NASA
Contractor
INTEGRATED
Report
182020
CONTROL/STRUCTURE
DESIGN
J. A. Bossi, K. S. Hunziker, and R. H. Kraft
Boeing
Defense
and Space
Aerospace & Electronics P.O. Box 3999 Seattle,
WA
Contract
Group Division
981242499
NAS118762
December,
1990
NqI151RO
(NAKACRIa2020) C_NTRNLSTRtJCTURE
INI_GRATED DFS[GN 2ind]
(_oein]
Co.)
Aerospdce
_5
9
ReDurt CSCL
OIC
G31oo
N/ A National Aeronautics Space Administration
and
Langley Research Center Hampton, Virginia 236655225
Integrated Joseph
Control/Structure
A. Bossi,
Boeing
K. Scott
Aerospace
Hunziker,
Design
and Raymond
& Electronics,
Seattle,
H. Kraft
Washington
Abstract A new The
approach
approach
extra action
integrates
opportunities
for avoiding
formulation
employed.
Control
design
based
methodology.
lated
and optimized
been
concerned
structure. results
Details are given
and
some
new,
based
a set of integrated
and control
the structure
and for discovering
control
duce
for the design
Boyd's
variables
A performance with respect
with achieving
whose
index
mission
involving
structures
processes
effects
reflecting
of Youla
spacecraft
requirements
of the integrated the integrated
mission
approach
2
PRECEDING
PAGE BI.ANK
NOT
Initial more
FILMED
is to pro
optimizationgoals
is formu
studies flexible
are presented
of a flexible
platform.
A linear
variables
through
with a lighter,
redesign
inter
parameterization
variables.
design
providing
design.
design
are selected
design
thereby
structural
with structural
values
is described.
of controlstructures
for future
to the integrated
of the formulation from a study
avenues
are coupled
space
design
implementation
parameters
design
controller
of the disastrous
unexpected
on
of flexible
have space and
geostationary
Introduction The "Integrated NASA
contract
18762.
The
computer from
Con_ol/Structure
"Aircraft
objective
code
of this effort
was
analysis
was
to provide
data base
system
as model
generation,
To accomplish the statement 1.
selection
element
methods
grated design
problem,
4.
study of the issues
documentation significant
differentiating
the
method
involved
we implemented
structure's
that
([AC)
provide
not only
to analysis
the multisoftware
[1]
an efficient
modules
and simulation
analysis
is normally
such
tools.
entailed
is the final report
problem
(as written
in
and the appropriate
IAC interfaces
for the pilot inte
design
of practical
problem,
performance
here.
the original several
were
made
the openIoop
approach
functions
By using
a control
and verified
structure
designs
objectives,
sensitivities
to avoid
the
By
without
difficult
formulation
themselves
and
(based
on
are parameters,
its usefulness
for the kind of
tool that can not only perform but also optimize
we demonstrated
its design
the for a
that such optimization
can result.
we have
areas that may warrant
of the task and describes
responses
efforts. of this research.
response
able
and constraints, better
modal
a software
controlled
in the course
we were
the controller
By implementing
objectives
and plans for followon
employed.
new
and that significantly
identified
performed
in which
of an actively
meeting
design
code for the selected
eigenvectors,
a promising
wide range of possible
and have
Capability
tasks were
in the formulation
for calculating
considered
Besides
facilitate
design,
of the necessary
accomplishments
the Youla parameterization)
is practicable
Recognized
and of the work
a method
integrated
structures.
software,
of the piIot computer
and constraints,
problems
control
the following
and computer
and enhancement
developing
of a pilot
that would
Analysis
pilot integrated
development
Several
and demonstration
but also easy access
analysis,
of a representative
demonstration
involved
analysis
No. NAS 1
of work):
3.
5.
environment
as Task #3 of the
Technology",
controlled
The IAC utilities
the stated objectives,
analytical 2.
this framework.
finite
and Control
of actively
The Integrated
for engineering
was performed
was the implementation design
involved.
study
Guidance
the need for a software
discipIinary chosen
and Spacecraft
for the integrated
the outset
Design"
learned
further
our objectives,
much
investigation. methods,
about the problem This document
and findings.
Analytical The usual approach tion
suppression
design
based
structural
for designing
is a twostep on structural
design,
closedloop
Approach
the
large,
flexible,
procedure. costs
second
dynamic
and Implementation
and
step
behavior.
The
first
openloop
is to design
These
space step
structures
is to determine
dynamic
behavior.
the control
two steps
may
with active
system
be iterated
the structural Then,
given
to obtain
until
vibra
this
the desired
a suitable
design
is
obtained. This sequential entirely
reliant
may naturally ables
approach
on the designer's exist between
and the control
design
is not available
this approach
to converge
optimization constraint;
uation
shown:
design
problems.
to a locally
lies above
variable
the design
in Figure
approach 1 has only
the constraint
under
study
if we are restricted
to moving
only
sequential
approach
were
then the design
constraint.
A clear
ables can be modified
used,
along
path to the minimum
there
simultaneously.
4
coupling
that
design
vari
information
regarding
is no reason
to expect
for most constrained
is easily
demonstrated.
two design line shown, design
the x or y axes,
exists,
(and
any
variables We might
variable.
lies on the constraint
cannot
it difficult
the structural
separately,
design
and that y is a control
currently
to exploit
Consequently, optimal
it makes
Because
are considered
of the sequential
space
however,
and intuition)
to the designer.
illustrated
its feasible
x is a structural
experience
variables
even
performance problem
straightforward;
the two design
this coupling
The poor
is fairly
but it can be taken
The simple and a single imagine
Consider
be the case
without
violating
only if the design
that
the sit
at the point A.
as would
be improved
problems.
Now if the the vari
Increasing object,/ve value
Possible design changes
Linear o0 Current design
,%,0° contours
Figure
For this reason, sidering many,
and
has been
a large
directed
of modern
likely
toward
formulating several
Unfortunately,
these
approaches
lems
they
apply.
to which
designs
could
often
variables.
are described
in the literature.
the combined
problem
examples
of which
tend to be quite
These
example.
set of design
of studies
control,
optimization
that improved
the complete
number
optimal
_
1: A simple
it seems
simultaneously
S
restrictions
typically
This has been Some
in a manner
are described
restrictive
be found
by con
apparent
of this work
similar
to those
in References
in terms
of the kinds
preclude
to
problems
[25]. of prob
of practical
complexity. An
alternative
approach
numerical
optimization.
This
restrictive
on the kinds
of design
ered.
References
[6,7] describe
to simultaneous approach
has
variables, several
optimization a distinct
objectives,
applications
5
is to use
advantage
in that
and constraints of this approach.
the
methods
of
it is much
less
that can be consid
Attemptsto usenumericaloptimization methods,however,havebeen hamperedby severaldifficulties. Becausethe objective andconstraintsare typically definedin terms of the closedloopresponseof the structure,a completestructural andcontrol analysis (with sensitivities) is required for every trial design. The effort involved can be prohibitive. This is particularly relevantto this multidisciplinary problem, sinceintegrated analyticaltools aregenerallyunavailable. This is exacerbatedwhen numericaldifferentiation is usedto obtainthe necessarygradients. Moreover,the dimensionsof the design spaceandthe complexity of the objective surfacemay alsofrustratetheseoptimization attempts. The approachtakenherefor the integrateddesignproblem is shown graphically in Figure 2. This approachwas implemented in a computer program called COSTAR. Using
a nonlinear
design
space,
various
constraints
programming
attempting
to minimize
on its design
structural
and control
We
attempted
to avoid
integrated
analysis
have
efficient,
that tend to reduce
method, the
an optimization specified
objective
and performance.
variables,
and the optimizer the principle tools,
the complexity
The design
gradient
of the objective
6
space
to this type calculations,
surface.
performs
function
is free to modify
obstacles
analytical
module
a search
while
is comprised them
of
satisfying of both
simultaneously.
of approach and design
by using variables
Mesh Generation objective, constraints, and their sensitivities responses
, /
/
I
\
l ....
iN
I c'l°seuc°°Pl ill
Analysis
/
Control
/
I\ /
"/_
_
]__.__.___
[ Structures
/
Matrix
/
_
lACy
I
Assembl___y, _1
I Design I i
/K,M,K',M'
i""
_
l
[ I I
Responses
The user.
functions
Although
from
the user,
constraint design
functions
variables.
torque" mass"
the hallmark
were
element
construct takes
'
conceptual
layout.
the objective
and constraints
of the problem
of this approach only
problem,
with more
they
must
functions
conventional
etc.
are defined
may require
is that the type and form
in that
"
by the
a significant
effort
of the objective
be continuous such as "peak
performance
a priori
functions error"
functions
and of the
and "peak
such as "total
error".
