A statistical methodology to enhance the Space Shuttle Main Engine (SSME) performance prediction accuracy is proposed. This methodology was to be used in conjunction with existing SSME performance prediction computer codes to improve parameter prediction accuracy and to quantify that accuracy. However, after a review of related literature, researchers concluded that the proposed problem required a coverage of areas such as linear and nonlinear system theory, measurement theory, statistics, and stochastic estimation. Since state space theory is the foundation for a more complete study of each of the before mentioned areas, these researchers chose to refocus emphasis to cover the more specialized topic of state vector estimation procedures. State vector estimation was also selected because of current and future concerns by NASA for SSME performance evaluation; i.e., there is a current interest in an improved evaluation procedure for actual SSME post flight performance as well as for post static test performance of a single SSME. A current investigation of analytical methods may be used to improve test stand failure detection. 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. Rogers used the Kalman filtering procedure to reconstruct the state variables of the Space Shuttle propulsion system. 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 reconstruction methodology for application to the reconstruction of the state vector of a single Space Shuttle Main Engine (SSME) by using static test firing data

Shipman, Jerry R.

Temple, Enoch C.

English

NASA Technical Reports Server

6 c 
STATISTICAL ANALYSIS OF SSME 
SYSTEM DATA 
Enoch C.  Temple 
and 
Jerry R .  Shipman 
Alabama A&M University 
Normal, Alabama 
Prepared for 
George C. Marshall Space Flight Center 
Under Grant No. NAGS-042 
October 2 4 ,  1988 
(hASA-CR-18Ub 50) S T A l I S T l C I L  AIIALYSIS OF b189- 1 5 166 
S2Bk S Y S l E H  C A T A  (Alabama A E ? Uaiv.) 
1 4  P CSCL 21H 
Unclas  
G3/20 01850E9 
https://ntrs.nasa.gov/search.jsp?R=19890005795 2020-03-20T04:52:12+00:00Z
TABLE OF CONTENTS 
Page 
1.  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .  1 
. . . . . . . . . . . . . . . . . . .  2.  Establishment of Notation 2 
. . . . . . . . . . . . . . . . . . .  3 .  Recursive Estimator of - X 3 
. . . . . . . . . . . . . . . . . . . . . . .  4 .  The Kalman Filter 6 
. . . . . . . . . . . . . . . . . . . . . . . . . .  5 .  Conclusions 10 
11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  
ACKNOWLEDGEMENTS 
The authors express their appreciation to Da1.e Blount for many helpful 
discussions of this work and related topics. This research was supported 
by the National Aeronautics and Space Administration through grant no. 
NAG8-042. 
with Klams Gross of the Propulsion Laboratory at the Marshall Space Flight 
The concepts presented here grew out of many discussions 
Center. 
. 
1. 
the 
the 
INTRODUCTION 
The task of this research, as originally proposed, was to focus on 
development of a statistical methodology that may be used to enhance 
Space Shuttle Main Engine (SSME) performance prediction accuracy. 
- 
This methodology was expected to be used in conjunction with existing 
SSME performance prediction computer codes to improve parameter 
prediction accuracy and to quantify that accuracy. However, after a 
review of related literature, the researchers for this project concluded 
that the proposed problem required a coverage of areas such as linear 
and nonlinear system theory, measurement theory, statistics, and 
stochastic estimation. 
Since state space theory is the foundation for a more complete study 
of each of the before mentioned areas, these researchers chose to 
refocus the emphasis of the project to cover the more specialized topic of 
state vector estimation procedures. State vector estimation was also 
selected because of current and future concerns by NASA for SSME 
performance evaluation. That is ,  there is a current interest in an 
improved evaluation procedure for actual SSME post flight performance as 
well as for post static test performance of a single SSME. Furthermore, 
Taniguchi (1985) and Cikanek (1985) described a current investigation of 
analytical methods that may be used to improve test stand failure 
detection. Many of the analytical tools that have been suggested for 
failure detection applications are based on a function of a state vector 
estimate. Hence, 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 
-1- 
2 
Kalman filtering procedure to reconstruct the state variables of the Space 
Shuttle propulsion system. 
An objective of this paper is to give the general setup of the Kalman 
filter and its connection to linear regression. 
examine the Rogers (1987) reconstruction methodology for application 
A second objective is to 
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. 
of any matrix or vector is denoted by using the letter T at the superscript 
position. 
