An analysis of methods for extracting aerodynamic coefficients from test data

MISSING IMAGE

Material Information

Title:
An analysis of methods for extracting aerodynamic coefficients from test data
Physical Description:
xv, 159 leaves. : ; 28 cm.
Language:
English
Creator:
Daniel, Donald Clifton, 1942-
Publication Date:

Subjects

Subjects / Keywords:
Aerodynamic measurements   ( lcsh )
Kalman filtering   ( lcsh )
Aerospace Engineering thesis Ph. D
Dissertations, Academic -- Aerospace Engineering -- UF
Genre:
bibliography   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis--University of Florida.
Bibliography:
Bibliography: leaves 152-157.
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 - 000868901
oclc - 14216826
notis - AEG5921
System ID:
AA00003545:00001

Full Text
















AN ANALYSIS OF METHODS FOR EXTRACTING
AERODYNAMIC COEFFICIENTS FROM TEST DATA



By



DONALD CLIFTON DANIEL


A DISSERTATION PRESENTED TO THE GRADUATE
COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMNT OF THE REQUIRPF.IEENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY.




UNIVERSITY OF FLORIDA
1973













To Pat.


Digitized by the Internet Archive
in 2010 with funding from
University of Florida, George A. Smathers Libraries with support from Lyrasis and the Sloan Foundation


http://www.archive.org/details/analysisofmethod00dani












ACKNOWLEDGEMENTS


Many people contribute in a variety of ways during

the course of most graduate programs. I would like to

express my sincere appreciation at this time to some of

those who have helped me.

First, I want to thank the members of my supervisory

committee for the assistance and encouragement they have

offered. In particular, this work would not have been

possible without financial aid procurred from the Air Force

Armament Laboratory by Dr. M. H. Clarkson.

I also want to thank Dr. T. E. Bullock for the time

he has spent teaching me to appreciate some aspects of

modern control theory as well as the use and convenience

of state variable notation.

I would also like to acknowledge the teaching efforts

of Dr. U. H. Kurzweg for his excellent courses in applied

mathematics.

Finally, Dr. W. W. Menke is due special thanks, not

only for giving me a great deal of encouragement, but also

for helping me to appreciate the importance of technological

management.


iii











TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS iii

LIST OF TABLES V

LIST OF FIGURES vii

NOMENCLATURE LIST xii

ABSTRACT xiv

CHAPTER

I. Introduction 1

II. The Coefficient Extraction Technique
of Chapman and Kirk 7

III. Analysis of the Chapman-Kirk
Coefficient Extraction Technique 13

IV. Development of the Extended Kalman
Filter for Estimating Parameters and
Their Uncertainties 45

V. Use of the Extended Kalman Filter for
Estimating Parameters and Their
Uncertainties 60

VI. Conclusion 111

APPENDIX

I. Chapman-Kirk Coefficient Extraction
Nomenclature List and Program Listing 119

II. Extended Kalman Filter Nomenclature
List and Program Listing 142

REFERENCES 153

BIOGRAPHICAL SKETCH 158











LIST OF TABLES


Page

I. Initial Conditions and Nonhomogeneous
Terms for Parametric Differential
Equations 15

II. Program Options 19

III. True Values of Constants Used in
Generating Data 22

IV. Summary of Results for Extracting Aero-
dynamic Coefficients from Dynamic Data
Containing Random Measurement Errors 24

V. Extracted Coefficient Uncertainty Ratios 27

VI. Summary of Results for Extracting Aero-
dynamic Coefficients from Dynamic Data
Containing Random Measurement Errors
and System Noise 30

VII. Effects of Variations in the Initial
Parameter Variances on Near Steady-
State Parameters and Their Variances
(Linear System, Measurement Errors Only) 67

VIII. Effects of Variations in the Initial
Parameter Variances on Near Steady-
State Parameters and Their Variances
(Linear System, Measurement Errors
and System Noise) 70

IX. Effects of Variations in the Initial
Parameter Variances on Near Steady-
State Parameters and Their Variances
(Nonlinear System, Measurement Errors
Only) 77

X. Effects of Errors in the Estimate of
the Measurement Error Variance on Near
Steady-State Parameters and Their
Variances (Nonlinear System, Measure-
ment Errors Only) 79









List of Tables, continued


Page

XI. Effects of Variations in the Initial
Parameter Variances on Near Steady-
State Parameters and Their Variances
(Nonlinear System, Measurement Errors
and System Noise) 83

XII. Effects of Errors in the Estimates of
Noise Variances on Near Steady-State
Parameters and Their Variances
(Nonlinear System, Measurement Errors
and System Noise) 84











LIST OF FIGURES


Page

1. Example of Program Fit to Data Containing
Measurement Errors 34

2. Variation of Percent Error in Extracted
C3 with Measurement Error 35

3. Variation of Percent Error in Extracted
C4 with Measurement Error 35

4. Variation of Percent Error in Extracted
C5 with Measurement Error 36

5. Variation of Percent Error in Extracted
C6 with Measurement Error 36

6. Variation of RMS Residual with Measurement
Error 37

7. Variation of Normalized Estimated Standard
Deviation of C3 with Measurement Error 38

8. Variation of Normalized Estimated Standard
Deviation of C4 with Measurement Error 38

9. Variation of Normalized Estimated Standard
Deviation of C5 with Measurement Error 39

10. Variation of Normalized Estimated Standard
Deviation of C6 with Measurement Error 39

11. Example of Program Fit to Data Containing
System 'oise and Measurement Errors 40

12. Variation of Percent Error in Extracted
C3 withI System Noise 41

13. Variation of Percent Error in Extracted
C4 with System Noise 41

14. Variation of Percent Error in Extracted
C5 with System Noise 42


vii








List of Figures, continued


Page

15. Variation of Percent Error in Extracted
C6 with System Noise 42

16. Variation of Normalized Estimated Standard
Deviation of C3 with System Noise 43

17. Variation of Normalized Estimated Standard
Deviation of C4 with System Noise 43

18. Variation of Normalized Estimated Standard
Deviation of C5 with System Noise 44

19. Variation of Normalized Estimated Standard
Deviation of C6 with System Noise 44

20. Variation of Percent Error in X3 with Time
(Linear System, Measurement Errors Only) 87

21. Variation of Percent Error in Xi with Time
(Linear System, Measurement Errors Only) 87

22. Variation of Near Steady-State Parameter
Error with Initial Parameter Variance
(Linear System, Measurement Errors Only) 88

23. Variation of Normalized Near Steady-State
Parameter Uncertainty with Initial Parameter
Variance (Linear System Measurement Errors
Only) 89

24. Variation of X3 Variance with Time for
Three Initial Estimates (Linear System,
Measurement Errors Only) 90

25. Variation of X4 Variance with Time for
Three Initial Estimates (Linear System,
Measurement Errors Only) 90

26. Variation of Percent Error in X3 with Time
(Linear System, Measurement Errors and
System Noise) 91

27. Variation of Percent Error in X4 with Time
(Linear System, Measurement Errors and
System Noise) 91


viii









List of Figures, continued


Page

28. Variation of Near Steady-State Parameter
Error with Initial Parameter Variance
(Linear System, Measurement Errors and
System Noise) 92

29. Variation of Normalized Near Steady-State
Parameter Uncertainty with Initial Parameter
Variance (Linear System, Measurement Errors
and System Noise) 93

30. Variation of Percent Error in X4 with Time
for Two Estimates of the Initial Parameter
Variance (Linear System, Measurement Errors
and System Noise) 94

31. Variation of X3 Variance with time for
Three Initial Estimates (Linear System,
Measurement Errors and System Noise) 95

32. Variation of X4 Variance with Time for
Three Unitial Estimates (Linear System,
Measurement Errors and System Noise) 95

33. Variation of Percent Error in X4 with Time
for Large Initial Parameter Estimate Error
and Two Initial Parameter Variance Estimates
(Linear System, Measurement Errors and
System Noise) 96

34. Variation of Percent Error in X3 with Time
(Nonlinear System, Measurement Errors Only) 97

35. Variation of Percent Error in Xi with Time
(Nonlimear System, Measurement Errors Only) 97

36. Variation of Percent Error in Xs with Time
(Nonlinear System, Measurement Errors Only) 98

37. Variation of Percent Error in X6 with Time
(Nonlinear System, Measurement Errors Only) 98

38. Variation of Near Steady-State Parameter
Error with Initial Parameter Variance
(Nonlinear System, Measurement Errors Only) 99









List of Figures, continued


Page

39. Variation of Normalized Near Steady-State
Parameter Uncertainty with Initial Para-
meter Variance (Nonlinear System,
Measurement Errors Only) 100

40. Variation of Near Steady-State Parameter
Error with Error in the Estimate of the
Measurement Error Variance (Nonlinear
System, Measurement Errors Only) 101

41. Variation of Normalized Near Steady-State
Parameter Uncertainty with Error in the
Estimate of Measurement Error Variance
(Nonlinear System, Measurement Errors Only) 102

42. Variation of Percent Error in X3 with Time
for Large Error in Initial Estimate of X5
(Nonlinear System, Measurement Errors Only) 103

43. Variation of Percent Error in X4 with Time
for Large Error in Initial Estimate of Xc
(Nonlinear System, Measurement Errors Oniy) 103

44. Variation of Percent Error in Xs with Time
for Large Error in Initial Estimate of Xc
(Nonlinear System, Measurement Errors Only) 104

45. Variation of Percent Error in Xg with Time
for Large Error in Initial Estimate of Xc
(Nonlinear System, Measurement Errors Only) 104

46. Variation of Percent Error in X with Time
(Nonlinear System, Measurement Errors and
System Noise) 105

47. Variation of Percent Error in X with Time
(Nonlinear System, Measurement Errors and
System Noise) 105

48. Variation of Percent Error in X with Time
(Nonlinear System, Measurement Errors and
System Noise) 106








List of Figures, continued


Page
49. Variation of Percent Error in X6 with Time
(Nonlinear System, Measurement Errors and
System Noise) 106

50. Variation of Near Steady-State Parameter
Error with Initial Parameter Variance
(Nonlinear System, Measurement Errors and
System Noise) 107

51. Variation of Normalized Near Steady-State
Parameter Uncertainty with Initial Parameter
Variance (Nonlinear System, Measurement
Errors and System Noise) 108