Modeling
In the COSTAR finite
'
2: The COSTAR
description
along
/
stresses,
In the demonstration
included
i t
_
is restricted
and "meansquare
Structural
I
Iit
_
that describe this explicit
[ OpenLoop[
IEigo.v l.o, j"
" A, _D
Figure
model
_desoription
_
/
___.L_/'
_
G
variables
/
I I
\
[
/'
) _sign " / '_
I Optimizer  i
I Pcrforrnance
sensitivities
H,H'
(not in loop)
mass
code,
method
equations
(FEM).
and stiffness
the form of several
erties, and material
of motion
A description matrices
matrices
for the structure of the structure
that characterize
that contain
properties.
J
7
are obtained
is used by the FEM
the structure.
the connectivity,
by using
geometry,
This
the
code
to
description
element
prop
Table
1 lists
the matrices
eral of the matrices for
the design
derivatives
that combine
axe functions
variable
of the design
values
with respect
result
to the design
Matrix
to describe variables
in different variables
OP
properties
Element
orientations
Element
connectivity
P
Element
property
pointers
Z
Material
property
pointers
Element
types
Ei" = _i
A
0 _vi
=0
Table
These
matrices
1: Structural
are themselves
preprocessor.
The input
form
so that changes
to the values
(The
MACSYMA
input
file.)
within add
For reasons
be required
numerical
errors
In many has been the design
report,
[8] 
a finite
The
sensitivities,
adding
is kept
can be easily
[9] is used
in a parametric accommodated.
to update
the
this kind of preprocessor
overhead time.
file")
involved
Moreover, greatly
in running a finite
PATRAN
cannot
be used
this code
would
difference
to the execution
procedure
time and to the
in the sensitivities.
previous
avoided
variables
program
however,
the use of PATRAN
(its "session
of the design
total execution
to obtain
this
Matrices.
through
to PATRAN
mathematics
procedure.
to the
Definition
generated
of efficiency,
the optimization
significantly
would
symbolic
choices
of the nodes
Material
_V i
OE
/c
sev
by a prime.
properties

different
Throughout
Element
Pi
E
element
v. Thus
structures.
Locations
_Vi
,
P
As indicated,
Description
_x Xi'=
vector
are indicated
Sensitivity
X
the structure.
approaches
by representing
variables.
For example,
to the simultaneous the mass
optimization
and stiffness
it is common 8
practice
matrices
problem, as explicit
to prescribe
them
this problem functions
of
to be linear
functionsof £hedesignvariables. Although this simplifies the analyticaltask,it imposes severerestrictionson thekindsof designvariablesthatcanbeconsidered.In particular,it precludesthe useof designvariablesthatcontrol theshapeof the structure. The COSTARimplementationavoidsthesedifficulties by representingthe structural model (that is, the descriptionmatriceslisted in Table 1) asa linear combinationof a set of model variables. The nodal locations, for example, are expressed as 
0X
0X
X : X + 0_1(/.tl_]) where/1
is the vector
specifying
the model
of model
variables
variables
as possibly
+
+ "'"
and the overbars nonlinear
refer
functions
(1)
to the baseline of the design
values.
By
variables,
= i(v) relatively
few
software (such The
restrictions
is used only
are placed
to generate
as for optimization) sensitivities
are obtained
on the models.
the model
proceed
Finite
Once
Updates
with only
simple
the modeling
for subsequent
matrix
analyses
operations
involved.
.,
OX r
7,
(3)
matrices.
Analysis
the model
generate
this approach,
as
for the other definition
Element
With
description.
rapidly
OX r
and similarly
(2)
description
the equations
is obtained,
of motion
the next step in the integrated
for the structure.
These
equations
analysis
of motion
is to
are given
by M_ + CYc + Kx = F The task of the structural
analysis
matrices
sensitivities.
as well
abound, tivities
few
as their
are designed
analytically.
for the integrated analysis
code
The analytical
SSA
(called routine
is to assemble
While
to calculate
Because design
module
sensitivities
we chose
the mass,
computer
of the importance
problem,
(4)
codes
and fewer of fast,
to develop
damping,
for finite
and stiffness
element
still can calculate
accurate our own
sensitivity special
analysis the sensi
calculations
purpose
structural
SSA). has only beam
sensitivities
of the mass
ous design
variables,
such as nodal
A lumped
mass
formulation
and concentrated
and stiffness locations,
is employed
matrices element
(resulting 9
mass
elements,
but it can provide
for any combination properties,
in a diagonal
of continu
and material mass
matrix),
properties. and
the
..a
stiffness
matrix
is stored
ces as calculated design
in a banded
in Equation
variable
form.
Using
(3), the sensitivity
the sensitivities of the stiffness
of the definition matrix
matri
with respect
to the
vi is
K;=Z g aEx}; ax [xr]j,+Z g a[plj ax [e,, .... j k j k and the mass
matrix
In COSTAR, damping
the damping
approach
This implies problem, mass
sensitivities
is used
are obtained matrix
wherein
is the case
and stiffness
C is never
ratio
of the undamped
when
way.
assembled.
the damping
that the eigenvectors
which
in the same
the damping
Instead,
of each
problem
matrix
(5)
mode
the common
modal
is specified
also diagonalize
can be expressed
directly.
the damped
in terms
of the
as C= M ___ ai(M1K)
i
(6)
i
with
arbitrary
special
scalar
coefficients
case of Equation
become
uncoupled
Usually, uncoupled below, tives
(6), with i={0,1 }. With
and their
solution
it is enough system.
to just
the demonstration
are obtained
in modal
However, Equation
problem,
this requires
(6) must
i=! was
used
damping
the equations
damping
sensitivities
Thus
is a
of motion
more
and
complicated.
form but without
the use
of C',
be differentiated in Equation
ratios
the
and used
then
solve
As described requiring
deriva
sensitivities
of the
to calculate
(6); this results
the
C'.
in damping
In
ratios
to frequency.
Extraction
As mentioned coordinates
modal
it is slightly
that are proportional Eigenvalue
the
of i. Rayleigh
simplified.
however,
of the eigenvectors. matrix.
is greatly
specify
values
modaldamping,
In COSTAR,
the response
damping
ai and for any integer
(with
above,
the openloop
mass normalization)
equations
of motion
are transformed
into modal
as
gli + 2_io2i_1i + 09_qi = _F This
transformation
modal
truncation.
both
uncouples
At the same
tionally
expensive
part
formed
repeatedly
for each
determine
the practicability
attempted
to perform
the problem
time,
eigenvalue
of the openloop new
design,
of searching
the eigenvalue
and
task.
efficiency
for an optimal
extraction 10
enables
extraction
analysis the
(7) model is often
Because with design.
as efficiently
reduction
through
the most
computa
this step
which
must
be per
it is performed
For this reason, as possible.
can
we have
To do this,
we
havetakenadvantageof the fact thatthe eigenvectorsfrom a previousdesign are good
approximations The
code.
to those
existing
EIGEN
Nevertheless,
verge
module
its Lanczos
of the eigenvectors. SS1) that makes
of the current
Instead, successive
design.
within
IAC
algorithm
we have
is a very
does
efficient
not benefit
implemented
improvements
to the true eigenvectors.
usually
eigenvalue
from
a subspace
good
a priori
iteration
to a set of starting
extraction estimates
approach
vectors
(called
to eventually
con
The set of equations w
KOk+I Kk.l
= MOt
= Ok+IKOk.I
Mk+ = Kk+I
IPrk+l = _'(k+l
I//k+lAk+l
Ok+l = Ok+l Vk+l are solved
for successive
By using
the
design,
eigenvectors
this convergence
OpenLoop
objective
One
and
from
the previous
is often achieved
implementation,
constraint
of the structure advantage
(finst order)
model
as starting
of eigenvectors.
vectors
for the present
in only a few iterations.
the closedloop are obtained
via the socalled
Qdesign
Since
coordinates,
transfer
directly
the openloop
needed
openloop
described
not require
the openloop
functions
from
approach
is that it does
of the structure.
into modal
equations
transfer
transfer
in the next
the assembly
funcsection.
of a statespace
of motion
functions
for the
can
have
been
be calculated
efficiently. In addition
to the transfer
the design
variables
functions.
