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NAVAL POSTGRADUATE SCHOOLMonterey, California

00

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THESISDEVELOPMENT OF REAL-TIME ERROR ELLIPSES

AS AN INDICATOR OF KAL"IANFILTER PERFORMANCE

by

CD2 Joseph JarosC.D)

March 1984

,__ Thesis Advisor: A. Gerba, Jr.

Approved for public release, distribution unlimited

8A " ', 0,12

SECURITY CLASSIFICATtON OF THIS PAGE (fhen Des Entered)

READ INStRUCTIONSREPORT DOCMENTATION PAGE BEFORE COMPLETING FORMI. REPORT NUME 2. GOVT ACCESSION NO: 3. RECIPIENT'S CATALOG NUMBER

4. TITLE (nd Subtitle) S. TYPE OF REPORT & PERIOD COVERED

Development of Real-Time Error Master's Thesis;Ellipses as an Indicator of Kalman March 1984Filter Performance 6. PERFORMING ORG. REPORT NUMBER

* AUTNOR(d) S. CONTRACT OR GRANT NUEBER(s)

Joseph Jaros

S. PER1FONMING ORGANIZATIOm NAME AND ADDRESS I*. PROGRAM ELEMENT. PROJECT, TASKAREA & WOAK UNIT NUMBERS

Naval Postgraduate SchoolMonterey, California 93943

IS. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

Naval Postgraduate School March 1984Monterey, California 93943 13. NUMBER OF PAGES

11914. MONITORING AGENCY NAME G AOOttESS(I* dillerent trai Controlling Office) IS. SECURITY CLASS. (of this report)

UNCLASSIFIED

D5E. DECLASSIFICATION, OOWNGRADING 0SCHEDULE

14. OISTRIGUTION STATIMENT (of this Repo")

Approved for public release, distribution unlimited

17. DISTMIOUTION STATEMEN (of te abstrato, o med in Sock 20. It dlliferen from Report)

IS. SUPP.LEETARY NOTES

19. KEY WORDS (Cmntine on reveo ole, It ndeeaey md Identil by block number)

Error Ellipsoids; Kalman Filter; Extended Kalman Filter

20. ABSTRACT (Codtwe an puweus aide 1 necesary mnd Id"tty, by block member)An error ellipse plotting routine was developed to provide real-

time indication of Kalman filter performance. The study includedan evaluation of the Hewlett-Packard HP-88 computer system'scapability for providing real-time tracking information and anevaluation of the computer's possible use on the three-dimensionalunderwater tracking range at the Naval Underwater WeaponsEngineering Station, Keyport, Washington. A series of tracking

DO I , ,, ,III 1473 coTON OF I NOV 66 IOBSOLETE 1S'N 0102. LF 0146601 SECURITY CLASSIFICATION Of THIS PAGE (When, Data Enhoroc

SECURITY CLASSIVlCATIOM OF THIS PAGE (Whm Dma EXeCOM0

runs were used to demonstrate both linear and extended Kalmanfiltering. Information obtained from the error ellipses wasused to modify filter parameters for improved filter performance.It was found that the error ellipse was useful as a tool forindicating filter performance and for making decisions regardingfilter parameter modification. The HP-86 provided accurate,reliable results and it could be used for on-line graphics.However, the computing speed of the HP-86 computer as used inthis study was too slow for on-line processing of the three-dimensional tracking problem.

Accession ForNTIS GI.&T

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of

S N 0102-LF-014-6601 2

SECURITY CLASSIFICATION OF THIS PAGtU a'l Daeft Entere)

Approved for public release, distribution unlimited

Development of Real-Time Error Ellipsesas an Indicator of Kalman Filter Performance

by

Joseph JarosCommander, United States Navy

B.S., University of Texas, 1967

Submitted in partial fulfillment of therequirements for the degree of

MASTER OF SCIENCE IN ELECTRICAL ENGINEERING

from the

NAVAL POSTGRADUATE SCHOOLMarch 1984

Author:

Approved by:Thesis Advisor

'0'0 *t" a-"m... Second Reader

Chairman, Depa mentof Electrical and Computer Engineering

I# dan of Science and Engineering

ABSTRACT

An error ellipse plotting routine was developed to

provide real-time indication of Kalman filter performance.

The study included an evaluation of the Hewlett-Packard HP-86

computer system's capability for providing real-time

tracking information and an evaluation of the computer's

possible use on the three-dimensional underwater tracking

range at the Naval Underwater Weapons Engineering Station,

Keyport, Washington. A series of tracking runs were used

to demonstrate both linear and extended Kalman filtering.

Information obtained from the error ellipses was used to

modify filter parameters for improved filter performance.

It was found that the error ellipse was useful as a tool

for indicating filter performance and for making decisions

regarding filter parameter modification. The HP-86 provided

accurate, reliable results and it could be used for on-line

graphics. However, the computing speed fo the HP-86 computer

as used in this study was too slow for on-line processing

of the three-dimensional tracking problem.

I4

TABLE OF CONTENTS

I. INTRODUCTION- ------------------------------------- 7

II. KALMAN FILTER THEORY----------------------------- 9

A. LINEAR MATHEMATICAL MODEL -------------------- 9

1. The Plant -------------------------------- 9

2. Noise Processes -------------------------- -0

3. Initial State Description ----------------- 11

B. DISCRETE TIME ESTIMATION --------------------- 12

1. The Estimator Equations ------------------ 12

2. Gain and Covariance Equations ------------ 14

C. NONLINEAR ESTIMATION - EXTENDED KALMANFILTER --------------------------------------- 17

1. Nonlinear Model -------------------------- 17

2. Extended Kalman Filter Equations --------- 17

III. ERROR ELLIPSOIDS --------------------------------- 20

A. THEORY --------------------------------------- 20

B. ERROR ELLIPSOIDS AND FILTER DIVERGENCE ------- 25

IV. PROBLEM DEFINITION ------------------------------- 27

A. PROBLEM DESIGN ------------------------------- 27

1. Linear Tracking -------------------------- 27

2. Nonlinear Tracking ----------------------- 32

B. COMPUTER SIMULATION -------------------------- 34

1. Computer Hardware/Software --------------- 34

2. Track Generation ------------------------- 35

3. Noise Generation ------------------------- 35

4. Gating Scheme ---------------------------- 36

5

5. Collection of Statistics- -- -- -- -- -- -- -- -- -- ----

V. TARGET TRACKING AND ERROR ELLIPSE ANALYSIS ------- 38

A. LINEAR 77RACKING--------------------------------- 38

B. NONLINEAR TRACKING----------------------------- 42

VI. CONCLUSIONS AND RECOMMENDATIONS--------------------8al

A. ERROR ELLIPSE----------------------------------- 81

B. COMPUTER PERFORMANCE--------------------------- 82

APPENDIX A - TRACK GENERATION---------------------------- 85

APPENDIX B - COMPUTER PROGRAM EXPLANATION---------------- 94

APPENDIX C - PROGRAM LISTING------------------------------ 98

LIST OF REFERENCES--------------------------------------- 117

BIBLIOGRAPHY---------------------------------------------- 118

INITIAL DISTRIBUTION LIST-------------------------------- 119

A

6

I. INTRODUCTION

The Kalman filter's importance as an estimator and

predictor is well documented. Providing real-time informa-

tion concerning filter performance so that on-line adjust-

ments to filter parameters can be made continues to be an

area of high interest. This study investigates the

usefulness of the error ellipse as a tool for providing a

real-time indication of filter performance.

Part of the investigation involves an evaluation of the

Hewlett-Packard HP-86 computer system's capacity to operate

in a real time tracking environment, and its capabilities for

providing information concerning filter performance. The

rationale behind this investigation is based on the require-

ment for the Naval Underwater Weapons Engineering Station,

Keyport, Washington, to accurately acoustically track

torpedoes on a three-dimensional underwater tracking range.

A knowledge of the range operation is not within the scope

of this study. It is sufficient to know that presently

the range receives four time measurements every 1.31 seconds,

and these measurements are nonlinear functions of the

torpedo position.

To gain a better understanding of the error ellipse, a

4-state tracking scenario was chosen for this study.

7

Initially, the linear tracking problem is discussed,

followed by an investigation of the nonlinear probie,. Of

primary interest are on-line methods to improve filter

performance using information provided by the error ellipse

for filter parameter modification.

L

8

ii KAL'., FILTER THEORY

A. LINEAR MATHEMATICAL MODEL

1. The Plant

For this model the state and measurement equations

for the plant are linear. Hence the discrete form is used.

The assumed plant model is described by a linear, vector

difference equation:

x(k+l) = cx(k) + Au(k) + rw(k) (State Equation) (2-1)

and a linear, vector measurement equation:

z(k) = cx(k) + v(k) (2-2)

where: Sx(k) is an n-dimensional column vector, denoting the

state of the plant at "time" k.

u(k) is the deterministic control input, an m-vector,at time k.

w(k) is a p-dimensional vector representing any randomforcing inputs at time k.

z(k) is a q-dimensional vector representing measure-ments made at time k.

v(k) is a q-dimensional vector representing randommeasurement made at time k.

, &, r, and c are assumed constant coefficient matrices of

dimension nxn, nxm, nxp, and qxn respectively.

9

2. Noise Processes

In order to place probabilistic structure on the

noise processes v(k) and w(k) the following assumptions are

made:

(a) v(k) and w(k) are individually white processes,

that is, for any k and 1, with k 9 1, v(k) and S

v(l) are independent random variables, and w(k)

and w(l) are independent random variables.

(b) v(k) and w(k) are individually zero mean, 4

Gaussian processes with known covariances.

(c) v(k) and w(k) are independent processes.