52. Variation of Near Steady-State Parameter
Error with Error in the Estimate of the
Measurement Error Variance (Nonlinear
System, Measurement Errors and System
Noise) 109

53. Variation of Normalized Near Steady-State
Parameter Uncertainty with Error in the
Estimate of Measurement Error Variance
(Nonlinear System, Measurement Errors
and System Noise) 110











NOMENCLATURE


Symbol Definition

A Reference area

d Reference length

I Vehicle moment of inertia about an
axis through the center of gravity and
normal to its pitch plane

q Dynamic pressure, 1/2 pv2

V Freestream velocity

C Static pitching moment coefficient
moo derivative, rad-1

C Static pitching moment coefficient
ma2 derivative, rad-3

C MStatic pitching moment coefficient
Cmt4 derivative, rad-5

C mqo Pitch damping coefficient
mqo -
C Pitch damping coefficient, rad-2

C Static pitching moment coefficient at
a=0

a Pitch angle

p Freestream density
B-1 The inverse of the matrix B

BT The transpose of the matrix B

E[g(X)] Expected value of g(X)
E[g(X)] = f. g(c)Fx(O)d

where Fx () is the probability density
function of the random variable X


xii








Symbol Definition

6.. Kronecker delta
t-t') Dirac delta function
6(t-t') Dirac delta function


xiii











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



AN ANALYSIS OF METHODS FOR EXTRACTING
AERODYNAMIC COEFFICIENTS FROM TEST DATA


By
Donald Clifton Daniel

March, 1973


Chairman: M. H. Clarkson
Major Department: Engineering Science, Mechanics, and
Aerospace Engineering


An analysis of numerical methods for extracting

aerodynamic coefficients from dynamic test data has been

conducted. The emphasis of the analysis is on the effects

that random measurement errors in the data and random

disturbances in the system have on the accuracy with which

the coefficients for linear and nonlinear systems can be

determined. Both deterministic and stochastic methods for

extracting the coefficients and determining their uncertain-

ties are considered.

The deterministic technique considered, due to Chapman

and Kirk, provides excellent estimates of both linear and

nonlinear static pitching moment coefficients for the range


xiv







of measurement errors and system noise considered. Somewhat

less accurate estimates of linear damping coefficients are

obtained. Nonlinear pitch damping coefficients extracted

using this deterministic technique are affected considerably

by both measurement errors and system noise. The estimated

standard deviations of the extracted coefficients obtained

using standard techniques are generally adequate when the

data being analyzed contain only measurement errors.

The stochastic approach considered demonstrates the

feasibility of using an extended Kalman filter, with a

parameter augmented state vector, for determining the values

of the aerodynamic coefficients and their uncertainties

from noisy dynamic test data.

The specific filter used generally reaches near steady-

state conditions in its estimates of the parameters in less

than one second. Variations in the initial parameter

variances or in the estimates of the noise statistics

essentially affect only the determination of the nonlinear

damping parameter.

Parameter estimates obtained with the extended filter

compare favorably with previously obtained results using

deterministic techniques. Estimates of the parameter

uncertainties provided by the filter are generally superior

to those obtained with deterministic techniques particularly

when system noise has corrupted the data.











CHAPTER I

INTRODUCTION


This dissertation presents an analysis of methods

for determining aerodynamic coefficients of flight vehicles

from dynamic test data. This determination is accomplished

by finding numerical values of the aerodynamic coefficients

appearing in the equations of motion such that the solutions

to these equations are adequate representations of the

given test data. When this determination has been made,

the coefficients are said to have been extracted from the

dynamic data.

The emphasis of the analysis is placed on the effect

that random measurement errors in the data and random

disturbances in the system have on the accuracy with which

the coefficients for linear and nonlinear systems can be

determined. Both deterministic and stochastic system

models for extracting the coefficients and determining

their uncertainties are considered.

Historical Background


The determination of aerodynamic coefficients from

dynamic data is generally agreed to have begun with the








work of Fowler, Gallop, Lock and Richmond1 in the first

quarter of this century. Their basic technique was con-

cerned with determining moment and damping characteristics

of artillery shells by firing these projectiles through

spaced cards and reconstructing the pitch and yaw angle

time histories by observing the obliqueness of the holes

that resulted when the, shells passed through the cards.

Their technique of data measurement is still in use at

some ballistic ranges to this date.2

Nielsen and Synge3 later clarified the linear theory

of Fowler et al. in their work during, and immediately

following, World War II.

Coefficient extraction techniques that are currently

in use at many ballistic and wind tunnel facilities evolved

from the work of Murphy4'5 and Nicolaides,6 both of whom

have considered various combinations of degrees of freedom

for a variety of flight vehicles. Their aerodynamic co-

efficient extraction techniques feature a least squares

fit to dynamic data of the exact solutions of linear

equations of motion, or approximate closed form solutions

of slightly nonlinear equations of motion using the quasi-

linearization technique of Kryloff and Bogoliuboff.7 Ex-

tensions of the work of Murphy and Nicolaides, particularly

for more complex nonlinear systems, have been made by

Eikenberry and Ingram.








The requirement for closed form solutions of the

equations of motion has recently been eliminated by the

formulation of least squares techniques which fit numerical

solutions of the equations of motion to dynamic data. The

minimization of the least squares criterion function involves

differential corrections, as do the techniques of Murphy

and Nicolaides; the required partial derivatives are

determined by numerically integrating parametric differ-

ential equations which are derived from the equations of

motion. Contributions to this numerical coefficient

extraction technique have been made independently by

Chapman and Kirk,10 Knadler,11 Goodman12'13 and
14
Meissinger, although the method is usually referred to

as the Chapman-Kirk technique. The computational require-

ments of this technique are sometimes extensive,15 but

this is usually outweighed by the fact that it can be

used to analyze highly nonlinear aerodynamic forces and

moments.16

All of the coefficient extraction techniques that

have been discussed to this point are deterministic in

nature in that the modeling of the equations of motion

does not account for random disturbances in the system.

Most angular or translational motion data obtained from

ballistic rac:ges or wind tunnel dynamic tests have, never-

theless, been affected by random system noise and random








measurement errors. The effects of noise of these types

on the accuracy of coefficients extracted from dynamic data

using deterministic techniques are of current interest.17

With the exception of Ingramn's partial analysis of noise
8
effects on Eikenberry's "Wobble" program, however, little

has been reported concerning these effects. One of the

primary goals of the research reported here is to show some

effects of random measurement errors and system noise on

the accuracy of aerodynamic coefficients extracted from

dynamic data using the most general of the various determi-

nistic techniques, that of Chapman and Kirk.

In recent years increasing attention has been given

to parameter and state variable estimation through the

use of stochastic modeling of the physical system of

interest. Most of this work has been done by optimal-

control specialists and is a direct result of pioneering

contributions in linear filtering theory of Kalman 8'19
19
and Bucy. The Kalman filter provides estimates of the

states of noisy, linear dynamic systems as well as estimates

of the state variable uncertainties. Bryson and Ho,20

among others, have extended the Kalman filter theory to

include estimates of the states for nonlinear systems.

Mehra21 has recently proposed a maximum likelihood

technique for the determination of aerodynamic coefficients

from dynamic data using the Kalman filter for linear systems





5


and the extended Kalman filter when the system is non-

linear. Results verifying his proposals are, as yet,

unavailable.

An additional maximum likelihood technique has been

developed by Grove et al.22 and has been used with modest

success by Suit.23

A stochastic approach to the coefficient extraction

problem using an extended Kalman filter with parameter

augmented state vector is developed in this dissertation.

Results obtained with this filter are also presented.


Scope


The following chapters are devoted to an analysis

of the effects of random system noise and measurement

errors on coefficients extracted from dynamic data using

the method of Chapman and Kirk, as well as the development

and evaluation of an extended Kalman filter for solving

essentially the same problem. The equation of motion

used in the analysis is one describing the one-degree-

of-freedom pitching motion of a rigid body. Pitching

moments that are both linear and nonlinear functions of

angle-of-attack are considered.

The theory of the Chapman-Kirk technique is recounted

briefly in Chapter II, followed by the analysis of noise

effects on this method in Chapter III.





6


Developments leading to the formulation of the extended

Kalman filter to be used are given in Chapter IV. Results

obtained with the extended filter are presented and discussed

in Chapter V.

Some concluding remarks are offered in Chapter VI.

Computer program listings and definitions of program

variables are given in the two appendices following the

body of the paper.











CHAPTER II


THE COEFFICIENT EXTRACTION TECHNIQUE
OF CHAPMAN AND KIRK


A brief reconstruction of the Chapman-Kirk algorithm

for extracting aerodynamic coefficients from dynamic data

is given in this chapter. The standard technique for

estimating uncertainties in coefficients determined from

a least squares fit of a given function to experimental

data is also presented.


Development of the Extraction Algorithm


The extraction technique of Chapman and Kirk has two

very basic requirements: (1) the different al equation

of motion for the body of interest must be given and

(2) a set of experimental data based on the observed motion

of the body must be available.

As an example, consider the nonlinear equation of

pitching motion for a rigid body

a + (C4 + CGa2)& + (C3 + C5a2)a = 0 (1)


subject to the initial conditions

a(0) = C1

& (0) = C2








where a indicates the derivative of a with respect to time.

Suppose, also, that the time history of the pitch angle

as recorded during some experiment, ae(ti), i=l,2,...m,

is also available.

The technique of Chapman and Kirk is used to determine

the values of the C. (j=1,2,...,6) in Equation (1) which

result in the solution to this equation of motion being

a best fit to the test data in a least squares sense.

Thus it is necessary to minimize the least squares cri-

terion function

m 2
S = ae(ti) ac(ti) (2)
i=l


where ac(ti) is obtained from the solution to Equation (1).

