a new eigenstructure fault isolation filter zhenhai li supervised by dr. imad jaimoukha internal...

A New Eigenstructure A New Eigenstructure Fault Isolation FilterFault Isolation Filter

Zhenhai LiZhenhai LiSupervised by Dr. Imad JaimoukhaSupervised by Dr. Imad Jaimoukha

Internal MeetingInternal MeetingImperial College, LondonImperial College, London

4 Aug 20054 Aug 2005

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OverviewsOverviews

11

IntroductionIntroduction

Model-based FDIModel-based FDI

Simple Example and ConclusionSimple Example and Conclusion

Fault Reverter - A Special CaseFault Reverter - A Special Case

Observer-based FI FilterObserver-based FI Filter

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22

Sensor Failures in Dynamic SystemsSensor Failures in Dynamic Systems

Modeling of sensor failures

SensorFaults

Actuator Component SensorsInput u Output y

Pseudo-actuator faults

)()()( tfDtCxty f

)()()( tfBtAxtx f

Representation of Sensor FaultsDirect representation (solid red line)

Indirect representation (dotted blue line)

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Fault Detection and Isolation (FDI)Fault Detection and Isolation (FDI)

Motivation

Occurring faults are always possible to be indicated by exploring deeper knowledge of the system inputs and outputs.

33

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Analytical RedundancyAnalytical Redundancy

44


Gd GfFilter F faults

residual

disturbances

‘small’ gain ‘large’ gain dFG fFG

Decisionthreshold (via online/offline

testing)

post-fault configuration

dFGfFGr df

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Residual GenerationResidual Generation

A LTI system

System input/output behaviour

where

General residual generator

55


)()()(

),()()()()(

tdDtCxty

tuBtfBtdBtxAtx

d

n

f

n

d

nn

yn

n

ufd

)()()()()()()( sfsGsdsGsusGsy fd

RΗFsomeforsfsGsdsGsFsr fd ))()()()()(()(

ff

ddd

BAsICsG

DBAsICsG

BAsICsG

1

1

1

)()(

)()(

)()(

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Residual EvaluationResidual Evaluation

Evaluation function

Threshold

Logic

66


0)()(

1)( dttrtrrJ T

0)()(

0)()(

tfforJrJ

tfforJrJ

ithi

ithi

i

i

0)())((

1dttdFGtdFGJ d

Tdth

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The error between the original output and the observer output is regarded as the residual signal.

The observer can provide diagonalized faults in the residual via a suitable selection of L and H.

The effect of disturbances is also attenuated by using Linear Matrix Inequality (LMI) techniques.

77

Observer-based FI Filter Observer-based FI Filter ApproachApproach

d f d f

y

-

u

B

B

BfBd Dd Df

C

A

A

C

L

-

H

r

x̂

xx

x̂

Real SystemReal System

Computer Aided Computer Aided ObserverObserver

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Fault Isolation Residual GenerationFault Isolation Residual Generation Problem 1Problem 1Assume that A is stable for simplicity. Let

find the matrices L and H, if they exist, such that

1. Trf(s) is a given diagonal transfer matrix

2.

3. A+LC is stable

The first condition is called isolation condition. This setup is also known as almost decoupling.

88


,)()()(

,)()(1

,

1,

ddddr

ffr

HDLDBLCAsIHCsT

BLCAsIHCsT

0)(,

givenforsT dr

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Observer-based FI Filter Observer-based FI Filter ApproachApproach Isolation ConditionIsolation Condition

Lemma Lemma Let all variables be as defined in above and assume that E:=CBf has full column rank, and denote E#=(ETE)-1ET. Let

and

Then,

idiag in f ,0),,,( 1

immmdiagM in f ,0),,,( 1

),,()(1

1

f

f

n

n

rf s

m

s

mdiagsT

ff BBLCA )(

.MHCB f

),()(

21

## LL

ff EEIREABBL

).( ##

1

EEISMEHH

fy nn

if L and H satisfy

andFurthermore,

and

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Isolation ConditionIsolation Condition ProofProof

There exists a completion such that is nonsingular. Let

Then,

where T-1T=I is used.

1010


B BBT f

TTTT 211

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Stability ConditionStability Condition Theorem 1Theorem 1 Let all variables be as defined in Lemma 3. Then the following

are equivalent.

