Multi-Axis Active Isolation from a Lightly
Damped Vibrating Structure
by
Francisco Aguirre III
Submitted to the Department of Electrical Engineering and
Computer Science
in partial fulfillment of the requirements for the degree of
Master of Science in Electrical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 1994
( Massachusetts Institute of Techn0logy 1994. All rights reserved.
Author.......
Department of Electrical Engineering and Computer Science
May 18, 1994
I
Certified by.
Michael Athans
Professor of Electrical Engineering
Thesis Supervisor
C'
Accepted
by.............
I
\v)
Frederic R. Morgenthaler
Chairman, Departmental Committee on Graduate Students
LISrwnico
Multi-Axis Active Isolation from a Lightly Damped
Vibrating Structure
by
Francisco Aguirre III
Submitted to the Department of Electrical Engineering and Computer Science
on May 18, 1994, in partial fulfillment of the
requirements for the degree of
Master of Science in Electrical Engineering
Abstract
This thesis applies multivariable Linear Quadratic Gaussian (LQG) control design
methodologies to provide simultaneous isolation of an active mirror mount in multiple directions. To achieve this goal, three single input single output (SISO) control
designs are implemented to determine the achievable performance in each of the directions of interest while neglecting the dynamical coupling within the active mount.
It is shown that, in general, attempts to isolate in any one direction by itself leads
to amplification of the disturbance in the other directions. In addition, a multiple
input multiple output (MIMO) control design which incorporates full knowledge of
the active mount coupling, and attempts to get simultaneous performance in all directions, is designed and implemented. The ultimate contribution of this work is to
show that the LQG multivariable control methodology can be successfully used to
achieve multi-axis vibration isolation for a highly complex system, in the presence of
significant experimental constraints.
Thesis Supervisor: Michael Athans
Title: Professor of Electrical Engineering
Acknowledgments
I would like to thank my thesis advisor Professor Michael Athans for supporting
my research, giving me the opportunity to help teach a semester of his course on
multivariable control theory, and introducing me to the world of research. I would
like to thank Gary Blackwood for his guidance, experience, and concern throughout
my research. I would like to thank my officemates for their support: Joel for his quirky
sense of humor, Alan for his randomness, Wesley for his not yet lost childhood, and
finally John and Steve for their outlooks on the future which lies before us. I'd like to
thank all at SERC who have helped in countless professional and personal ways: Rob
J., Dave W., Simon C., Simon G., Leonard, Charissa, Carl, Roger, Brett, Tupper,
Becky, and the many others who gave SERC its personality.
This research was conducted at the MIT Space Engineering Research Center with
support provided by NASA grant NAGW-1335.
In memory of my father,
on whose shoulders I stand.
Contents
1
10
Introduction
1.1 Interferometer Testbed ............
. . . ..... . .
10
1.2
The Concept of Active and Passive Isolation
.......
. . .
13
1.3
Thesis Contributions
. . . ..... . .
15
.......
16
1.4 Thesis Outline.
.............
................
2 Experimental Setup
18
2.1
Interferometer Testbed .......
................... . 18
........
2.2
Multi-Axis Active Isolation Device
. . . . . . . . . . . . . . . . . . . . 19
2.3
Disturbance Source .........
. . . . . . . . . . . . . . . . . . . .. 22
2.4
Performance Metric .
. . . . . . . . . . . . . . . . . . . .26
2.5
Real Time Computer.
. . . . . . . . . . . . ......
. . . . ..
27 ..
. . . . . . . . . . . . . . . . . . .
28
2.6 Chapter Summary .........
3
. . .
Control Design
29
3.1
30
System Identification ...........................
3.1.1
Significance
3.1.2
Experimental Loop Setup ....................
34
3.1.3
Augmented Plant Model .....................
35
3.2 Disturbance Characterization ......................
38
3.3 Experimental Constraints ........................
40
3.4
41
of Sensor-Actuator
Pair . . . . . . . . . . .....
Control Design Approach .........................
3.4.1
SISO LQG Problem
Formulation
5
. . . . . . . . . . . .....
33
42
3.5
3.4.2
MIMO LQG Problem Formulation
3.4.3
Prediction of Performance
3.4.4
Compensator Order Reduction and Discrete Representation
Chapter Summary
...............
45
....................
48
48
............................
50
4 Experimental Results
52
4.1
Multivariable
4.2
SISO D-Direction Performance Results .................
55
4.3
SISO A-Direction Performance Results .................
58
4.4
SISO B-direction Performance Results
61
4.5
MIMO Performance Results .......................
64
4.6
Summary of Results
64
Nature
of Plant
. . . . . . . . . . . . . . .
. . .
.
.................
...........................
5 Conclusions and Suggestions for Future Research
5.1
Conclusions
5.2
Suggestions
. . . . . . . . . . . . . . . .
for Future
Research
.
A Full and Reduced Order SISO Compensators
6
68
. . . . . . . . .....
. . . . . . . . . . . . . . .
52
. .
68
.
69
71
List of Figures
1-1 The SERC Interferometer Testbed ........
11
1-2 Typical passive force transmissibility
14
......
2-1 Scale drawing of the active softmount isolator..
........... . .20
2-2 Vertical actuation model of the actuator.....
. . . . . . . . . . . .
21
2-3 Transverse actuation model of the actuator. . .
. . . . . . . . . . . .
22
2-4 Disturbance source driving signal autospectrum
. . . . . . . . . . . .
23
2-5 Disturbance autospectrum in D-direction .
. . . . . . . . . . . .
24
2-6 Disturbance autospectrum in A-direction ....
. . . . . . . . . . . .
24
2-7 Disturbance autospectrum in B-direction .
. . . . . . . . . . . .
25
3-1
Overall control problem formulation ......
........... ..30
3-2
Example of quality of model fit to the data (magnitude) .......
32
3-3
Example of quality of model fit to the data (phase).
32
3-4
Comparison of full model and data singular values..........
33
3-5
Depiction of control loop.
34
3-6
Singular values of augmented mirror mount dynamics compared against
.......................
true displacement output mirror mount dynamics.
3-7
Disturbance autospectrum fit: D-direction.
3-8
Disturbance autospectrum fit: A-direction.
3-9
Disturbance autospectrum fit: B-direction.
3-10 H 2 problem formulation
..........
...........
37
39
...............
.........................
39
40
43
3-11 Normalized control frequency weighting function ............
44
3-12 Normalized noise frequency weighting function .............
45
7
3-13 D-direction modeled loop sensitivity ...........
3-14 MIMO H2 problem formulation.
.....
.
....................
46
47
3-15 Comparison of singular values of MIMO full and reduced order compensators
..................................
49
4-1 Illustration of Multivariable Plant Coupling. ..............
54
4-2 D-direction SISO controller:D-direction performance.
.........
56
4-3 D-direction SISO controller:A-direction performance.
.........
57
4-4 D-direction SISO controller:B-direction performance.
.........
57
4-5 Illustration of Coupling of D-direction commands into Other Directions
58
4-6 A-direction SISO controller:A-direction performance ..........
59
4-7 A-direction SISO controller:B-direction performance.
.........
59
4-8 A-direction SISO controller:D-direction performance.
.........
60
4-9 Illustration of Coupling of A-direction disturbance into Other Directions 60
4-10 Illustration of Coupling of B-direction disturbance into other directions
61
4-11 B-direction SISO controller:B-direction performance.
.........
62
4-12 B-direction SISO controller:A-direction performance.
.........
62
4-13 B-direction SISO controller:D-direction performance.
.........
63
4-14 MIMO
Controller:A-direction
performance.
.
. . . . . . . . . . . . .
4-15 MIMO Controller:B-direction performance. .........
...
65
.
65
4-16 MIMO Controller:D-direction performance ................
66
A-1 Comparison of D-direction SISO full and reduced order compensators.
72
A-2 Comparison of A-direction SISO full and reduced order compensators.
72
A-3 Comparison of B-direction SISO full and reduced order compensators.
73
8
List of Tables
4.1
Thesis
performance
metric
summary.
9
. . . . . . . . . . . . . .....
66
Chapter 1
Introduction
The purpose of this thesis is to show, experimentally, the benefits of using the Linear
Quadratic Gaussian (LQG) multivariable control design methodology to achieve isolation in multiple directions. In order to accomplish this objective, this first chapter
will introduce the motivating problem of space based optical interferometry. A summary of the field of passive and active isolation will follow. Having fully described
the necessary background areas, the overall contribution of the thesis will be fully
described, and an outline of the thesis will be given.
1.1 Interferometer Testbed
The Interferometer testbed is in the Space Engineering Research Center (SERC),
which is a multidisciplinary laboratory of the Aeronautic and Astronautic department
of the Massachusetts Institute of Technology. A schematic of the testbed is shown in
Figure 1-1. Large optical telescopes, such as the Hubble space telescope, are limited
in size by the ability to manufacture and launch large, optical quality mirrors. This
size limit leads to a limit in angular resolution. Interferometry is a method of imaging
astronomical objects to a high degree of angular resolution, using smaller spatially
separated light collecting optics, called siderostats.
These siderostats collect light,
which is then reflected to a single location, and combined in such a way, that an
image of the astronomical object of interest is formed. Critical to the clarity of the
10
/Z///
Suspensio
Directic
interei
this th
3.5 Meters
Figure 1-1: The SERC Interferometer Testbed
image is the knowledge of the locations of the siderostats with respect to each other
and the light collecting optics. Errors in the assumed locations of the siderostats
translate directly to errors in the measurements of the stellar images of interest.
Therefore, any structure which connects these spatially distributed siderostats must
not allow the vibrations of the siderostats, with respect to each other or the imaging
optics, to exceed very tight tolerances on the order of , where A is the wavelength
of light of interest.
