An analysis of a new nonlinear estimation technique

MISSING IMAGE

Material Information

Title:
An analysis of a new nonlinear estimation technique the state-dependent Ricatti equation method
Physical Description:
xi, 125 leaves : ill. ; 29 cm.
Language:
English
Creator:
Ewing, Craig M
Publication Date:

Subjects

Subjects / Keywords:
Aerodynamic load -- Mathematical models   ( lcsh )
Nonlinear control theory   ( lcsh )
Aerospace Engineering, Mechanics, and Engineering Science thesis, Ph.D   ( lcsh )
Dissertations, Academic -- Aerospace Engineering, Mechanics, and Engineering Science -- UF   ( lcsh )
Genre:
bibliography   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (Ph.D.)--University of Florida, 1999.
Bibliography:
Includes bibliographical references (leaves 122-124).
Statement of Responsibility:
by Craig M. Ewing.
General Note:
Typescript.
General Note:
Vita.

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 030472549
oclc - 43460574
System ID:
AA00012992:00001


This item is only available as the following downloads:


Full Text










AN ANALYSIS OF A NEW NONLINEAR ESTIMATION TECHNIQUE:
THE STATE-DEPENDENT RICATTI EQUATION METHOD
















By

CRAIG M. EWING


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA














ACKNOWLEDGMENTS

I would like to thank my advisor, Dr. Norman Fitz-Coy, for his efforts in

continually pushing me to develop a better dissertation. I agree it is now something I can

look upon with pride. I also thank Dr. James Cloutier, the man behind the idea. Without

his help and guidance this work could never have been written. I would especially like to

thank my wife, Darsi, for her never-ending support through these years of work. She

never let me give up.














TABLE OF CONTENTS



ACKN O W LED G M EN TS............................... .................................... ....................ii

L IST O F T A B LE S .............................................................................. .................... v

L IST O F FIG U R E S .................................................................... ..................................vi

A B ST R A C T ............................................... ................................................................... x

CHAPTERS

1 INTRODUCTION....................... ............................................. 1

2 LITERA TU R E SU R V EY ............................................................ ............................4

3 SCOPE OF W O RK .................................... ....................................................... 8

4 EXOATMOSPHERIC-GUIDANCE-PROBLEM ENGAGEMENT SCENARIO ..... 10

5 EXOATMOSPHERIC-GUIDANCE-PROBLEM FILTER DEVELOPMENT.......... 13

5.1 EK F D erivation ................................................................. ........................ 14
5.2 Bootstrap Estimator Derivation............................................... 19
5.3 SD REF D erivation ............................................................ ........................ 22

6 EXOATMOSPHERIC-GUIDANCE-PROBLEM SIMULATION RESULTS........... 49

6.1 B aseline Perform ance.................. ...................... ................................................. 49
6.2 PDF Distribution Comparison................................................... 54
6.3 Sensitivity A analysis .............................................................. ..................... 64
6.4 SDREF Closed Loop Performance............................................... 79
6.5 Summary of Exoatmospheric Guidance Problem.............................................. 87

7 PENDULUM PROBLEM FILTER DEVELOPMENT.............................. .............. 88

7.1 SD REF D erivation .......................................... .......................................... 90
7.2 EKF D evelopm ent.............................................................. ....................... 91

8 PENDULUM PROBLEM SIMULATION RESULTS................................ ....92


iii








9 MSDREF DEVELOPMENT ................................................................. ............ 103

9.1 M SD REF D erivation........................................................................................103
9.2 Stability Proof................ .................................. .............................. 104

10 MSDREF PROBLEM SIMULATION RESULTS............................................. 112

11 CRITICAL ANALYSIS OF RESULTS ........................................ .................... 119

REFERENCES .................................................................................................... 122

BIOGRAPHICAL SKETCH................................................................................... 125














LIST OF TABLES


Table page

1. Filter simulation parameters baseline conditions............................. ............ 50

2. EKF and SDREF parameters one-percent initialization error.................................. 65

3. EKF and SDREF parameters five-percent initialization error ................................ 69

4. EKF and SDREF measurement noise parameters level 1 ....................................... 72

5. EKF and SDREF measurement noise parameters level 2..................................... 72

6. M aneuvering target parameters................................... .............................. 76

7. Head-on simulation parameters.......................... ....................... 81

8. 90-degree beam shot simulation parameters ......................... .......... ........... .. 81

9. Tail-chase simulation parameters.......................................... 82

10. Miss distance performance for varying scenarios................... ........................ 83

11. Miss distance performance for stressing target maneuvers.................................... 84

12. Miss distance performance for varying measurement noise................................. 85

13. Miss distance performance for varying initialization error...................................... 86














LIST OF FIGURES


Figure page

1. G generic intercept vehicle.......................................................... ...................... 11

2. Guidance system block diagram.......................................... 12

3. EKF position error with 3a standard deviations ................................................... 52

4. SDREF position error with 3a standard deviations ................................................52

5. Bootstrap position error with 3o standard deviations .......................... ............. 52

6. Comparison of EKF and SDREF covariance 3o standard deviations
for Y position .................................................................................................... 53

7. Expanded view of Figure 6 .................................................................................. 53

8. Posterior probability density functions at time = 1, 5, and 9 seconds
for X position .................................................................................................... 55

9. Posterior probability density function at time = 1, 5, and 9 seconds
for Y position ...... .............. ............................................................ 56

10. Monte Carlo posterior probability density functions at time = 1, 5, and 9
seconds for X position................................................................ ....................... 58

11. Monte Carlo posterior probability density functions at time = 1, 5, and 9
seconds for Y position............................................................... .......................... 59

12. EKF estimated and true acceleration............................................ ....................... 61

13. SDREF estimated and true acceleration......................................... ..................... 61

14. Bootstrap estimated and true acceleration..................... ......................61

15. Monte Carlo posterior probability density functions at time = 2, 2.5, and 3
seconds for thrust cut-off X position...................... ........................ 62








16. Monte Carlo posterior probability density functions at time = 2, 2.5, and 3
seconds for thrust cut-off Y position............................................ .................... 63

17. EKF position errors with 3o standard deviations for 1-percent
initialization errors .................................................... .. .... ....................... 67

18. SDREF position errors with 3o standard deviations for 1-percent
initialization errors ................................................................ .... ....................... 67

19. EKF position errors with 3a standard deviations for 5-percent
initialization errors ................. .... .. ....................... 68

20. SDREF position errors with 3o standard deviations for 5-percent
initialization errors ..................................................... .. ... ....................... 68

21. SDREF position errors with 3a standard deviations after new initialization
method, -percent error .................................................................................... 70

22. SDREF velocity errors with 3O standard deviations after new initialization
method, 1-percent error .... .......... ..... ....................... 70

23. SDREF acceleration errors with 3o standard deviations after new initialization
method, 1-percent error ........................ ....................... 70

24. SDREF position errors with 3o standard deviations after new initialization
method, 5-percent error................ ............................................................. 71

25. SDREF velocity errors with 3o standard deviations after new initialization
method, 5-percent error ...................... ...................... 71

26. SDREF acceleration errors with 3a standard deviations after new initialization
method, 5-percent error .............................................................. ...................... 71

27. EKF position errors with 3o7 standard deviations 1-percent error in X,
100 microradian error in Y and Z............................................. ....................... 74

28. SDREF position errors with 3a standard deviations 1-percent error in X,
100 microradian error in Y and Z.............................................. ....................... 74

29. EKF position errors with 3o standard deviations 10-percent error in X,
500 microradian error in Y and Z............................................... ...................... 75

30. SDREF position errors with 3cr standard deviations 10-percent error in X,
500 microradian error in Y and Z.............................................. ....................... 75








31. EKF position errors with 3a standard deviations for 10 deg/sec rotating
target m maneuver ................................................... ............... ...................... 77

32. SDREF position errors with 3a standard deviations for 10 deg/sec rotating
target m maneuver .......................................................... ................................ 77

33. EKF position errors with 3a standard deviations for dogleg target maneuver...... 78

34. SDREF position errors with 3u standard deviations for dogleg target maneuver. 78

35. A aspect angle description ........................................................ ............................ 80

36. Pendulum set-up ............................................. ............................................... 88

37. EKF and SDREF estimates vs. true theta with no initialization error..................... 93

38. EKF and SDREF estimates vs. true theta-dot with no initialization error .............. 93

39. EKF and SDREF estimation error for theta with no initialization error ............... 94

40. EKF and SDREF estimation error for theta-dot with no initialization error............. 94

41. Monte Carlo probability density function for theta, with no initialization error,
tim e = 0.5 sec ........................................ .............. ................................... 95

42. Monte Carlo probability density function for theta-dot, with no initialization
error, tim e = 0.5 sec......................... ... ............................ ....................... 95

43. Monte Carlo probability density function for theta, with no initialization error,
tim e = 2.0 sec ........................................ .................................................... 96

44. Monte Carlo probability density function for theta-dot, with no initialization
error, tim e = 2.0 sec..................................... ............................................... 96

45. Monte Carlo probability density function for theta, with no initialization error,
time = 3.0 sec ................................................97

46. Monte Carlo probability density function for theta-dot, with no initialization
error, tim e = 3.0 sec............................................................. ...................... 97

47. EKF and SDREF theta estimation error with initialization error............................ 99

48. EKF and SDREF theta-dot estimation error with initialization error ..................... 99

49. Monte Carlo probability density function for theta, with initialization error,
time = 0.5 sec ................................................ 100








50. Monte Carlo probability density function for theta-dot, with initialization
error, tim e = 0.5 sec............................................. ............... 100

51. Monte Carlo probability density function for theta, with initialization error,
tim e = 2 .0 sec ....................................................... ............. ......................... 10 1

52. Monte Carlo probability density function for theta-dot, with initialization error,
tim e = 2 .0 sec ............................................................. ................................... 10 1

53. Monte Carlo probability density function for theta, with initialization error,
tim e = 3 .0 sec ...................................... ......................................................... 102

54. Monte Carlo probability density function for theta-dot, with initialization error,
tim e = 3.0 sec ...................................................... .............. ......................... 102

55. True vs. estim ated X 1 state ....................................................................... 117

56. True vs. estimated X2 state ............................................................. 117

57. True vs. estimated X1 state, non-zero initial conditions....................................... 118

58. True vs. estimated X2 state, non-zero initial conditions.......................................118














Abstract of Dissertation Presented to the Graduate School
of The University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

AN ANALYSIS OF A NEW NONLINEAR ESTIMATION TECHNIQUE:
THE STATE-DEPENDENT RICATTI EQUATION METHOD

By

Craig M. Ewing

August 1999

Chairman: Norman Fitz-Coy
Major Department: Aerospace Engineering, Mechanics, and Engineering Science

Research into nonlinear estimation techniques for terminal homing missiles has

been conducted for many decades. The terminal state estimator, also called the guidance

filter, is responsible for providing accurate estimates of target motion for use in guiding

the missile to a collision course with the target. Some form of the extended-Kalman filter

(EKF) has become the standard estimation technique employed in most modem weapon

guidance systems. EKF linearization of nonlinear dynamics and/or measurements can

cause problems of divergence when confronted by highly nonlinear conditions. The

objective of this dissertation is to analyze a new nonlinear estimation technique that is

based on the parameterization of the nonlinearities. This parameterization converts the

nonlinear estimation problem into the form of a steady-state continuous Kalman filtering

problem with state-dependent coefficients.








This new technique, called the state-dependent Ricatti equation filter (SDREF),

allows the nonlinearities of the system to be fully incorporated into the filter design,

before stochastic uncertainties are imposed, without the need for linearization.

The SDREF was investigated in three problems: an exoatmospheric, terminal

homing, ballistic-missile intercept problem; a highly nonlinear pendulum example; and

an algorithmic loss of observability problem.

The exoatmospheric guidance problem examined nonlinear measurements with

linear dynamics. To investigate the SDREF when used with a combination of nonlinear

dynamics and nonlinear measurements, a highly nonlinear, two-state pendulum problem

was also examined.

While these problems were useful in gaining insight into the performance

characteristics of the SDREF, no formal proof of stability could be determined for the

original formulation of the estimator. The original SDREF solved an algebraic SDRE

that arose from an infinite-time horizon formulation of the nonlinear filtering problem. A

modification to the SDREF formulation was developed that led to a differential SDREF,

and a proof for local asymptotic stability was achieved. The modification removed the

infinite-time horizon assumption and integrated the coupled state-dependent state and

covariance equations. This new form of the estimator is called the modified SDREF

(MSDREF). A problem involving algorithmic loss of observability was then examined.

This problem shows a performance advantage when using parameterization versus

linearization as in the EKF technique.














CHAPTER 1
INTRODUCTION

The extended Kalman filter (EKF) has become an accepted standard for both

tactical and strategic interceptor guidance problems. While some problems lend

themselves to this type of estimation strategy, systems with highly nonlinear dynamic

environments or misunderstood uncertainties can cause divergence of the filters. Many

alternative nonlinear estimation techniques have been researched in an attempt to

overcome this problem. The linearization of the nonlinear dynamics and/or

measurements is one area of concern. The EKF linearizes about the current estimate of

the state. If the state estimate is fairly accurate, this method has been shown to be

effective. If, however, the estimates are poor, linearization can cause errors in the filter,

which can lead to degraded performance or possible divergence of the state estimate.

