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6.1991-2737

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IdeI
J Ramakrishnan", A. Hutand R. VanderVoort"
J. Berg+ and Lt. D.F. Cosseyt
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2737
bstract
Identification experiments on the Air Force Phillips
Laboratory's (formerly the Astronautics Laboratory)
Advanced Space Structure Technology Research Experiment facility (ASTREX) is described in this paper.
A finite element Model (FEM) of ASTREX was constructed to provide modal vector information (necessary for control law development) and also to provide
a comparison for the identified models derived from
experimental data. The q-Markov COVER (short for
Covariance Equivalent Realization) and Frequency Response Functions (FRFs) were used for the identification. The identified models show good correlation with
the time history and frequency response data obtained
in the laboratory.
Figure 1: ASTREX Facility
The ASTREX facility provides the hardware to test
and validate emerging space structures technology. The
large precision structures laboratory provides three
axis large angle slew capability, realistic missionrepresentative test article, the ability to change key
structural members, actuators and/or sensors and a
fully programmable real time computer. The test article is supported by a 0.5 meter airbearing on top of a
5 meter pedestal.
The pedestal comprises of pneumatic components, electronics, power supplies and a computer at its base. A
gimbal system, located alongside the airbearing, is part
of a cable follower system that minimizes the boundary
effects of all the electronic lines running from the floor
to the slewing test article. The ASTREX test article
is a dynamically scaled model of a 5-meter class Space
Based Laser (SBL), consisting of primary, secondary
and tertiary components (Figure 1). For now, the mirrors are simulated by masses placed at the appropriate
locations. Identification of ASTREX facility is crucial
from both the controller design and the finite element
"Dynacs Engineering Co., Palm Harbor, FL 34684
t Airforce Phillips Lab, Edwards AFB, CA
OCopyright 01991 by the AIAA, Inc. All rights reserved
1219
model validation perspective. The purpose of the laboratory experiments described herein is to obtain a precise characterization of the ASTREX structure. Some
of the parameters of interest were the frequency spectrum, the damping associated with the different modes
and for control design the linear time invariant state
space models. The q-Markov COVER identification
method was used to generate the state space models
of ASTREX. The q-Markov COVER approach can be
used on data generated by an applied pulse or white
noise input and seeks to generate covariance equivalent
realizations matching the first q covariance and (q 1)
Markov parameters of the data. Frequency response
characterizations were found using a disturbance electromagnetic shaker subjected to either a sine dwell or
periodic chirp (fast sine sweep) source input.
+
The paper is organized as follows: a brief formulation
of the g-Markov and the FRF methods is presented in
section 2 followed by a description of t!ie
setup in section 3. Results are presente
section followed by conclusions from the s
orrnulat ion
where
The q-Markov COVER approach for discrete systems
is based on the results from reference [I], [2]. A brief
description of the approach is presented here for the
sake of completeness.
2.1
(5)
q-Markov COVER Theory
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2737
The Markov and autocorrelation parameters of the system to be identified are determined from the experimental data. There are two approaches to this problem:
(i) Excite the system by a pulse input to obtain the
Markov parameters and by a white noise input to
obtain the autocorrelation factors.
(ii) Obtain both the Markov and autocorrelation parameters from the pulse response.
The second approach is used in this work since one experiment yields both the Markov and autocorrelation
data. Let the Markov and autocorrelation parameters
obtained from the pulse response data be {Hi}? and
{Rl}?.
Further, the finite data sequences { H i } $ and {Ri}$-l
are consistent, in the sense that it can be generated by
a linear system, if and only if
D, 2 D9 2 0
It can be shown [l]that D, and D q satisfy the paranieter identities
d
D,
Then the multivariable discrete time system
3(k
+ 1)
= A q k ) 4-B u ( k )
d(k) = & ( k ) + B u ( k ) k >_ 0
with a state covariance
2 = limk,,
=
A
0,XO;
A
De = O,AXA*Oi
(1)
2(k)i*(k)
(9)
where the set { A ,B , C, D ,X} is the multivariable disCrete time system generating the output data used for
realization and 0, is the corresponding observability
satisfying
X=AXia+BB*,
x>o
is a q-Markov COVER of the system generating the
data sequences {Hi}:, and {&}:
if
Hi
&
= H i , i = O , 1 , 2 , . . . ,Q
= &, i = o , l , . . . , q - l
(3)
The realization problem states that given the finite
find all minimal
data sequences { H i } : and
stable q-Markov COVERS.
The realization problem is based on the factorization
and projection of two data matrices. Using the pulse
response data, the two data matrices D, and De are
constructed.