All vectors are of the column type and the transpose 
The letter E denotes the expectation operator, N(U, C ) denotes 
the multivariate normal probability distribution with mean vector U and 
covariance matrix C , The caret symbol '? ,. 11 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 - Z tdenote the observed values of 
vector - Z at time t. 
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. 
components for vector Zt and 71 components for vector X t  where - X t  is 
defined below. 
-t 
Each component of - Z t  represents a relevant ouptut 
Rogers (1987) lists 35 
Vector X 
It is assumed that the observation vector - Z 
is a state vector of parameters to be estimated at time 
is a function of the t. 
state vector Xt . That is, 
3 = & at , t )  + 3 (2.1) 
3 
where h - is some function and xt 41 N(O - , R t ) .  
to change with respect to time according to the equation 
State vector 5 is known 
T where f - is some function, Kt % N(O - ,Qt) and E(W, xt 1 = 0. 
I f  it is assumed that f and h in equations ( 2 . 1 )  and ( 2 . 2 )  are - - 
linear, then numerical procedures allow u s  to transform these equations 
to the form 
S = H  Y + v  K % -K ( 2 . 4 )  
where K represents discrete values of t ,  W 
N(O ,RK). 
'L N ( g ,  QK) and xK ?r 4 
-After a Space Shuttle flight has taken place, the post flight reconstruc- 
tion procedure seeks to use the observed values of & and the equations in 
( 2 . 3 )  and ( 2 . 4 )  to reconstruct (estimate) the parameters zK so that the cstima- 
tion error is minimized. The estimated value of 5 is denoted by ZK. A 
3. A RECURSIVE ESTIMATOR OF X 
In order to complete the development of the reconstruction process 
mentioned in section 2 ,  w e  consider a system of equations 
where (3.1) is a system of linear equations that has been generated by 
making n observations on the single equation 
4 
where Z *  and E* are vectors or  scalars. 
j = number of components in vector Z *  and X is fixed. 
being fixed means that X does not change with time or does not change 
H* is a jxp matrix where 
Vector X 
- - 
- - - - 
- 
as the number of observations on Z increase. 
det ( 1) f 0, then the weighted least squares estimate of kector X is 
given by 
I f  Cov( E) = C and - - 
- 
( 3 . 3 )  
and the covariance of the estimate is given by 
Cov(k> = E [(i - X )  - - -  (2 - X ) T ]  = (HT 2 - l  H)-' = P . ( 3 . 4 )  
The reader should be reminded that w i t h  the appropriate assump- 
tions, the max imum 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 ) .  
measure of the quality of the estimates in vector X. 
cussion of these estimators may be found in Elbert (1984). 
- 
We also remark that Cov(X) is a - 
A 
A detailed dis- - 
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 
- 
-K 2 = H K X +  - ZK (3.5) 
where Cov L~ = Z K  and det (2,) # 0. B y  equation ( 3 . 3 )  
is the estimate of X where observations up to and inciuding observation 
I< are used. 
- 
The covariance of the estimator is denoted by PK and 
5 
Assume that an additional measurement gK+l has been made. Let 
T T 
cmd E (sK E ~ + ~ )  = 0 .  Note that E ( L ~  - E ~ + ~ )  = O ’K+1 where Cov (&K+l)  = 
means that the (K+l)th observation is independent of the 1st K observa- 
tions. 
the system 
Putting the K observations with the (K+l)th observation yields 
By substituting into equation ( 3 . 3 )  we get 
( 3 . 1 0 )  
Note that w e  can write 
Substituting results from ( 3 . 6 )  and ( 3 . 1 0 )  into ( 3 . 1 1 )  and using matrix 
algebra, we get , 
A A - 1  ) - 1  T - 1  
) . ( 3 . 1 2 )  T &+1 -  - -K x + (P-l K + H K + l  ‘K H K + l  H K + l  ‘K+1 (&+1 - H K + l  -K 
B y  substituting into equation ( 3 . 7 )  and using matrix algebra, we get 
I 
Substituting PK+l into equation ( 3 . 1 2 )  yields 
(3.13) 
and 
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 
I 
by using the symbol Zn. 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 ) .  
changes with time where there is one observation per time interval, 
then the recursive equations are quite useful. 
and (3.14) when combined with equations (2.3) and (2.4) form the 
Kalman filtering process. 
However, if X - 
In fact, equations (3.13) 
4.  
used 
THE KALMAN FILTER 
It is worthwhile to mention that Brown (1983) and Gelb (1974) 
matrix theory to express equations (3.13) and (3.14) as 
where 
K K + l  = P  K H K + 1  ( 'K+1 + H K + l  P K H T  K + 1  >-' 
and KK+l  is called the gain matrix. 
(4.3) 
7 
For the convenience of the reader, equations ( 2 . 3 )  and ( 2 . 4 )  are 
reprinted here. That is, 
&+1 = F K -K X + W K  - 
and 
( 4 . 4 )  
When equations (4.1) and (4.2) are combined with equations ( 4 . 4 )  and 
(4.51, two types of estimates of XK+l are possible for each K .  
notation by Gelb (1974) allows for the two estimators to be distinguished. 
That notation is 
A - 
A 
X -K+1 = the estimate of XK+l using all observations up to and - 
including observation K . 
= the estimate of P using all observations up to and including 'K+1 
observation K . 