Now it is well known that in order to determine para-

meters directly by a least squares fit of a given function

to test data, the parameters must appear linearly in the

function. This requirement is met in the problem at hand

by expanding aC in a truncated Taylor series about the

numerical solution resulting from some initial estimates

of the parameters of interest, Cjo, i.e.,


"c(ti) = aco(ti) + AC. (3)
j=l 3ji "








where Da are evaluated for Cj=C. and AC.=C.-C
acjj 30 o J 3jo

Substituting (3) into (2) yields

M 6 2
S = e (t) a (ti) ( a AC. (4)
i= co1 i j= aCJ 3


Now, assuming that the, [a- are known, Equation (4) is a
function of the AC.'s only. Therefore, to determine the
values of these coefficients that will minimize this equation,
it is necessary to take the partial derivative of Equation
(4) with respect to each AC., set each of the resulting
equations to zero and solve for the AC.'s. Carrying out
these operations yields the following matrix equation

[A] [AC] = [B] (5)

where A is, in this case, a 6 X 6 matrix with elements
given by


A C J Ci (6)
jk Xi=l ac i Ck


AC is a 6 X 1 column matrix, or vector, and B is a 6 X 1
column matrix with elements given by

m (
B. = a (ti) ac t.i) 1, (7)
3 il i co I
i








The solution to Equation (5) is

[AC] = [A]-1 [B] .(8)

The solutions for the AC.'s obtained from Equation (8)

are exactly correct only if a is a linear function of the

C.'s as assumed by Equation (3). This condition of linearity

is seldom the case and the process must be repeated with

new initial guesses,


C = C. + ACo (9)

until the change in the criterion function [Equation (2)]

from one iteration to the next is sufficiently small.

The algorithm just presented requires that time

histories of the influence coefficients, a, be available.

These time histories are determined by numerically inte-

grating parametric differential equations which are derived

by differentiating the equation of motion with respect to

each of the parameters of interest. As an example, the

parametric differential equation for C1, the initial

condition of a is



L--[a + (C4+C6a2)& + (C3+Csa2)a] = 0
aLn








as aCs 3Cs Ba a 2Cs
3a + aC4- + 2C6 a2& + 2Ca& a + C6a2 3& + 3C a
C1 aC, aC 6 ac, aC1 aC1


3a 3C5 aa
+ C3 -a + -- a3 + 3C5d2 0
3R+1 a CI


Assuming that the parameters are independent of each other

and that the order of differentiation can be reversed,

the final form of the desired equation is


d2 fa d fa at
dt2 C a2 d 1 + (C +3C a2+2C a&) J- 0 (10)
dt2 to th dtinitial cons


subject to the initial conditions


aa(0) aC,
aC aC1


ac(O) aC2
aC1 aC1


The complete set of parametric differential equations for

the given equation of motion [Equation (1)] is given in

Chapter III.

In summary, the process for extracting numerical

coefficients from test data given the system model [Equa-

tion (1)] and criterion function [Equation (2)] is

1. Estimate the numerical values of the C.'s.
2. Integrate Equation (1) to obtain aco(ti).


(11)








3. Determine L-


4. Solve Equation (8) for the AC 's.


5. Repeat the process with C. =C. +AC. until the
31 Jo Jo
change in Equation (2) is sufficiently small.



Estimation of Extracted Parameter Uncertainties


The estimation of the uncertainties, or standard

deviations, of parameters that have been determined by the

least squares fitting of a given function to test data is

a well-known result available in a variety of references

(see, for example, References 24 or 25).

The least square parameter uncertainties are estimated

in this paper by



A.. S (12)
m-K
j jj -K (12)


where A. is the jth diagonal element of the inverse

Grammian matrix [Equations (6) and (8)], S is the sum of

the squares of the residuals as given by Equation (2), m

is the total number of data points and K is the total number

of parameters being determined by the fit.












CHAPTER III

ANALYSIS OF THE CHAPMAN-KIRK
COEFFICIENT EXTRACTION TECHNIQUE


Coefficient Extraction Computer Program


This section provides a detailed description of a

one-degree-of-freedom coefficient extraction computer

program based on the previously described iterative process

of Chapman and Kirk. This program considers the pitching

motion of a symmetric missile about a fixed point. It

is used to determine values of static pitching moment

coefficient derivatives, pitch damping coefficients, and

a trim term so that the solution to the appropriate differ-

ential equation of motion is a best fit to test data in a

least squares sense.

The program was developed to be used as an economical

tool in the evaluation of the sensitivity to noise and

convergence sensitivity of the Chapman-Kirk technique when

operating in its least complex mode.


Computational Equations

Equation of motion

The complete equation of motion that is used in the

program is





14


S+ (C +C a2)& + (C3+C a2+C7 a)a + C = 0


with initial conditions


caO) = C1


where


(Cmao)q
3 I


&(0) = C2




Ad C (Cmqo)qAd2
S-2VI
2VI


C (Cma2)qAd
sI



S(Cma4)qAd
C7 -


(Cmq2)qAd2
6 2VI




(CM6 )qAd
8 I


Parametric differential equations

The eight parametric differential equations for the

given equation of motion [Equation (13)] and initial con-

ditions [Equation (14)] are of the form


P. + APj + BP. = F.
3


j = 1,2,...,8


(15)


3 a3C
3j


S d 3a
' J dt 3Cj
3


d2 'ca
3 dt32 -9


A = (C4 + C6a2)


(13)


(14)


where









and

B = (C3 + 3C5a2 + 2C6a& + 5C7a4).


The values of the initial conditions P.(0) and P (0) as
J J
well as the functional forms of the nonhomogeneous term

F. are give in Table I for j=1,2,...,8.


Table I.


INITIAL CONDITIONS AND NONHOMOGENEOUS
TERMS FOR PARAMETRIC DIFFERENTIAL EQUATIONS


Pj (0)


P (0)


0

-Ci



3

a

-a5

-1








Program Description


The program is written in Fortran IV for use primarily

on an IBM 360/65. The paragraphs that follow describe the

functions of the main program and its four subroutines,

the required input data, and the program options. A listing

of the complete program is given in Appendix I.


Main program functions

The functions of the main program in their approximate

order of occurrence are as follows:

(a) Reads and writes input data.

(b) Computes initial parameter values (these are

the C.'s that appear in Equation (13).

(c) Calls the numerical integration subroutine ADDUM.

(d) Writes current parameter values.

(e) Computes the matrix elements required for incre-

menting the parameter values.

(f) Calls the matrix inversion subroutine MINV.

(g) Computes the sum of the squares of the residuals

between the calculated and experimental data

points.

(h) Computes the root-mean-square residual and root-

mean-square error (Reference 24) of the current

fit to date.









(i) Computes the estimated standard deviations

(Reference 24) of the current parameter values.

(j) Tests the difference between the current and

previous values of the root-mean-square error

to determine if the iteration process has

converged.

(k) Computes the incremental changes for the para-

meters (if convergence has not occurred).

(1) Computes the updated parameter values (if

convergence has not occurred).

(m) Returns to (c) (if convergence has not occurred).

(n) Computes the values of the extracted coefficients

from the current parameter values (after con-

vergence is achieved).

(o) Computes the estimated standard deviations of

the extracted aerodynamic coefficients (after

convergence is achieved).

(p) Writes extracted aerodynamic coefficients,

estimated standard deviations, and the pitch

angle output that represents the final fit to

the experimental data (after convergence is

achieved).


Subroutine functions

The names and functions of each of the four subroutines

that are used in the coefficient extraction program are








given below.

ADDUM.--Subroutine ADDUM integrates the equation

of motion and parametric differential equations using a

fourth order Runge-Kutta method for starting and a fourth

order Adams-Bashforth predictor-corrector method for

running. It calls subroutines XDOT and OUT. This sub-

routine is described in detail in Reference 26.

XDOT.--Subroutine XDOT computes current values of

first derivatives that are required by ADDUM.

OUT.--Subroutine OUT stores the results of the

numerically integrated equation of motion [Equation (13)]

and parametric differential equations [Equation (15)].

MINV.--Subroutine MINV inverts an K x K matrix using

a standard Gauss-Jordan technique and is described in

detail in Reference 27.


Required input data

The program reads six categories of input data.

These categories and the specific data in each are de-

lineated in Appendix I. The format and units of the

entries on a specific data card can be determined from

the program listing and nomenclature list provided in

this appendix.








Program options

This program has options for extracting various

combinations of aerodynamic coefficients from the given

test data, in addition to the option of extracting no

coefficients and merely integrating the equation of motion.

These options are controlled by the numerical value of the

number of first order ,equations to be integrated, N, which

is read by the program on the first data card. The various

options are given in Table II.


Table II. PROGRAM OPTIONS


N Coefficients to be Extracted


2 No coefficients are extracted. The program
integrates the equation of motion for the
given input data and prints the results.

8 ao, &,o Cmao

10 ao, o', Cmao, Cmqo

12 ao, &o, Cmao, Cmqo, Cma2

14 ao, ao, Cmao, Cmqo, Cma2, Cmq2

16 ao &0o Cmao' Cmqo' Cma2' Cmq2' Cma4

18 ao, &o, Cmao' Cmqo' Cma2' Cmq2 Cma4' Cm6a



.1.alysis of the Sensitivity to
Noisej of the Chlapman-Kirl- Technique


The evaluation of the sensitivity to noise of the

Chapman-Kirk coefficient extraction technique has been








approached from several directions. Coefficients have

been extracted from numerous sets of computer program

generated data containing only measurement errors as well

as data containing system noise and measurement errors.

In addition to the use of artificially generated data,

some very preliminary work has been done with noisy wind

tunnel data obtained from actual experimentation. The

design of the experiment used to produce the dynamic

data is described by Turner in Reference 28 along with some

preliminary results obtained with the one-degree-of-freedom

program described earlier in this chapter.

Since the wind tunnel experimentation is still in a

developmental stage, however, the results presented in the

remainder of this chapter are those obtained from arti-

ficially generated data.


Generation of Noisy Dynamic Data

The computer program UFNOISE described in detail in

Reference 29 was used to generate the dynamic data from

which the aerodynamic coefficients were extracted. This

program simulates the pitching motion of symmetric missile

oscillating in a wind stream about a fixed point, for any

given initial pitch angle displacement and initial pitch

angle rate, by numerically integrating the equation of

motion. The program has two options for simulating system

noise: it considers the magnitude and direction of the








freestream velocity vector as normally distributed random

variables, with programmer set mean values and standard

deviations, to determine the system noise perturbation

accelerations or it simply selects random perturbation

accelerations which have zero mean, are normally distributed

and have a programmer set standard deviation. Both methods

are essentially equivalent. New values of the system

noise perturbations are randomly selected at each numerical

integration step. The program also has the option of

simulating random measurement errors of the pitch angle

by superimposing normally distributed random noise on the

output of the numerical integration.