Thus for the measurement noise: 0

MEAN: E[v(k)] = 0 (k=0,l,2,3,...) (2-3)

T TCOVARIANCE: E[v(k)v (1)] E[v(k)]E[v (1)1

0 k lA: k k = 1

or E[v(k)v T (1)] = Rk kl (k,l=0,1,2,...) (2-4)

where 6kl is the Kronecker delta function defined as:

: 0, k 1 1

: 1, k I 1

10

0

Likewise for the random forcing input:

< = (z l,2, 3,...)

E[w(k)w ( 1)] k 6 kl k,l=0,l,2,3,...) (2-6)

and R k are nonnegative definite symmetric for all

k. Also since v(k) and w(k) are zero mean and independent

then:

E[v(k)wT (1)] = 0 (2-7)

For the purposes of this study, unless otherwise

specified nk and R are considered to be known and constant,-k k

although both may be time varying.

3. Initial State DescriDtion

For the initial state of the difference equation (2-1)

it is unlikely that x will be available. Hence, it is

assumed that x is a Gaussian random variable of known mean

x and known covariance P., i.e.,

E[x(0)] x

-0 --0 - 0 --

11

This choice for the initial state has the advanz_:,

of causing the subsequent estimation scheme to be unbiased

for all z. [Ref. 1] Further it is assumed that the initial

state and the measurement noise are uncorrelated:

E[x(O)v T (k)] : E[v(k)x T(0)] : 0 (k:0,!,2,3,...)

Also the initial state and the random forcing input are

uncorrelated:

E[x(0)w T (k)] E[w(k)xT (0)] 0 (k:0,l,2,3,...)

B. DISCRETE-TIME ESTIMATION

1. The Estimator Equations

The estimation problem involves generating an

optimal estimate for x(j) for the system described by the

difference equation (2-1) from the noisy measurements

z(0),z(l),...,z(j). This estimate will be denoted by

x(j/j), which means the estimate of x at time j given

measurements at times up to and including time j. The

estimate must be optin.al in the sense that the expected

value of the sum of the squares of the error in the estimate

is a minimum, i.e.:

TAE{[(x(k/k) - x(k)] [x(k/k) - x(k)]} minimum

12

I_ __ ,,. . . . . .. ..__ .. .. .._. . .. . . . . . . . . . .

The estimator is characterized by the linear relationship:

x(k/k) = x(k/k-i) + G(k)[z(k) - cx(k/k-l)] (k C,,,..)

(2-8)

where

x(k/k) is the optimal (minimum variance) estimate ofx(k) given observations at times up to andincluding k.

x(k/k-1) is the optimal one-step prediction of x(k)given observations at times up to and -including k-l.

G(k) is the optimal estimation gain matrix whichwill minimize the variance of estimationerror.

For the initial estimate ;(0/0), the estimator

equation (2-8) is initialized with x(O/-i), which is not a

random variable. If x(O/-l) is selected such that: 0

;(0/-1) = E[x(O)] =

it can be shown that this choice of x(O/-l) makes the

estimator unbiased for all k. [Ref. 1] The estimators best

available information concerning x(k-l) is the estimate

x(k-1/k-l), therefore it is reasonable to assume that

;(k/k-l) = x(k-l/k-l) + Au(k-l) (2-9)

is the best prediction.

1

13

l r I .. .... ' . . . m . . .. . .

In summary, equations (2-8) and (2-9) are the estimator

equations, with x(O/-l) = x as the initial cond-:ir

2. Gain and Covariance E;uations

Without going into detailed derivations, tne optimal

estimator gains, G(k), used in the estimator equation (2-8),

are those which satisfy:

G(k) = P(k/k-I)CT [cP(k/k-)C T + R] - I (2-10)

P(k/k) = [I - G(k)C]P(k/k-1) (2-11)

P(k+l/k) = P(k/k) T + Q (2-12)

with the initial conditions:

P(0/-l) P 0 E{[x(0) - ][x(O) ---- o --- o - --<

where

P(k/k) = Ef[x(k/k) - x(k)][x(k/k) - x(k)]T }

is the covariance of estimation error matrix.

0L

P(k/k-1) E{[x(k/k-1) - x(k)][x(k/k-1) - x(k)]}

is the covariance of one-step prediction error matrix.

Q E[r(k)-w(k).wT (k)'rT (k)]

is the state excitation matrix.

P(k/k) and P(k/k-l) are symmetric, positive definite matrices.

Several observations can be made concerning the linear

Kalman gain (2-10), covariance (2-11, 2-12) and estimator

(2-8) equations.

(a) The estimator gains, G(k), do not depend on the

measurement data and hence can be precomputed, stored, and

used as the processing measurements become available.

(b) Although not obvious from the equation, the

time-varying gain, G(k), depends in time as:

1G1k) =k-+ 1) [Ref. 21 (2-13)

Thus the effect is to weight the correction term,

[z(k) - cx(k/k-l)], in the estimator equation (2-8) less

heavily as time progresses. The advantage of a greater

initial weight allows for possibly large differences

between z(k) and x(k/k-1) during the initial observations,

and a large gain will result in a significant change in the

- I

next estimate. This advantage is also borne out in that

there is less confidence in the quality of the estimates

during the early observations compared with the quality

after numerous observations. Hence the later an observation,

the less drastic an estimate will be altered or affected by

an isolated observation discrepancy.

(c) In general, the variance of estimation error

decreases in a manner analogous to the gain schedule (2-13),

i.e., it decreases as k grown larger, reflecting greater

confidence in the estimate as the number of observations

increases. Selection of the proper initial condition, P ,

is important when studying the effect of measurement errors

on the behavior of the estimate. So P should be assigned-0

pessimistic values which would correspond to a lack of

information about the initial state. In cases when the 5

initial state is completely unknown, then P --I. [Ref. 3]-- o

(d) The Q matrix serves to compensate for model

errors and prevents the covariance matrix from becoming too 0

small or optimistic. A small covariance matrix would result

in a small filter gain, and subsequent observations are

essentially ignored, which could result in filter divergence. 0

The Q matrix prevents G(k) from approaching zero by adding

uncertainty to the system which is reflected in a

degradation of certainty (increase in P(k+l/k)).

16

F-

0

C. NONLINEAR ESTIMATION - EXTENDED KALMAN FILTER

in many practical applications, the state equations

and/or measurement equations are nonlinear. Before the

Kalman filter equations can be used, the problem must be

linearized and the Kalman filter equations are applied with

some modification.

1. Nonlinear Model

Consider a nonlinear discrete system of state and

observation equations given by: 0

x(k+l) = f(x(k), u(k),k) + w(k) (2-14)

and

z(k) = h(x(k), k) + v(k) (2-15)

In these equations f and h are nonlinear functions

of the state variable x, w(k) is the plant excitation noise,

and v(k) is the measurement noise. The plant noise and

measurement noise are assumed to be uncorrelated, zero-mean,

and white. The same equations (2-3 thru 2-7) apply as for 0

the linear model.

2. Extended Kalman Filter Equations

In order to apply the linear filter equations,

equations (2-14) and (2-15) are expanded about the best

estimate of the state at that time and only the first-order

terms are kept.

17

That is, defining A(k) as:

A~k) afA(k) a (x(kix), u(k), k)

and

H(k) (x(k/k-1))

Ux

As can be seen from the above equations, the filter

estimates, x(k/k) and x(k/k-l) are used as the "best"

estimates about which the linearization is performed. The

matrices Ak) and H(k) must be used to generate G(k) so it

is available to process z(k) when it is obtained. The

modified extended Kalman filter equations are then:

Gain Equation:

'(k) = P(k/k-l)H T (k)[H(k) • P(k/k-l) • HT (k) + R] -1

(2-16)

Filter Update Equation:

x(k/k) = x(k/k-l) + G(k)[z(k) - h(x^(k/k-l))] (2-17)

18

. . . I . . . . . . . . . .

Prediction Equation:

x(k+i/k) f(x(k/k), u(k), k) (2-18) 5

Covariance of Estimation Error Equations:

P(k/k-1) : A(k-l)P(k-1/k-1)A (k-1) + Q(k-1) (2-19)

P(k/k) = [I - G(k)H(k)]P(k/k-l) (2-20) 4

For the initial estimate x(0/0), equation (2-17) is

initialized with x(0/-l) with

x(o/-1) = E[x(0)] x

x(O/-l) is also used to initially evaluate H(k).

As in the linear case:

19

III. ERROR ELLIPSOIDS

A. THEORY

Since the estimate x(k/k) is unbiased in the Kalman

filter equations, the P(k/k) matrix represents the covariance

of the error in the estimate. If the estimate were biased,

P(k/k) would represent the second-moment matrix rather than

the covariance matrix. Hence, P(k/k) provides significant

information about the accuracy of the estimate. If the

physical model is accurately described by the state and

measurement equations (2-1, 2-2), then P(k/k) can be used

to describe the manner in which the estimate converges

(or diverges) to the true state. Examination of the P(k/k)

matrix directly, element by element, is not a realistic

2approach, since the matrix contains n elements, where n is

the number of state variables. To simplify the situation

the concept of the error ellipsoid is used. [Ref. 4]

As discussed earlier, the assumptions are made that the

initial state of the plant x is Gaussian, as are the random-o

processes v(k) and w(k). Using these assumptions it follows

that x(k) and x(k/k) are also Gaussian since they are linear

combinations of Gaussian variables and deterministic

quantities. Using the same rationale, the estimation error,

defined as:

20

e(k/k) x x(k/k) - x(k)

is also Gaussian. Using the fact that the mean of the

estimation error is 0, the probability density function for

e(k/k) is:

pe [(k/k)] = [( 2 )n/2 L?(k/k)1i/2-1

exp[-1/2eT (k/k)P- (k/k)e(k/k)] (3-1)

The density function, p_ [e(k/k)] will have a constant

value whenever the exponent has a constant value. That is:

-1/2e T (k/k)P- 1 (k/k)e(k/k) = c

or

T -l2

e (k/k)p- (k/k)e(k/k) = c (3-2)

where c is an arbitrary constant.

