In view of the complex flight operation of the Space Shuttle propulsion system together with an expected launch rate increase, the flight performance reconstruction process needs to be performed by automated computer programs. These programs must have the capability to quickly and reliably determine the true behavior of the various components of the propulsion system. For the flight reconstruction, measured values from the solid rocket motors, liquid engines, and trajectory are appraised through the Kalman filter technique to identify the most likely flight propulsion performance. A more detailed data collection program for the single SSME engine captive test firing evaluation is scheduled for startup in September of 1988. Engine performance evaluation for the captive test firing requires a reconstruction process that is similar to the process that is used for the flight reconstruction. Analytical tools that may be used to reconstruct a propulsion system's true performance under flight and/or test conditions are described

Temple, Enoch C.

English

NASA Technical Reports Server

Prepared by : 
Academic Rank : 
ss 4E PROP LSION PERFORMANCE 
RECONSTRUCTION TECHNIQUES 
University and Department : 
NASA/MSFC 
Laboratory : 
Division : 
Branch : 
MSFC Colleague : 
Date : 
Contract No. : 
N09 - 21 7 5  2 
Enoch C. Temple, Ph.D. 
e 
Associate Professor 
Alabama A&M University 
Department of Mathematics 
Propulsion 
Propulsion System,s 
Performance Analysis 
Klaus Gross 
August 1988 
NGT- 01- 022- 099 
The University of Alabama 
XXVI 
https://ntrs.nasa.gov/search.jsp?R=19890012381 2020-03-20T02:22:42+00:00Z
SSME PROPULSION PERFORMANCE 
RECONSTRUCTION TECHNIQUES 
Enoch C. Temple 
Associate Professor of Mathematics 
Alabama A&M University 
Normal, Alabama 
ABSTRACT 
In view of the complex flight operation of the Space Shuttle propul- 
These programs must have the capebility to quickly 
sion system together w i t h  an expected launch rate increase, the flight 
performance reconstruction process needs to be performed by automated 
computer programs. 
and reliably determine the true behavior of the various components of 
the propulsion system. For the flight reconstruction, measured values 
from the solid rocket motors, liquid engines, and trajectory are appraised 
through the Kalman filter technique to identify the most likely flight 
propulsion performance. 
A more detailed data collection program for the single SSME engine 
captive test firing evaluation is scheduled for startup in September of 
1988. 
a reconstruction process that is similar to the process that is used for 
the flight reconstruction. 
be used to reconstruct a propulsion system's true performance under 
flight and /or test conditions. 
Engine performance evaluation for the captive test firing requires 
This paper describes analytical tools that may 
XXVI-i 
ACKNOWLEDGMENTS 
I would like to thank NASA, MSFC, and ASEE for providing this 
excellent research opportunity. 
of the Propulsion Laboratory for the help willingly given during this 
ten-week period. Special thanks go to Klaus Gross, my NASA counter- 
part, for suggesting and discussing the topic addressed in this report. 
Further appreciation is extended to Judy Maples who typed this report. 
Thanks are also extended to the staff 
XXVI-ii 
1. INTRODUCTION 
During the summer of 1987, Rogers Engineering and Associates (REA) 
submitted a technical report to Marshall Space Flight CeEter. This report 
summarized the results of the Propulsion Estimation Development Verifica- 
tion performed by REA under contract NAS8-36152. During that contract 
period, REA modified an existing program, which was developed under 
a previous contract, to include improved models of the Solid Rocket 
Booster, the Space Shuttle Main Engine (SSME) gain coefficient model, 
the vehicle trajectory using quaternions, and an improved Kalman filter 
based on the U-D factorized algorithm. 
program will be identified by the symbols PROG1. 
In this report, this modified 
Under a current contract, REA has proposed to design, evaluate, 
and refine a model for a single SSME that does not include the influences 
of other Space Shuttle components. The refined SSME model is expected 
to permit the collection of high quality measurements and to provide an 
improved quality of SSME performance estimation. 
model should be available in the Fall of 1988, and this report refers to 
that program by the symbols PROG2. 
