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Investigation of digital controller for a superspring

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Title:
Investigation of digital controller for a superspring
Creator:
Buxton, William Kip
Publication Date:
Language:
English
Physical Description:
ix, 67 leaves : illustrations ; 29 cm

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Subjects / Keywords:
Digital control systems ( lcsh )
Seismic waves -- Damping ( lcsh )
Digital control systems ( fast )
Seismic waves -- Damping ( fast )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references (leaf 67).
General Note:
Submitted in partial fulfillment of the requirements for the degree, Master of Science, Electrical Engineering.
General Note:
Department of Electrical Engineering
Statement of Responsibility:
by William Kip Buxton.

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Source Institution:
|University of Colorado Denver
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Auraria Library
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
38324219 ( OCLC )
ocm38324219
Classification:
LD1190.E54 1997m .B89 ( lcc )

Full Text
INVESTIGATION OF DIGITAL CONTROLLER FOR A
SUPERSPRING
by
William Kip Buxton
B.S., University of Colorado at Denver
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Electrical Engineering
1997


This thesis for the Master of Science
degree by
William Kip Buxton
has been approved
by
Miloje Radenkovic
Jan Bialasiewicz
Tamal Bose


Buxton, William Kip (M.S., Electrical Engineering)
Investigation of Digital Controller for a Superspring
Thesis directed by Associate Professor, Miloje Radenkovic
ABSTRACT
In this thesis, the use of digital controllers is investigated for the purpose
of improving the performance of a long period seismic isolation device, the
Superspring, over the current analog controller. A Superspring is a seismic
isolation device that is used in a FG5 absolute gravimeter. In this investigation a
model of the Superspring is first developed and then checked against an actual
Superspring. Next, the current controller is modeled and then checked against an
actual controller. The closed loop system of the model and actual Superspring are
compared. The comparisons of closed loop systems include simulating ground
motion and feeding it into the closed loop model. This data is then fit for gravity,
using a least squares algorithm. To accomplish the ground motion simulation a
filter is developed to simulate ground motion.
After the model of the current system is developed, two different digital
controllers are adapted for use with the Superspring. The first controller is a pole
111


placement controller. The second controller is a state space controller with
observer.
The closed loop system is then checked by standard procedures such as
step response and initial state response. The system is also examined by inserting
simulated ground motion into the closed loop system and fitting the output for
gravity.
A comparison of the systems is made to determine if improvements occur
when a digital controller replaces the current analog controller.
This abstract accurately represents the contents of the candidate's thesis. I
recommend its publication.
Signed
Miloje Radenkovic
IV


DEDICATION
I dedicate this to my wife Donna for her love, understanding and support
through the many years of school.


ACKNOWLEDGEMENT
This thesis would not have been possible without the cooperation and
support of many people, to whom I will always be grateful.
I would like to thank everyone at Micro-g Solutions, Inc. for their help and
support. In particular, I appreciated the advice and support of Dr. Tim Niebauer
and Fred Klopping.
I would also like to express a sincere thanks to my advisor Miloje
Radenkovic for all his help and advice. Furthermore, I would like to show
appreciation to the other committee members Jan Bialasiewicz and Tamal Bose for
their support.
Mike Kotar was helpful in editing this thesis.
Finally, I would like to thank my entire family for their patience and
understanding.


CONTENTS
Chapter
1 Introduction.......................................................1
2 Superspring Plant..................................................4
2.1 Theoretical Analysis...........................................4
2.2 Physical System................................................7
2.3 Analysis.......................................................9
3 Current Controller and Closed Loop System.........................12
3.1 Current Controller............................................12
3.2 Closed Loop System............................................15
3.3 Ground Noise..................................................18
4 Lag-Lead controller...........................................21
4.1 Design........................................................21
4.2 Simulation....................................................25
5 Pole Placement....................................................30
5.1 Design........................................................30
5.2 Simulation....................................................35
6 State Feedback Controller.........................................52
6.1 Design........................................................52
6.2 Simulation....................................................56
7 Conclusion........................................................64
References ...........................................................67
vii


FIGURES
Figure 1.1-1 FG5 System [1]..........................................................2
Figure 2.1-lSuperspring schematic....................................................4
Figure 2.1-2 Superspring Block Diagram...............................................6
Figure 2.2-1 The Superspring Drawing [1].............................................8
Figure 2.3-1 Bode Plot of Superspring...............................................10
Figure 3.1-1 Superspring UAF Filter Schematic.......................................13
Figure 3.1-2 Bode Diagram of Current Superspring Controller.........................14
Figure 3.2-1 Bode Diagram of Closed Loop System.....................................15
Figure 3.2-2 Gravity Data without Superspring...................................... 16
Figure 3.2-3 Gravity Data with Superspring........................................ 17
Figure 3.3-1 Theoretical Ground Motion..............................................19
Figure 3.3-2 Ground Noise fit to g..................................................20
Figure 4.1-1 Block Diagram of Closed Loop System....................................21
Figure 4.1-2 Bode Diagram Closed Loop System with Proportional Gain.................22
Figure 4.1-3 Bode Diagram of Closed Loop System for Lag-Lead Controller.............24
Figure 4.1-4 Bode Diagram of Lag-Lead Controllers...................................25
Figure 4.2-1 Initial State Offset Plot for Lag-Lead Controller......................26
Figure 4.2-2 Step Response Plot.....................................................27
Figure 4.2-3 Closed Loop Diagram....................................................27
Figure 4.2-4 Ground Motion Simulation Lag-Lead Systems..............................29
Figure 4.2-5 Ground Motion Simulation g Plot........................................29
Figure 5.1-1 Closed Loop Block Diagram Pole Placement Controller....................30
Figure 5.1-2 Closed Loop Block Diagram with out Tracking............................34
Figure 5.2-1 Step response for pole placement varying C,\...........................37
Figure 5.2-2 Step response for pole placement varying Q, transient..................37
Figure 5.2-3 Initial offset response for pole placement varying Q...................38
Figure 5.2-4 Initial offset response for pole placement varying C,u transient.......38
Figure 5.2-5 Residual plot for pole placement varying ..............................39
Figure 5.2-6 Gravity values for pole placement varying Q............................39
Figure 5.2-7 Initial offset responses for pole placement varying ooi................41
Figure 5.2-8 Step responses for pole placement varying i...........................41
Figure 5.2-9 Residuals for pole placement varying i................................42
Figure 5.2-10 Gravity values fore pole placement varying i.........................42
Figure 5.2-11 Step responses for pole placement varying h..........................44
Figure 5.2-12 Initial offset responses for pole placement varying coh...............44
Figure 5.2-13 Residuals for pole placement varying h...............................45
Figure 5.2-14 Gravity values for pole placement varying h..........................45
Figure 5.2-15 Step responses for pole placement varying Q,..........................47
Figure 5.2-16 Initial offset responses for pole placement varying £h................47
viii