It is this linearization that motivates the following research. This dissertation

investigates a new nonlinear estimation algorithm, which originates from a recent

nonlinear regulator design technique. It uses parameterization to bring the nonlinear

system to a linear-like structure with state-dependent coefficients (SDC). The filtering

equivalent to the regulator technique can be realized in a recursive nonlinear estimation

algorithm, which has the structure of the steady-state continuous Kalman filter. The

Kalman gain is found by solving the state-dependent Ricatti equation at each filter update

time. The filtering technique, called the state-dependent Ricatti equation filter (SDREF),







2
allows the incorporation of nonlinear dynamics and/or measurements in the filter design

without the need to linearize the equations.

The SDREF was analyzed in a several ways to determine the benefits and issues

associated with this type of structure. By using a sampled-data Bayesian filter, known as

the bootstrap filter, the "true" probability density function (pdf) of the SDREF states was

monitored. This allowed an investigation into the differences in pdf evolution for

linearizing versus parameterizing filters. Finding the true pdf can be difficult; however

recent work, by the author on the bootstrap estimator, proved valuable for this analysis.

The bootstrap technique uses thousands of samples of the pdf along with Bayes' rule, to

estimate the true pdf of any given state. With this filter and a large number of samples,

the "true" state pdf can be closely approximated. An EKF was evaluated in a similar

manner for comparison purposes.

The SDREF was exercised in three problems. A realistic and detailed evaluation

of the filter was accomplished by examining an exoatmospheric guidance problem. The

SDREF, bootstrap filter, and EKF were coded into a six-degree-of-freedom (6-DOF)

FORTRAN simulation for analysis in an exoatmospheric terminal homing engagement

using passive and active sensor information. The sensor measurements included azimuth,

elevation and range. The characteristics of the SDREF and EKF were evaluated in

several different engagement scenarios. The SDREF was developed in a Cartesian

coordinate frame, which causes the measurements to be nonlinear, while the dynamics

remain linear.

The second problem was that of a highly nonlinear, two-dimensional pendulum

problem. Evaluation of the SDREF when used with nonlinear dynamics and nonlinear







3
measurements was examined. The SDREF and EKF were coded into the visual

simulation environment VisSim. The performance of the filters and their pdf's were

compared. This academic problem was well suited to exploring any differences that

nonlinear dynamics may have on the performance or pdf characteristics of the SDREF.

No formal proof of stability could be determined using the above formulations of

the SDREF, so an alternate filter formulation was sought. The original SDREF was

based on an infinite-time horizon formulation of the nonlinear filtering problem that

resulted in an algebraic SDRE. A MSDREF was developed that utilizes a differential

SDRE. Because the Ricatti equation is state dependent, it must be integrated in

conjunction with the state estimates. A proof of local asymptotic stability was able to be

determined. The new MSDREF was examined in a problem consisting of loss of

algorithmic observability; and the results indicated a significant performance advantage

over the EKF.

This dissertation fully develops the requirements for the SDREF and MSDREF. It

includes an investigation of controllability and observability, which are point-wise and in

the linear sense and pertain to a valid SDRE solution. An exhaustive investigation of the

pdf characteristics, noise sensitivity, initial-error robustness, and closed-loop performance

of the SDREF using the EKF as a comparative source was performed. A theoretical basis

for the MSDREF, when used as an observer, was shown. The following research

provides theoretical and practical information demonstrating that the SDREF and

MSDREF are viable candidates for nonlinear estimation, and for some problems can out-

perform the EKF.














CHAPTER 2
LITERATURE SURVEY

The literature abounds with research of various nonlinear estimation techniques,

most of it mathematical and not application oriented. An elegant presentation of the

history of air-to-air target state estimation techniques by Cloutier et al.' outlines a wide

variety of techniques, which include the Kalman filter, EKF, second-order Gaussian filter

(SOGF), recursive nonlinear filter, recursive maximum-likelihood filter, and the assumed

density filter. These filters have also been examined in the context of single- and

multiple-model adaptive filters. Adaptive filters may make use of certain properties of

the residual or covariance to continuously change filter parameters to gain better filter

performance. On the other hand, multiple-model filters use banks of filters to model

many possible target dynamic situations. Various methods are then used to either select

the results of one filter or combine that of several.

With the advent of the Strategic Defense Initiative (SDI), a new set of problems

arose when estimating the motion of a boosted intercontinental ballistic missile (ICBM)

with the use of passive-only measurements. While the EKF is still the most widely

accepted estimator for this type of engagement, it has been shown to have degraded

performance when stressed by highly maneuverable targets.24 This is the same problem

that tactical filters experience in maintaining track of highly maneuverable aircraft or

missiles. When tracking a highly nonlinear maneuver, state estimates can become poor

and the linearization assumption begins to break down. The specific problems associated

4









with guidance filters for the exoatmospheric intercept problem have been addressed in

several research programs noted in the following paragraph.

Schmidt2 investigated an extension of work by Daum in which a closed-form

solution to the Fokker-Planck equations for propagating the pdf was developed. A target-

maneuver-detection algorithm was added to this formulation to enhance performance.

The emphasis in this research was to more accurately model the target acceleration based

on the rocket equation, instead of a Markov-process. The target model caused the state

equations to become nonlinear as well. The results of the work did show an improvement

over the EKF technique; however this was accomplished, in part, through the use of the

maneuver-detection algorithm. Gido3-4 developed a nonlinear estimation algorithm,

which numerically implements the theoretical work of Kolmogorov. The algorithm

places a grid over the pdf and uses the Fokker-Planck equations coupled with Bayes' rule

to provide the state estimates. While this method showed improved performance over the

EKF, it is computationally intensive. In the author's masters' thesis at the University of

Florida,5 another novel nonlinear estimation technique was investigated. It involved

using thousands of samples of the pdf, along with the state dynamics and Bayes' rule to

propagate and update the pdf .7 The technique places no restrictions on the type of

statistics used to model noise. It also makes no linearization assumptions. While

computationally intensive, it is directly applicable to a massively parallel processing

architecture since the same operation is done to each sample.

While the bootstrap method may not be seen in any missile guidance system

design in the near future, it does provide a brute-force method for obtaining a reasonable

representation of the true state PDF for a given problem. It is in this context that the filter






6

is used for the current research. While many of the above filters were shown to have

performance superior to that of the EKF, realistic implementation may be difficult.

Other techniques that have been proposed to solve the bearings-only guidance

problem include the modified gain pseudo-measurement filter (MGPMF)8 and modified

gain EKF (MGEKF).9 Both use pseudo-measurements to avoid the need for linearization

around a "nominal" trajectory, such as is found in the EKF. These methods are restricted

to a special class of nonlinear functions, which can be manipulated into a pseudo-

measurement form.

The construction of implementable nonlinear filters for nonlinear stochastic

systems remains a challenge. Numerous filtering algorithms, based upon series

expansions to approximately realize the conditional mean, have been suggested (see the

bibliography of Bucy et al.10) Once again, these techniques represented a heavy

computational burden, even for low-order dynamical systems.

The SDREF may alleviate some of the problems discussed so far. The technique

is derived from a new nonlinear regulator design methodology known as the state-

dependent Ricatti equation (SDRE) control method.11-12 The objective is to take the

nonlinear system described by


x= f(x)+g(x)u (2-1)

and transform it to the state-dependent coefficient form

x= A(x)x + B(x)u (2-2)

where f (x)= A(x)x B(x)= g(x) (2-3)







7

using direct parameterization. Once in this form the SDREF is obtained by considering

the state-dependent coefficients to be fixed and applying the linear continuous Kalman

filter to the "frozen" problem. The result is an algebraic SDRE whose structure is the

same as the steady-state Ricatti equation. The emphasis of this research is to develop and

analyze a guidance filter based on this technique. Initial work in the development of the

SDREF'3 demonstrates the filter in a simple two-dimensional pendulum problem. Other

applications include induction machine estimation,14 simultaneous state and parameter

estimation, 5 and gyro-less satellite angular rate estimation.16 The current research

extends their work to include an exhaustive investigation into the internal properties of

this filter in a strategic interceptor guidance problem, as well as in the friction-pendulum

problem. Insight into the internal representation of the pdf when using parameterization

versus linearization is examined, as well as the necessary conditions for stability. The

first example presented here uses only nonlinear measurements, while the pendulum

problem discussed herein adds the complexity of nonlinear dynamics as well. A third

example demonstrates the unique parameterization property of the SDREF when

confronted with a loss of algorithmic observability.














CHAPTER 3
SCOPE OF WORK

This chapter outlines the scope of the work in this dissertation. Throughout the

design and analysis of the SDREF, the EKF was used to compare performance issues

between linearized and parameterized methods. Similarly a sampled-data bootstrap filter

was used to compare probability distribution functions between the different nonlinear

filters in the exoatmospheric guidance problem. The three filters are theoretically derived

and application-specific equations are formulated for each problem. Controllability and

observability requirements for the SDREF algorithm are analyzed and determined.

Several performance measures are investigated that include sensitivity to initial errors,

noise parameters, and target maneuvers. In addition, a somewhat nonstandard

methodology, the bootstrap filter, is used to investigate the generally unobservable

evolution of the pdf within the filter. The pdf was examined at distinct time steps to

determine whether the parameterized filter pdf is different from the linearized filter. All

three filters were coded in FORTRAN and installed in a 6-DOF simulation for evaluation

in an exoatmospheric terminal homing engagement.

To examine the effects of both dynamic and measurement nonlinearities, the

SDREF and EKF were also coded in VisSim (a visual simulation tool for a simple, but

highly nonlinear, pendulum problem). Performance was measured in a manner similar to

the exoatmospheric guidance problem. Because no formal proof for stability could be

found for the SDREF, modifications were made that allowed a detailed proof of local

8







9
asymptotic stability to be found. The MSDREF is fully derived and an examination of its

performance against the EKF for a problem consisting of loss of algorithmic observability

is also explored.














CHAPTER 4
EXOATMOSPHERIC-GUIDANCE-PROBLEM ENGAGEMENT SCENARIO

The filters were initially analyzed in the context of an exoatmospheric terminal

homing engagement of an interceptor tracking a thrusting ICBM for approximately the

last 10 seconds before impact. The interceptor was closing on the target at approximately

10 kilometers per second. The target may perform thrust cut-off or other evasive

maneuvers, which provide for a stressing nonlinear engagement scenario in which to test

the filters. At the start of the engagement, the guidance filter was given erroneous initial-

state and covariance estimates. The sensor is assumed to have acquired the target and the

guidance filter is ready to estimate the relative motion between the two vehicles using

passive-only measurements. The estimates were used by a standard augmented

proportional-navigation (PRO-NAV) guidance law to command accelerations that keep

the interceptor on a collision course with the target. No errors associated with the attitude

control system were used as this would complicate the analysis and cloud the

performance comparison.

Figure 1 shows a model of the interceptor, a generic 4-kilogram vehicle

configured with divert and attitude control thrusters. A generic, SS-18-like ICBM model

was used to generate the target motion. The interceptor may have many error sources,

some of which include hand-over errors, pointing errors, thrust errors, center-of-gravity

location errors, and seeker noise. Initially many of these noise sources were not used so

that the filter characteristics could be more easily discerned. Several engagement closing

10







11

angles were examined to allow for more stressing conditions (i.e., tail chase, nose-on,

beam shot). Once proper filter operation was assured, noise sources were added to more

realistically emulate the true intercept scenario, as well as to identify weaknesses in filter

performance.



Back View
SDivert Thrusters



Top View Z
Divert Thrusters Back View





Attitude Control Thrusters
X Z Z

Figure 1. Generic intercept vehicle.


The simulation used was the Guided Simulation Program Exoatmospheric (GSP-

EXO).' It is a FORTRAN simulation developed by the General Electric Company for

the Air Force. Figure 2 shows a block diagram representation of the subsystems

modeled.

























Figure 2. Guidance system block diagram


To allow for rapid and efficient evaluation of filtering concepts, the author has

modified the guidance system. It has a Monte-Carlo capability with an outer loop around

the Monte-Carlo loop, to allow for parameter trades. Analysis was performed using a

personal computer; however, analysis can be accomplished on other platforms, including

the SUN, VAX, and CRAY.














CHAPTER 5
EXOATMOSPHERIC-GUIDANCE-PROBLEM FILTER DEVELOPMENT

The following paragraphs outline the formulations of the EKF, bootstrap filter,

SDREF, and MSDREF that were used for this research. A brief theoretical description of

each filter is given, followed by the application-specific equations.

There are three filter types under consideration in this research paper. The

mathematical derivations for all of the estimators are fully developed and all assumptions

regarding noise characteristics and linearization are explained. The filters are presented

in coordination with the problems that were used in examining their performance. The

SDREF theory and derivation used in the exoatmospheric problem is shown in this

Chapter, the SDREF pendulum problem and derivation is presented in Chapter 7, while

the MSDREF stability proof and analysis is found in Chapter 8.