This requirement, although stringent, would not be very difficult if the connecting
structure were allowed to be very massive. Its sheer inertia and stiffness could be
designed to make the disturbances which are expected to impinge upon the orbiting
interferometer negligible with respect to the scientific requirements.
However, be-
cause of the large cost per unit mass associated with launching any object into space,
such massive designs are unacceptable.
As a result, stiff, yet lightweight, designs
must be considered and the associated increase in siderostat motion due to lightly
damped closely spaced structural modes must be handled in other ways. It is pre11
cisely the methods of addressing this last issue which are being investigated on the
interferometer testbed in SERC.
This testbed, shown in Figure 1-1, is
th the scale of a possible lightweight design
of an orbiting space based stellar interferometer. The shaded regions at points A, B,
and C represent the three siderostat positions. The top vertex contains a shaker which
injects disturbance energy whose spectrum might be typical of that experienced by an
orbiting interferometer [5]. Point D houses a laser metrology system which represents
the single location at which the light collected at A, B, and C would be combined
to form an image of the stellar object of interest. As described above, motions of
points A, B, C, and D are of concern for the success of the motivating scientific
mission. Figure 1-1 shows the direction from each of the siderostats to point D, and
the directions from point C to the other siderostats and point D. It is motion in this
second set of directions which are of concern for this thesis. Further details regarding
the scientific mission motivation and testbed design can be found in [5].
As mentioned earlier, it is the flexibility of the structure, and the associated ease
with which disturbances propagate to the siderostats, which is of direct concern to
the scientific mission. There are four primary methods of minimizing the effect of
these structural vibrations upon the siderostat positions. First, there is the option of
isolating the disturbance source at the top of the structure from the rest of the structure. This option, although feasible, has yet to be investigated on the Interferometer
testbed.
Second, there is the active and passive damping approach, in which the damping
of the structure is enhanced, through active and passive means, so that propagation of
disturbances is attenuated. Varying aspects of this approach have been investigated
on this testbed by Anderson [2], Spangler [18], and MacMartin [15]. Anderson investigated the optimal placement of actuators and dampers for global structural control
and damping. MacMartin and Spangler investigated the use of collocated and dual
sensor actuator pairs for active damping.
A third method is global structural control, in which the structural shape is actuated in such a way that the effect of the disturbances on the siderostat vibra12
tions is minimized. Work related to this approach has been done by Jacques and
Lublin [12, 14]. Jacques presented a global structural control design with on-line system identification. Lublin did an empirical comparison of the Hoo and LQG design
methodologies for global structural control.
The fourth method concerns isolating the siderostats themselves from the vibrations of the structure.
Blackwood addressed part of the advantages of this method
[4]. He concentrated on the tradeoffs involved in mounting an isolation device upon
a flexible structure. He designed a siderostat which combines passive and active isolation concepts in order to illustrate these tradeoffs experimentally on the testbed. It
is this fourth method which is being extended in this thesis. In particular, whereas
Blackwood was concerned with isolation from each of the siderostats to point D, this
thesis examines the possibility and tradeoffs of isolating one of the siderostats towards point D and towards the other two siderostats simultaneously. The siderostat
designed by Blackwood, or mirror mount as it will be referred to from now on, is fully
described is Chapter 2. The next section describes the concepts of active and passive
isolation.
1.2 The Concept of Active and Passive Isolation
The goal of any vibration isolation approach is to reduce the force transmitted through
the mounts which connect a device to its structural foundation. Historically, a primarily passive approach has been taken in achieving this goal. The passive approach
consists of using a soft spring or rubber mount between the device and foundation.
The mount is designed such that its resonance, as prescribed by its stiffness, damping,
and inertial characteristics, is low enough that the frequency energy content of the
vibrations lie above the resonant frequency. A characteristic plot of the ratio of force
exerted at the foundation to force transmitted to the device through a passive mount
is shown in Figure 1-2. The main problem with this approach is that lower bounds on
the realistic resonant frequencies of the mount are defined by the necessity of carrying
a static load and perhaps transmitting low frequency motions of the foundation, i.e.
13
o
a)
0
LL
'0
CD
E
U,
3
(a
c0o
0
LL
a,
C,
M
la
m
0
.o
n-
)2
Normalized mount natural frequency
Figure 1-2: Typical force transmissibility as the mount damping ratio is varied.
if the foundation is a maneuvering ship, or space structure. This lower bound could
be a problem if there is disturbance energy before or at the resonance, because for
this situation the passive stage is either useless or actually amplifies the disturbance.
Depending on the disturbance, the resulting amplification could easily outweigh, in
terms of overall performance, the high frequency attenuation which is gained.
In
addition to the tradeoff involved in choosing the passive resonance frequency, there is
a tradeoff in choosing the damping ratio for the passive stage as shown in figure 1-2.
A lower damping ratio results is faster rolloff just after the resonance, but results in
large amplification around resonance. Therefore, there is a penalty associated with
attempting to get very good high frequency attenuation. These rather severe tradeoffs could be modified if, somehow, the damping and stiffness characteristics of the
mount could be made to vary as a function of frequency, rather than be constant.
This would clearly give the designer greater flexibility, although at the cost of greater
complexity.
Active feedback control answers these needs. It allows the designer to shape the
damping and stiffness properties of the mount dynamics as functions of frequency by
14
modifying the forcing properties of actuators in the mount. This ability makes the
tradeoffs described above much less severe and results in improved performance. The
disadvantage of active isolation is due to the active nature of the isolator, inaccurate
modeling or changing system parameters could result in degraded performance or
even instability of the system. In addition, the greater complexity, associated with the
need for sensors and actuators, makes the active isolation approach more expensive.
Normally, however, the performance improvements which can be gained through the
use of feedback outweigh the disadvantages mentioned above. In fact, in practice
what is often used is a combination of active and passive methodologies as described
in [21], and implemented by Blackwood [4]. This way active control can be used to
get performance in the region of the passive resonance and the passive stage provides
the performance for higher frequencies. Various applications and implementations of
active vibration control are described in [16, 22, 6, 17, 20].
Blackwood, as mentioned earlier, designed an isolation mount which combines
the active and passive methodologies described above. This mount is described in
detail in Chapter 2. The purpose of this device is to isolate a rigid mirror in a
desired direction from vibrations of its flexible foundation. The passive components
were designed to have low frequency resonances around 30Hz thereby providing high
frequency vibration isolation. However, the disturbance source which needs to be
attenuated has most of its energy between 20 and 250 Hz, as will be shown later.
Therefore, the passive modes around 30Hz will actually amplify the disturbance over
that frequency range. Although this would be a problem for a purely passive mount,
the active stage allows the designer to counteract this through the use of feedback.
1.3 Thesis Contributions
Blackwood, in his experimental work, used active control to minimize the motion
of each of the three siderostats in the laser directions shown in Figure 1-1. This
addressed the need to have the mirrors remain a fixed distance away from point D
in spite of structural disturbances. However, it ignored the motion of the siderostats
15
in the plane perpendicular to this direction which is also important to the scientific
mission. In particular, it is vibrations in the directions towards the other siderostats
which are the most important of the off-axis directions.
This thesis focuses on the mirror mount located at C and develops a multivariable
control law which provides enhanced isolation of the mirror in the above described
directions. In order to show the benefits of a multivariable control law, three SISO
control designs, one for each direction of interest, is designed and implemented. The
performance of each of these control designs in all three directions is compared against
a multivariable control design which attempts to isolate in all directions simultaneously. It is shown that, at the cost of greater complexity, a multivariable control
design can yield significantly better results for systems with dynamic coupling between the control variables. It is futher shown that if the individual SISO control
loops were simultaneously closed, the closed loop system is predicted to be unstable. The ultimate contribution of this work is to show that LQG based multivariable
feedback control can be successfully used to experimentally achieve multi-axis vibration isolation for a highly complex system, in the presence of significant experimental
constraints.
1.4
Thesis Outline
This thesis consists of a total of five chapters. Chapter 1 has introduced the motivating
problem, previous work, and overall field of isolation.
Chapter two will present, in detail, the hardware upon which the experiments
were implemented. Included in this discussion will be relevant details of the interferometer, the multi-axis active isolation device, the disturbance source, and the real
time computer. In addition, the performance metric being used in this thesis will be
formulated and explained.
Chapter three will discuss the control design process from the system identification
aspects to the controller reduction and implementation issues. In particular, this
discussion will describe how the stringent experimental constraints were incorporated
16
into the LQG control design methodology.
Chapter four contains the experimental results which were obtained by implementing the controllers formulated in Chapter three. Three of the control designs are
SISO designs, each of which focuses on isolation in one of the three directions of interest. The isolation results of these three SISO controllers are compared against the
results of a MIMO controller which has as its performance objective the simultaneous
isolation of all three directions.
Finally, Chapter five will conclude the results obtained and discuss future areas
of research related to this topic.
17
Chapter 2
Experimental Setup
This chapter describes, in detail, the primary hardware components which were used
in the experiments. First, the interferometer testbed will be discussed. Second, the
active isolation mirror mount designed by Blackwood [4] will be presented. Third,
the disturbance source to be used will be explained. Fourth, the performance metric,
by which this work will be measured, will be established. Finally, the real time digital
computer with which the controllers were implemented will be presented.
2.1
Interferometer Testbed
Having introduced the scientificmotivation for the Interferometer testbed in section
1.1, a more detailed physical description of the testbed will now be given.
The
structure is shown in figure 1-1, and is a 3.5 meter tetrahedron suspended from three
of its four vertices. The suspension was designed to give a pendulum mode of .3
Hz, a bounce mode of 2.5 Hz, and a spring surge mode at 100 Hz. Each leg of the
tetrahedron is a 13 bay triangular truss beam employing aluminum tubing for the
members and 3cm steel balls for the nodes. The frequency of the fundamental mode
of the truss is about 26 Hz.
