H filtering with inequality constraints for aircraft ... · H∞ filtering with inequality...
Transcript of H filtering with inequality constraints for aircraft ... · H∞ filtering with inequality...
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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
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Embedded Control Systems Research Lab 2
Outline
• H∞ filtering– Unconstrained– State equality constraints– State inequality constraints
• Turbofan engine health estimation• Simulation results
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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)
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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
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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.
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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
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Embedded Control Systems Research Lab 7
H∞ with equality constraints
DDVIDD
dxDdDxmCxy
BwAxx
T
Tkk
kkk
kkkk
≡=
=→=+=
++=+
~
1 δ
Same problem statement as before.
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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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Embedded Control Systems Research Lab 9
Turbofan health estimation
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Embedded Control Systems Research Lab 10
Turbofan health estimation
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Embedded Control Systems Research Lab 11
Turbofan health estimation
Benefits of health estimation:• Intelligent maintenance scheduling• Better closed loop control
temperaturespressuresengine
controlsestimator
(efficiencies &flow capacities)
engine health$$$
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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:
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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
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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
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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)
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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)
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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