Many variations of SDREF are possible due to the infinite number of

parameterizations available. Initially, simple parameterizations were used. Another

important factor in evaluating performance differences between the three filters was the

selection of the noise characteristics. The EKF performance may be enhanced in some

cases by artistically "tuning" the noise parameters within the filter. To assure an unbiased

comparison between the filters, noise parameters were set to the same nominal values for

all filters. Due to several limitations on the bootstrap filter, developed in a thesis by this

author, it was used only for pdf comparisons in the analysis of the strategic guidance

problem. The bootstrap filter needs further development to be useful in stressing

13






14
engagements as the number of samples required for adequate performance makes it

intractable to run for large numbers of comparison.

The SDREF and EKF were analyzed in great detail to assess key performance

characteristics including convergence, stability, and state estimation error. A key

performance critique was the comparison of the evolution of the SDREF and EKF pdf's

to that obtained by using the bootstrap filter. Interesting insight into the differences

between how the SDREF and EKF model nonlinearities was observed.

There are many issues involved with the SDREF development, including correct

parameterization of the nonlinear measurement equations, investigation into the

requirements for solution of the SDREF, and the stability of the SDREF. These areas

were explored to determine an accurate set of state and measurement equations and a

feasible method to solve them.

An initial non stressing scenario was selected to establish a benign comparative

baseline performance assessment. The pdfs were compared using this baseline scenario

as well as a nonlinear maneuvering target scenario. Once the baseline was established,

the filters were tested against more stressing engagements and intercept angles.

Performance was measured by examining the state estimate error and 3-sigma standard

deviations, as well as comparing the state estimate to the true state. The author has found

that this is an accurate method of evaluating filter performance.


5.1 EKF Derivation

The derivation for the EKF can be found in many sources,18-19 and is presented

here for completeness. The dynamics for the intercept problem at hand are linear, while








the measurement equations are nonlinear. For this reason a combination of linear and

EKF equations were used and implemented in discrete form. Given a linear, time-varying

differential equation for a dynamic system

x(t) = F(t)x(t) + G(t)u(t) + L(t)w(t) (5.1)

where x(0) is given, u(t) is a deterministic input, and w(t) is a random disturbance, the

corresponding discrete time solution is

t,
x(t) = D(t, t_,)x(t_) + fQ(tr .,)[G()u(r) + L(r)w(r)dr. (5.2)
1.-

If u and w are assumed piece-wise constant over the sampling interval, this can be

written as

Xk = ~ k-Xk-I + Tk-luk- +Ak ,_,wk (5.3)


where rk_ k = J (tk,r)G(T)udr (5.4)



and Ak-IW, =j (tk,r)L(r)wdr. (5.5)
1t-i

However if F, G, and L are also considered constant over the sampling interval, AT, then

4(At) = e" (5.6)

F(A) = D(At)[l, D-'(At)]F-'G (5.7)

A(Ar)= 'D(At)[,, -'(At)]F-'L. (5.8)

Assuming then that

E{xo) =m, E{(x-xo)(x-xo)T) = P

E{w)= 0 E{wwwkT)=Q; and

E{ww,_,} =0








the state propagation equation becomes

xk = (-), xk- + rk-,Uk_ + A,_(0) (5.9)

The covariance matrix can be propagated in a similar manner by

P. = l-,P- k-, l'r+ k- (5.10)

with Qk-i = Ak-Q-Ak. (5.11)

To relate the covariance matrix of random sequences to the spectral density of the random

process for the continuous disturbance input w(t) we note that

E{w(t)wT ()}= Q'c (t)S(t- z) (5.12)

The equation for Qk,_then becomes


Qk-1= i (tk')L(r)Q'c ( r)r(t t)dr (5.13)
k-;

This now represents the integrated effect of the continuous disturbance input w(t).

Thus the two equations necessary for the propagation of the state and covariance

are

xk = k-l,_l- + k_-Uk-_, (5.14)

P'k = kIPk-_ IT-i' + Qk-i (5.15)

where Ok-1 is defined by eFM and 1_, is define in equation (5.7).

The measurement process for the filter is now determined. Given the continuous,

nonlinear measurement equation

z(t) = h(x(t))+ v(t). (5.16)

The discrete version can be written as

z, = h(xk(t))+v (5.17)







17
A complete derivation of the EKF update equations can be found in,18 however the final

solution for the calculation of the Kalman gain, and the update of the covariance and state

are

k (+)= (-)+ K [zk -hk(k(-))] (5.18)

K, = 'k(-)HkT (i(-))[Hk(ik(-))Pk (-)H'(k ((-))+Rk (5.19)

Pk(+) = [I- KkH,((-))]Pk(-)[ KkHk (k (-))I + KRKkT (5.20)

where ik (-) and Pk (-)come propagation equations (5.14) and (5.15) and


Hk (xk = (5.21)


with E{v } =0 and E {vkvk } = Rk. Now (5.14) and (5.15) and (5.18) through (5.20)

define the full set of EKF equations.

For the exoatmospheric intercept problem the state consists of

sxR
sYR
SZR
xR
xk =- YR (5.22)
VZR
axr
ay,
az,

where the states are relative position, relative velocity, and target acceleration. The

system equations are

sR =VR (5.23)

vR = AT A,, (5.24)

ar, = -Aa + w (5.25)








Putting this in state space notation,

Ss 1 0 S 0 0 0 000
S1=0 I0 v, + 00 -I 0 a co 0 + 0 0 (5.26)
a, 0 0 -AJla, 0 0 o0 0 0 L o wr

From this equation the state transition matrix is computed as

I IAt I(e "+ At- 1)/A'
0(At)= 0 1 I(l-e-)/IA (5.27)
0 0 le-'

where I is the 3x3 identity matrix.

Assuming the commanded acceleration remains constant over the filtering interval

-IAt2A ,I
2
SD(tk,,r)G(r)udT= -AtAcoA (5.28)
0


The state propagation equation becomes


At AcaI
+ tAO1 (5.29)
0


The covariance propagation proceeds in a similar fashion. The covariance can then be

propagated by

P = [ <]_, []T + Q. (5.30)

The measurement equation is


zk = hk (,(tk) + v,


(5.31)








Range
where h (i(tk))= -zE / Range Range = x + + R (5.32)
y, I Range

and vk = N(O, R) (5.33)

is the Gaussian (normal), random vector with zero mean, and covariance Rk.

This completes the derivation and equation formulation for the nine-state EKF

needed for the interceptor guidance system. Given an initial estimate of the state x(0)

and the covariance matrix P(0) equations (5.14) and (5.15) and (5.18) through (5.20) can

be solved in a recursive fashion to achieve an estimate of the relative vehicle motion.


5.2 Bootstrap Estimator Derivation

The bootstrap filter is derived using the recursive Bayesian estimation

relationships. To begin the discrete dynamic state equation is given by

Xk =fk- (Xk-, iWk-) (5.34)

where fk- is the system transition function and w,_, is a random, zero mean, white-noise

sequence representing a known calculable probability distribution function. At discrete

times, an observation becomes available

z, = h(x,,vk ) (5.35)

where vk is a zero mean, white-noise sequence representing a known distribution

function. It is further assumed that an initial pdf p(xolYo) of the state vector is available.

The objective of this process is to construct the pdf of the current state xk given all the

available information; i.e., p(xkIY).






20
The prior distribution before the update is defined as

p(xk I Y-) = p(xk I x) p(xk -1 k-)dxk- (5.36)

where the probabilistic model of the state evolution can be obtained through the use of

the system state equations and the known statistics of wk as

P(Xk xk-1) )=p(xk Ixl, wk) p(wk-_xk-I) dw,_ (5.37)

Because the process noise is assumed to be independent of any previous state values,

p(wk_ xk-1) = p(wk-_). If x,_1 and wk-_ are known, then x, can be arrived at

deterministically.

The update stage occurs when a measurement becomes available. The a priori

density may be updated according to Bayes' rule as


p(xIY )= P(Yk Ix) P(xklYkI) (5.38)
P(YlY -,)

where the normalizing denominator is

P(YlYk-) )= p(yYk x,) p(xk IYkY) dx (5.39)

and the conditional pdf, p(ylxk), is defined by the measurement model and the known

statistics of v,.

The method by which the following theory is implemented in the bootstrap filter

will now be defined. The bootstrap filtering algorithm provides the mechanism for

defining a set of random samples {x, (i); i = 1,..., N} from the pdf and obtaining the

distribution p(xk5, ,,_) .5-7 The propagation stage assumes a number of N random

samples have been taken from some initial pdf p(xk- _, lY). Each random sample for








each state is then passed through the system model to obtain a set of a priori samples at

time step k

xk (i)=k 1 f (i),wk-l(i)),i= 1,... N (5.40)

where the asterisk represents the a priori sample.

The update stage of the filter takes each of the a priori samples and computes a set

of a posteriori samples from which the estimate can be obtained. To do this, the

likelihood of each a priori sample l(xk'y ) is calculated using the measurement

equations and a normalized weight is given to each sample by



P(yk Xk(j))

This defines a discrete distribution over the a priori density. The normalized values of q,

can be resampled using a uniform distribution. A larger value of q, allows more samples

from that bin to fall into the a posteriori pdf. The state associated with each new sample

selected is assigned to the new a posteriori state {xk (i); i = ...,N. This a posteriori set

of samples is approximately distributed as p(xk lY). From this the mean and variance for

each state estimate can be computed. Because this filter makes no assumption of linearity

or Gaussian statistics, it was useful in providing a close approximation of the true pdf.

This semi-true pdf can then be used to investigate the effect of linearization and

parameterization on the evolution of the pdf.






22
5.3 SDREF Derivation

The SDREF is the nonlinear "dual" of a nonlinear regulator control design

technique, which uses a parameterization technique to bring the nonlinear dynamic and

measurement equations


x=f(x) (5.42)

z = h(x) (5.43)

and transform them into a linear-like state-dependent-coefficient (SDC) form

x= F(x)x (5.44)

z = H(x)x (5.45)

The distinctive assumption in the SDREF technique is that the SDCs can be

treated as "frozen" over a small interval to give adequate piece-wise performance. The

algebraic Ricatti equation is solved at every filter measurement update and the linear

Kalman filtering algorithms applied.

The technique was first applied to the following nonlinear regulator problem.13

The task is to minimize


I= 1 xQ(x)x + uTR(x)u dt (5.46)


with respect to the state x and control u, subject to the nonlinear differential constraint

S= f(x) + g(x)u (5.47)

The SDREF approach for obtaining a sub optimal solution is to use direct

parameterization to bring the nonlinear dynamic equation to the SDC form

i= A(x)x+B(x)u (5.48)






23
where f(x) = A(x)x and B(x) = g(x)

From this point the SDREF can be solved for P as

Ar(x)P + PA(x)- PB(x)R-~(x)BT(x)P+Q(x)=0 (5.49)

and the nonlinear feedback controller can be constructed as

u= -R-(x)BT(x)P(x)x (5.50)

The SDREF technique is shown'3 to have desirable properties of local asymptotic stability

and suboptimality given mild conditions of stabilizability and detectability.

It also shows that if A,(x) and A,(x) are distinct SDC parameterizations, they can be

combined to yield a third parameterization by

A(x, a) = a4, (x)+(l- a)A2(x) (5.51)

This technique of combining parameterizations allows an infinite number of

parameterization possibilities to enhance controller design.

An example'3 of the SDC form and SDREF is repeated to introduce the method.

Consider the stochastic nonlinear system

= f(x) + r(w) (5.52)

z= h(x) + v (5.53)

where w is Gaussian zero-mean white process noise with E[w(t)wT(t + r)]= Q(t)8(r)

and v is Gaussian zero-mean white measurement noise with E[v(t)v'(t + )] = R(t)8(r).

After bringing the system to the SDC form

x= F(x)x+Fw (5.54)

z= H(x)x +v (5.55)







24
the SDREF is given by the steady state, linear, continuous Kalman filter equations as

x = F()i+ + K (i)[z(x)- H(i)5] (5.56)

where K (j) = PHT ()R-' (5.57)

and P is the positive definite solution to the SDREF

F(i)P+ PFT(i)- PHT(i)R-'H(i)P+F'Q = 0 (5.58)

The preceding method was applied to a simple two-dimensional pendulum

problem.13 Results were comparable to those from an EKF. The main thrust of the

research was to develop a SDREF for the linear dynamic and nonlinear measurement

equations associated with the strategic, terminal homing guidance problem and

investigate filter characteristics using the EKF and bootstrap filter for comparison and

analysis.


5.3.1 Angle-Only Measurement SDREF

The following derivations are performed for the bearing-measurements only

SDREF. It shows that this form of the SDREF is not detectable and therefore violates the

necessary conditions to assure the positive definite solution of the Ricatti equation. Both

the continuous and discrete forms of the filter are given, as well as stabilizability and

detectability analyses for each type. The filter derivation begins with the definition of the

state equations

x= Fx + Gu+ Lw (Continuous form) (5.59)

Xk =lkIX_-1 +rFk_-l-1+Ak_Iwk_1 (Discreteform) (5.60)









and the measurement equations

z=h(x)+v (Nonlinear form) (5.61)

z = Hx+v (Linear form) (5.62)

The states to be estimated are


SXR
SYR
SZR
VXR
xk = VYR (5.63)
VZR
vz,


ayr
TaZ
az,

where the states are relative position, relative velocity, and target acceleration.