As demonstrated by Sorenson [Ref. 1] and Kirk [Ref. 21,

it can be shown that the locus of points e(k/k) which

satisfy equation (3-2) are hyperellipsoids. For the

two-dimensional case which is of concern, equation (3-2)

describes an ellipse. This can be seen by fixing time and

rewriting (3-2) as:

21

e~we

e we c( -)

where

w = P- (k/k) (a 2 x 2 symmetric matrix)

S

Expanding the left side of (3-3) gives:

2 2 2w1 1 e + W1 2W 2 1 e1e 2 +w 2 e2 c

which because of symmetry gives:

2 2w e + 2w 2e e2 + w22 e2 = c (3-4)

Since wl>0, w22>0, and W w 2 2 >w 1 2 , equation (3-4)

describes an ellipse, in which the principal axes do not

coincide with the coordinate axes. The ellipse is rewritten

in terms of z(1) and y(2) as the coordinate axis, and it can

be shown that y(l) and y( 2 ) are the eigenvectors of w with

1 and X2 defined as the corresponding eigenvalues. [Ref. 2]

(See Figure 3-1). The equation for the ellipse can be

rewritten in terms of a coordinate system having unit vectors

in the directions of y(1) and y(2) as the basis vectors.

The ellipse equation becomes:

Xil2(1) + X2 2(2) = c 2 (3-5)

22

*

e 20

Y(2)1<1

0

Figure 3-1 Error Ellipse

Remembering that w =P- (k/k), it can be shown that the

-II

corresponding eigenvectors and eigenvalues for w- 1P(k/k)

Y (1( )

2, 2

y ( 1 z ( 2 z c ( 3 -6 )1 2

23

Fi g ur 3 -1 E rro Ell p se

2 23

Z (i) Z (2

For the purposes of this study, the term error ellipsoi:

refers to the specific case when c 1. So equation <-,9)

becomes:

~2y2 (1) y (2)

+ -a 2

In terms of the Cartesian coordinates x', yt, which use e1

and e2 as basis vectors:

xf = xcosO + ysine

y I =-xsine + ycose

where e is the angle of rotation of the axes and can be

computed from:

2cov(e e)= 1/2 tan-l[ xve x ] [Ref. 5]

var(e) - var(ex y

For a given value of c it is possible to integrate the

probability density over the surface of the error ellipse to

obtain the probability that a particular sample point will

lie within the ellipsoid. For this study, n = 2, c = i,

and the probability the error is inside the ellipse is 0.394.

In summary, the error ellipsoid can be used to

characterize the concentration of the estimate abouz the

24

.. . 0 . . .

true value of the state. A decrease in the magnitude of an

axis cf the ellipse is an indication that the error 4n -ne

estimate is decreasing in that direction.

One important item that needs to be pointed out is that

often the components of the state vector represent entirely

different types of variables, for example the components

might represent range, velocity, and depth. Since the two-

dimensional ellipses are determined by using two components

of the state vector, it is reasonable to examine submatrices

relating state variables of the same character. Doing so

will preclude most scaling difficulties when plotting the

ellipses, and provide more meaningful insight in to the S

resulIts.

B. ERROR ELLIPSOIDS AND FILTER DIVERGENCE

Thus far the discussion has centered around using

submatrices of the P(k/k) covariance of estimation error

matrix as the input for the error ellipse to indicate

filter performance. As proposed by Heffes [Ref. 6] and

Nishimura [Ref. 7], P(k/k) can be considered as a "design"

covariance matrix P d, using the assumption that the only

errors are in R and P with the followina inecualities

holding 5or all k:

a d >a pd > .a-K • k k _ k --o -:-o (3-7)

2

- ' • , , , , • . . . . . . .

with the subscript "i" indicating designed and "a" indicating

actual. Equation (3-7) implies more input noise, more

measurement noise, and more initial state uncertainty in the

design than actually exists. This conservative filter design

results in a somewhat pessimistic design error covariance

Pd(k/k). The actual error covariance Pa(k/k) resulting

from using a filter designed with Q d, Rd , and Pd is related> --- o

to the design error covariance in the following manner:

P d(k/k) > P a(k/k)

This result is particularly useful when one simply does

not know accurately the noise covariance of the input or

output, but an upper bound is known. Designing assuming

the noise covariance is at its upper bound will result in

Pa(k/k) being upper bounded by Pdk/k). In some sense a

worst case design results. Filter divergence exists when

the design error covariance P d(k/k) remains bounded while

the error performance matrix Pa(k/k) becomes very large

relative to Pd(k/k) or is, in fact, unbounded.

2

IV. iRQBLEM DEL:i3I

A. PROBLEM DESI&N

The purpose of the tracking problem is to study the use

of error ellipsoids as real-time indicators of filter

performance. In order to keep the design model realistic

albeit reasonably simplified for ease of study, a

two-dimensional tracking problem using several different

tracks has been selected.

1. Linear Tracking

All tracks are based on an x-y coordinate system

with the target moving in the x or y direction relative to

the sensor located at the origin. Thus for aircraft

tracking, altitude is considered constant, as is depth for

torpedo tracking.

Defining the state variables as:

xi = x x-coordinate of the target location.

x2 = x velocity of target (vx) in x-direction.

x3 = y y-coordinate of the target location.

x4 =y velocity of target (v y) in y-direction.

2'7

resultinq in a state vector:

X (4-1- y

*Y

The following are the state equations:

1 (t) x2 (t)

x 2 (t) = W1 (t) S

(4-2)

x3 (t) = x4(t)

4(t) = w 2(t)

where w (t) and w 2(t) are assumed to be uncorrelated,

random processes that account for unknown target accelerations

and nonlinear target motions. Writing the discrete form of

the state equations gives:

28

xi(k+l) x!(k) + T x.2(k) + M1 2

x 2(k+l) x2 (k) + T w (k)

(4-3)-2x (k~l) =x1)(k) + T x4(k) + ±-2 w k

x4(k+l) x 4(k) + T w 2(k)

or

x(k+l) ox(k) + rw(k) (4-4)

With T, the sampling period equal to 1 second, the matrices

_ and r are:

1 1 0 0 .5 0

S4- ( 5)0 0 1 1 0 .5

0 0 0 1 0 1

It is assumed that the sensor gives noisy, but

uncorrelated measurements of x and y. Hence the discrete

measurement equations are:

23

zl(k) = x,(k) + v,(k)

LS

z2 (k) x3 (k) + v 2(k)

with v (k) and v (k) uncorrelated random noise.1 2

Thus for the measurement equation:

z(k) = cx(k) + v(k) (4-7)

the matrix c is:

1 0 0 0_IC:

For the initial run and unless otherwise noted the

following values for the Gaussian random processes will be

used:

E[v(k)] = 0 for all k

-20xl10 0

T Wl2 RfoalkE[v(k)v (k)] m R 'or all k

0 20x!O

30

rm the standard deviation of

_15 0 measurement noise.

E[w(k)] = 0 for all k

100 0

E[w(k)w (k)] (m/sec 2) = coy w for all k

0 100

10

2C : m/sec the standard deviations of the-w S

_10- random forcing input.

The covariance of estimation error matrix is initialized:

-2210 2 0 0 0

0 10 20 0? =P(/-l) 0 o2 0

-0- 0 0 10 0

0 0 0 102

31

Since the filter is to be unbiased, the initialization:

x(O/-l) = x = Initial condition of the problem.

2. Nonlinear Tracking

The state equations are the same a3 f.r The linear

tracking problem. The measurement equation is considered

as a noisy range measurement by the tracking sensor and is

characterized as:

z(k) [x2(k) + x2(k)] 1 2 + v1 (k) (4-8)

Thus z(k) is a nonlinear function of the states. Using

equation (2-14) and with T = I second:

1 I (k) + x2(k)

f(x(k), u(k), k) 2

x3 (k) + x4(k)

4(k)

32

I q

Taking partial derivatives of f with respect to x i_

1 1 0 0-

A(k) = = __0 1 0 0

(x(k/k), u(k), k) 0 0 0 0

using equation (2-15):

h(x(k), k) = [x2 (k) + x 2 k) 11/21 3

and taking the partial derivative with respect to x gives:

H(k) -102 - 2 ]12-- 1 (k)+x2(k)] /

(x(k/k-1))

x3(k) 02 -2 1/2

[x 2(kl+x32(k)] x1k/-l

1 3

33

I

Using the above results with the same vali- s :: 7-i

random noise Drccesses as previously stated for the liear

case, thie exended Kal.an filter equations can noD; ae o pc lea

to the nonlinear tracking problem.

B. COMPUTER SIMULATION

1. Computer Hardware/Software

a. Hardware

All computer simulations were run on the Hewlett-

Packard HP-86 personal computer. This particular model was

chosen to evaluate its capabilities in determining its

usefulness in actual torpedo tracking at the underwater

tracking range at Naval Underwater Weapons Engineering

Station, Keyport, Washington. The HP-86 system used included

keyboard, 9 inch CRT monitor connected through an integrated

monitor interface, and two HP Flexible Disc Drives connected

through an integrated disc interface. Plotting was done on

a HP-7225B Plotter and printing on a HP-2631G Printer. These

peripherals were interfaced using a HP-IB Interface module.

Because the system uses interface select codes, the HP-IB

factory preset code was set at 7, which is the select code 5

for the printer/disc interface. This code however did not

work when interfacing with the external plotter and printer,

since duplicate select codes are not allowed. So the 5

internally set select code of the HP-IB was set to 8 for

proper system operation.

34

S

b. Software

The HP-86 has 60K built-in, useable bytes of

computer memory, expandable to 572K using either 32K, 64K,

or 128K Memory Modules. A HP-86 pilg-in ROM was required

to operate the external plotter. Also a Matrix ROM was used

to reduce program length and computer run time.

