Optimal Estimation in Optimal Estimation in Systems Health ManagementSystems Health Management

Dimitry GorinevskyConsulting Professor of EE

Information Systems Laboratory Stanford University

2November 2005

Acknowledgements

• Stanford: Sikandar Samar, Stephen Boyd

• Honeywell: Manny Nwadiogbu, Ossie Harris, HanifVhora, Suvo Ganguli, Shardul Deo, John Bain

3November 2005

Topic

• What is systems health management? – Estimation of system fault and damage states

from the operation data – Decision support: advisory information – Automated response: feedback control – Knowledge management: where and how we

get models for the analysis

4November 2005

Motivation

• Why system health management is important?– Maintenance servicing involves 30% of revenue

and 50% of profit across manufacturing industries (including hi-tech) and growing

– Emergence of autonomous systems. Autonomy requires automated contingency management

5November 2005

IVHM - Integrated Vehicle Health Management

• On-board fault diagnostics • Vehicles: space, air, ground, rail, marine

– Complex systems, many subsystems

6November 2005

Aircraft Fleet Maintenance

Airline EnterpriseJoint Strike Fighter

7November 2005

Semiconductor Manufacturing

• E-diagnostics and e-Manufacturing initiatives of SEMATECH, since 2001

• Maintenance of semiconductor capital equipment

8November 2005

Automotive Telematics

• Telematics– Embedded diagnostics and connectivity– Remote diagnostics and repair management

• GM’s OnStar: broadly licensed – IBM, Motorolla IS backbone– $600M revenue for 2006

• Toyota, BMW, Daimler• Everybody else follows

9November 2005

Key Problems

• Diagnostics– On-line: BIT/BITE, FDIR – Off-line: maintenance automation– Estimation and detection

• Prognostics– Trending for Condition Based Maintenance – OR: reliability-based maintenance – Prediction, statistical optimization

10November 2005

Some Prior/Other Work

• Related, but not discussed today• Discrete computational models, AI methods

– Model-based reasoning (graph models)– Case-based reasoning (machine learning)

• Pattern recognition – Discriminate discrete fault cases – Data driven approaches

• Diagnostics

11November 2005

Our Focus

• Parametric (continuous) data– Sensor data: raw or pre-processed

• Continuous faults – Deterioration trend for prognostics– Relaxation approach to discrete faults

• Controls and estimation approaches • Diagnostics and prognostics

12November 2005

Estimation Approaches

• Traditional univariate monitoring: SPC• Multivariate SPC • Multivariate fault estimation• Trending• Estimation based on embedded

constrained optimization

13November 2005

SPC - Statistical Process Control

• EPC (Engineering Process Control) -‘normal’ feedback control

• SPC - warning for an off-target manufacturing quality parameter

• Three main methods of SPC: – Shewhart chart (1920s)– EWMA (1940s)– CuSum (1950s)

14November 2005

SPC: EWMA

• Exponentially Weighted Moving Average• First order low pass filter

Detection threshold

)()1()1()1( txtyty λλ +−−=+

)2(2 λλ −= cZ

15November 2005

Estimation Approaches

• Traditional univariate monitoring: SPC• Multivariate SPC • Multivariate fault estimation• Trending• Estimation based on embedded

constrained optimization

16November 2005

Multivariate SPC: Hotelling's T2

• Multivariate normal distributed data

• Empirical parameter estimates

))()()((1,)(111

Tn

t

Tn

ttYtY

ntY

nµµµ −−≈Σ≈ ∑∑

==

),()( Σ≈ µNte

• The Hotelling's T2 statistics is

• T2 can be monitored using univariate SPC (almost)

( ) ( )µµ −Σ−= − )()( 12 tYtYT T

)()()( tftetY +=

17November 2005

-3 -2 -1 0 1 2 3

-2-1

01

23-2

-1

0

1

2

Multivariate SPC - Abnormality Detection

Multivariable envelope of scatter determined from normal operation data

Data point outside of this envelope implies abnormal behavior, a fault

18November 2005

• MSPC for cyber attack detection

• Y(t) consists of 284 audit events in Sun’s Solaris Basic Security Module

Multivariate SPC Example

Ye & Chen, Arizona State

19November 2005

Estimation Approaches

• Traditional univariate monitoring: SPC• Multivariate SPC • Multivariate fault estimation• Trending• Estimation based on embedded

constrained optimization

20November 2005

Multivariate Fault Estimation

• Fault parameters are explicitly modeled• Estimate and trend the fault parameters f