The system equations are

sR = vR (5.64)

-R = AT (5.65)

a, = -A + WT (5.66)

Putting this in state space notation



R = 0 L v + -I aco, + O WT (5.67)
Ir 0o JAlaJ /

i = [F] Ix)+ [G] u + [L]w

The measurement equation is


Zk =h(x)+v,


(5.68)






26
where, based on the small-angle approximation,



h(x) = Rane Range = J2 +y2 R2 (5.69)

Range

and vk = N(0,R,)

The SDREF approach for obtaining a sub optimal solution is to use direct

parameterization to bring the nonlinear dynamic equation to the SDC form

x= F(x)x+G(x)u (5.70)

where f(x) = F(x)x

and z= H(x)x + v (5.71)

From this point the SDREF can be solved

F(x)P+PFT(x)-PHT(x)R-l(x)H(Hx)P+LQLT(x)=0 (continuous) (5.72)

QPTgr-P-OPHr(R+HPHT)-'HPOT+Q_, =0 (discrete). (5.73)

Since the system dynamic equations are linear, the only parameterization needed is for the

measurement equation. From the definition

z = Hx + v (5.74)

the linear form for the SDREF can be cast as


H [ R 0 0 0 0 0 0 J (5.75)
01/R 0 0 0 0 0 00


5.3.1.1 Continuous SDREF

Using the above-defined state and measurement equations with A =0.1, the

following matrices are used for the filter equations







27
0 0 0 1 00 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0 0
0 0 0 0 00 1 0 0
F= 0 0 0 0 0 0 0 1 0 (5.76)
0 0 0 0 00 0 0 1
0 0 0 0 0 0 -.1 0 0
0 0 0 0 0 0 0 -.1 0
0 0 0 0 0 0 0 0 -.1

The time constant for the Markov process A, was based on an expected target

acceleration of approximately 981 (m/s2).

-0 0 0
0 0 0
0 0 0
-1 0 0
G= 0 -1 0 (5.77)
0 0 -1
0 0 0
0 0 0
0 0 0

The process noise used for this problem is 1 g on the x-target acceleration and 3 g's on

both y and z target accelerations. The disturbance input is

E[w(t)] = 0 (5.78)
[(000)= (( )] (579)
000
000


G= 0 -1 0 (5.77)
0 0 -1
000
000
000

The process noise used for this problem is 1 g on the x-target acceleration and 3 g's on

both y and z target accelerations. The disturbance input is

E[w(t)]=0 (5.78)

E[w(t),w'(V)= Qc(t)3(t-r)] (5.79)






28
which for small values of At can be expressed as

0 0 0 0 00 0 0 0
0 0 0 0 00 0 0 0
0 0 0 0 00 0 0 0
0 0 0 0 00 0 0 0
Qk1 =LQcL At= 0 0 0 0 0 0 0 0 0 At (5.80)
0 0 0 0 00 0 0 0
0 0 0 0 0 0 96 0 0
0 0 0 0 0 0 0 866 0
0 0 0 0 0 0 0 0 866

where 866 (m/s2) is the variance of 3 g's of acceleration noise in the y and z directions.

The parameterized measurement matrix can be computed as


0 0 0 0 0 0 0 0
(x)= Range (5.81)
0 a 0 000000
Range

The standard linear, continuous Kalman filtering equations follow as

= F(i) + K, (G)[z() H(i)i] (5.82)

where K, (i) = PHTR,'- (5.83)

and P is the positive-definite solution to the state-dependent Ricatti equation

F(i)P + PFT()- PHT(i)RK,_-1H(i)P+Q, = 0 (5.84)

where Qk-i = LQc'LrAt. Ricatti equation theory provides for a legitimate positive semi

definite solution if the following conditions are met

(TD,Q,PER"-,HT ER-m,ReR RrnQ=QT>0,R=RT>o,m
and (F,H) is point-wise detectable in the linear sense and(C, F) is point-wise






29
stabilizable in the linear sense for all x e 9, where C is a full-rank factorization (FRF)

of Qk- given by

(C'C)= Q and rank(C)= rank(Qk-) (5.86)

For the given linear dynamics we can construct the controllability matrix as

Con=[C FC ...F"'C] (5.87)

If Con is of full rank, controllability and therefore stabilizability is assured. If it is not

fully controllable, a minimum condition of stabilizability must be met. Putting the

system equations into the controllability canonical form, the uncontrollable poles can be

checked for stability.20

Using a similarity transformation, the controllability canonical form can be found

as

F=TFT', C =TC (5.88)

where T is unitary and the transformed system has a staircase form

=- 0] C_ 0o


where (F,,G.) are controllable and (F,,) is uncontrollable. The poles of the

uncontrollable subspaces are then checked to insure stability. If they are found to be

stable, then stabilizability is verified.

Using the noise matrix in equation (5.13) and performing the FRF for C, it can be

shown that the controllability matrix has full rank and therefore stabilizability is verified.






30
Similarly, the observability Grammian matrix is constructed as

H
HF
Obs= (5.90)

HFn-1

If Obs is of full rank for all x, then full point-wise observability in then linear sense is

assumed. Once again point-wise detectability needs to be assumed for the Ricatti

equation solution to be positive semi definite. If it is not of full rank the similar

observability canonical form can be used to again check the point-wise stability of the

unobservable poles. This canonical form is given by


=[F FI, H=[0 Ho] (5.91)

Using the above referenced matrices for Fand H, it can be shown that the observability

matrix is not of full rank. Therefore, point-wise stability of the unobservable poles must

be checked. It was found that the poles were not point-wise stable and therefore point-

wise detectability was not verified. A similar analysis was then performed on the discrete

form of the filter. From this point on, only the discrete form of the SDREF is analyzed.

This form was chosen due to the use of discrete measurements that are only available at a

subset of the integration step size of the simulation. The discrete filter is better suited for

this type of simulation and would most likely be used in any realistic guidance software.








5.3.1.2 Discrete SDREF

The discrete form of the SDREF is derived using the following formulas. Since

F is a constant matrix, the state transition matrix can be found using Laplace transforms

as

I IAt J(e +At-1)/1
D(At) = 0 I I(1-e -')/A (5.92)
0 0 le-

When using 2 = 0.1 and At = 0.025, the following transition matrix is computed

1 0 0 .025 0 0 .000312 0 0
0 1 0 0 .025 0 0 .000312 0
0 0 1 0 0 .025 0 0 .000312
0 0 0 1 0 0 .0249 0 0
S= 0 0 0 0 1 0 0 .0249 0 (5.93)
0 0 0 0 0 1 0 0 .0249
0 0 0 0 0 0 .9975 0 0
0 0 0 0 0 0 0 .9975 0
0 0 0 0 0 0 0 0 .9975

Assuming the commanded acceleration remains constant over the filtering interval


At2A co
2
Fuk- ,,= (tk,T )G()ud= -AtAcoml (5.94)
0


The state propagation equation becomes


[- At21
2
x [E]x, + -AtI Ao (5.95)
0









Also,

E[wwT] = QS(t ) (5.96)

and


Qk,- = I(tk ,T)L()Q'c (r)L (r)c'(tk ,r)dv (5.97)
1-I

Using 1 g of acceleration noise on the x target acceleration and 3 g's in y and z and

performing the above integration, the process noise matrix is obtained.

9.46e-9 0 0 9.36e-7 0 0 4.99e-5 0 0
0 8.54e-8 0 0 8.44e-6 0 0 4.45e-4 0
0 0 854e-8 0 0 8.44e-6 0 0 4.45e-4
9.36e-7 0 0 9.98e-5 0 0 .006 0 0
Qk- = 0 8.44e-6 0 0 9.0e-4 0 0 .054 0
0 0 8.44e-6 0 0 9.0e-4 0 0 .054
4.98e-5 0 0 .006 0 0 .478 0 0
0 4.5e-4 0 0 .054 0 0 4.32 0
0 0 4.5e-4 0 0 .054 0 0 4.32

The discrete Ricatti equation solver requires the following conditions to be met to insure

that a legitimate positive semi definite solution to the Ricatti equation is obtained:

(O, Q,Pe Rn',HT R Re Rm", Q=Q > R = R >0,m
and (D, H) is point-wise detectable in the linear sense, and (C, () is point-wise

stabilizable in the linear sense for all x e Q, where C is a FRF ofQ,_ and is defined as

(CTC)= Qk and rank(C)=rank(Qk_) (5.99)

It is also must assumed that Q is invertible without need for further modifications to the

solver. If all assumptions are met, then the Ricatti equation will have a unique non-

negative definite solution throughout 2.







33
Using the above referenced matrices, the same analysis used for point-wise

stabilizability of the continuous system was performed. The controllability matrix of the

pair (C,D) was found to be of full rank and therefore the system is stabilizable.

Again an analysis similar to the continuous version was performed using the pair

((D, H) and the point-wise observability matrix was again found to be not of full rank.

The canonical form was calculated and again the poles of the unobservable subspace were

determined to be point-wise unstable and therefore the system could not be declared

point-wise detectable in the linear sense.

Because the system has been shown to be point-wise undetectable, the Ricatti

equation solver cannot provide the positive semi definite solution to the Ricatti equation.

Consequently, another approach to the problem must be examined. One way to make the

system detectable is to allow a measurement in the range or x-direction. This is shown in

the next section.


5.3.2 SDREF With Range Measurement

This section analyzes the same filter described above except that a range

measurement has been added to the measurement equation. By adding the range

measurement the system becomes both stabilizable and point-wise detectable under the

conditions set forth above. Two types of range measurements were evaluated to provide

adequate observability for the system.








5.3.2.1 Intensity range measurement

The range measurement used is the source radiant intensity of the target arriving

at the focal plane and is


I= Range k (5.100)
Range

where

Ik =intensity (watts)
( watts i
J, = source radiant intensity watts
(sr/pm)
A = bandwidth (,um)
r = radius of seeker aperature (m)
R = range from source to seeker (m)
v, = white Gaussian noise

This gives the measurement equation:

-z/Range
h(x)=- ylRange Range=-x x/+yR2 +zR (5.101)
JA2tmr2 /Range2

Putting equation (5.101) into the SDC form


0 0 000000
Range
H= 0 1 0 0 0 0 0 0 0 (5.102)
Range
Rang 2r 0 0 0 0 0 0 00
Range2 *XR

The measurement noise for the range measurement is somewhat more complicated as it is

range dependent. The actual intensity measurement has Gaussian noise from the focal

plane electronics, which can cause error in the measurement. However, it is assumed this

error is small in comparison to the fluctuations of the source radiant intensity and is






35
assumed to be zero. Jz is, however, comprised of both the deterministic source radiant

intensity and a random part given by

JA = J, + I_ (5.103)

The resulting intensity contribution from the random fluctuations in the plume

core can be calculated from equation (5.101) as

I = J (5.104)
Range2

The variance of this random component of intensity is obtained by taking the expected

value of the square of I, (i.e., E{I, }). The equation for the expectation operation

gives


E i(J,,AAIr2' 2 a E{J' 1 (5.105)
Range1 J Range2

If p= E(J2,, ), then the range dependent measurement variance term for the filter

becomes
(pAArr2 2
SRange 2 (5.106)

Using appropriate values for these constants, a measurement matrix can be given by

2.5e-9 0 0
Rk= 0 2.5e-9 0 (5.107)
0 0 8.9e 7

Using the above-determined matrices, the SDREF was again calculated. The filter

responded well during the first three to four seconds of flight and then diverged. The







36
problem lies in the instability of the Ricatti equation solution. In examining the point-

wise closed-loop eigenvalues of the system (i.e., the spectrum of

DT HT(R + HPHT)- HPQ (5.108)

with x, in this case range, frozen), the system would be asymptotically point-wise stable

in the linear sense if the closed-loop eigenvalues all lie inside the unit circle. The

eigenvalues of the SDREF were computed for each filter update stage. They began near

the unit circle (i.e., 0.9992) and grew to become unstable as the simulation ran. The final

step in the Ricatti equation solution is the solution of an nt order linear matrix equation.

A number was computed to determine the condition number for the inverse matrix. It

could be seen that the condition number increased (i.e., ill-conditioning) as the closed-

loop eigenvalues approached the value of one. The relatively small size of the range

variance causes R-' to become very large, and therefore ill conditioned. Once the closed-

loop eigenvalues became larger than one, the problem diverged quickly. Because of this

numerical instability, a different range measurement was tried.


5.3.2.2 Distance range measurement

The new range measurement is the distance measurement in meters. This value is

assumed to be available from the signal processing algorithms. The new measurement is

Range =x, + yR + (5.109)







37
Therefore, the new measurement equation becomes

-z
Range
h(x) (5.110)
Range
Range


and the measurement matrix becomes


0 0 -1 0 0 0 0 0
Range
H= 0 --0 0 0 0 0 0 (5.111)
Range
Range 0 0 0 0 0 0 0 0
X,

Using these values in the SDREF, satisfactory results were obtained. (These results are

given in Chapter 7.) It was decided that this would be the formulation that would be used

to further test this new filtering technique. Before these results are presented, however,

an investigation into the theoretical stability of the SDREF is presented.


5.3.3 Theoretical Investigations

One of the main theoretical criteria for filtering techniques is the stability of the

algorithm. The following section reviews several theoretical stability investigations.