D,
A
= R,-H,H;
D, A=
R,-H,Hi
(4)
1220
The solution of the realization problem depends on the
key observation that D, and Dq can be factored in
terms of the observability matrix 0 , , the system matrix A and the state covariance matrix X . The full
factorization of the first data matrix yields
D, = P A P " , A > O
A
T
=
P
E
rank (D,) = rank (P)= rank ( A )
V y q X r ,
Our goal is to force P and A to play the role of the
observability matrix 0 , and the state covariance ,f of
the identified state space realization (1). In order to
force P and A to play the roles of Oq and
we partition
s,
dimension ny x r and define two new
where G is an ny x r matrix to be determined such that
Dq = PAPe
P will play the role
of the matrix product
and the second parameter
identity in (9) will be satisfied. The matrix Da can be
If this G can be determined, then
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6,a
Figure 2: Laplace Plane Representation
leading to the relations
Dq-l
= FhF*
Eq =
FAG*
Dqq = GAG"
The relation (15) is satisfied by virtue of the construction of p . Solution of relations (16) and (17) yields G
and hence p . The identified matrices are
a
-
A
- A
Po, D=H,, X = A
A
Mq = [H;,
H ; , . . . , H;]"
The interested reader is directed to references [l],[2] for
more details and proofs.
2.2
Frequency Response Function
It is well known that a system's transfer function correiy in.put/output pair can be represf mted
the Laplace plane by plotting the system's complex poles and zeros (Figure 2).
Because the poles of a 2nd-order, viscously damped system are intimately related to the modal natural frequencies and modal damping ratios as described in Equation
19, knowledge of a system's poles is sufficient for the
identification of those modal parameters. The modal
parameters of frequency and damping are global properties of the system; consequently, all input/output pairs
d
Figure 3: Bode Plot
consist of identical poles. However, the modal amplitudes depend upon the choice of input and output and
may be too small for the identification of some of the
poles. This means that one would investigate multiple
sets of input/output pairs in order to flush out all of
the poles of interest.
Unfortunately, it is not possible to measure the traasfer function directly. The best we can do is to measure the semi-infinite slice of the Laplace domain corresponding to the positive imaginary axis. The resulting
2-dimensional curve, known as the Frequency Response
Function (FRF), is simply the ratio of the output to
the input of a system with respect to the frequency of
excitation. There are several ways of plotting FRFs,
the most popular being the Bode Plot (Figure 3). The
Bode Plot is actually two curves with a common abscissa being the log of the frequency: one curve is the
magnitude of the F R F (in decibels) and the other is
the phase of the FRF. Figure 4 displays a linear, timeinvariant system subjected to noisy sensor readings at
both input and output. The FRF, symbolized as H ( u ) ,
is the ratio of output, y, over input, 2.
are generally plagued by noisy sensor r
make the exact measurements of z and
A method of minimizing the effe
1 2 21
= cxy + c , u
mu
Therefore, the F R F estimates H1 and H:!become:
nu
Hi
M
cy, + cup
cx, + c,,
-
k+%
W
X
e+%
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2737
Figure 4: System Subjected to Noisy Signals
out auto- and cross-correlations which are then used for
F R F estimates. There are two common F R F estimates,
often symbolized by H1 and Hz. H1 is the ratio of
with both the numerator and denominator multiplied
by the complex conjugate of u. Similarly, Hz is the
ratio of
with both the numerator and denominator
multiplied by the complex conjugate of w.
Hi
cwu.
cuu
Hz
-.
C W W
cuw
The subscripts of C are the frequency dependent sequences which are correlated. For example, Cab is the
correlation between A ( w ) and the complex conjugate of
E ( w ) , E* ( w ) . Expanding the correlations of Equations
(20) and (21) in terms of the actual signals and noise,
and assuming that all noise is uncorrelated with the input and output, we can see how these two estimates
compare with the true FRF:
cwu
E
W(w)U*(w)
= (Y+ N)(X" + M")
= cy, cy, c u ,
+
cuu
+
+ cup
CY,+C",.
2 U(w)U*(w)
= (X M)(X* + M")
= cxx cxp c
,x 4- c,,
(22)
+
+ +
= cx, + c,,.
cw,
=
W(w)W*(w)
+
+
-
We know that as we approach resonance, the required
load to drive the system decreases rapidly yet the system maintains its relatively large vibrations. Because
the input signal falls off, the noise within the input sensor signal becomes
influential. Likewise, the noise
within the output sensor signal becomes
influential. Away from resonance, the opposite occurs. Using
this insight with Equations 25 and 26, we can construct
guidelines on the use of the two FRF estimates. We
see that H1 is generally a better estimate of the FRF,
except at resonance where H2 is the better estimator.