The siderostat plates located at points A, B, and C each supports a mirror mount,
which together represent the spatially separated apertures necessary for interferometry. The cat's eye optic, which is located in each of the mirror mounts, ideally needs
18
to be inertially fixed with respect to each other and point D. As mentioned in chapter
1, the mirror mounts were designed as a combination of the concepts of active and
passive isolation so as to achieve this goal and will be described next.
2.2
Multi-Axis Active Isolation Device
The active mount which is being used in this thesis for multi-axis active isolation was
designed by Blackwood [4]. Figure 2-1 shows a cross sectional scaled drawing of the
mirror mount. Although only two actuators and sensors are visible in the drawing,
there is one additional actuator and sensor which are obscured by the base of the cat's
eye holder and the cat's eye optic respectively. As a further note of explanation, the
magic point referred to in figure 2-1 is the single point of the cat's eye optic from which
all reflected light seems to come. Reduction of the motion of this point due to motions
of the base, therefore, is the performance objective. The device can be divided up
into three main parts. The darkest shaded areas represent the rigid connection to
the structure.
The medium shaded region is a one kilogram mass connected to the
structure through flexures which are essentially parallel spring/damper
pairs. The
wire flexures between the reaction mass and the piezoceramic actuators serve two
purposes. First, they help ensure that most of the transmitted force is in the vertical
direction, and second, that the upper stage and the reaction mass can tilt freely
relative to each other. The piezoceramic stacks are actuators which have a stroke
of 13.5 m at 150 volts, a capacitance of 6500 nF, and axial stiffness of 350 N/m.
The very large axial stiffness of the stack with respect to the inertial load of the cat's
eye and reaction mass, results in actuator elongation which is proportional to the
voltage applied over the frequency ranges of interest in this thesis. The actuators
were originally designed to be biased at 75 volts so that they could provide ±6.75/Im
of stroke. However, due to a failure of the amplifier which could provide that voltage
to a capacitive load, the experiments executed in this thesis biased the actuator
about 50 volts, thereby providing only +4.5/m of stroke. This decrease in controller
authority proved to be a small problem due to the fact that the effective stroke in the
19
M: magic point
SOFTMOUNT
G: c.g. of active stage
t z
cat's eye
C: c.g. of reaction mass
A: hinge point
L'~x
~\
-
accelerometer
r
-
,k
7 i;C
wire flexure
w
I
,.
I
I
-
F-1
i
-
(
a
-
0
M
G
-~~~~-------I~~~~~~~~~
I
collar (base) -
A-
0
vertical blade
P
-
T!!T---~
C]-
flexure
-[2r2
lateral blade
flexure
reaction mass
L 1 inch
I --5
Figure 2-1: Scale drawing of the active softmount isolator. Active stage (white) is
mounted to reaction stage (medium shade) which enables high frequency dynamic
decoupling from base (dark shade) due to soft damped blade flexures; figure from
Blackwood
[4].
20
M
M
active stage
f
F·
reaction mass
z
x
///////
///////
(a)
(b)
Figure 2-2: Vertical actuation model of the actuator. At low frequency (a) lower
spring is rigid and point M moves full commanded stroke S = f/kl. At high frequency
(b) lower spring is soft and point M moves 6/2.
lateral directions proved to be less than the needed to compensate for the standardized
disturbance, which will be described later.
In order to understand the mechanism by which multi-axis actuation can be
achieved given this design, figures 2-2 and 2-3, which illustrate this mechanism quite
clearly, have been taken from [4].
Figure 2-2 illustrates how a common voltage
command to all actuators results in a motion of the cat's eye in the vertical (or z)
direction. In addition, it shows that the gain above the passive stage resonance is
roughly half that of the gain below the passive stage resonance. This change in actuation authority also applies to commands in the lateral directions, as shown in
figure 2-3. As should be clear from figure 2-3, lateral motion of the magic point is
achieved through coordinated elongation of one actuator and contraction of the other
actuator.
This motion is achievable, albeit with less gain, at high frequency due to
the design which placed the center of gravity below the magic point.
The sensors used in this thesis are a triaxial arrangement of Kistler 8302A2S1
accelerometers mounted about the cat's eye such that the accelerations in three or21
M
active stage
reaction stage
C
C
z
e
(a)
(b)
Figure 2-3: Transverse actuation model of the actuator. Piezoceramic actuators
produce torque which pivots active stage about point A. At low frequency (a) reaction
stage spring is rigid; at high frequency (b) both active stage and reaction mass pivot
about their respective centers of gravity (points G and C). High frequency stroke is
reduced by
IMGI/IMAI compared to low frequency.
thogonal directions of the magic point can be resolved through a suitable coordinate
transformation.
2.3
Disturbance Source
The disturbance source is located at the top of the structure and consists of three
440 gram masses mounted in orthogonal directions on piezoceramic stacks.
The
disturbance source is meant to emulate the vibrations which would normally be caused
by reaction wheels in a space structure. For the purposes of this testbed, the driving
signal to the disturbance source was created to have an energy spectrum which forms
an envelope of possible reaction wheel spectrums. In this way, successful attenuation
of this worst case spectrum will imply good performance under the actual reaction
wheel disturbances.
The driving signal of the amplifier channels which drive the
disturbance source is created by driving a four pole low pass bessel filter with a
corner frequency of 70 Hz by a pseudo-random signal. The autospectrum of the signal
22
Standard Disturbance Input Autospectrum
.- 1
I
r4
Cj
C,,
Hertz
Figure 2-4: Autospectrum of signal used to drive piezoceramic amplifiers for disturbance source.
driving the amplifiers is shown in figure 2-4, and is referred to as the standardized
disturbance signal.
It is important to understand the effect of the disturbance source on the position
of the cat's eye at point C. Using the accelerometers mentioned above, and the fact
that the signal which is driving the disturbance source is known, a transfer function
from disturbance signal to accelerometer output can be taken in which all sensor noise
and extraneous effects are averaged out. This transfer function can then be multiplied
by
to convert from acceleration to displacement. This result, when squared and
normalized by the autospectrum of the driving spectrum, forms the autospectrum
of the position of the cat's eye optic due to the disturbance signal. In fact, all
autospectrums to be shown in this thesis will be calculated in this manner.
This
was done so as to maintain focused on how the motion due to the disturbance was
reduced by active control. Figures 2-5, 2-6, and 2-7 show the effect of the disturbance
source upon the motion of the cat's eye in the three directions of interest as defined
23
,-5
N
a)
"r
<;
101
102
Frequency (Hertz)
Figure 2-5: Position Autospectrum in D Direction.
-5
N
I-r
E
C
102
Frequency (Hertz)
Figure 2-6: Position Autospectrum in A Direction.
24
-5
14
1(
1(
1(
N
a)
1'
c\J
El1
C
1
1
1
10
102
Frequency (Hertz)
Figure 2-7: Position Autospectrum in B Direction.
in figure 1-1. Note that, although the overall motion in all the directions is on the
same order, the motion towards D is the smallest. This observation makes sense
because the D direction is the closest aligned of the three directions with the vertical
direction of the mirror mount, and hence benefits the most from the passive isolation.
Greater evidence of this assertion can be found by focusing on the second hump of the
disturbance spectrum around 150 Hz. The overall magnitude of this hump, which is a
direct result of the passive isolation stage's effect on the disturbance, is smaller in the
D-direction than in the other directions. The second thing to notice is that a large
amount of the energy lies between 26 Hz and 45 Hz. This is mostly due to the passive
stage resonances and partly due to the truss structural modes in this frequency range.
Finally, the last issue to realize is that most of the disturbance energy lies between
10 and 250 Hz, thereby implying that the active control needs to have high gain over
these frequency ranges, to provide effective disturbance rejection.
25
2.4
Performance Metric
A standard performance metric for the Interferometer testbed has been established
at SERC [5]. It consists of maintaining the testbed performance objective while
the disturbance source is being driven by a signal with autospectrum described in
section 2.1. The testbed performance objective is to keep each of the three differential
pathlength errors to within 50 nm RMS. Mathematically,
(E [(DA - DB)] )
< 50nm
(E [(IDB-IDC)]
< 50nm
)
(E [(IDC- lIDA) )
<
50nm
(2.1)
where lij is the distance from point i to point j measured by the laser metrology
system mentioned in chapter 1. In practice these evaluations of the RMS value of the
differential pathlength errors are limited to integration of their autospectrums from
10-500 Hz, and taking the square root. As mentioned earlier, this thesis is concerned
with displacements of the mirror at point C in the line of sight directions to points
A, B and D. Due to the lack of laser metrology measurements for two of these three
directions, the differential position measurements for these directions are unavailable.
Therefore, only local displacements are discernible from a double integration of the
measurements made by the accelerometers. As a result, the performance metric which
is being used to evaluate the success of the control designs to be presented in this
thesis is somewhat different. Mathematically the goal is to approach the value of,
min
2
500
'DD(j
2f)df)
2
500
+ 2
AA(j27rf)d)
2
+ (2
500
BB(i27rf)df)
2
(2.2)
through active control, where ii(t) is the Fourier transform of the autocorrelation
function of the local displacement towards the ith point due to the disturbance source.
26
2.5
Real Time Computer
The real time computer in the SERC laboratory, which was used for the experiments in
this thesis, is made up of two distinct parts: a digital vector parallel processing board
for the real time computations, and the board which controls the analog to digital
and digital to analog inputs and outputs. Although the compensator could have been
designed in the discrete domain, this would have required fixing the sampling rate to
be used at a much earlier stage in the design process than was feasible for this work.
Therefore, characterizations of the maximum sampling rate achievable as a function of
number compensator states and inputs and outputs was done. This characterization
is crucial, due to the need to incorporate the limiting sample rate and limited number
of implementable states into the control design problem.
