NASA Technical Reports Server (NTRS) 20160001158: Sensor Selection for Aircraft Engine Performance Estimation and Gas Path Fault Diagnostics

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NASA/TM—2016-218926 GT2015–43744
Sensor Selection for Aircraft Engine Performance
Estimation and Gas Path Fault Diagnostics
Donald L. Simon
Glenn Research Center, Cleveland, Ohio
Aidan W. Rinehart
Vantage Partners, LLC, Brook Park, Ohio
January 2016
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NASA/TM—2016-218926 GT2015–43744
Sensor Selection for Aircraft Engine Performance
Estimation and Gas Path Fault Diagnostics
Donald L. Simon
Glenn Research Center, Cleveland, Ohio
Aidan W. Rinehart
Vantage Partners, LLC, Brook Park, Ohio
Prepared for
Turbo Expo 2015
sponsored by the the American Society of Mechanical Engineers (ASME)
Montreal, Quebec, Canada, June 15–19, 2015
National Aeronautics and
Space Administration
Glenn Research Center
Cleveland, Ohio 44135
January 2016
Acknowledgments
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Sensor Selection for Aircraft Engine Performance
Estimation and Gas Path Fault Diagnostics

Donald L. Simon
National Aeronautics and Space Administration
Glenn Research Center
Cleveland, Ohio 44135

Aidan W. Rinehart
Vantage Partners, LLC
Brook Park, Ohio 44142

Abstract performance, which are typically relatively rapid or abrupt in
nature (Refs. 1 and 2). A notional illustration of the observed
This paper presents analytical techniques for aiding system measurement shifts caused by gradual deterioration compared
designers in making aircraft engine health management sensor to an abrupt fault is shown in Figure 1.
selection decisions. The presented techniques, which are based Although performance estimation and gas path fault
on linear estimation and probability theory, are tailored for gas diagnostics typically apply different algorithmic approaches,
turbine engine performance estimation and gas path fault both are conducted using the same engine sensor measurement
diagnostics applications. They enable quantification of the data—primarily data acquired from the available engine
performance estimation and diagnostic accuracy offered by control sensor suite. In general, adding additional engine
different candidate sensor suites. For performance estimation, sensors will improve performance estimation and diagnostic
sensor selection metrics are presented for two types of accuracy, but this does add to the overall engine life cycle
estimators including a Kalman filter and a maximum a cost. Therefore, the decision to add sensors should be made
posteriori estimator. For each type of performance estimator, judiciously.
sensor selection is based on minimizing the theoretical sum of Several researchers have presented sensor selection
squared estimation errors in health parameters representing approaches for engine health management applications.
performance deterioration in the major rotating modules of the Mushini and Simon (no relation to the author) proposed a
engine. For gas path fault diagnostics, the sensor selection sensor selection approach for Kalman filter-based
metric is set up to maximize correct classification rate for a performance estimation applications (Ref. 3). In this work, a
diagnostic strategy that performs fault classification by performance metric was defined as a function of the steady
identifying the fault type that most closely matches the state error covariance and the cost of the selected sensors.
observed measurement signature in a weighted least squares Three separate metrics were considered for searching for the
sense. Results from the application of the sensor selection optimal sensor suite, including a random search, a genetic
metrics to a linear engine model are presented and discussed. algorithm search, and an exhaustive search. The study by
Given a baseline sensor suite and a candidate list of optional Mushini and Simon assumed that the estimation problem was
sensors, an exhaustive search is performed to determine the over-determined (i.e., there are more sensors than unknown
optimal sensor suites for performance estimation and fault parameters to be estimated), which is usually not the case for
diagnostics. For any given sensor suite, Monte Carlo engine performance estimation applications. Borguet and
simulation results are found to exhibit good agreement with Léonard approached the problem of sensor selection for
theoretical predictions of estimation and diagnostic accuracies. engine performance estimation within the scope of linear
information theory (Ref. 4). They defined performance
metrics based on the Fisher information matrix, and an
Introduction
exhaustive search was conducted to identify the best sensor
Aircraft operators rely on engine performance estimation suite. Sowers et al. introduced a systematic framework for
and gas path fault diagnostics to ensure the safe and efficient automating sensor selection decisions for diagnostic
operation of their gas turbine engine assets. Performance applications. This framework enables incorporation of factors
estimation enables the estimation and trending of gradual of merit commonly considered in the sensor selection process
performance deterioration that the engine will experience over including diagnostic accuracy, diagnostic criticality, and cost
time due to fouling, corrosion, and erosion of turbomachinery (Ref. 5). The framework relies on the end user to specify the
components. Gas path fault diagnostics enables the detection merit function used by the optimal search algorithm.
