a new eigenstructure fault isolation filter zhenhai li supervised by dr. imad jaimoukha internal...
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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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IntroductionIntroduction
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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IntroductionIntroduction
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
Model-based FDIModel-based FDI
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
Model-based FDIModel-based FDI
)()()(
),()()()()(
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
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Model-based FDIModel-based FDI
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
Observer-based FI Filter Observer-based FI Filter ApproachApproach
,)()()(
,)()(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
Observer-based FI Filter Observer-based FI Filter ApproachApproach
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
Observer-based FI Filter Observer-based FI Filter ApproachApproach
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
:)(
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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
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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.
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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
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Simple Example and ConclusionSimple Example and Conclusion
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
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Simple Example and ConclusionSimple Example and Conclusion
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.
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Simple Example and ConclusionSimple Example and Conclusion
Conservatism in stability conditions.
Only suboptimal solution achieved at this moment.
Further research needed to handle unstructured modelling errors.
BenefitsBenefits
LimitationsLimitations
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Thank YouThank You