors in the system parameter . Both
solutions reach the Ž nal position without any residual vibration.
equations is given as
p
X
£
m
for clarity, but it can be
0
Figure 9 does not show the position of
³ t
¡a t
u t D
. /
C
t C
e
C
e¡³ t
C
e
b t
sin. /
i
i
¸
¸
¸
¸
¸
1
2
3
4
.1 C 4i/
m
inferred from the displacementof 1 that the residualenergyof the
systemafter the completionof the maneuveris zero becausethe two
masses are statically coupled. In the following plots the position of
i D 1
¡a t
i
i
C
C
e
b t C .3 C 4i/te¡a t
b t
sin. /
i
¸
¸
cos.
/
¸
.2 C 4i/
i
m0
will not be shown for the aforementionedreason.
¤
.4 C 4i/te¡a t
b t
/
i
Figure 10 is a plot of the same control input for the nonrobust
and robust solutionsshown in Fig. 9, this time when there is a 20%
i
(
)
50
cos.
1
k k D
¤
a
i
b
high error in [.
1:2 . 2 /]. The Ž gure shows a reduction in the
where is the real part of the th complex conjugate pole and is
i
i
i
k
the imaginary part of the th complex conjugatepole of the system.
This generalclosed-formsolution is only valid when the sensitivity
residual vibration as a result of the error in the system parameter
with the robust solution.
Figure 11 is a plot of the residual energy in the system at the
(
)
is taken with respect to the stiffness or damping for the system
considered.The nextsectionusesthisclosed-formsolutionto obtain
theweightedminimumpower/jerkcontrolfortheexamplediscussed
inSec. IIIwiththeinclusionofthesensitivityequationsderivedhere.
Both the undamped and damped systems will be addressed.
k
k
speciŽ ed Ž nal time vs the actual system value using a design
1
2
value of for both the nonrobustand robustsolutions. The residual
energy in the system at the Ž nal time is a scalar quantity deŽ ned as
q
¡
¢
xP
T
1
2
xr KxrT C xP MxP
( )
53
D
V. Numerical Example 2
residual energy
(
)
Using the general closed-form solution given in Eq. 50 , the
robustweightedminimumpower/jerkcontrolwillbe determinedfor
thebenchmarkproblemalreadyconsideredwithsystemvaluesgiven
by Eq. 23 and the additionof the sensitivityequationderivedin the
preceding section. First the undamped system will be considered,
followedby the dampedsystem.Followingthe same procedureused
inSec. III,thefollowing¸ valuesaregivenfortheundampedsystem:
K
M
where is the stiffnessmatrix, is the mass matrix, is the vector
x
of velocities, and is a vector of residual positions deŽ ned as
r
(
)
( )
54
x D x t ¡ x
t
2
. 2/
. /
r
desired
2
3
0:106792
From this Ž gure the robustsolution for this system reduces residual
6
7
¡
0:016996
k
vibration when the actual system values are less than approxi-
6
6
6
6
6
6
7
7
7
7
mately 0.405 and/or greater than approximately 0.48, with the ex-
E¡
0:14
6
ception that both the nonrobust and robust solutions will have zero
¡
¡
0:042876
7
5
k D
residual vibration at the design value of
12 . In the region where
(
)
51
D
¸
7
6 0:0267397
· k ·
0:405
0:48, the nonrobust solution has a smaller residual vi-
6
6
6
4
7
7
bration as compared to the robust solution.
¡
0:005400
7
Next, the robust solution for the damped system considered in
Sec. III will be considered.The values for this system are given by
0
0:000859
(
)
c D
c D
Eq. 23 with the value of
0:25 instead of
0 for this damped

824
HINDLE AND SINGH
Fig. 10 Undamped system nonrobust and robust control with 20% error in k.
(
)
Fig. 11 Residual energy at Ž nal time vs k value for nonrobust and robust control undamped .
system. Following the same procedure as just stated, the ¸ values
for this system are
and the control is given as
u t D
¡
t C
E¡ et ¡
6/
e¡t
. / 0:115933 0:023130
.0:83
.0:011278/
2
3
0:115933
£¡p ¯ ¢ ¤
6
7
C
£
¡
e.t=4/
t C
e.t=4/
e.t=4/
0:011957
sin
15 4
0:005039
¡
0:023130
6
6
6
6
6
7
7
7
7
p
p
£¡
¯ ¢ ¤
£¡
¯ ¢ ¤
E¡
0:83
6
t ¡
t
t
cos
15 4
0:001404
sin
15 4
¡
0:011278
7
(
)
55
p
D
£¡
¯ ¢ ¤
¸
6
7
e.t=4/
t
t
6 0:011957 7
(
)
0:000987
cos
15 4
56
6
6
6
4
7
7
7
5
0:005039
(
)
(
out-
m
Figure 12 is a plot of the control input and position of
1
¡
0:001404
0:000987
)
(
)
put for both the nonrobust solution [Eq. 34 ] and robust solution
¡
(
)
[Eq. 56 ] for this damped system when there are no errors in the