While these investigations were unsuccessful in proving stability, it is hoped that they

provide valuable insight into the SDREF. It is these attempts that led to the development

of the MSDREF shown Chapter 9. While the methods outlined below approach the

problem from several different theoretical aspects, all were unsuccessful in showing local

stability for the SDREF. Because the true and estimated states x and i both appear in the

error equation






38
e = (F(x)x F(.)! K(x)[H(x)x H(i)]) (5.112)

each of the approaches fail when presented with the cross terms that are created. It is

hoped that this section will prove useful to future stability investigations. The approaches

are shown starting with simple linearization techniques and progress through more

complicated and sometimes restrictive theorems. The first of these is a simple Taylor

series expansion of the parameterized equations.


5.3.3.1 Taylor series expansion

The following stability analysis will exploit a Taylor series expansion about the

local zero point. Consider the stochastic, nonlinear system

= f(x)+g(w) (5.113)

z= h(x)+v (5.114)

where w is Gaussian zero-mean white process noise with E[w(t)wr(t + r)] = Q(t)8(T)

and v is Gaussian zero-mean white measurement noise with E[v(t)v'(t + )] = R(t)8(T).

After bringing the system to the SDC form

= F(x)x+Fw (5.115)

z = H(x)x+v (5.116)

the SDREF is given by the linear, continuous Kalman filter equations having state-

dependent coefficients

S= F(i) + K(i)[z(x)- H(i)i] (5.117)

where K(i)= P(i)HT(i)R-' (5.118)

and P is the positive semi definite solution to the algebraic SDRE

F(i)P(i)+P(i)FT(i)-P(i)HT(i)R-'H(i) P(i)+ FQF =0 (5.119)






39
Conjecture: Assume that R > 0, Q >0, f(x), and h(x) RK, and that the

parameterizations F(x) and H(x)can be shown to satisfy that (F(x), H(x))is a point-wise

detectable pair in the linear sense and (C(x), F(x)) is a point-wise stabilizable pair in the

linear sense, where CC = Qk-,. Also assume F(x) e C" and H(x) e C~ are analytic,

where C~ is the class of functions whose derivatives of all orders are continuous, and that

f(0) = h(0) = 0. Then the SDREF is locally asymptotically stable.

Outline of Attempted Proof:

For ease of notation the proof will be examined in scalar form. Given that the

scalar system dynamic equations are

x = F(x)x+Gu+ Lw
z=H(x)x+v and (5.120)
x = F(i)i + Gu +K[z H(x) ]

and defining the estimation error rate as e = x the above equations can be written as

= (F(x)x- F(i) K(x)[H(x)x- H(i)i]) (5.121)

Now expanding h(x) and H(x)x in a Taylor series expansion about zero

h(x) = H(x)x (5.122)

A(0) d h -) h (0)| i (x0)
(0)-- (0) + -H.O.T H(0)+--- -)+- +.O.T x
x= 10 A X=o 2! L Ix=0 ax=0 21

Also for f(x)= F(x)x

f(x) F(x)x (5.123)
3() d2f ( 2-O)2 F dF(O) d2F (x-0)2
Po)+ (x-o)+ if + 0) o L H O.T-
(0-o )+- +H.o.T.= F(O)+- ((x-)+" 2 o.r. x
SI o= 2 L A X=O a 0 2!
Similar equations can be written for h(I) = H().i and f (x) = F(i)i.








Equating like terms

A(0) A(0)
h() = H(0) -


and O(o) -F(O)= (O)


Using this and recalling that K = PHTR-',

E(t) = (F(0) PHT(O)R-'H(O)))E(t)+ H.O.T. (5.124)

Herein lies the problem, the higher order terms contain both x and i. These terms cannot

be guaranteed to approach zero, much less go to zero faster than the first term. If the

error dot equation could be expressed in this linear form a local stability proof could be

achieved in the following manner. For ease of notation, let F(0) = F, H(0) = H, and

e(t) = e. Following a similar derivation,'9 a Lyapunov function is chosen

V(e) = eIE (5.125)

where I is the positive definite information matrix, the inverse of P. From the definition

of the derivative of the inverse,

I = P- = -IPI (5.126)

By substituting this into the matrix Riccati equation, a differential equation

-j = IF + FI +ILWLTI HTN-'H (5.127)

is yielded that has the steady-state solution

0= IF +FrI +ILWLI HTN-'H (5.128)






41
The time derivative of the Lyapunov function is

V= 2erI = 2eT[F- KH]e
= 2eT[F PHTN-'H]e (5.129)

= er[(IF-HTN'-H)+(IF- HTN-H)]e

Using (5.128) this can be written as

V= -4ILWLrI+H TN -H]E (5.130)

As the term in square brackets is positive definite at all times, V < 0 at all times. This

would verify the local asymptotic stability of the SDREF for some small neighborhood

about the origin. The estimation error would approach zero as time approaches infinity.


5.3.3.2 Using a Taylor series expansion about the true states

This approach varies slightly from the above, in that a Taylor series expansion of

the estimated dynamics and measurement equations are linearized about the true states.

Using the same state and measurement equations (5.113-5.120) and the error equation

(5.121) in the preceding proof, F(i) and H(i) are expanded as

F"(x)
F() = F(x)+F'(x)(i-x) + (.-x)2+...H.O.T. (5.131)
2

and H()= H(x)+H'(x)(-x)+H-(x -x)2+...H.O.T. (5.132)
2

Using equations (5.131) and (5.132) equation (5.121) is written as

S= {F(x) KH(x)}e+ {F'(x) KH'(x)} d t(e2) (5.133)

Once again the problem with this approach can be seen. The only way it can be shown

that the second term disappears or is negative is for ex to be zero. This can only be

shown to be true for the linear case.







42
5.3.3.3 Taylor series expansion about the error

Again, using the same system equations (5.113-5.121) as in the first Taylor series

approach, this second approach linearizes about the error. Linearizing the dynamic and

measurement equations

F"(E)
F(x)= F(e)+F'(e)(x )+ (x-E)2+...H.O.T. (5.134)
2

F"'(e)
F'(x)= F'(e)+F"(E)(x- e) + (x- )2+...H.O.T. (5.135)
2

H"(e)
H(x)=H(e) +H'(E)(x ) + (x e) +...H.O.T. (5.136)
2

H"(e)
H'(x)= H'(e)+ H"(e)(x-e)+ (x-e)2+... HO.T. (5.137)
2

After substituting these into error equation (5.121),

= (F KH) +(F'- KH')(x- e)e
+(F'- KH')ei +(F" KH")(x E),

+(F"-KH")(x E)2 (5.138)
2
+(F"'-KH"')(x- e)2 + (e2)
2

Again, the error equation is only stable if it can be shown that ex -> 0. This can only be

shown to be true for the linear case.


5.3.3.4 Residual error stability approach

The following approach illustrates, under conditions of point-wise detectability in

the linear sense that ensure a legitimate solution of the SDRE, that the SDREF can be

shown to be asymptotically stable.21 The system dynamics and filter equations are given

in discrete form as








xk+i = 4kXk +Wk (5.139)

Zkt+ = h(xkl)+ Vk+ (5.140)

Xk+i/k b (5.141)

"k+i =k+k + Kk+I[Zk+ Hk+- (k+Ik)k+l/ ] (5.142)

Kk+I Pk+1 (k+I)Hk+T (k+,)Rk+,-' (5.143)

with E[vk+l vk+I ]= Rk+l (5.144)

E[wwk+WT] = k+ (5.145)

where P, (-ik+) is the solution of the SDRE. These are the equations necessary for the

discrete SDREF. It is also assumed that vk+5 and w.k+ are random, zero mean inputs and

therefore, the stability of the SDREF is independent of these error sources.

The state estimation error both after and before the update is defined respectively

as

k k+ k+ (5.146)

xk+/k = Xk+-l Xk+/k (5.147)

Also, for ease of notation

Hk+ (-k+= Hk (5.148)

Pk+ (-k= P+. (5.149)

e,+ = Zk+l- Hk+ (k+ )k+l (5.150)

And finally a candidate Lyapunov function Vk, is defined as

vi = pk+ pk Xk+ (5.151)

The conditions for which Vk+~ is a decreasing sequence must be determined, therefore

showing the exponential stability of the SDREF.







44
In the preceding stability approaches, linearization of the filter equations has not

shown stability of the filter. This attempt constrains the state transition matrix and

measurement noise matrix so as to characterize the error due to linearization. The

measurement residual error can be characterized as

ek+1 =zk+ -h(ik+1/k) (5.152)

e+ = Hk+(l k+ )Xk+l Hk+ (k+ l)k+l (5.153)

Rewriting H,, (i+,) using a Taylor series expansion, for each output error prediction

component of e+,, namely e, ,k+, there always exists a residual r,,k+jdue to the first order

linearization in order to obtain an exact equality. Therefore,

e,.k+1 H,,k (x+ k+I)x+ Hi.k+ (xk+) + H'i,+ (xk+1 )(X+ X )+. (5.154)

and the residual error is

rk+= {Hi, (Xk+1)(ikl -Xk)+. } +l (5.155)

Then the exact measurement error can be defined as

e,.k+ = Hik+1 +1k + r.,k+ (5.156)

or written in another form

Hik+3lk+1k = ai.k+lei,,k+ (5.157)

where a,,,k is an unknown time-varying scalar related to ri,k+ by

,k++ (1- a,k+l )e,k+l (5.158)

Introducing the signal vector

ak+lek+ = Hk+15k+/k (5.159)

where ak+, e RP' is an unknown time varying diagonal matrix defined as

ak+1 = diag(a,k+1, .. a.,~k+1) (5.160)








If both sides of equation (5.142) are subtracted from xkl,,

+ = k+I/k Pk+IHk+ITRk+I-'ek+1, (5.161)

Substituting this into equation (5.151), and using the fact that Pk+1 =Pk

Vk+1= .k+k iT Pk+1-1 k+l/k Ktk T Hk+RIT+ k+1 (51
-ek +lt R+1- Hk+1 k+l/tk ++l tR+ -Hk+,Pk+H l R,+ ek+l

Because the covariance matrix is not propagated during the time between updates, it is

assumed that the a priori covariance before update is equal to the previous updated

covariance,

Pk+,1k = P (5.163)

Therefore,

Pk+-' =Pk' (5.164)

Before proceeding further, the inverse relationship is defined as

Pk+- = Pk+,,1-1 + Hk+,TRk+,-'Hk+5 (5.165)

Once again a similar problem arises. Equation (5.165) is not derivable from the basic

premise of the SDREF update equations.21 If this equation was available, it could be

shown that Vk,, is a decreasing sequence. Then

Vk+I = Vk+I1, Xk+U III T H Rk+1- ek+1
(5.166)
-ek+, Rk+1, Hk+,,k+11k +e,, +RLHk+ Pk+ Hk+IT Rk+-1 ek+

and reviewing that

Hk+lk+1= ak+, (5.167)

and


XTk+IHHTk+i = eTk+laT k+


(5.168)








Equation (5.166) is written as

Vt+l = Vk+l/k ek+,1 t+a k+l atk+lek
ek+ ak+lRk+-lHk+lek+1/k -k e+TRk1 -'ak+ek+ (5.169)
+ek+l Rk+l -lHk+I Pk+lHk+I Rk+,-lek+l

And using the fact that

Vk+,1/ = TFkTPk+,,k-I'Fk (5.170)

with Equations (5.167-168), Equation (5.169) is rewritten

Vk+1 V = eV+lT k a+l Rk+,-1ak+l akt+iRk+., Rk+,-,ak+,
+Rk+1-lHH+lP+1Htk+TRk+-'I }ek+, (5.170)
+;5FT(FkTP-IF Pk-1

Therefore under the condition of point-wise detectability in the linear sense, and certain

restrictions on R and Q (i.e., Rk > 0 and Qk >0) then,

VI, <0

and the system is asymptotically stable as the values in parentheses are < 0.


5.3.3.5 Lyapunov approach using the expectation operator

This stability approach uses a similar method.9 Given the system equations

x+, = A, ,+w, (5.171)

= + K,[H(x,)x, H(x,)x,] + Kv, (5.172)

the error can be updated and propagated by the equations

e, x, i (Updated error) (5.173)

x -x, (Propagated error) (5.174)








Using equations (5.171-5.172)

e, = x, x, K,[H(x,)x, H(,),]- K,v (5.175)

= A,-_e, + w,_, (5.176)

After some manipulation of these equations, the error update equation can be developed

eF1= A, -, Kix,+A,K,HA, H -K,v,+w, (5.177)

where H, = H(x) and H, = H(()

If the following Lyapunov function is defined

V,+ = e,j.e+, (5.178)

it can be shown that

E{V, (eF.)-V,(e,)} 0 (5.179)

Substituting equation (5.177) into equation (5.179), the following equation can be

computed

E{V,,,((,) -V()}= eE{ATA I}e,
+E{2"ATAKH,x-2xTATAKHx}
+E{.-2x-TArAKH2x + 2xTATAKH2Y}
T T KT AT Y1 (5.180)
+E(-2xTHTKTATAKH (5.180)
+ ExTHTKTATAKHlx}

+ E{xTH2,TKTAAKH2x}

In this equation several terms exist that cannot be shown to be negative definite. These

terms are brought about by the same problem found in the previous approaches evaluated

to show stability. They are cross products created by the presence of both xand i in the

error equation.