REA'S single SSME 
Propulsion performance estimation procedures applied in programs 
PROGl and PROG2 are basically the same. 
that a correct model of the Space Shuttle's propulsion systeni exists and 
the model involves two main equations. They are an equation that models the 
dynamics of a state vector and a second equation that defines the rela- 
tionship between an observation vector and the state vector. Also, the 
engine performance estimation consists of using collected data on the 
observation vector to obtain a corresponding estimate of the state vector. 
This state vector estimate is called an estimate or a reconstruction of the 
engine's performance. 
optimization criterion. 
quality of the estimated state vector are judged through an application 
of statistics tools. 
properly apply and correctly interpret the results of programs PROGl 
and PROG2, an intuitive review of estimation procedures utilized by these 
programs is needed. 
Each requires the assumption 
The state vector estimate is based on the satisfaction of a statistical 
Furthermore, data quality determination and the 
Hence, in order for the Propulsion Laboyatory to 
Therefore, the objectives of this paper are to: 
1. Examine the assumptions and limitations of the Kalman filtering 
technique for the intended application to flight performance reconstruc- 
tion of propulsion systems and single engine performance reconstruction 
based on static testing data. 
2. Select specific topics from the area of statistics and probability 
for discussion and better understanding with EP55 personnel. 
XXVI- 1 
3 .  Identify some additional applications of contractor's programs. 
Throughout this paper, underlined capital letters are used to denote 
vectors, capital letters denote matrices, and the identity matrix is denoted 
by the capital letter I. 
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  , I) denotes the multivariate normal probability distribution w i t h  mean 
veFtor U and covariance matrix 1. written directly 
above ascalar  or  vector denotes a statistical estimator of that scalar or 
vector. 
All vectors are of the column type and the 
The caret symbol 
2.  NOTATION 
For any Space Shuttle flight or single SSME test, let 3 denote the 
Each component of 5 represents observed values of vector - Z at time t. 
a relevant measurable output of the Space Shuttle Propulsion System or 
a relevant navigation measurement. For example, the components of Z -t 
may be oxygen pressurant flow, fuel volume flow, hydrogen pressurant 
f l o w ,  fuel f low pressure, fuel flow temperature, and chamber pressure 
whenever ct represents the output of a single SSME static test. 
actual flight of the Space Shuttle, Zt will  contain all of the aforemen- 
tioned measurements for each of the three main engines plus additional 
components that represent navigational measurements and one measure- 
ment for each of the two Solid Rocket Motors (SRM) . For the actual 
flight, Rogers (1987) lists 35 components for vector 3 and 71 compo- 
nents for state vector 5 where 5 is defined in the next paragraph. 
However, for the single engine static test setup, vectors Ct and Kt have 
fewer components because the SRM's and navigational components become 
inactive. 
For an 
Vector zt is a state vector of parameters to be estimated at time t .  
It is assumed that the observation vector 5 is a function of the state 
vector Et. That is ,  
Z 4 - -  -t = h (Xt,t) + V 
where h is some function and It Q N(0, Rt) .  
to change with respect to time according to the equation 
State vector X is known - -t - 
x = f (Xt,t) + Et -t - -  
XXVI-2 
T where f is some function, W 'L N(O,Qt) - and E[Wt Kt ] = 0. - -t 
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 
into the form 
(k,k-1)  5 - 1  + %c x = Q  -k ( 2 . 3 )  
-k Z = H  k -k X +s ( 2 . 4 )  
where k represents discrete values of t ,  $ Q, N ( O , Q k ) ,  x k  Q N(O,Rk) - 
and 
E(?, sT) = E(Wi W . T )  = E(Vi V.T) = 0 , - -3 - -1 
provided i # j. 
connects vector 5 to &-l. 
