THE EFFECTS OF NOISE ON MEASUREMENTS
MADE IN A DISTRIBUTED PARAMETER SYSTEM
by
Rodney Robert Hersh
,
B. S. E. E.
Pennsylvania State University
(1968)
SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June, 1972
Signature Redacted
Signature of Author
Department of ~EectifAalEngineering, May 12, 1972
Certified by
Redacted
Signature-~-----------
Thesis Supervisor
Accented by
Chairn -n
Signature Redacted
Oep t e-nal
- __ ---- - - -- - - ----
mm tee on Graduate Students
Archives
JUN 27 1972
klRA IE9
THE EFFECTS OF NOISE ON MEASUREMENTS
MADE IN A DISTRIBUTED PARAMETER SYSTEM
by
Rodney Robert Hersh
Submitted to the Department of Electrical Engineering on
May 12, 1972 in partial fulfillment of the requirements for
the Degree of Master of Science.
ABSTRACT
It is the objective of this thesis to explore some of the problems
associated with a distributed system driven by noise. Three main areas
are investigated. First, the correlation of two sensors placed along a
stochastically driven distributed system is considered and an expression
relating correlation coefficient, sensor spacing and sensor position is
found. Second, the noise bandwidth of a distributed system is examined.
Finally, state estimation using the Kalman-Bucy filter and finite-dimensional approximations to the state-equation of a distributed system is
explored. The effects of filter order (modeling accuracy) and multiple
sensors are considered, and a lower bound on estimation accuracy is
found for the single sensor case.
A one-dimensional diffusion equation with two different sets of
boundary conditions is used for the distributed parameter system.
THESIS SUPERVISOR: Leonard A. Gould
TITLE: Professor of Electrical Engineering
-3ACKNOWLEDGEMENTS
The author wishes to thank his thesis advisor, Professor
Leonard A. Gould, for his suggesting and supervising this problem.
His comments and criticisms throughout this work were most useful.
Thanks are also due to the author's colleagues at the Electronic
Systems Laboratory.
In particular, discussions with Dr. Gervasio
Prado and Dr. John Fay were most fruitful, and the computer program
donated by Mr. Michael Warren was invaluable.
The author is also
grateful to the E. S. L. Drafting Department for their reproduction of
the many figures in this work.
Finally, the author wishes to thank Mrs. Rita Baughman for her
patience in typing this manuscript.
This thesis was sponsored in part by the National Science Foundation grant NSF 72917.
-'4TABLE OF CONTENTS
CHAPTER 1
page
INTRODUCTION
1. 1
Motivation for this Study.
. . . . . . . . . . . . . . . . .
5
1.2
1. 3
Background...........
.........................
Outline of Thesis . . . . . . . . . . . . . . . . . . . . . .
6
6
CHAPTER 2
CORRELATION OF SENSORS
2.1
Introduction.
. . . . . . . . . . . . . . .
8
2. 2
2. 3
2. 4
2. 5
2.6
Analysis of Fixed and Free End Systems . . . . . . . . . .
Discussion of Function . . . . . . . . . . . . . . . . . . .
Computation and Results . . . . . . . . . . . . . . . . . .
Interpretation of Results . . . . . . . . . . . . . . . . . .
Summary. . . . . . . . . . . . . . . . . . . . . . . . . . .
8
11
12
CHAPTER 3
3.
3.
3.
3.
1
2
3
4
. . . ..
15
16
BANDWIDTH CONSIDERATIONS
Introduction . . . . .
Analysis of Fixed and
Noise Bandwidth . . .
Summary. . . . . . .
CHAPTER 4
. . ..
. . .
Free
. . .
. . .
. . . . . . .
End Systems
. . . . . . .
. . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
. . . . . . . 17
. . . . . . . 17
. . .. .. 21
. . . . . . . 22
ESTIMATION OF A DISTRIBUTED PARAMETER
SYSTEM
4. 1
Introduction . . . .
4. 2
4. 3
4. 4
Mathematical Preliminaries . . . . . . . . . . . . . . . . .24
Truncation of State . . . . . . . . . . . . . . . . . . . . . 27
Multiple Sensor Estimation . . . . . . . . . . . . . . . . . 29
4. 5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .38
CHAPTER 5
. . . . . . . . . . . . . . . . . . . . . 24
CONCLUSIONS AND RECOMMENDATIONS FOR
FURTHER STUDY . . . . . . . . . . . . . . . . . .
39
APPENDIX A
MODAL ANALYSIS OF FIXED END SYSTEM. . .
40
APPENDIX B
POTTER'S METHOD FOR SOLVING STEADY
STATE RICCATI EQUATION. . . . . . . . . . . . 44
1.0
INTRODUCTION
1. 1
Motivation for this Study:
The purpose of this thesis is to investigate the behavior of sensors
placed in a distributed parameter system and by using the outputs of
these sensors to estimate the state of a distributed system excited by
a stochastic source.
Problems of this type occur in many physical
situations such as the motion of an aircraft wing passing through turbulent air, the temperature distribution in a furnace being heated by
an uncertain source, or the voltage and current distributions along a
power transmission line terminating in a randomly varying load.
