NASA Technical Reports Server (NTRS) 19910005867: Integrated control-structure design

A new approach for the design and control of flexible space structures is described. The approach integrates the structure and controller design proce...

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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

98124-2499

NAS1-18762

December,

1990

NqI-151RO

(NAKA-CR-Ia2020) C_NTRNL-STRtJCTURE

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 23665-5225

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 control-structures

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 open-Ioop

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 follow-on

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 two-step on structural

design,

closed-loop

Approach

the

large,

flexible,

procedure. costs

second

dynamic

and Implementation

and

step

behavior.

The

first

open-loop

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

[2-5]. 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 closed-loopresponseof 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°seu-c°°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 "mean-square

Structural

I

Iit

_

that describe this explicit

[ Open-Loop[

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

pre-processor.

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 pre-processor

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(M-1K)

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 open-loop

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 open-loop 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

Open-Loop

objective

One

and

from

the previous

is often achieved

implementation,

constraint

of the structure advantage

(fin-st order)

model

as starting

of eigenvectors.

vectors

for the present

in only a few iterations.

the closed-loop are obtained

via the so-called

Q-design

Since

coordinates,

transfer

directly

the open-loop

needed

open-loop

described

not require

the open-loop

functions

from

approach

is that it does

of the structure.

into modal

equations

transfer

transfer

in the next

the assembly

funcsection.

of a state-space

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 right-hand 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 well-known

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'Oii-C'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 fight-hand

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 Q-design, based

on the Q-parameterization

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 closed-loop

for our purposes. co-prime

factorization

= _-1_

(16)

= ,_-I

the set of all stabilizing

Furthermore,

all achievable

Q parameter

IE ] D

B

QN_)-I(X

closed-loop

-X

N

controllers

K = {(Y-

affine

[11].

such that

I In that case,

by Youla

and input/output

the stable,

Knom,

Knom = y-Ix be chosen

developed

this formulation,

property

controller

Pyu = N D-1

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 closed-loop

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 Q-parameterization 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

Q-design specify

variety

approach an objective

of closed-loop

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

2-norm density

Functions

(LQG)

of a transfer

and undershoot

function

at specified

frequencies

for a step input

margins

• Infinity-norm • Transfer

of impulse

function

and step responses

infinity-norm

(H-Infinity)

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

Q-design

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

infinity-norms

QDES

tractable

of design

and a solution

Admittedly,

response

optimization

the number

and undershoot

infinity-norm

2: Sample

this approach

the resulting

and controls

margins

Table

What

overshoot

of states

from

of a truss-type

in length

known

design as the

the Ford Earth bus with

and the antennas

two

are 15

15 m---------_

7.5m-_

(typical)

t-Figure

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 y-coordinate

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 y-coordinate 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

open-loop 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: Z-direction 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 uQ---uo

=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 Q-design

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.

Q-Design

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 z-direction

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 1000N-m. The minimum Nyquistdistancewasrequiredto beat least0.5 s-1. 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

10-12,

response

of

three

orders Figures

that

violates

thruster

disturbance.

actuators.

The

constraints

were

illustrate

closed-loop imposed

error

right

Of the four The

the pointing

gr26, the pointing

13-15

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 result9-18.

a constraint into

on

a 1.7x10 -6

can see that the openof--6.5x10

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 closed-loop 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 open-loop

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

closed-loop

z-direction which

in high order

in about

0.5

071 is the most

responses,

for the open-loop 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

16-18

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 N-m 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-

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23

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_

Res_nse

12

I _-

:E

-_

__

off¢42 to a unit fz45:lrmpulse, 26 OR1G!NAL

PAGE

IS

OF POOR QklALITY

03

0 2

,31

&

-0_

-

,32

-

r)3 0

5

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 twice-continuously

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 closed-loop

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

finite-dimensional

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

Q-design.

It has proven

to be a very flexible that meet these

to Work well--inthe

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

non-intuitive

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, controllersdevelopedusingQ-designtendto 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 Q-designin the combined optimization does not preclude the use of other control design tools for the optimized structure; the benefits

of Q-design

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 Q-filter

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 pre-filtering

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 Q-filter. 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. 1368-74.

and

Optimization",

Shim,

J., "Gradient

Journal

of Guidance,

Haftka,

R.T.,

Based Control,

Combined

Structural

and Dynamics,

and

Vol.

Control

10, May-June

1987, pp. 291-8.

[5]

Onoda,

J. and

Optimization

for Large

"An

Approach

Flexible

to Structure/Control

Spacecraft",

AIAA

Simultaneous

Journal,

Vol.

25, August

1987, pp. i133-8. [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,

Control-Structure

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,

D-6767,

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. 486-91.

Bongiorno, The

"Modem

J.J.,

Multivariable June

V.; Barratt,

Wiener-Hopf

Case,"

IEEE

Design

of

Transactions

on

1976.

C.H.;

S.A., "A New

CAD

Method

Technical

Report

No.