Figure 5.2-17 Residual plot for pole placement varying £h...............................48
Figure 5.2-18 Gravity values plot for pole placement varying .........................48
Figure 5.2-19 Step responses for pole placement systems...............................50
Figure 5.2-20 Initial offset responses for pole placement systems.....................50
Figure 5.2-21 Residual plots for pole placement systems...............................51
Figure 5.2-22 Gravity plot for pole placement system....................................51
Figure 6.1-1 State Control Block Diagram................................................53
Figure 6.1-2 Block Diagram of State Observer............................................55
Figure 6.2-1 Step response for state feedback system, observer Vs controller..........58
Figure 6.2-2 Step response for state feedback system, observer Vs controller, transient.59
Figure 6.2-3 Step response for state feedback system varying C,.........................61
Figure 6.2-4 Step responses for state feedback system varying transient.................61
Figure 6.2-5 Initial offset responses for state feedback system varying C,............62
Figure 6.2-6 Initial offset responses for state feedback system varying £, transient..62
Figure 6.2-7 Residual plots for state feedback system varying Q.........................63
Figure 6.2-8 Gravity plot for state feedback system varying C,..........................63
IX


1 Introduction
The Superspring is an active long-period isolation device. It is used in a
commercially available gravity meter, the FG5. The FG5 is a transportable device
that measures the vertical acceleration of gravity (g). The measurement starts with
a test mass dropped vertically inside a vacuum chamber, dropping about 20cm.
The mass is tracked by a laser interferometer to accurately measure the acceleration
of the free-falling mass. The fringes are scaled and timed with an atomic clock to
develop the basis for acquiring accurate time and distance pairs. The time and
distance pairs are then fit using a least squares algorithm to a parabolic trajectory to
give a measured value of g. The FG5 is an absolute gravity meter because the
determination is purely meteorological and relies on standards of length and time.
A block diagram of the FG5 is shown in Figure 2.1-1.
As can be seen in the Figure 2.1-1, the laser is split in the interferometer.
The test beam travels through the dropper and super spring, while the reference
beam travels directly through the interferometer. The beams recombine to create
fringes that are used to make the time and distance pairs. The fringes are detected
by an avalanche photo diode (APD).
1


Figure 2.1-1 FG5 System [1]
When the test mass is in free-fall, it is free from all forces except for the
force of gravity. This is not true of the other interferometer arm. This arm is
susceptible to seismic motions. It is the purpose of the Superspring to isolate the
comer cube from seismic motion. The current Superspring system gives a 10 to 30
times reduction in seismic motion. This is appropriate for a seismically quiet site in
which the noise from the spring is not the limiting factor of the measurement. For
gravity sites that are seismically noisier, the Superspring is the limiting factor on
drop to drop measurements. For these sites, a controller that would decrease the
2


seismic noise by a greater factor would then allow for decreased measurement time
with a greater precision.
This thesis will examine two different digital controller designs along with
the current analog controller to determine if either of the digital controllers can
improve the performance of the Superspring. The current controller is a band pass
with some lag-lead compensation. The first digital controller to be examined is the
pole placement controller [2]. The second digital controller to be explored is the
state feedback controller [3].
The different control systems are examined with the normal tools such as
step response, bode diagrams and initial conditions. The control systems will also
be examined by running simulated ground motion into the system and fitting the
output for gravity (g).
3


2 Superspring Plant
2.1 Theoretical Analysis
To create a model of the Superspring it is necessary to go back to the basic
equation of a mass on a spring.
mx(t)= fix(t)-kx(t) (2.1)
Where m is the mass, k is the spring constant, P is the damping coefficient,
and x is the position of the mass. The schematic diagram of the Superspring plant
is shown in Figure 2.1-1.
ml
£Jb1
^l)
:k2
l~Jb2
-H(t)
m2
x2
Figure 2.1-lSuperspring schematic
From (2.1) the following equations for the Superspring system is obtained.
K
(t)+x2(t)+f(t)
m, n\ n\
n\ m, tm, m,
x2(t)=x, (0-*2(0+---*i (0-^(0
TMj Mj HI}
(2.2)
4


Where Xj is the position of x2 is the position m2, and xe is the position
of the ground. While k, and k2 are spring constants, P, and P2 are spring damping
coefficients and kc (N/Amp) is the coil force calibration. If xe = 0 (2.2) can be
rewritten to give the following transfer functions.
F(s)
F(s)
K
m,

m.
h
ftln
(2.3)
j4 +
P\ | Pi | A
ml mi m2
Is3 +
k\ k2 k2 P2
+-^
2 >
mx ml mn m,m
I 2
ly2 +
ic
+
m, w,
k1P2_+k1P}_
Km\m2 rnxm2 J
y+
kxk2
m.m.
V 1 yj
54 +
Pi Pl\i '
w, w, m2
A, A:, A, /?,
+-^
2 '\
m, /, m, m.
1"*2
hr2 +
KPl ^2 P\
^ f kxk2 ^
& +
mxm2 mxm2
mxm2
From the above equations the transfer function of the sphere output is.
(2.4)
5(5) yfefrMifr))
F(s) F
'V

54 +
P\ Pi | Pi
rt\ n\ ntj
' 'KA.h.PA'' r
1?3 +
n\ w, rtij
i?2 +
k[P2 ^ kjPi

?+
A,A2
M
Where A (V/m) is the sensor gain, which gives the difference between x,
and x2. The block diagram shows another possible way to represent the super
spang.
5