Of particular interest in this work is the problem of correlating
measurements made in a one-dimensional diffusion system driven by
white noise at one end.
Physically,
this situation would correspond
to the taking of temperature measurements along a metal bar heated
at one end by a random source.
Also of interest is the accurate esti-
mation of the state of the one-dimensional diffusion system using the
outputs from sensors placed along the system.
The concept of white
noise is used throughout this work, and although white noise is not
physically realizeable it is useful because it represents a "worst case"
phenomenon and because it makes the problems considered here analytically tractable.
academic,
The particular physical example, although somewhat
is adequate for demonstrating the key issues which arise.
-61. 2
Background:
Previous work on the subject of noise in distributed parameter
systems is rather limited.
A digital simulation of a diffusive system
was mdde by Hisiger (4) and the degree of correlation between two
sensor placed along the system was studied.
This study provides a
qualitative feel for sensor correlation.
The analytic description of distributed systems using modal
analysis techniques has been presented by Murray-Lasso (5), Gould
(2),
and Prado (7), while the use of Laplace transformation and fre-
quency domain methods is discussed in Could (2) and Hildebrand (3).
State estimation methods for continuous time systems were developed
by Kalman and Bucy (5) and these methods have been extended to
infinite-dimensional systems by Falb (1).
Estimation of the modes of a distributed system was considered
by Prado (7) but in the context of pole-allocation and three-mode con-
trol of a diffusion system where the uncertaintly was in the initial
conditions and not the input.
1. 3
Outline of Thesis:
This thesis is organized into three main chapters which treat,
first, the degree of correlation between sensors placed along a distributed system, second, the noise bandwidth of a distributed system,
and,
finally state
estimation
of a
distributed
system . In each chapter
the same distributed system described by a diffusion equation was considered,
although two cases with different boundary conditions were studied in the
first two chapters.
-7Chapter 2 develops an analytic expression for the correlation
between two sensors placed along the diffusion system for each set of
boundary conditions.
This expression is then evaluated numerically
and a simple formula relating correlation coefficient,
sensor spacing,
and a measure of sensor position along the system is developed.
In Chapter 3 the bandwidth of the diffusive systems is considered.
The transfer function for these systems is developed and using communication theory techniques a measure of the noise bandwidth is found
giving an indication of the behavior of a diffusion system when driven
by noise.
Chapter 4 is concerned with state estimation of a diffusion system
being driven by white noise.
The trade-offs between modeling accuracy
and number of sensors used are considered.
For the single sensor
case a lower bound on estimation accuracy (for a common comparison
among different order models) is developed.
Appendices are included to derive fully one of the important results in Chapter 2 and to explain an interesting and useful way to solve
the steady state Riccati equations that arise in the design of the Kalman-
Bucy filter.
-82. 0
CORRELATION OF SENSORS
2. 1
Introduction:
In this chapter the question of how the outputs of two sensors in
a noisy distributed parameter system are correlated is considered.
Perfect point sensors were assumed and the distributed system of
interest was described by a diffusion equation with two different sets
of boundary conditions.
Analytical results are derived for the corre-
lation coefficient between two sensors, but numerical techniques were
employed to evaluate the expressions.
2. 2
Analysis of Fixed and Free End Systems:
The two diffusive systems studied are both described by the
normalized diffusion equation
2
_
3ax(zt)
x(zt)
2.1
3Z
at
The systems are driven at the
z = 0 end by a stationary white noise
process, w(t), with zero-means and variance
T~
S
.
Figure 2. 1
shows these two systems with their respective boundary conditions
w(t)
w(t)
z. = 0
x(o,t) = w(t)
z
; x(1,t) = 0
1
z = 0
x(o,t) = w(t)
z= 1
;
iz x(z,t)j
= 0
z=1
x(z,0) = 0
x(z,0) = 0
Fixed End Case
Free End Case
Figure 2.1.
Two Diffusion Systems with Boundary Conditions
-9Using the modal analysis techniques presented by Prado (7) for
the fixed end case, the partial differential equation was represented
by an infinite-dimensional set of ordinary differential equations which
will be refered to as the state equation.
.02O
e
o
o
For the fixed end case
,,,'~
Tr
XW
-7Tr
+
[
J T3
W~f AX W)+
applying the variation of constants formula for semi-groups from Prado
(7), the state equation was solved to give
cktc
(Ym
Hence, the equation for the output of a point sensor, y1 (t), placed at a
point zf along the system is
-10It is obvious that y1(t) is a zero-mean random process because
E
= 0 , and by squaring yi (t), taking expectations,
W(t)j
and evaluating
integrals the variance of the steady state output is found to be
A complete derivation of Eq. 2. 7 is found in Appendix A.
Using exactly the same techniques, the variance of the output of
a second sensor, y2, placed at the point z , is
2
4T2v
I
The cross-correlation between the two sensors,
E
y, (y) y(03)
for the steady state case is found to be
Then from the definition of the correlation coefficient
[E { y }- E f Ya-
}'A
It is worth noting that if a non-normalized diffusion equation
C
E fYi
were assumed, the expressions for
, and
E
Y,% ya
would each be divided by
G<
FiIY,
and
hence the correlation coefficient would remain the same as in the
normalized case.