The basic equation of motion used to generate the

noisy dynamic data was a slightly simplified form of

Equation (13):

a + (C4 + C6a2)& + (C3 + C5a2)a = w(t) (16)


subject to

a(0) = Ci (O) = C2 (17)


The true values of the physical and aerodynamic con-

stants used in generating the data are given in Table III.

The various noise level standard deviations and mean values

are discussed in the following paragraphs to which they

are pertinent.








Table III. TRUE VALUES OF CONSTANTS
USED IN GENERATING DATA


Constant


Value


d(Ft)

A(Ft2)

I(slu.g ft2)

V (ft/sec)
m
q(lb/ft2)

Cmoo(rad- 1)

C (rad 3)
ma2
C
mqo
Cmq(rad-2)

a0 (rad)

a (rad/sec)
0


0.333

0.0873

0.1080

500.0

297.0

-2.00

-24.5

-60.0

-163.0

0.5235

0.0


Effects of Random Measurement Errors

Aerodynamic coefficients were extracted from a total

of nine sets of data containing only measurement errors.

Each data set was made up of 201 discrete points which re-

sult from integrating the equation of motion [Equation (16)]

in increments of 0.005 second for a total of 1.000 second.

The desired mean value of the measurement errors for

each of the mine data sets was -0.0 radian. The desired

standard deviations of the random errors were 0.00146








radian, 0.00582 radian, and 0.01745 radian,with three

different sets of data being generated at each noise level.

These standard deviations are typical of measurement un-

certainty for data of this type (Reference 9).

The above standard deviations correspond to percent

noise levels of 1.1, 4.4, and 13.2, respectively where

percent noise level is defined in Reference 30 as


Percent Noise = (18)
Approximate average peak amplitude


with a being the standard deviation of interest. The

approximate average peak amplitude can be determined in a

variety of ways. The value used here is the mean positive

peak amplitude for the three cycles that result when the

nominal equation of motion is integrated for 1.000 second.

The results of this portion of the analysis are given

in Table IV and on Figures 1 through 10.

Table IV is primarily a summary of the percent error

in the extracted aerodynamic coefficients together with

their normalized estimated standard deviations for the

various actual measurement errors, om. The percent error

in the extracted coefficients is defined by


Percent Error (C) = (C-C) 100 (19)
J r









r--f
Ci
: J



Id
t, I Cd "


E-
Z




0









0
U


J M
a0
U E-














Z
U






Sw



;2Fo







P4d
u

< u





O M
U




O
m










oltfr


I O

O
9*#
V )i



*iJ 0)
4-)












S.r-t














4- ,.-4
0 i
u d

a3

u n


tJ


OV)
md
X c




0




rO-




S$
a z


:: O .0 r-o co 0)


%O tn
'0 a



0 0






N O
0 f
-4 r







c o
I
V-, r-





'd -
in





N N
*





0 0





* *











r a
0 0


0 0








oo oo
0 0















0 0
*




ra- 0














r-4
0 0
* *0

0 0
CD' C)
iCl C)


*
L/ *=


C14 OC
cM cn


0 0
0 0

0 0





C14 -1

l il
N t












) 00
*











O-H-


. -4
4 -
-C4 -4q




0 r-4







c M
v v
, 0

a cc


0 N








. v

* *

0 CM


0 0
0 0

0 0
C> C
C) C


oi 0


O.


fr-






4-)
cd U)
m








where C. is the coefficient estimate determined with the
3
program in the fitting of the data and C. is the true

value of the coefficient of interest. The normalized

estimated standard deviations are the estimated standard

deviations calculated from Equation (12) divided by the

true value of the parameter of interest,


Percent = 100 (20)
Jn Cj



Table IV also contains the RMS residual for each fit to the

noisy data and the percent noise levels based on the values

of am.

An ex3 ple of one of the program fits to a noisy data

set is shown on Figure 1. Figures 2 through 5 show the

variation of the percent error in the individual extracted

coefficients with measurement error. The variation of RMS

residual with measurement error is shown on Figure 6. The

variations of the normalized estimated standard deviations

of the extracted coefficients with measurement error are

shown on Figures 7 through 10.

An analysis of Table IV and Figures 2 through 10

reveals the following.

(1) The extracted static pitching moment coefficient

derivatives show little or no error for the

entire range of measurement errors considered.

( 'e Figures 2 and 3.)








(2) The extracted pitch damping coefficients show

some error for the lower noise values (om 0.005

radian) and deviate significantly for am=0.018

radian. (See Figures 4 and 5.)

(3) The variation of RMS residual with measurement

noise is essentially linear with a slope of unity

(Figure 6). ,This indicates that the RMS residual,

as calculated by the coefficient extraction pro-

gram, is a good estimate of the amount of measure-

me-t noise in the data when this is the only

type of noise present.

(4) All variations of the normalized estimated standard

deviations of the extracted coefficients, ajn.
in
increase linearly with measurement error (Figures

7 through 10). Approximate values of the slopes

of these variations are given in Table V. These

normalized uncertainty ratios give the relative (to

each other) uncertainty which can be expected when

extracting coefficients from data containing

measurement noise using the given coefficient

extraction program.

(5) The true value of a given coefficient is contained

within the interval difined by its extracted

value 3ac for every coefficient extracted from

the nine sets of data.





27




Table V. EXTRACTED COEFFICIENT UNCERTAINTY RATIOS


Coefficient
(Cj)


Approximate Normalized
Uncertainty Ratio
(Aj /A m)








Effects of Random Measurement Errors and System Noise

For this portion of the analysis, attempts were made

to extract aerodynamic coefficients from twelve basic

sets of dynaunic data. Each data set was made up of 201

discrete points resulting from integrating the equation

of motion ina increments of 0.005 second for 4.00 seconds;

the pitch augle was output every four integration steps

so that the time between data points was 0.020 second.

The time between data points used in this portion of the

analysis is different from that previously used. This

change was made so that the time between data points would

correspond to that used by Turner28 in actual experiment-

ation.

Of the twelve basic data sets, nine contain system

noise and measurement errors, whereas three contain only

measurement errors. Coefficients were extracted from these

latter three to determine if the above-mentioned change in

the time increment between data points had any appreciable

effect on the extracted coefficients or their uncertainties.

The estimates of the uncertainties were improved due pri-

marily to the fact that a longer data record was being

analyzed.

The desired mean values of the freestream velocity

vector direction fluctuations and measurement errors for

all data sets were zero. The desired standard deviation








of the measurement errors, am, was 0.00582 radian

(30m = 1.0 degree) for all sets; the desired standard

deviation of the velocity magnitude fluctuations, av,

was 5.0 ft/sec for all nine data sets containing system

noise. The desired standard deviations of the velocity

direction, oa, were 0.02910 radian (3 a = 5 degrees),

0.05820 radian (3aa = 10 degrees), and 0.11640 radian

(3oa = 20 degrees), with three different sets of data
being generated at each noise level.

The random velocity fluctuations act as a forcing

function for the equation of motion and cause oscillations

even if the vehicle has no initial displacement. The

resultant maximum amplitudes of these forced oscillations

are usually within a certain magnitude or noise band width.

The widths of the noise bands for the data used in this

analysis are generally equivalent to the 3aa values, and

these have been used to calculate the percent system noise

values. The approximate average peak amplitude used in the

percent noise calculations is the mean positive peak ampli-

tude for the 10 cycles that result when the nominal equation

of motion is integrated for approximately four seconds.

The results of this portion of the analysis are given

in Table VI and on Figures 11 through 19.

Table VI is a summary of the percent error in the

extracted aerodynamic coefficients and their normalized

estimated standard deviations along with the various noise











Table VI. SUMMARY OF RESULTS FOR EXTRACTING AERODYNAMIC
COEFFICIENTS FROM DYNAMIC DATA CONTAINING
RANDOM MEASUREMENT ERRORS AND SYSTEM NOISE


a
(Rad)


0.00

0.00

0.00

0.02L- .

0.03038

0.02828

0.005516

0.06120

0.05662

0.11262

0.11706

0.12058


Percent
System
Noise


0.00

0.00

0.00

41.4

43.1

40.2

78.4

86.8

80.3

152.9

158.9

163.7


Data
Set


a m
(Rad)


0.30527

0.00589

0.00574

0.00611

0.00597

0.00587

0.00603

0.00562

0.00600

0.00572

0.00547

0.00571


ov
(Ft/sec)


0.00

0.00

0.00

5.05

5.18

4.88

5.01

4.90

4.96

5.08

4.85

4.96


~











(Extended)


Percent Error in Extracted Coefficients and
Normalized Estimated Standard Deviations

C3(03) C4(04) C5(05) C (as)


RMS
Residual
(Rad)


-0.27(0.25)

-0.34(0.28)

0.42(0.27)

0.55(0.64)

4.19(0.63)

2.87(0.55)

2.71(1.24)

14.44(1.32)

1.41(0.70)


-0.84(0.93)

0.29(1.04)

-0.36(1.01)

8.25(2.75)

5.84(2.41)

20.73(2.10)

-42.80(3.95)

-0.90(5.35)

-35.67(2.33)


0.68(0.65)

1.51(0.73)

-0.94(0.69)

9.55(1.88)

1.35(1.63)

-2.80(1.24)

6.54(3.64)

5.44(3.87)

14.59(2.30)

DIVERGED

DIVERGED

DIVERGED


5.72(12.92)

-3.60(14.43)

4.40(13.88)

329.59(43.58)

-44.80(34.40)

-247.20(26.39)

605.77(66.79)

197.10(90.61)

588.34(44.19)


0.00520

0.00574

0.00563

0.01258

0.01244

0.01026

0.02806

0.02627

0.01488


Table VI.








level standard deviations, percent system noise levels,

and the RMS residual of each fit to a noisy data set. The

percent system noise levels are similar to those encountered

experimentally by Turner in Reference 28.

Figure: 11 is an example of one of the program fits

to a noisy data set. Figures 12 through 15 show the varia-

tion of percent error for the extracted coefficients with

system noise. The variations of the normalized estimated

standard deviations of the extracted coefficients with

system noise level are shown on Figures 16 through 19.