– MSPC detects anomaly for Y (unknown faults)

• Need models for fault signatures S(t)

)()()()( tetftStY +=

21November 2005

Engine Fleet Monitoring• Honeywell LF-507 Engine• Two-spool turbofan engine• Used in regional jets

Ganguli, Deo, & Gorinevsky, IEEE CCA’04

22November 2005

Parametric Faults - Sensitivities• Prediction residuals using detailed engineering model

• If f = 0 (nominal case) we should have Y = 0.• Nonzero residuals Y indicate faults

• A model is obtained by linearization (assume f is small)

)()()( teftStY +=

Flight cycleSensitivity

scatter = ‘noise’ ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

driftsensor EGTleak band Bleed

iondeteriorat Turbinef

Fault parameters

),( fUFXY −=

23November 2005

Normalize Data: Prediction Model Residuals

• Ambient condition data– Altitude– Mach – Ambient Pressure– Air Temperature

• Engine input data– Speed of Spool #1

• Engine output data– Speed of Spool #2 – Exhaust gas temperature – Fuel Flow– Pressures, temperatures

Prediction Model Simulink Model

Ambient condition data

Engine input data

Engine output data predictions

Faults Actual engine output data,

Residuals

Collected Data

Y

X

24November 2005

Optimal Estimation

• Consider estimation at fixed time moment

• Bayes rule

• Optimal estimate – MAP/ML

eSfY += ),0(~ RNf ),0(~ 0QNe

min)( log)|( log →−−= fPfYPL

c fPfYPYfP ⋅= )()|()|(

min)()( 110 →+−− −− fRfSfYQSfY TT

25November 2005

Optimal Estimation

• Optimal estimate

• This is a GLS estimate • The prior for f effectively serves for

regularization. – important if S is ill-conditioned

[ ] YQSSQSRf oT

oT 1111ˆ −−−− +=

26November 2005

Fault Estimates

• Deployed: modeling, simulation, estimation

LF507 Engine

Estimates of (fault)performance parameter deterioration

1550 1600 1650 1700 1750 1800 1850 1900-1

-0.50

0.51

1.52

2.53

HP

Spo

ol D

eter

iora

tion

(%)

Sample No.

1550 1600 1650 1700 1750 1800 1850 1900-10-505

1015202530

Ble

ed B

and

Leak

age

(%)

Sample No.

1550 1600 1650 1700 1750 1800 1850 1900-40

-20

0

20

40

60

EG

T S

enso

r Offs

et (d

eg C

)

Sample No.

6-10-03

27November 2005

Estimation Approaches

• Traditional univariate monitoring: SPC• Multivariate SPC • Multivariate fault estimation• Trending• Estimation based on embedded

constrained optimization

28November 2005

Trending Functions• What is trending?

Filtering Smoothing

Present FuturePast

Prediction

29November 2005

Trend Estimation

• Data:

• Bayes rule

• Random walk model of the trend )()()( tetxty +=

)}(),...,1({}{)}(),...,1({}{

NxxXNyyY

==

Observed

Underlying trend

c XPXYPYXP ⋅⋅= })({}){|}({}){|}({

})({XP

}){|}({ XYP

)()()1( tvtxtx +=+ ),0(~ RNv

),0(~ QNe

30November 2005

Trend Estimation

• Observation model:

• Prior (trend model)

• ML loss index

• Batch least square estimation[ ] [ ] min)1()()()( 2121 →−−+−= −−∑ txtxRtxtyQL

( ) ( ) [ ]∏∏=

−

=

−==n

t

n

t

txtyQtxtyPXYP1

21

1

)()()(|)(}{|}{

( ) ( ) [ ]∏∏=

−

=

−−=−=n

t

n

t

txtxRtxtxPXP1

21

1

)1()()1(|)(}{

( ) min}{|}{log →−= YXPL

31November 2005

First-order Kalman Filter

• Recursive filter form of the LS KF• Stationary KF yields a recursive update

• The gain a comes from the Riccati equation• This is a EWMA filter – a basic SPC tool

)()(ˆ)1()1(ˆ taytxatx +−=+

32November 2005

Monotonic Regression - concept

• Monotonic deterioration, never an improvement

• Palmgren-Miner rule: additive damage accumulation, counting load cycles

• Modified random walk model

• Quite fundamental model!