-11In the free end case a different set of eigenfuctions must be used
in the modal analysis giving slightly different results for the variance
The steady state variance of the first sensor yj
is
(
The corresponding expressions for
Y.10
L
,
T
E Ky, yz
}
,
of the sensor outputs.
and e follow easily.
2. 3
Discussion of Doubly-Infinite Series
At this point some comments on the doubly-infinite series
51mrAv7Tj
WMv\
SjvA-n2
are in order.
finite for all values of z in the interval
2 + 0
.
(o , I
This funct ion is
, but diver ges as
To observe this behavior consider Figure 2. 2 which 3hows
the sign behavior of the arguement of the series as a function of the
indices m and n for a constant value of z.
A
-
d
C
0
Fig. 2. 2
+
+
M
L~1.
a
b
e
Sign of the Series Argument
-12a
The points
and b occur when
VIT\
Z
and
respectively, giving 6..
equals
.
Qr
and
IT
The points
c and
d
are determined similarly.
Now, when summing over a finite number of indices, the series
should be summed over an integer number of cycles of
'WTT Z
SIVN
because the doubly-infinite series is oscillatory about its actual value
as index of summation increases.
This means that the function should be
summed to either n = b and m = d, or to the end of the next cycle
n=e and m=f.
where
For example, let z=0. 20.
a and b will correspond to n=5 and n=10
Then the points
respectively.
the series should be summed over an equal multiple of
Thus,
10 indices
along each axis.
As
z
goes to zero, the first sign change points
infinity leaving only positive terms.
5 1n ;Z
)I
a and c go to
The main diagonal terms are then
- CoSaln 7r2)which clearly diverges when summed
from m=1 to infinity.
The divergence of the series as
z
goes to zero should be no
surprise because the system is being driven by a white noise process
which has infinite variance.
All the preceding analysis in this section also holds for the free
end case where the doubly-infinite series is slightly different.
2. 4
Computation and Results:
To examine the degree of correlation between two sensors as a
function of their location,
some measure of their spacing and position
1.0
b
0.9
q
0
*
0.5
0.6
I
I.-'
I
K.
0.7
Ir
I
0
.08
I
.16
I
.24
I
.32
I
40
I
.48
I
.56
AZ about a point Zc
Fig. 2.3
I f1m
,
p
End
Correlation between Sensors Equi-spaced about Z - Fixed
F
i
.64
I
.72
1.12
NV
N0
0.9-
N0
0.8P
0.6
-0
0.5-
0)
.08
.A6
.24
.32
.40
.48
.56
.64
AZ about a point ZC
Fig. 2.4 Correlation between Sensors Equally Spaced about Z - Free End
.72
-15along the distributed system must be devised.
was used.
I,
ZC ,
Zj .
Let z 1
The following method
and z2 denote the position of two observers,
Then A 7 , the spacing between them is 2,- ZI , and
their midpoint
is (Z, -2,+2
.
Numerically evaluating the
correlation coefficient for increasing values of A Z about a center
point zc yields the curves shown in Figure 2. 3 for the fixed end case
and those shown in Figure 2. 4 for the free end case.
Figure 2. 3 and 2. 4 reinforce the intuitive feeling that for a given
spacing between sensors,
their outputs will be more correlated as
they are placed farther from the driving noise because of the rapid
attenuation of inputs to a diffusive system as one moves into the system.
The similarity of the two figures also indicates the relatively small
effect that different boundary conditions have on the diffusive system.
2. 5
Interpretation of Results:
Observation of Figures 2. 3 and 2. 4 reveals that over the interval
0. 5
F < 0. 9 the correlation curves can be approximated relatively
well by straight lines.
on Figure 2. 4.
Three straight line approximations are shown
Furthermore, when these straight lines are extended
to the vertical axis they intersect in a common point, e =1. 12.
In
addition, if the slope of these straight line approximations versus the
center point zc is plotted on a log-log scale, a straight line relationship is obtained.
The exact relationship for the free end case is
Combining this expression with the slightly different one for the fixed
-16-
sensor spacing AZ , and sensor midpoint
~
/~
52
-
,
end case, the following relationship among correlation coefficient
zc is obtained.
,3 O54O. Q413
While this expression is an approximation for both the fixed and
free end cases of the diffusive system, it is accurate to within 5% over
This means that for a fixed value of g
,
the specified range for '.
the sensor spacing is directly proportional to the center point zce
Furthermore, for ? = 0. 6 the sensor spacing A 2 equals the midpoint
zc.
The linear relationship between L. and Z c indicates that as
zc increases, i. e., the sensors are placed farther from the random
driving noise, that their outputs become more linearly dependent.
2. 6
Summary of Results:
In this section the problem of relating sensor correlation to
sensor placement in a distributed parameter system driven by white
noise has been solved.
It has been demonstrated that sensor correlation
is relatively independent of the boundary conditions for a diffusive
system.
The results support intuitive feelings about distributed systems
and provide quantitative relations for the qualitative results shown by
Hisiger (4).