As mentioned earlier, three single input single output (SISO) designs and one
multiple input multiple output design were implemented. For the real time computer
one of the main factors which reduced the achievable sample rate was the number of
analog to digital (A/D) and digital to analog (D/A) conversions which need to be
done. Clearly then, for a given number of states, a SISO controller could be run at a
faster rate, thereby allowing a compensator with greater bandwidth, and contributing less phase loss over the bandwidth of the disturbance, than a MIMO controller.
This inequity implies that a fair comparison of performance of the SISO and MIMO
methodologies, for the given equipment limitations, should allow the SISO controller
to utilize the fastest sample rate possible. This rate would necessarily be greater
than for the MIMO design. Although the faster sample rate would probably imply
improved performance in the axis of interest for the SISO design, the deterioration
of off-axis performance would still be present, and might even be more significant.
Since it is the general off-axis behavior of the SISO systems which is of interest in this
thesis, and this will not change through an increase in the SISO sample rate, a single
sample rate will be used for all designs. Through trial and error it was determined
that the maximum number of states which would be required would be 32, knowing
that three inputs and three outputs would be worst-case, a maximum sample rate of
27
2600 Hz was used.
2.6
Chapter Summary
The have been five key elements of this chapter. First, a description of the physical
properties and dimensions of the interferometer testbed were given.
Second, the mechanism by which the mirror mount achieves multi-axis isolation
through actuation of the piezoceramic stacks, was explained. Isolation in the "piston",
or z direction, was shown to result from common command signals to all actuators.
However, isolation in the lateral directions, requires a coordinated elongation and
contraction of the three actuators, such that a pivoting motion is achieved in which
the magic point moves in the lateral direction.
Third, the standardized disturbance source was defined, and its influence on the
motion of the mount at point C was described. It was shown that the overall disturbance has an effective bandwidth of 250 Hz. In addition, most of the energy was
found to be focused about the passive mount modes of the mirror mount around 30
Hz, and the structural modes around 150 Hz.
Fourth, the performance metric of this thesis was established.
The difference
from the overall Interferometer testbed performance metric was explained and justified. Whereas the testbed performance metric focused on minimization of differential pathlengths, this thesis' performance metric focused solely upon the localized
displacement motion of point C. This was done due to the lack of differential displacement sensors for the directions to the other siderostats from point C.
Fifth, it was shown that a single sample rate could be used, for all control designs, in order to illustrate the overall benefit of the multivariable control design
methodology.
28
Chapter 3
Control Design
The purpose of this chapter is to describe the method by which the active isolation
problem, already discussed, gets posed and solved as a control problem. Recall that
the influence of the disturbance source on the position of the cat's eye is known
(see section 2.3) through direct stabilized double integration of the measurement
at the output.
Because of this information, the control problem will be posed as
a loop sensitivity minimization problem as suggested by figure 3-1. Ki(s) is the
compensator to be designed for the generic direction(s) of interest represented by i,
Gj(s) is the corresponding complete plant dynamics, and Di(s) is the disturbance for
the i output(s). The figure could apply for the SISO control loop, in which case i is
either A, B, or D, and d(t) is a scalar white noise disturbance. Figure 3-1 could also
apply for the MIMO control loop, in which case i represents all three directions and
d(t) is a vector of three disturbance inputs. In addition, Di(s) becomes a diagonal
matrix whose diagonal elements are the autospectrums for each of the directions.
This chapter will be organized as follows. First, the system identification step of
the active mirror mount and the development of Gi(s) will be described. Second, the
method of incorporating the knowledge of the disturbance energy will be presented.
Third, the experimental constraints will be discussed. Finally, the LQG formulation
will be presented. Through this sequential discussion of the steps required to create
an effective active control algorithm, it is hoped that an appreciation of the difficultly
of this isolation problem will be conveyed, and the power of the LQG methodology
29
t
Figure 3-1: Overall Control Problem Formulation: Ki(s) is the compensator to be
designed for the generic direction(s) of interest represented by i, Gi(s) is the corre-
sponding complete plant dynamics, and Di(s) is the disturbance for the i output(s).
will become apparent.
3.1
System Identification
Before any type of active isolation can be considered, a full understanding of the
dynamic relationship between the actuators and sensors of the mirror mount must
be achieved. For the purpose of control design, it is desirable for this understanding
to be quantified in the form of a state space model. Due to the linearity and timeinvariance of the systems under consideration, transfer functions of sensor outputs
due to actuator inputs capture all the relevant dynamic information necessary for
control purposes. Therefore, experiments were conducted which measured the transfer
functions from each actuator to each sensor in order to fully characterize the system.
In order to simplify interpretation of the data, an output static decoupling matrix was
implemented which took the three accelerometer signals as inputs and transformed
them into the A, B, and D line of sight accelerations. The same was done for the
inputs so that commanded directions could be specified in terms of A, B, and D line of
sight directions. Having acquired the transfer function data relating commands in the
30
A, B, and D lines of sight, and actual motion in those directions, a methodology was
required which would give a linear state space realization, in the form of equation 3.1,
which mimicked the input-output relationship described by the data.
)(t) Axct) + Bu(t)
(3.1)
=(t) = Cx() + Du(t)
An identification methodology provided by Jacques [12, 13] was used to find accurate
values for the matrices A, B, C, and D. Part of this methodology was an algorithm called FORSE which takes a set of multivariable data and comes up with an
initial state space realization which coarsely approximates the magnitude and phase
characteristics of the individual transfer functions.
This first guess is then tuned
through the minimization of a logarithmic weighted cost functional of the ratio of
the model and data transfer functions. Issues such as model over parameterization
were dealt with through user engineering judgment in order to prevent against fitting
noise. These two identification algorithms provided a seamless and straight forward
approach to developing state space models from experimental data which replicated
to a great degree of accuracy the individual magnitude and phase transfer functions
of the multivariable plant.
A fifty-six state model was developed to represent the
input-output behavior of the active mount. A sample of one of the fits overlayed with
the data can be found in figures 3-2 and 3-3. As is clear by the near indistinguishable
plots, Jacques' parametric identification provides a high fidelity fit.
Above three
kilohertz, the fit is no longer accurate. There was no need to model those dynamics
since no control was expected to be done at that frequency. The knowledge of the
high frequency modeling errors was incorporated later in a stability robustness test to
ensure that not modeling those high frequency dynamics does not affect the stability
of the active isolation loop. A plot of the model and data singular values overlayed
can be found in figure 3-4 as a summary of the overall quality of fit to the data.
31
2
0
Hertz
Figure 3-2: Example of quality of magnitude fit of the model fit to the data for the
D-direction command to D-direction output transfer function.
(a,
a,
(n
L
10
10'
10
Hertz
Figure 3-3: Example of quality of phase fit of the model to the data for the D-direction
command to D-direction output transfer function.
32
-2
0
0'
)4
Hertz
Figure 3-4: Comparison of full model and data singular values.
3.1.1
Significance of Sensor-Actuator Pair
As described in chapter 2, the disturbance energy of interest lies between 10 and 500
Hz. The plant transfer functions over that frequency range ramp upwards with a slope
of 40dB/decade as is evidenced in figures 3-2 and 3-4. This positive slope is due to the
sensor-actuator combination. As was discussed in chapter 2, the actuators, over the
frequency range of interest, are essentially displacement actuators. Therefore, since
the sensors are accelerometers, the output looks like the second derivative of the input
and hence the 40dB/decade ramp upwards. At one kilohertz the accelerometers have
their resonance and roll off at 40dB/decade. For this reason, the transfer functions
level off after one kilohertz. The main difficulty which was encountered with respect
to the given sensor-actuator pair was it's high bandwidth property. Since the plant
had no natural rolloff near the bandwidth of the disturbance, the compensator was
forced to provide the rolloff so that the high frequency unmodeled dynamics didn't
destabilize the system in the closed loop. The implementation of such a controller is a
33
d(t)
error
FY21- ;;
/15Hz
Hi Pass
Notch
F17HzInt
Low pass
Sensor
Noise
Figure 3-5: Depiction of control loop.
problem when it is being implemented on a digital computer because the controller can
only provide rolloff up to the Nyquist rate. In order to handle the rolloff requirements,
analog components must be used in order to continue the attenuation beyond the
Nyquist frequency. In fact, since the Nyquist rate was 1300 Hz for this thesis, it is
quite clear that the analog anti-aliasing filter dynamics must lie within the bandwidth
of desired disturbance rejection. The problem with the addition of analog dynamics
within the disturbance bandwidth was the large amount of phase loss whichhad to be
accepted. This large amount of phase loss was handled by incorporating the analog
dynamics into the overall plant dynamics, as will be described shortly. Before doing
this, the experimental loop arrangement with the associated analog filters will be
presented.
3.1.2
Experimental Loop Setup
The experimental loop setup is depicted pictorially in figure 3-5. The analog elements
referred to above are broken up into two groups, those appended to the accelerometer
outputs, and those appended to the control inputs. Although the figure depicts a
SISO system, the analog dynamics shown apply to all three inputs and outputs. The
dynamics at the outputs consist of three filters per channel with roughly the same
break frequencies. First there is a high pass filter which removes the DC signal which
comes out of the accelerometer due to measurement of the gravity field. The high pass
34
filter also has a gain of 100 so as to increase the loop gain. Second, there is the notch
filter. The purpose of this filter is to attenuate the lightly damped 2.14 kHz mode such
that it does not alias down into the control bandwidth. Third, there is a first order
rolloff filter to prevent aliasing of the accelerometer noise spectrum. The dynamics at
the plant inputs consist of two first order low pass filters per channel. Their purpose
is to smooth out the D/A signal coming out of the real time computer and to provide
loop rolloff to prevent aliasing of control inputs through the real time computer. The
mount dynamics, represented by G(s), which were modeled in section 3.1, represent
the input-output transfer functions from the command directions in the A, B, and D
directions and the measured motions in the A, B, and D directions. The disturbance
dynamics, D(s), represent a disturbance adding into the output of the mirror mount
as discussed at the beginning of this chapter. Finally, n(t) is added in to represent
the accelerometer sensor noise.