0)(),()()1( ≥+=+ tvtvtxtx

Gorinevsky, IEEE ACC’04

33November 2005

Nonlinear Filtration

• Exponentially distributed driving noise

• Yields a batch ML estimate

• This a QP problem! – Can be solved using off-the-shelf solvers

• This is a nonlinear batch-mode filter

( )

),...,2(,0)1()(

min)1()())()(())()((21 1

1

1

Tttxtx

txtxqtxtyRtxtyJT

t

T

=≥−−

→−−+−−= −

=

−∑

⎩⎨⎧

<≥−

0,00,

~/

xxqe

vqx

)()()1( tvtxtx +=+

34November 2005

First-order Monotonic Regression

0 10 20 30 40 50 60 70 80-1

-0.5

0

0.5

1

1.5

2

2.51st ORDER MONOTONIC REGRESSION VS. EWMA FILTER

SAMPLE NUMBER

RAW DATAMONOTONIC REGRESSIONEWMA FILTERUNDERLYING TREND

35November 2005

2nd Order Random Walk

• Linear gaussian model

• Linear trend in the absence of the ‘noise’ v• ML estimate Hodrick-Prescott filter

)()()()()()()1(

)()()1(

2

2212

111

tetxtytvtxtdxtx

tvtxtx

+=++=+

+=+

[ ] [ ]2

221

1

21

ˆ

min)(ˆ)(ˆ)(21

xy

tyQtytyRJN

t

=

→∆+−= −

=

−∑

36November 2005

2nd Order Random Walk

• Hodrick-Prescott batch filter (trending)

0 50 100 150 200

0

2

4

6HP SMOOTHING, SECOND−ORDER REGRESSION MODEL

SAMPLE NUMBER

0 50 100 150 200

−0.1

0

0.1

0.2

SMOOTHING KERNELS FOR H−P FILTER

SAMPLE NUMBER

37November 2005

2nd-order Monotonic Regression

• Gaussian noise e• Exponential v1(t)≥ 0, describes primary damage • Exponential v2(t)≥ 0, secondary damage• Optimal MAP estimate yields a QP problem

)()()()()()()1(

)()()1(

2

2212

111

tetxtytvtxtdxtx

tvtxtx

+=++=+

+=+

[ ] [ ] [ ] min)()()()(21

21

11

1

22

1 →∆+∆+−= −−

=

−∑ txptxqtxtyRJN

t0)(,0)( 21 ≥∆≥∆ txtx

38November 2005

Monotonic Regression Trending of LF507 Performance Loss

39November 2005

Nonlinear Filtering

• Fault evolution models lead to nonlinear filtering problems

• A known systematic approach is particle filters – not scalable

• Embedded convex (QP) optimization is scalable

40November 2005

Shock and Wear Models

• Long studied, reliability theory motivation – e.g., Esary et al, Annals of Probability, 1973

• Shocks = jumps in deterioration parameter • Wear = gradual change on the parameter • 2nd-order monotonic regression

automatically provides estimates of the shocks and wear

41November 2005

Mono Regression Properties

• Accurate reproduction of jumps• Exponential noise well describes jump

processes.

0 50 100 150 200

0

2

4

6SECOND−ORDER MONOTONIC REGRESSION

SAMPLE NUMBER

42November 2005

Analysis of Mono Regression

• Increments as decision variables

• The QP problem can be presented as

• An equivalent of

⎥⎦

⎤⎢⎣

⎡

∆∆

=−

−

)()(

)(2

11

1

txqtxp

tX

min)1( 2 →+−−⋅ XBxAXY T1λ0 subject to ≥X

min→XT122)1( ,0 subject to ε≤−−≥ BxAXYX

43November 2005

Convex Relaxation of L0 Optimality

• Related recent work in signal processing:

is equivalent to

(under some technical conditions)• Sparse representation by overcomplete

dictionaries (basis pursuit regression)

min→XT1AXYX =≥ ,0 subject to

AXYX =≥ ,0 subject tomin)( components nonzero ofnumber →X

min1→X

min0→X

44November 2005

L1 Relaxation of L0 Optimality

• Donoho, Elad; Tropp; Candes - 2004/05 • Problem variations, e.g., noise

• Despite apparent similarity these results do not work for monotonic regression – Results are based on assuming that ‘mutual