To determine
the modal
equations
in order
functions,
we also need
to compute
the sensitivities
these
of motion
sensitivities, (Equation
it has been 7) to form
?li" + 2_io9i?ii" + o0?qi" = (_F)'that characterize cies
to the true matrix
design
evaluations
of this approach
transformed very
of k until Ok converges
Responses
In the COSTAR
tions
values
the modal
and damping
the forcing system.
function
ratios
response
as the original
sensitivities
In this way, the response
sensitivities system,
11
sensitivities
of the objective common
practice
an additional
with
respect
and
qi'.
to differentiate
set of equations
This system
but the righthand obtained
can be obtained
to
constraint
2( ¢i_i)'gli  2COi_" qi
but also the responses sensitivities
their
has the same
frequen
side contains
not only
from solving from
(9)
the original
x" = O'q Unfortunately, eigenvector exist 
there
derivatives
a common
formed
[10].
condition
For COSTAR, advantages.
are wellknown
difficulties
implemented coordinates
arbitrary
difficulties
(10) associated
are manifested
with when
the calculation
repeated
of
eigenvalues
in practice.
we have
If the physical
by some
These
+ Oq"
constant
an approach of the original
matrix
that appears problem
to offer
in Equation
significant (4) are trans
O, so that x = Or/
then the equations
of motion
may be written
(11)
as
oTmoii + oTcoi7 + oTxor/= By differentiating the matrix
Equation
O is constant, OVMOi_"
We have
to the design
(12)
variables,
keeping
in mind
that
we obtain
+ OrCOil"
not specified
the eigenvectors reduces
(12) with respect
OTF
+ OrKOrl'=
O'(F'M'OiiC'O_K'Or/)
the transformation
of the original
system.
matrix
With
(13)
O; let us now consider
this choice,
it equal
we can see that Equation
(13)
to }1i" + 2_io)iT?i" + O)_r/i'=d(F'M'eP_C'¢i?K'ePr/}
and the sensitivities
of the physical
responses
(14)
are
x'= or/' The
to
coordinates
r/are
equal
Comparing
Equations
to the modal
(15)
coordinates
q, but their
sensitivities
are differ
ent. (9) and
the latter approach
are obvious.
which
responses
the original
however, derivatives. when
the physical Based
• contains
on our
a truncated
with Equations
In both cases, appear
response
(10)
on the fighthand
sensitivities
limited
a second
are obtained
experience,
set of mode
shapes.
12
(14)
and
(15),
set of equations sides.
With
without
this approach
the advantages
of
must
in
the second computing
seems
to work
be solved approach, eigenvector well
even
Controls Analysis A recently developedmethodfor controlssynthesis,thatis herecalled Qdesign, based
on the Qparameterization
With
this parameterization,
be expressed transfer
all possible
as functions
functions
of a stable
are affine
For a given
(or stabIefactorization) stabilizing
parameter
in Q P
r
X
maps
can
the closedloop
for our purposes. coprime
factorization
= _1_
(16)
= ,_I
the set of all stabilizing
Furthermore,
all achievable
Q parameter
IE ] D
B
QN_)I(X
closedloop
X
N
controllers
K = {(Y
affine
[11].
such that
I In that case,
by Youla
and input/output
the stable,
Knom,
Knom = yIx be chosen
developed
this formulation,
property
controller
Pyu = N D1
may
controllers
Q. With
an important
plant Pyu and nominal
theory
is
= I
(17)
ff
is given
by
+ QD_] Q stable}
input/output
maps
(18)
are described
in terms
of the
via nzw = T1 + T2QT3
where
the nominal
system
is characterized
(19)
by
T1 = Pzw  PzuDXPyw = Hzw _=o 7"2 = PzuD = nzv 7"3 =/_Pyw The Q parameter Figure
3 depicts
Knom represents w, regulated
its connection the nominal
outputs
such as commands, the designer
may be thought
controller.
z, and measured disturbances,
has chosen
to regulate,
= Hew
of as a tuning
to the nominal The outputs
and sensor
parameter
controller. plant y. noise.
and that reflect
13
(20)
Here P represents
has control The The
for the nominal
inputs
controller. the plant
u, exogenous
inputs
exogenous
inputs
include
regulated
outputs
are signals
the performance
of the system.
and
signals that
P
z
Y
3: Nominal
Figure
Q is connected tions
above)
Thus,
to the nominal
that it sees
if Q is stable,
Q to range
over
all stable
and Q span the space
Q parameter
has been
implemented
is represented
functions,
within
by Boyd
function
of the closedloop
subset
to the equa
from
v to e is zero.
system.
By allowing
of the nominal
controller
compensators.
can be used in a design
a finite
(according
the combination
stabilizing
theory
by Q.
a manner
is, the transfer
the stability
of all possible
modified
in such
That
affect
transfer
This Qparameterization suitable
controller
no feedback.
it cannot
controller
of the stable
[12] in a computer
code
procedure
transfer
by searching
functions.
called
QDES.
for a
This method
The
Q parameter
by Q = ___ viQi
(21)
i
a linear
combination
of finite search
impulse
of a finite
response
for a set of design
set of fixed,
(FIR)
filters.
variables
stable
In QDES,
v that minimize
maps
Qi.
Thus
numerical
Q is restricted
optimization
the objective
function
to a set
is then used while
to
satisfying
the constraints. The directly wide capable
Qdesign specify
variety
approach an objective
of closedloop
of producing
several
function
and
system
a wide
[15] and H** [16] methods
offers
variety
attractive strict
characteristics. of controllers,
as subsets.
14
equality
features. and (See
including
It allows
inequality Table those
2.)
the
user
to
constraints
on a
In addition,
it is
obtainable
with LQG
Objective • Transfer
function
• Power
spectral
• Overshoot • Stability
2norm density
Functions
(LQG)
of a transfer
and undershoot
function
at specified
frequencies
for a step input
margins
• Infinitynorm • Transfer
of impulse
function
and step responses
infinitynorm
(HInfinity)
Constraints • Limits
on impulse
and step responses
• Limits
on step response
• Limits
on transfer
• Limits
on step and impulse
• Limits
on stability
function
makes
variables, even
though
tively
simple
Qdesign
has some
controllers
are of very
experience
with it so far, the computational controllers
have
been
methods
and software
Earth Pointing Observation flexible
problem
System Sciences
antennas.
and 7.5 meters
The
(EPS)
amount
the
one. large,
Because
This
structural
design
is a great
benefit;
the problem
it is based
of computation.
Nevertheless,
remains
rela
has been
to reduction
does
show
This
in Figure
15
promise.
In our and
techniques.
example 4.
the integrated
structure,
It is derived
and consists 25 meters
in diameter.
the resulting
Problem
for this task.
bus is approximately
opti
large but not prohibitive,
on which to demonstrate
platform
on numerical
In addition,
the method
effort
amenable
[13] is shown
geostationary
excluding
if it exists.
was selected
developed
that,
may be very
Demonstration A representative
specifications.
is a convex
drawbacks.
it can require
high order.
is the fact
variables
will be found
a significant
control
problem
mization,
the high order
infinitynorms
QDES
tractable
of design
and a solution
Admittedly,
response
optimization
the number
and undershoot
infinitynorm
2: Sample
this approach
the resulting
and controls
margins
Table
What
overshoot
of states
from
of a trusstype
in length
known
design as the
the Ford Earth bus with
and the antennas
two
are 15
15 m_
7.5m_
(typical)
tFigure
The
total
mass
and
cross
section
structure
mass
548.32
mostly
modes
Baseline
EPS
structure
is 1027.95
kg of nonstructural
rotation
of the first involve
mass
Structural
local
kg, including
tubes.
of the antennas
16 flexible
Model
on the antennas.
of 135 graphite/epoxy
involves
and frequencies these
4: The
consists
=i
26.25 m
of the baseline
and
_ ,m"'"T'"'_ .,J"\ I /\
normal
deflections
within
considered
here
The
The
150 kg of actuator truss
first
are shown
the antennas
a 3.0 meter
flexible
and has a frequency
modes
has
mode
of the
of 0.24 Hz.
in Appendix
A.
with insignificant
Shapes Many
motion
of
of the
truss. The
design
problem
flexible
structure
structure
about
modes. line
has no articulated some
nominal
A set of reaction
system
and
is used
of the structure.
actuators,
and linear
Dynamic bus module. developed, fact,
however,
satisfactory
the present
do not restrict
practical
time and the frequency
along
radii
for various
its length,
sensors
are
would
the
stability
the center
the reaction
to require
body
of the base
colocated
from
the
rigid
due
to the flexible wheel
of each antenna. a thruster
located
were considered.
or characteristics
seem
of the
of gravity
with
The
to regulate
of the antennas
no other loads
the number
suppression.
is required
at the center
is considered
study,
vibration
many
on the main The methods
of the external load
cases,
loads.
In
in both
the
domains.
A total of 19 structural bar
designs
near
errors
are included
of the structure
To expedite
is located
rate
accelerometers
excitation
enhancing
the pointing
Angular
of active
the controller
while
actuators
to control
response
elements;
attitude
wheel
is one
groups
design
variables
of bars,
the
were chosen width
and the size and orientation
and
depth
of the antenna
16
for the EPS structure, of the supports.
truss
including
at several
These
design
points vari
ables are describedin Table 3. Figure 5 showsan exampleof the structure for which arbitrary values havebeenchosenfor eachdesignvariable. Name
Description
Baseline Value
c W'tDTH
Width
of the truss at its center.
3.0 m
C_DEPTH
Depth
of the truss at its center.
3.0 m
L_WIDTH
Width
of the truss at the large
antenna
end.
3.0 m
L_DEPTH
Depth
of the truss at the large
antenna
end.