All programs were written in BASIC programming

language using REAL (full) precision, which provides 15 digit

precision. Appendix B provides an explanation of the program

options and Appendix C contains the program listings used for

this study.

2. Track Generation

To evaluate the real-time use of the error ellipse as

an indicator of filter performance, Monte Carlo simulation

runs were made. Four tracks were generated by separate

programs and one second incremental values of x, y, v , and

v were stored in data files. Appendix I contains an

explanation of the generation of tracks three and four.

3. Noise Generation

In order to simulate the random noise processes, the

computer's random number generator was used and the generated

numbers scaled accordingly. For each track and each

different value of noise sigma, a different generator "seed

number" was used. These noise values produced were added to

the applicable true track values to simulate a sensor

measurement corrupted by independent ]aissian noiSe. For all

3:

Icases where filter parameters were varied for a parzicu!er

track under a specific noise condition, one noisy trez< waC

genera-ed, stored, an: used zhrughoLut that arti1-::

simulation. This was done for ease of filter performance

comparison.

4. Gating Scheme

In order to preclude catastrophic filter failure due

to excessive measurement noise, a bound was established for

the maximum acceptable limits of measurement noise. A

three-sigma gate was designed using the covariance of the

measurement noise, R, and the predicted covariance of error

matrix P(k/k-1). The gate is defined as:

Gate(k) = 3 (pii (k/k-i) + R )/2

max

This gate is the maximum error allowable for the measurement

at time k. If the absolute difference between the actual

measurement received and the predicted measurement is

greater than the three-sigma gate, then that particular

measurement data is rejected as unacceptable. When this

occurs, the filter gain, G(k), is set to zero, resulting in

that measurement being ignored and the prediction of the

states set equal to the estimate, that is:

x(k/k) = x(k/k+l)

36

5. Collection of Statistics

In order to study er,-or elipses as an indica .

filter oerformance, statisticc were calculated, D -

after each measurement during the Monte Carlo run. The

statistics computed were relative error (in some cases the

error was normalized), error mean, error variance, and error

covariance for the positional variables. The following

equations apply:

Relative Error = e(k/k) = x(k) - x(k/k)

(filter error residual)

kError Mean = e(k/k) - i/k E e(j/j)

j=l

kError Variance =Var[e.(kWk)] 1/k Z Ee.i(j/j)]-ELe (k/k)12

'xI, x?, X 3 , x 4

Var[e (k/k)] 1/k E [e (j/j)][e (j/j)]

Positional j=1 x3

Error -[x (k/k) . (k/k)]rxCovariance

= 1 x 3

Mat rix

i/k Z [e (j/j)][ex(j/j)] VarEe (k/k)]

j=l1 xIX 3 3

-[ xl(k/k)' x (k/k)]

37

V. TARGET TRACKING AND ERROR ELLIPSE AN4ALYSIS

A. LINEAR TRACKING

Track 1 depicts a target approaching at a constant

velcciry of 223.6 feet Der second. The solid line of Figure

5.1 indicates the true track of the target and the numbers

along the track indicate the time in seconds. The target

was tracked in a measurement noise, G = 150, with the

random forcing noise a = 1. R was set for 20,000. The

dots indicate the filtered track using the linear Kalman

filter equations. Figures 5.2 and 5.3 are the filter error

ellipses for this run, computed at increments of 10 seconds.

As can be seen the ellipse size decreases with increasing

time indicating filter convergence. The computed ellipse

surface areas shown on the figures confirm this. Figure 5.4

shows the filtered track for the case where a has been

increased to 300 and all other parameters remain the same.

The error ellipses of Figure 5.5 computed for 10-second

increments show increasing area indicating filter divergence.

Figure 5.6 shows the error ellipses for the same track run

but this time the ellipses were computed using a "statistics

window" of 10. By this is meant that the ellipses were

derived from statistics computed for the last 10 data

measurements. All previous data is disregarded. Usin7 this

3 e

0

method, the ellipses of Figure 5.6 show filter conver ence

from iterations 15 tz, 25. The fil tered zrack of Fi.::-

5 con r rs t1I. .zo're 5.7 shows e r'ess z z f

same tack parameters, except in this case a statistics

window of 5 was used. The window 5 ellipse area at time

25 (5179 so ft) is much less than the area of either the

run with the statistic window of 10 (11,540 so ft) or The

run with no window at all (65,500 sq ft). This is expected

since the filter is essentially "locked on" at time 21,

and the window 5 ellipse at time 25 disregarde all data

previous to tLme 20. Figure 5.8 shows the error ellipses

for the same track but the measurement noise was increased

to 0 = 40C, while R was kept at 20,000. The error ellipses

indicate filter divergence, and indeed the filtered track

headed off in the wrong direction.

Track 2 depicts a target approaching at a constant speed

of 500 feet per second in the -y direction. Figure 5.9

depicts the solid line track and the octs indicate the linear

filtered track with ' 150, G = 1 and R = 20,000. Figure

5.10 and 5.11 are the error eloo'ses cr the run. No

statistics window was used. Other than the fact that the

ellipse area is decreasin7, t_e shase of -he ellipse proviCes

little additional infoma-ion. Figure 3.12 and 5.13 show the

ellipses for the ncrma!i-e :'rcr. -ih the same measurement

noise si-ma for both the x-oosi-oin snh V-00siton measurements,

the normalized error eliho and D reflect

0..ne he de~=-o elc

0

the target track proximity to either axis. > Fin ' .

the ellipse major axis indicates a large i:r.alize: -r-i:

t-ne ZCecsn i -o e u-Ce-- --

maintains a constant x-position of 500 feet, while the

y-position is initially 20,000 feet. However, near time 40

as the target aporoaches the x-axis, the normalized v error

increases and becomes so large at the x-axis crossing that

the normalized x error becomes insignificant in comparison.

This can be seen in Figure 5.13 for the ellipses at time

45 and 55 compared with the ellipse at time 35. The

normalized error ellipse is an excellent indicator of

target proximity to an axis, but the rapid shifts in ellipse

surface area make it difficult to determine filter

convergence.

Track 3 depicts a target approaching on a parabolic

track at a speed of 203 feet per second. Figure 5.14 is the

true track, with a linearly filtered track indicated by the

dots. For this run a =150, a =1l, and R=20,000. Figure 5.15

are the error ellipse plots for times 40, 50, and 60, the

period of the highest rate of change in x- and y-velocity.

The ellipse areas increase with time indicating divergence.

Figure 5.16 and 5.17 are the ellipse plots using a 10-data

point and a 5-data point statistics window respectively.

In both cases the ellipse areas for time 60 are less than

time 50 indicating the filter has tracked around the curve.

40

.. 6

Using error ellioses based on a statistics window in tis

tracking situation provides a better indicator of filter

performance.

For the second run of track 3, a was increased to 300

and all other parameters remained the same. Figure 5.18

shows the resulting filtered track, which obviously didn't

track around the curve. With the large amount of measurement

noise and considering that the maximum random forcing input,

i.e. the maximum acceleration in the x- and y-direction,

occurs between times 40 and 60, it is a logical place to lose

track. Another factor to be considered is the decrease in

gain as k increases. By time 40 the gains have little

influence. So a system was incorporated in the program to

"reinitialize" the filter by setting the gains to G(0), if

certain conditions were met. After several trial and error

runs, it was determined that if a statistics window of 10

were used, and the gains reinitialized if the error ellipse

area increased consecutively a certain number of times,

that the filtered track would follow around the curve.

Figure 5.19 is the filtered track using a statistics window

of 10 and reinitializing the filter if the error ellipse area

increased consecutively during five 1-second increments.

The error ellipses for that run are shown in Figures 5.20

and 5.21. As indicated on the figures, the filter was

reinitialized four times and the ellipse areas for the

10-second increments increased, until reinitializing the

.L

filter at time 52 locked the filter in. Consequently e

error ellipse areas fr t-t Le and 70 decreased, indio4 -Cng

convergence.

In the final run, the filter was reinitialized after 10

consecutive 1-second error ellipse increases, with all other

parameters remaining tihe same. ig _re 5.22 is th filtered

track; Figures 5.23 and 5.24 are the error ellipse plots

for this run. As indicated on Figure 5.24 the filter was

reinitialized only once at time 57. A comparison of ellipse

areas for the last two runs shows that, with the exception

of time 70, the areas were larger in the first run when the

filter was reinitialized 4 times. This is expected since

reinitializing results in larger gains producing more widely

vary estimates, and hence greater error variance. At time

70 the error ellipse area of the first run is less since in

the first run the last filter reinitialization occurred at

time D2 versus time 57 in the second run. The first

filtered track had more time to settle out by time 70,

resulting in less error.

0B. NO:ILI-AR TRACKING

Track 4 depicts a target moving at a constant velocity

of 50 feet per second (30 kts) in the x-direction for 15

seconds, at which time the target turns and travels in the

-y-direction. (See Figure 5.25) Using the extended Kalman

filter with a =30 and R=900, a series of tracking runs were

4 2

made for various values of COVW (a 2=from 20 to 200). In nonew

of these runs did the filter successf' Ily track the tar--

around the turn. Figure 5.26 shows the results for CaSe

when CVW=ISO. In this instance the filter lost -rack as -he

target came out of the turn. Trying to track through a turn

using a constant COVW (and hence, a constant Q) did not work.

So a scheme was devised to vary COVW dependent on information

derived from the error ellipse. After several trial runs for

this particular track, it was determined that if the error

ellipse area increased consecutively for 7 iterations of

k, COVW would be doubled, and if the ellipse area decreased

consecutively for 5 iterations of k, COVW would be halved.