coherence’ of A is small– Not so for a Toeplitz impulse response A

min→XT1 2 ,0 subject to ε≤−≥ AXYX

45November 2005

Mono Regression Sparsity

Theorem• Consider a 2nd order mono regression

• Assume that , where the jumps described by V are at least 2 steps apart

• Then the sparse solution is almost exactly recovered:

min→XT122)1( ,0 subject to ε≤−−≥ BxAXYX

1BvAVY +=

)(εOVU +=Gorinevsky 2005 (in preparation)

46November 2005

Estimation Approaches

• Traditional univariate monitoring: SPC• Multivariate SPC • Multivariate fault estimation• Trending• Estimation based on embedded

constrained optimization

47November 2005

Multivariable Model

• Model of fault evolution

)()()( teftStY +=

)()()()()1(

tCxtftvtAxtx

=+=+ ⎥

⎦

⎤⎢⎣

⎡=

1101

kA

Trend model

Exponentially distributed noise v• ML estimation is a QP

0)1()( ≥−− tAxtx[ ] [ ] min)1()()()()(

21

1

21 →−−+−= ∑=

−N

t

T tAxtxqtCxtStYRJ

48November 2005

ModelModel

Moving Horizon Estimation

• MHE

time

)()()( teftStY +=

)()()()()1(

tCxtftvtAxtx

=+=+

MHE Algorithm:Solve QP

Based on modelUsing the data

TrendLast point is the current estimate

49November 2005

Rocket Flight Control Example• Shuttle-class vehicle ascent

eir

bir

wivr

ir

Q

kgme

rbkr

ekr

θ

γ

α

δ

mg

vib

L

D

Ti

ω

ωe

le

Model

Gorinevsky, Samar, Bain, & Aaseng, IEEE Aerospace, 2005

50November 2005

Rocket Flight Control Example

0 20 40 60 80 100 120 140 1600

0.5

1

1.5ACCELERATION [ft/s2]

0 20 40 60 80 100 120 140 1600

1

2

3x 10

-3 FLIGHT ANGLE RATE [rad/s]

0 20 40 60 80 100 120 140 160-5

0

5

10x 10

-4 PITCH ACCELERATION [rad/s2]

0 20 40 60 80 100 120 140 160-5

0

5x 10

-3 ENGINE ANGLE RATE [rad/s]

TIME [s]

5 0 1 0 0 1 5 0-1 .4 0 5

-1 .4

-1 .3 9 5

-1 .3 9

D o w n ra n g e a n g le ( ra d )

5 0 1 0 0 1 5 0

5

1 0

1 5x 1 0

4 A l titu d e (f t)

5 0 1 0 0 1 5 0

2 0 0 0

4 0 0 0

6 0 0 0

8 0 0 0V e lo c ity (f t /s )

5 0 1 0 0 1 5 00 .40 .60 .8

11 .2

F lig h t p a th a n g le ( ra d )

5 0 1 0 0 1 5 00

5

1 0

1 5x 1 0

-3 E n g in e g im b a l a n g le ( ra d )

5 0 1 0 0 1 5 00

5

1 0

1 5

x 1 0-3E n g in e ro ta tio n a l ra te ( ra d /s )

5 0 1 0 0 1 5 00

5

1 0

1 5

x 1 0-3V e h ic le ro ta tio n a l ra te ( ra d /s )

5 0 1 0 0 1 5 0

0 .60 .8

11 .21 .4

P itc h a n g le (ra d )

5 0 1 0 0 1 5 0-0 .0 5

0

0 .0 5

G im b a l c o m m a n d ( ra d )

T im e (s )5 0 1 0 0 1 5 0

-0 .0 4-0 .0 2

00 .0 20 .0 4

G im b a l p o s itio n (ra d )

T im e (s )