Since the concept of white noise was used exclusively, the results
of this section provide a lower bound on the correlation of two sensors
(if a deterministic input were present the outputs of two observers
would be perfectly correlated, (p = 1).
-17-
3. 0
BANDWIDTH CONSIDERATIONS
3. 1
Introduction:
This chapter is concerned with the bandwidth of a diffusive distributed parameter system.
The bandwidth of a distributed system is
of interest because it will affect the propagation of noise through the
system and reveal some interesting facts about sensor placement in a
diffusive system.
Analytical expressions for the transfer functions for
both the fixed end and free end cases described in Chapter 2 are derived
and communication theory techniques are employed to obtain a measure
of the bandwidth of these systems.
Numerical techniques were again
necessary to evaluate the analytic results.
3. 2
Analysis of Fixed and Free End Systems:
To obtain a measure of the bandwidth of the diffusive systems
being considered the transfer functions for these systems were derived
using Laplace transformation techniques.
The application of Laplace
transforms to partial differential equations is discussed by both Could
(2) and Hildebrand (3).
Starting with the diffusion equation and boundary conditions for
the fixed end case the Laplace transform is taken,
55Xz)
-
s
0) X (SX
X(zo
(Zi0 =s0
391
3
-18Selecting a general solution for this equation and applying boundary
conditions yields,
v
)
1
X(Z s)= C' e Z
e
-'"Z
3.3
-5A(
e
e
Ce
-e
Hence the transfer function is
C
X (Zs>-
H(z s)
- e
3.
- e
ef
Because the diffusion equation is stable,
the real part of
s
may be
set equal to zero to obtain the frequency response,
- e-VF(I -7)
-7)
I-i (z, ~
e- Vj
Some limiting cases of interest are
as
I-z
W -3.(
H -+
H
,
0
I
-
Of0
H
For the free end case the frequency response was found to be
H (z,j
F I--Z)
Z)
+
e
-
3-L
6
with the limiting cases
-F+
e-v,
H-+ I
as
z -+o
H
I
f e -V
a
1 0
.
r
x=0.1
0.9o
o..0
0*1
0.2
0.70-
IHI 0.5
I
0.3
o -
-
0
m-W...- 0--
*-
Ib
*
0
0.5
-o.
0
" k a0
0
N0ft
0
I
0
a
0.7
0.3
0.1
0*N
ON
~0
I.
0.9
0I
0.1
on-o
m0mw
o
I
1.0
10.0
O0
*%eN
4
h
100.0
FREQUENCY (rad/sec)
Fig. 3.1 Magnitude Transfer Function vs. Frequency for D.P.S. Input One End, Other End Fixed
1.0
0.8
x.0.1
-
-0A
I
0.41-
0.3N
0.4
0
I
010
0.2
-
H|
0 -6
0.8
V
1v
0.
0. 1
I
1.0
%1
0*
I
10 .0
100.0
FREQUENCY (rad/sec)
Fig. 3.2
Magnitude of Transfer Function vs. Frequency for D.P.S. Input One End, Other End Free
-21The frequency response of the fixed end case is shown in
Figure 3. 1 with that for the free end case shown in Figure 3. 2.
Both
are shown for several values of z.
3. 3
Noise Bandwidth:
In communications problems it is often of interest to know the
mean-square value of the output of a network driven by noise.
This
desire has led to the noise-equivalent bandwidth defined as the bandwidth of an ideal rectangular pass network having the same mean-square
output as the actual system.
_H
Mathematically,
(:3wj)j
AL.W
3.7
In distributed systems one is also concerned with how uncertainty
propagates in a system and hence how uncertainty will affect measurements of the system.
However, the noise-equivalent bandwidth concept
that is useful for lumped systems must be modified for use with distributed systems.
The reason for this necessary modification is that the
frequency response is a function of both frequency and distance.
diffusion system being considered here when z
fixed case
Since Hma
approaches
1
In the
in the
H goes to 0, as does the intergral in the formula for Wn
also goes to zero, by
and in this case Wn
L 'Hopital's rule Wn may be defined,
indeed does approach a non-zero limit as was
shown by numerical evaluation of Wn , a result difficult to interpret
-22physically.
Therefore,
the measure of bandwidth used in this work
was the integral of the magnitude squared of the frequency response.
00
IF(z U)i 01 W 3.
E3W (Z)
H
It is easy to show that for high frequencies the magnitude of H decays
exponentially with
W
and hence the above integral exists.
A
numerical integration scheme using Simpson's rule was employed to
evaluate the bandwidth integral and the bandwidth versus distance is
shown in Figure 3. 3 for both the fixed and free end cases.
indicates that higher frequencies,
This figure
and hence higher modes of the sys-
tem, are attenuated rapidly along the system.
3. 4
Summary:
From the work in this section a qualitative feel for the bandwidth
of a diffusive system has been developed.
An important inference from
this work is the location of sensors used to estimate the state of a system.
Because of the rapid attenuation of higher modes with increasing distance
from the driven end, one would expect to place sensors nearer the driven
than the fixed or free end.
Furthermore,
one would expect the "optimal"
sensor location to move closer to the driven end as an estimation of more
modes is attempted.
This qualitative feel for sensor placement will be
confirmed in the next chapter.