Figure 2.1-2 Superspring Block Diagram
The equations for Figure 2.1-2 are shown below
Gl =
A f 2 Pi kl s +^s +
m2 m2
.y4 + 'a+a+an mx m2} y3 + (k k *1 | K2 m, m \ 1 i k2 | P\Pl m2 mxm2
|y2 +
k\Pi | kiP\
mxm2 mxm2
y +
k]k2
mxm2
G2-
P2 k2
s +
m2 m2
2 Pi k2
S H--
(2.5)
N1 =
A s3 + (P\Pi ' |y2 + f KPi KPx 1
m{m2 mx r mxm2 mxm2
y+-
k.k
lfv2
m,m
l'"2
f P\ Pi Pi 1 1 ( kx k2 k2
s + m. m, m2 K 1 1 2 J y + h ~ H -h mx mx m2
P\Pl
m\m2j
Is2 +
(KPi kiP\A
m.m.
1"2
mxm2 j
y +
f kxk2 ^
m[m2
yj
The sphere output S(s) is the only directly measurable output of the
Superspring. X2(s) is only measurable from looking at gravity data of the physical
system.
6


2.2 Physical System
The Superspring system is shown in Figure 2.2-1. The main spring is a
20cm spring with a natural period of approximately lsec with the main mass
attached. The mass is made out of brass and houses a comer cube, and a sphere.
The outside spring is made from three support springs and 5-6 flexures. The
support springs are overstressed and are primarily used to support the middle mass
and the inner spring system. The flexures have a natural frequency of about 3 Hz.
The middle mass is made up of a tube, 2 triangular mounts, lever assembly and
detector block. The detector block is attached to the bottom of the tube, while the
lever arm assembly is attached to the top of the tube. Included in the detector
block is a base, which the support springs are connected, along with the coil of the
actuator.
The triangular supports attach to the tube and the flexures. The lever arm
system is made up of two levels that both adjust the position of the spring. One
level is attached to an aneroid that is tuned to adjust for length changes caused by
temperature changes. The other lever is attached to the focus motor and is used to
manually zero the main spring. The emitter block houses the emitter and detector
used to sense change in position between the main mass and support mass. The
emitter is an inferred LED that is then directed at the sphere on the main mass.
The sphere has the properties of a lens and focuses the light. The light is then
directed at a split detector, which gives out the position S(s). The base piece holds
7


die coil. When assembled the coil will hang in about the center of the magnet. The
coil gives the ability to move the support mass F(s).
Figure 2.2-1 The Superspring Drawing [1]
8


2.3 Analysis
To model the system, all the system constants must be found. To find the
spring constant (k) and damping coefficient (P) x is defined to be:
x(f)= x0e~m sin(atf)
Where xQ is the initial position of the spring, (0 is the undamped natural
frequency of the spring and a is the damping of the spring. Substitute (2.6) into
(2.1) to get
x0e~a [(a2 + eo2 )sin(atf)- 2aa cos(&tf)]= (2.7)
-xge~a sin{cot)-x0e~at [(Mcos(wf)-asin(ftrt)]
m m
Solving (2.7) for k and P gives
k = ?n(a2 +ty2) ^2'8^
P = lam
With the above equations, each spring was set up separately and the
position of the mass was monitored. From this data a and co were obtained.
Plugging a and (0 into (2.8) the following spring constants and damping
coefficients were obtained.
it, =149.17 k2= 24.33
Px = 0.3249 p2 = 0.0235
The weight of the main mass was obtained by direct measurement while
the support mass structure was obtained by calculation. The coil constant was
9


obtained from manufacture specification and verified by direct measurement.
While the sensor constant was obtained by measurement.
m, = .518£g m2=\kg
A = 6200 V/m kc=4.67N/Amp
Figure 2.3-1 shows the Bode plot of the simulated plant and two actual
systems. The natural frequencies vary from system to system as is seen in the bode
plot.
Figure 2.3-1 Bode Plot of Superspring
There are many factors that this model does not take into account. This
model assumes the spring mass system has only a single degree of freedom. A real
spring has 6 degrees of freedom [4]. The 5 additional degrees of freedom are not
10


controllable by this system, so will not be included in this analysis. To minimize
the effect of these additional modes some mechanical damping is used. This model
also does not take into account noise added by the sensor and electronics [5].
11


3 Current Controller and Closed Loop System
3.1 Current Controller
The current controller is made up of a low pass and high pass filter with
some lead and lag compensation. The high pass filter is needed since the sphere
signal will have an offset and drift due to temperature. With out the high pass filter
the controller attempts to compensate for the drift and eventually becomes
unstable.
To minimize the electronics to accomplish the filter two universal active
filters (UAF) were used [6]. The UAF gives three outputs, low pass, high pass, and
band pass filter. The current controller uses two of these filters. The controller
uses one filter for the high pass filter and lag compensation. The other filter is used
for the low pass filter and lead compensation. The two filters are hooked in
parallel. The band pass outputs of each UAF are used for the lead and lag
compensators. The output of the filter is then fed into a current driver.
The transfer function of the current systems controller is
12


Figure 3.1-1 Superspring UAF Filter Schematic
The bode plot of the current controller is shown in Figure 3.1-2
13


f(Hz)
Sm---------SS111
SSI 08
Figure 3.1-2 Bode Diagram of Current Superspring Controller
14


3.2 Closed Loop System
The closed loop Bode diagram of an actual system compared to the closed
loop model is shown in Figure 3.2-1. The phase plot of the model was shifted by
+180 to the make comparison easier to view. Figure 3.2-1 shows that the basic
shape of the two systems are the same, but the model has a wider bandwidth. The
actual system has a resonance at approximately 30Hz. This resonance is probably
the running mode of the spring [4]. It is one of the other modes of the spring.
After the low frequency hump the actual system magnitude increases, this is
probably due to measurement and sensor noise [5].
Bode Diagram Closed Loop System
Figure 3.2-1 Bode Diagram of Closed Loop System
15


All the plots and data, previously presented is direcdy measurable from the
sphere output s(t) which is the difference between x,(t) and x2(t). The important
point in the system for the gravity meter is x2(t). Since the test mass is in free fall
during the measurement the other arm of the interferometer also should be
isolated. If the second arm of the interferometer is attached to the floor the data
gives a scatter of 100-150 pGal (1 Gal = 1 cm/secA'2) for one data set. A data set
with the comer cube on the floor is shown in Figure 3.2-2. Figure 3.2-3 shows a
data set taken at that same site with the super spring.
Gravity Mean: 9.79 622 863 2
Std Dev (uGa 1) : 150.1
Gravity Median: 9.79 622 854 3
Ualues Accepted: 98
Last g Ualue: 9.79 622 709 3
Error (uGal): 131.
Tide (uGal): -14.58
Drop 100 of 100 Set 1 of 2
BOUL 40.1308 N -105.2328 E
DROP MODE- PLAYBACK
BOULDER AO FG5 108 (spring locked doun)
Figure 3.2-2 Gravity Data without Superspring
SENSOR DATA
Air Tenp (C): 24.75
Air Pressure (nbar) : 837.89
Channel 4 (U) : -.2367
Uacuun Level (utorr) : 1.870
Superspring Position (nU): 42.43
Update: 1997:892 17:32:00 GMT
MEAS. HEIGHT cn: 100.00
E Peak Lock
04/02/97 1997 092 17 32 00
16