An analysis of Table VI and Figures 12 through 19

reveals the following:

(1) The errors in the linear static pitching moment

coefficient derivative parameter, C3, are gen-

erally less than 5 percent for the entire range

of system noise considered. The errors in the

corresponding nonlinear term, C5, are generally

less than 10 percent. (See Figures 12 and 13.)

(2) ThMe pitch damping parameters show significant

error for all nonzero system noise levels. The

e:xtracted values of the nonlinear term, C6, are

sometimes in error by several orders of magnitude

at the higher noise levels. (See Figures 14

ard 15.)

(3) The estimated standard deviations of the extracted

coefficients generally increase as the system noise









increases. The estimated standard deviations

of coefficients extracted from dynamic data

containing both system noise and measurement

errors are generally too small and do not reflect

the true uncertainty of the extracted coeffi-

cients as was true in the previous case when

the data contained only measurement errors.

(See Table VI and Figures 16 through 19.)

(4) All attempts at extracting the complete set of

coefficients from data with a system noise band

of approximately 20 degrees failed (see Table VI).

The failures resulted when a value of C6 was

eventually calculated by the iterative process

which caused the solution to the equation of

motion to diverge.

(5) The divergence problem can sometimes be circum-

vented by not attempting to extract Cm from

extremely noisy data. The resulting coefficients

that are extracted, Cmo, Cm2, and Cmqo have

accuracies comparable to those extracted from

noisy data with a 10 degrees system noise band.

























N4 N



I +


to


It
Ia II
E E
0 C.


00 4)
*4
c+


0 0


0 0
I I






30-


20-0


10-


0



Figure 2.


0.005


0.010
0.010


om(Rad)
Variation of Percent Error
with Measurement Error


I
0.015


000
0.020


in Extracted C3


30-





20 -


10-


Figure 3.


0

0.605


0.i10


om(Rad)

Variation of Percent Error
with Measurement Error


o.d1s 0


0.020


in Extracted C4






30-





20-





10-


0




Figure 4.






0


-100





-200-


-300-


Figure 5.


_~h rO0


0.010
om(Rad)


0.015


Variation of Percent Error in Extracted C5
with Measurement Error


0.005


I
0.010


0.015


I
0.020


o (Rad)
.m


Variation of Percent Error in Extracted C6
with Measurement Error



0


r-

Q
O-i


o.oosO
0.0050














0.020-





0.015 -


0.010 -


0.005 -


0.000


GP


0.005
0.005


0.010


0.015
0.015


0.020


a (Rad)


Variation of RMS Residual with Measurement Error


I __ I __ _


Figure 6.
C,








15 -


10-


Figure 7.



15 -


10 -





5-


0.005
0.005


I
0.010
m (Rad)


0.015
0.015


0.020
0.020


i I E i da


Variation of Normalized Estimated Standard
Deviation of C3 with Measurement Error








o@


0.005
0.005


0.010
0.010


0.015
0.015


0.020


Figure 8. Variation of
Deviation of


om(Rad)
Normalized Estimated Standard
C4 with Measurement Error








15-


10-


Figure 9.



120 -


80-


40-


0.005
0.005


0.010
om(Rad)


0.015


0.020


0 0


Varitio ofNraie stmtdSadr


Variation of Normalized Estimated Standard
Deviation of C5 with Measurement Error





O


0.005
0.005


0.010


0.015


0.020


Figure 10.


m (Percent)
Variation of Normalized Estimated Standard
Deviation of C6 with Measurement Error










cq 0
S4)
s-U


r4

'-4

0r
'4


(p~U) 70


c0 t N9


00 S


N O
cq
C> to

I 0

II U
i 0
U 0


41 0
o. &



cM O
0 '
C9i



















I
0.04
a (Rad)
a


Figure 12.


10-


-10-


-20-


-30-





-40-


Figure 13.


Variation of Percent Error in Extracted C3
with System Noise


0.02


0.04


0.06


0.08


o (Rad)
a


0
Variation of Percent Error in Extracted C4
with System Noise


20-


10-


0
0


0.02
0.02


I
0.06


1
0.08


e


p






20-


10-


Figure 14.






600 -


S0
0


in


0


4-J
U

Pc)


0.04


0.06


0.08


o (Rad)

Variation of Percent Error in Extracted C5
with System Noise


400 -I


200 -


0






-200 -


0.02 Q 0.04


0.06


o.b8


a (Rad)
a


Figure 15. Variation of Percent Error in Extracted CG
with System Noise


0.02
0.02







ci)
ci)


1 _


Y



























0.02


Figure 16.







10-

4J

U



-r
&S


0.04


0.06


0.08


a (Rad)

Variation of Normalized Estimated Standard
Deviation of C3 with System Noise











O


0.02


0.04


0.06


0.08


Figure 17.


a (Rad)

Variation of Normalized Estimated
Deviation of C4 with System Noise


Standard


L-.


- ---- ---- -. --1 -7











10 -


0



Figure 18.






80-


40 -


0.02


Variation
Deviation


0.04
o (Rad)
a


0.06
0.06


of Normalized Estimated
of C5 with System Noise


0.08


Standard


O


I
0.02


0.04


0.06


I
0.08


Figure 19.


a (Rad)

Variation of Normalized Estimated
Deviation of C6 with System Noise


Standard


_ __











CHAPTER IV

DEVELOPMENT OF THE EXTENDED KALMAN FILTER FOR
ESTIMATING PARAMETERS AND THEIR UNCERTAINTIES


This chapter presents an abridged derivation of the

Kalman filter for discrete and continuous linear systems,

followed by a general statement of the extended Kalman

filter for continuous nonlinear systems. The extended

filter is then used as a base for the development of a

specific algorithm for estimating states and parameters

of the second order equation of pitching motion for a

missile being forced by random disturbances.

Development of the Kalman Filter


The original derivation of the Kalman filter was

presented by Kalmanl8 in 1960 for multistage systems

making discrete linear transitions from one stage to

another. Kalman and Bucy9 gave the analogous development

for continuous linear systems approximately one year after

the first work was published. The purpose of the linear

filter is to provide estimates of the state of a system

by making use of measurements of all, or some, of the

state vector components of the system. The system is

assumed to be operating in the presence of random distur-

bances, the statistical properties (i.e., mean and variance)









of which are known. The measurements of the state vector

components of the system are also assumed to have random

errors of known statistics.

In addition to the original derivations of Kalman

and Bucy, a number of other methods offering various degrees

of insight but leading to the same results are available.

A brief development taken primarily from Bryson and Ho20

is presented here. Other derivations or developments of
31 32
the filter equations are given by Jazwinsky,3 Sorenson3

and Barham.33

The treatment given here starts with a static system,

is extended for a single-stage linear transition which

leads directly to linear multistage processes. Finally,

by making use of a limiting process, the desired form of

the equations for a continuous linear dynamic system is

given.


Static System

The problem at hand is to estimate the n-component

state vector X of a static system using the p-component

measurement vector, z, containing random errors, v, which

are independent of the state. The measurement vector can

be represented as


z = HX + v


(21)








where H is a known p x n matrix. Conditions on the measure-

ment errors are


E(v) = 0 (22)


E(vvT) = R (23)


where R is a known matrix of dimension p. It is assumed

that a prior estimate of the state, designated as X, is

available and also that the covariance of the prior esti-

mate, M, is known. Thus

-T
M = E[(X-X)(X-X) T] (24)


where M is of dimension n.

The desired estimate of X, taking into account the

measurement, z, is the weighted-least-squares estimate,

X, which minimizes


J = [(X-))T M-1 (X-() + (z-HX)T R-1 (z-HX)]. (25)


Differentiating Equation (25) with respect to X, setting

the resulting equation to zero and solving for X yields

S= X + PHTR-1 (z-HX) (26)


where


p-1 = M-1 + HTR-1H


(27)








The quantity P is the covariance matrix of the error in

the state estimate after measurement, X, i.e.,


P = E[(X-X)(X-X)T] (28)

Single-Stage Transitions

It is now desired to estimate the state of a system

that makes a discrete transition from state 0 to state 1

according to the linear relation


X1 = 0oX + ro (29)

where o is a known transition matrix of dimension n and

r is a known n x r matrix. The mean and variance of the

random forcing vector are assumed to be known and are

given by


E(w ) = wo E[(w -wo)(wo-w ) Qo (30)

The statistical properties of the random state vector are

assumed to be known initially as

E(Xo) = X0 E[(Xo-X )(X -X )] = Po (31)

It is also assumed that Xo and w0 are independent. From

this information X1 is a random vector whose mean value

and covariance are


x = OoXo + ro.w (32)









M1 = T + rQ T (33)


Suppose now that measurements of the state are made after
the transition to state 1; then from Equations (26) and
(27) the best estimate of X is

X, = X + PiHIR-1 (z-HTX) (34)

where

P, = (M 1 + HT1R1H )-1

= M M H (HM HT + R-1) HM, (35)


Linear Multistage Processes
The developments of the two preceding sections can

be extended for linear, stochasitc, multistage processes.
Given the following difference equation system model

Xi+ = iXi + r.w. (36)

where

E(X ) = X (37)

E(wi) = wi (38)

E[(X -X )(X o- o)T = M (39)
00 0









E[(wi-wi)(wj-w.j)] = Qi6ij

E[(wi-wi)(Xo-Xo) ] = 0

and measurements of the state

zi = HiXi + vi
1 1 1 1


where


E(vi) = 0

E(v v) = Ri6ij


E[(w.-wi)vj] = 0
11


, E[(Xo-Ro)V] = 0
00 1


the estimate of the state is

Xi = X + Ki(zi-HiX)

where


.i+ = ii + riwi


K. = P.HTR.-
1 i l 1

P. = (Mi + HTR. lHi)-1
1 i 11 1

= Mi M.HT(H.M.HT + R.-1)H.M.
1 1 1 1 1 1 1 i 1


Mi+ = iPii + iQir
i+1 1 1r1.Q 1.