3.0 m
Value
of the ycoordinate
SMALL
Y
1
end nearest
of the
and on the same
side
truss
at the
1.0 m
as the small
antenna. SMALL
Y 2
Value end
of the ycoordinate nearest
small
and
of the
on the
opposite
truss
at the
side
as the
1.5 m
antenna.
S_DEPTH
Depth
of the truss at the small
antenna
ALPHA
Angle
between
and the plane
the truss
axis
0.5 m
end. of
0.0
the antennas. Radius
of the longerons.
0.0255
m
BATTEN_R
Radius
of the battens.
0.0255
m
DIAG_EXT_R
Radius
of the external
diagonals.
0.0255
m
DIAG_INT_R
Radius
of the internal
diagonals.
0.0255
m
L_SUP_R
Radius
of the large
antenna
supports.
0.0255
m
Radius
of the small
antenna
supports.
0.0255
m
ACT_SUP_R
Radius
of the actuator
0.0255
m
ACT_MASS
Actuator
LONGERON_R
S_SUP
R
Table
supports.
mass.
3: Design
150.0 kg variables
17
for the EPS structure.
Figure
The EPS 16 flexible used,
5: Structure
equations
modes,
with arbitrary
of motion
ranging
in every
openloop Figure
cycle
impulse
from 0.24 Hz to 3.53 Hz.
of the highest
responses
6 illustrates
The
sample
frequency
is sufficient
the locations
in the design
that were used for the control
with 20 Q taps and 500 samples.
points
changes
design
A sample
and the 500
to capture 6 full cycles
in the baseline
18
model
relevant
contained
frequency
rate is sufficient
mode,
variables.
the first
of 20 Hz was
to capture samples
of the lowest
at least 5
used for the frequency.
to the control
design.
Node #42: Reaction wheel actuators
Node #71: Antenna Pointing
_
D_r
Y
Node #45: Zdirection ThrusteNode
[_
Direction and Accelemmeters
X
Figure Rotations actuators
are designated control The
task,
regulate
is located outputs
initial
exogenous configuration
located
at node
#45
is depicted
A thruster,
configuration
in Figure
Figure
torques
by
4 regulated
outputs,
7: The original
rate at node #42 and the linear The
conla'ol objective wheel
structural
model
and 7 measured
YS=[{_42,
plant defined 19
_r26,
Ovl,
consists
of
outputs.
This
_r71]
;42,'42, N26,_26,'71,'71]
by the structural
is to
actuators.
Y0
026,
for the
z direction.
7.
z_=[ f_g42]
a disturbance
the reaction
the
The
by the actuators
fz45, in the positive
#26 and #71.
V¢71using
V, respectively.
exerted
represents
of angular
at nodes
wo T=[ f.5] f042,
The
p
u0
u0T=[ f042,
as 0, 4_, and
which
provided
inputs,
structure.
a thrust
components
026, _26, 071, and
3 controller
#42.
and exerts
and z directions
angles
EPS
are designated
at node
andf_¢42.
system
input,
z axes
are the three
in the x
the antenna
The
wheels
asfo42,f_2,
measured
accelerations
6: The baseline
about the x, yand
are reaction
#26: Antenna Pointing/
model.
1
To accommodate were
added
the design
to the system.
As shown
added.
Also,
the exogenous
angular
rates
measured
variety
WQ
of control
_a
=
of a robust
output
in Figure
several
8, actuator
was augmented
at the reaction
specifications,
system,
wheels.
including
additional
inputs
noise
sensor
and
with the noisy
actuator
This configuration
robustness
and outputs noise
signal
now allows
and noise
were
and the
for a wide
sensitivity.
w° ctuator noise sensor norse uQuo
=ZQ
y0 + sensor noise = yQ
Figure
8: Augmented
plant.
Findings The COSTAR was
applied
implementation
to the EPS
structural
design
variables
held
design
variables fixed.
of the integrated
The
held
fixed,
first
analysis
baseline
structure,
analysis
is an integrated
design
sensitivity
computations
analytical time,
the latter
which
problem.
is useful
analysis
this is a severe
capability
of COSTAR
Optimal
Control
By fixing QDES adjusted
control
the
to minimize
design
is for a specific
thruster
at node #45.
amounts
described
in two
all but one
ways:
the Qdesign
only
a single
in the
demonstrated
control
concurrent
problem,
for
The
design
the integrated
the
second
Because
in COSTAR
structural
all the design
design
structure.
implemented
optimization
(1) with
methodology.
of the controlledE_'S incompletely
previously
of the structural
to an optimal
were
limitation
the at the
variable. design
adequately.
QDesign
structural
design
done
(2) with
optimization
is nevertheless
Using
and
was
for evaluating
considered
Although
This
designmethod
code.
design The
the objective disturbance
variables,
control
is essentially
design
variables
shown
while
satisfying
the constraints.
function 
COSTAR
a 100 N.s impulsive
20
force
identical
in Equation
to the (21)
are
This control
in the zdirection
by the
The objectivefunction chosenfor this demonstrationcontainedcontrol effort, pointing error, and stability margincomponents.Specifically,this objectivewascalculatedas the sumof 103times the meansquaretorque, 1016times the meansquarepointing error, and 104times the inverseof the minimum Nyquist distancet. (All quantitiesare measuredin SI units.) Thesecostswerechosenfor a disturbanceenvironmentwherethe ratio of thrustdisturbanceto sensornoiseis approximately10,000. Constraintswereappliedto the antennapointing error, the actuatortorquelevels,and robustnessto actuatornoise. The maximum allowable pointing error of either antenna was 0.01% The torque applied by the reaction wheel actuatorswas limited to at most 1000Nm. The minimum Nyquistdistancewasrequiredto beat least0.5 s1. Theseobjective and constraintfunctions, along with the impulse responsesof the baselinesystem,weregiven asinput to QDES. The resultwas an optimal controller that minimized the costs without violating the imposedconstraints. As mentioned earlier, QDES
often
results
36 state baseline the controller
system, order
that the controller To
the
the controller
impulsive
radian
loop response
clearly
On the other
hand,
but never
sensitive
to the
Figures
1012,
response
of
three
orders Figures
that
violates
thruster
disturbance.
actuators.
The
constraints
were
illustrate
closedloop imposed
error
right
Of the four The
the pointing
gr26, the pointing
1315
goes
the constraint.
do not violate
of magnitude
response
about
other
by 100.
antenna,
translates
It has a peak response
pointing
constraints
even
the z axis
depicted
The result918.
a constraint into
on
a 1.7x10 6
can see that the openof6.5x10
angles,
angle
for
071, for this
up to the constraint pointing
in the
the disturbance
included
One
[14],
performance.
in Figures
at the small
This
using
disturbance
is scaled
specifications input.
For the
balancing
upon
is precisely
in the figure.
this constraint.
the closedloop violates
is shown
impact
are depicted
the x axis
the control
which
This
system
intenaal
impulsive
that its magnitude
of 0.01 ° for a 100 N.s impulsive
limit for a unit impulse,
seconds,
except
#45.
no exception.
It was demonstrated
weighted
a 1 N.s
at node
error about
Recall
was
with minimal
obtained,
of the openloop
the pointing
disturbance. error
thus
was designed,
9 depicts
of frequency
to 30th order
by the thruster
and the response
Figure
pointing
be reduced
and this design
a 146 state controller.
technique
system
was applied
ing response
found
reduction could
controllers,
QDES
closedloop
zdirection which
in high order
in about
0.5
071 is the most
responses,
for the openloop in Figure
6 radians.
shown case.
in The
12, is roughly
less than that of 071. the effect
of the impulse
response
exhibits
upon
these
angular
21
greatly rates.
on the angular improved
rates dynamic
measured behavior.
at the No
Finally, bance.
The
Figures
1618
actuators
were
line structure,
however,
of approximately mization, this peak cedure
torque
the response
constrained
none
2 N.m
however,
depict
was
could
from designing
to have
a 10 Nm peak
of the actuators seen
it is entirely
limit
of the actuators
in actuators
possible
f042 andf_2.
Placing
that would
torque
level.
this limit.
might
excessive
torque
the combined
evolve
this limit prevents
require
distur
For the base
A peak
During
that the structure
be reached.
a system
approached
to the impulsive
opti
into one where
the optimization
actuator
level
pro
authority.
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Impulse
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23
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Figure Combined
presented
variables
variables,
were
in COSTAR,
truss.
considered. design
forcing
to a unitfz4s
"7
f 4
impulse.
was
In many
costs
optimization
design
augmented cases,
such
as manufacturing the results
In addition applied
to the structural
and C_DEPTH the objective a term
assembling
account It cannot the
as good
objective
described
variables.