3.1.3
Augmented Plant Model
Having characterized the dynamics of the plant, it is also critical to exactly determine
the dynamics of the analog dynamics to be added into the loop. The two sets of analog
dynamics depicted in the previous section were characterized in terms of a state-space
representation of the following forms. First, the full input dynamics were represented
by equations 3.2.
-(t)
=
Adyil(t) + Bdyl(t)
y(t) = Cd.l x(t)
(3.2)
Second, the full output dynamics were represented by equations 3.3.
x(t) = Ad,.2 (t) + Bd2.u(t)
y(t) = Cd,,2x(t)
35
(3.3)
This complete characterization of the analog dynamics will be useful for creating
a new augmented plant model, for which the control inputs are the outputs of the
digital computer and the outputs are the inputs to the digital computer. In this way,
the augmented plant model would represent the Gi(s) from figure 3-1. Before this
can be done however, the effects of the digital computer need to be approximated
and incorporated into the plant model.
Due to the fact that sample rate is within only a factor of 5 of the expected
bandwidth of the controller, accurate predictions of the time delay and effect of the
zero order hold from the digital computer need to be incorporated into the plant
model. The time delay through the real time computer was measured to be 1.6T,
where T = pi, and f
is the sample rate which equals 2600 hertz.
Of this, T/2
was due to the zero order hold on the plant output, and r = (1.1)T was due to a
pure time delay within the real time computer. These two elements are incorporated
into the model through the use of a state space representation of a fourth order Pade
approximation with time delay equal to r/2 and a first order approximation of the zero
order hold represented by
'1
[8]. In this way the outputs of the model feed into these
delay and rolloff states, to get a new delayed model with transfer function: Gd(s).
The state space representation of this system is then combined with the state space
representations of the analog input and output systems represented by equations 3.3
and 3.2 respectively. The resulting plant is the Gi(s) shown in figure 3-1. The singular
values of this new plant are shown in figure 3-6. The main issue to note is that the
augmented system is fairly close to the ideal case of a mount with displacement
actuators and displacement sensors. This fact adds credibility to the summation
in figure 3-1 where the displacement autospectrum is being directly added to the
output of Gi(s). The importance of this will be more apparent with the presentation
of the control formulation, but first the characterization of the disturbance must
be addressed, now that the characterization of the mirror mount plant has been
established.
36
-2
1
0
'O
'0
C
(!)
0
cm
a)
.N
co
E
0
Z
)4
Hertz
Figure 3-6: Singular values of mirror mount dynamics augmented by analog circuit
dynamics and real time computer effects compared against true displacement output
mirror mount dynamics.
37
3.2
Disturbance Characterization
The effect of the disturbance source upon the motion of the mirror was shown in
section 2.3. In order to design an effective controller, the energy information contained
in the autospectrums shown in figures 2-5, 2-6, and 2-7 needs to be summarized. One
option would be to model all the dynamics evident in the autospectrums using the
same methods which were used to create a state space model of the plant. This would
give a lot of information regarding the disturbance. In addition, it is likely that the
dynamics of the active mount would be part of the disturbance spectrum and hence
would not need to be added. Jacques, in his work [12], used this fact to his advantage
by fully characterizing the disturbance spectrum, and thereby targeting his control
very efficiently.
The problem with this approach for the work presented here is that the dynamics
of the mirror mount are, to a large extent, decoupled from the dynamics of the truss
except around the passive resonances. As a result, a large number of dynamics evident
in the disturbance would have to be added to the state space model of the system.
This associated increase in states of the model would lead directly to an associated
increase in the number of states in the controller which LQG would provide, hence
making the ultimate reduction of the compensator order more difficult. In addition,
any errors in the assumed dynamics of the disturbance would tend to result in greatly
reduced performance due to the lightly damped nature of the modes.
For all of these reasons, another approach of summarizing the disturbance spectrum was utilized in this thesis. Rather than capturing every mode in the autospectrums, the rough backbone associated with the spectrums were fit using simple eight
state systems, as was done by Blackwood [4]. The idea was to provide performance
robustness to disturbance dynamics variations by summarizing where most of the
energy was, on average, without fitting the lightly damped dynamics exactly. The resulting fits overlayed with the appropriate autospectrum data are shown in figures 3-7,
3-8, and 3-9. These fits were normalized such that the area from 10 to 500 Hz under
the two autospectrums are equal. The final choice of shape of fit was done primarily
38
._4
r4
E
C
101
10
10
Hertz
Figure 3-7: Disturbance autospectrum fit: D-direction.
._4
c'J
r'
E
C
10
10o
10'
Hertz
Figure 3-8: Disturbance autospectrum fit: A-direction.
39
.,4
E
C
Hertz
Figure 3-9: Disturbance autospectrum fit: B-direction.
by iteration with the control design stage to get the best predicted performance.
3.3
Experimental Constraints
In designing a dynamic compensator to attenuate the disturbances described in the
previous section, it is crucial to understand the experimental constraints whichrestrict
the type of compensators which can be implemented. There have been three primary
constraining issues encountered in this thesis. The first concerns the use of the real
time computer. Due to the issues discussed in section 2.5, the sample rate was fixed
at 2600 Hz, which in turn fixed the implementable controller bandwidth to 1300 Hz.
The second concerns the sensor-actuator pair which necessitated the use of analog
filters, and was discussed earlier in this chapter.
The third constraint was caused by the need to use actuator amplifiers which were
less powerful than those originally intended for this work due to failure of the original
amplifiers, as discussed in section 2.2. This constraint reduced the effective actuation
40
by thirty-three percent from what was required to compensate for the standard disturbance source discussed in section 2.3. This reduction in authority affected attempts
to control displacements in the lateral directions, namely in the A and B directions,
due to the pivoting mechanism used by the mount in order to achieve displacements
in those directions (see section 2.2). There were two possible ways of dealing with this
misfortune. First, the control authority could be reduced in order to reduce the overall loop gain and hence control signals. The problem with this was that the associated
closed loop performance would correspondingly degrade, and because over a third of
the control authority had been lost, it was felt that this performance loss would be
prohibitive. Therefore, instead of lowering the control authority and driving performance to zero, control designs were performed assuming the original amplifiers were
to be used. Then, upon implementation, the pseudo-random signal which drove the
disturbance source was decreased in intensity until the resulting closed loop control
signals did not saturate the amplifier. Transfer function data between the disturbance
source signal and accelerometer response were taken. In order to make common performance comparisons, the experimental input autospectrum data was scaled up to
the standard disturbance source level so that the autospectrums of the performance
variables, as derived from the transfer functions and scaled input autospectrum measurements, could be meaningfully compared. This was allowable due to the linearity
of the system. With this approach, the failure of the high powered amplifier channels
didn't affect the comparison of a SISO versus MIMO approach.
3.4
Control Design Approach
The Linear Quadratic Gaussian (LQG) control methodology was used to come up
with controllers for the active isolation. This choice was made due to LQG's property of minimizing weighted root mean squared (RMS) errors from a white noise
disturbance source, to performance variables. For this application the performance
variables are closely related to position (see 3.1.3). Thus, the LQG methodology minimizes these RMS errors, an obvious advantage given the statement of the problem
41
shown in equation 3.4. However, because of the many non-idealities in the control
formulation, and implementation, no claims concerning the optimality of the implemented compensator can be made. Therefore, LQG must be seen as a design tool
which, in the ideal case, would minimize the desired quantities. For ease of reference,
the overall performance metric for this thesis represented by equation 2.2 is repeated
in equation 3.4.
min
2
500
X~
10
2
DD(j27f)r
+ (2
500
10
'AA (j
2f)d)
a
+ (2
j
500
10
BB(j 27f)df)
a
(3.4)
A presentation of the overall control design approach is first presented for the SISO
controller design. Then a presentation of the MIMO design approach, which is parallel
in many ways to the SISO approach, will be presented.
3.4.1
SISO LQG Problem Formulation
It should be recalled that the purpose of each of the SISO control designs is to
attempt to minimize the 10-500 Hz RMS displacement due to the disturbance source
in the direction of interest for that control design, while seeing what happens to the
disturbance seen in the other directions. This overall isolation goal of attenuating the
effect of the disturbance source on the motion of the cat's eye optic in one direction
will be approached from the point of view summarized in figure 3-1 at the beginning of
this chapter. The following formulation applies to all the SISO designs with the i being
replaced by the direction of interest (ie. A, B, or D). Let the state space representation
of the augmented multivariable plant be parameterized by the following state space
representation where the subscript "a" labels them as being from the augmented
dynamics:
x(t) = Aax(t) + Bu(t)
yi(t
= C x(t)
42
(3.5)
I--------------- ------------------- ------------
I
sensor;
noises
z(t)
w(t)
process,
noisesI
l
u(t)
Figure 3-10: H2 problem formulation: Find a compensator Ki(s) such that ui(s) =
Ki(s)yi(s) and IIT.ll12 is minimized.
Then, the state space representation for the SISO design in the ith direction is given
by equation 3.6:
x(t) = Ax(t)+Bjui(t)
yi(t) =
Cix(t)
(3.6)
where Ba and Ca from equation 3.5 are represented by,
CA
Ba
[BA
B
BD]
Ca =
CB
(3.7)
CD
Having a state space representation of the SISO augmented plant labeled Go(s), the
LQG control problem solves for the dynamic linear compensator which minimizes
IIT ll2 where T,, is the matrix transfer function relating the exogenous, assumed
to be white, stochastic signals represented by w(t), to the performance variables
represented by z(t).
