Linear Matrix Inequality Linear Matrix Inequality Solution To The Solution To The Fault Fault

Detection ProblemDetection ProblemEmmanuel MazarsEmmanuel Mazars

co-authors Zhenhai li and Imad Jaimoukhaco-authors Zhenhai li and Imad JaimoukhaImperial CollegeImperial College

IASTED International ConferenceIASTED International Conference

CancunCancun19 May 200519 May 2005

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OverviewOverview

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IntroductionIntroduction

Problem definitionProblem definition

Solution using LMIsSolution using LMIs

Numerical exampleNumerical example

ConclusionConclusion

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IntroductionIntroduction Target identification and tracking systemsTarget identification and tracking systems

involve a large number of actuators and sensors.involve a large number of actuators and sensors.

An actuator failure implies actuator output isAn actuator failure implies actuator output isdegraded by bias, drift or physical damage.degraded by bias, drift or physical damage.

Actuator or sensor failures can cause rapidActuator or sensor failures can cause rapidbreakdown in control systems.breakdown in control systems.

Design objective :Design objective :Design and implement a fault detection and isolation (FDI) filter for large scale systems that is insensitive to disturbances

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IntroductionIntroduction To enhance the reliability of sensor systems in tough conditions. To enhance the reliability of sensor systems in tough conditions. To act as an aid to human operator in fast changing situations. To act as an aid to human operator in fast changing situations.

Domain of applications :Domain of applications :Noisy control and monitoring systems that involve a large number of sensors when :

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The dynamic model is knownThe dynamic model is known The sensors are prone to failureThe sensors are prone to failure Disturbance are acceptable, but faultsDisturbance are acceptable, but faultsmay cause performance degradationmay cause performance degradation

Pitch angle

Elevon deflector

wind gusts

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Problem definitionProblem definition A LTI systemA LTI system

System input/output behaviour

Where

44

)()()()(

),()()()()(

tfDtdDtCxty

tuBtfBtdBtxAtx

fd

n

f

n

d

nn

yn

n

ufd

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

0)(

C

BAsG

d

dd DC

BAsG )(

f

ff DC

BAsG )(

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Problem definitionProblem definition Fault detection and isolation observer/filterFault detection and isolation observer/filter

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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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Problem definitionProblem definition State estimation error :State estimation error :

The residual dynamics are given by :The residual dynamics are given by :

By taking Laplace transforms, we have :By taking Laplace transforms, we have :

wherewhere

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)(ˆ)()( txtxte

)()()(

)()()()()()()(

tdHDtHCetr

tfLDBtdLDBteLCAte

d

ffdd

)()()()()()( sfsGsdsGsFsr fd

yf nns

HHC

LLCAsF

R(s))(

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Problem definitionProblem definition Problem :Problem :

Assume that and that has full column rank on the Assume that and that has full column rank on the extended imaginary axis. Findextended imaginary axis. Find

and an optimal filter (which has the previous and an optimal filter (which has the previous form) that achieves the infimum.form) that achieves the infimum.

Remark :Remark :

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fy nn )(sG f

dFG

o FG

ynfnRHsF

f

)(

1inf

yn fnRΗF(s)

))((inf

jGGR

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Solution using LMIsSolution using LMIs Problem 1: Problem 1: Assume that the pair (C,A) is detectable and

is a co-outer function. The optimal FDI filter design is to find L and H to minimize a such that

(stability) (stability) is stableis stable

(detection)(detection)

(isolation) (isolation)

WhereWhere

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)( LCA

)(sTrd

IsTrf )(

f

ffs

frf HDHC

LDBLCAsGsFsT )()()(

d

dds

drd HDHC

LDBLCAsGsFsT )()()(

0

fG

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Solution using LMIsSolution using LMIs Lemma 1: Lemma 1: Let . There exist and such that

is stable and if and only if there exist , and such that and

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)( LCA

)(sTrd

fy nn

nnR TPP 0PL

L

H

H

0)(

)()()(

IHDHC

HDIPLDB

HCLDBPLCAPPLCA

d

TTd

Tdd

TTdd

T

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Solution using LMIsSolution using LMIs

We want to achieve isolationWe want to achieve isolation

Assume that has full column rankAssume that has full column ranklet (Moore Penrose Generalized inverse)let (Moore Penrose Generalized inverse)

LetLetWithWith

1010

#1

#2

#1 fffff DHDDILDBL

0 ff LDB

2121 SLHHRLLL

Tff

Tff DDDD 1# )(

IHD f We get andWe get and

and are free matricesand are free matricesS

IHDHC

LDBLCAsT

f

ffs

rf

)(

fD

R

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Solution using LMIsSolution using LMIs Theorem 1: Theorem 1: Assume that is detectable, and

has full column rank, let as defined previously. There exist and such that the problem 1 is solved if there exist , and such that and

If these LMIs are solved, we can construct and as

1111

),( AC

nnR TPP0P

ynnZ R

121 ,, HLLH

0)()(

)()()()(

2121

21

21

IDSLDHCSLCH

IZLDPDLB

CZLPCLA

dd

TTTd

Tdd

T

fy nn

L

L H

21

21

1

SLHH

ZLPLL

yf nnS R

fD

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Solution using LMIsSolution using LMIs Remark 1 : Remark 1 : In the case that

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CLA 1H

fy nn

L and are unique Isolation if is stable

Remark 2 : Remark 2 : The assumption that is co-outer can be relaxed by effecting a co-outer-inner factorization fG

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Numerical exampleNumerical example Randomly generated state-space plant with :Randomly generated state-space plant with :

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1,2,3,4 dfy nnnn

The solutions given by LMIs are :The solutions given by LMIs are :

5129.543501.322807.23

3318.544962.414676.33

9259.654846.369407.22

8954876927.540974.37

L

4440.46966.52151.6

8429.26913.41507.4H

3029.00

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Numerical exampleNumerical example Simulation with :Simulation with :

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fault in actuator1 simulated by a soft bias at the 2th second fault in actuator2 simulated by a negative jump at the 6th second

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ConclusionConclusion Optimal FD filter scheme is maximally

insensitive to disturbances with acceptable sensitivity to faults

We have incorporated fault isolation into our scheme without the need for using a bank of observers.

The numerical algorithm is much simpler than solving a model-matching problem

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Thank YouThank You