Gravity Mean: 9.79 622 838 2
Std Dev CuGal): 5.435
Gravity Median: 9.79 622 838 7
Ualues Accepted: 99
Last g Ualue: 9.79 622 852 5
Error (uGal): 4.69
Tide (uGal): -14.70
Drop 99 of 100 Set 14 of 24
TMGO AH 40.1308 N -105.2328 E
DROP MODE- PLAYBACK
TMGO U APD
APPLIED CORRECTIONS (uGal)
Speed of Light: -11.3
Gradient Height: -26.7
Uertical Transfer: 98.3 6 100.00 cn
Ocean Load Corr: .000
Baronetric Pressure: -.900E-01
Polar Motion: -2.38
MEAS. HEIGHT cn: 100.00
E Peak Lock
11/30/95 1995 334 07 31 20
Figure 3.2-3 Gravity Data with Superspring
To analyze different controllers, an understanding is needed of how the
data is actually used. The time and distance pairs are fit to a parabola using a least
squares fit In Figure 2.3-2 the upper left graph is of the residuals of the least
squares fit The lighter line shows the residual of the last drop, while the darker line
shows the mean residual of all the drops in that set. The plot under the residual
plot is the gravity plot in pGal offset by the mean value. The graph on the right
hand side of the figure is the histogram plot of the gravity values. The text in the
lower left comer is the only other data that is useful to this analysis. The gravity
mean (m/s^2) is the mean value of all accepted drops in this set. The Std Dev
17


(pGal) is the standard deviation of all the gravity values in this set. Values accepted
indicate the number of drops that have been accepted by the boxcar and 3 sigma
filters. The last g value gives the result for the last drop. While the Error (pGal)
gives the error of the last fit.
For the following analysis of each control scheme, not only look the Bode
plots and the step response of each system are examined, the ground motion is
simulated and fed in to each system. The output of each system is fit to a parabola
and then compared to the actual system. For this analysis to succeed, a filter is
needed to create a simulation of ground noise.
3.3 Ground Noise
For the following analysis, a model of ground motion is needed. A graph
given by Dr. David Newell gives a straight-line approximation of ground vibration
as measured at JILA [9]. From the straight-line approximation a filter which
approximates ground noise was created. Figure 3.3-1 shows the straight-line
approximation and the filter is shown below.
Gn(s)=. 5e-4
(
.1(2 jt)
s + .\(2a)
1
s + .15(2^)
(s + 2nJ
1
s + 30(2;r)
(3.2)
18


Theoretical Ground Motion
Figure 3.3-1 Theoretical Ground Motion
Using the filter above and running the noise through the closed loop
system and using a typical sampling scheme used by the FG5 of taking one gravity
(g) measurement every 10 seconds. One measurement is made up of
approximately 200 time and distance pairs equally spaced in distance. For
simulation purposes, the data is spaced equally in time. In Figure 3.2-2 is an actual
data set where the Superspring is not used. The data was taken at TMGO (Table
Mountain Gravity Observatory) which is a seismically quiet site. Figure 3.3-2
19


shows gravity data simulated from the theoretical ground noise and actual data
from Figure 3.2-2.
2000
1500
1000-
-1000
Ground Noise g plot Actual vs. Simulated
simulated
4-----4 actual
_l_____________
_1____________I____________l_
0 10 20 30
Figure 3.3-2 Ground Noise fit to g
40 50 60
drops
70 80 90 100
20


4 Lag-Lead controller
4.1 Design
To begin, the system is examined with proportional feedback. The plant is
defined as G. The transfer function of G is shown below. This was obtained from
(2.4).
k.
V
G(,)=

(4.1)
s4 +
rB & aV (
rt\ n\
y
K k2 K PA
k2 +
KPi M
------1-----
v
m,
A ' kh 'i
.?+
y JJ
rt\ n\ m^ntz
All the systems in this section will have the closed loop block diagram
given by Figure 4.1-1. The sign of the feedback is positive since the output was
defined as x^-x^t) it would be negative feedback if this were reversed. The closed
loop transfer function of Figure 4.1-1 is shown below.
S(s) G
cl R{s) 1 -GH
(4.2)
Figure 4.1-1 Block Diagram of Closed Loop System
21


Figure 4.1-2 shows the Bode diagram of G and Hd for unity feedback and
a gain of 24. The unity feedback separates the resonant frequencies of the system
to .078Hz (12 sec. period) with damping ratio C, of 7.67e-004 and 27.2 Hz has a C,
of 1.15e-003. Increasing the gain of the system to 24 spreads the resonant
frequencies to 62.8-seconds and 133Hz with even lighter damping. The system
should have at least a 20-second period not to have a major effect on the fit of g.
Bode Diagram
Figure 4.1-2 Bode Diagram Closed Loop System with Proportional Gain
Viewing the phase in bode diagram shown in Figure 4.1-2 shows the
system phase is 180 degrees out of phase at the low and high frequencies. This
22


means noise at these frequencies will couple in and make the system unstable. To
increase stability and damping of the low frequency response, a lag compensator is
added [7] [8]. The transfer function of a lag compensator is shown in (4.3).
H(s)=K
1
s +
____T_
1
(4.3)
Where K is the gain, T is the time constant and P is the adjustment of the
lag and it is always greater than 1. Choosing T to be 6, P to be 10 and K to be 24
changes the closed loop system from having an con at 62.8 seconds to 78.5, while
increasing the damping C, from 1.58e-004 to .878 as can be seen in Figure 4.1-3.
The phase margin at the low frequency was also increased. The period of the
spring is now low enough for accurate measurements.
The upper frequency should have its phase margin increased. To increase
the phase margin some lead compensation is added. The transfer function of the
lead compensator is shown in (4.4).
1
S H--
H(s)=K------(4.4)
S H---
aT
23