-239590-
80-
70-
60-
z50I-
Z
40-
30-
20-
10
FREE END
FIXED END
0.
0
0
0.2
0.4
0
0.6
0.8
z
Fig. 3.3
Bandwidth vs. Distance
1.0
-
-2h.
4.0
ESTIMATION OF A DISTRIBUTED PARAMETER SYSTEM
4. 1
Introduction:
Chapter 4 is concerned with the problems of state estimation
in distributed parameter systems.
Estimation is of interest because
it is a necessary prerequisite for the control of a system.
The appli-
cation of modern estimation theory to distributed systems creates
problems not normally encountered when working with finite dimensional systems,
such as truncation of state (accuracy of model) and sensor
placement along the system.
Both of these problems are considered
here along with multiple sensor estimation of a diffusive system.
For
the single sensor case a lower bound on the estimation error is developed.
4. 2
Mathematical Preliminaries:
A necessary requirement for estimation of any system is that
it be observable.
Strict conditions for observability of distributed
systems are given by Prado (7) which for the problems being considered
here reduce to the condition that for a system to be observable a sensor
must not be placed at the zero-crossing point of any mode being considered.
For a three mode estimator of a diffusive system of unit length
this means that sensors may not be placed at z = . 25, . 333, . 50, . 667,
or
. 75.
In general,
all zero crossings.
sensors must be placed at irrational points to avoid
This placement of observers is intuitively correct
-25because no information about a mode is gained if it is always zero at
the sensor point.
Because of the finite dimensional limitation of numerical tech-
niques, the infinite dimensional state equation of a distributed system
must be truncated for any practical considerations.
As has been seen
in Chapter 3, the higher modes of a diffusive system are likely candidates
for truncation since they are attenuated rapidly in the system.
However,
when studying estimation schemes of various orders, a common basis
for comparison is needed.
The trace of the error covariance matrix
provides a measure of the magnitude of the errors in estimating the state
and will be used in this chapter.
Furthermore, because different numbers
of modes are estimated by different order filters, the sum of the estimation errors of the first three modes will provide a common basis for
comparing estimators.
For the remainder of this chapter this criterion
r
will be denoted
.
This choice is also motivated by the fact
that three modes are often observed when trying to estimate the behavior
of a distributed system.
The K.alman-Bucy filter (5) was used for the estimation because it
provides a minimum mean-square estimate of the state.
[()
interest here is the error covariance
Of particular
which is propogated by the
Riccati equation
(t) = A
(t) +
(k)
+ BQB' - $_(t)
M (z) R
A and B from state equation
M (z)
Q
observation matrix from
Driving noise covariance
2. 2
2.
6
M (z)
(t)
4.1
-26-
C(t) Observation Noise
SYSTEM
w M)
Driving-
Noise
t)
~Vz
x
A
+
7, M'(z)_R
A
+f
M~z)
E {w(t)} = 0, E jw(t)w (t +r)
E {g(t)} = Q
Fig. 4.1
7-
Q8(r)
E{
f(t)C'(t+r)}
R8(r)
Structure of System and Kalman-Bucy Filter
R
Sensor noise covariance
N
Order of system, i. e., A is N X N matrix
To be considered here is the steady state case of the Riccati
equation,
(t) = 0.
The existence of the steady state solution is
guaranteed by the controllability and observability of the system.
Potter's method (8), an algebraic routine, was employed to solve the
steady state Riccati equation.
A description and derivation of Potter's
method appears in Appendix B.
Figure 4. 1 shows the structure of the
system and the Kalman-Bucy filter.
4. 3
Truncation of State:
To determine the sensor placement that minimizes the mean-
square estimation error, a sensor was moved along the fixed end diffusion
system, thereby changing the observation matrix M(z) and the Riccati
equation for the error covariance was solved to yield the steady state
estimation error.
v ~ 3
A plot of steady state error for the first three modes,
, versus sensor placement for six different order estimators
is shown in Figure 4. 2.
These results reinforce the intuitive feel that
a higher order Kalman-Bucy filter (better model of the system) will yield
a better estimate than a lower order filter.
However, with a higher order
filter the best estimate is much more sensitive to small deviations in
position about the optimal point.
Of considerable interest is the change in optimal sensor location as
the order of the filter varies.
Plotting the optimal sensor location versus
the order of the Kalman-Bucy filter on log-log paper yields a straight line
-28-
3000
-
25
3rdORDER
20t
4h.
69
o
0
M,15 -N%
10 -
4:
R ='.0
12th
5-
01
0
.05
.10
.15
.20
.25
.30
.35
Z DISTANCE ALONE SYSTEM
Fig. 4.2
Estimation Error vs. Sensor Placement for Various Order Filters
-29relationship as is shown in Figure 4. 3.
zopt = 5.
N-0.855
Analytically,
36 N6 12.
,
4. 2
Strictly this equation is valid only over the interval stated but the author
sees no reason why it shouldn't hold for all N ', 3.
Vr
Figure 4. 4 shows
23
of the best estimate for each order fil Ler
versus the sensor location zopt where that estimate was made.
A
definite relationship between the error of the best estimate and sensor p osition appears; giving
+f
Substituting Eqn.