Simulated telemetry data Model prediction residuals

51November 2005

Estimation Results

20 40 60 80 100 120 140 160

15

20

25

30GIMBAL SLUGGISHNESS, PERCENT

20 40 60 80 100 120 140 160

2

3

4THRUST REDUCTION, PERCENT

20 40 60 80 100 120 140 1600

2

4

6DRAG INCREASE, PERCENT

20 40 60 80 100 120 140 1600

0.2

0.4

0.6

0.8

PITCH MEASUREMENT OFFSET, PERCENT

Seeded Fault

Seeded Fault

Seeded Fault

Seeded Fault• Simulated flight

with seeded faults• Accurate

estimation achieved

• Estimates improve with time

• Constant, monotonic, and arbitrary faults

52November 2005

Aircraft Fault Estimator Design• Aerosonde UAV

Moving Horizon Estimation

Using Monotonic RegressionFault

Signatures

Fault Trends

Tuning ParametersPredictionResiduals

PredictionResiduals

Flight Data Model-

Based PredictionNo fault

Data Prediction ModelFault 1

Samar & Gorinevsky, 2005 (in preparation)

53November 2005

AeroSim simulation

Flight Control MPC

Guidance

Flight Dynamics, Wind Gusts

VHM Estimation

MS Flight Simulator

54November 2005

0 50 100 150 200 250 300

−0.20

0.20.4

SMOOTH (5sec smoothing) PREDICTION RESIDUALS

a x [m/s

2 ]

0 50 100 150 200 250 300−0.2

0

0.2

a y [m/s

2 ]

0 50 100 150 200 250 300

00.5

11.5

a z [m/s

2 ]

0 50 100 150 200 250 300−2

024

α x [rad

/s2 ]

0 50 100 150 200 250 300−0.6−0.4−0.2

00.20.4

α y [rad

/s2 ]

0 50 100 150 200 250 300

−0.20

0.2

TIME [s]

α z [rad

/s2 ]

Model Prediction Residuals

LinearAccelerations

AngularAccelerations

55November 2005

0 50 100 150 200 250 300

−505

1015

x 10−3

a x [m/s

2 ]

SENSITIVITY TO LEFT WING CL CHANGE %

0 50 100 150 200 250 300

−0.01

0

0.01

a y [m/s

2 ]

0 50 100 150 200 250 300

0.03

0.04

0.05

0.06

a z [m/s

2 ]0 50 100 150 200 250 300

−1.2

−1

−0.8

−0.6

α x [rad

/s2 ]

0 50 100 150 200 250 300

5

10

x 10−3

α y [rad

/s2 ]

0 50 100 150 200 250 300−0.12

−0.1−0.08−0.06−0.04

α z [rad

/s2 ]

TIME (s)

0 50 100 150 200 250 300

−4−2

024

x 10−3

a x [m/s

2 ]

SENSITIVITY TO LEFT WING CD CHANGE %

0 50 100 150 200 250 300−4−2

024

x 10−3

a y [m/s

2 ]

0 50 100 150 200 250 300−3−2−1

0

x 10−4

a z [m/s

2 ]

0 50 100 150 200 250 300

−4

−2

0x 10

−3

α x [rad

/s2 ]

0 50 100 150 200 250 300

3.5

4x 10

−3

α y [rad

/s2 ]

0 50 100 150 200 250 300

−0.044

−0.042

−0.04

−0.038

α z [rad

/s2 ]

TIME (s)

Fault Signatures (Sensitivities) left wing drag left wing lift

56November 2005

Fault Estimation

• 50 point MHE• 2Hz sampling

0 50 100 150 200 250 300

0

2

4

6RIGHT WING CL CHANGE %

0 50 100 150 200 250 300

0

2

4

6RIGHT WING CD CHANGE %

0 50 100 150 200 250 300

0

2

4

6LEFT WING CL CHANGE %

0 50 100 150 200 250 300

0

2

4

6LEFT WING CD CHANGE %

TIME [s]

0 200 400 600 800 1000 1200 1400 16000

500

1000

1500

2000

NO

RT

H P

OS

ITIO

N [m

]

EAST POSITION [m]

UAV TRAJECTORY IN NORTH−EAST PLANE

N-E trajectory

57November 2005

Vision: MHE of Faults • Related to MPC process control technology

– Estimation is dual to control– Closely related math– 1000 times faster computers

• Step response models in MPC

• Fault signature models

58November 2005

Conclusions

• ISHM estimation is the key to– service and maintenance automation– autonomous systems

• Estimation based on embedded constrained optimization – performs extremely well in ISHM problems– is a scalable and flexible technology– has great practical potential