50
Bode Diagrams Closed Loop System
Figure 4.1-3 Bode Diagram of Closed Loop System for Lag-Lead Controller
Where K and T are the same as in the lag compensator and a is between
0 compensator is used. The lag-lead transfer function is shown below.
1 1
S-\--S H----
H(s)=K---------{-----T-f- (4.5)
5 H----S +-----
prx aT2
For the lag-lead compensator the same values are used for the lag portion
as before. While T2 = 0.02 and a = 0.01 the Bode plots for these are show in
Figure 4.1-4.
24


Bode Diagrams Controllers
Frequency (Hz)
Figure 4.1-4 Bode Diagram of Lag-Lead Controllers
4.2 Simulation
Figure 4.1-3 shows the Bode diagrams of the dosed loop systems between
R(s) and S(s) for the current controller, lag and lag-lead compensator. The step
response and initial offset plots are shown for the current controller and lag-lead
compensator in Figure 4.2-1 and Figure 4.2-2. The current closed loop system is an
8-pole system, all poles being complex. This is seen in the step response and initial
offset plot. Each plot shows two different periods. While the lag-lead system has 6
poles only 4 are complex. The lag-lead system has only one dominant low
frequency. By increasing the gain, the period of each closed loop system is
25


x2 (meters)
lowered. A problem with increasing the gain is that the systems can become
unstable due to unmodeled noises.
0.1
0.05
0
-0.05
-0.1
0 20 40 60 80 100 120
Initial Offset to Analog System
Figure 4.2-1 Initial State Offset Plot for Lag-Lead Controller
26


Step Response to Analog System
Figure 4.2-2 Step Response Plot
Next the ground noise is simulated and input it into the system shown in
Figure 4.2-3.
Figure 4.2-3 Closed Loop Diagram
The current Superspring controller and lag-lead controller are compared,
by inserting noise into each closed loop system. The filter developed in (3.2) is
27


used to generate Xe. This is then fed into the closed loop systems to generate the
output X2 The output of X2 is then fit for gravity using a least squares algorithm.
The residual output of the least squares fit is shown in Figure 4.2-4. While the
gravity values are shown in Figure 4.2-5. Table 4-1 shows the mean g value and the
standard deviation of g. From the standard deviations listed in Table 4-1 and the
data shown in Figure 4.2-5 it is seen that the current system controller has a
smaller standard deviation. This is probably due to the current system having a
lower period.
Table 4-1 g Data for Lag-Lead Controller
9 (f^Gal) Std Dev.
Current -0.0020 0.0668
Lag-Lead -0.0034 0.0890
28


Cumulative Residual (nm)
Ground Motion Simulation with Analog System
Figure 4.2-4 Ground Motion Simulation Lag-Lead Systems
0.3
0.25
0.2
0.15
0.1
f. 005
05
0
-0.05
-0.1
-0.15
-0.2
0 10 20 30 40 50 60 70 80 90 100
drop
Figure 4.2-5 Ground Motion Simulation g Plot
Ground Motion Simulation with Analog System

Current Lag-Lead -
I
J___________I_________ i__________I_________i i t
29


5 Pole Placement
5.1 Design
The Pole placement design allows the controller to be designed without
any restrictions on the location of the zeros [2]. This system does not simplify the
zeros as many other controllers do. Not simplifying the zeros is required since the
Superspring has a zero on the unit circle in the z plane or at 0 in the s plane. This
design does require that there be no common factors in the plant. The system
block diagram is presented in the following form.
Figure 5.1-1 Closed Loop Block Diagram Pole Placement Controller
For this design process the plant needs to be characterized by a pulse
transfer function shown below.
Where d is the number of delayed sampling periods in the system. A(q )
and B(q ) are shown in (5.2).
30


(5.2)
^fe')=1 + ai q~l +---+an B(q )=blq~l + + amq~m =q~]B*(q~')
The closed loop diagram is shown in Figure 5.1-1. From the Figure 5.1-1
the closed loop transfer function is given by:
h G-'~l
^ 4^"Pki~')+^kr' W?"1) 5
Now the closed loop poles are defined to be P(q1) so this means:
<5'4)
where
p(?-)=i
(5.5)
+ Prt~l + Pl Where p; can be zero. The degree of P(q4) needs to be
degP(^_1 )< 2r -1 (5-6)
for (5.4) to have a unique solution. While r is defined as
r = max deg(, m + d) (5.7)
P(q4) controls the behavior of the regulation (closed loop poles). So
inserting (5.4) into (5.3) gives equation (5.8).

(5.8)
S(q_1) and R(q') have the form shown below
S(q~' )= 1 + sxq~x + + Vi R(q~] )= ro+w~' + +rr-i^_r+1
31


To solve (5.4) the solution will be defined by
where
and
where
Mx p
(5.10)
M =
*r=l [l,5,,..
PT=1 Plr-\
1 0 ... 0 0 ... 0
ax 0 0 :
Qr-1 1 K-x 0
ar &r-\ ... ai K K_x - b[
... o ,-i 0 K-
0 0 ar 0 0 b[
(5.11)
(5.12)
at = 0 for i > n
bj = Ofori = 0,1,...,d where bj = fori>d + i (5.13)
b; = Ofori > n
To have 2ero steady state error for a step input or disturbance an
integrator must be added to S(q_1) so the equation then becomes
S{q~' )= J) q-' )= S(g-' )ff2 ( For digital robustness a filter can be added to R(q )
R'(9-,)=f^(9-1)p-(l-a9"1)=f^(^-1)f^1(^',) 0 < 1 (515)
32


Inserting R'(q') and S'(q_1) into the dosed loop transfer function R(q'1) and
S(q_1) it becomes
q-T(q-'W') f516)
So P(q'1) becomes
p(?-')= A(q-')s(a-')H2(q-')+q-'B(q-')((q-%(S-') <517>
When solving the above equation include H2(q_1) with A(q_1) and H, (q1)
with B(q_1). When the system is implemented indude H2(q) with S(q_1) and H^q1)
with R(q ')
For most systems the output should track the input. In this system only
regulation is needed. A method for having the output track the input is to run it
through a model with the desired response, a reference modd. The transfer
function of the reference model is
(5.18)
where
I?-' )= 1 + amlq~l + amlq-2 + (5.19)
Bm{q )=q~'(pma+bmXq~' +-)
So with Hm(q') this gives the desired response between r(t) and y(t). In this
design the system zeros are preserved and the time delay d cannot be
33


compensated. Since the delay cannot be corrected the reference trajectory
becomes:
y*(<}=9 Tf'-i ^
Am\g )
Removing the delay from the reference model (5.20) becomes
(5.20)
(5.21)
The predicted output y*(t+d+l) is fed into T(q4). T(q_1) is chosen to
ensure unitary steady state gain between y*(t) and y(t) and to compensate for the
closed loop poles P(q .,). This leads to defining T(q.,) to be
T(q-')= GP(q-1) (522)
where
G =