Additionally, H1 tends to behave as a lower bound for
the true F R F while Hz behaves as the upper bound. In
the experiments conducted here it is found that the difference between H1 and H2 are slight. The coherence,
which is the ratio
remains very near 1.0 throughout the frequencies 11 Hz and greater. When a periodic
chirp is used for the input source, there are shallow dips
in the coherence at the anti-resonances, which is to be
expected. To estimate the F R F to an acceptable degree
of resolution, the poles are first found using a periodic
chirp (fast sine sweep). One does this by using different actuator and sensor locations in order to miss as
few poles as possible. Once the pole frequencies have
2,
(23)
= (Y+ N)(Y*+ N")
= cy, + cy, cuy
The terms N i l N o , Neil and, N,, are power ratios which
represent the degree of noise within the sensor readings.
Ni is the ratio of the power of the input sensor noise
m e r the power of the input sensor signal. Similarly, N o
corresponds to the output sensor signal. N,, and N,,
are the ratios of the noise cross-correlation (input sensor
noise and output sensor noise) over the power of the input sensor signal and output sensor signal, respectively.
cuu
cy,+ c , u .
cuw = c,zu
1222
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Figure 5: W i n d o w of Curve Fit
estimated, the SISO Sine Dwell procedure is used on
each of the individual poles keeping a frequency resolution of approximately 5-mHz. Various methods have
been tried to determine the modal damping ratios, but
the one used herein is the steady-state method of curve
fitting the peaks of H I across the frequency band which
encloses a 3-dB spread in the FRF amplitude (Figure
5). This is a straightforward sequence of steps using an
HP-3562A Dynamic Signal Analyzer.
Figure 6: Selected S e n s o r S e t
Experiment
3.1
q-Markov ID
The identification experiment was conducted at the
AFAL ASTREX facility. The control data acquisition computer (CDAC) has 32 analog input and output
channels with a maximum sampling rate of 1000 Hz.
From an admissible set of sensors (accelerometers) 31
tical sensor locations were selected using the sensor
d actuator selection algorithm [3]. The 32"d channel
used to record the input pulse. The selected sensor
et is shown in Figure 6. The accelerometers used are of
he piezoelectric type with an eighth-order Butterworth
ost-filter. The actuator locations for the q-Markov ID
vperiment are given in Figure 7 and Table 1. The six
ctuators include four thrusters and two roll actuators;
n the absence of these actuators impulse hammers were
to excite the ASTREX structure. The actuator
ions are all on the primary truss. The sensors, on
ther hand, have been installed on the primary, secondary and tertiary mirrors and the connecting tripods.
I
t
A sampling frequency of 500 Hz w a s used in identification. This sampling frequency choice w a s dictated by
( i ) Sampling of the 90 Hz mode
Figure 7: A c t u a t o r Locations
(ii) Using the 500 Hz data, data at 250 Hz and 100 Hz
could be constructed by deleting the appropriate
1.22' 3
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samples.
The experiment was repeated 10 times for each input
channel. The repetition was done from statistical considerations to obtain well averaged data. The averaging
process helps in smoothing measurements corrupted by
high and low frequency noise. The experiment is repeated for all the input channels. For each input, 10 sets
of sensor output data were collected for approximately
10 seconds. The data was detrended, scaled and averaged to produce a processed data file. The processed
data files have 8 seconds (4000 samples) of data for each
of the 31 output channels due to each input pulse. The
input intensities are determined from the input channel
(Output 32) to be:
W
=
I-,
I
1
I
1
I
Apply pulse
L
collect Output
data
diag ([0.2305 0.2695, 0.2776, 0.2337
0.3065, 0.31611)
pulse strength for
each experiment
Table 1: Input D a t a
Input #
I
Type
Node
15
Direction
Pulse
Location
Lower Right
Pulse
Upper Right
14
+z
Pulse
Upper Left
13
+z
Pulse
Lower Left
16
+z
+z
Scale all outputs
associated with
i=i+l
b
Pulse
TOP
27
+Y
Pulse
Bottom
28
-Y
The data collection and processing procedure is shown
in Figure 8. For modal testing, an electromagnetic
shaker is attached to node 14 of the structure and the
response is measured at node 23. The electromagnetic
shaker has a stinger unit that transmits the force to the
structure. A force transducer on the stinger sends the
signal to channel 1 of a HP 3562A dynamic signal analyzer. The input signal is controller from the analyzer.
The accelerometer is connected to channel 2 of of the
analyzer and the transfer function is obtained from the
input-output relationship.
4
seconds of scaled
data for each
Average the data
from the 10
Processed data
esults
The results from the q-Markov COVER identification
are compared with the results from modal testing in
Table 2. The frequencies from modal testing, finite element modeling and ID experiments are presented. Table 2 shows that the frequencies of the identified model
are close to the frequencies obtained from modal testing.