Initially the the trial runs were made without a statistics

window, and the filter did not track successfully. Without

the statistics window, the old data weighted down the statis-

tics, and the error ellipses were not indicative of what was

currently happening. So it was decided to use a statistics

window. Windows of 5, 10, and 15 were tried. Window 5

proved to be too responsive and window 15 not responsive

enough. So a statistics window of 10 was chosen for the

tracking run. With P =102, 230, R=900, and COVW initially

set at 20, the tracking run was made. Figure 5.27 depicts

the filtered track output, and Figures 5.28-5.30 are the

10-second incremental error ellipses for the run. As

can be seen, the filter did track around the curve. Figure

5.29 show3 the ellipse areas are becoming less between times

55 and 65. Also indicated below the plots are the values of

4S

COVW for the k time the plot was computed. During this

particular run COVW varied from an initial value of 23 up

to 160, and then decreased to 40 by the end of the run.

Using the same filter parameters as above except a was

increased to 200, and R to 40,000, another run was made. The

filter did not track at all. During this run COVW varied

from an initial value of 20 up to 40 and decreased to 5 by

the end of the run. Obviously, the criteria for increasing

and decreasing COVW was not effective. More trial runs were

necessary to determine the optimum consecutive increases

or decreases of the ellipse areas before adjusting COVW

accordingly.

A second approach to the nonlinear tracking problem, one

that was used earlier for the linear case, is to reinitialize

the filter if certain conditions are met. Again, using

trail and error runs with and without statistics windows,

it was determined that using a statistics window of 10 gave

the best results. Using initial conditions of COVW=I50,

P =102, a =30, and R=900, several runs were made, reinitializ--o< -v -

ing the filter if the error ellipse area increased consecu-

tively for a certain number of iterations of k. Of the runs

attempted, the best results were obtained when the filter was

reinitialized if the area increased for 5 consecutive

iterations of k. Figure 5.31 is the filtered track for this

run, and FiguresS.32-5.34 are the error ellipse plots.

Indicated below the plots are the times, k, when the filter

4 L

was reinitialized. For this particular run the filter was

reinitialized 5 times. It should be noted that whern2

reinitializing the filter, P(k/k-l) was reset to i X P .-o

P was not large enough to be effective in getting the--o

filter back on track. The initial value of 102 was used for

P to reflect the high confidence in the initial state--0

conditions. Other values of P did not work as well.-0

Using the same parameters as above except the filter was

reinitialized after 7 consecutive error ellipse area

increases, another run was made with the resulting track

depicted in Figure 5.35. A comparison with Figure 5.31

reveals that using 7 area increases as the criterion for

filter reinitialization resulted in poorer filter performance,

as can be shown from the error ellipses.

The final nonlinear filter tracking runs attempted

involved simultaneously varying COVW and filter reinitializa-

tion. The results were disastrous, and highly unpredictable.

It was an interesting experiment in futility, and no

meaningful results could be obtained.

45

SY-POS Tn

10

400

39

200

Figure 5.1 Solid Line Track 1, Vel 223.6 ft/sec; DotsIndicate Filtered Track, a =150, a =1, Rz20,000

46

50(

Y-POS (FT)155

78

~X-POS (FT)

-78

-155LO ~ r- LO

COVW-1 SIGV-15 R2 000 TIME AREA(SQ T) LEO

10 1057E+001 -20 8790E+000 ....

30 7540E+000 - - -

Figure 5.2 Filtered Track 1 Error Ellipses at 10 SecondIncrements, G 'i50

47

Y-POS (FT)

78

-78

-155

COVW-1 SIGV-150 R-20000 TIME AREA(SQ FT) LEG

40 6681E+00050 5741E+000 ..

Figure 5.3 Filtered Track I Error Ellipses at 10 SecondIncrements, a v =1Q

43

Y-PO amT

am

10

20

39

aF i II I-2M0

404

Figure 5.4 Filtered Track i, a 0300, a :i, R:2OJ00

'I .

Y-POS (FT)326

163 -/ I

/ z01

-326Co CY CCVJ Co Co C'JCTn -4 mI I

COVW-10 SIGV-300 R-20000 TIME AREA(SQ FT) LEG

5 2858E+000 -15 3480E+001 ....

25 6550E+001 - - -

Figure 5.5 Filtered Track 1 Error Ellipses for a =3OO

50

--

Y-POS (FT)326

163

X-POS (FT)

-163

-326

C\J D(D C'jI I

COVW-10 SIGV-300 R-20000 TIME AREA(SQ FT) LEG

5 2858E+000 -

15 1745E+001 ....

25 1154E+001 - - -

Figure 5.6 Filtered Track 1 Error Ellipses for % =330,Using Statistics Window of 10

S

Y-POS (FT)326

S

163

0!1~X-POS (FT)

-326 tN C Co N~

I I

COVW-10 SIGV-300 R-20000 TIME AREA(SO FT) LEG

5 2858E+000 -

15 7225E+000 ....

25 5179E+000 - - -

Figire 5.7 il-tered Track 1 Error' Ellipses =Using Stazistics 'indow of 5

S q

52S

Y-POS (FrT)617

308

0 X-POS (FT)

-308 -

-617CO- CV) O -

I I

COVW-10 SICVY400 R-2000 TIME AREA(SO FT) LEG

5 5825E+001

9 8489E+001 ....

13 9507E+001 - - -

.iit ered raci < EIrror 7-liise fo- - =40

5":S

I 4

Y-POS (FT)20000

18010

12000

20

8000

*304000

-4000

U, In m

Figure 5.9 Solid Line Track 2, Vel 500 ft/sec, Dots IndicateFiltered Track for u =150, a =i, R=20,000

-il IIII I i i . . . .

Y-POS (FT)93

47

0p

11) X-POS (FT)

-47

-93--

I I

COVW-1 SIGV-150 R-20000 TIME AREA(SQ FT) LEG

5 3868E+00015 4604E+000 ....

25 3132E+000 - - -

Figure 5.10 Filtered Track 2 Error Ellipses, = 150

. . . ..4. . . . . . .

0

Y-POS (FT)93

47

0 \ .. X-P0S (FT)

-47

-93I I

COVW-1 SIGV-150 R-20000 TIME AREA(SQ FT) LEG

35 2550E+00 -

45 2137E+000 ....

55 1950E+00 - - -

Figure 5.11 Filtered :rack 2 Error -pses, u =l5C-V

Y-POS (FT)1E-001 T

SE-002

X-POS (FT)

-5E-002

-1E-001._.

I lU

I

CDVW-I SIGV,150 R-20000 TIME AREA(SQ FT) LEG

5 402BE-00715 5156E-007 ....

25 4521E-007 - - -

Figure 5.12 Filtered Track 2 Error Ellipses, =150, ErrorNormalized

5-I

E- Y-POS (FT)

5E-002 I.

1.

-5E-002I"-I

"l I.

II.

-5E35 7401. 00

l I.

] I.

I.4 l -- 1I.

I N.

I I

I I

COVW-I SIGVm15B R-n22B00 TIME AREA (SO FT) LEG

35 740 E-007 --

45 3869E-005 ....

55 3416E-005 - - -

Figure 5.13 Filtered Track 2 Error Ellipses, a z!50, ErrorNormalized

58

Y-POIT113N TFD

4MF

2m 4

04

50X-POSITION Tn

-2m9

-409

90 I

Figure 5.14i Solid Line Track 3, Vel 200 ft/sec, DotsIndicate Filtered Track for a. =150, a V =10,R=20 000

59

Y-POS (FT)150

75

6 0

-75

-150

CDVW-100 S IGY- 150 R-20000 T IME AREA (SQ FT) LEG

501291E+001 ..

60 1388E+001 - - -

Figure 5.15 Filtered Track 3 Error Ellipses, a-=150

Y-POS (FT)

56

-56

CDVW-100 SIGV-150 R-20000 TIME AREA(SQ FT) LEG

40 3284E+00050 1420E+001 ..

60 32.57E+000 - - -

Figure 5.16 Filtered Track Error --!ises, a., 1BSt at _4st -.c s Wr

Y-POS (FT)114

57

-57

-114

.4

COVW-100 SIGV 150 R=20000 TIME AREA (SQ FT) LEG

40 1448E +00 -50 2945E+00 ....

60 5517E-001 - - -

Figure 5.17 Filtered Track 3 Error Ellipses, a :15,Statistics Window 5 /

62

S

Y-POSITION T)

4M0

07

X-PSeIO IOMso

F16gure 5.1S Filtered Track 3 for azOO 4AI =, R=20,000

Y-P STION GT)

410

0 30

X-PQSITION T

-209

*170

II

Figure 5.19 Filtered Track 3 for u :300, a =i0, R=20,300,

Statistics Window 10, Reinitialized at 5Consecutive Ellipse Area Increments

54 I

. . . . . . . ,. , . . . . . . . . . .

