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 Â£h................47

viii

Figure 5.2-17 Residual plot for pole placement varying Â£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 Â£, 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

Â£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, = .518Â£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 + 2Â£y + to2)

Where CD's are in (radians/sec) and Â£'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 -Â£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)

Â£

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)

-2Â£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.

Â£=0.2 -0.0054 0.1783

Â£=0.5 -0.0127 0.4455

Â£=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 Â£

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