(5.23)
1 i iM(i)=o
As stated earlier, only the regulation of the system is important So the
closed loop block diagram that is used in this set of simulations is shown in Figure
5.1-2.
r(t) +
H(q-') 4
i L r *(?")
4J)
Figure 5.1-2 Closed Loop Block Diagram with out Tracking
34


The closed loop pulse transfer function of Figure 5.1-2 is

(5.24)
(5.25)
5.2 Simulation
To begin with the following model is used to place the closed loop poles:
(s2 + 2 Js2 + 2£y + to2)
Where CD's are in (radians/sec) and £'s are the damping coefficients. After
discretization of (5.25) equation (5.26) is obtained, where T is the sampling
frequency.
(l + a,^"1 + a2lq~2^ + alhq+ a2hq~2)
au = -2e~^'T cos^1-C/2;T
a2l = e~2<'a>lT cos^-y/l -C,2j
.* = 2e_f*<8*T cos^/l -£h2o)hT
a2h = e-2i"a"T cosfVWV *T
(5.26)
The above equation means P(q4) is given by:
P(q-')=l+(au +aVl)q-' +(a2l +auau, +a2h}f2 +{aua2h +a2la]h)q'2 +a2la2hq^ (5.27)
For Figure 5.2-1-Figure 5.2-6 CO, is 0.1047 (60-second period) with C,x
varied from .2, .5, and .9 and ooh is 628.3 with <^h 0.9. Figure 5.2-1 and Figure 5.2-2
shows the step response of the above system. Figure 5.2-3 and Figure 5.2-4 shows
the response of the system to an initial offset in the plant. Figure 5.2-5 shows the
35


residuals and cumulative residuals from the simulated gravity values. Figure 5.2-6
shows the gravity values. Table 5-1 shows the gravity values and their standard
deviations. From the gravity plot in Figure 5.2-6 and the residuals in Figure 5.2-5 it
can be seen that the lighter damping improves the g fit. This is probably due to less
acceleration being introduced.
Table 5-1 g Data for Pole Placement Controller
g (pGal) Std Dev.
5=0.2 0.0010 0.1790
5=0.5 0.0031 0.4475
5=0.9 0.0063 0.8055
36


0.05
Step Response to Pole Placement System
Figure 5.2-1 Step response for pole placement varying Q
x IQ-4 Step Response to Pole Placement System
Figure 5.2-2 Step response for pole placement varying Q, transient
37


Initial Offset to Pole Placement System
Figure 5.2-3 Initial offset response for pole placement varying ^
Initial Offset to Pole Placement System
Figure 5.2-4 Initial offset response for pole placement varying Q, transient
38


Ground Motion with Pole Placement System
t (sec)
Figure 5.2-5 Residual plot for pole placement varying Q
Ground Motion with Pole Placement System
Figure 5.2-6 Gravity values for pole placement varying Q
39


For Figure 5.2-7-Figure 5.2-10 (0, is varied with of 0.2 and toh is 628.3
with Cfo 0.9. to, is varied, so the closed loop system will have a period of 60, 90 and
120 seconds. Figure 5.2-8 shows the step response of the above system. Figure
5.2-7 shows the response of the system to initial offset in the plant. Figure 5.2-9
shows the residuals and cumulative residuals from the simulated gravity values.
Figure 5.2-10 shows the simulated gravity values. From this data it can be seen that
the system with the longest period has the least error. Table 5-2 shows the gravity
values and their standard deviations.
Table 5-2 g Data for Pole Placement Controller
9 (uGal) Std Dev.
Period 60 0.0010 0.1790
Period 90 0.0007 0.1193
Period 120 0.0006 0.0894
40


Initial Offset to Pole Placement System
Figure 5.2-7 Initial offset responses for pole placement varying cog
Step Response to Pole Placement System
Figure 5.2-8 Step responses for pole placement varying CO)
41


Ground Motion with Pole Placement System
x10'4
Figure 5.2-9 Residuals for pole placement varying CO)
Ground Motion with Pole Placement System
Figure 5.2-10 Gravity values fore pole placement varying (0|
42


For Figure 5.2-11 and Figure 5.2-14 CO, is set to a period of 120 seconds
with <^, of 0.2 and C0h is varied with <^h 0.9. Where coh =27lfh and fh is varied, so the
closed loop system will have a frequency of 10, 50 and 100 Hz. Figure 5.2-12
shows the step response of the above system. Figure 5.2-11 shows the response of
the system to initial offset in the plant. Figure 5.2-13 shows the residuals and
cumulative residuals from the simulated gravity values. Figure 5.2-14 shows the
simulated gravity values for this controller. It appears that wider bandwidth offers
a small improvement for gravity mean measurements. While the lower bandwidth
makes a major reduction in the transient spike as can be seen Figure 5.2-11 and
Figure 5.2-12. Table 5-3 shows the gravity values and their standard deviations.
Table 5-3 g Data for Pole Placement Controller
9 (pGal) Std Dev.
f 10 -0.0077 0.0835
f 50 -0.0006 0.0894
f 100 0.0006 0.0894
43


Step Response to Pole Placement System
Figure 5.2-11 Step responses for pole placement varying Initial Offset to Pole Placement System
Figure 5.2-12 Initial offset responses for pole placement varying (O),
44


Ground Motion with Pole Placement System
Figure 5.2-13 Residuals for pole placement varying cdh
Ground Motion with Pole Placement System
Figure 5.2-14 Gravity values for pole placement varying coj,
45