and isolation of gas path system faults affecting engine Kamboukos et al. proposed sensor selection for performance

NASA/TM—2016-218926 1
C-MAPSS40k Commercial Modular Aero-Propulsion
Rapid shift System Simulation 40k
(potentially due to
D Mahalanobis distance
M
a fault event)
FPR false positive rate
hift H influence coefficient matrix relating
s
nt Gradual deterioration changes in health parameters to changes
e in sensed measurements
m
e Hf fault influence coefficient matrix relating
r
u faults to changes in sensed measurements
s
a I identity matrix
e
M
Gradual deterioration MAP maximum a posteriori
N number of fault types
Measurement shift
PMC probability of misclassification
Measurement shift moving average
P health parameter covariance matrix
h
Time (flights) R measurement noise covariance matrix

Figure 1.—Gradual versus rapid performance shifts. SSEE sum of squared estimation errors
T fault detection threshold
estimation applications based on the condition number of the TPR true positive rate
influence matrix that relates changes in health parameters to V* transformation matrix relating h to q
changes in sensed measurements (Ref. 6). Here, a determined WSSE weighted sum of squared errors
health parameter estimation problem was considered where WSSM weighted sum of squared measurements
there are as many sensors as parameters to be estimated. h health parameter vector
The contribution of this paper will be to introduce separate f fault vector
sensor selection metrics for performance estimation and fault k number of additional sensors to add
diagnostic applications. In terms of performance estimation, m number of tuning parameters
the problem is assumed to be underdetermined (i.e., fewer Number of additional sensors to choose
n
sensors than unknown health parameters to be estimated), and from
two separate estimators will be considered—one applying a p number of health parameters
Kalman filter designed for processing dynamic sensed q reduced order tuning parameter vector
measurement information, and a second applying a maximum u actuator command vector
a posteriori estimator for processing quasi-steady-state v measurement noise vector
measurement data. In terms of fault diagnostics, a single fault wk, wh,k, wxh,k process noise vectors
diagnostic strategy applying a weighted least squares x state vector
hypothesis test will be considered. y measurement vector
The remainder of this paper is organized as follows. First, Γ gamma function
metrics are defined through analytical derivations of the γ lower incomplete gamma function
performance estimation accuracy and gas path fault diagnostic residual vector (estimate minus its
ε
accuracy based on linear system theory. These analytical expected value)
functions can be directly used to theoretically predict the Φ standard normal distribution function
estimation or diagnostic accuracy offered by a given sensor λ mean value of the WSSM signal
suite. Next, example application of the sensor selection μ mean value of ith sensed measurement
i
techniques is presented by applying the approaches to a linear Subscripts
engine model. Theoretically predicted results are calculated a fault type index
and compared against empirical results obtained through b misclassified fault type index
Monte Carlo simulation analysis. This is followed by k sample index
discussions and conclusions. xh augmented state vector (x and h)
xq reduced order state vector (x and q)
Superscripts
Nomenclature
† pseudo-inverse
A, A , A , B, B , B , system matrices ^ estimated value
xh xq xh xq
C, C , C , D, L, M ~ error value
xh xq
CCR correct classification rate – mean value
NASA/TM—2016-218926 2
Operators matrices A, B, C, D, L, and M are of appropriate dimensions.