These
design As men
implemented
at a time. section
in Table
We per
of the EPS 3.)
described
in the previ
to the total mass
of the struc
for the cost of the materials account,
structure.
however,
As with
as the objective
function,
be twicecontinuously
constraints
described
proportional
to orbit.
are only
design
variable
function
may adequately
a very simple
to the control
design
the width and depth of the center
and
is that it must
in COSTAR
only one structural
the structure
tation
structural
were incompletely
problem,
used
to include
only control
computations
CWIDTH
this demonstration
that analysis
in which
sensitivity
this mass term
procedure,
optimization
can be demonstrated.
to include
of transporting
to a COSTAR
of COSTAR
including
CI'hese are the variables
and the cost
were
off_2
2 1
1_
capability
us to consider
two such analyses,
ous section
other
15
By extending
the analytical
For this integrated
ture.
Response
above pertain
the integrated
tionexl previously,
formed
18:
12
Optimization
The resuhs design
_,
;'
any
function.
the only objective
for many structural Although
function
limi
differentiable. previously,
simple
constraints
specified
side
constraints
commonsense
27 ORIG!NAL
PAGE IS
OF POOR
Q_JALrT'Y
restrictions
such
Far more
complicated
In Figure depth.
point
an optimal
and
for
approximately the total mass over
the range
location able
the particular Significant
and mean
square
of the resulting
as a function
a particular The
value
widths
shown,
design.
error terms.
the mass
it does not play a significant incorporation
on the closedloop
in
has a depth
which
function
truss
by COSTAR
structure
from
Because
section
and constraint
to the objective
Nevertheless, effect
analyzed
EPS
space
and depth. if desired.
of the center
design
baseline
width
be included
(for the objective
contributions pointing
a reasonable
etc.) could
finitedimensional
2.6 m.
has an important
have
strains,
is obtained.
that its optimum
of truss
must
is shown
represents
design
of the optimal
clearly
value
on the curve
shows
section
(on stresses,
objective
control
3.0 m; the figure specified
constraints
19, the
Each
which
as that the truss cross
functions
Q is chosen) value varies
come
is from
by only 7%
role in determining
of the structural
behavior
of
the
design
vari
and on the performance
design.
2600 2400 2200 ov.I
¢.2
O
2000 1800 1600 1400 1200
I
1
'
i
2
'
3
Figure
20 shows
the combined
2.6 m, the center
section
that significantly
decreases
ter
section
decreases
obviously the combined
19:
depth
decreases
vs. truss
control/structure
the bending the
costs associated
stiffness structural
5
6
depth.
optimum
from
configuration
the baseline
of the truss. cost
with control
28
I
(meters)
Objective
has decreased
'
4
Depth Figure
I
(total energy
EPS
design
This
narrowing
mass)
slightly,
and performance.
obtained. 
At
a change of the cenbut
it also
Figure
A similar the structural Figure been
21.
20:
Optimized
analysis design
was performed variable.
Here again,
designed
ure shows
with
structure
control
with the center
section
A plot of the objective
each point
COSTAR.
that its optimum
considering
represents
The baseline
value
EPS
verses
for which
structure
is approximately,
width,
value
a system
and
truss
width.
rather this width
an optimal
has a depth
than depth,
as
is shown
in
controller
has
of 3.0 m; the fig
1.4 m.
3000
>
2000 ©
1000
ii I
0
Because represents strates,
the optimal a globally
however,
the
I
1
2 Width
Figure
21:
control
problem
optimal
design
optimization
"
I
3 (meters)
Objective
vs. truss
is convex,
4
are introduced.
Several
local minima
optimum
is not generally
assured
for the combined
29
that each point
structure.
is no longer
variables
5
width.
we know
for that particular problem
I
convex
are seen to exist. problem.
As the when Clearly,
in Figure
21
figure
demon
structural
design
finding
a global
Figure 22 showsthe combinedcontrol/structureoptimum configuration considering the trusswidth structuralvariable. Notethat, as with the center sectiondepth, the optimum centersectionwidth is alsolessthanthat of the baselineEPS structure again a changethat significantlydecreases the bendingstiffnessof the structure.
Figure
22:
Optimized
Structure
Considering
Control
and Structural
Depth
Conclusions In conducting
a simultaneous
it is possible
to investigate
optimization
would
mization designs.
reduction
optimal
In the process
lating
involving
design
stiffness
either
finite
tool
Qdesign.
It has proven
to be a very flexible that meet these
to Work wellinthe
30
represents
a
optimization,
a method
of analytically over
other
calcumethods
of eigenvectors.
As implemented
specifications.
This
improvement
was accomplished
tool for making
optimal
to intuition.
This method
or the differentiation
opti
in this report found
section.
control/structure
a significant
and structure
nonintuitive
described
truss
parameters,
control/structure
for generating
of the center
of the optimization
control
to combined
was developed.
represents
has beendemonstrated
controllers
promise
this combined
meth0d
thesizing
approach
a separate
of the EPS structure
sensitivities
called
which
and control
 a result which might run counter
differencing
aspect
synthesis
holds
the size
of developing
sensitivities
The control control
to decrease
structural
structural
the overall
the optimization
in structural
for determining
Thus,
of both structural
configurations
in this report
In particular,
that it was
design
miss.
presented
optimization
control
using
the relatively
by the program combined design
QDES,
optimization specifications
new this
context. and syn
With this increasedflexibility alsocomesincreasedcomputationalcost. In addition, controllersdevelopedusingQdesigntendto be of very high order  much higher than that of the systemsthey arecontrolling. On theother hand,it hasbeendemonstratedthat the controller order reduction techniqueof frequency weighted balancingcan greatly reducecontroller order. Moreover,the use of Qdesignin the combined optimization does not preclude the use of other control design tools for the optimized structure; the benefits
of Qdesign
which
would
it is theoretically While
convex
the
optimization
will most
that achieves
Areas
for Future
likely
input.
have
specified,
the This
and
should
be completed
tural
design
variables.
from
one
make
design
truncation,
should
be studied
further.
associated
this
area
approach necessary
by
to find
a
points.
Kosut
involves to obtain
area
next
be advantageous
and
a good
of supervi
communication
between
of future
free"
work
model
would and
functions
until
only been
be to
its design have
an optimal
the inclusion
have
optimization
procedure
of analytic
been
design
of a larger
optimized
for research
is
num
over
does
is included
of the Qfilter
control
not change
a means
one
design.
in an attempt
Another 31
as Appendix
approach
research
number
of strucmethod,
structures, a great
would
since
deal.
The
problem,
the computation has been
B in this
to reduce
much
computa
optimization
of reducing
Some
for
extraction
for previous
to this simultaneous
to develop
and
of a larger
calculated
room
sensitivity
is the eigenvalue
the structure relate
have
design
the inclusion
part of the optimization.
Kabuli,
a prefiltering
deal
and constraint
designs
to facilitate
as they
with the control
a great
the structural
also facilitate
the implementation
to the
of modal
time
would
goals
"hands
use of the eigenvalues
iteration
it would
to run
in the
Another
effects
Finally,
control/structure
variables.
in order
better
requires
once
the objective
Present
subprocesses
In particular,
might
for
it is not
of attempting
starting
to facilitating
Ideally,
be able
all control
procedure
of the primary
and once
variables.
tions
which
a system
problem,
the combined
multiple
primarily
One
of COSTAR
design
of the
optimization
should
streamlining
improvement.
is a convex
and the process
will require
procedure.
defined,
optimizer
variable
Some
variables
local minima,
by designing
performance.
Consequently,
minimum
components.
been
ber of structural structural
design
This is related
the optimization
variables
have
exploited
the desired
parameters.
the COSTAR
software
streamline
to achieve
been
Research
sion and designer the various
already
the control
a global
At this point,
found.
over
structural
design
then
possible
optimization
over
have
done
report.
the number
in
This of taps
be to incorporate
a
nominal controller for the baselinesystem. This would reducethe numberof samples requiredto characterizethe impulseresponses,andwould alsoreducethe numberof taps requiredin the Qfilter. References [1]
Vos,
R.G.;
Manual
Beste,
(Level
D.L.;
and Gregg,
2.5)",
Boeing
J., "Integrated
Aerospace
Analysis
Capability
& Electronics,
flAC)
Seattle,
User
Washington,
1989.
[2]
Messac,
A., "Optimal
Structures",
Ph.D.
Massachusetts, [3]
thesis,
Venkayya,
V.B.;
to Enhance
the
Vibration
Miller,
D.F.
and Control
Institute
Design
of Large
of Technology,
Space
Cambridge,
1985.
K.hot, N.S.;
Active
Structural
Massachusetts
November
Vol. 24, A0gust [4]
Simultaneous
and Eastep,
F.E.,
Control
"Optimal
of Flexible
Structural
Modifications
Structures",
AIAA
Journal,
1986, pp. 136874.
and
Optimization",
Shim,
J., "Gradient
Journal
of Guidance,
Haftka,
R.T.,
Based Control,
Combined
Structural
and Dynamics,
and
Vol.
Control
10, MayJune
1987, pp. 2918.
[5]
Onoda,
J. and
Optimization
for Large
"An
Approach
Flexible
to Structure/Control
Spacecraft",
AIAA
Simultaneous
Journal,
Vol.