The symbol Q (k,k- 1) denotes the transition matrix that 
After a Space Shuttle flight has taken place or after a static test, 
the post flight /test reconstruction procedure seeks to use the observed 
values of ck and equations ( 2 . 3 )  and ( 2 . 4 )  to reconstruct (estimate) the 
vector 5 so that the estimation error is minimized. The reader should 
see Gelb (1974) for a review of minimization procedures. The estimated 
value of zk is denoted by 5. A 
3 .  THE KALMAN FILTER SETUP 
Equations ( 2 . 3 )  and ( 2 . 4 )  along with assumed properties of vectors 
$ and xk allow for the development of the below Kalman filtering 
process. 
for each k. 
to be distinguished. 
During the process, two types of estimators of 5 are possible 
A notation used by Gelb (1974) allows the two estimators 
That notation is 
A s(-' = the estimate of 5 using all observations up to and 
Pk(-) = the estimate of the error covariance matrix for zk using 
including observation (k- 1) .  
all observations up to and including observation (k- 1) . 
xxv1.- 3 
f; (+) = the estimate of 5 using all observations up to and includ- 
P (+) = the estimate of the error covariance matrix for 5 using 
-k 
ing observation k .  
k 
all observations up to and including observation k .  
A 
The values of 5 
The 5 (-I vector is often called the projected ahead value of ?& and 
Pk(-) is the projected ahead variance. Vector &k (+) is called the up- 
dated estimate of 5 and Pk denotes the updated variance. 
equations ( 3 . 1 )  through ( 3 . 5 )  describe the discrete Kalman filtering 
process. 
and Pk(- )  are obtained by using equation ( 2 . 3 ) .  
A 
A 
+ Computation 
k = cP(k,k-1) A (+I 
-k 
Pk(-) = cP(kyk-l) P F ) 1  Q T ( k Y k - l )  + Qk-l 
(Hk Pk(-) Hk + Rk) -1 
Hk K = Pk k 
Pk + = ( I  - Kk Hk)  Pk ( - 1  
( 3 . 1 )  
( 3 . 2 )  
( 3 . 3 )  
( 3 . 4 )  
( 3 . 5 )  
( - 1  and Po 
A 
To start the computation procedure, initial estimates Xo 
are needed and are usually determined by the users of equations ( 3 . 1 )  
through ( 3 . 5 ) .  
The linear system described in equations ( 2 . 3 )  and ( 2 . 4 )  is essen- 
tial to the development of the discrete Kalman filtering process given in 
equations ( 3 . 1 )  through ( 3 . 5 ) .  Therefore, when functions f and h are 
nonlinear, as they are in the case of the flight/test reconstruction-model, 
vector 5 has to be estimated through a linearization process. 
different linearization procedures are described in the next paragraph. 
The first procedure is called the linearized filter and the other is known 
as the extended Kalman filter. 
Two 
XXVI- 4 
* . 
Let be some known vector such that &* = f (zk*,k). 
times Zk* is called a reference solution or a reference nominal trajectory. 
That is ,  5 is a known solution to equation ( 2 . 2 )  where the influence 
of VIt is not considered. I f  a discrete solution to equation ( 2 . 2 )  is 5 
where &k = Kk* + A X ,  then the linearization process may be used to 
estimate AX.  Here, we see that for any AX and for any k value of t ,  
a first degree Taylor series approximation Gf functions ( 2 . 1 )  and ( 2 . 2 )  
may be determined by 
Often - 
* 
- 
* --k + A ?  - % - -  f (Xk ,k) + [sIIx=xk* AX - + yk 
- -  
and 
Since 5 = - -  f (xk ,k)  , equations ( 3 . 6 )  and ( 3 . 7 )  reduce to 
and 
(3'. 6) 
Equation (3 .8 )  is called the linearized dynamics equation and ( 3 . 9 )  is 
the linearized measurement equation. These two equations are linear and 
are equivalent to the linear equations ( 2 . 3 )  and ( 2 . 4 ) .  Hence, for each 
discrete t i m e  k, AXk can be estimated and error covariance matrices can 
be determined. 
