H ∞ filtering with inequality constraints for aircraft turbofan health estimation Dan Simon Cleveland State University IEEE CDC December 14, 2006 Supported by the NASA Glenn Research Center, Cleveland, Ohio
H∞ filtering with inequality constraints for aircraft turbofan
health estimationDan Simon
Cleveland State UniversityIEEE CDC
December 14, 2006
Supported by the NASA Glenn Research Center, Cleveland, Ohio
Embedded Control Systems Research Lab 2
Outline
• H∞ filtering– Unconstrained– State equality constraints– State inequality constraints
• Turbofan engine health estimation• Simulation results
Embedded Control Systems Research Lab 3
H∞ filtering
• Various approaches have been proposed• Yaesh and Shaked’s approach (1992)
kkk
kkkk
mCxyBwAxx
+=++=+ δ1
wk and mk are uncorrelated, zero mean, white, unity varianceδk is an adversary’s input (non-random)
Embedded Control Systems Research Lab 4
H∞ filtering
• Use the following predictor/corrector structure:
])ˆ([)ˆ(ˆˆ 1
kkkkkk
kkkkk
nxxGLxCyKxAx
+−=−+=+
δ
• Gk is a specified matrix (tuning parameter)• Lk is the adversary’s gain matrix• nk is zero mean, white, unity variance,
uncorrelated with wk and mk
Embedded Control Systems Research Lab 5
H∞ filtering
• Filtering goal:
1sup22
,<
+ mw
eG
mw
The solution of a certain saddle point problem ensures that this bound is satisfied.
Embedded Control Systems Research Lab 6
H∞ filtering solution
kT
kkkk
TTkk
kT
kkTkkk
T
kkkkkkT
kk
kkkkkT
kk
QxxxxE
BBAAPQ
QCCQGGQIP
xxxxEQ
nxxGLGAPL
xCyKxAxCAPK
≤−−
+=
+−=
−−=
+−==
−+==
+
−
+
])ˆ)(ˆ[( :bound Covariance
)(
])ˆ)(ˆ[(
])ˆ([
)ˆ(ˆˆ
1
100000
1
δ
Embedded Control Systems Research Lab 7
H∞ with equality constraints
DDVIDD
dxDdDxmCxy
BwAxx
T
Tkk
kkk
kkkk
≡=
=→=+=
++=+
~
1 δ
Same problem statement as before.
Embedded Control Systems Research Lab 8
kT
kkkk
TTkk
kT
kkTkkk
T
kkkkkkT
kk
kkkkkT
kk
QxxxxE
BBVIAPAVIQ
QCCQGGQIP
xxxxEQ
nxxGLGPAVIL
xCyKxAxCPAVIK
~])~)(~[( :bound Covariance
)(~)(~
~)~~(~])~)(~[(~
])~([~,~)(~)~(~~~,~)(~
1
1
00000
1
≤−−
+−−=
+−=
−−=
+−=−=
−+=−=
+
−
+
δ
H∞ with equality constraints
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Turbofan health estimation
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Turbofan health estimation
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Turbofan health estimation
Benefits of health estimation:• Intelligent maintenance scheduling• Better closed loop control
temperaturespressuresengine
controlsestimator
(efficiencies &flow capacities)
engine health$$$
Embedded Control Systems Research Lab 12
Turbofan health estimation
),,(),,,(1
kkkk
kkkkk
vpxhywpuxfx
==+
x = state vector (16 elements)u = control input vector (6 elements)p = health parameter vector (8 elements)y = measurement vector (12 elements)w = process noisev = measurement noise
DIGTEM:
Embedded Control Systems Research Lab 13
Turbofan health estimation
Augment the state vector with the health parameter vector, and linearize:
kkk
kkk
vCxywAxx
px
x
+=+=+=
←
+ ctorelement ve 8)(161
Embedded Control Systems Research Lab 14
Health parameter constraints
0 100 200 300 400 500 0
0.5
1
1.5
2
2.5
3 Expected Health Parameter Degradation
flight number
perc
ent h
ealth
par
amet
er d
egra
datio
n
flight m flight (m+1)
expected change
upper constraint
lower constraint
Embedded Control Systems Research Lab 15
Simulation results
0.1850.2330.1490.170Average0.2350.3040.2200.249LPT enthalpy0.1370.1690.1230.146LPT flow0.1250.1590.1130.126HPT enthalpy0.1450.1700.1550.167HPT flow0.1230.1680.1010.124Comp. eff.0.2200.2510.1830.194Comp. flow0.2610.3660.1450.164Fan eff.0.2320.2740.1540.192Fan flow
Constr H∞
UnconstrH∞
ConstrKalman
UnconstrKalman
Health Parameter
Ave RMS est errors under nominal conditions (500 flights)
Embedded Control Systems Research Lab 16
Simulation results
1.6842.0062.5512.716Maximum0.5740.7022.2182.436LPT enthalpy0.8651.4690.8531.000LPT flow0.5440.5390.9980.990HPT enthalpy1.6841.7781.3181.512HPT flow0.8651.2281.1191.200Comp. eff.1.5801.8711.2761.405Comp. flow0.8640.9882.5512.716Fan eff.0.5782.0061.2141.489Fan flow
Constr H∞
UnconstrH∞
ConstrKalman
UnconstrKalman
Health Parameter
Max RMS est errors with measurement bias ±σ / 2 (500 flights)
Embedded Control Systems Research Lab 17
Conclusion
• Kalman filtering gives better RMS performance, H∞ filtering gives better worst-case performance under off-nominal operating conditions
• Constrained filtering gives better performance than unconstrained filtering
• Future work: Moving horizon estimation with an H∞ performance function