This same approach was also tried with the Lyupanov function

V=eTe (5.181)

and Equation (5.182) was developed.

E{V,, (e,)-V(e,)} = e,TE{ATA e,
+ E{2A'rAKH,x 2x ATAKHx}
+ E(-2 TArAKH2i + 2xTA'AKH2i}
+E{-2xTHTKTATAKH2} (5.182)

+E{XTH,TKTKKH,x}

+ E(IrH2T'KKH2z)

Once again cross terms are generated and cannot be shown to be negative definite. All of

the previous attempts at stability proofs fail for the same reason: The cross term xi,

produced in the error equation, raises significant difficulties in finding a closed form

differential equation for e. Attempts to limit the problem to a scalar were not able to

resolve the cross coupling problem. While several standard and non-standard approaches

were tried, a satisfactory stability proof could not be verified. It is hoped that these

investigations can prove useful in future attempts to determine filter stability.














CHAPTER 6
EXOATMOSPHERIC-GUIDANCE-PROBLEM SIMULATION RESULTS

Chapter 6 investigates the comparative performance of each filter against a variety

of engagement conditions and noise parameters. Before any study on filter performance

was done, a baseline performance analysis was completed, which allowed a detailed

investigation of the SDREF against one engagement scenario. Once the baseline

performance was established, further investigation into filter sensitivity, robustness, and

closed-loop performance was performed. A study of pdf evolution for each filter is also

shown.


6.1 Baseline Performance

To establish a baseline performance measure, an engagement representing the

final 10 seconds of a closing interceptor/boosting ICBM engagement was chosen. Table

1 details the major initial filter parameters. The aspect angle is defined as the initial Euler

rotations between the axes of the interceptor and the target. For example, an aspect angle

of (30,30,30) would signify that the interceptor's initial velocity vector has been rotated

30 degrees about each axis. An aspect angle of (0,0,0) would be a head-on engagement.

The (30,30,30) aspect angle was chosen so that all axes of the filter would be exercised in

the engagement. SIGP and SIGVP are the standard deviations for the random

initialization errors on the "true" position and velocity states respectively. When SIGP






50
and SIGVP are zero, the filter has been initialized with the "true" initial-intercept

conditions. SIGXHAT are the values used to initialize the covariance matrix for the EKF

and the bootstrap filter. XHAT is the initial state given to the filters. The measurement

noise for the x-direction is a range-dependent noise that is given as a percentage of the

range (i.e., as the interceptor gets closer to the target the noise on the range measurement

decreases). All filters were made to operate with the same initial conditions and statistics

so that comparisons could be made.

A perfect filter was used to guide the vehicle to impact, while the EKF, SDREF,

and bootstrap filter were run open loop. The bootstrap filter was run with 10,000

samples.


Table 1. Filter simulation parameters baseline conditions


Parameters (units) X Y Z

Aspect Angle (degrees) 30 30 30
SIGP (m) 0 0 0
SIGVP (m/s) 0 0 0
a,2 (m2) 1.0E5 1.0E5 1.0E5
SIGXHAT O 2 (m2/s2) 1.0E3 1.0E3 1.0E3
aA_2 (m2/s4) 1.0E2 1.0E2 1.0E2
R (m) 1.0E5 10.0 10.0
XHAT V (m/s) -1.0E4 -335.9 223.9
A (m/s2) 55.2 57.40 -38.2
SIGQ (EKF) (m2 /s4) 196.0 196.0 196.0
SIGQ (SDREF) (m2 /s4) 1.0E6 1.0E3 1.0E3
Meas Noise (%,pm,/pm) 0.1 50.0 50.0







51
The following graphs reflect filter performance. Figures 3 through 5 show the

position-state estimate errors with 3v+. standard deviations for each filter. Velocity and

acceleration performance was similar between the EKF and SDREF. The bootstrap

acceleration estimates are poor due to an inability to forward measurement information to

the second derivative states. The problem can be remedied by the use of millions of

samples or other ad-hoc improvement schemes; however, computer limitations and the

scope of this dissertation do not allow investigation at this time. Figures 3 through 5 give

initial insight into the relative performance of each of the filters for the given scenario.

All of the filters adequately estimate the state to the precision necessary for an intercept to

occur. With state estimate errors remaining within 3a standard deviations, the filters

perform well. The bootstrap filter does show a tendency to keep a wider variance than

the other two filters. However, as discussed in paragraph 6.2, when the EKF and SDREF

are run in Monte Carlo fashion, they too have wider variances. The results shown here

are for one run, while noticing that the bootstrap filter is inherently Monte Carlo, even for

one run. The performance curves in Figures 3 through 5 verify proper filter operation and

establish a base-line performance from which more in-depth analysis proceeds.

It is interesting to note in Figures 6 and 7 that the standard deviations for the EKF

and SDREF are approximately identical after the initial transients in the EKF die out.

This indicates that once the EKF has settled and the estimates are being computed

accurately, the EKF covariance propagation and update are equivalent to the algebraic

Ricatti solution being computed in the SDREF. This relationship is only shown for one

state; however, similar results were found for all states.





































Figure 3. EKF position errors with +3" standard deviations.











S. 1n


,-~~0--~~~ I I I


-=I


Figure 4. SDREF position errors with 3ustandard deviations.


1 & I I I I


'*

E
n -~m
Pm






i


0-.


Figure 5. Bootstrap position errors with 3astandard deviations.


'


"


I <
-!


'


r


k


10


- -









































Figure 6. Comparison of EKF and SDREF covariance
3cr standard deviations for Y position.


5 EKF
EKF
1 SDREF
SSDREF














15 _____ ___
15 -0. 0 1


5 0.0 0.5 1.0
Time (sec)


Figure 7. Expanded view of Figure 6.


1.5 20







54
6.2 PDF Distribution Comparison

Using the baseline scenario, a comparison of the posterior (after the measurement

update) probability density functions was investigated, to determine whether

parameterizing versus linearizing the nonlinearities has an effect on the shape of the pdf.

The scenario given in Table 1 was run once and the state estimate and standard deviation

were computed three times during the engagement. Samples were taken at 1, 5, and 9

seconds. The pdfs for the EKF and SDREF were computed using a Gaussian

distribution function based on the mean and standard deviation at each time increment.

The resulting curves, Figures 8 through 9, do not represent the "true" pdf as nonlinear

dynamics are involved. However, they do show the basis of the theoretical filter

formulation. ("True" pdf functions, calculated using Monte Carlo simulations, are

discussed in following paragraphs.)

The pdf for the bootstrap filter was generated by computing a histogram of 1,000

samples, curve fitting a function using a cubic spline, and normalizing the area under the

curve. Figures 8 through 9 show the one-run posterior pdf functions for the filters. The

first time increment is taken 1 second into the flight. The filters are still trying to

converge and therefore the distributions are large. This is demonstrated in Figure 8 by

comparing the narrow spike of the highly converged bootstrap filter to the wide, non-

converged pdfs for the other filters. If consideration is given to the scaling on the x-axis,

no concern should be given to the fact that the distributions are not near the true state. It

can be seen that the estimates are close to the "truth". The x-position state shows a more

accurate representation of the pdf than the y- and z-position states. This is due, in part, to

measurement noise decreasing as the engagement progresses, and a more accurate












True Sta p
0.025 -- oostrap
Time= 1 Second EKF
SDRE
0.020


0.015


S0.010


0005


0.000

88500 89000 89500 90000 90500 91000
X Position (m)

0.05

Time =5 Seconds True sate
0,04


0.03


S0.02


0.01


0.00 ... -. ...

50000 50200 50400 50600 50800 5
X Position (m)

0.16

0'14 Time 9 Seconds True St -
-- EKF
0.12

0,10-

t 0.08
008

0 0.06

0.04


1000


12180 12200 12220 12240 12260 12280
X Position (m)

Figure 8. Posterior probability density functions at
time = 1, 5, and 9 seconds for X position.


I



































Y Position (m)

True State
5 T t Boolstrap
S-- EKF
Time = 5 Seconds SDREF
1














-421 -420 -419 -418 -417 -416 -415
Y Position (m)


-- Bootstrap
EKF
Time = 9 Seconds SDREF



True State


-174.0


-1735 -173.0
Y Position (m)


Figure 9. Posterior probability density functions at
time = 1, 5, and 9 seconds for Y position.


0.


0.


0.0






5


4


3


2








knowledge of the true state is obtained. Another interesting factor shown in Figure 9, is

the compression of the EKF and SDREF distributions. This is typical of the "falling

asleep" problem associated with many filters. As the state estimation error becomes

small, the covariance matrix also becomes small, and the filter reacts slowly to new data,

which takes the estimate away from the converged answer. The bootstrap filter, however,

keeps the distribution open wider. Even using the one-run pdfs, the EKF and SDREF

pdf shapes appear to be similar.

To determine the "true" pdf functions for the EKF and SDREF, the filters were

run in a 100-run Monte Carlo experiment. Filter state estimates were saved at the same

time increments. The samples from each of the runs were used to generate the pdf curves

in Figures 10 through 11. The same histogram method described in the previous

paragraph for the bootstrap filter was used to plot the curves. (The bootstrap filter did not

need to be examined in this fashion, as its formulation is inherently Monte Carlo.) The

largest differences can be seen in Figure 11. In the later time samples (5 and 9 seconds),

the previously tight distributions have flattened out, while the wide distributions have

tightened up. This is to be expected, as the one-run case is only one realization of the

Monte Carlo runs.

Summarizing the above analysis shows that the EKF and SDREF pdf's do not

resemble the "true" pdf. However, they are similar to each other. This may be explained

by the fact that the EKF linearization method provides an accurate filtering methodology

as long as the filter state estimate is close to that of the "true" state. If a target maneuver

is introduced that causes the state estimate to be far from the "truth", then it is assumed
































0 2798 89740 897 0


89780 89800
X Position (m)


89820 89840 89860


X Position (m)


12220 12240 12260
X Position (m)

Figure 10. Monte Carlo posterior probability density functions at
time = 1, 5, and 9 seconds for X position.


True Sale BoosIrap
EKF
Time = 1 Second SDRE







,4 '


0.05


0.04


0.03


0.02


0.01


0.00













0.35

0.30 Trne State oottp
Time= 1 Second SORE
0.25

020

0.15

0.10

0.05 -

0.00

-270 -260
Y Position (m)

0.5
True Slate

0.4 Time = 5 Seconds Sa


03


S0.2


0.1


0 0 .. .

-425 -420 -415 -410 -40
Y Position (m)



--Bootsrap
EKF
Time = 9 Seconds SDREF




True State





0


-175.0 -174.0 -173.0 -1720
Y Position (m)


-171,0 -170.0


Figure 11. Monte Carlo posterior probability density functions at

time = 1, 5, and 9 seconds for Y position.







60
that the distributions between the EKF and SDREF differ due to the different theories of

linearization versus parameterization. The following paragraphs explore this scenario.

The filters were once again run in a Monte-Carlo simulation. However, this time

the target performs a thrust cut-off maneuver, 2 seconds after launch, which causes the

target acceleration, in all axes, to drop to zero for the rest of the flight. This is

representative of timing the engagement too close to booster burnout. The filters are still

operating under the assumption of a Gauss-Markov target acceleration model, even

though no target acceleration is present after 2 seconds. The same initial conditions listed

in Table 1 were repeated.

Figures 12 through 14 show the divergence from the "true" acceleration-state for

the filters. For previously discussed reasons, Figure 14 shows the bootstrap filter

performs poorly when tracking the acceleration states. The deviation from the "true"

state is significant after the maneuver occurs. This is a good test to determine the

differences between linearization and parameterization. Once again the Monte Carlo pdf

functions were plotted in Figures 15 through 16 Samples were taken at 2, 2.5, and 3

seconds into the engagement. This allows the pdf to be examined before and during the

most stressful time of the maneuver. Suprisingly, the EKF and SDREF pdf shapes do not

differ significantly. It can be seen in Figure15 that although the "true" probability has

become bimodal, the shapes are still similar. While the single-run EKF and SDREF

distributions can not be bimodal, Figures 15 through 16 represent Monte Carlo results.

The results from the pdf comparison experiments indicate that there is no

appreciable difference in pdf shapes between the EKF and SDREF. The conclusion may






61








- I I







Figure 12. EKF estimated and true acceleration.















Figure 13. SDREF estimated and true acceleration.















Figure 14. Bootstrap estimated and true acceleration.






























0 0997 79920 799 0


74900 75000 75100
X Position (m)


75200 75300


70100 70200
X Position (m)


Figure 15. Monte Carlo posterior probability density functions at
time = 2, 2.5, and 3 seconds for thrust cut-off X position.