The block diagram of the formulation used in this thesis is
shown in figure 3-10. This presentation was more generally developed by Doyle et al.
in [7]. The Di(s) used in figure 3-10 is the autospectrum fit for the ith direction as
described in section 3.2. The Wc(s) and W,(s) are design weights, which, through
43
o4
Hertz
Figure 3-11: Normalized control frequency weighting function (We(s)): notice the
high control penalty at low frequencies.
weighting of the control and sensor noise respectively, shape the compensator. Plots of
the normalized design weights can be found in figures 3-11 and 3-12. In determining
the proper weighting functions, two main issues were addressed. First, due to the
bandwidth constraint imposed by the Nyquist rate, the compensator had to have
rolled off by 1.3 KHz. The sensor weighting function took care of this by increasing the
relative noise at high frequencies. Second, due an increasingly large sensor noise at low
frequencies, the control weighting was needed to discourage the application of control
below 10 Hz. The choice of these weights, which were directly driven by the hardware
limitations of the experimental setup and through iteration, provided a fundamental
limitation on the achievable performance due to the Bode integral. Freudenberg and
Looze do a good job of explaining the "waterbed" effect which becomes particularly
pronounced once the bandwidth of the controller is constrained [9]. As mentioned
earlier, for performance, the sensitivity of the loop needs to be less than one over the
10-500 Hz bandwidth. However, due to the fact that the compensator can only act
over that bandwidth, it must necessarily amplify the disturbance over parts of the
frequency range, in order to satisfy the Bode integral. An example of a typical loop
sensitivity plot which resulted from the SISO control design for the D-direction is
44
4)
cm
2co
C
4
30
Hertz
Figure 3-12: Normalized noise frequency weighting function (W,(s)): notice the high
sensor noise at high frequencies.
shown in figure 3-13. This "push-pop" phenomena could be disastrous if the model
of the disturbance is not accurate. As figure 3-13 shows, the controller shapes the
loop such that the sensitivity function is the inverse of the disturbance spectrum
weight, DD(s), which was shown in figure 3-7. Since the fit by DD(s) captures the
relevant autospectrum information, this control design methodology results in good
predicted and actual performance, in spite of the disturbance amplification above 180
Hz and below 15 Hz, as will be seen. Best designs were found through iteration of
hypothetical process noise levels and static control penalty gains, for each of the three
directions of interest.
3.4.2
MIMO LQG Problem Formulation
An almost identical procedure as described above for the SISO LQG problem formulation, applies to the MIMO LQG problem formulation, with one important clarification. This clarification is concerned with the manner in which the disturbance source
is modeled. Part of the power of the MIMO methodology is that the Bode integral
type tradeoffs involve not just magnitude tradeoffs in frequency but also magnitude
tradeoffs in directions. Therefore, for the sensitivity minimization problem, it is pos45
101
°
' 10
'
(U
10
il* 101
102
10
Hertz
Figure 3-13: D-direction modeled loop sensitivity: notice the increased sensitivity to
disturbances above 180Hz and below 15 Hz.
sible to decrease the sensitivity in one direction (as defined by the singular value
directions) in a manner which would be impossible for a SISO design, however, the
excess amplification spills over into the other directions under control by the MIMO
controller. A more in depth treatment of the MIMO equivalent of the Bode integral
can be found in Freudenberg and Looze [9]. The point is that, if the disturbance
occupied a single frequency varying direction, and the compensator has exact knowledge of this fact, the MIMO compensator could achieve much better attenuation of
that disturbance than a comparable SISO controller. For the experimental setup described in this thesis, there is one disturbance signal which affects the motion of the
cat's eye optic in multiple directions, which can all be controlled. Therefore the above
argument should work for this setup. The problem with this approach is that, the
controller does not have exact information of the disturbance source due to the simplified disturbance fits described extensively in section 3.2. The answer is to incorporate
robustness into the system by telling the mathematics that the disturbance comes in
all directions and hence there is no detailed directional information available. This is
done by formulating the problem such that there are three independent process noises
each driving the appropriate set of simplified dynamics which then get individually
46
I-
- - - -
- - - -
-
- -
-
- -
-
- - - -
- - - -
- - - -
-
- -
-
- -
z(t)
w(t)
,
Figure 3-14: MIMO H2 problem formulation: Find a compensator K(s) such that
u(s) = K(s)y(s) and IITIwlI
2 is minimized.
added to the three output directions of interest. The disadvantage is that with this
formulation, the sensitivity function is being pushed down in all directions, hence
the MIMO Bode integral constraint begins to look a lot like the SISO Bode integral
constraint in terms of achievable performance in each of the directions. However, the
MIMO approach still retains its ability to isolate in multiple directions at once.
For the MIMO LQG formulation, the dynamics of the augmented multivariable
plant were described by equations 3.5. The resulting transfer function matrix is then
called G,(s).
The minimization of IITZrI2 formulation is used, where in this case
the weighting matrices, We(s), and Wa(s) are diagonal three by three matrices with
repeated diagonal dynamic systems. As before, the LQG control design is done by
finding the dynamic linear compensator matrix which minimizes ITzll2 where Tz,
is the matrix transfer function represented in figure 3-14, following the generalized
formulation described by Doyle et al. in [7]. The D(s) used in figure 3-14 is diagonal
and has as its three diagonal elements the autospectrum fits for the A, B, and D
directions, respectively, as described in section 3.2 and in the previous paragraph.
The W,(s) and Wn(s) are design weights, which, through weighting of the control
and sensor noise respectively, shape the compensator, just like in the SISO case.
Nearly equivalent weighting functions were used in the MIMO case as in the SISO
case, in order to address the same constraints. Again, similar design variables were
47
varied in order to get the best overall design. It was found, that the primary limit
to performance was the bandwidth constraint.
However, a best design, given the
constraints, was found, and that became the full order design which needed to undergo
dynamic reduction in order to be implementable with the available digital computer.
3.4.3
Prediction of Performance
In order to predict the performance and stability of both the MIMO and SISO designs, a procedure was used in which the transfer function of the compensator was
combined with the actual transfer function data of the mirror mount plant and disturbance source, to form predicted loop, sensitivity, and complementary sensitivity
transfer functions. This aided in the stability prediction through a construction of
the Nyquist plot (either SISO or MIMO) from these transfer functions. Predictions
of the performance were achieved by multiplying the magnitude of the sensitivity
squared, by the displacement disturbance autospectrum, and calculating the relevant 10-500 Hz RMS values. This procedure allowed fine tuning and quick iteration
rates in the development of controllers, as well as predicting the stability of reduced
compensators, which are going to be discussed next.
3.4.4
Compensator Order Reduction and Discrete Representation
The solution of the LQG problem above yielded compensators of 97 states for the
SISO designs and 147 states for the MIMO designs. These compensators were too
large to implement on the real time computer where the maximum number of states
was 32. Clearly, a large amount of compensator order reduction was required. It
was found that a lot of the compensator states were relatively extraneous and a
Hankel singular value reduction could remove two-thirds of the states with little
or no performance/stability
costs. This reduced the SISO designs down to within
the implementable limits, however the MIMO design still had 50 states. Further
reduction through Hankel singular value truncation resulted in large performance
48
,2
1
1(
1
0
1
1
1
)4
Hertz
Figure 3-15: Comparison of singular values of MIMO full and reduced order compensators.
deterioration and, often times, instability of the reduced compensator when applied
to the model. In order to further reduce the number of states in the compensator,
the system identification software of Jacques [12] was used. A lower order model
was created by using a Hankel singular value reduction, and then the software was
used to fit the lower order model to the data representing the full order compensator.
This method allowed further reduction of the MIMO compensator down to 32 states
with minimal predicted performance degradation and continued predicted stability on
the data. An overlay of the full order compensator singular values and the 32 state
reduced order compensator can be found in figure 3-15. Although there are clear
disparities, the reduced order compensator was still predicted to stabilize the plant
and provided predicted performance values within dB of the full order compensator.
Overlays of the full and reduced SISO compensators can be found in appendix A.
Both the MIMO and SISO reduced continuous time compensators described above
were converted to discrete time using the Tustin transform, with prewarping set at
49
500 Hz in order to correctly match the dynamics of the compensators about crossover.
In order to compensate for the large shift in compensator poles and zeros due to the
low sample rate relative to the controller bandwidth (a factor of 5), the frequency
responses of the discrete controllers were tuned to the original continuous time controllers using Jacques' system identification software. This method resulted in almost
exact replication of the reduced continuous time compensator below the Nyquist rate
for all designs.
3.5
Chapter Summary
This chapter has presented the mathematical framework within which the multi-axis
active isolation problem has been posed. Focus was given to four main areas.
First there was the area of system identification, in which it was shown that highly
accurate models of the dynamics of the isolation mount were developed. In addition,
the use of analog circuits to limit the high bandwidth nature of the sensor-actuator
pair were highlighted. It was further demonstrated that proper engineering insight
was used to choose the analog dynamics. Due to the choice, the resulting augmented
dynamics looked very similar to an ideal system with a displacement sensor and
actuator.
Second, the use of a simplified disturbance model improved performance robustness due to disturbance autospectrum modal frequency and damping variations. Fur-
thermore, additional robustness to disturbance energy directionality for the MIMO
formulation was incorporated. This was done through the artificial assumption that
there were three independent disturbances acting on the outputs of the system, rather
than three correlated disturbances.
Third, the method by which the lack of actuator authority was bypassed for the
purpose of performance evaluation was fully described. In addition, the performance
limitations due to limited sample rate and the sensor-actuator pair were mentioned.
Fourth, the LQG control formulations for all the control designs were described.
The use of weighting functions to limit controller bandwidth was presented.
50
The
Bode integral constraint was given as an explanation for the fundamental limits on
performance for both the SISO and MIMO designs due to the extreme bandlimited
nature of the possible control bandwidth. The reduction of the MIMO compensator
using the system identification software allowed a much further reduction in dynamics
than simple hankel singular value reduction. The usual frequency warping problems
associated with the conversion of a continuous time system to a discrete time system
were overcome through the use of the system identification software to shift the discrete poles and zeros in order to fit the transfer functions of the desired continuous
time compensators up to the Nyquist rate.