E[•] expected value of argument Through algebraic manipulation, Equation (1) can be rewritten
tr{•} trace of a matrix such that h is concatenated with x to form an augmented state
vector, x , as shown in Equation (2). Since engine
xh
performance deterioration is very slowly evolving relative to
Sensor Selection Metrics
other engine dynamics, h is here modeled without dynamics.
Here, and throughout the remainder of this section, the ∆
As previously mentioned, aircraft engine performance
symbols are omitted for simplicity.
estimation and gas path fault diagnostics pose different
problem formulations. Analytical formulations of each are
x  A Lx  B  w 
introduced below along with derivations of performance  k+1=   k+ uk + k 
estimation and diagnostic accuracy for a given sensor suite. hk+1 0Ihk 0 wh,k
The performance estimation problem assumes the application Axh xxh,k Bxh wxh,k
of two separate estimators—a linear Kalman filter and a
= A x +B u +w
xh xh,k xh k xh,k
maximum a posteriori estimator, while the gas path fault
(2)
diagnostic problem assumes the application of a single fault
isolator applying a weighted least squares hypothesis test. yk =[CCxhM]hxkk+Duk +vk
Kalman Filter-Based Health Parameter Estimation xxh,k
=C x +Du +v
xh xh,k k k
In the aircraft engine community, Kalman filters are
commonly applied for on-board performance estimation or
The vector w is zero-mean white noise associated with the
post-flight analysis of full-flight streaming measurement data. xh
augmented state vector, [xT hT]T. w consists of the original
In this subsection, Kalman filter health parameter estimation xh
state process noise, w, concatenated with the process noise
accuracy is discussed following a derivation previously
associated with the health parameter vector, w .
introduced by Simon and Garg as part of an optimal tuner h
Once the h vector is appended to the state vector as shown
selection methodology for Kalman filter-based performance
in Equation (2), it may be directly estimated by applying a
estimation applications (Ref. 7). This optimal tuner selection
Kalman filter as long as the system is observable. However,
methodology is designed to minimize the Kalman mean
the number of health parameters that can be estimated is
squared estimation error in the parameters of interest when
limited to the number of sensors, the dimension of y (Ref. 9),
facing underdetermined estimation problems, but can readily
and typically an aircraft gas turbine engine has fewer sensors
be extended to also calculate the mean squared estimation
than health parameters. To enable Kalman filter formulation
error offered by different sensor suites, as was shown in
for an underdetermined estimation problem, a reduced-order
Reference 8.
state space model is constructed. This is accomplished by
The formulation begins by considering the following
defining a model tuning parameter vector, q, which is a linear
discrete linear time-invariant state space equations
combination of all health parameters, h, given by
representing engine dynamics about an operating point
q=V*h (3)
∆x = A∆x +B∆u +L∆h +w
k+1 k k k k
(1)
∆y =C∆x +D∆u +M∆h +v
k k k k k where q ∈ m, h ∈ p, m < p, and V* is an m × p
transformation matrix of rank m, which relates h to q. Given
where k is the sample index, x is the vector of state variables, u
an estimate of q (i.e., qˆ), an approximation of the health
is the vector of control inputs, and y is the vector of measured
outputs. The vector h, where h ∈ p, represents the engine parameter vector, hˆ, can be obtained as
health parameters, which induce shifts in other variables as the hˆ=V*†qˆ (4)
health parameters deviate from their nominal values. The Δ
symbols denote parameter deviations relative to the linear where V*† is the pseudo-inverse of V*. Substituting Equation (4)
operating point trim condition. The vectors w and v are into Equation (2) yields the following reduced-order state space
uncorrelated zero-mean white noise input sequences. The equations, which may be used to formulate a Kalman filter

NASA/TM—2016-218926 3
covariance matrix reflecting the expected distribution in the
qxkk++11= A0LVI*†qxkk+B0uk +wwqk,k hcthoeean lstithde ecprha nrKaiqamulmeet aenrcs a fntiol t ebbree h eersaetliatmhd ialptyea dra.e mxWteehtneidlre e edtsh tiistmo pa atiopopentr i mawciiczlule r oantchlyye,
xxq,k+1 Axq xxq,k Bxq wxq,k estimation accuracy of any unmeasured performance
= A x +B u +w
xq xq,k xq k xq,k parameters such as thrust, airflows, or metal temperatures.