iKiI' = the estimate of XK+l using all observations up to and - 
including observation ( K + 1 ) .  
The values of XKcl - - (-1 and p K l i )  are obtained by using equations (4.4) 
and (4 .5 ) .  The _Xg+l vector is often called the extrapolated o r  pre- 
dicted value of XK+l and PK+l ( - I  is called the extrapolated variance of 
a ( -1  
CI 
- 
X K + 1  * .  
. < - I  
(+) are computed by substituting &+l The values of &+l ,. (+I and PKcl 
for XK in equation (4.2) and PKcl (-1 for pK in equation (4.1). - 
The computation summary is 
a 
( + I  T 
FK + QK = F p ’K+1 K K  
( 4 . 7 )  
( 4 . 1 0 )  
Equations ( 4 . 6 )  to (4.10) completely describe the K a l m a n  filtering process 
when the original functions f and h of equations ( 2 . 1 )  and ( 2 . 2 )  are 
linsar . 
- - 
. 
However, when functions f and h are nonlinear, the state vectors - - 
XK ( K  = to, t l ,  . . . t T )  are estimated through the application of an 
extended K a l m a n  filtering procedure. 
iznlioll of functions f and h about some known state value X K .  The 
next paragraph provides a brief overview of the extended K a l m a n  filter 
concept . 
- 
This procedure requires a linear- 
. *  
- - - 
* 
Let XK be some known value of X and assume that AX is s m a l l .  - - - 
A first degree Taylor series approximation of functions ( 2 . 1 )  and ( 2 . 2 )  
may be 
and 
( 4 . 1 1 )  
(4.12) 
. *  * If it is assumed that X K  * is selected so that X K  = f ( z K ,  IC), then - - 
equations ( 4 . 1 1 )  and ( 4 . 1 2 )  become 
9 
and 
( 4 . 1 3 )  
(4.14) 
Equation (4.13) i s  called the linearized dynamics equation and ( 4 . 1 4 )  
is the linearized measurement equation. Note that equations ( 4 . 1 3 )  and 
( 4 . 1 4 )  may be transformed to equations that are equivalent to equations 
(2.3) and ( 2 . 4 ) .  
and error covariance matrices can be determined. 
estimate at time K is then given by 
Hence for each discrete t i m e  K ,  AXK can be estimated - 
The state vector 
where A"+) is computed by substituting from equations ( 4 . 1 3 )  and 
( 4 . 1 4 )  into equation ( 4 . 8 ) .  
A 
Vector XK+I is computed by letting - 
A 
X* = X and repeating the above procedure. -K+1 -K 
The extended Kalman filter has performed well in a large class of 
However, there are occasions when divergence occurs in applications. 
the state vector estimates. 
entries of the error covariance matrix PK become s m a l l  as compared to 
the actual error in the estimate of the state vector. 
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 sys.tem being studied. 
Divergence occurs when the computed 
The cause of this 
Additional 
possibIe causes of divergence are mentioned by Celb (1974). 
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. 
code has implemented the U-D factorized algorithm as a means of con- 
The Rogers (1987) computer 
trolling numerical roundoff error that may lead to divergence in the 
state vector. 
are conducted through computer evaluations. 
Checks for other sources that may generate divergence 
5. CONCLUSION 
At this point w e  have reviewed the general setup for a regression 
problem where the parameters to be estimated are fixed. 
recursive equations ( 4 . 1 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. 
by h e a r i n g  h and f about some known vector and treating the 
This lead to 
- _- 
If f and h are nonlinear, X K  can be estimated - - - 
- - 
linearized equations as if they are equations ( 2 . 3 )  and ( 2 . 4 ) .  
The problem of applying the extended Kalman filtering procedure 
to tke static test firing data remains. The basic approach for the 
application is identical to the presentation given in Section 4. 
fore, the computer codes that have been prepared by Rogers (19871, 
which are currently operational on the MSFC computer system, may be 
modified so that static test data may be analyzed. 
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 wil l  be necessary to evaluate the 
Performance of the computer code for the divergence of parameter 
estimates. 
There- 
The static test 
11 
REFERENCES 
Brown, R.  G. (1983), Introduction to Random Signal Analysis and Kalman Filtering, 
New York: John Wiley and Sons, Inc. 
Cikanek, H.  A. (1985), "SSME Failure Detection" in the Proceeding of the American 
Control Conference. June. 
Elbert, T. F. (1984), Estimation and Control of Systems, - New York: Van Nostrand 
Reinhold Company. 
Gelb, A. (Editor, 19741, Applied Optimal Estimation, Cambridge: MIT Press. 
Rogers, R. M. (1987), Space Shuttle Propulsion Estimation Development Verfication , 
Final report for Contract NAS8-36152. 
Taniguchi, R. M. (Editor, 1985), Failure Control Techniques for the SSME. 
Final Report, NAS8-36305, Rocketdyne Division. 
VerHaegen, M. and Van Dooren, P. (1986),  "Numerical Aspects of Different 
Kalman Filter Implementations, IEEE Transactions (AC)  , October 1986, 
pp. 907-917. 


Statistical analysis of SSME system data

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