Y-POS (FT)548

274

X-POS (FT)

-274*i-548

O N

COVW-IO SIGV-300 R"20000 TIME AREA(SQ FT) LEG RESET

30 4734E+001 - 2940 7035E+001 .... 35

50 2698E+002 - - - 44

Figure 5.20 Filtered Track 3 Error Ellipses, o =300,Statistics Window 10, Reini ±-*aze' 5

35

,I . . . . . _ -

Y-POS (FT)548

274

-- 74

I I

CCVW-100 SIGV-300 R-20000 TIME AREA (SQ FT) LEG RESET

62 1617E+002 -- 5272 6563E=-000 .... 52

F'igu-re 5.21 Filtered Track 3 Error Ellipses, o_= 3Di,S:tatz*s-ics kndow 13, Reini:ae

Y-POSITION (FM)

2M0

40

* X-POSITION Tn

CS)

Slaiitics Wi-icw V, Reirnt1.:e 11

Y-POS (FT)292-

1461

0 I X-POS (FT)

-146

CD

COVW=100 SIGV-300 R=20000 TIME AREA (SO FT) LEG RESET

30 1024E+001 - 0

40 llelE+00I .... 0

50 91886E+0z1 - - - 0

ie5.23 Filtered Tracl, 3 7Error £1iiseS, (7 =300,

Sta,:--stics Windlow 10, Reinitiali7ze 10

Y-POS (FT)22 -

146

X-POS (F-T)

-146

I4

-44

COVW-100 SIGV-30 R-20000 TIME AREA(SQ FT) LEG RESET

60 1380E+002 - 57

72 1478E+001 .... 57

Figure 5.24 Filtered Track 3 Error Ellipses, a =300,

Statistics Window 10, Reinitiaiize 10

Y-POSITION FM)

30

5050

400*

70

300

so

23900

X-POSITIDN (FD)

Figure 5.25 Track 4, Vel 50 ft,/sec

7,I

Y-POSIrION TD

4

acn in cX-POSITION CT)

Fig ure 5.26 Filtered Tr-ack 4, a,=30, R=900, a = 150--w

0

Y-POSIT!ON OFr)

flBr

v 2

X-P ITION (FT)

Figure 5.27 Filtered Track 4, a -3 O, R=900, Statistics

Window 10, Varying a 2-72

72

34T

Il

Y-POSS (FT)

-3-4

7;

1 / .1

°. 4

-68wcc C') co

SIGV -30 R 90 TIM~E AREA(SQ FT) LEG. COYW

15 9688E-002 - 225 8097E-001 ....

-35 2740-+0- - - - 160

Figure 5.29 Fiitered Trac-- 7rror~ 71-iDses, s3Zatis-icsil in w 12, 7ari;D

7 3

Y-POS (FT)

68 --

/

34/S /

//X-POS (FT)//

-34

- 8t I!I

SIGV -30 R -900 TIME AREA(SQ FT) LEG. COVW

45 2239E+000 - 16055 6303E+000 .... 16065 4625E+000 - - - 80 q

Figure 5.29 Filtered Track 4 Error Ellipses, StatisticsWindow 10, Varying COVW

7L

* 4

Y-POS (FT)68

34

. X-POS (FT)

-34

CO CT)m COI I

SIGV - 30 R 900 TIME AREA(SQ FT) LEG. COVW

75 I163E+0 0 - 4085 7916E-001 .... 40

Figure 5.30 Filtered Track 4 Error Ellipses, StatisticsWindow 10, Varying COVW

75

Y-PMITION (Fn

I

SM r

4

2W, 1

X-PSITION FT)

2

Figure 5.31 Filtered Track 4, a =30, R=900, w =150,Reinitialize 5 V - -w

76

. . .. . . ....

103 Y-POS (FT)

10

Y-POS (FT)51

7" /

/13c ..

/ /

i//

SIGV-30 R-900 CCVW-150 TIME AREA(SQ FT) LEG. RESET

15 IB8SE-001 - 025 1211E+000 .... 23

35 5743E+000 - - - 33

Fgure S.32 Filtered Track 4 Error lli-ses, =Reinitialize 5

7-

S

Y-POS (FT)103

51

B/ /

/ , X-POS (FT)

ftf

-51

-103 -

SIGV"30 R-900 COVW-150 TIME AREACSQ FT) LEG. RESET

45 4383E+00 - 3855 6852E+000 .... 55

65 6495E+000 --- 55

Figure 5.33 Filtered Track 4 Error Eliioses,Reinitialize 5

78

S

Y-POS (FT)103

51

• X-POS (FT)

-1I

-03

-4

SIGV-30 R-900 COVW=150 TIME AREA(SO FT) LEG. RESET

75 1348E+000 - 6965 2452E+000 .... 81

Fi.ure 5.34 Filtered Track E Error 3liise, 0

Reiniti~ilze 5

7 I

Y-POSITION (MT

44

304

7u5,35 ril-er,,J Track< 4, a =3 ,=R900, ~1

Reinitializ-e 7

I, T 7 1 ___ __

A. ERROR ELLIPSE

The f4ilter error ellipse proved useful as a tool for

indicating filter er.crmance. The information "r by

the ellipse, particularly surface area changes, was used to

make decisions concerning the alteration of the filher

parameters. Several approaches for using the error ellipse

were applied to both the linear and nonlinear tracking

oroblem and the results are summarized below:I

Procedure Applicable CommentsFilt er

( Linear orx t e nd ed)

tais-ics Both sUseful in keepin: the errorWiindcw elliose current and resocnsive

to cresent data. -he ellipsereflects most recent data; 3lidata is disregarded. eneficialwhen making a decision concerninzfilt7er Daramecer modification.Normally a better indicator ofi!ter c .nve...ce or dv_ _ ncethan without a statistics window.

:!ormalized Both Aid in displaying error trends

--rror as tarD-et ar-oacne o3or nate4 ~ize axis or origin. 'r -racti:-a

in ceterMnnng fiicr conver-7ence/diver-ence l:e t: scsdellicse changes in 'ocinicy ofaxes.

1 ncreas 4rir ~x e n ei- S ceisu-n- C S IEll1ipS e A r ea ?(Klkr ren -: ef .

,and '-(-e / ) ee st t -Mocre e ewnen us-_c in.con_ i .no ion wi tnh s _t i 3 cs dno w.

Err or Extended This 3'mi-u varya y -3 CQ"

Expansion! crease or 'decrease is DarticularlyCompression tuserFl in a :_raokir-Iz environmentto vary COVW cont-aining lrc variations in the(Adaptive Q) random fcrcin- input. The proce-

lure is more effective when uisedin conju nction with a st atisticswindow. Using this t eonniquewith filter reinitialiZation Wasunsuccessful.

B. COMPTER PEF.FORMANCE

The HiP-86 proved to be an extremely reliable computer

w,;-tn no "jowntime exo-erienced jur~ng the 5 month Deriod of

,Dreration. Full(Real) :)recision was use,-- throughout the

suyprovidiing iS d'igit precision, which was more th.,an

a'e quane. The use of the Ma _ri x RCM redu-ced Drogram lengt-

J: 96 n ncreased, com-.uti ng speed by a :actor o: bu 6.

~.ug ntused in this study1 a Statist cs ROM would h!ave

uincoub)tedly ru,,rther increased cmuigspeed_-. For any

rrerstudi-y using t he HP-36, it- Is recommend'ei that_ a

7tatistics ROM be Drocured.

-cctoo arprox=inately 2 secOnds f=r eachn 4ncre-e7ntcl

tme m ea7,,rement -,:ao e cuce c ' te fil1,

'Ka nns, bhfrte liear and- nornline -I- e. T_3

IL

computing time also included all statistics computations.

With an incoming data rate -F I set of measurement Uc a cer

second. this computing speed is not sufficient for n-iine 4e

processing. As previously mentioned, the Naval Underwater

Tracking Range receives a series of 4 measurement times

sequentially every 1.31 seconds. The range's three-

dimensional tracking problem will necessarily involve more

than the 4 state variables used in this study. Hence

greater matrix dimensions resulting in longer computing •

times can be expected.

The HP-86 CRT graphics were used extensively to provide

error ellipse plots during the tracking runs. Using a "no *

frills" approach to plotting, i.e. plotting without x-y

axis or labelling, it took approximately 2.5 seconds per

ellipse plot. The ellipse plotting routine used involved D

sines and cosines, plotted point by point in 30 degree

increments for a total of 360 degrees. This method was

somewhat slow. Had there been available a graphics program *

that would sketch in the ellinse around the intersected

major and minor axis, the graphics presentation could have

been speeded u-3. But since ellipse plotting for every 3 to S "

5 increments of time Drovided sufficient "real-time"

information, the time of 2.5 seconds per plot was tolerable.

Summarizing, the HP-36 could be used to comoute •

statistics and Drovide 7raDhics in -he real-time underwater

tracking environment, if the graphics were required not moreS 4

S q

often than 3 to 5 seconds. However, before the HP-86 can e

considered feasible for real-time Kalman filtr procssinz,

more investigaz ion i: needi in findin wa173 to swe-v

computer processing time such as parallel processing,

additional use of manufacturer-provided ROMs and machine

language programming.

I8

AI

APPEfIiX A

TRC G,.ZNB.FATIO I

1. TRACK THREE

Target ovement is in the x-y plane with the tracking

sensor located at the origin of the cartesian axes. The

target follows a parabolic track (see Figure A-i) at a

constant speed of 200 feet per second.

y

h x

-V .Vx

Figure A-i Track 3

3?

0

The parabolic equation is:

!v

y 4pD(x-h)

where p = 1000 and h = 1000, resulting in:

2y 4 4000(x-1000)

Initial target location is (x,y) = (8000, 5291.5). Target

x-direction velocity is given by:

v = v cos(Angle) (A-1)

and target y-direction velocity is:

v = v sin(Angle) (A-2)Y

where the Angle is obtained from:

Angle = tan-1 (-X)LX

and v 4s obtained from:

S(v 2 + v2 ) 1/2

x 7

03

4

where the argument of the inverse tangent is the slope _f

small increment (less than 1 second) at each successiu:e sa=a

point. Usinlg a sampling period of one second, the 3a

points (x,y) can be obtained from:

x(k+l) = x(k) + v (k)X

y(k+l) = y(k) + v (k)y

Table A-i gives the numerical values for the four states.