For Figure 5.2-15-Figure 5.2-18 (D, is set to a period of 120 seconds with 5i
of 0.2 and (Dh is set to 623 (rad/sec) with 5h set to 0.2, 0.5 and 0.9. Figure 5.2-15
shows the step response of the above system. Figure 5.2-16 shows the response of
the system to initial offset in the plant. Figure 5.2-17 shows the residuals and
cumulative residuals from the simulated gravity values. Figure 5.2-18 shows the
simulated gravity values for the above controller. The higher damping makes a
reduction in the transient spike as can be seen in Figure 5.2-15 and Figure 5.2-16.
It also appears to improve the g mean value. Table 5-4 shows the gravity values
and their standard deviations.
Table 5-4 g Data for Pole Placement Controller
g (pGal) Std Dev.
?0.2 0.0015 0.0894
5=0.5 0.0011 0.0895
5=0.9 0.0006 0.0894
46


Step Response to Pole Placement System
Figure 5.2-15 Step responses for pole placement varying
x 10
a
0)
£
CM
X
Initial Offset to Pole Placement System
0.005 0.01
0.015
0.02 0.025 0.03 0.035 0.04
Figure 5.2-16 Initial offset responses for pole placement varying
47


Ground Motion with Pole Placement System
Figure 5.2-17 Residual plot for pole placement varying
Ground Motion with Pole Placement System
Figure 5.2-18 Gravity values plot for pole placement varying
48


The next system to be viewed, takes its poles from the closed poles of the
lag-lead controller in chapter 4 and is designated sys2. It is compared to a system
where CO, has a period of 120 seconds with of 0.4 and coh of 2ti100 and Q of 0.9
designated sysl. The fits to gravity values are shown below. The step response and
initial offset for each system is shown in Figure 5.2-19 and Figure 5.2-20. Figure
5.2-21 shows the residual plots for the fit to g and Figure 5.2-22 shows the g plot.
Table 5-5 shows the gravity values and standard deviations for the 2 systems.
Table 5-5 g Data for 2 Different Pole Placement Controllers
g (pGal) Std Dev.
Sys1 0.0006 0.0894
Sys2 -0.0028 0.0832
49


Step Response to Pole Placement System
Figure 5.2-19 Step responses for pole placement systems
Initial Offset to Pole Placement System
Figure 5.2-20 Initial offset responses for pole placement systems
50


Ground Motion with Pole Placement System
Figure 5.2-21 Residual plots for pole placement systems
Ground Motion with Pole Placement System
Figure 5.2-22 Gravity plot for pole placement system
51


6 State Feedback Controller
6.1 Design
In this chapter, a state feedback controller is designed. It is also called a
pole placement controller [2]. The digital version of this controller is the sole focus
examined here. For the first part of this design, all the states are made measurable.
From these states the closed loop poles are placed to a preset position.
To create a state feedback controller, the system must first be modeled in
state space. The basic state space model is:
jc(A: +1) = Ax{k) + Bu(k) (g i)
y(k) = Cx(k) + Du(k)
Where
x(k)= state vector (n x 1)
u(k) = control signal (scaler)
y(k) = output signal {scaler) (6.2)
A = n x n matrix
B = n x 1 matrix
C = 1 x n matrix
D = 0
The closed loop system to be examined is shown in Figure 6.1-1.
52


Figure 6.1-1 State Control Block Diagram
This controller requires the plant to be completely state controllable. To be
completely state controllable, the system can be taken from any initial state to any
other desired state in a finite number of samples. Checking the rank of the
controllability matrix checks the system for controllability. If controllability matrix
rank is n, then the system is completely state controllable. The controllability
matrix is shown below.
M = [B\AB\---\An~xB'\ (63)
If the system is completely state controllable, the following method and a
few other methods can find K the state feedback matrix.
K = [<*n "- T is given by
T -MW
M is the controllability matrix (6.3) and W is given by
(6.5)
53


(6.7)
an-1 an-2 1
an- 2 an-3 1 0
ax 1 0 0
1 0 0 0
The a,s are the coefficients of the plants characteristic equation
|zl A\ = z + h-----------1- an_xzn +an = 0 (6.8)
While the a/s are defined by the coefficients of the desired characteristic
equation of the state feedback controller
\zl -A + BK\ = zn + axzn^ + i- at_xzn + an =0 (69)
Once the desired controller model is determined, a state observer is
required. The state observer is required since none of the states are directly
measurable. The state observer predicts the state of the actual system and is then
feed into the state feedback matrix. The closed loop system with observer is shown
in Figure 6.1-2.
For the system to be completely observable the rank of the observability
matrix must be n. The observability matrix is:
N =
C
CA
CAn~]
(6.10)
54


Figure 6.1-2 Block Diagram of State Observer
If the system is completely observable then the observer gain matrix Kc
can be found with equation
Ke=Q
an ~an
an-\ ~an-1
1
(6.11)
Where Q is defined as
Q = (WN)~' (6'12)
W is the same as in the state controller design as given in (6.7). N is the
observability matrix (6.11). The a;s are the coefficients of the plants characteristic
55


equation and is shown in (6.8). While the a;s are defined by the coefficients of the
desired characteristic equation of the state observer
\zl -G + KeC\ = z" + axzn~x ----------1- o.n_xzn + an = 0 (6.13)
6.2 Simulation
The above equations allow the development of a state feedback controller.
To begin, the system is represented in a state space representation. This model was
derived from (2.2). For design purposes xc will be made 0 since this input is not
controllable.
*(0 =
0 0 1 0
-(*,+ , o k2) k2 -ta+A) 1 h.
m] m, w,
k2 Pt_
m2 m2 m2 m2
A A 0 0]x(f)
' 0 '
0
K
m,
0
(6.14)
f(t)
Where
x =
(6.15)
After discretizing the system with a zero-order hold the system will
represented by (6.16).
x(k +1) = Ax(k) + Bf(k) (6.16)
y(k) = Cx(k)
56


Where
x(k) = state vector (4 x 1)
u(k) = control signal {scaler)
y(k) = output signal (scaler) (6-!7)
A = 4x4 matrix
B = 4x1 matrix
C = 1x4 matrix
To start with the desired characteristic equation is derived from
s2 +2^(0 +(D2 (648)
Where co is the natural frequency (rad/ sec) and C, is the damping
coefficient. After discretization of (6.18), (6.19) is obtained
z2 +,z2 + a2
a, = -2e~CmT cos
a2=e 2itoT cos

(6.19)
Since the plant has 4 poles the model requires 4 poles. This makes the
characteristic equation from (6.19) become
|zl A + BK\ = z4 + a,z3 + a2z2 + a3z] + a4
a, = -2e~ieoJ cos
>(Vl -A2 t)
-2£uT (ft TT rp^ (6.20)
a2=e 4 cos^Vl-a an)
a3=a4 = 0
The same process is used to find the characteristic equation of the
observer, but different CO and C, are chosen. The settling time of the observer
should be at least 20 times less than the controller [2]. To begin co = 2tc(1/60) and
57