= 3 0 zolo
3
4. 3
4. 2 into 4. 3 yields
IrI
Og_.
+ 3_: = /400 N - 0,QQN
4. 4
This expression gives a lower bound on the estimation error of the first
three modes as a fuction of the order of the filter
0.l
+V
2
4 00 N
3
'
4.5
the equality holding only when the sensor is placed at the optimum point
given by Eqn. 4. 2.
Again one should note that this lower bound has been
developed only for
34 N
4. 4
12.
Multiple Sensor Estimation:
From the previous section it can be seen that the system of a distribu-
ted system driven by a stochastic source can be estimated to a very high
degree of accuracy provided a large enough model of the system is used in the
-30-0
1.0;
0
4-
0.
0
0\\\
z
w
'M.10
w
Zp>.0
0
z
w
(I)
.5
zop =5.0 N
CL
0
I
.01
I
2
1
I
I
I
4
I
8
(ORDER OF FILTER) N
Fig. 4.3
Optimum Sensor Placement vs. Order of Filter
12
D
00
-
15
-
-
20
trls =8
0
pt
*
5 -
0
.05
.10
.20
.15
.25
z
Fig. 4.4
Optimal Estimate Error vs. Sensor Position
.30
4-32-
estimator.
However,
it is usually not practical to work with systems
of large dimension giving rise to the question:
Is there a point of
Figure 4. 5 shows
diminishing returns in the modeling of the system?
the error of the best estimate of the first three modes versus the dimension
of the filter used to make the estimate.
around
N = 8
A "knee" in the curve appears
indicating that the per unit gain in the estimation accuracy
for increases in filter order is steadily diminishing.
further work using more than one sensor the value
the dimension of the filter.
Therefore, for
N=8
was chosen for
It should also be noted that the computation
time needed to solve the algebraic Riccati equation is proportional to
N
3
, thus there results a savings of one-half in the time needed to solve
an 8th order Riccati equation as compared to a 10th order one.
When using more than one sensor to estimate the state, the obvious
problem of sensor location arrises.
Since this problem for the single
sensor case has been solved, a reasonable extension is to place the first
sensor at the position deemed optimal in the previous section and vary
the second along the system until a minimum of fY ]I 3
is obtained.
For the eighth order Kalman-Bucy filter being considered,
point for the first sensor is zj = 0. 08485.
the optimal
Figure 4. 6 shows the effect
of using two sensors to estimate the state of the diffusive system.
A
definite improvement in the estimation error of the first three modes
occurs when two sensor are used as can be seen by comparing the two
curves in Figure 4. 6.
-33-
20-
16-
12-
K
0
8
4-
01
0
4
12
8
N
Fig. 4.5
Optimum Estimate vs. Filter Dimension
16
a
a
asa
0
a
25-
20Q
= 10.0
R =1.0
15ro
NI
100
a -ONE
5 -
0
0
SENSOR
a -TWO SENSORS,ONE
FIXED AT Z =.08485
.05
Fig. 4.6
.10
.15
Z
.20
.25
.30
Comparison of Estimate Error Using One and Two Sensors
.35
However, the most interesting facet of the two sensor case is
that both sensors should be placed at the same point,
z1 = z 2 = 0. 08485.
This phenomenon is explained by noting that the best sensor location
maximizes the signal-to-noise ratio for all the modes being considered.
Because the two sensors were assumed independent of each other, their
individual errors can be "averaged out" by the Kalman-Bucy filter.
Hence, intuitively one might suspect that each sensor should be placed
at the point of maximum signal-to-noise ratio.
Furthermore, when more
than two sensors are used, the same result still appears, namely that
the optimal location for all observers in a diffusive system remains at
the maximum signal-to-noise ratio point.
When using M sensors (M> 2) in the estimation process the following scheme was used for sensor placement.
The first M-1
sensors
were placed at the optimal point and the Mth one moved along the system
to find a minimum for
'r 23
.
In each case the optimal point for all
M sensors was the same, 0.-08485, for the eighth order filter.
Estimation of the fixed end diffusion system with multiple sensors
yields the plot of estimation error versus the number of sensors shown
in Figure 4. 7.
Both the total error in estimating all eight modes, +r 2.
and in estimating only the first three modes,
Y
3
, are shown.
While increases in accuracy for all eight modes occur, a much larger
increase in estimation accuracy is evident for the first three modes.
Also, a definite "knee" appears in both curves indicating that a point of
-36-
40 F
35 h
10
0
HI 30 F
a)
8
FO
0
C.)
0
0
251-
6
NA
20
-
4
0
15
2
-
0
)
0
iI
I
Fig. 4.7
Ii
2
Ii
Ii
4
3
NUMBER OF SENSORS
Estimation Error vs. Number of Sensors
I
5
0
-37diminishing returns for the addition of more sensors.
For an accurate
comparison of the multisensor cases Table 4. 1 gives the optimum
error covariance of the first three modes as a function of the number of
sensors.
Number of Sensors
2
1.429
0.752
0.547
4
0.423
V- 222
2. 441
1. 607
1. 270
1. 046
0. 932
332
3.590
2.637
2.177
1.855
1.677
11
Estimate
Error
3
Table 4. 1.