(5)
Maximum A Posteriori Health Parameter Estimation
yk =[CCMxqV*†]xqxxqkk,k+Duk +vk apMpliaexdi mfourm g roau npdo-bstaesreido raii rc(MrafAt Pe)n geinseti mgaast ipoant hi sa nacloymsims. oInt liys
based on quasi-steady-state engine snapshot measurements
=C x +Du +v
xq xq,k k k
acquired in flight (Refs. 2 and 10). Unlike the Kalman filter,
which is a recursive estimator designed to process dynamic
The reduced-order equations introduced in Equation (5) will
measurement data, the MAP estimator provides a point
enable a Kalman filter to be formulated that can estimate the
estimate based on an assumed quasi-steady-state measurement
augmented state vector, [xT qT]T. The resulting Kalman filter-
process. The MAP estimator incorporates a priori knowledge
produced tuner parameter vector estimate, qˆ, can be inserted
regarding the distribution of the parameters to be estimated,
into Equation (4) to produce an estimated health parameter
which enables it to provide an estimate when facing
vector, hˆ However, this does not circumvent the underdetermined estimation problems. To introduce the MAP
ˆ estimator, consider the following linear steady-state
underdetermined nature of the h estimation problem, and the
measurement process
ˆ
fact that the produced h estimates will contain errors is
unavoidable. However, estimation accuracy is directly ∆y = H∆h+v (7)

dependent on the available sensor suite and the selection of the
transformation matrix, V*. This gives rise to an optimization where H is an influence coefficient matrix that relates the
problem of selecting the best sensor suite and the effects of the health parameter vector changes, Δh, to changes
corresponding V* that minimizes the estimation error in the (i.e., residuals) in the sensed measurement vector, Δy. Here, v,
parameters of interest. For a given sensor suite, an optimal is zero-mean white noise with covariance R. As with the
iterative search can be conducted to select a V* matrix that previously introduced Kalman filter equations, the Δ symbols
minimizes the theoretical mean sum of squared estimation denote parameter deviations relative to the operating point
errors (SSEE) in the parameters of interest trim condition at which Equation (7) was generated. For
simplicity, the Δ symbols are omitted throughout the
( )
argminSSEEV* remainder of this section on the MAP estimator and the terms
(6)
V*∈Rm×p y and h are used to indicate measurement and health parameter
changes, respectively. The maximum a posteriori (MAP)
where the above statement indicates the V* matrix that
estimator follows the closed form expression
minimizes the SSEE function. Once V* is obtained, it can be
inserted into Equation (5) to construct the reduced-order state hˆ=(P−1+HTR−1H)−1HTR−1y
space equations. Here, it is important to emphasize that the V* h
matrix and q vector are unique to each sensor suite considered. Gh (8)
Therefore, Equation (6) is individually applied to each sensor hˆ=G y
h
suite, and the suite that provides the lowest SSEE is identified
as optimal. where P is a matrix containing a priori knowledge of the
h
Due to page limitations, a complete derivation of the expected health parameter covariance. As with the Kalman
Kalman filter SSEE metric is not provided in this document. filter introduced above, the MAP estimator produces a biased
However, readers are referred to Reference 7 for this estimate due to the underdetermined nature of the estimation
derivation. Some notable aspects regarding the derivation are problem. However, its accuracy depends on the available
that it focuses on linear Kalman filter estimation accuracy sensor suite, thus giving rise to a sensor selection problem. As
under steady-state operating conditions, and that the error of with the Kalman filter, the MAP health parameter estimation
each estimated parameter comprises mean squared bias and error will be defined in terms of the sum of squared estimation
variance terms. Additionally, the derivation incorporates user- errors (SSEE), which consists of the sum of two components:
specified a priori knowledge regarding the health parameter mean squared bias and variance, as defined below.
NASA/TM—2016-218926 4
MAP Estimation Mean Squared Bias ˆ
where the vector ε is the residual between h and its expected
The bias of an estimator is the expected difference between value. By combining Equation (7) and Equation (8), ε can be
the estimator’s estimated value and the true value of the written as
parameter being estimated. For the MAP estimator, the
( [ ])
~ ε= hˆ−E hˆ
estimated health parameter bias vector, h , is defined as
[ ]
=G y−E G y
h h
h~ = E[hˆ−h] =Gh(Hh+v)−E[Gh(Hh+v)] (12)
[ ] [ ]
=G Hh+G v−G HE h −G E v
  h h h  h
= EG y−h h 0
h  =G v
 hˆ  h
 
Then, by substituting Equation (12) into Equation (11) the
= EG (Hh+v)−h (9)
 h  covariance matrix of the MAP estimate becomes
 y 
= E[(GhH−I)h+Ghv] Phˆ = E[εεT]
= (GhH −I)E[h]+GhE[v] = E[GhvvTGhT]
= (G H −I)hh 0 =GhE[vvT]GhT (13)
h
R
=G RGT
where the operator E[●] represents the expected value of the h h
argument, and the expected value properties E[h]=h and
E[v]=0 are leveraged in Equation (9). The estimation error Diagonal elements of Phˆ will reflect the variance of
bias equation given in Equation (9) is a function of an individual health parameter estimates, while off diagonal
arbitrary health parameter vector h. The mean sum of squared elements reflect the covariance between estimates.
biases across a fleet of engines is given as The overall sum of squared estimation errors (SSEE) can be
{ } obtained by combining the estimation mean squared bias and
~ ~ ~  ~~ 
h2 = E hTh =E trhhT variance information as
   
= E[tr{(GhH−I)hhT(GhH−I)T}] SSEE(hˆ)=tr{(GhH −I)Ph(GhH −I)T}+tr{GhRGhT} (14)
  (10)
=tr(GhH−I)E[hhT](GhH−I)T Mean squared bias and variance are equally weighted in the
  above equation. However, end users may weight them
 Ph 
{ } differently if they so choose.
=tr (G H−I)P (G H−I)T
h h h
Weighted Least Squares Single Fault Diagnostic
where tr{●} represents the trace (sum of the diagonal Approach
elements) of the matrix. Here, the E[hhT] reduces to the health
parameter covariance matrix, Ph, which is leveraged in Gas path fault diagnostics poses a different problem than
Equation (10). that of performance estimation. Unlike performance
deterioration, which is assumed to occur gradually and affect
MAP Estimation Variance all health parameters simultaneously and somewhat
independently, gas path faults are assumed to primarily occur
The variance of the MAP estimate is found by constructing
abruptly and in isolation. In other words, it is rare to have
the estimation covariance matrix,Phˆ, which is defined as multiple unrelated gas path system faults occurring
simultaneously. Applying the single fault assumption
Phˆ = E(hˆ−E[hˆ])(hˆ−E[hˆ])T (11) ttroa nasnf oormvesr g-daest epramthi nfeadu lte dstiiamgnaotisotnic sp frroobmle man. uTnhdiesr dseutbersmecitnioedn
 ε ε  will present a single fault isolator that applies a weighted least
squares hypothesis test to diagnose faults. Additionally, the
NASA/TM—2016-218926 5
accuracy offered by this diagnostic approach is analytically where H is the column of the H matrix corresponding to the
f,l f
derived. lth fault type, and the scalar fˆ is the estimated magnitude of
l
The fault diagnostic approach considered in this study, like
the lth fault type that produces the best match of the observed
the previously described MAP estimation approach, is ground-
vector of sensor measurement residuals, y, in a weighted least
based, and designed to process snapshot engine measurements
ˆ
acquired in flight. To introduce the diagnostic approach, first squares sense. The resulting f estimate is then combined
l
assume the following linear steady-state sensor measurement with H to produce an estimated measurement residual vector,
f,l
process yˆ , for the lth fault type:
l
∆∆y=Hf f +v (15) yˆ =H fˆ (18)
l f,l
where ΔΔy is a vector of residuals reflecting recent shifts in
engine sensor measurements, for example, the change The difference between yˆl and y defines the estimation error
measurements have undergone within the past one or two vector for the lth fault type, ~y , defined as
l
flights. Also shown in Equation (15) is f, a vector of gas path
fault magnitudes, and H, a fault influence coefficient matrix ~
f yl = yˆl −y (19)
relating fault magnitudes to sensor measurement residuals.
Furthermore, v denotes zero-mean normally distributed sensor
The weighted sum of squared errors for the lth hypothesized
measurement noise of covariance R. The measurement
fault type is calculated as
residuals, ΔΔy, are regularly updated as new snapshot data
become available. They are referred to as “delta-delta” WSSEl = ~ylTR−1~yl (20)
measurement shifts, as they will reflect fault induced shifts
relative to the gradual deterioration induced shifts the engine
After WSSE’s are calculated for each potential fault type
has experienced up until the time of fault initiation (Ref. 2).
they are compared, and the hypothesized fault type that
Since faults are assumed to occur abruptly and cause relatively
produces the minimum WSSE is classified as the fault cause.
large measurement shifts, the ΔΔy residuals will be small in
Theoretical predictions of fault detection and fault
the case of no fault, and larger once a fault has occurred (see
classification performance for the single fault isolator are
Fig. 1). For simplicity, the ΔΔ symbols are omitted throughout
given below.
the remainder of this section and the term y is used to indicate
recent observed shifts in the sensor measurements. Given
Fault Detection Performance
Equation (15), a fault detection and classification (isolation)
approach can be formulated. Here, it is assumed that fault
For any diagnostic system, fault detection performance is
detection is performed by calculating and monitoring a
directly related to the applied fault detection threshold. Larger
weighted sum of squared measurement (WSSM) signal:
thresholds will result in fewer false alarms in the absence of a
fault (false positives), but will also result in fewer true
WSSM = yTR−1y (16)
detections when a fault is actually present (true positives),
while the opposite is true for smaller thresholds. In order to
If the WSSM signal exceeds an established detection threshold
facilitate a common basis of comparison, each sensor suite
(T), a fault is assumed to be present and the diagnostic logic
considered in this study applies a WSSM fault detection
proceeds in attempting to isolate the most plausible single
threshold, T, necessary to achieve a user-specified target false
fault root cause for the fault. Here, fault classification is
positive rate (FPR). The FPR of a system monitoring a WSSM
performed by applying a weighted least squares approach.
signal for fault detection purposes can be approximated if it is
Each possible gas path fault type is evaluated individually, and
assumed that all sensed measurements are independent in
the hypothesized fault whose signature best matches the
addition to being zero mean and normally distributed. With
observed measurement residuals in a weighted least squares
this simplification, the distribution of the WSSM signal under
sense is classified as the fault. For the lth fault type, the
the no-fault case will be the sum of the squares of k
estimated fault magnitude is calculated as
independent standard normal random variables, which is a chi
square distribution with k degrees of freedom. The cumulative
fˆl =(HTf,lR−1Hf,l)−1HTf,lR−1y (17) distribution function of a chi square distribution is given as
(Ref. 11)
NASA/TM—2016-218926 6