2. TRACK FOUR

Target movement simulates a 30-knot torpedo (velocity 50

feet per second) at a constant depth. The target's initial

position is (x,y) = (3250, 5000), and it is moving in the v

direction with v =0. (See Figure A-2.) The target remainsy

on a straight course for 15 seconds, and then executes a 90

degree turn and travels in the -y direction at v = -50 ft/sec.y

The trajectory of the target turn is described by a 90 degree

arc of a circle of radius, R = 1000, with the circle centered

at (x,y) = (4000, 4000). The 90 deg arc will be traversed

in:

27rR/4v sec. 107 sec.(where v = 50 ft/sec)

37

TABLE A-i

NUMERICAL VALUES OF STATES FOR TRACK THREE

K X X-VEL Y Y-VL

1 eO00. CO -186.90 5291.50 -71.152 7813.10 -186.93 5220.38 -71.133 7626.17 -186.60 5148. 27 -71.98

•4 7439.5E -186.25 5075,26 -72.87c 7253.33 -185.89 5001.33 -73.79

6 7067.44 -185.51 4926. 43 -74.747 6e81.93 -185.11 4850.54 -75.73

8 6696. 62 -184.68 4773. 60 -76.76

9 6!12.14 -184.24 4695.59 -77.83

10 6327.90 -183.76 4616.45 -78.94

11 6144.14 -183.26 4536. 14 -80.09

12 5960.87 -182. 73 4454.60 -81.30

13 5778.14 -182.17 4371.79 -82.56

14 5595.98 -181.57 4287.65 -83.8715 5414.41 -180.92 4202.10 -85.24

16 5233.49 -180.24 4115.09 -86.68

17 5C53.25 -179.51 4026.54 -88.19

18 4873.74 -178.72 3936.37 -89.78

is 4695.C2 -177.87 3844.49 -91.44

20 4517.15 -176.96 3750.81 -93.1921 4340.19 -175.97 3655.24 -95.04

22 4164.22 -174.91 3557.65 -97.00

23 3989.31 -173.74 3457.93 -99.06

24 3815.57 -172.48 3355.93 -101.25

25 3643.0c, -171.09 3251.52 -103.57

26 3472.00 -169.57 3144. 52 -106.05

27 3 02.43 -167.89 3034. 75 -108.68

28 3134.54 -166.04 2922.01 -111.50

25 2S68.5C -163.98 2806. 06 -114.51

30 2804.52 -161.67 2686.65 -117.7431 2642.E5 -159.09 2563. 47 -121.21

32 2483.76 -156. 17 2436. 20 -124.94

33 2327.59 -152.86 2304.42 -128.98

34 2174.74 -149.07 2167. 70 -133.3335 2C25.66 -144.72 2025.50 -138.05

36 1880.95 -139.67 1877. 18 -143.15

37 1741.28 -133.78 1721.95 -148.67

38 1607.50 -126.83 1558.85 -154.64

39 1480.67 -118.59 1386. 61 -161.05

40 1362.08 -108.70 1203. 46 -167.88

41 1253.38 -96.73 1006.73 -175.05

42 1156.65 -82.01 791.58 -182.4143 1C74.6 4 -63.44 546. 41 -189.67

44 1011.20 -37.24 211. 63 -196.50

45 1coo. CC 0.00 0.00 -200.00

88

TABLE A-1 (CONT.)

K X X-VEL Y

46 1011.20 10.57 -211.63 -199.724-0 1C21.77 25.14 -295.07 -198.4148 1046.90 35.82 -433. 13 -196.7749 1082.72 48.89 -575. 23 -193.9350 1131.61 61.85 -7251 57 -190.2051 1193.46 74.49 -879. 69 -185.6152 126 7 .95 86.36 -1035.28 -18C.3953 1354.31 97.25 -1190.48 -174.7754 1451.56 107.04 -1343.96 -168.9455 1558. 6C 115.75 -149U. 80 -163.1056 1674.35 123.43 -1642.38 -157.3757 1797.7E 130.16 -1786.37 -151.3558 1927.94 136.06 -1926.60 -14o.5859 2C64.01 141.24 -2063. 01 -141.6160 2205.24 145.78 -2195.67 -136.9261 2351.02 149.78 -2324. 67 -132.5462 2500.80 153. 31 -2450. 14 -128.436 3 2654.11 156.45 -2572.25 -124.6364 2810.56 159.24 -2691.14 -121.0165 2969.8C 161.73 -2806.99 -117.6666 3131.52 163.97 -2919.95 -114.5267 3295.4S 165.98 -3030. 17 -111.5868 3461. 47 167.80 -3137. 1 -108.826 3E29.2 7 169.46 -3243. 01 -106.2370 3798.73 170.96 -3345.88 -103.7971 3S69.6s 172.34 -3446.56 -101.4972 4142.03 173.6C -3545. 15 -99.327-Z 4315.62 174.76 -3641.77 -97.2674 4490. 3 175. 82 -3736. 51 -95.3275 4666.21 176.81 -3829. 47 -93.4876 4843. 02 177. -3 -3920. 72 -91 .7377 5C20.74 178.57 -4010.36 -90.0678 5199. 32 179. 36 -4098.45 -88.487S 5378. 6 180.10 -4185. 06 -86.9780 5558. 78 180. 79 -4270. 26 -85.5381 5139.57 181.44 -435U. 11 -84.1582 5S21. 01 182.04 -4436. 67 -82.8383 E 103.05 182.61 -4517.99 -81.5784 6285.66 183. 14 -4598. 11 -80.3685 6468.80 183.65 -4677. 09 -79.2086 6652.45 184. 13 -4754. 98 -78.0987 6E36.5E 184. 58 -4831. 80 -77.0188 7021.16 185.00 -4907.61 -75.98e9 7206. 16 185.41 -4982. 43 -74. 990 7391. 57 185.79 -5056.31 -714.03

89

*Yx

I ..y.. V

A-

0

Figure A-2 Track ~4

So each second:

2-74 =.05 radians will be traversed 4

a Using the trigonometric identity:

sin 2 (A) + cos 2 (A) 1

39

And the equation for a circle:

2 n 2X +y =C2

It follows that the arc of Figure A-2 is described by:

x(k) = 4000 + 1000sin(.05k)

y(k) = 4000 + 1000cos(.05k)

where the angle argument is in radians and k=0,1,2,...31

seconds. The velocities v and v can be obtained from:x y

v (k) = SOcos(.OSk)

yv (k) =-50sin(.05k) 4

using the same angle argument.

The track values for the track arc are contained in Table

A-2.

91 3

J I

TABLE A-2

NUe1ERIAL VALUES CF STATES FOR TRACK FOUR

K X-VEL Y Y-VEL

12 3049.9S 49.98 4999. 38 -1.2513 3099.S6 49.94 4997. 50 -2.501a 3149.46 49.86 4994.38 -3.7515 3199.E7 49.75 4990. 01 -4.99 S16 3249."5 49.61 4984. 40 -6.2317 3298.88 49.44 4977.54 -7.4718 3348.22 49. 24 4969. 45 -8.7119 3397.34 49.00 4960. 13 -9.9320 3446.2 1 48.74 4949. 59 -11.1621 3494 .1 48.45 4937. 82 -12.37 •22 3543.C . 48. 12 492U. E5 -13.5323 3591 .C4 47 77 4910. 67 -14. 7824 3638.62 47.38 4895. 30 -15.972 3685.8C 46.97 4878.75 -17.1426 3732.55 46.53 4861.02 -18.3127 3778.E4 46.05 4842. 12 -19.4728 3824.64 45.55 4822. C8 -20.622s 3869.93 45.02 4800. 89 -21.7530 3914.68 44. 46 4778.59 -22.8731 -958.E5 43.88 4755.17 -23.9732 4002.43 43.27 4730.65 -25.0633 4045.37 U2.63 U705.05 -26.1334 4087.67 41.96 4678.38 -27.1935 4129.28 41.27 4650.67 -28.2336 4170. 19 40.55 4621. S3 -29.2537 4210.3- 39.80 4592.17 -30.2638 4249.7S 39.04 4561. 41 -31.2439 4288.44 38.24 4529.68 -32.2140 4326.27 37.42 4497.C0 -33.1641 4363.28 36.58 4463. 38 -34. 0842 4399.43 35.72 4428. 84 -34.994- 4434.1 34.84 4393.41 -35.87u4 4469. 10 33.93 4357. 11 -36.7345 4502.56 33.00 4319.97 -37.5646 4535.C9 32.05 4281.99 -38.3841 4566.65 31 .09 4243. 22 - 9. 17

:2*

11 A-2 (CONI.)

X X-VEL Y Y-VEL

48 4597.24 30.09 4203.67 -39.9349 4626. E3 29. 0 4163. 37 -40.6750 4 655. 4C 28.06 4122. 34 -41 .3951 4682.S4 27.02 4080.60 -42.07

52 4709.43 25.95 4038. 20 -42.7453 4734. E5 24.88 3995. 14 -43.3754 4759. 18 23. 79 3951. 46 -43.9855 4782.41 22.68 3907. 19 -44.56

56 4804.54 21. 56 3862. 35 -45.1157 4825.53 20.42 3316. 97 -45.6458 4845.38 19.28 3771.09 -46.1359 4864. C8 18. 12 3724..72 -46.6060 4881.61 16. 95 3677. E9 -47.0461 4897.S7 15.77 363). 64 -47.4562 4913. 14 14.5E 3583. CO -47.83E3 4927.12 13.37 3535.00 -48.1864 4939.eS 12. 17 3486.66 -48.50

j E5 4951 .45 10.95 3438. 01 -48.7966 4961.79 9.73 3389. 10 -49.04

67 4970.SC 8.50 3339.93 -49.2769 4978.78 7. 26 3290. 56 -49.4769 4985.43 6.03 3241.01 -49.6470 4990.83 4.7E 3191.20 -49.7771 4 994. 3.54 3141. 47 -49.3772 4997.90 2.29 3091.56 -49.9573 4999.57 1. 04 301. 59 -49.9974 4999.98 -. 21 2991. 59 -50.00

9

A , P -H

COMP'JTER PROGRAM EXP LANATIDN

1. i±HMAL

The L7, KAL ro-gram comcutes the filter ?A:;(.2)

for the 4-state system and stores the gains in "LINGAIN

.STORAG". The theoretical covariance of error matrix, 7I_,(4,4),

is also computed and stored in "LINCOV.STORAG". Several

different sets of gain and covariance values were conmputed

and stored for various values of measurement noise, RMAT(2,2),

and random forcing noise, COVW(2,2).