True Sta


Time = 2 Seconds / :'



.i

-


0.030

0.025

0.020

N 0.015

0.010


0.005

0U000


I8ZU 79 /840 79860 79880
X Position (m)


Time= 2.5 Seconds -- ooltrap
True Bate. E











\


74800


-














0.35

0.30

0.25

0.20

, 0.15
oto
010 -

005

0.00

390


Tue
Time = 2 Seconds


-- ootstrap
EKF
--REF


Y Position (m)


-405


-400
Y Position (m)


0.40
Bootstrap
0.35 Time =3 Seconds EKF
SDREF
0.30

0.25

True State
0.20

S0.15

0.10 /

0.05

0.00 -

-404 -402 -400 -398 -396 -394
Y Position (m)


Figure 16. Monte Carlo posterior probability density functions at

time = 2, 2.5, and 3 seconds for thrust cut-off Y position.








be drawn that the two filters operate with identical statistical models although the filters'

method of handling and operating on the stochastic processes are significantly different


6.3 Sensitivity Analysis

The following section examines the SDREF sensitivity to measurement noise,

initialization error, and the robustness properties in the presence of stressing target

maneuvers. Only the EKF and SDREF are used for this area of the study. Further

bootstrap filter development is needed to make it useful in less than ideal conditions. The

EKF and SDREF were tuned separately to give them the best chance of performing well

under these varying conditions. Therefore each filter may have been started with different

initial conditions and statistical information. The following studies are accomplished

using perfect state information to guide the vehicle, while the EKF and SDREF run open

loop. Allowing the guidance algorithms access to perfect state information removes the

closed-loop guidance algorithm effects from the analysis of filter performance.


6.3.1 Initialization Error

To test the response to initialization errors, the filters were given erroneous initial-

state estimate errors of 1 and 5 percent. The EKF initial covariance was set to

approximate this error. The SDREF, however, has no direct means of initializing the

covariance matrix as it is solving for the covariance matrix at each filter update time.

This will be shown to be a detriment for this particular implementation of the SDREF.

Table 2 gives the initial filter conditions for the EKF and 1 percent initial error.

Measurement noise was held constant for both cases, as was the measurement noise

matrix given to both filters. Range measurement noise is again range dependent and is






65
given as a percentage of "true" range. The initial setup for the SDREF used the same

parameters as those of the EKF except that higher values of process noise were tried to

increase the covariance response. The only tuning parameters in the SDREF are Qk, R,

and the nonlinear parameterization methodology. The measurement noise matrix,R, is

not directly related to the initialization error; therefore, only the other two tuning

parameters can be used. Because this implementation of the SDREF has linear dynamic

equations the only method available for tuning is to adjust Q,-,. This is an unacceptable

way of tuning the filter for initialization errors as it causes the covariance estimate to

remain artificially high for the entire flight. The SDREF as previously derived, does not

permit any initialization of the covariance matrix, as it is simply calculated from the

solution of the SDRE. This deficiency is examined later in this section.


Table 2. EKF and SDREF parameters one-percent initialization error


Parameters (units) X Y Z

Aspect Angle (degrees) 30 30 30
SIGP (m) 1000 35 30
SIGVP (m/s) 100 3.4 2.2
o72 (m2) 1.0E6 1.0E3 1.0E3
SIGXHAT 0,2 (m2 /s2) 1.0E4 50.0 50.0
oA2 (m2/s4) 1.0E1 1.0E1 1.0E1
R (m) 1.0E5 10.0 10.0
XHAT V (m/s) -1.0E4 -335.9 223.9
A (m/s2) 55.2 57.4 -38.3
SIGQ (EKF) (m2 / s) 196.0 196.0 196.0
SIGQ (SDREF) (m2 /s4) 1.0E6 1.0E3 1.0E3
Meas Noise (%, im, tm) 0.1 50.0 50.0








Figures 17 and 18 show the position-state response of the EKF and SDREF to a 1-

percent initial error. (Note the oscillations in the SDREF estimate during the initial

settling portion of the flight.) Velocity and acceleration trends were similar.

The filters were then run with a 5-percent initial error. (The initialization

parameters for the EKF and SDREF are given in Table 3.) Because of the type of

engagement and large distances and closing velocities involved, a fixed-percentage type

of initial error induces large disturbances in the x-direction. Therefore, little attention is

given to the performance in this direction, as these magnitudes of error are not likely to

occur. Figures 19 through 20 show the performance for a 5-percent initial error. The

inability to properly adjust the initial covariance matrix in the SDREF causes a large

oscillatory error in the early portion of the estimation problem. This error was

unacceptable and a new initialization procedure was investigated.

The basic problem with the initialization of the SDREF is that the initial

covariance matrix is computed using the solution of the algebraic Ricatti equation. This

indicates that no large errors are expected. Therefore, there is no method of supplying the

SDREF with an initial covariance error of the magnitude of the expected initialization

errors. Because of this the SDREF computes a smaller magnitude covariance matrix and

therefore a smaller gain to be used in the state update. This is why the filter oscillates

during the initial portion of the flight. If the SDREF could be "jump-started" with a high

initial covariance, the majority of the initialization error could be taken out in the first

time step.







67










-----------PPL




0 2 4 6 8 10 12







F1













0 2 4 8 10 1

Time (sec)


Figure 17. EKF position errors with 3ustandard deviations

for 1-percent initialization error.





-2.


1000

0 2 4 8 10










0 2 4 6 0 10


4o0


20





0 2 4 8

Time (sec)


Figure 18. SDREF position errors with 3astandard deviations

for 1-percent initialization error.


-


--~YU
-INU
------- 111~1






II 10





E400



01s

1)0
so
S o
g 0

-150


.60
40
S so


-100


F


o 0 4 5 a 10










04 6 0




Time (sec)

Figure 20. SDREF position errors with 13standard deviations
for 5-percent initialization error.


I I I


I


~


uD

E 'm
m


0 0 I -0 12

-Io

o 4 e S 10






0 0 0 10 12




Time (sec)

Figure 19. EKF position errors with 3cstandard deviations
for 5-percent initialization error.


^








Table 3. EKF and SDREF parameters five-percent initialization error


Parameters (units) X Y Z

Aspect Angle (degrees) 30 30 30
SIGP (m) 5.0E3 168.0 112.0
SIGVP (m/s) 5.0E2 16.8 11.2
C2 (m2) 25.0E6 3.0E4 3.0E4
SIGXHAT o-2 (m2 / 2) 25.0E4 3.0E2 3.0E2
CA2 (m2 / 4) 1.OEl 1.LEl 1.OEI
R (m) 1.0E5 10.0 10.0
XHAT V (m/s) -1.0E4 -335.9 223.9
A (m/s2) 55.2 57.4 -38.2
SIGQ (EKF) (m2/s4) 196.0 196.0 196.0
SIGQ (SDREF) (m2 / s4) 1.0E6 1.0E3 1.0E3
Meas Noise (%,am,Aum) 0.1 50.0 50.0



If the gain calculation used in both the EKF and SDREF is examined,

K= = p (-_)Hk[H p, (_-)H + R,]

it can be seen that the covariance used at time step zero, Pk (-), is needed in the

calculation. If it is assumed that the value of Pk that the SDREF computes is Pk(+) (i.e.,

after the updated measurement has occurred), then the SDREF is initialized with P, (-) to

be used at time step zero. Using this methodology, the SDREF was initialized with the

same covariance as the EKF and the SDREF process noise was also reduced to the same

level as that of the EKF. This change in methodology had positive impacts on the

performance of the SDREF when confronted with initialization errors. Figures 21

through 26 show the new performance of the SDREF using the same initialization errors

as in the baseline runs. It can be seen that the initial oscillatory motion in the state

estimate errors has been corrected and the filter compares favorably with that of the EKF.















"I I i





Time (sec)

Figure 21. SDREF position errors with 3astandard deviations
after new initialization method, 1-percent error.





I I






Time (sec)

Figure 22. SDREF velocity errors with 3oastandard deviations
after new initialization method, 1-percent error.













Time (sec)

Figure 23. SDREF acceleration errors with 3astandard deviations
after new initialization method, 1-percent error.



















Time (sec)
Figure 24. SDREF position errors with 3cstandard deviations
after new initialization method, 5-percent error.



I: I




It o I I


Time (sec)
Figure 25. SDREF velocity errors with 3astandard deviations
after new initialization method, 5-percent error.


j ^I I---


ITime (s )
- I ... m I Jc

Time (sec)


Figure 26. SDREF acceleration errors with 3ostandard deviations
after new initialization method, 5-percent error.


i








This new initialization methodology not only improves the performance in the presence

of initial errors, but also allows a convenient way of initializing the SDREF.


6.3.2 Measurement Errors

This section investigates the performance of the EKF and SDREF when subjected

to various degrees of measurement noise. The relative performance of the two filters

using the baseline measurement noise was shown in Figures 3 through 5. Using the same

baseline initial conditions given in Table 1, two other levels of measurement noise were

added (Tables 4 and 5).


Table 4. EKF and SDREF measurement noise parameters level 1


Parameters X Y Z

Measurement Noise (%,rad,rad) 1.0 100.0E-6 100.0E-6

Filter Noise Variance (.01* Range)2 1.0E-8 1.0E-8



Table 5. EKF and SDREF measurement noise parameters level 2


Parameters X Y Z

Measurement Noise (%,rad, rad) 10.0 500.0E-6 500.0E-6

Filter Noise Variance (0.1 Range)2 2.5E-7 2.5E-7








The second level of noise is large for a typical engagement; however, it exercises

the robustness of each filter. Both filters were given measurement noise covariance

matrices that matched the noise characteristics given to the sensor. The position state

estimate plots are given in Figures 27 through 30. There is not a significant difference in

filter performance using the large noises. As expected, the estimates for both filters are

noisier and more erratic. The acceleration state estimate has become even worse under

these conditions. This estimate was initially poor, and with noisy measurements the

filters do not have enough information to estimate the target acceleration to the desired

degree of accuracy. It can also be noted that the SDREF estimation error under the

second level of noise seems to be worse than the EKF estimate. While these figures only

represent one run, the SDREF seems capable of adequate performance even under

stressing noise conditions.


6.3.3 Target Maneuvers

This section investigates the open-loop performance of the EKF and SDREF

against two stressing target maneuvers. The capabilities of the filters against a thrust cut-

off maneuver have been previously discussed. In this section two other nonlinear

maneuvers were tested. The first is a 10-degrees per second rotating target maneuver.

This maneuver allows the target acceleration vector to rotate approximately 110 degrees

during the engagement. This is a moderately high target maneuver, but it is possible for

an ICBM to perform this type of action in an active countermeasure scenario. The second

stressing engagement consists of a dogleg maneuver in which the target acceleration

vector rotates 90 degrees out of plane and then reverses itself to rotate back inward 90















2OO


-300
soo


10
x -100

-BOO




30




-20


-3O

10

i o
N -0


-30


Time (sec)


Figure 27. EKF position errors with 3oastandard deviations

1-percent error in X, 100 microradian error in Y and Z.


Time (sec)


Figure 28. SDREF position errors with 3ostandard deviations

1-percent error in X, 100 microradian error in Y and Z.















I ?A..S


I I


t0 0 iiiir i- -
sQ ---- ------------ ------------ .
mo=^--^
E m ; -- --------------
a 2 -- *l .
S. .ao ---,-'---- ------
x ) : --== -= -
o ---- ----
.80- --- ---------------
.100 *_ '


. ............ ..


0 2 10 1
-so





























-2w
015



,0 2--- ^ ------ ------ ------------- ----- 4 ---S 1





Time (sec)


Figure 29. EKF position errors with +3" standard deviations

10-percent error in X, 500 microradian error in Y and Z.























0 2 4 6 6 10 1o


Time (sec)


Figure 30. SDREF position errors with 3astandard deviations

10-percent error in X, 500 microradian error in Y and Z.


E


1 -IcT I 1
a ra 12







76
degrees. The initial conditions used for this analysis were the same as the baseline

scenario except for the process noise characteristics given to the filters (see Table 6).


Table 6. Maneuvering target parameters


Parameters (units) X Y Z

Aspect Angle (degrees) 30 30 30
SIGP (m) 0 0 0
SIGVP (m/s) 0 0 0
o2 (m2) 1.0E5 1.0E5 1.0E5
SIGXHAT o2 (m2 / 2) 1.0E3 1.0E3 1.0E3
Cr2 (m2 / s4) 1.0E2 1.0E2 1.0E2
R (m) 1.0E5 10.0 10.0
XHAT V (m / s) -1.0E4 -335.9 223.9
A (m/s2) 55.2 57.4 -38.2
SIGQ (EKF) (m2 /s4) 866.0 866.0 866.0
SIGQ (SDREF) ( / s4) 1.0E3 1.0E3 1E3.0
Meas Noise (%,nm, rn) 0.1 50.0 50.0



The process noise levels were tuned to allow the filters to open up and "accept" the

unexpected target maneuvers. While this allows for better tracking, it does lead to a

noisier state estimate in both filters. Figures 31 through 34 show the relative performance

of each filter against these two scenarios.

As can be seen from these figures, there are no appreciable differences in the state

estimates between the two filters. They each track this maneuverable target, however the

state estimate error is noisier and larger, as expected. It should be noted that once the

initial transient terms in the EKF covariance have settled, they are similar in shape and

value to the "steady-state" covariance terms of the SDREF.





















45 ..8 0 12








0 2 4 6 S 10 02











Time (sec)


Figure 31. EKF position errors with 3astandard deviations
for 10 deg/sec rotating target maneuver.






- 20



0 2 4 0 6 10











... ..-- _
0 4 6 -









Time (sec)


Figure 32. SDREF position errors with 3ostandard deviations

for 10 deg/sec rotating target maneuver.





