5
0.368
Estimate Error of First Three Modes
By observing Fig. 4. 7 and Table 4. 1, one can conclude that by
using three independent sensors the "knee" of the curve is reached.
Hence, any further addition of sensors buys little more as far as increases in estimation accuracy are concerned.
To determine if the optimal sensor location were affected by the
strength of the driving noise, the estimation process was rerun using
larger driving noise covariances,
previous work where Q equaled 10.
remained fixed at 1. 0.
Q, of 20 and 50 as compared to the
The observation error covariance
No change in optimal sensor placement occured
for either the one or two sensor cases indicating that sensor placement
is independent of the driving noise strength.
-384. 5
Summary:
In this chapter a study of estimation of a diffusion system has
been presented.
The consideration of model accuracy reaffirmed the
fact that better models make better estimators but also revealed the
fact that there is a point of diminishing returns beyond which large
increases in the state of the model used in the estimator are necessary
to produce modest increases in estimation accuracy.
If the performance
criterion is an accurate estimate of the first three modes of the system,
a reasonable choice since the first three modes cover a 10 to 1 frequency
range, this point of diminishing returns occurs when a finite dimensional
system of order eight is used to approximate the diffusion system.
Further-
more, a lower bound on estimation error with the order of the filter as a
parameter was developed.
This bound is attainable only if the single sensor
is placed correctly.
For estimation with multiple sensors again one is faced with a
diminishing returns situation.
While the accuracy of estimation can be
improved with the addition of more sensors at the optimal point,
for the
performance criterion considered here the use of more than three sensors
gains little.
-.395. 0
CONCLUSIONS AND RECOMMENDATION FOR FURTHER STUDY
It has been the goal of this thesis to explore the problem of
measuring the behavior of a distributed parameter system driven by
a stochastic input.
While the limited case of a one-dimensional
diffusion equation was considered, the results shed light on the problems
peculiar to distributed systems.
Chapters 2 and 3 provide insight into
the propogation of uncertainty in a diffusion system and develop a relationship among the variables of sensor correlation, sensor placement
and sensor spacing.
The work in these chapters also shows the relative
independence of the diffusion equation to different boundary conditions.
Chapter 4 considered the problems of state estimation of a diffusive
system using a finite-dimensional model.
sensor location,
The questions of model size,
and multiple sensor estimation were explored.
Much work remains in the area of estimation of distributed systems.
Obvious extensions of this work would be the estimation of systems described by hyperbolic or elliptic partial differential equations.
Another
area open for study is that of the trade-offs between the dimension of the
filter used and the number of sensors used.
A better knowledge of esti-
mating distributed systems will make the job of control much simpler.
-440APPENDIX
A
Consider the following diffusion equation with its related boundary
conditions,
S(Z,,
~)
ETt~L
A.1
YL(z , o)
IL(I,
L)
)
t( =
w
).
where w(t) is a white stochastic process with mean zero and variance
U -'-)
)
.
Because both ends of the system are zero, an
orthonormal set of eigenfunctions for this system is
Then
i
xh Ui)
i0
2(Z) f)
J
and
I
( (Z) = F
Y(Z)
YD
SA7nT
A9
I Y -
A .3
integrating by parts
I
.
T
262.
51 ',Y
aZ ALZ
I
But upon application of the boundary conditions the first term on the
AAl
-):1right-hand side is zero.
Integrating by parts again
I
I'
Fa s; vt h -rz dz
Y (Z
=-
t)
a 1T C 0 S
- -Z
J
I
,
'3
Jr
P
1 2 -rid, X(Zj)
Observing that the last integral is
A5
S;V% YOTZ A 7.
- V2TT n
, and applying
)
f
boundary conditions gives
~V(Z)v
f-I
= A2 7T)-
flt)L
TZC
cZ aY
M\ + V7hi-T
wU-m.
Combining all the xn's into an infinite dimensional vector yields the
equation
0
X(
)
d
0
-14 -1
0
0
9..
J
F
It
.4
2
3
j
A.7
From the variation of constants formula for semi-groups presented by
Prado (7),
(i),Z
S (VO(t Xo
wcwv
+f
0
-42where
F-rr
e
S
(vi')
5
U~z
e
a' Tir
,J--4)'9
0
-4-PaL
(n')
IQ~Jrj
-\
X
Since
1
VT
(0)
V-ITT
0
4 na -v)
C
2N Tr
Wv&)
0
017-
I
e
rd W (T) cOr
e-s)
26
With a point sensor located at
o, as;n ITZ, I
M
z
/~(z.)
y 1 (Q)=
W
r-T
A. IO
(q)d C
the output
1
XW
OF, s ;V%
y 1 (t) is given by
)
FTf0
- --1
z,
A.11
Combining M (zl) with the above expression for the state gives
**O
S; VIA -Mz I
E I
y,
(,)}
=
T2
-)V
Y) n W(7-) O(q-.
0
,
Clearly
Te
Y1
(~)z.
