A brief overview of statistical tools needed to perform post flight/test reconstruction of state variables is given. Linear regression, recursive linear regression, and the exact connection between the Kalman filter and linear regression are discussed. The regression connection is expected to serve as an aid in the application of a recently developed analytical method of flight reconstruction to single engine test firing data

Temple, Enoch C.

English

NASA Technical Reports Server

N88-15632
PROPULSION ESTIMATION TECHNIQUES
Prepared by : Enoch C. Temple, Ph.D.
Academie Rank: Associate Professor
University and Department: Alabama A&M University
Department of Mathematics
NASA/MSFC Laboratory : Structures and Propulsion
MSFC Colleagues : James Crafts
Klaus Gross
Date: August 1987
Contract No. : NGT-01-008-021
The University of Alabama in
Huntsville
XXXlI
https://ntrs.nasa.gov/search.jsp?R=19880006250 2020-03-20T07:38:23+00:00Z
ACKNOWLEDGMENTS
I would like to thank NASA, MSFC, and ASEE for providing this
excellent research opportunity. Thanks are also extended to the staff
of the Propulsion Laboratory for the help willingly given during this
ten-week period. Special thanks go to James Crafts and Klaus Gross,
my NASA counterparts, for suggesting and discussing the topic
addressed in this report. Further appreciation is extended to Judy
Maples who typed this report.
XXXII-i
PROPULSION ESTIMATION TECHNIQUES
by
Enoch C. Temple
Associate Professor of Mathematics
Alabama A&M University
Normal, Alabama
ABSTRACT
This paper gives a brief overview of statistical tools that are
needed to perform post flight/test reconstruction of state variables.
Linear regression, recursive linear regression, and the exact connection
between the Kalman filter and linear regression are discussed. The
regression connection is expected to serve as an aid in the application
of a recently developed analytical method of flight reconstruction to
single engine test firing data.
XXXII-ii
1. INTRODUCTION
This paper considers the issue of post flight/test state variable
reconstruction through the application of observations made on the
output of the Space Shuttle propulsion system. The incentive for the
concepts discussed here is rooted in a NASA contract report by Rogers
(1987). Rogers used the Kalman filtering procedure to reconstruct the
state variables of the Space Shuttle propulsion system.
Since its development by Kalman (1960), the Kalman filtering metho-
dology has been a popular procedure with control engineers. Within recent
years, it has seen increased applications in other engineering areas as well
as in nonengineering areas. Although the applications have been diverse
and research on some of its defects has been quite extensive, applied statis-
ticians have not taken full advantage of the tool. This lack of use by some
applications areas may be because many publications on the topic have
obscured statistical simplicity. Often times the basic statistics concepts are
embedded in a discussion of numerical solutions to differential equations.
An objective of this paper is to give the general setup of the
Kalman filter and its connection to linear regression. A second objective
is to examine the Rogers (1987) reconstruction methodology for applica-
tion to the reconstruction of the state vector of a single Space Shuttle
Main Engine (SSME) by using static test firing data.
Throughout this paper, underlined capital letters are used to
denote vectors, capital letters denote matrices, and the identity matrix
is denoted by the letter I. All vectors are of the column type and the
transpose of any matrix or vector is denoted by using the letter T at the
superscript position. The letter E denotes the expectation operator,
N(U,_) denotes the multivariate normal probability distribution with
XXXII-i
mean vector U and covariance matrix _.. The caret symbol "^" written
directly above a scalar or vector denotes a statistical estimator of that
vector or scalar.
2. ESTABLISHMENT OF NOTATION
For any Space Shuttle flight, let Zt denote the observed values of
vector Z at time t. Each component of Z_t represents a relevant output
of the Space Shuttle propulsion system that can be measured. For
example, the components of Z_t may be chamber pressure, oxygen flow
rate, hydrogen flow rate, etc., for each of the three SSME's. Rogers
(1987) lists 71 components for vector Zt and 35 components for vector
X__t where _ is defined below•
Vector X t is a state vector of parameters to be estimated at time t.
It is assumed that the observation vector _-t is a function of the state
vector X__t. That is,
Z_t = h (X t) + V4
where h is some function and Vt_ N(O__,Rt).
(2.1)
State vector X__t is known
to change with respect to time according to the equation
Xt = f (Xt,t) + W__t (2.2)
where f is some function, Wt _ N(O,Q t) and E(W__t v tT) = 0.
If it is assumed that f and h in equations (2.1) and (2.2) are
linear, then numerical procedures allow us to transform these equations
to the form