25, August
1987, pp. i1338. [6]
Morrison, "Integrated Guidance,
[7]
Milman,
S.K.;
Ye, Y.; Gregory,
Structural/Controller Navigation, M.;
Propulsion
[8]
PATRAN
M.;
Users'
A
California
Guide,
Conference,
Scheid,
Design:
Laboratory,
Kosut,
Optimization
and Control
Salama,
ControlStructure
C.F.,Jr.;
PDA
R.; Bruno,
Multiobjective
R.L.;
for Large
and Space
Minneapolis, R.;
and
of Technology,
Engineering,
Santa
Ana,
M.E.,
Structures", Minnesota,
Gibson,
Approach",
Institute
Regelbrugge,
A/AA 1988.
J., "Integrated
JPL January,
D6767,
Jet
1990.
California,
September
1986.
[9]
MACSYMA
Reference
Manual,
Burlington,
Massachusetts,
Document
November 32
No. SMI0500030.013,
1988.
Symbolics,
Inc.,
[10]
[11]
Dailey,
R.L.,
Journal,
Vol. 27, No. 4, April
Youla,
D.C.;
Optimal
Boyd, D.G.;
Control,
S.P.;
Belvin,
i989,
H.A.;
and

Part II:
Balakrishnan,
with
Repeated
Eigenvalues",
AIAA
pp. 48691.
Bongiorno, The
"Modem
J.J.,
Multivariable June
V.; Barratt,
WienerHopf
Case,"
IEEE
Design
of
Transactions
on
1976.
C.H.;
S.A., "A New
CAD
Method
Technical
Report
No.
Controllers,"
December
Derivatives
Vol. 21, pp. 31938,
and Norman,
Linear
[13]
Jabr,
Controllers
Automatic
[12]
"Eigenvector
Khraishi,
N.M.;
Li, X.M.;
and Associated L104861,
Meyer,
Architectures
Stanford
for
University,
1986. K., "EPS
Structure
Description",
NASA
Langley
Research
Spacecraft
Models
Center,
June
1989.
[14]
[15]
Gregory,
C.Z.,
Balancing
Theory",
Tennessee,
August
Skelton,
Francis, Criterion",
and
SIAMJ.
Flexible
Guidance
and
Control
Conference,
Using
Internal
Gatlinberg,
1983.
and Sons,
B.A.
of Large
AIAA
R.E., Dynamic
John Wiley [16]
"Reduction
Systems
Control:
Linear
Systems
Analysis
and Synthesis,
1988.
Doyle,
J.C.,
Cont.
Optim.,
"Linear Vol.
33
Control
Theory
25, No. 4, July
with
an H** Optimality
1987, pp. 81544.
1,
Appendix This appendix of the
baseline
contains
EPS
plots
structure.
showing These
A
the mode
are screen
session.
/
34
shapes copies
of the first 16 flexible from
an interactive
modes
PATRAN
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49 ORIGINAL OF POOR
P,_,CiE!? _UALiY'y
Appendix
On
QDesign
Prepared
Dr. Dr.
Robert Gfintekin
Integrated 2500 Santa
B
Mission Clara,
by:
L. Kosut M. Kabuli Systems
College California
50
Inc. Boulevard 950541215
1
Introduction
Consider transfer lated
a linear function
outputs,
the
P
function account
unstable
and
u
sufficient
dynamic the
set
1); and
sensor
original
of
inputs
and
P
in
condition
are
plant
compensation)
the
set
and
y
respectively.
Py_
block
with
(the
stable
Py_ . With
this
stabilize
used
of the
P_
exogenous
P
as weights (which
that
transfer
taking
is in fact
stabilize
regu
is required
system P
the
represents the
when
all that
by
inputs,
representing
closedloop C
described
Typically, P
assumption
stabilization
the
In general,
of all compensators
that
of
blocks
outputs.
plant
denote
outputs,
regulated
for internal
of all compensators
discretetime
w , z , u
y ) is augmented
exogenous poles
Figure
the
to
finitedimensional
inputs
plant;
from the
(see
actuator
augmented
the
timeinvariant
into is that
a necessary
in Figure
1 by
in Figure
1 is
.
W
Z
P u
y
K} Figure
For ments,
a given the
(Pu_(oo) system
lightlydamped
transfer
= 0 ).
function Hence,
in Figure
the
1 is stable { Q(I
Using
the
tions
Hz_
parametrization is given
1: Closedloop
flexible
space
Pu_ is stable set
is given
the
with
in the
position
open
compensators
unit C
and/or
disk) such
and that
rate
measure
strictlyproper the
closedloop
by
Py,,Q)_ in (1),
structure
(poles
of all proper
system
I Q is proper set
and
of all achievable
stable stable
} closedloop
(1) transfer
func
by { Pz,,,  PzuQP_
[ Q is proper
51
and
stable
}
(2)
2
Problem
Once
the
one
complete
can
the
number
order
FIR
The
problem plant
desired
Qdesign
can
As
long
lem,
need the
design
All three
of the
openloop
plant
parameter In the the
EPS
against
rest
the
Qdesign). by a fin_:e
of real desired
input
no
parameters,
a
improvement output
functions
minimization
in
plant
{Qq)q
over
Pv ,
e _2,.
q C IR N
methods,
(the
do not
of this Pointing
until
the
for the achie',es
strong
poin,=s
weights
on the
plant
any
we focus
Satellite).
compensator taken
during
feedback
on
a case
As a benchmark
_2design.
52
care
helps).
Such
a
methods. responses,
there
of, each law
is no Since
objective
function
is calculated
using
con
model. Pzw,
Pz_ and
augmenting
paths). are
prob
constrai::s
is to be implemented.
feedback
identified
parameters
definitely
by impulse
in (2) (namely,
introduced
introduce the
the
optimization
frequencydomain
_oodesign
given
a stabilizing
relying
and
parameters
is already
after
functions
report,
is available,
as a parameter
7/2 and
is already
problem
be done
transfer
Q
time
in the
conventional
identification not
both
convex
description
optimized,
(Earth
n_
of stable
include
function
stabilization need
which
number
transfer
and
to as the
an
parameters.
is approximated
can
in the
a model
ventional
blocks
problem
objective
inputoutput
feedback
of stable
is "small"
reducing
as follows:
function
suitable
evaluation
for an ni • no
parametrization
design
objective
for
until
performed N.
by
can not be parametrized
be
that
is parametrized,
improvement.
be summarized
the
now on referred the
Note
functions
performance
by increasing
class
N
closedloop
functions
requires
a "suitable"
is not
If the
can
that
transfer
Q (from
transfer
a "suitable"
such
as the
the
(choosing mix
stable
approximation
performance that
the
the parameter
(as in QDES).
is to find
Provided
possible)
optimizations
at hand
closedloop
Nevertheless,
is achieved
Nth
specific
over
parameters.
of parameter
performance
of
of Hz_
(if
set of M1 realizable
of real
sequence
to all achievable
improve
function
However
the
solution
conceivably
objective
an
Description
study
Py_ are
Hence
simply and
Pv_)
if the
appended
stable
plant
Qdesign
on the cascaded
is also
to those
a simplified
comparison,
depend
of
SISO
to be Q .
model
is compared
of
3
Method
Suppose
that
be defined
there
used
for
is one
actuator
SISO
Puu
and
one
T1
:=
sensor.
Let
the
transfer
matrices
T1
and
T2
as follows: Pzw
T2:=G.G,. _ihe following
steps
parametriza_ion
are
Qq
taken
to
compute
• q 6 IR N ) norm Hz,.
1.
Determine the
2.
the
transfer
number
the
of the
transfer
= T1
QqT2
of samples
matrices
T1
and
(in
sense,
over
a
matrix :
; get
Nsarnples
7_2norm
the
the
impulse
response
sequences
for
T2 •
Choose
a numerator
polynomial
d of the
same
Determine
order.
optimal
n and the
a strictlyHurwitz
number
of taps
denominator
polynomial
Nt_p_ . Set
Wraps n
c2q
d F_, q_z' ; i=l
we choose
the
FIR,
term
strictlyproper
since
the
optimal
Q
for a proper
plant
is
strictlyproper. 3. Filter
the
pulse
sequences
of
T2
with
_
: n
4.
Determine
a solution
q
of Wraps
argminllT_T2
q,z'll2
_ i=1
from
the
minimum
norm
least
squares
solution
of
I ql]
y=A
"
,
qNtaps
where find
y the
and
A
solution,
w,
z C IR7
and
ATy
and 6
can there N_mples
]R Ntaps
be
obtained
is no
need
= 5000 Get
the
the
leastsquares
error
construct and
=
A_[[2 53
and
. Instead,
norm
(A TA)
][y 
the y
10000
minimum q
Compute
by reordering
entries A form
leastsquares
of
T1
and
completely;
in our
ATA
6 ]R Nt_psxNtar_
solution
\ (ATy) without
T2 . To
constructing
A
cases
5. Plug the compensator
c = Q_(1 p_Q_)I in the
feedback
6. Compute loop
map
the
loop. H2norm
H_w and
check
analytically against
from step
4 .
