H filtering with inequality constraints for aircraft ... · H∞ filtering with inequality...

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


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)


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


H∞ filtering

• Filtering goal:

1sup22

,<

+ mw

eG

mw

The solution of a certain saddle point problem ensures that this bound is satisfied.


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


H∞ with equality constraints

DDVIDD

dxDdDxmCxy

BwAxx

T

Tkk

kkk

kkkk

≡=

=→=+=

++=+

~

1 δ

Same problem statement as before.


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


Turbofan health estimation



Benefits of health estimation:• Intelligent maintenance scheduling• Better closed loop control

temperaturespressuresengine

controlsestimator

(efficiencies &flow capacities)

engine health$$$



),,(),,,(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:



Augment the state vector with the health parameter vector, and linearize:

kkk

kkk

vCxywAxx

px

x

+=+=+=

←

+ ctorelement ve 8)(161


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


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)


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)


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

H filtering with inequality constraints for aircraft ... · H∞ filtering with inequality...

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