HINDLE AND SINGH
825
Fig. 12 Damped system nonrobust and robust control without error in k.
Fig. 13 Damped system nonrobust and robust control with 20% error in k.
k
system parameter . Both solutions reach the Ž nal position without
Fig. 14, with the exception that both the nonrobust and robust
any residual vibration. Figure 13 is a plot of the same control input
solutions will have zero residual vibration at the design value of
1
k D
for the nonrobust and robust solutions shown in Fig. 12, this time
.
2
1
k k D
¤
when there is a 20% high error in
[
1:2 . 2 /]. The Ž gure shows
This section has demonstrated the beneŽ t of the derived sen-
a reductionin the residualvibrationcausedby the errorin the system
sitivity equations when there are errors present in system param-
eters. Both the undamped and damped systems considered dis-
play reduced residual vibration with the addition of sensitivity
equations. Though this technique may not reduce residual vi-
k
parameter with the robust solution.
Figure 14 is a plot of the residual energy in the system at the
(
)
k
4¼ vs the actual system value using a
t D
speciŽ ed Ž nal time
2
1
(
)
k
k
design value of for both the nonrobust and robust solutions.
bration for all actual system values as seen in Fig. 11 , it
does prove to be a useful method to locally reduce the residual
vibration.
2
From this Ž gure the robust solution for this damped system re-
k
duces residual vibration for all actual system values shown in

826
HINDLE AND SINGH
(
)
Fig. 14 Residual energy at Ž nal time vs k value for nonrobust and robust control damped .
–
VI. Conclusions
of Guidance, Control, and Dynamics, Vol. 12, No. 1, 1989, pp. 71 81.
3Ben-Asher, J., Burns, J. A., and Cliff, E. M., “Time-Optimal Slewing of
Flexible Spacecraft,” Journal of Guidance, Control, and Dynamics, Vol. 15,
A systematic procedure to obtain the closed-form solution for
the rest-to-rest maneuver of the benchmark problem has been in-
troduced, which minimizes the weighted power/jerk cost function.
It has been shown that using the closed-form solution obtained for
the minimum power/jerk control the minimum jerk and minimum
power solutions are approached as the weighting parameter ³ ap-
proaches zero and inŽ nity, respectively. The concept of sensitivity
equations has been introduced, which, when added to the system
state equations, gives a control that is robust to errors in system
parameters.It has been shown that this robustcontrolreducesresid-
ual vibration when the actual system parameter is in the vicinity
of the design parameter used to derive the control. Though the ro-
bust solution obtained may not be an improvement for all possi-
ble errors in the system parameters, particularly when the param-
eter error is large, it does prove to be a useful technique for small
perturbations thus giving a locally robust solution. Extensions of
this work will include a study of the effect of varying damping
as well as the application of this technique to more complicated
systems.
–
No. 2, 1992, pp. 360 367.
4Farrenkopf, R. L., “Optimal Open-Loop Maneuver ProŽ les for Flexible
Spacecraft,” Journal of Guidance, Control, and Dynamics, Vol. 2, No. 6,
–
1979, pp. 491 498.
5Hablani, H. B., “Zero-Residual-Energy, Single-Axis Slew of Flexible
Spacecraft with Damping, Using Thrusters: A Dynamics Approach,” Pro-
ceedings of theAIAA Guidance,Navigation,andControlConference, Vol. 1,
–
AIAA, Washington, DC, 1991, pp. 488 500.
6Swigert, C. J., “Shaped Torque Techniques,” Journal of Guidance, Con-
–
trol, and Dynamics, Vol. 3, No. 5, 1980, pp. 460 467.
7Liu, Q., and Wie, B., “RobustiŽed Time-Optimal Control of Uncertain
Structural Dynamic Systems,” Proceedings of the AIAA Guidance, Naviga-
tion, and Control Conference, Vol. 1, AIAA, Washington, DC, 1991, pp.
–
443 452.
8Singh, T., and Vadali, S. R., “Robust Time-Optimal Control: Frequency
Domain Approach,” Journal of Guidance, Control, and Dynamics, Vol. 17,
–
No. 2, 1994, pp. 346 353.
9Bhat, S. P., and Miu, D. K., “Solutions to Point-to-Point Control Prob-
lems Using Laplace Transform Technique,” Journal of Dynamic Systems,
–
Measurement, and Control, Vol. 113, No. 3, 1991, pp. 425 431.
10Bhat, S. P., and Miu, D. K., “Minimum Power and Minimum Jerk Con-
trol and Its Application in Computer Disk Drives,” IEEE Transactions on
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vers, Elsevier, Amsterdam, 1986.
Magnetics, Vol. 27, No. 6, 1991, pp. 4471 4475.
–
11Singh, Tarunraj, “Fuel/Time Optimal Control of the Benchmark Prob-
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