- 
The state vector estimate at time k is then given by 
X X V I - 5  
( 3 . 1 0 )  
A +  e where AXk - is computed by applying the Kalman filtering process to equations (3.8) and (3.9). * When each X++l (k = O , l ,  ..., T) is known 
prior to the beginning of the application of the Kalman filtering process 
to equations (3.8) and (3.9),  the estimate in equation (3.10) is called 
the linearized Kalman estimate. When each X z + l  is determined frcm the 
previous estimate of 5, i.e. X* = & , the estimate in equation 
(3.10) is called the extended Kalman filter estimate. 
+ 
-k+ 1 
4. FAILURE DETECTION 
The current repalcement cost of a single SSME is about $50 million. 
This cost, combined wi th  analyses of data from static test failures and 
shutdowns, suggest that there is a need for a more advanced state test 
failure detection system for the SSME. 
emphasizes the importance of SSME failure detection improvements and 
identified some possible analytic designs for a failure detection system. 
Each of Taniguchi's failure detection designs assumed that a correct 
model of the single SSME exists. 
tion of a statistics test as a decision tool. 
sion of REA'S SSME static test model would be into the area of failure 
detection during static testing. 
undergoing refinements and is expected to be available to Marshall Space 
Flight Center by the Fall of 1988. Also, according to Taniguchi (1985), 
data is available on 1200 static test firings of a single SSRIE. 
overview of a failure detection setup is provided in the next paragraph. 
A 1985 report by Taniguchi 
Each design also required the applica- 
Therefore, a natural exten- 
REA'S static test model is already 
A brief 
Assume that equations (2.3) and (2 .4)  are modified to yield 
where vectors 
additional states with dynamics such that 
are called biased vectors. If biases are regarded as 
%+1= % 
then equations (4.1) and (4.2) may be transformed into the form 
XXVI- 6 
and 
where 
( 4 . 3 )  
If Kalman filtering is applied to equations ( 4 . 3 )  and ( 4 . 4 ) ,  the bias 
vector 3 is estimated along wi th  the other components of the state 
vector. 
If no failure has occurred at time k ,  then it is intuitively reasoned 
that & = 0. - is really zero 
is determined by a multivariate statistical test. 
Of course, the judgement as to whether 
For additional details on bias vector estimation, the reader should 
consult Friedland (1983) .  
5 .  CONCLUSION 
The assumptions and limitations of the Kalman filtering process have 
In particular, it was pointed out 
been discussed with NASA personnel. 
vant topics from the area of statistics. 
in this paper that equations ( 2 . 3 )  and ( 2 . 4 )  are the essential ingre- 
dients for the discrete variable reconstruction. 
and h I of equations ( 2 . 1 )  and ( 2 . 2 )  are nonlinear, gk can be estimatgd 
by linearizing f - and h - about some nominal vector solution 5 . 
to form a failure detection procedure. 
that the intuitive failure detection approach outlined from equations ( 4 . 3 )  
and ( 4 . 4 )  is based on an assumption that f and h of equations ( 2 . 1 )  and 
(2 .2)  are linear functions. Since functions f a d  h are generally non- 
linear for SSME applications , it is worthwhile to investigate the effective- 
ness of bias estimation techniques as an SSME static test failure detec- 
tion device. 
These discussions included rele- 
Hence , if fmctions f 
* 
This report also points out that REA'S SSME model may be modified 
The reader should be reminded 
XXVI-7  
REFERENCES 
Gelb, A.  (Editor, 1974) , Applied Optimal Estimation, Cambridge : MIT 
Press. 
Friedland, B . ( 1983), "Separated-Bias Estimation and Some Applications 
in Control and Dynamic Systems, Advances in Theory and Applica- 
tion, - Edited by C. T. Leondes, New York: Academic Press. 
Rogers, R .  M. ( 1987), Space Shuttle Propulsion Estimation Development 
Verification. Final reDort for Contract NAS8- 36152. 
Taniguchi, M. H. (Editor, 1985), Failure Control Techniques for the 
SSME. Final Report, NAS8- 3'6305, Rocketdyne Division. 
XXVI-8 


SSME propulsion performance reconstruction techniques

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