51
Chapter 4
Experimental Results
The four controllers were implemented in sequence, starting with the three SISO
controllers and ending with the MIMO controller. Before closing any of the loops,
the loop transfer functions were measured and checked against the predicted loop
transfer functions. Having verified the model and compensator, the loops were closed.
Because understanding of the results requires an understanding of dynamic coupling,
the next section discusses what is meant by the multivariable nature of the plant.
The following four sections each describe the performance of one of the controllers,
with a summary of all results at the end.
4.1
Multivariable Nature of Plant
Understanding of the dynamic coupling between the commanded directions and the
resulting directions of motion is crucial to explain the off-axis performance degradation which is generally observed for the SISO control designs. Equation 4.1 shows a
representation of the input-output transfer functions in order to clarify what is meant
by dynamic coupling.
YA(S)
YB(S)
YD(S)
=
GAA(S)
GAB(S) GAD(S)
UA(S)
GBA(S)
GBB(S)
GBD(S)
UB(S)
GDA()
GDB(S)
GDD(S)
UD(S)
52
dA(S)
+
dB(s)
dD(s)
(4.1)
where:
GAA(S)
G(S)=
GAB(S)
GAD(S)
GBA(S) GBB(S) GBD(S)
=C(sI-A)-
1 B+
D
(4.2)
GDA(S) GDB(S) GDD(S)
where, A, B, C, and D are from equation 3.1. The outputs, yi(s), represent the
Laplace transform of the acceleration outputs in the ith direction.
Likewise, the
Uj(s) represent the Laplace transforms of the input commands in the jth direction.
The Gij(s) element of the 3x3 matrix is the transfer function relating u1 (s) to yi(s). In
general, an analysis of the coupling of a plant is accomplished though an examination
of the directions which result from a singular value decomposition of the transfer
function matrix. Because G(s) is diagonal dominant over most of the frequency range,
comparisonsof the relative magnitude of the off-diagonalterms to the diagonal terms
give a reasonable indication of what it being called the input-output coupling of the
plant dynamics. If the off-diagonal elements are zero, then the system is considered
to be completely uncoupled. If the ratio of the off-diagonal elements to the diagonal
element in its row is constant over a given frequency range, the controls are said to be
statically coupled over that frequency range. If the ratio of the off-diagonal elements
to the diagonal element in its column is constant over a given frequency range, the
outputs are said to be statically coupled over that frequency range. If the above
mentioned ratios vary as a function of frequency, the plant is said to be dynamically
coupled.
Due to the fact that the line of sight directions to A, B and D form an ellipsoidal
cone, there will be some static interdependence due to the non-zero inner product
between the directions. The reason dynamic coupling is of interest is that if there is
some frequency range over which actuating in what is being called the A direction,
for example, actually results in significant motion in the D direction; this knowledge
can be used in the design of a compensator, using a multivariable design technique
(such as LQG), which would actuate in the A direction in order to get performance in
the D direction over that frequency range. It turns out that the active mirror mount
does indeed have high dynamic coupling particularly about the passive stage modes
53
4,3
0
.,..
O)
Hertz
Figure 4-1: Illustration of Dynamic Plant Coupling through plots of GDA, GDB,
and GDD: notice large degree of coupling between commanded directions about the
mounts modes at 30 Hz., and again at high frequency.
54
around 30 Hz. as figure 4-1 illustrates.
The figure also illustrates that below the
passive stage resonance, there is static coupling between the variables, as one would
expect.
The presence of the dynamic coupling discussed above will influence the disturbance spectrums of the off-axis directions when the SISO loops are closed. Simple matrix manipulations show the relevant relationships. Given, for example that the SISO
loop which is trying to isolate in the A-direction is closed, ie. UA(s) = -KA(S)YA(s)
where KA(s) is the SISO compensator with the analog dynamics. The resulting closed
loop performance is represented by the equations in 4.3.
1 dA()
YA(S)
=
(I + GAA(S)KA(S))-
YB(S)
=
-GBA(S)KA(S)(I + GAA(S)KA(S)) dA (s) + dB(s)
YD(s) = -GDA(s)KA(s)(I
(4.3)
+ GAA(S)KA())>dA(S)+ dD(S)
where, dA(S), dB(S), and dD(s) are the square roots of the acceleration autospectrums, which are artificially assumed to be independent, as seen by each of the axes.
Therefore, in general, whereas it is the sensitivity transfer function which determines
the performance in the direction of interest, it is the product of the off-diagonal coupling matrix, the compensator, and the loop sensitivity which determines the off-axis
performance degradation in those directions. The only clear conclusion which can be
made from these equations is that the magnitude of the off-axis coupling matrices
has a direct effect. However, the size and shape of the compensator and sensitivity
have a strong influence on the off-axis amplification due to feedback.
4.2
SISO D-Direction Performance Results
The achieved isolation in the D-direction can be found in figure 4-2. As is evident
from the plot, most of the performance comes from attenuation of the mount modes
around 30 Hz.. This is exactly what is expected from the active isolation, as described
in Chapter 1. Some performance is also gained about 100 Hz, showing the additional
55
,-5
D-dir PerformanceAutospectrum
N
I
0
E
c
Frequency(Hertz)
Figure 4-2: D-direction SISO controller:D-direction performance - notice the successful attenuation of disturbance energy which had been amplified by the mount modes.
benefit of conveying to the control problem that there is disturbance energy at those
frequencies. Although the performance in the D-direction is satisfactory, recall that
the ultimate objective is to minimize the sum of the RMS positions in all three
directions. Figures 4-3 and 4-4 show the performance in the other two directions.
These figures show a significant performance degradation in the off-axes. If the only
coupling between the directions were due to the inner product of the direction vectors,
successful isolation in one of the directions would actually result in improvement
in the other directions.
The degradation in performance is a direct result of the
dynamic coupling in the plant. Figures 4-3 and 4-4 show that the disturbance gets
amplified between 30-60 Hz., and 200-300 Hz.. Figure 4-5, which shows the transfer
function between the disturbance in the D direction and motion in the A and B
directions, shows that over these frequency ranges disturbances in the D-direction get
amplified and added to the disturbance in both the A and B directions. Therefore,
control actions which act to compensate for disturbances in the D-direction over
that frequency range spillover into commands in the A and B directions due to the
56
.- 5
A-dir PerformanceAutospectrum
It
N
0I)
c'J
E
a
Frequency(Hertz)
Figure 4-3: D-direction SISO controller:A-direction performance - notice the deterioration in performance due to isolator mount dynamic coupling.
. -5
B-dir PerformanceAutospectrum
N
t:
a)
E
;r
Frequency(Hertz)
Figure 4-4: D-direction SISO controller:B-direction performance - notice the deterioration in performance due to isolator mount dynamic coupling.
57
.-
0
0
)3
Hertz
Figure 4-5: Illustration of Coupling of D-direction commands into Other Directions: solid line denotes plot of -GADKD(I + GDDKD)- 1 , dotted line denotes plot of
-GBDKD(I + GDDKD)-.
dynamic plant coupling. If, there were no disturbance energy over those frequency
regions where there is dynamic coupling, this off-axis deterioration behavior would
not occur.
4.3
SISO A-Direction Performance Results
The closed loop results which show the implementation of the SISO controller designed
to improve isolation in the A-direction can be found in figures 4-6, 4-7, and 4-8.
Figure 4-9 explains why there is much less deterioration in the performance of the
off-axis directions. As is evident from the figure, the portion of the plot which is
greater than one is fairly small and the amplification is not much greater than one.
Therefore, although there is some amplification of the A-direction disturbances in
the other directions, it is not nearly as pronounced as it was in figure 4-5. Hence, it
makes sense that the deterioration is not as pronounced as for the D-direction SISO
58
.A5
A-dir Performance Autospectrum
N
0)
cJ
E
C
Frequency (Hertz)
Figure 4-6: A-direction SISO controller:A-direction performance - notice the successful attenuation of disturbance energy which had been amplified by the mount modes.
I
5t
B-dir Performance Autospectrum
1
1
1
a)
E=l
1
1
1
1
Frequency (Hertz)
Figure 4-7: A-direction SISO controller:B-direction performance - notice the deterioration in performance due to isolator mount dynamic coupling.
59
D-dir Performance Autospectrum
-5
N
I
=;
cE
C
Frequency (Hertz)
Figure 4-8: A-direction SISO controller:D-direction performance - notice the deterioration in performance due to isolator mount dynamic coupling.
.- 1
10
100
0
'
102
-2
3
1 AN
10'
102
3
10
Hertz
Figure 4-9: Illustration of Coupling of A-direction disturbance into Other Directions: solid line denotes plot of -GBAKA(I + GAAKA) - 1 , dotted line denotes plot of
- 1
-GDAKA(I + GAAKA)
60
. at
0
)3
Hertz
Figure 4-10: Illustration of coupling of B-direction disturbance into other directions:
solid line denotes plot of -GAB KB(I + GBBKB)-1 - (filter to the A-direction), dotted
line denotes plot of -GDBKB(I + GBBKB) - 1 - (filter to the D-direction).
loop, discussed above.
4.4
SISO B-direction Performance Results
As was described in section 4.1, the feedthrough of the disturbance in one direction
to another, due to the implementation of control, is directly related to the product of
the relevant part of the overall system matrix, the compensator, and the SISO loop
sensitivity. A plot of this product is shown in figure 4-10. This plot represents a filter
through which the disturbance seen in the B-direction passes before adding to the
response in the other directions. Contrary to the previous two SISO designs, figure 410 shows that, for the most part, the disturbance is attenuated before adding into the
other directions. The isolation experimental results are shown in figures 4-11, 4-12,
and 4-13. Figure 4-11 shows good levels of performance. The expected performance
penalty due to the need for the sensitivity to amplify the disturbance over some region
occurs above 300 Hz., and below 15Hz., however, overall performance remained good.