54
the
statespace
description
of the
closed
4
Case
Let
G
Study
denote
1
the
Let
82 +O.ls+l
zeroorder
P
hold
in Figure
equivalent
1 be given
We
0
100G
Z2
0
0
0.1
y
G
0.01
G
H2optimal
chose
at 50 Hz 1 ) of tile transfer
design
Ns_mvl¢s = 5000
constants).
(3)
%0 2 IO I It
I
IIHz tl=,opt =
gives
( 5 time
function
by
100G
Zl
Discretetime
(sampled
0.8225
Let
n=d=l
For
a given
Ntaps
,
let
i=1
For different using
the
of taps,
executables)
the
closedloop
are
listed
Table
number
in Table
IIH_ll2,svs
and
map
H_,
IIH_,I[2,EXE
obtained
(the
computed
( ?/2norm
by plugging
leastsquares
from in the
the
system
error
matrix
compensator
Qq(1
IlY
Aq]12
description  P_uQq)
of 1
.)
1 .
1: Optimal
FIR
1
31.5830
31.5839
5
2.1230
2.1241
10
1.6210
1.6215
20
0.8513
0.8514
apprommationsfor
different
number
oftaps;optimM
7(2norm
is
0.8225.
Note
that
responses
of
in Figure results
column Pz_,
1, where are
due
computations. rate
and
sample
,
2 in Table P,,,
the
and
plant
Py,, P
to considerably For a larger size,
trading
1 is computed Column
by the
sampling
problem,
the
5000
3 is obtained
is represented high
scale
.
from
one
off computational
rate
and
should burden
samples
from
the
statespace long
be cautious versus
of the
closedloop
description.
pulse
impulse
sequence
in assigning
system The
used the
close in the
sampling
aiia.sing.
1Continuoustime 7/2optimal design for the plant description where G in (3) is replaced by s't0!l,+1 results in a compensator eigenvalue magnitude of 41.56 rad/s. The sampling rate was chosen apprommately 7.5 times faster to get the discretetime optimal design performance similar to the continuoustime frequency plots. 55
In order to reduce the number of parameters (
Nt_ps
), an initial
check
was
made
by
assigning _.
Z 4 4
d = II(zp,) i=1 4
Qq where
the
preassigned
parameter
Q
pi's
denote
representation

n _ d.=
the
qiz i
optimal
=
0.8306
IIH= ,ll ,sY =
0.8306
Clearly,
this
4 parameter
representation
of
rameter
FIR
representation
in Table
1 . However,
locations
to start
A couple poles
were
of other spread
re +j°k
IIHz_]12,V.XE = I]Hz_]12,sYs these
trims The
factor
better
motivation of the
smaller the
did
number in the
not
where
Ok e
[0,
1.0645,
0.8969
objective
be useful
and
If the desired
of the
approach due
achieves
Solving
one
a better
does
not
as follows:
result
have
for
than
access
for
the
to the
plant
and
in assigning to the
constraint
for
a feedback depend the
only
numerator
that
the
56
0.8,
this
4
20 paoptimal
the
FIR
factor,
law,
this
preassigned
on
the
plant
parameter
a stable
is stable.
over
into
structure
parameters.
of
.
hopefully
denominator Q
Both
(2.89)
taking
fo=r
, we obtained
3 (typically
is to be designed
and
r , the
respectively.
approximation
as in Section
with
plant
and
FIR
Q
tune
openloop
from
order
radius
[01 02] = [0.1 0.5]
r = 0.95
4th
fine
for a fixed
0m_x] • For
a structure
plant)
of parameters.
an identification may
done
a straightforward
of the
order
Q
were
is to preassign
performance
at most
=
than
inverse
locations.
with. d assignments
as
pole
( Qq ), we obtained IIH_II2,EXE
pole
Q
account will
Note
coefficients
a
be
that of
Q
5
EPS
In order original
Model
to try
out
32state
Consider
the
EPS
the
idea
involving
model
32state
the
plant
is obtained.
state
space
The
inversion, procedure
description
2,
=
Ax
an SISO
approximation
is explained
of the
below.
of EPS:
t Bthrdthr
t BactUact
[Y'°_l] Cl°sxylos2 =
(0
dthr
:=
fz45
u_ct
:=
[f042
f¢42
fc4J
y
y_o_l := [026 ¢_6]T y_os2:= [07, ¢711T There
are
16
modes
ranging
from
0.2423
Hz
to
3.5317
Hz
.
The
damping
ratio
in Figure
2 .
is
¢= 0.02 The
5.1
four
singularvalue
Simplified
We used
the
plots
EPS
following
of the
3 mode
reduced
_t Ylos
by the
following
SORIG32=[A [SBAL [AA
SIG
BTHR
plant
in (4) are
shown
Model model

Az

ClosX
:_
fo42
::
071
Ylos
obtained
openloop
MATRIXx
in (5) for the
+/)thrdthr
T
EPS
•
Bactu
(_)
commands:
BACT;CLOS
0*EYE(4)];
T]=BALANCE(SORIG32([l:10
35],[1:10
34]),10);
BB CC DD]=SPLIT(SBAL,10);
BTHR=T\SORIG32(1:10,33); ABAR
= AA(1:6,1:6);
BBARTHR
= BTHR(I:6)
BBARACT
= BB(I:6);
CBARLOS
= CC(:,1:6);
The (sampled
discretetime at 300
Hz)
model of the
;
used 6state
for plant
the
EPS description 57
model
is the in (5).
zeroorder
hold
equivalent
The magnitude plots for the discretetime 6state EPS model axeshown in Figure 3. Compared with plots C and D in Figure 2 , plots A and B in Figure 3 match the frequency responsesup to approximately 6 rad/s .
58
40 4O
2O
20
0
0 d B
20
d B
20
iiiiiiiii iiiii ili
iiiiii iiiiii i
40
40 60
60
..:.L_.i.U_!i....'..'..; ;.":::....:.i.:;;:::: ....i..:
80
,
.1 R
1
10
Frequency,
,,,
.....
,
100
,,,
......
°,o°,,,
!!iiii_ii
...........
100
1000
! !!ii!iii
, ......
•1
1
radls
,
6O
4O
4O
2O
2O
0
0 d B
° _'
° °,,°,,, ; ::'":"
; ;:::;::
10
Frequency,
6O
° :
° ,,°,,,, , ::,:;::
,
o.,
,
:_
! !ii_..
100
1000
rad/s
, ,
° ;
°°,,,,, :;'":'
° :
° ° ..... : :';;:;
:::ii:iiiiiiii::
i i iiiiiii i iiiiiiii ! iiiii!ii i!iiiiii
20
....F'.÷"m..'..'4".mr.... "..g4gg_i.o.o_4.g 4_4i
40
40
.... ;_'t:;;;;i_..'"H; L_..b .'." H h'_4..';4 I:'_
60
60
....i.._..H._.iiii....;..i.4.,_.i_iii._
80
80
100
100
20
.1
C
1
10
Frequency,
100
four
singularvalue
plots C
dthr H
:
'':':;:
plant
: ; :':I:;
:
! :;:t::;
10
Frequency,
;
'
;:;,
100
000
rad/s
2:
in (4):
Ylos2 , D
59
:
1
l)
rad/s
of the
:
.1
1000
Figure
The
,
!!ii!ii_i
100
d B
80
A u_:t _
dt_
_
Ylo_2 •
91o.1 , B 
u._t
_
ylo,1 ,
'OPCNL00P 60
:
5O
: •
SDMOOEL :
; .
;
= i
;
OTHR
>
:;=:;
+ = ; : = : " _'i'!
YL(_ .:
i
i
_ i'ii_
•
"
i
"
"_ i
= !
= : i i
: = _: :::_:
" 'i
J
+;
'
i
=
i +=_.'_ = ..=
.L_..._..__..,.L.L,_ { : = :
:
_":'i'
: ..... i
::'::
40
30
..................... ._'":"t',''_,:., _..........!
. . .i
t"I"THI'tII
.............. p......
i
!
i
iiiiii
i
I
i f
!
i
i
iiiiii
i
i
i
i
i iiiii
i
i
i i'!iii
i
'
i i'iili
!
i
i
i .=iiiii
i
i
_ ii_iii
_
i
i ili'!i
i
i
i
i
i
! ! i!iii
_
!
! i!l!!i
i
!
i
!
_,
i
i
:
!
i
i i!!!ii
i
i
i !iiili
i
i
20
10
0
10
ii!!!!
20 l
Eli!
!
iilTi
i
iiii_
!!