(40)

(41)


(42)




(43)

(44)


(45)


(46)


(47)


(48)


(49)


(50)









Equations ('46), (47) and (48) are the discrete Kalman

filter with the state variable covariances given by

Equations (49) and (50). As can be seen from the above

equations, the Kalman filter is essentially the same as

the system model [Equation (36)]. The differences are

(1) the actual system noise, which is random from one stage

to the next in Equation (36), is replaced by its mean or

expected value, and (2) there is a correction term based

on the difference between the actual measurement of the

state and its predicted value. The difference term is

multiplied by a gain, Ki, which is essentially the ratio

of the uncertainty in the state to the uncertainty in

the measurement. If the covariances of the measurement

errors are large, the gain will be relatively small and

the corresponding difference term will have little effect

on the estimate of the state; if, however, the system

noise is relatively large or the measurement errors are

very small, the gain will be large and differences between

the actual and predicted measurements of the state at a

given stage have increased significance in the state

variable estimates.


The Continuous Kalman Filter

By applying a limiting process to the discrete filter

with the time between stages tending to zero, the linear

system model becomes








X = F(t)X + G(t)w(t) (51)

and the continuous Kalman filter is given by
SHT ^
X = FX + Gw + PH R-l(z-HX) (52)

P = FP + PFT + GQGT PHTR-HP (53)

The Extended Kalman Filter

The extension of the Kalman filter for estimating
the states of nonlinear systems in the presence of noise
has been given by Bryson and Ho20 and Jazwinski,31 among
others, as

X = f(X,t) + G(t)w(t) + P[ R-1[zt)-h(,t)] (54)


) aa af (hah
S P + P fX + GQG' PTJ R-1JP (55)
-1x 1X axj P, ax]

for the nonlinear system

X = f(X,t) + G(t)w(t) (56)

with measurements

z(t) = h(X,t) + v(t) (57)

where
E[w(t)] = w(t) (58)

E[w(t)-w(t)][w(t')] = Q(t)6(t-t') (59)

E[X(t )] = o (60)

E[X(t)- j[X(to)-Xo I = P0 (61)








E[X(t )-X ][w(t)-w(t)]T = 0 (62)

E[v(t)] = 0 (63)

E[v(t)vT(t')] = R(t)6(t-t') (64)

E[X(t )-Xo lv(t)]T 0 (65)

E[w(t)-w(t)][v(t')] = 0 (66)


The Parameter Estimation Algorithm

In this section a specific parameter and state esti-

mation algorithm using the extended Kalman filter is

developed for the basic equation of pitching motion pre-

viously analyzed with the Chapman-Kirk technique.

The nonlinear equation of motion, or system model,

is thus

a + (C4+C6a2)& + (C3+C5a2)a = w(t) (67)

where

E[w(t)] = 0 E[w(t)w(t')] = q6(t-t') (68)

The measurements of the state of the system are assumed

to be given by

z(t) = a(t) + v(t) (69)

where


1 ;, Ct) =o0


E[v(t)vT(t')] = r6(t-t')


S (70)









Now to reduce Equation (67) to the required system of

first order differential equations, let


a = X1


& = X2


At this point, to estimate the aerodynamic parameters

appearing in the equation of motion in addition to the

state variables, the state vector is augmented by setting


C3 = X3

C4 = X4


C5 = X5

Ce = X6


(71)


(72)


with the constraints


= 0

= 0

= 0

= 0


(73)


Making use of Equations (71), (72) and (73), Equation (67)

now reduces to the following nonlinear system of first

order equations:








Xi = X2

12 = -(X4+X6X12)X2-(X3+X5X12)X1 + w(t)

X3 = 0

X, = 0

is = 0

Xi = 0


with linear measurements


z = Xi + v


Comparing Equations (74) and (75) with (56) and (57) the

following matrices may be identified:


C(t) =


H(t) =


X1
0
0
0
0
0


f (X, t) =


X2
-(X4+X6Xi2)X2 (X3+X5Xi2)XI
0
0
0
0
4


(76)


Now, since the mean value of the system noise is assumed

to be zero JEquation (68)] and because the measurements


(74)


(75)








of the state are related linearly to the state [Equation

(75)], the extended Kalman filter, previously given by
Equations (54) and (55), can be simplified somewhat to


X= (X,t) + PH [z-HX] (77)


P f P + PI fHT q T
X= p + +1 GqGT PHT 1HP (78)
P x +


where


H = [ 1 0


Applying Equations (76)


Xl X2

X2 -(X4+XX12) X2 -

X3 0
X4 0
Xc 0

Xs 0


So 0 0 ]

and (77) yields


(X3+XsX1 2)X


P13 P14 P15 P16

P23 P24 P25 P26

P33 P34 P35 P36

P34 P44 P45 P46

P35 P45 P55 P56

P36 P46 P56 P66


f(z-[1 0 0 0 0 0]
r


PI1 P12

P12 P22

P13 P23

P14 P24

PIs P25

P16 P26









Carrying out the prescribed multiplications and equating

like components of the resulting two column matrices

yields the desired filter equations,


= P + -r- (z-)
2 r1

= -(4+X6, X2 (X3+Xi25) + r (zX)

P13
- (z-X1)


r- (z-X)

Sp15
r- (z-X1)


r 16


(79)


The necessary covariance equations are obtained from the

matrix Ricatti equation [Equation (78)] making use of

Equations (76) and the fact that



xx=


0

-2XiX2X6-X3-3XiX5

0

0

0

0


1

-(X4+X6eXi)

0

0

0

0


0

-X2

0

0

0

0


0
_3
-xl

0

0

0

0


0

-XiX

0

0

0

0
J


(80)








The results of substituting Equations (76) and (80) into

(78), carrying out the rather tedious matrix multipli-
cations and equating like terms are

P11 = 2PI2- P11
r 1
12 = P22-AP1i-BP12-XIP3i-Xz2Pl-CPIs-DPiS- P zlP12

P13 = P23-P1P13
1
P14 = P24-P11P14
P15 = P25- llP15
1
P16 = P26_-P11P16
22 = q-2(AP12+BP22++X1P23+X2P24+CP25+DP26)-P12
P23 = -AP13-BP23-X1P33-X2P34-CP35-DP36-ppl2p13

P24 = -AP14-BP24-X1P34-X2P44-CP4S-DP46-P1l2Pl4
P25 = -AP15-BP25-XIP35-X2P45-CP55-DP56- P12P15

P26 = -API6-BP26-X1P36-X2P46-CP56-DP66-1pl2P16
P33 = -P13
P34 = P13P14
1
P35 = -1I1315
P36 = -P13P16
1 2
P44 = Pl4


P45 = -IP14P15
I
P46 = PI4P16

Ps 1 = -Pl5
P56 = -PP15P16
P66 -1P162 (81)









where
^2^
A = 2XIX2X6 + X3 + 3XIX5


B = X4 + XiX6

^3X
c = Xi


D = X1X2 (82)


The desired estimates of the states and parameters

are obtained by numerically integrating the filter equations

[Equations (79)] and their covariances [Equations (81)].

Initial estimates of the states and their covariances as

well as the variances of the system noise and measurement

errors are assumed to be available. In the event that the

data consist of discrete measurements, which is usually

the case, the constant time between data points should be

equal to the numerical integration step size. This is

necessary because the state estimates are updated at each

integration increment and in doing this the filter requires

knowledge of the difference between measured and estimated

state values.

If the filter successfully adjusts the true states and

the imbedded parameters so that XI is an adequate match to

the data, z, the time derivatives of the filter equations

for the parameters become small [see Equations (79)] and

the parameters reach steady-state or near steady-state

conditions.












CHAPTER V

USE OF THE EXTENDED KALMAN FILTER FOR
ESTIMATING PARAMETERS AND THEIR UNCERTAINTIES


This chapter presents an analysis using the extended

Kalman filter with parameter augmented state vector to

determine aerodynamic coefficients and their uncertainties

from noisy dynamic data. A description of the digital

computer program used in the analysis is given first,

followed by results obtained with this program from a

variety of noisy data sets.


The Extended Kalman Filter Program


The extended Kalman filter program is written in

Fortran IV for use primarily on an IBM model 360/65

digital computer. The basic function of the program is

to integrate numerically the 27 first order differential

equations which are given in the previous chapter, and

which comprise the extended Kalman filter with parameter

augmented state vector. The specific functions of the

main program and its subroutines, as well as the required

input data, are described in the following paragraphs.

A listing of the complete program is given in Appendix II.









Main Program Functions

The main program reads and writes the input data

and calls the numerical integration subroutine, ADDUM.


Subroutine Functions

The names and functions of each of the three sub-

routines that are used in the extended Kalman filter

program are given below.

ADDUM

Subroutine ADDUM integrates the 27 filter and

covariance equations using a fourth order Runge-Kutta

method for starting, followed by a fourth order Adams-

Bashforth predictor-corrector method for running. It

is essentially identical to the subroutine of the same

name used in the Chapman-Kirk coefficient extraction

program and is described in detail in Reference 26.

Subroutines XDOT and OUT are called from ADDUM.

XDOT

Subroutine XDOT computes values of the time deri-

vatives for the differential equations being integrated

by ADDUM.

OUT

This subroutine writes the output of the numerical

integration.








Required Input Data

The program reads four categories of input data. These

categories and the specific data in each are delineated in

Appendix II. The format and units of the entries on a

specific data card can be determined from the program

listing and nomenclature list, which are also given in

Appendix II.


Analysis of the Extended Kalman Filter


The use of the extended Kalman filter for estimating

the parameters of interest is analyzed for linear and non-

linear systems with data containing only measurement errors

as well as data containing both system noise and measurement

errors. The analysis begins with a linear system and data

containing only measurement errors and progresses through

increasing stages of difficulty up to nonlinear systems and

data which have been corrupted by both system noise and

measurement errors. The previously discussed computer

program and algorithm require no modifications to consider

the linear case; by initially setting the parameters that

are the numerical coefficients of the nonlinear terms in

the equation of motion (and their variances) equal to zero,

the extended filter for a nonlinear system reduces to the

one required for a linear system.









All data used in the analysis were generated by the

previously mentioned computer program, UFNOISE (Reference

29). The true values of the constants used in the data

generation are those previously given in Table III with the

exception of the initial value of the pitch angle for the

linear system. This initial displacement is a more realistic

0.1745 radian for the linear cases considered.