1. There exist L and H such that .

2. The following transfer matrix is co-outer

3. The matrix CBf has full column rank and the pair (A+L1C,L2C) is detectable.

4. CBf has full column rank and Gf(s) has no finite zeros in .

5. CBf has full column rank and there exists such that P>0 and

1111


i,C- )( LCAi

CynnnnT ZPP RR ,

0)()( 2211 TTTT ZLCCZLCLAPPCLA

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Observer-based FI Filter Observer-based FI Filter ApproachApproach Stability ConditionStability Condition

ProofProof(1=>2) Let . Note that

(2=>1) Co-outer implies (A,C) is detectable, i.e., (A+L1C,L2C) is detectable. This can be shown via a contradiction.

C

f

f

ff

f

ff

f

ff

nn

CBC

BBLCAILCArank

CBC

BABIA

I

LIrank

CBC

BABIArank

)(

0

:)(


Observer-based FI Filter Observer-based FI Filter ApproachApproach Fault Isolation FilterFault Isolation Filter

Theorem 2Theorem 2 Let all variables be as defined in Lemma 3 and Theorem 3, and assume that any of (2)-(5) is satisfied. Let . Then there exist R and S, with L and H, respectively, such that specifications in Problem 1 are satisfied if and only if that there exist

such that P>0 and

where R=P-1Z.

0

ynnnnT ZPP RR ,

0(*))(

(*)(*))()(

2121

21

2211

IDSLDHCSLCH

IZLDPDLB

ZLCCZLCLAPPCLA

dd

TTTd

Tdd

TTTT


Observer-based FI Filter Observer-based FI Filter ApproachApproach Summary of the AlgorithmSummary of the Algorithm

Problem Formulation

Eigenstructure Assignment(Lemma)

Observer Design with Gain Tuning

LMI(Theorem 2)

Stability Checking (Theorem 1)

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Problem 2Problem 2If Gf is co-outer, which means we can always find a filter F such that

Then, the original problem can also be simplified to the following objectives:

(stability) The closed-loop system is stable.

(detection) The –norm of the sensitivity to disturbances is bounded by a small value.

(isolation) Each potential fault signal is indicated by a unique component in the residual signal.

1515

Fault Reverter – A Special CaseFault Reverter – A Special Case

Bounded by LMI

control input uu

disturbance dd

output yy

fault ff

FDI Integrated

Plant residual rr

d

nT

f

n

r

nd

rd

ff

d

d

f

f

f

r

r

r

1

2

1

2

1

Η

IFG f

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AlgorithmAlgorithm1. Choose with suitable choice of L1, L2 and H1 to ensure

Here, R and S are free matrices.

1616

Fault Reverter – A Special CaseFault Reverter – A Special Case

0(*))(

(*)(*))()(

2121

21

2211

IDSLDHCSLCH

IZLDPDLB

ZLCCZLCLAPPCLA

dd

TTTd

Tdd

TTTT

21

21

1 ,

SLHH

ZLPLL

2121 , SLHHRLLL

ITrf

ZPR 1

3. Construct the observer gain

2. Let all variables be defined as before. Then there exist R and S, with L and H, respectively, such that Problem 2 are satisfied if there exist Z and S such that P>0 and

with



Analysis of An AircraftAnalysis of An Aircraft A modified F16XL aircraft sensor fault detection and

isolation system (Douglas and Speyer, 1995) can detect pitch angle sensor failure and pitch rate sensor failure .

These faults may be difficult to distinguish from each other and the effect of wind gusts and deflector bias.

Pitch angle

Elevon deflector

wind gusts



Analysis of An Aircraft Analysis of An Aircraft (contd.)(contd.)

Suppose there are failures in pitch sensors. The FDI system will raise the alarm immediately despite the existence of disturbances from wind gusts.

0 1 2 3 4 5 6 7 8 9 10-3

-2

-1

0

1

2

3Wind Gust Noise

Time

Am

plitu

de

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5Elevon Deflector Fault

Time

Am

plitu

de

Longitudinal Dynamics

Control System

wind gust

FDISystem

Indicator

a constant deflector

bias

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Fast response to incipient faults.

An intuitive realization and numerically reliable.

Incorporate detection into a single observer without using banks of observers.

The robustness issue is partially covered by bounding the effect from disturbances to the residual.

Residual signal can be used for post fault handling or fault tolerant control.

1919


Conservatism in stability conditions.

Only suboptimal solution achieved at this moment.

Further research needed to handle unstructured modelling errors.

BenefitsBenefits

LimitationsLimitations


Thank YouThank You

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