.10 4--- 10------ -------1------2-------____________

B8 10
-lo










lo








0 0 4 4 8 10 12














o 04 6 l 0
Time (sec)


Figure 33. EKF position errors with 30 standard deviations

for dogleg target maneuver.






8 o


. Ii K ;. / ,---- Av-v '. A
.1


Time (sec)


Figure 34. SDREF position errors with 3+o standard deviations

for dogleg target maneuver.


IL1


---w~
----~nu
----iraaoa







I to







79
6.4 SDREF Closed Loop Performance

6.4.1 Other Engagement Aspect Angles

Up to this point the EKF and SDREF performance in an open-loop environment

have been examined. The following analysis examines the closed-loop performance of

each filter against various engagement scenarios. This includes a variety of engagement

aspect angles, target maneuvers, initial condition errors, and measurement noise levels.

In this section, however, performance in terms of probability of hit and percentage of fuel

used to achieve the hit are measured. Measuring the percentage of fuel used, allows a

comparison of how noisy the state estimates are, as increased error in the beginning of the

engagement results in fuel being wasted performing unnecessary maneuvers. This

analysis was performed using 31 Monte Carlo run statistics. The first analysis focused on

three engagement scenarios. Up to this point a 30,30,30-degree closing aspect angle was

used. A head-on, 90-degree beam shot, and a tail-chase engagement are now added.

Figure 35 shows the three engagements and the corresponding simulation parameters are

given in Tables 7 through 9.












Y

(0,0,0) Aspect Angle Head On
(0,0,0) Aspect Angle Head On


13*


(0,0,-90) Aspect Angle Beam Shot


Y


(0,150,0) Aspect Angle Tail Chase
Figure 35. Aspect angle description.


x -r
t--








Table 7. Head-on simulation parameters


Parameters (units) X Y Z

Aspect Angle (degrees) 30 30 30
SIGP (m) 0 0 0
SIGVP (m/s) 0 0 0
r2 (m2) 1.0E5 1.0E5 1.0E5
SIGXHAT O2 (m2 / s2) 1.0E3 1.0E3 1.0E3
oaA (m2/s4) 1.0E2 1.0E2 1.0E2
R (m) 1.0E5 10.0 10.0
XHAT V (m/s) -10000.0 -335.9 223.9
A (m/s2) 55.2 57.4 -38.2
SIGQ (EKF) (m2 / s4) 866.0 866.0 866.0
SIGQ (SDREF) (m2 /s4) 1.0E3 1.0E3 1.0E3
Meas Noise (%,/pm, mn) 0.1 50.0 50.0



Table 8. 90-degree beam shot simulation parameters


Parameters (units) X Y Z

Aspect Angle (degrees) 0 0 -90
SIGP (m) 0 0 0
SIGVP (m/s) 0 0 0
62 (m2) 1.0E5 1.0E5 1.0E5
SIGXHAT O,2 (m /s2) 1.0E3 1.0E3 1.0E3
___ (m /s4) 1.0E2 1.0E2 1.E2
R (m) 1.0E5 10.0 10.0
XHAT V (m/s) -1.0E4 498.2 0.0
A (m/s2) 0.0 -88.3 0.0
SIGQ (EKF) (m2/s4) 196.0 196.0 196.0
SIGQ (SDREF) (m2 /4) 196.0 196.0 196.0
Meas Noise (%,mn, Um) 0.1 50.0 50.0







82
Table 9. Tail-chase simulation parameters


Parameters (units) X Y Z

Aspect Angle (degrees) 0 150 0
SIGP (m) 0 0 0
SIGVP (m/s) 0 0 0
a,2 (m2) 1.0E5 1.0E5 1.0E5
SIGXHAT O,2 (m / s2) 1.0E3 1.0E3 1.0E3
o2 (m2 / s) 1.0E2 1.0E2 1.0E2
R (m) 1.0E5 10.0 10.0
XHAT V (m/s) -1.0E4 0.0 -237.9
A (m/s2) -76.5 0.0 -44.1
SIGQ (EKF) (m2 /s4) 196.0 196.0 196.0
SIGQ (SDREF) (mi / s4) 196.0 196.0 196.0
Meas Noise (%,/Um, pm) 0.1 50.0 50.0



The 90-degree beam shot is the most stressing of these engagements as the entire

target acceleration profile is out of plane. The head-on case is one of the easier

engagements when using a range measurement, and the tail chase lies in-between these

two in terms of estimation difficulty. Table 10 displays the results from the Monte Carlo

analysis against these engagements. Probability of hit (PHIT) is the probability that the

interceptor hit the target. Circular error probable (CEP) is the probability that 50-percent

of the misses lie within this bound, 95-percent miss includes ninety-five percent of the

misses, and worst miss displays the worst of the Monte Carlo runs. The mass usage

numbers reflect the same quantities except in terms of percent of divert fuel used. The

vehicle has a mass of 4 kilograms, 1.5 of which is devoted to divert fuel. When this limit

is exceeded, the interceptor will continue to fly towards the target with no further

guidance corrections, typically resulting in large miss distances. Since these engagements








had good initial-state estimates and the target is not maneuvering, all of the runs were

hits; however as shown in the following sections that this is not always the case.


Table 10. Miss distance performance for varying scenarios


Parameter (units) 00 90 1500
EKF SDREF EKF SDREF EKF SDREF
PHIT 1.0 1.0 1.0 1.0 1.0 1.0
CEP (m) 0.185 0.262 0.432 0.625 0.315 0.265
Mean Miss (m) 0.213 0.272 0.546 0.667 0.347 0.354
95% Miss (m) 0.364 0.426 1.212 1.209 0.654 0.671
Worst Miss (m) 0.421 0.599 1.828 1.679 0.803 0.895
Mass CEP (%) 16.70 27.61 67.38 72.67 43.42 48.06
Mass Mean (%) 16.90 28.27 66.98 72.76 43.11 48.40
Mass 95% (%) 21.03 32.78 69.25 75.22 46.10 51.62
Mass Worst (%) 22.70 33.16 71.20 75.51 48.57 52.59



From Table 10, no significant difference in the closed-loop performance of these two

filters under the above scenarios is shown. All performed adequately and ensured a hit

against the target in all cases. Fuel usage was also not a decisive factor as both filters

differed by only a few percentage points.

6.4.2 Stressing Target Maneuvers

To further test filter performance, two other target maneuvers were considered.

These maneuvers are on the outer edge of filter capability of properly tracking the "true"

state. They were included to investigate whether the parameterization of the SDREF

allows an advantage over the linearization of the EKF in highly nonlinear environments.

The first maneuver is a rotating target, however in this engagement the engagement

aspect angle was a 90-degree beam shot and the rotation rate was increased to 20 degrees








per second. Therefore, the target will have completed an approximately 200-degree turn

during the length of the engagement.

The second maneuver consisted of a dogleg evasive maneuver. This maneuver is

similar to the previously described dogleg maneuver except that it was performed in the

90-degree-beam-shot scenario, causing it to become more difficult for the filter to

estimate.

Table 11 displays the results of Monte Carlo runs. The process noise levels for

each filter were tuned to give the best performance against these scenarios.


Table 11. Miss distance performance for stressing target maneuvers


Parameters (units) 200/Sec Dogleg
EKF SDREF EKF SDREF
PHIT 0.935 0.806 0.871 0.742
CEP (m) 1.61 2.20 1.14 2.85
Mean Miss (m) 1.83 200.20 1.50 3.05
95% Miss (m) 4.30 7.44 3.94 6.95
Worst Miss (m) 6.00 6121.89 5.59 7.36
Mass CEP (%) 100 100 100 100
Mass Mean (%) 99.94 99.99 99.87 99.98
Mass 95% (%) 100 100 100 100
Mass Worst (%) 100 100 100 100



As shown in Table 11, the SDREF performed slightly worse against the targets than the

EKF. The one large miss distance for the SDREF was caused by poor tracking, which

caused the interceptor to run out of fuel early and coast to a large miss distance.

The purpose of this section was to stress the filters to their extreme operational

limits and investigate the behavior. These types of target maneuvers would be considered







85
unlikely. However, they do provide a convenient way to determine the robustness of each

filter design. Even at these extremes, the SDREF was almost able to meet the EKF

performance measures.

6.4.3 Measurement Errors

This paragraph discusses filter sensitivity to various levels of measurement error.

For this analysis the range measurement error was held constant at 1 percent due to the

size of the range error when dealing with large engagement distances. Azimuth and

elevation noises were included for 100, 500, and 1000 microradian errors. The results

displayed in Table 12 were done using a 31-run Monte Carlo experiment.


Table 12. Miss distance performance for varying measurement noise


Parameters (units) 00rad 500rad 1000rad
EKF SDREF EKF SDREF EKF SDREF
PHIT 1.0 1.0 0.968 1.0 0.935 0.806
CEP (m) 0.455 0.460 1.39 1.41 3.09 2.22
Mean Miss (m) 0.468 0.568 1.47 1.45 3.17 3.14
95% Miss (m) 0.775 1.218 2.47 2.60 5.66 6.29
Worst Miss (m) 0.935 1.569 2.55 3.08 6.03 8.18
Mass CEP (%) 77.13 76.86 84.19 81.83 89.79 86.99
Mass Mean (%) 77.57 76.96 84.66 82.36 90.23 87.05
Mass 95% (%) 80.67 78.95 88.27 85.12 94.46 89.40
Mass Worst (%) 81.23 79.55 90.92 86.31 96.66 92.67


Once again, the results between the two filters are similar. Even with high-measurement

noise levels, both filters performed adequately, missing only one or two times in the

Monte Carlo run. The fuel usage numbers are also comparable. These results verify the

results of the pdf comparison between the filters, as the probability distributions are

similar between the EKF and SDREF. Therefore, it can be assumed that the filters are








Monte Carlo run. The fuel usage numbers are also comparable. These results verify the

results of the pdf comparison between the filters, as the probability distributions are

similar between the EKF and SDREF. Therefore, it can be assumed that the filters are

using approximately the same set of distribution functions, even though they are achieved

in different ways.

6.4.4 Initialization Errors

Using the new initialization method described in section 6.3.1, a closed-loop

analysis of the EKF and SDREF was performed. The initial conditions used for the open-

loop initialization study were used for this investigation. The filters were run closed-loop

in a 31-run Monte Carlo simulation. Table 13 displays the results of these runs. Using

this new initialization method not only shows better performance in the single-run, open-

loop case, but performs equally as well in the closed-loop analysis. Probability of hit,

miss distance, and fuel-usage numbers show that the SDREF performance is similar to

that of the EKF.


Table 13 Miss distance performance for varying initialization error


1% Error 5% Error
Parameters (units) E SDREF EKF5% Errr

PHIT 1.0 1.0 1.0 1.0
CEP (m) 0.436 0.377 0.336 0.389
Mean Miss (m) 0.464 0.380 0.423 0.432
95% Miss (m) 0.849 0.750 0.858 0.770
Worst Miss (m) 1.01 0.793 1.13 1.12
Mass CEP (%) 75.13 79.18 72.53 92.03
Mass Mean (%) 75.00 79.45 72.53 92.03
Mass 95% (%) 77.34 81.92 74.56 93.20
Mass Worst (%) 78.12 84.78 75.55 93.42







87
6.5 Summary of Exoatmospheric Guidance Problem

The exoatmospheric guidance problem analysis demonstrated, with some minor

caveats, the SDREF was capable of satisfying the interceptor guidance problem. The

assumptions of detectability in solving the SDRE required the addition of a range

measurement. An ad hoc initialization algorithm needed to be developed to properly

initialize the SDREF. With these two restrictions in place, however, the SDREF

performed similarly to the EKF for the exoatmospheric guidance problem for several

engagements, noise sources, and initial-error combinations.












CHAPTER 7
PENDULUM PROBLEM FILTER DEVELOPMENT

To evaluate the effect of nonlinear dynamics on the SDREF performance, a simple

two-dimensional pendulum problem was constructed. The scenario consists of a simple

rigid pendulum with friction about the connection point, see (Figure 36). The

measurements are assumed to come from an accelerometer attached to the bob, measuring

g.




\0












/.


Figure 36. Pendulum set-up.







89
The pendulum is given some initial position and velocity and the friction term

damps the oscillations. This system, while simple, demonstrates nonlinear dynamics and

nonlinear measurements together in the same problem.

The system dynamics for the pendulum problem are given by the 2"d order

nonlinear differential equation

B=-gsinO -BO (7.1)
L

where BO is the friction term. B was set to be 0.5.

Both the SDREF and EKF were derived using the stochastic nonlinear system

.i = f(x) +w
z= h(x)+v

where w and v are Gaussian, zero-mean white-process and measurement noise,

respectively, given by

E[w(t)w'(t + r)= W(t)3(r)]
(7.3)
E[v(t)v(t + r)= R(t)S()]

If

x, =
x (7.4)


then the state equations for the pendulum system are

xi = x
g (7.5)
2 = in(x,)-Bx2 (7
L

The measurement equation is


z = sin(x1) Bx2
L




Full Text
xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E3A4DIPYE_PKLNE8 INGEST_TIME 2013-01-23T15:27:35Z PACKAGE AA00012992_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES