Document
Sensor Selection for Aircraft Engine Performance
Estimation and Gas Path Fault Diagnostics
Paper GT2015 - 43744
Donald L. Simon Aidan W. Rinehart
NASA Glenn Research Center Vantage Partners, LLC
21000 Brookpark Road 3000 Aerospace Parkway
Cleveland, OH, 44135
Brook Park, OH 44142
ASME Turbo Expo 2015 June 15 - 19, 2015 Montreal, Canada
Outline
• Background – aircraft engine performance
estimation and gas path fault diagnostics
• Application - Specific Sensor Selection Metrics
– Kalman filter - based health parameter estimation
– Maximum a posteriori health parameter estimation
– Weighted least squares single fault diagnostic approach
• Linear Turbofan Engine Model Example
• Conclusions
Background – aircraft engine performance
estimation and gas path fault diagnostics
• Performance Estimation: – Estimation and trending of gradual performance deterioration due to fouling and erosion of turbomachinery – Entails the estimation of health parameters such as efficiency and flow capacity scalars, which reflect deterioration in major engine components – Poses an underdetermined estimation problem — more unknowns than available sensor measurements • Gas Path Fault Diagnostics: Measurement shift – Detection and isolation of gas path system faults Measurement shift affecting engine performance such as sensor faults, Measurement shift moving average actuator faults, turbomachinery damage – Faults are relatively abrupt or rapid in nature Time (flights) – Single - fault assumption makes the diagnostic
Gradual versus rapid
problem overdetermined
performance shifts
Sensor Selection
• Problem: In general, additional sensed
measurements will improve estimation
and diagnostic results, but which
sensors are best and how much
improvement will they provide?
Sensor Sensor Sensor #1 #2 #3
• Objective: Develop techniques to aid in
engine health management sensor
selection decisions, tailored to the
specific estimation or diagnostic
System method applied.
Designer
• Approach: Develop analytical metrics
based on linear estimation and
probability theory to quantify
theoretical accuracy enabled by
different candidate sensor suites.
Kalman filter - based performance estimation
Linear dynamic measurement process: Sensor selection methodology designed to determine sensor suite that minimizes the mean SSEE w h L u B x A x k k k k k 1 v h M u D x C y k k k k k Kalman filter mean sum of squared estimation errors (SSEE) is the sum of the following components: k = discrete time index • Mean squared bias y = sensed output vector • Variance h = health parameter vector x = state vector Kalman filter mean squared bias and variance are u = actuator command vector functions of: v = measurement noise ( N (0, σ ) with covariance R ) • Linear state - space model w = process noise ( N (0, σ ) with covariance Q ) * • Choice of V • Process noise, Q Optimal tuner selection methodology* applied to • Health parameter covariance, P h produce reduced - order system and enable Kalman • Available sensor suite and corresponding filter estimation when facing underdetermined measurement covariance, R estimation problem: * • Define q = V h * where V is a transformation matrix * • V is selected through an optimal iterative search to minimize Kalman filter mean squared estimation error in the parameters of interest * ˆ • Health parameter estimation : ˆ q V h *Reference: Simon, D.L., Garg, S., (2010), “Optimal Tuner Selection for Kalman Filter - Based Aircraft Engine Performance Estimation,” Journal of Engineering for Gas Turbines and Power , Vol. 132 / 0231601 - 1.
Maximum a posteriori (MAP)
performance estimation
Linear steady - state measurement process: Sensor selection methodology designed to determine sensor suite that minimizes the v h H y mean SSEE Δ y = residuals in the sensed measurement vector MAP estimation mean sum of squared H = influence coefficient matrix estimation errors (SSEE) is the sum of the Δ h = health parameter vector following components: v = measurement noise ( N (0, σ ) with covariance R ) • Mean squared bias: ~ T
I H G P I H G tr h
h h h Maximum a posteriori estimator : • Variance : T T 1 1 1 1 ˆ
y R H H R H P h
h T
RG G tr var
G h h h ˆ y G h MAP mean squared bias and variance are h functions of: • Linear state - space model P is the health parameter covariance matrix.