30
A
3O
40 .0001
.001
.01 TH_rA
.l _" ilRD3
B Figure
The
two magnitude
plots
of the 6state B
3:
discretetime _ __ _os
60
EPS
model
:
A
dt_
_1o_ ,
6
Closedloop Time EPS
Let
G denote
description
the
in (5)
Performance Model zeroorder
hold
of SISO
equivalent
(sampled
the
augmented
plant
the
feedback
100G1
As
the
entries
0
100G2
Wl
z2
0
0.1
W2
y
GI
0.01
G2
U
3, the
range
in Figure
Sample As in the taps
performance
optimal
Q
(G)
study
as
is the
the
4 , FIR
( Qq = EN_ aps qiz _ ). Results
are
The
magnitudeplots
negative
approximations
are
in Table
is de
for
D in Figure
of the due
P_
to the
(approximately
listed
compensator
4. Comparing
60 dB shift
Ns_mples = 10,000
in Section
.
in Figure
plot
account
_2optimal
2.284
=
axe shown
magnitude into
size is chosen case
plant
G2Qq)'y
discretetime
][Hzwll2,opt
Hz_
3; taking

measure,
yields
7/woptimal
Figure
6state
is to be determined.
Qq
closedloop
of the
of the
law
1 , where
a reference
signed;
Hz)
+ G2fi
0
u = Qq(1 as in Figure
300
model
Zl
Apply
at
Discrete
where ylos = Gldthr
Consider
6State
plot
1 time made
four
4 and
(over
weights
the
the
0.1 and
constant
B in
0.3 tad 0.01).
at 300 Hz).
for different
number
of
2:
[ lv,,p, I IIH.,.II_,Ex_IIH.,.ll_,_¥s
Table
2:
Optimal
FIR
10
225.5873
277.0512
20
225.5419
276.6257
100
224.1206
269.1206
approximations
for
different
number
of taps;optimal
"Hwnormis
2.2840.
Note due
that,
unlike
to 1 time
is not
constant
a considerable
Instead,
Table
1,
IIHz_IIe,EXEand
truncation
of the
improvement
original
in the
IlH_ll2,sYs (see pulse
sequence.
closedloop
section Despite
4)
are
large
not
close,
Ntaps
there
7/2 norm.
we chose
Qq =
12
4
_ E
qiz'
,
(7)
i1
where (the
n has two
zeros
all six poles at zero
are
of
G2
chosen
and
d has
all four
to make
relative 51
stable degree
zeros
of
zero;
G2 no
and specific
two
at
zero
reason
for
assigning them at zero). Hence _ represents the stable factor the
The
four
FIR
magnitude
in Figure
parameters
plots
are
introduced
to fine
]IH,_[[2,EXE
=
7.7974
I[//.w]]2,SYS
=
3.2685
of the
four
entries
of the
tune.
Calculations
(optimal final
5.
62
run
of the inverse show that
of
G2
and
is 2.284) H_
with
Qq
as in (7) are
shown
9
15
i iiiiii !iiiiiiiii iii!!!!i
10
i iiiiiiii i iiiiiiii i iii!i!iii iiiiiii!
...:..i i;.::.:i;. :.; i::._:;i .:.; ; L:.:.:: : ; i_::.; .......................
5
3
0
0
..
,:, :,"
• ...°,,
_.:.:,:,'; • ..:.
.........
m
5
, . ; .:.$W'
,
• • • .'... ;. ; .:;;M_.
......
,, '
....
• ,,,,,,,
;
;
;,';;:;
;
:
;,:;::;
; ;;,;;;;
;
;
;;;::::
:
',;:;::;
3
10
....
15
.... !..i. i _.:,'_ii..._..i. i _.;';!i;....';._.i._:,:,:,!i ....
Oo. i.
i ,.,vii,......
r. _ _.,,,i
Q....,..
a, i ._l_.l....*.
° s. _._ i _ _,J.
o•
6 ..........
iii!iiii'ii
9 20 12
25 .0001
.001
.01
.1
1
_
.0001
.001
.01
.1 rod
rod 20
3
0 6 20 9 m
40
m "13
12
60
15
80
18
100
:
: :::::::
:
: :::::::
:
: :::::::
:
: :':::::
."
: :::""
,
, :,;;:;:
;
, ,,:,:,:
,
; '.':::::
iiiiiiiiiiiiiii ii!iiiiil iiiiiii i iiiiiiiil; i;_;;_;
21 C. o0001
.001
.01
120 _ .0001
I
.I rod
in (6):
plots for _he four entries A
w_ _
z_
, B
: "
::
: : : :'::" : :'..":':
: :
. "._;...:..i.;.;_.;_'i..:'.._i.!.;;_:_ .... : : : ::::: ;:.';;'2;
.001
; "
: : : :::7:. " :':::::
.01
: '.
: : : :::': '.':'::::
. rod
Figure
The magnitude
1
4:
of the
w2 _
z_
63
7/zoptimal , C
H,_ for the plant
w_ __ z2
, D
w2
description _
z2
I
4O 3O 2O 10
m "D
m Q
0
!iiii'"'M'i':'i'i'ii'"'_"i'i_'i'_iii""i"i'i'i'i'i'i'ii'"'i"'iii i! i!! 2
_iiiiii! !iiiiiii! !iiiiiiii i iii!iiii ....'...i.; ;.'.::_;.•;.i. i ".:,_.!_;....;, i,i.;;;_.!i...._.. ;.i.!,;;;i'.
10
i!iilili iiiiiiiii!iiiiii iiiiiiii
20
30 R .0001
.001
,01
0
2
.I rod
,
.
.
°..,.,
,
•
• H,,.
,
.
....
•
•
• ,,.,.,
.
•
.
.
•
• ....,.
•
•
• .,°.,°
...........
.,•..°
,
.001
6.oooI
•..
....
.01
.
.
.
•
•
° • .....
.
,
°.
•
..,..°
• .....
,1
1
rod
2O
iiiiliiiiii!il i!iii!i !ili}iiiiiiiiili!iiiiil iiii!iiii
2
;
;;;;:::
;
:
;;::;;;
:
;
:;;:;'4
;
:
;;';:;:
!i!!iiiii iiiiiiiii i iiiiiiii !!iiiiT
0
4
:
20
6
40
....i'!iiii_!!'"_"!i+iiiiiii__""
m "o
60
1D
8 " ..... :"
80 10
12
...._..;.;;!;._..... _.,._.!.::,,....;. i._.__,._ii!..
100
!i!!!i!i i!i!!ii_!ii!!!ilii!!!!_i_i!_! ,0001
,001
.01
D
Figure
The
magnitude in (7):
plots A
Wl
of the I4
120
.I rod
G
four Z 1
,
entries B
w2
:: ". :_;...:..i.i.;.:.,_._i....;.M.;;;;_ ....
.0001
,001
.01
.I rod
I
5:
of
H_
_
zl
64
for the , C
plant wl
_
description z2
, D
in (6) w2
_
and z2
Qq
N/ A Na_aJ
Ae,ro_u'_cs
Report Documentation
and
Page
Space Mrninis_'ation
2. Government
1. Report No.
3. Recipient's
Accession No.
Catalog No.
NASA CR182020 5. Report Date
4. Title and Subtitle
January Integrated
ControlStructure
15, 1991
Design 6. Performing Organization Code
8. Performing Organization
7. Author(s)
Report No.
K. Scott Hunziker and Raymond H. Kraft 10. Work Unit No.
506145101
9. Performing Organization Name and Address
Boeing Aerospace P. O. Box 3999 Seattle, WA 921242499
11. Contract or Grant No.
NAS118762 13. Type of Report and Period Covered
Contractor Report
12. Sponsoring Agency Name and Address
National Aeronautics and Space Administration Langley Research Center Hampton, VA 236655225 15. Supplementary
i14,
Sponsoring Agency Code
Notes
Langley Technical Monitor: Dr. Ernest S. Armstrong Task 3 Final Report
16. Abstract
A new approach for the design and control of flexible space structures is described. The approach integrates the structure and controller design processes thereby providing extra opportunities for avoiding some of the disastrous effects of controlstructures interaction and for discovering new, unexpected avenues of future structural design. A control formulation based on Boyd's implementation of Youla parametrization is employed. Control design parameters are coupled with structural design variables to produce a set of integrateddesign variables which are selected through optimizationbased methodology. A performance index reflecting spacecraft mission goals and constraints is formulated and optimized with respect to the integrated design variables. Initial studies have been concerned with achieving mission requirements with a lighter, more flexible space structure. Details of the formulation of the integrateddesign approach are presented and results are given from a study involving the integrated redesign of a flexible geostationary platform.
17, Key Words (Suggested
18, Distribution Statement
by Author(s))
Controls, Flexible Structures, Vibration Suppression, Optimizationbased design, Integrated Design, ControlStructures Interaction, Youla parametrization, Nonlinear programming 19. Security Classif. (of this report)
Unclassified NASA FORM 1626 OCT 86
20.
UnchL_sifiedU.limited Subject Category 18
Security Classif. (of this page)
21,
No. of pages
65
Unclassified For sale by the National
Technical Information
Service,
Springfield,
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