'~J
cc:
tooI
-
w () O V I T:)
0
Sf4Vi
rI (t~)
r
I
t
- Y1 2 Tr
- (f -7)
A, 13
-43Then the variance of y 1 (t),
value of yi
2
t>O, may be found by taking the expected
(t)
El: Y, (0)1
n Si M~
z, S;VV1
r j!
7~r~
Wi Y)$~,v11rZ
f0
7r
- -0 d 7- Cti)..
T2 X (V-
A. Iq
Evaluating the integrals and cancelling out terms leaves
00
i
.
=
YA nSn
2
MTr, s
-(d+ Vna T M
C
Y%
TrI7
Ii
[
I
A.15~
Ignoring the transient term reduces the expected value of yj 2 to
y121
sYAiw
OI
LH,
a
7rZ, S'V n TrZ,
%2
- A.16
By the same method the cross-correlation between two sensors
located at
z1 and
z2
is found to be
M Trz
Z
hI P1 S;In
V" 1 +
=kY
.51i, qVT Z
A, 1
The correlation coefficient is then found using the definition
(E
-Y
I
,
E:=
yV
yY1jF
A4Ia
-144APPENDIX B
In finding the steady state estimation error using the Kalman
filter it is necessary to solve the following algebraic Riccati equation
=
('M R M Z)2
8' - I 1Q
A 2 + 2 A' f-
0
13.
The algebraic method of solution outlined here is due to Potter (8).
For this solution to be valid the following conditions must be met,
[A,8]])
Cofri'oila6e
t) (z M)] 06seyvcxle
Q>0
3)
t)
R >o.
Assuming that the state is an n-dimensional vector, the
dv'%x2n
matrix
v =~.
V is formed
~Z) R- M(-Z)
-A'
First it will be shown that if
X
-A
is an eigenvalue
This condition is known as quadrilateral symmetry.
of V so is
-
A
Consider the
%X 2n vimatrix
~.3
Q
T
Observe that
.
TV J
11
Then by direct multiplication
3
-M45Multiplying each side by J again gives
e
= a
o V
13.5-
'Te
If XA is an eigenvalue of V with an associated eigenvector
e
V
showing that
-
=
-
is an eigenvalue of V
Since V and V
e
Te
with eigenvector Je
have the same eigenvalues,
-A
is therefore also
an eigenvalue of V . At this point the conditions of controllability
and obserability guarantee that no eigenvalues will have zero real
parts.
Now consider the n eigenvalues of V with positive real parts
Arranging the eigenvectors in column form yields a
which can be partitioned into two Vxvn matrices
Q, ,a
,
with their respective eigenvectors
X VA
X andY
matrix
by
X
Notice that the following equation holds
where
A-
is the
wVxn
Jordan canonical matrix of the eigenvalues
with positive real parts.
It is now the claim that the solution to the Algebraic Riccati equation
can be written
=
xY-1
B.
-W46The proof follows easily.
From Equation B. 8
AX + C Y
DX
-
XJ\
A-"Y
/0.
Yr _
Postmultiplying both of these equations by Y
(Y
can be shown
to be invertible) and premultiplying Equation B. 11 by XY 1
ANY-' +C =
A
_
X Y'DX)Y' -XY'-1A I
Subtracting Equation B. 13 from B. 12
XY_'
AXWFurthermore,
least
C
or
D
-
4
for C
and D
non-singular,
gives
91
Xi2Y
A)/1
leaves
XY-1D XyYc
0.
semidefinite and symmetric with at
it can be shown that
,
the solu-
tion to B. 1 , is positive definite.
In the actual algorithm, first the
Qyt X%
matrix
and then its eigenvectors and eigenvalues are found.
V is formed
Next the eigenvectors
corresponding to eigenvalues with positive real parts are placed in a
SA X
Y X V
V1
matrix.
matricies.
This matrix is then partitioned horizontally into two
The lower matrix
tiplied by the upper matrix
to the
X
to give
algebraic Riccati equation.
Y
is then inverted and premul,
the steady state solution
-147REFERENCES
1.
Falb, P. L., "Infinite-Dimensional Filtering: The Kalman-Bucy
Filter in Hilbert Space", Information and Control, 11, 1967.
2.
Gould, L. A., Chemical Process Control Theory and Applications,
Addison-Wesley, Reading, Mass., 1969.
3.
Hildebrand, F. B., Advanced Calculus for Applications, PrenticeHall, Engelwood Cliffs, N. J., 1962.
4.
Hisiger, R. S.,
"An Investigation into a Measurement tJncertainty
Principle for a Distributed Parameter System Driven by Noise",
S. M. Thesis, M. I. T., May 1971.
5.
Kalman, R. and Bucy, R., "New Results in Linear Filtering and
Prediction Theory", Trans. ASME, ser. D, Vol. 83, 1961.
6.
Murray-Lasso, "The Modal Analysis and Control of Distributed
Parameter Systems", Ph. D. Thesis, M. I. T., 1965.
7.
Prado, G., "Observability, Estimation, and Control of Distributed
Parameter Systems", Ph. D. Thesis, M. I. T., Aug. 1971.
8.
Potter, J. E., "Matrix Quadratic Solutions", SIAM Journal of Applied
Mathematics, Vol. 14, No. 3, May 1966.