Controllers,"

December

Derivatives

Vol. 21, pp. 319-38,

and Norman,

Linear

[13]

Jabr,

Controllers

Automatic

[12]

"Eigenvector

Khraishi,

N.M.;

Li, X.M.;

and Associated L-104-86-1,

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. 815--44.

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,_,C-iE!? _UALiY'y

Appendix

On

Q-Design

Prepared

Dr. Dr.

Robert Gfintekin

Integrated 2500 Santa

B

Mission Clara,

by:

L. Kosut M. Kabuli Systems

College California

50

Inc. Boulevard 95054-1215

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

closed-loop 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

discrete-time

w , z , u

y ) is augmented

exogenous poles

Figure

the

to

finite-dimensional

inputs

plant;

from the

(see

actuator

augmented

the

time-invariant

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

lightly-damped

transfer

= 0 ).

function Hence,

in Figure

the

1 is stable { Q(I-

Using

the

tions

Hz_

parametrization is given

1: Closed-loop

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-

strictly-proper the

closed-loop

by

Py,,Q)-_ in (1),

structure

(poles

of all proper

system

I Q is proper set

and

of all achievable

stable stable

} closed-loop

(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

Q-design

can

As

long

lem,

need the

design

All three

of the

open-loop

plant

parameter In the the

EPS

against

rest

the

Q-design). 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

_2-design.

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

frequency-domain

_oo-design

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

input-output

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

closed-loop

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

closed-loop

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

Q-design

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,. _i-he 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_2-norm

the

the

impulse

response

sequences

for

T2 •

Choose

a numerator

polynomial

d of the

same

Determine

order.

optimal

n and the

a strictly-Hurwitz

number

of taps

denominator

polynomial

Nt_p_ . Set

Wraps n

c2q-

d F_, q_z-' ; i=l

we choose

the

FIR,

term

strictly-proper

since

the

optimal

Q

for a proper

plant

is

strictly-proper. 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

"

,

q-Ntaps

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

least-squares

error

construct and

=

A_[[2 53

and

. Instead,

norm

(A TA)

][y -

the y

10000

minimum q

Compute

by reordering

entries A form

least-squares

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. H2-norm

H_w and

check

analytically against

from step

4 .

54

the

state-space

description

of the

closed-

4

Case

Let

G

Study

denote

1

the

Let

82 +O.ls+l

zero-order

P

hold

in Figure

equivalent

1 be given

We

0

100G

Z2

0

0

0.1

y

G

0.01

G

H2-optimal

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

Discrete-time

(sampled

0.8225

Let

n=d=l

For

a given

Ntaps

,

let

i=1

For different using

the

of taps,

executables)

the

closed-loop

are

listed

Table

number

in Table

IIH_ll2,svs

and

map

H_,

IIH_,I[2,EXE

obtained

(the

computed

( ?/2-norm

by plugging

least-squares

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-(2-norm

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

state-space long

be cautious versus

of the

closed-loop

description.

pulse

impulse

sequence

in assigning

system The

used the

close in the

sampling

aiia.sing.

1Continuous-time 7-/2-optimal 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 discrete-time optimal design performance similar to the continuous-time 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(z-p,) 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

32-state

Consider

the

EPS

the

idea

involving

model

32-state

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

singular-value

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

discrete-time at 300

Hz)

model of the

;

used 6-state

for plant

the

EPS description 57

model

is the in (5).

zero-order

hold

equivalent

The magnitude plots for the discrete-time 6-state 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

singular-value

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"TH-I'tI-I

.............. 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 6-state B-

3:

discrete-time _ _-_ _os

60

EPS

model

:

A-

dt_

_1o_ ,

6

Closed-loop Time EPS

Let

G denote

description

the

in (5)

Performance Model zero-order

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

_2-optimal

2.284

=

axe shown

magnitude into

size is chosen case

plant

G2Qq)-'y

discrete-time

][Hzwll2,opt

Hz_

3; taking

-

measure,

yields

7-/woptimal

Figure

6-state

is to be determined.

Qq

closed-loop

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

6-State

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.

closed-loop

section Despite

4)

are

large

not

close,

Ntaps

there

7-/2- norm.

we chose

Qq =

12

4

-_ E

qiz-'

,

(7)

i----1

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 ..........

--i--i-i!iiii'-i-i

-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-/z-optimal , 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-'!ii-ii_!!'-"_"!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 I----4

-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 CR-182020 5. Report Date

4. Title and Subtitle

January Integrated

Control-Structure

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.

506-14-51-01

9. Performing Organization Name and Address

Boeing Aerospace P. O. Box 3999 Seattle, WA 92124-2499

11. Contract or Grant No.

NAS1-18762 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 23665-5225 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 control-structures 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 integrated-design variables which are selected through optimization-based 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 integrated-design 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, Optimization-based design, Integrated Design, ControlStructures Interaction, Youla parametrization, Nonlinear programming 19. Security Classif. (of this report)

Unclassified NASA FORM 1626 OCT 86

20.

UnchL_sified--U.limited Subject Category 18

Security Classif. (of this page)

21,

No. of pages

65

Unclassified For sale by the National

Technical Information

Service,

Springfield,

Virginia

22161-2171

22. Price

P

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