Linear Systems with Measurement Errors Only

After the extended Kalman filter computer program had

been constructed and checked, it became apparent that the

technique would probably be best understood by considering

relatively simple cases at first and then progressing to

more difficult ones as confidence in the technique was gained.

To this end, the first success with the program was realized

when analyzing data containing only measurement errors for

a linear system.

The pitch angle data used in this part of the analysis

were generated by integrating the equation of motion

a + Cq4 + C3a = 0 (83)


and superimposing random Gaussian errors on the output.

The standard deviation of the errors is 0.00588 radian,

which corresponds to 3c measurement errors of approximately

1.0 degree. The numerical integration step size used in








generating the data was 0.005 second and the pitch angle

was output every integration increment. The equation was

integrated for a total of 2.00 seconds.

Basic filter performance

The results of using the extended Kalman filter to

identify the correct values of the parameters of interest,

X3 and X4, are shown on Figures 20 and 21. These figures
show time histories of the percent errors in the estimated

values of the parameters that are computed by the extended

filter program when analyzing the noisy data described

previously. The percent error is defined by


(X.-X.).O00
PERCENT ERROR (X.) = j-J (84)
X.


The input for the measurement error variance was

the one previously computed by the simulation program

based on the errors that were actually put into the data.

The initial values of the parameter variances were computed

from knowledge of the true value of the parameters and the

arbitrarily selected initial values of the parameters.

These sample variances are defined by


P.jj(0) = [X.(O)-Xj(O)]2 (85)









This method of initializing the parameter variances was

chosen merely to generate the required initial numerical

values. In actual experimentation, the initial variance

values would depend on the method of selecting the initial

parameter estimates and prior knowledge as to how accurately

these initial parameter estimates were in relation to the

true values of the parameters.

The initial values of all covariances were chosen as

zero. This implies that errors in the estimates of the

individual state vector components are initially uncor-

related. The initial values of the variances for the

pitch angle and pitch angle rate were also chosen as zero.

For wind tunnel dynamic experimentation where the model

is initially held rigid at some given displacement to the

wind stream, this seems to be a valid assumption.

As can be seen from Figures 20 and 21, the filter

does an excellent job of identifying the parameters of

interest. The X3 parameter, which corresponds to the

Cm term, and which is initially in error by 25 percent,

is identified with approximately zero percent error in

less than 0.3 second. The correct identification of the

damping parameter takes slightly more time but the results

are of equal quality.

Effects of variations in the initial parameter variances

This section presents some effects on the near steady-

state estimates of the parameters of interest and their









near steady-state variances for various values of the

initial parameter variances. The parameter variances

were initialized to values that were 25 percent higher

than those used in the previous section and also to

values that were 25 percent lower than the referenced

values. The effects of these initializations are shown

on Figures 22 through 25 and in Table VII for the linear

system of interest.

Table VII is a summary of the percent errors in the

parameter variances and the normalized parameter uncertain-

ties for the above mentioned variations in the initial

parameter variances.

The same results are depicted graphically on Figures

22 and 23. The numerical results in Table VII are based

on the near steady-state parameter and variance values

that result after the filter has integrated for 1.5

seconds. These correspond to the last point shown on the

supporting figures. The normalized parameter uncertainties

are defined by

P?.
PERCENT (P/P-) = -- 100 (86)
J3 n x.j


Time histories of the parameter variances, P33 and P44,

are shown on Figures 24 and 25. All of the information

presented on these figures indicates that initial values












Table VII. EFFECTS OF VARIATIONS IN THE INITIAL
PARAMETER VARIANCES ON NEAR STEADY-
STATE PARAMETERS AND THEIR VARIANCES
(LINEAR SYSTEM, MEASUREMENT ERRORS ONLY)


Variation in
Initial Parameter
Variance*
(Percent)




-25


0


25


Errors in Near
Steady-State
Parameters
(Percent)
X3 X4


0.17


0.19


0.22


0.55


0.46


0.38


Normalized Near
Steady-State
Variances
(Percent)


0.12


0.11


0.11


0.91


0.91


0.91


* relative to sample variance
I









of parameter variances in the range investigated have no

effect on final parameter estimates or their variances

when using the extended Kalman filter to analyze data for

a linear system containing only measurement errors.


Linear Systems with Measurement Errors and System Noise

Data for this portion of the analysis were generated

with the computer program UFNOISE by numerically integrating

the following equation of motion,


a + C4& + C3a = w(t) (87)


and superimposing random Gaussian errors on the output.

The forcing function, w(t), is also random in nature and

Gaussian with zero mean. The standard deviation of the

system noise was 4.80 rad/sec2, which is of sufficient

magnitude to drive the oscillations in a noise band of

approximately 0.0872 radian (5 degrees) for zero initial

displacement. The standard deviation of the measurement

errors was 0.00556 radian. Also, as before, the equation

of motion was integrated in increments of 0.005 second for

a total of 2.00 seconds.

Basic filter performance

The basic performance of the filter is demonstrated

on Figures 26 and 27 which show time histories of the percent

error in X3 and X4. As can be seen from these figures, the

initial estimates of X3 and X4 are 25*percent in error and








these estimates are corrected to within approximately 1 per-

cent and 3 percent of their respective true values. The

noise variances used are those given by the simulation

which generated the pitch angle data. The initial para-

meter variances are calculated from Equation 85.

Effects of variations in the initial parameter variances

The sensitivity of the filter to variations in the

initial parameter variances is given in Table VIII and on

Figures 28 through 32. Table VIII is a summary of the

percent error in the parameters of interest and their

normalized uncertainties for variations in the initial

parameter variances of -25 percent, 0.0 percent and 25

percent relative to the sample variances.

Figures 28 and 29 show the variation of the near

steady-state errors in the parameters and their normalized

uncertainties as functions of the percent variations in

the initial parameter variances. Figure 30 shows the two

time histories of the error in the estimate of X4 that

result for two different initial values of the parameter

variances. One of the initial variances is the sample

variance based on the initial and true values of the

parameters [equation (85)] and the other is 25 percent

less than this value.

The X3 error time histories are essentially identical

for both cases and are not presented. The variations of












Table VIII. EFFECTS OF VARIATIONS IN THE INITIAL
PARAMETER VARIANCES ON NEAR STEADY-
STATE PARAMETERS AND THEIR VARIANCES
(LINEAR SYSTEM, MEASUREMENT ERRORS
AND SYSTEM NOISE)


Variation in
Initial Parameter
Variance*
(Percent)


-25


0


25


Errors in Near
Steady-State
'Parameters
(Percent)


0.72


0.88


0.98


-0.47


2.66


4.95


Normalized Near
Steady-State
Variances
(Percent)


2.42


2.42


2.42


14.39


15.18


15.81


* relative to sample variance








the parameter variances with time, for the three initial

values considered, are shown on Figures 31 and 32.

From the information given on the figures mentioned

above, it is obvious that neither X3 nor P33 are affected

by variations in the initial variances within the 25

percent range considered. The estimates of the damping

parameter as well as its variance are affected, however,

by these variations. The near steady-state damping para-

meter estimates vary almost linearly with initial variance

values from a low of 0.47 percent to a high of 4.95 percent

(see Figure 28). The variation in the normalized uncertainty

is less pronounced, ranging from a low of 14.39 percent to a

high of 15.81 percent (see Figure 29). Nevertheless, the

near steady-state uncertainties are of sufficient magnitude

so that the true values of the coefficients are within

less than one standard deviation of their estimated values

regardless of the initial parameter variance, when the

standard deviation is taken as the square root of the near

steady-state parameter variance. Attention is also called

to the fact that since system noise is now present in the

data, the near steady-state parameter variances are of

greater magnitude than was the case when only measurement

errors were present (see Figures 24, 25, 31 and 32).









Filter response to large errors in the initial parameter

estimates

The results of using the extended Kalman filter to

identify the damping parameter, X4, when the initial esti-

mate of the parameter is in error by 100 percent are shown

on Figure 33 for two initial parameter variance estimates.

When the initial parameter variance is based on the initial

estimate of X4, the near steady-state parameter estimate is

approximately 14 percent high; for an initial parameter

variance that is greater by approximately 25 percent,

the resulting near steady-state value of X4 is nearly 19

percent high. The corresponding parameter variances 1.5

seconds after the filter begins integrating are such that

the estimated values of X4 are within approximately one

standard deviation of the true value.

Comparison of filter performance to Chapman-Kirk technique

The parameters of interest and their uncertainties

were extracted from the given noisy data set using the

previously described technique of Chapman and Kirk in order

to be compared to the filter results. The percent error

in the parameters and their normalized estimated standard

deviations obtained with the Chapman-Kirk program are


X3:12.5 percent (1.8 percent)

X4: 3.1 percent (0.3 percent)









The above results are similar to those presented in Chapter

III (Table VI) in that the estimated parameter uncertain-

ties are not of sufficient magnitude to reflect the true

error in the extracted parameters. Whereas the extended

filter estimates the parameters and variances so that the

true value of the parameter is generally within one standard

deviation of the estimate, it is shown that such is not the

case when using the Chapman-Kirk technique. The errors in

the above parameters are also slightly greater than those

obtained with the extended filter, although this is probably

of less consequence than the differences in the uncertainties

computed by the two techniques.


Nonlinear System with Measurement Errors Only

The pattern used in the previous two sections of

generating a data set, investigating the basic response

of the filter in identifying the parameters of interest

and checking the sensitivity of the filter to variations

in the initial parameter variances is essentially repeated

here for the more complicated system model

a + (C4+Ca2)E + (C3+C5a2)a = 0 (88)


In addition, some effects of errors in the measurement

error variance are also included and discussed. Some

comparisons of results achieved with the extended filter

and the Chl: aLnn-Kirk technique are also made.









The standard deviation of the measurement errors in

the generated data is 0.00569 radian. The mean value of

the errors is zero.

Basic filter performance

The time histories of the four parameters of interest

as computed by the extended filter while analyzing the data

just described are shown on Figures 34, 35, 36 and 37.

The initial errors in the parameters are -25 percent for

X3, X4 and X, and -5 percent for X5. The measurement
error variance is that obtained from the simulation. The

initial values of the parameter variances are based on

the difference between the true values of the parameters

and the above mentioned initial estimates.