h • Health parameter covariance, P h • Available sensor suite and Note: Incorporating a priori knowledge through P h corresponding measurement allows estimates to be produced when facing covariance, R underdetermined estimation problems
Weighted Least Squares
Single Fault Diagnostic Approach
Estimated fault Linear steady - state measurement process : signatures for each v f H y f candidate fault type ΔΔ y = vector of measurement residuals reflecting y Actual recent abrupt shifts in sensor measurements measured H = fault influence coefficient matrix f fault signature f = vector of gas path fault magnitudes v = measurement noise ( N (0, σ ) with covariance R ) Fault diagnostics performed via a two - step process: Legend: y = fault type “a” = fault type “b” 1) Fault detection performed by monitoring a = fault type “c” Fault detection weighted sum of squared measurements ( WSSM ): threshold T - 1 WSSM = ΔΔ y R ΔΔ y Illustration of Single - Fault Diagnostic Approach in Two - Dimensional Measurement Space If WSSM > Threshold, T , fault declared • Fault signature, exceeding defined failure threshold, is detected (indicated by red “ x ”) 2) Single fault isolation performed by comparing • Fault signatures of three system fault types (a, b, and c ) are known system fault signatures to observed vector individually compared to measured fault signature • In this example, fault type “b” will be classified as the fault of measurement residuals, ΔΔ y . Fault type that as it most closely approximates the observed fault signature most closed matches observed signature in a weighted least squares sense is isolated as fault.
Weighted Least Squares
Single Fault Diagnostic Approach (continued)
Sensor selection methodology determines sensor suite 3) Applying a two - fault class assumption, the probability that maximizes the correct classification rate (CCR) of misclassifying fault type “a” as fault type “b” is approximated as: 1) WSSM signal fault detection threshold, T , is set to yield a common target false positive rate (FPR): D PMC 1 M a b | T k , Where : 2 2 Φ = Standard normal distribution
1 , k T FPR
k D = Mahalanobis distance M Where: The probability of misclassifying fault type “a” as any T = WSSM detection threshold other single fault type is: k = Number of sensors N PMC PMC λ = Mean value of the WSSM signal
a b a |
b 1 Γ = Gamma function a b Where: 2) In the presence of a fault, the WSSM signal is N = number of fault types distributed as a non - central chi - square distribution, and the true positive rate (TPR) is calculated as: 4) Correct classification rate (CCR) for fault type “a” and all fault types is: j T k
, j
2 2 2
1 , , e k T TPR PMC TPR CCR 1
a a a
k ! j
j
0 j N CCR a CCR
Where : N a 1 λ = Mean value of WSSM signal in the presence of a fault
Linear Turbofan Engine Model Example
The sensor selection approaches were applied to a linear point model extracted from the NASA Commercial Modular Aero - Propulsion System Simulation 40k (C - MAPSS40k) high - bypass turbofan engine model.
7 State variables, x 3 Actuators , u 10 Health Parameters, h 6 baseline + 4 Optional Sensed outputs, y Nf – fan speed Wf – fuel flow FAN efficiency Nf – fan speed Nc – core speed VSV – variable stator vane Nc – core speed FAN flow capacity Hs_LPC – LPC metal temp VBV – variable bleed valve LPC efficiency Ps30 – HPC exit static press Baseline Sensors Hs_HPC – HPC metal temp LPC flow capacity T30 – HPC exit total temp Hs_burner – burner metal temp HPC efficiency P50 – LPT exit total pressure Hs_HPT – HPT metal temp HPC flow capacity T50 – LPT exit total temp Hs_LPT – LPT metal temp HPT efficiency P14 – Bypass duct total pressure Additional HPT flow capacity T14 – Bypass duct total temp (Optional) LPT efficiency P25 – HPC inlet total pressure Sensors LPT flow capacity T25 – HPC inlet total temp.