The outputs sampled a t 500 Hz were used to identify
the ASTREX facility. The q-Markov procedure w a s executed with different values for q ( q = 10, 20 and 30).
1224
Figure 8: Data Collection and Processing Procedure
am
io
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405
Figure 9: Singular Values of D, (Ny = 6, q = 30,
Hz 1
. . . .- .
For the identification, the first six sensors alone were
used. One of the decisions made by the user is the selection of the rank of the data matrix D,. This rank
determines the order of the state space realization. In
this work, the singular values of D, were plotted and
rank determination was made based on the relative
nitudes of the singular values. The results from the
ces of q of 10 and 20 were not very good. In order to
ve the model, and to obtain better low frequency
teristics a q parameter value 30 was used. The
ot of the singular values of the data matrix is shown
Figure 10: Time response comparison y1 due to
= 30, 500 Hz)
~1 (9
A
Based on the singular values, an identified model of
der 100 was obtained. The time responses of output
annels 1 and 2 due to the first input and the correonding lab response are plotted in figures 10 and 11.
The correlation between the experiment and the identified response is very good. Next, the Fast Fourier
I (FFT) of the two data sets were compared.
e FFT of outputs 1 and 2 are shown in figures 12 and
. The figures show poor agreement below 50 Hz and
ement in the higher frequency ranges. The
r this may be the use of accelerometers as senhe high sampling rate used. With acceleromeeven substantial low-frequency vibration displacet may result only in small accelerations, since acration is proportional to the square of the natural
1225
Figure 11: Time response comparison y2 due to
( q = 30, 500 Hz)
211
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L M D Diu W 1
Figure 14: Time response comparison yl due t o
( q = 40, 250
Figure 12: FFT of y1 due to
u1
(q
Hz)
= 30, 500 Hz)
Increasing q will result in better low frequency characfkistics. However, this also results in higher computational burden and introduces more noise modes in the
model. The q-Markov procedure was repeated at 250
Hz to alleviate the problem.
The processed lab data file is converted to a 250 Hz sampling rate by retaining every other data point. Using a
smaller sample rate increase the response time window
for the same number of samples. The first 500 samples
of the data at 250 Hz (T = 2s) were used. Using a q
of 40, a 120th order identified model was obtained. T h e
time history response of outputs 1 and 2 in figures 14
and 15 show good correlation with the lab data.
Further the FFT of these outputs (figures 16 and 17)
show good results for the low frequency spectrum. The
identified model's frequencies match the results of other
ASTREX modal identification experiments.
Figure 13: FFT of
y2
due to
u1
(q
= 30, 500 Hz)
1226
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Figure 17: FFT of yz due to
rb
u1
( q = 40, 250 H z )
Table 2
Frequency Comparison (10-20 H z ) :
Experiment, FE Model, ID Model
Sine Dwell
Spectral
Curve Fit
Periodic Chirp
Curve Fit
(0-20 Hz)
I
Curve Fit
(0-100 Hz)
Models
ID Hz
F E Hz
I
10.5
onclusion
Identification experiments on ASTREX are described
in this paper. The q-Markov COVER approach is used
to identify models at 500 Hz and 250 Hz. The identified
model at 500 Hz has poor low frequency characteristics.
This is caused by the choice of accelerometers as sensors and the high sampling rate used. The identified
model at 250 Hz shows a significantly better response
at the lower frequencies. The comparison between the
tion. The modal characteristics of
finite element models are compared
modal data well. The results
of q-Markov COVER metho
122 1
6
rences
1. A.M. King, U.B. Desai and R.E. Skelton, A Generalized Approach to q-Markov Covariance Equivalent Realizations for Discrete Systems, 1987 American Control Conference, Minneapolis, MN.
2. U.B. Desai and R.E. Skelton, Partially Nested
q-Markov Covariance Equivalent Realizations,
CTAT, Vol. 3, No. 4, pp. 323-342, December
1987.
Downloaded by UNIVERSITY OF ADELAIDE on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2737
3. A. Hu,R.E. Skelton, G.A. Norris and D.F. Cossey,
Selection of Sensors and Actuators with Applications to the A S T R E X Facility, Proceedings of
the 41h NASA/DoD CSI Conference, Orlando, FL,
Nov. 1990.
cknowledgements
Prof. Robert Skelton was a consultant for this work.
His help and suggestions a t different stages of development are gratefully acknowledged. Thanks are also
due to Dr. A Das for his helpful suggestions. Ms.
Linda Large’s dedication to typesetting and preparing
this document is appreciated.
1228
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