XK = FK X.K_I+ W__.K (2•3)
XXXII-2
_K = HK _-K + VK (2.4)
where K represents discrete values of t, WK _ N(O__, QK ) and V K
N(O ,RK).
After a Space Shuttle flight has taken place, the post flight reconstruc-
tion procedure seeks to use the observed values of ZK and the equations in
(2.3) and (2.4) to reconstruct (estimate) the parameters X K so that the estima-
^
tion error is minimized. The estimated value of X__K is denoted by X K.
, A RECURSIVE ESTIMATOR OF X
In order to complete the development of the reconstruction process
mentioned in section 2, we consider a system of equations
Z = H X + _ (3.1)
where (3.1) is a system of linear equations that has been generated by
making n observations on the single equation
Z* = H* X + E* (3.2)
where Z* and E* are vectors or scalars. H* is a jxp matrix where
j = number of components in vector Z* and X is fixed. Vector X
being fixed means that X does not change with time or does not change
as the number of observations on Z increase. If Cov(_) = 7. and
det (7) # 0, then the weighted least squares estimate of vector X is
given by
= (H T Z-I H) -I H T 7.-1 Z (3.3)
and the covariance of the estimate is given by
XXXII-3
Cov(X) = E [()_ - X) (,X - X) T] = (HT r -1 H) -1 = P . (3.4)
The reader should be reminded that with the appropriate assump-
tions, the maximum likelihood, the minimum variance unbiased linear as
well as the minimum mean-squared error estimator of X are identical to
^
the estimator given by equation (3.3). We also remark that Cov(X) is a
A
measure of the quality of the estimates in vector X. A detailed dis-
eussion of these estimators may be found in Elbert (1984).
In the sequel that follows, the estimator in equation (3.3) will be
rearranged so that X can be estimated by using a recursive process.
The development of the recursive process is started by assuming that
there are K observations on equation (3.2) which form the system
ZK =HK X+ EK
where Coy -_K = r K and det (r K) _ 0. By equation (3.3)
(3.5)
, (3.6)
is the estimate of X where observations up to and including observation
K are used. The covariance of the estimator is denoted by PK and
^
PK = E[(XK - X) (_X - X)T] = (HK T ZK 1 HK)-I " (3.7)
Assume that an additional measgrement _K+I has been made. Let
Z_K+I = HK+ 1 X + _eK+1 (3.8)
T
where Cov (eK+ I) = 7.K+1 and E (eK eK+ 1) = 0. Note that E (5-K 5_+1 ) = 0
means that the (K+l)th observation is independent of the 1st K observa-
tions. Putting the K observations with the (K+l)th observation yields
the system
XXXII-4
(3.9)
where
Cov ( jl):K! 0\o i_:
By substituting into equation (3.3) we get
I I
,, I ZK
-_K+I = H ]H +1 ) ..... (H T : T
' , -1 K : HK+I) -O-- "
:_K+{\nz+1'J
(3.i0)
Note that we can write
-XK+I :-_-K + (XK+I - -_-K) " (3.11)
Substituting results from (3.6) and (3.10) into (3.11) and using matrix
algebra, we get
-X-K+1 = _ + (P + H +1 HK+I) HK+I ZK+I (ZK+I - HK+I -XK) " (3.12)
By substituting into equation (3.7) and using matrix algebra, we get
PK+I
' \o i II I HK+I)-I
= (pK 1 + HK+I 7K+I-1
Substituting PK+I into equation (3.12) yields
XXXII-5
XK = XK + PK+I HT+I -1^ ^ _'K+I (ZK+I HK+I -XK ) (3.13)
and
PK+I (PK 1 + HT+I -1 -1= _K+I HE+I) " (3.14)
Equations (3.13) and (3.14) provide recursive equations for estimating
X and its corresponding covariance matrix P.
Recall that the estimator given in equation (3.3) may be identified
^
by using the symbol X.Xn. Therefore, if X is fixed as it is in equation
(3.1), the recursive equations of (3.13) and (3.14) have no statistical
advantage over the estimator given in equation (3.3). However, if X
changes with time where there is one observation per time interval,
then the recursive equations are quite useful. In fact, equations (3.13)
and (3.14) when combined with equations (2.3) and (2.4) form the
Kalman filtering process.
4. THE KALMAN FILTER
It is worthwhile to mention that Brown (1983) and Gelb (1974)
used matrix theory to express equations (3.13) and (3.14) as
PK+I = (I - KK+ 1 HK+ 1) PK (4.1)
^ ^
XK+ 1 : X K + KK+ 1 (ZK+ 1 - HK+ 1 X K) (4.2)
where
T -i
KK+I = PK HK+I (ZK+I + HK+I PK HK+I) (4.3)
and KK+ 1 is called the gain matrix.
XXXII-6
For the convenience of the reader, equations (2.3) and (2.4) are
reprinted here. That is,
XK+ 1 = Fx X K + _WK (4.4)
and
ZK+ 1 = HK+ 1 XK+I + VK+ 1 • (4.5)
When equations (4.1) and (4.2) are combined with equations (4.4) and
(4.5), two types of estimates of XK+ 1 are possible for each K. A
notation by Gelb (1974) allows for the two estimators to be distinguished.
That notation is
(-)
= the estimate of XK+ 1 using all observations up to and"-'K+I
including observation K.
PK = the estimate of P using all observations up to and including
observation K.
= XK+ 1 using all observations up to and
(+) the estimate ofK+I
including observation (K+I).
The values of _XK(1 ) and PK(1)are obtained by using equations (4.4)
and (4.5). The -XK+I vector is often called the extrapolated or pre-
dicted value of XK+ 1 and PK(+I ) is called the extrapolated variance of