Objective: assess the estimation and diagnostic improvements that can be gained by adding sensors individually or in combination to the baseline sensor suite
Sensor Selection for
Health Parameter Estimation
C - MAPSS 40 k Health Parameters Health Parameter Estimation : Objective is to minimize sum of squared Health parameters estimation errors (SSEE) across all 10 health 1 η Fan efficiency FAN parameters 2 γ Fan flow capacity FAN 3 η Low pressure compressor (LPC) efficiency Health parameters are assumed to exhibit LPC 4 γ Low pressure compressor (LPC) flow capacity LPC simultaneous, uncorrelated, normally 5 η High pressure compressor (HPC) efficiency HPC distributed random variations with a 6 γ High pressure compressor (HPC) flow capacity HPC standard deviation of 2% 7 η High pressure turbine (HPT) efficiency HPT Health parameter covariance matrix, P , is a h 8 γ High pressure turbine (HPT) flow capacity HPT 10 10 diagonal matrix with elements of 4.0 9 η Low pressure turbine (LPT) efficiency LPT along the diagonal.
10 γ Low pressure turbine (LPT) flow capacity LPT Analytical techniques applied to predict theoretical SSEE for each candidate sensor suite Monte Carlo simulations conducted to validate theoretical predictions Uses C - MAPSS40k linear point model Health parameters randomly assigned according to health parameter covariance matrix, P h Random sensor measurement noise added in accordance with sensor measurement covariance matrix, R Kalman Filter Estimator – 200 trials, each 30 seconds in duration MAP Estimator – 400,000 trials, each a single sample in time
Sensor Selection Results for
Health Parameter Estimation
Kalman Filter Estimator Results MAP Estimator Results
sensors added sensors added Sum of Squared Estimation Errors Sum of Squared Estimation Errors to baseline to baseline Monte Carlo Monte Carlo Theoretical Theoretical T14 T25 T14 T25 # Sensors P14 P25 Simulation # Sensors P14 P25 Simulation 6 17.21 17.35 6 16.35 16.36 7 x 13.66 13.94 7 x 12.86 12.86 7 x 13.81 14.44 7 x 14.54 14.55 7 x 12.83 13.19 7 x 12.36 12.37 7 x 12.58 12.90 7 x 12.36 12.36 8 x x 22.45 23.74 8 x x 12.38 12.38 8 x x 9.14 9.94 8 x x 8.87 8.86 8 x x 8.78 9.60 8 x x 8.87 8.86 8 x x 10.44 11.47 8 x x 10.55 10.55 8 x x 9.27 9.84 8 x x 10.55 10.54 8 x x 8.60 8.90 8 x x 8.40 8.41 9 x x x 10.07 11.66 9 x x x 8.39 8.38 9 x x x 6.13 7.29 9 x x x 8.39 8.38 9 x x x 4.79 5.45 9 x x x 4.91 4.91 9 x x x 4.95 5.75 9 x x x 6.59 6.60 10 x x x x 4.47 4.98 10 x x x x 4.43 4.43 Sensor Selection Results (Kalman filter and MAP estimator select the same sensors) : Baseline + 1 sensor, choose: T25 Baseline + 2 sensors, choose: T25 and P25 Baseline + 3 sensors, choose: T25, P25, and P14
Sensor Selection for
Gas Path Fault Diagnostics
Gas Path Fault Types Fault Fault Health parameters and ID type actuator biases Gas Path Fault Diagnostics : 1 Fan fault η = - 1 % , γ = - 2 % FAN FAN Objective is to maximize the correct 2 LPC fault η = - 1 % , γ = - 2 % LPC LPC classification rate (CCR) across all 8 gas 3 HPC fault η = - 1 % , γ = - 2 % HPC HPC path faults 4 HPT fault η = - 2 % , γ = + 1 % HPT HPT 5 LPT fault η = - 2 % , γ = + 1 % Each fault considered to occur in isolation, LPT LPT 6 Wf bias Wf bias = - 2 % and to be of equal criticality and probability 7 VSV bias VSV bias = - 1 degree stroke of occurrence 8 VBV bias VBV bias = + 20 % Analytical techniques applied to predict theoretical CCR for each candidate sensor suite Monte Carlo simulations conducted to validate theoretical predictions Uses C - MAPSS40k linear fault influence coefficient matrix Random sensor measurement noise added in accordance with sensor measurement covariance matrix, R Monte Carlo simulation study consisted of 80,000 no fault cases and 10,000 cases for each of the 8 gas path fault types Detection threshold set to achieve theoretical false positive rate of 0.01 (1 %)
Sensor Selection Results for
Gas Path Diagnostics
Gas Path Diagnostic Results sensors False Positive Correct Classification added to Rate (%) Rate (%) Monte Carlo results baseline • Confirmed theoretical target false positive rate Monte Carlo Monte Carlo of 1% Theoretical Theoretical # Sensors P14 T14 P25 T25 Simulation Simulation • Theoretical correct classification rate (CCR) found to under - predict Monte Carlo CCR. This is 6 1.00 1.04 84.60 88.40 attributed to the 2 fault class simplifying 7 x 1.00 1.02 85.51 89.06 assumption made in calculating the theoretical 7 x 1.00 0.99 85.84 89.18 CCR.