2. 1,!NE ST

T :he LI.EST program retrieves the appropriate ja nn

sc.qedule from storage and computes the optimal estimate,

X.AT(4,l), and the optimal one-step prerczion, X.H!l('-I)

The following capabilities are contained in the LINST program:

a. Gating Scheme

If the absoluze difference (DIF-) betw een the one-s* ....

Drediction, XHKIK and the noisv track value, ZHAT, is greater

than tne three-s 7ma gate, then GAIN matrix is di..sre.:.

and XHAT is set ecual to XHh1IK.

. ra ck Noise

'i, :ITA AiK 1, the r-ndom rumber aenerator,

,ID, and the ..- utin.. simulated, n'i2z - v1 n: ;,

4

are bv!,:,a ssed a nd th-e -r ack -. alIue s corotd;iti7, noisl'UZ

arc etr e f-_crn" dft fil 3 RiKS " NT: 7-7.

programn run, producing a different set of noise values

resulting in a unique noise corrupted track for each run.

C. Stati:-s t ic s W indow

When WINDOW is set to 0, the filtcer stat-e statistzocs

are computed after each Monte Carlo run. The error mean,

variance and position covariance are computed. If WINDOW

is set to an integer, I, such that max k.>I>0, the stat istics

will be computed based on the data compiled during the 'Last

I iterations of the simulat:ion. If, fo r e xam ple , 'AlINJD 1~

the computati on of statistics will be based on the c ata

obtained during the last ten iterations of k, and al!

:)rev 4ous data is disregarded.

d . Error 'TormaliZation

By setting NORM = , the error (ERR), whico is

de fi-Ln ed as the difference between the tru-e t-rac , valu.e (TR.AY)

adthe e stimate (XHAT), is normaliz7ed For all other value; s

of NORMI the relative error is S Sed in comp utn te

statistics.

e. F ein it ial1i z ing EO7a_4n s

T he program has the opt ion D." reinit iacizccg toe .

-to G0) T h is c an b e don e by7 set-_tin -L~ n- Hn nt et

suco, toat max JI .. If the sur* ace a-rea

ILiPNOT OF NIL-flU ERROR ELLIPSEW AN M fW-OF KELIMN FILTER PEWFORUANCECU) NAVAL POSTIRROUNTESCHOOL MONTEREY CR J JANOS NA 64

UNCLASIFIED F/G M4 ML

EEEEEEEEEE-E

-~~~~ - - -- - - - -

------ i 332 1112 20

L

1.25 ~ 1.

MICROCOPY RESOLUTION TEST CHARTNATIONAL BLIR4AU OF SANDARO 19bi A

increases I consecutive times, indicating filter divergence,

the gains are reinitialized to G(O). If reinitializing

gains is not desired as an option, then HIGH is set :o some

arbitrary large number greater than max k.

3. EXTKF

The EXTKF program computes the optimal estimate, XHAT,

and the optimal one-step prediction, XHKIK, for the 4-state

nonlinear tracking problem. The EXTKF program has the

options: gating scheme, track noise, statistics window,

and error normalization as described for the LINEST program.

EXTKF has the following additional options:

a. Reinitializing the Filter

The program has the option of reinitializing the

covariance of one-step prediction error matrix, PKIK, by

setting it to 10nxPo, where n is zero or some small integer.

If HIGH is set to an integer I, such that max k>I>0, and the

surface area of the ellipse increases I consecutive times,

then the PKIK matrix will be reset to 10 .xP--0

b. Adaptive Q

EXTKF has the option of automatically increasing or

decreasing the state excitation matrix, 2, under certain

conditions by changing the value of the covariance of

excitation noise, COVW, INCREASE and DECREASE are set to

integers I and J respectively, such that max k>I,J>0. If

the surface area of the error ellipse increases I

96

consecutive times, COVW is doubled or increased by some

factor. Conversely, if the surface area of the error ellipse

decreases J consecutive times, COVW is halved or decreased

by some factor. if COVW is changed, increased for example,

the value of COVW can be changed again later in the run,

if the criteria for either increasing or decreasing is met.

If the adaptive Q is not desired then INCREASE and DECREASE

are set to some large number greater than k.

97

APPENDIX C

PROGRAM LISTING

*N

=#-4 %%w 4*

mw t t - * *.

o)g %* 04. * *-uON-0 ft) * 0

U) iii a a* *0 W

trk 04'-* W4 4.N 4

~O) %4 %*Z 0 4t 41.v lb to. (o

4pra %C4 O OjOc %M, IC

+10 404 Q') %MM 01 o*AI(

06 0 %4 W 0* k q-4

aL w 41 0 41SrqCV~- 11.4 SJ4 llt1 *0 4- WO*aLI-i

"q4. o~'.r 0v a1 Mv* rv 40WW6L 4 0- % . n LH a aa* 0 1 JM J b40401 41c 04 is0 4=I . i C79 # m Inr* VV4 M thw

w (U 44 - ull l b 1M- 0ql p a0 MntPE "-4J o 0 00 CQ : tU1= U * Q .) *W0f~ 4 t (C W1 %"CV %c jA .1 * * V1 * M140+G) m 0 . i~l3= V ) ft %, a. AU (*.,I 0lP A4)L=

=. U.i'JO4 - 00o% % % + =0a -40 Wi )fiU to ' 4 .4 %0 %0. 3 4 '.1-44 #UO 4 J Q'. ~ 11 W

orf~ fC*'U)4 0 04.40%- * 4-4 *M AC-. 4J 4w.7- %"* *zz 0 Uor %aK tifB 04-rI' o+J(NBWBq 1.4714 0"1 f

+' o %24 Wr i - C.0- V%0 k%0 %* 11 4J It CM V '_ rl u* r''

o~~~~ -o CM' 3 ON*QO. 41*4 NUUJ-' **Y).4 4

1.4 V _ aU U4-1-aQ (d. %to P 3 to mu) to * co 8 a V * 1 ci wai 0) T4. 04* *70-4.C0 0'4) 4)bv Wv+i '.q W. 0 (1) 4 CO*0 0WB* M. W. (04) 00 4J "D40*V

toWr U&4 0%+'.-) C W W 04- Wt $ 4 Q 041 B~i-W A*4~tW

ig. u. 0 4 4- toc- &' MU to * . gg Ig Li *r.Q+.JJ4+* U4 tilU Q(3 4-)0 4* 0 - 4N 4-*W

-A ~ ~ ~ ~ ' o4 10 d o1(4U)t(0M1 0CCQto M o * ' (0 0I* 0-4CC o( t~-o o* *.-1t- 0 MO 1 0)U 1USU 110 fV V q VA1 v 0 0)4 U)4 NIB it)4 6- %4# l 8 I 0c: 0f

U.0.c-'-UG) 0 .O .U(0tf0410.-UUO1oc93a400 o4*

'- .*0'4b41'W' W'0 C40* ~~0)* .~C j49.80

as ,)4J 140 4 ' U10 t - *-414. 4- Q CU +J fl

Wr-4 M *14 -Cs 0! .40~ adq 0 (11 -

0J44 Ii*$- V 4 2 c.) (1 l 0 C: 0 ) Uwo a) ( Ou*0Oc: 0 IU .1 %, U.

(d0fd 4 .;P di H. f-4wO V 'D - ONof-I 050 0 4Q to .

-A X r. % %* WS 3110 1) 4.4( f/r4 0ri 9 W.m0W

W..-A 94 t W *i.44-4 Ad 011 " a ) -P-tI-W+)S.4 41-M~( m0O o 1t U 1tI M54W W0 144r4 fV.0 fl ( CLS-IS9% a,4 ('4.U A Or4 W,4 -CO

0 M0( M l etX4 1 I 4J-r4 W0) 0 Fw 4 V -W+ -a4 o '0+) :PC w t:0 u f0 'P a W'4f W~P tr 014

*4 Ara - t cuoo NfV 0 D W) 10 3k -'P.X P..+'

CUP Ar" T o0uS0gm .,tg (CO 4 U4 ' U.~ i u

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LIST OF REFERENCES

1. Sorenson, H.W., "Kalman Filtering Techniques", Advancesin Control Systems, v. 3, pp. 219-292, Academic Press,1966.

2. Kirk, D.E., EE4413 Class Notes "Optimal Estimation: AnIntroduction to the Theory and Applications", (unpublished),Naval Postgraduate School, 1975.

3. Jazwinski, A.H., Stochastic Processes and FilteringTheory, Academic Press, 1970.

4. Cramer, H., Mathematical Methods of Statistics, PrincetonUniversity Press, 1961.

5. Mitschang, G.W., An Application of Nonlinear FilteringTheory to Passive Target Location and Tracking, PhDDissertation, Naval Postgraduate School, 1974.

6. Heffes, H., "The Effects of Erroneous Models on theKalman Filter Response", IEEE Trans. Automatic Control,v. 11, pp. 541-543, 1966.

7. Nishimura, T., "Error Bounds of Continuous Kalman Filtersand the Application of Orbit Determination Problems",IEEE Trans. Automatic Control, v. 12, pp. 268-275, 1967.

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BIBLIOGRAHY

Anderson, B.D., and Moore, J.B., Optimal Filtering,Prentice Hall, 1979.

Meditch, J.S., Stochastic Optimal Linear Estimation andControl, McGraw-Hill, 1969.

Sage, A.P., and Melsa, J.L., Estimation Theory withApplications to Communications and Control, McGraw-Hill,1971.

Ladas, J.E., Interactive Monte Carlo Simulation Programfor Evaluation of Estimators and Predictors for FireControl Systems, Master's Thesis, Naval Postgraduate

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