The reason the initial estimate of Xs is only 5 percent

low instead of 25 percent, as is the case with the other

parameters, is due to its magnitude and a weakness in the

numerical scheme. The true value of X5 is 1960 based on

the data in Table III of Chapter III. A 25 percent initial

error in this parameter results in an initial parameter

sample variance of


Pss(0) = (1960-(0.25)1960)2 = 240,000

which is of sufficient magnitude to cause the filter to

diverge for the numerical integration step size of 0.005

second. The divergence occurs because the large initial








value of P55 results in large values of P25 and P15 which

in turn cause the values of P55 that are fed into the

numerical integration subroutine to be very large [Equa-

tions (81) and (82)]. The result of this cascading effect

is a rapid decrease in P55 in large increments until it

becomes negative (a physical impossibility since this term

is the variance of a parameter). The structure of the

covariance equations is such that a negative variance

causes numerical divergence in the solutions to several

of the equations.

To circumvent the problem, an initial error of -10

percent in X5, with a corresponding adjustment in P55(0),

was tried. The result was again divergence. Finally an

initial estimate of -5 percent was found to be successful.

The results shown on Figures 34, 35 and 36 indicate

that the filter does an excellent job in correctly identi-

fying the linear and nonlinear static restoring moment

parameters, X3 and X5, as well as the linear pitch damping

term, X4. The results for the nonlinear damping parameter,

Xg, are not as gratifying as can be seen by referring to

Figure 37. The error in this term is still approximately

-10 percent after 1.5 seconds. The reason for this problem

is related to the magnitude of this parameter as compared

to others in the equation of motion. The true value of

X3 and XS are 160 and 1960 respectively, although the








magnitude of X5 is effectively reduced by at least an order

of magnitude when it is multiplied by a3. X4 and X6, on

the other hand, have true values of 1.60 and 4.35 respec-

tively. After X6 is multiplied by a2& it is effectively

the smallest parameter in the equation of motion by an order

of magnitude and it is naturally more difficult to identify,

since it has the least effect on the trajectory.

Effects of variations in the initial parameter variances

The effects of variations in the initial parameter

variances are summarized in Table IX and on Figures 38

and 39. As can be seen from these results, the near

steady-state values of the parameters and their variances

are relatively insensitive over the range of initial para-

meter variances considered. Only the X6 parameter estimate

and its variance show a variation of more than 1 percent.

As is obvious from the referenced table and figures, no

data are available for initial parameter variances of 50

percent above the sample variances; the reason for this is,

again, that the magnitude of the P55 variance caused the

filter to diverge.

Attention is called to the fact that for the range

of initial variances considered, the near steady-state

estimates of the X3, X4 and Xs parameters are all within

one standard deviation of their true values when the

standard deviation is taken as the square root of the near
















S9



Zf to to o ;
(t) H Cd a)





O 4 Uo man
(d *n *
<- | 0 0 0 0
UL4 p1 r- 4 Lnr-





0-0 nI r-*


H- 0 -"H $ ,
Cr M r-1 4 "






Z






SH rHt 0 .
H 0 0< t H








4- ,--4 o A


;2 P0 o oD










0 m n H H--c H *
U 4 00 0 0 0
0 C a (

















00
E- -4 04U )
u< wU -el CI) C) C
PZ z 1












*.dC
C%: 1dr O r0 )O
C Q













( ; 0 cd
i4 ( C r* r-4
Ft;_ d J () c -) r) rl r
















t-l *K








steady-state variance for the parameter of interest. The

Xg parameter is always within two standard deviations of

its true value.

Effects of errors in the estimate of the measurement error

variance

Up to this point, exact values of the measurement

error variance, r, have been used in all runs made with the

extended filter. Exact knowledge of this quantity is avail-

able because the pitch-angle data being used with the extended

filter are computer generated with specified noise statistics.

In actual test situations, true values of the noise para-

meters may not be known exactly and, in fact, methods for

determining both measurement error variances and system

noise variances from noisy dynamic data are of current

research interest.3

Some effects of errors in the estimates of the measure-

ment error variance on the near steady-state parameter

estimates and their uncertainties are given in Table X

and on Figures 40 and 41. These results show that errors

in the measurement noise variance of 25 percent have

essentially no effect on estimates of the four parameters

of interest. The uncertainties of the X3, X4 and X5

parameters are changed by less than 1 percent over the

range considered. The normalized Xg uncertainty varies from

approximately 5 percent to 12 percent.





















SV)
O

CO0


















- C/)
2 0
03 CO)



O O



CO


















MH
o






E


0a

DiM
K; U


ca
te






r,.


- 4 c-1
r-I 4--. IJ
C .t I -


0a

or.c)


F:

DD
0I









04



t4
t



C!
I "
in









ro


|^


z c! ><
s)











m >.


(a)
*-a






4- ;V-4
m c m



*r4 P







$-4 i
O -
o~g ce-

W: r
Md
wf
0P


CLn


M
cn








Filter response to larger error in the initial estimate

of Xs

The difficulty associated with.numerically large initial

parameter variance estimates has been discussed earlier.

The sample variance of P55 which corresponds to a -5 percent

error in the initial X5 parameter estimate appears to be an

approximate upper limit for the integration step size and

physical constants used in this analysis. A run was made,

however, with the previously described nonlinear data set

containing only measurement errors where all initial parameter

estimates were in error by -25 percent; the P55 variance,

however, was set equal to the previously used sample

variance which corresponds to a -5 percent initial estimate

of X5. The parameter estimate time histories resulting from

this initialization are shown on Figures 42 through 45.

These results indicate that the near steady-state values

of the parameters are very nearly equal to those presented

when the sample variances based on the initial parameter

estimates were used initially.

Comparison of filter performance to Chapman-Kirk technique

The linear and nonlinear static restoring moment and

damping parameters were extracted from the noisy data set

being considered in this section using the Chapman-Kirk

program. Th:e percent errors in the parameters and their

normalized estimated uncertainties are given below:









X3:0.36 percent (0.35 percent)

X4:0.69 percent (0.43 percent)

X5:0.41 percent (0.49 percent)

X6:4.60 percent (11.27 percent)


Comparing the above results with those previously given

in Table IX reveals that the estimates obtained using the

extended filter are slightly more accurate for all but the

X4 parameter. Since the errors are, for the most part,

less than 1 percent the differences are essentially

insignificant. The normalized uncertainties obtained

with both techniques are equally good.


Nonlinear System with Measurement Errors and System Noise

In this section, as before, the basic filter performance,

sensitivity of the filter to initial parameter variance vari-

ations and sensitivity of the filter to errors in the esti-

mates of the noise variances are investigated. The data

used in the analysis were generated by numerically inte-

grating the following nonlinear equation with a random

forcing function,

E + (C4+C6a2)& + (C3+C5a2)a = w(t) (89)


Approximately the same noise variances used previously are

repeated once again; the variance of the measurement errors

is 0.00576 radian and the variance of the system noise is

4.75 rad/sec2.








Basic filter performance

The time histories of the errors in the parameters

of interest obtained with the extended filter are given on

Figures 46 through 49. The response time required for the

parameters to reach their near steady-state values is

slightly greater than in the simpler cases that have been

considered previously. The accuracy of the parameters is

similar to that which was achieved for the nonlinear system

with only measurement errors in the data with the exception

of X4, which is approximately 5 percent higher.

Effects of variations in the initial parameter variances

The effects of variations in the initial parameter

variances are given in Table XI and are shown on Figures

50 and 51. Only the nonlinear damping parameter and its

uncertainty vary by more than 1 percent for initial variance

variations of up to 50 percent relative to the sample

variances.

Effects of errors in the estimates of the noise variances

The effects of errors in the estimates of r, the measure-

ment error variance, and q, the system noise variance, are

given in Table XII and are shown on Figures 52 and 53.

These figures show the variation of the near steady-state

parameter errors and normalized parameter uncertainties as

functions of the error in the estimate of the measurement

error variance for various values of error in the system

















U 1w

c*
*r


>
1d


(1 4-1 a

*r-H C Q)
r-4 4J- U




Z ci)
0



-(

cI

m
(^1


' t






I-








14


CO 2
P-- 0
U Z
2Z


<




HZ







< z



I -
U) CO















MH




PCu)>-
dHZ
S-r4 ~C









ci)D




































cn


H


? 4: ,-1

O(UN
*8 cd O 4
04 c u r:
+- cd rd U
Cd c l *
*H -q N 4 )
N Cd Cd a4
cd *H >

*H
'-4<


9


4-3


cdCd
0



Cd c)


24 4-J 12
cd
e 0)
*rl X





+-> (1,
U) > (


(M W
N N





r-4 1-1







f-N 00





4* *







. *Co




4 co
(* *










0ct 00






S-l 0
0

cd

00 0
rt



C)l CO




4
In cd


0 0


N *r4
ci









4C
ov, 0





1-1
^i






















zWZ

P4 m
P4 :2:
S<
P4

= :-1 En
ZO4





E<-
0 E

OSH







0 0<
W 0

0Z2
O (.
m 0












c oo p-
0 i
P< d 0


U)

0



cd
F4





r-4 4 U
rt







0r 3 !














U)


cU a
9 0
Z c $

0 CS 0


0 0
to

COd



W Cd
o










4J

ri C













U) -


4 4-l4


LO
(M


0 t)


o LC ) L)n


o in
i








noise variance estimate. From these results it is obvious

that only X6 is affected by errors of 25 percent in the

estimate of the system noise variance or measurement error

variance.

Comparison of filter performance to Chapman-Kirk technique

The percent errors in the parameters and their normalized

estimated standard deviations that are obtained using the

Chapman-Kirk program with the data currently being analyzed

are given below:


X3:0.62 percent (0.40 percent)

X4:2.63 percent (1.23 percent)

Xs:0.41 percent (0.61 percent)

X6:46.3 percent (14.8 percent)


Comparing these results with the results in Table XI

indicates that both techniques yield essentially the same

accuracy in their determination of X3 and X5. The filter

does considerably better in estimating X6 but is slightly

worse in its estimate of X4.

The uncertainties computed using the filter are of

sufficient I.::gnitude so that the true value of the X3,

X5 and X6 parameters are within one standard deviation

of the filter estimates; the true value of X4 is within

two standard deviations.