7 x 1.00 1.00 88.98 90.39 7 x 1.00 1.07 91.58 92.59 8 x x 1.00 1.03 86.29 89.47 Sensor Selection Choices: 8 x x 1.00 1.03 89.73 91.11 • Baseline + 1 sensor, choose: T25 8 x x 1.00 1.05 92.28 93.23 • Baseline + 2 sensors, choose: T25 and T14 8 x x 1.00 1.02 89.97 91.19 • Baseline + 3 sensors: 8 x x 1.00 1.02 92.42 93.30 o Theoretical choose : T25, T14, and P25 8 x x 1.00 1.01 92.04 92.57 o Monte Carlo choose: T25, T14 and P14 9 x x x 1.00 1.02 90.28 91.48 9 x x x 1.00 1.02 92.69 93.55 9 x x x 1.00 1.01 92.70 93.29 9 x x x 1.00 1.02 92.83 93.32 10 x x x x 1.00 1.00 93.07 93.57
Conclusions
• Sensor selection methods introduced in this paper were found to
perform well in identifying optimal sensor suites
• Results are application - specific to engine model considered and
measurement noise, health parameter variations, and fault types
assumed
• Kalman filter and MAP estimator based sensor - selection methods
found to yield good agreement between theoretical predictions and
simulation results. Also found to yield same sensor selection choices.
• Weighted Least Squares Single Fault Diagnostic sensor selection
methods found to slightly under - predict correct classification rate
• Follow - on recommendations
– Incorporate other factors of merit such as sensor life cycle cost (cost,
weight, reliability, etc.) and criticality of different fault types
– Extend to additional operating points beyond single linear point analysis
Acknowledgments
• This work was conducted under the NASA Aviation Safety
Program, Vehicle Systems Safety Technologies Project and the
NASA Transformative Aeronautics Concept Program,
Convergent Aeronautics Solutions Project
Backup Slides
Linear Turbofan Engine Model Example
Sensor Selection Approach: • Optional sensors are evaluated for estimation accuracy or diagnostic improvement they provide if added individually or in combination to baseline sensor suite.
• Given n sensors to choose from, and a target number, k , of additional sensors, the total number of sensor suite combinations will be:
n
! n
k
! ! k n k
• Thus, the number of sensor combinations when adding 1, 2, 3, or 4 sensors to the baseline 6 sensors are: o Baseline sensors 1 combination o Baseline + 1 sensor 4 combinations o Baseline + 2 sensors 6 combinations o Baseline + 3 sensors 4 combinations o Baseline + 4 sensors 1 combination • Analytical metrics are applied to calculate theoretical performance for each sensor suite • Monte Carlo simulation analysis is then conducted to verify theoretical predictions
Performance estimation and gas path fault
diagnostic methods considered in this study
• Performance Estimation Methods:
– Kalman filter estimation
• Applied in dynamic, streaming (continuous) engine measurement process • Underdetermined estimation addressed by combining health parameters into a reduced set of optimal tuners
– Maximum a posteriori estimation
• Applied in steady - state measurement process as available through “snapshot” measurements • Underdetermined estimation addressed by leveraging a priori knowledge regarding health parameter covariance • Gas Path Fault Diagnostic Method:
– Weighted - least squares single fault diagnostic approach
• Fault detection performed by monitoring for abrupt shifts in measurements • Fault isolation performed by identifying the known fault signature that most closely matches observed measurement signature in multi - parameter measurement space in a weighted - least squares approach