C, 0.2 for the state controller is chosen. For the observer co = 27t(30) and C, = 0.9
is chosen. In Figure 6.2-1 and Figure 6.2-2 a comparison between a system that
allows all the states to be directly available and one using an observer to produce
the internal states can be seen. The primary difference between the two systems is
the transient response, peak overshoot and steady state response of x2. Since the
system requirements are for noise rejection, the offset is not important. The state
observer causes the differences in peak overshoot and transient response
Step Response Observer vs State Controller
Figure 6.2-1 Step response for state feedback system, observer Vs controller
58


3
Step Response Observer vs State Controller
-4
x 10
Figure 6.2-2 Step response for state feedback system, observer Vs controller, transient
Next, three different systems with three different C, factors for the state
controller characteristic equation are examined. The step responses of these
systems are shown in Figure 6.2-3 and Figure 6.2-4. The figures show that the
controller changes the damping as predicted. The controller also changed the
transient response, due to the error generated by the observer. Using initial
parameters for the plants states would approximate the actual system when the
feedback was engaged. Figure 6.2-5 and Figure 6.2-6 shows the output verses the
initial condition. This shows that the transient response is smaller as the damping
in the main system decreases. Figure 6.2-8 shows the simulated gravity data. Figure
6.2-7 shows the residuals and cumulative residuals from the simulated gravity
59


values. From this data, it can be seen that the controller with the least damping is
preferable. Table 6-1 shows the gravity values and there standard deviations.
Table 6-1 g Data for State Controller
g (uGal) Std Dev.
£=0.2 -0.0054 0.1783
£=0.5 -0.0127 0.4455
£=0.9 -0.0222 0.8020
From Table 6-1, Figure 6.2-7 and Figure 6.2-8 again it can be seen that
lighter damping is preferred. It also can be seen comparing Table 4-1 from the
pole placement that the standard deviations are essentially the same. This should
be true since both systems place the poles or states of the system. Also Figure
6.2-7 and Figure 6.2-8 is very similar to Figure 5.2-5 and Figure 5.2-6 from the pole
placement plots for similar damping. The mean g value is probably different due to
the high frequency response differences in the systems.
60


Step Response of State Feedback Systems
Figure 6.2-3 Step response for state feedback system varying C,
Step Response of State Feedback Systems
Figure 6.2-4 Step responses for state feedback system varying Q, transient
61


Initial Offset to State Feedback Systems
Figure 6.2-5 Initial offset responses for state feedback system varying C,
Initial Offset to State Feedback Systems
Figure 6.2-6 Initial offset responses for state feedback system varying C,, transient
62


Figure 6.2-7 Residual plots for state feedback system varying £
Ground Motion Observer State Feedback Systems
Figure 6.2-8 Gravity plot for state feedback system varying Q
63


7 Conclusion
This thesis has examined the possibility of replacing the current analog
Superspring controller with a digital controller. An analytical model of the
Superspring was created. The analytical model was then verified by comparing it to
an actual Superspring. The current controller was then modeled and compared to
an actual controller. The models were combined to create a closed loop model,
which was also compared to the actual Superspring closed loop system. A filter to
simulate ground motion was created for a more comprehensive analysis. The
output of the ground motion filter was fit for g and compared to actual gravity
data taken without a Superspring. The ground motion was then applied to the
closed loop system and the motion of x2(t) was fit for g and this was compared to
actual data taken with the FG5 absolute gravity meter.
The model of the Superspring and controller accurately simulated the
actual Superspring components. The closed loop system showed the same form
but the bandwidth of the actual system was diminished and showed some of the
unmodeled portions of the system. The gravity data of the simulated ground
motion and actual gravity data showed good equivalence. While the agreement of
the simulations of ground motion through the closed loop system and actual FG5
data were not in equivalence. The simulated data standard deviation was 100 times
better than the actual measurements, while the residuals of the modeled system
64


were 1000 times smaller than the actual measurements. These differences are likely
due to unmodeled dynamics in the Superspring and electronic, mechanical and
optical limitations in the FG5. However the closed loop model provides a good
base line for comparison between alternative systems.
With the models developed above, a lag-lead controller was developed and
compared to the current system. This comparison showed that the current
controller exhibited superior characteristics to the lag-lead controller.
The digital simulations showed that both the pole placement and state
feedback work for placing the closed poles. Through simulations with the pole
placement controller it was shown that the closed loop system should have a long
period with very little damping. The high frequency dynamics of the modeled close
loop system were shown to be less critical. The high frequency dynamics of the
actual system needs to be checked since the actual system induces vibrations to the
Superspring when the dropper is active.
The digital controllers both show promise of becoming replacements for
the current controller, but for this to occur, a better closed loop model needs to be
developed. Comparing the two digital controllers, the pole placement would be
much easier to implement, since it does not require observing the internal states of
the Superspring in the controller. To verify the superior control system the
Superspring model parameters would need to be varied to determine which
controller, has the better response. This is needed since each Superspring plant is
65


slightly different and not all the dynamics of the system are included in the model
developed in this thesis.
66


REFEREN CES
Drawing provided by Micro-g Solutions, Inc.
I. D. Landau, System Identification and Control Design, Englewood Cliffs New
Jersey: Prentice-Hall, 1990
K. Ogata, Discrete-Time Control Systems, Englewood Cliffs New Jersey: Prentice-
Hall, 1987
R. L. Rinker, "Super SpringA New Type of Low-Frequency Vibration Isolator",
Ph.D. Thesis, University of Colorado, 1983
P. G. Nelson, An active vibration isolation system for inertial reference and
precision measurements, Review Scientific Instruments, Vol. 62, No. 9
pp. 2069-2075, September 1991
Burr-Brown, Integrated Circuits Data Book, Tucson, Burr-Brown
K. Ogata. Modem Control Engineering 2 Edition. Englewood Cliffs New Jersey:
Prentice-Hall, 1990
J. J. D'Azzo and C. H. Houpis, Linear Control System Analysis and Design:
Conventional and Modem Third Edition, New York: McGraw-Hill, 1988
Straight-line approximation supplied by Dr. David Newell private correspondence.
67