The values of _ (+) and P (+) _tK+l"--'K+I -K+I are computed by substituting _ (-)
for X K in equation (4.2) and PK(1 ) for PK in equation (4.1).
The computation summary is
(4.6)
' XXXII-7
(4.7)
(+) (-)+ ^ (_)
K+I = "-'K+I KK+I (ZK+I HK+I XK+I ) (4.8)
v(+) Hz+1)vz(+ )K+I = (I - KK+ 1 (4.9)
KK+I pK(+l) T T -1= HK+ 1 (HK+ 1 PKI1 ) + .HK+ 1 RK+I) (4.10)
Equations (4.6) to (4.10) completely describe the Kalman filtering process
when the original functions f and h of equations (2.1) and (2.2) are
linear.
However, when functions f and h are nonlinear, the state vectors
X K (K = t 0, t I .... t T) are estimated through the application of an
extended Kalman filtering procedure. This procedure requires a linear-
ization of functions f and h about some known state value X K. The
next paragraph provides a brief overview of the extended Kalman filter
concept.
Let X K be some known value of X and assume that AX is small.
A first degree Taylor series approximation of functions (2.1) and (2.2)
may be
and
X K + AX % f(X K, K) + ,
_ _ _ X=X K
• AX + _Wt (4.11)
Z t % h(XK) + ,. AX + V t .
- X=XK -
(4.12)
If it is assumed that _XK is selected so that _XK = f(XK,_ K), then
equations (4.11) and (4.12) become
XXXII-8
and
--- x=  X+WK
(4.13)
ZK - h(X K) = . AX + V K (4.14)
X=X K
Equation (4.13) is called the linearized dynamics equation and (4.14)
isthe linearized measurement equation. Note that equations (4.13) and
(4.14) may be transformed to equations that are equivalent to equations
(2.3) and (2.4). Hence for each discrete time K, AX K can be estimated
and error covariance matrices can be determined. The state vector
estimate at time K is then given by
^ ^
=
where AX (+) is computed by substituting from equations (4 13) and
--K " "
A
(4.14) into equation (4.8). Vector XK+ 1 is computed by letting
, ^
XK+ I = X K and repeating the above procedure.
The extended Kalman filter has performed well in a large class of
applications. However, there are occasions when divergence occurs in
the state vector estimates. Divergence occurs when the computed
entries of the error covariance matrix PK become small as compared to
the actual error in the estimate of the state vector. The cause of this
l
divergence is not due to a defect in the filtering procedure, but may
be caused by the linear approximation procedure, numerical rounding
error, or an inadequate model of the system being studied. Additional
possible causes of divergence are mentioned by Gelb (1974).
XXXII-9
A publication by Varhaegen and Van Dooren (1986) is representa-
tive of the theoretical and experimental analyses that are currently
being done on the divergence problem. The Rogers (1987) computer
code has implemented the U-D factorized algorithm as a means of con-
trolling numerical roundoff error that may lead to divergence in the
state vector. Cheeks for other sources that may generate divergence
are conducted through computer evaluations.
5. CONCLUSION
At this point we have reviewed the general setup for a regression
problem where the parameters to be estimated are fixed. This lead to
reeursive equations (4.1), (4.2), and (4.3) which allowed for the
development of an estimation procedure for a time varying parameter.
When functions f and h are linear, it has been clearly stated that
equations (2.3) and (2.4) are the essential ingredients for state
variable reconstruction. If f and h are nonlinear, X K can be estimated
by linearing h and f__about some known vector and treating the
linearized equations as if they are equations (2.3) and (2.4).
The problem of applying the extended Kalman filtering procedure
to the static test firing data remains. The basic approach for the
application is identical to the presentation given in Section 4. There-
fore, the computer codes that have been prepared by Rogers (1987),
which are currently operational on the MSFC computer system, may be
modified so that static test data may be analyzed. The static test
setup differs from the actual flight data in that many flight associated
modules will become inactive. Therefore, after adjustments are made
for the analysis of static test data, it will be necessary to evaluate the
performance of the computer code for the divergence of parameter
estimates. XXXII- 10
REFERENCES
Brown, R. G. (1983), Introduction to Random Sisal Analysis and
Kalman Filtering, New York: John Wiley and Sons, Inc.
Elbert, T. F. (1984), Estimation and Control of Systems, New York:
Van Nostrand Reinhold Company.
Gelb, A. (Editor, 1974), Applied Optimal Estimation, Cambridge- MIT
Press.
Kalman, R. E. (1960), "A New Approach to Linear Filtering and
Prediction Problems," Journal of Basic Engineering, March 1960,
pp. 35-46.
Rogers, R. M. (1987), Space Shuttle Propulsion Estimation Development
Verification, Final report for Contract NAS8-36152.
VerHaegen, M. and Van Dooren, P. (1986), "Numerical Aspects of
Different Kalman Filter Implementations," IEEE Transactions (AC),
October 1986, pp. 907-917.
XXXII-11


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