61
-5
B-dir Performance Autospectrum
1
N
a)
E
c
Frequency (Hertz)
Figure 4-11: B-direction SISO controller:B-direction performance - notice the improvement in performance.
,' 5
A-dir Performance Autospectrum
N
a)
E
C
Pk, I
Frequency (Hertz)
Figure 4-12: B-direction SISO controller:A-direction performance - notice the lack of
deterioration in performance.
62
.,5
D-dir Performance Autospectrum
N
t
CD
E
C
Frequency (Hertz)
Figure 4-13: B-direction SISO controller:D-direction performance - notice the lack of
deterioration in performance.
A lot of the performance was achieved above 100 Hz., and some of the performance
was achieved through attenuation of the passive mount modes around 30 Hz..
The A and D direction results, shown by figures 4-12 and 4-13, show a marginal
improvement of performance due to the SISO control of the B-direction. Although
this is different behavior from the off-axis results from the other loop closures, it is explainable from figure 4-10. As discussed before, almost none of the disturbance which
influences the B-direction get passed to the A or D directions. Note though, that the
feedthrough to the D-direction is most pronounced, and the performance improvement
is correspondingly worse for that direction. The general lack of feedthrough disturbance allows the performance achieved in the B-direction to be reflected, through
a vector inner product, to small improvements in the isolation of the other directions. This possibility is allowable because of the physical correlation which exists in
actuality between dA(s), dB(s), and dD(s).
63
4.5
MIMO Performance Results
The previous three SISO control designs showed that, due to the dynamic plant coupling, in general, serious off-axis performance degradation occurs when only a single
direction of isolation is considered part of the performance metric. Although an inexperienced control designer might suggest closing all the SISO loops together, in
order to get simultaneous performance in all directions, this would result in instability, and hence would be unacceptable.
The predicted instability was inferred from
two sources. First, a check of the closed loop eigenvalues of the model with all the
SISO loops closed revealed four unstable poles. Second, a check of the number of encirclements of the critical point in a MIMO Nyquist diagram, with the compensator
applied to the data, revealed four encirclements.
Therefore, in order to attack this problem, a MIMO control design methodology
was required in order to find a stabilizing compensator which provided multiple axis
performance. A design was developed, which took into account all the dynamic coupling, as described in Chapter 3, and was implemented. The results can be found in
figure 4-14, 4-15, and 4-16.
As can be seen in the figures, good performance was
achieved in all directions. It is interesting to note, that as expected, the primary
source of the performance improvement was attenuation of the mount modes. Significant additional performance was achieved through attenuation of the energy around
150 Hz.. The amplification of the disturbance above 300 Hz. was not very noticeable
due to the lack of disturbance energy around that frequency.
4.6
Summary of Results
Recall that the overall goal has been to minimize the sum of the RMS disturbances
in all three directions, as described by equation 3.4. Therefore, a summary of the
results, with this metric, is shown in table 4.1. This table makes clear the already
verified statement that a MIMO control design methodology, which takes into account
the dynamically coupled nature of the plant, can provide good vibration isolation in
64
A-dir Performance Autospectrum
--5
N
a,
I
101
102
Frequency (Hertz)
Figure 4-14: MIMO controller:A-direction performance - notice the improvement in
performance due, primarily, to attenuation of the mount modes.
-- 5
B-dir Performance Autospectrum
N
I
a)
E
C:
Frequency (Hertz)
Figure 4-15: MIMO controller:B-direction performance - notice the improvement in
performance due, primarily, to attenuation of the mount modes.
65
D-dir Performance Autospectrum
105
104
103
102
N
r
a,
101
E 100
C
1U
-2
10
10
1
-3
-4
102
Frequency (Hertz)
101
Figure 4-16: MIMO controller:D-direction performance - notice the improvement in
performance due, primarily, to attenuation of the mount modes.
Table 4.1: Thesis performance metric summary.
Direction of motion
A-direction
B-direction
D-direction
Total
Mirror C motion (nm RMS) 10 - 500 Hz
Open Loop
Closed Loop
SISO-A SISO-B SISO-D MIMO
136
126
87
62
151
106
118
66
85
226
242
41
74
59
40
349
319
269
509
173
66
multiple directions. The slight overall performance improvements for the A and B
direction SISO loops is due to a combination of the lessening of dynamic coupling for
those directions into the other directions, and that the majority of the disturbance
was observed in those directions.
This chapter has shown the relationship which exists between dynamic coupling
of a plant and the possible performance degradations which could result from closure
of a SISO loop relating one of the inputs to one of the outputs.
In addition, the
degree of off-axis performance deterioration was shown to be directly related to the
energy distribution of the disturbance source and the product of the off-axis coupling
term, SISO compensator, and loop sensitivity. Successful implementations of all four
compensators demonstrated the overall superiority of the MIMO controller for the
purpose of multi-axis vibration isolation.
67
Chapter 5
Conclusions and Suggestions for
Future Research
5.1
Conclusions
The purpose of this thesis has been to show, experimentally, the benefits of using
the Linear Quadratic Gaussian (LQG) multivariable control methodology to achieve
isolation in multiple directions. In developing the controllers it was found that the
considerably stringent implementation issues limited achievable performance.
sample rate of 2600 Hz., for a controller with a required bandwidth
The
of 500 Hz., and a
plant which ramps up with a slope of 40db/decade over the control bandwidth, places
an extraordinary requirement on accurate modeling and prediction of all systems in
the control loop. This is due to the fact that every "rule of thumb" of digital design
regarding, choice of sample rate depending on bandwidth and plant considerations,
was violated [8]. In the presence of these obstacles, successful control designs were
implemented. It has been shown that, at the cost of greater complexity, a multivariable control design can yield significant results for systems with dynamic coupling
between the control variables. Attempts to isolate in individual directions yield good
results in the directions of interest, yet tend to increase the disturbance in the off-axis
directions. For the single case where the off-axis disturbances did not increase, it was
shown that this was a direct result of the lack of significant dynamic plant coupling
68
for those directions. It was found that the primary increases in complexity which the
MIMO designs incurred over the SISO designs were twofold. First, a more complicated state-space model which captured, accurately, the plant dynamic coupling was
required. Second, the sample rate of the digital computer which was to implement the
system had to be slowed down to compensate for the increased number of A/D and
D/A conversions. In general, if the application has a multiple performance objective
and the dynamic plant has significant coupling, a multivariable control design will
result in significant performance improvement, given the above costs.
5.2
Suggestions for Future Research
As described in section 3.1.1, one of the main limitations to performance was the
sensor-actuator pair. This limitation was due to the inherently wide bandwidth of
the transfer function relating the sensor and actuator, and the resulting need to add
analog dynamics to limit the bandwidth. This limitation could be eliminated through
the use of displacement sensors. The interferometer tetrahedron has the potential to
have a laser metrology system measure the differential pathlength between all the
siderostats. Such a system, if it were implemented, could be used to get differential
displacement measurements of the mount at point C towards the other siderostats.
For such an arrangement, the measurement and the performance variables would be
the same. This would result in a system whose fundamental performance limitations
could be analyzed without many of the tight experimental constraints encountered in
this research. In addition, it would give a concrete example of how the sensor-actuator
pair used in this research limited performance.
This work could also be extended to minimize the motion of all the siderostats in
the three directions corresponding to the D-direction and to the other two siderostats,
through the use of local multivariable controllers. Blackwood established in [4] that
the actuations of the mounts were relatively uncoupled through the interferometer
structure for actuations towards point D. This decoupling was due to the passive
stage, and the fact that actuations towards point D correspond to a mostly pistoning
69
motion for all the mounts, and therefore benefits the most from the passive stage. It
is not clear, however, that the decoupling of all the mounts would apply for actuations
towards each other because, as was mentioned earlier for point C, such actuations tend
to have a large lateral component. It should be recalled that the lateral directions
benefit the least from the passive isolation stage; hence there might be non-negligible
coupling through the interferometer structure. This, possibly weak, coupling could be
used to test theories regarding decentralized control and the degree to which coupling
needs to be understood for stability and performance purposes.
70
Appendix A
Full and Reduced Order SISO
Compensators
This appendix shows overlay plots of the full order and reduced order transfer functions of the three SISO compensators. There are two main issues to note in examining
the plots.
First, the plots show that, although some dynamic details were removed through
the state reduction process, the overall fit remained quite good between reduced order
and full order compensators. This translated directly to very little loss in predicted
performance values and to continued prediction of stability when the reduced order
compensator was applied to the data.
Second, when comparing the D-direction controller and the A and B direction
controllers, it is immediately apparent that whereas the D-direction controller has
relatively equal gain in the 30 Hz and 120 Hz regions, the A and B directions have
most of their gain in the 120 Hz region. This is a direct manifestation of greater
controllability of the mount modes from the D-direction since it is closer to the z
direction of the mount. The A and B directions, being more aligned with the lateral
directions, do not have as much influence over the mount modes. Hence, due to the
limited control resources which results due the control weighting, more effort is put
into controlling the disturbance at 120 Hz than at 30 Hz.
71
I
,2
1(
1
1
)4
Hertz
Figure A-i: Comparison of D-direction SISO full and reduced order compensators.
102
i
_i
ii
ri
,
I
T
I
i
i
T
i
101
100
0
C~C1
10 -1
10-2
-!Ill -3
'
10
26 state Reduced Order Compensator
97 state Full Order Compensator
|
.
-1
*
i
i(11
i
100
I
i
,I i
1
I
102
101
, ,
il
I
3
10
I I,
,
104
Hertz
Figure A-2: Comparison of A-direction SISO full and reduced order compensators.
72
j 2
0
)4
Hertz
Figure A-3: Comparison of B-direction SISO full and reduced order compensators.
73
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74
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