identification and model updating in condition of distributed ...identification and model updating...
TRANSCRIPT
Identification and Model Identification and Model Updating in condition of Updating in condition of distributed uncertaintiesdistributed uncertainties
A. De StefanoA. De Stefano
Politecnico di Torino, Dipartimento di Ingegneria Politecnico di Torino, Dipartimento di Ingegneria Strutturale e Geotecnica, Strutturale e Geotecnica, TurinTurin, , ItalyItaly;;
Workshop on Operational Modal Analysis, Campobasso, 27.05.2008
Unknown inputUnknown input
It is the case of all the identification techniques It is the case of all the identification techniques based on ambient vibrations. Generally they apply based on ambient vibrations. Generally they apply to to OPERATIONAL MODAL ANALYSISOPERATIONAL MODAL ANALYSISThose techniques operate often on low energy Those techniques operate often on low energy signals. signals. The response signals are almost, The response signals are almost, but not but not necessarilynecessarily, linear; response parameters are , linear; response parameters are influenced by preinfluenced by pre--load conditions. load conditions. Unknown input techniques are important because Unknown input techniques are important because they allow a cost effective test exploitation and a they allow a cost effective test exploitation and a continuous oncontinuous on--line monitoring action.line monitoring action.
VibrationVibration basedbased IdentificationIdentification::whatforwhatfor??
NumericalNumericalNumerical model model model assessmentassessmentassessment (OMA)(OMA)(OMA)PredictionPredictionPrediction of of of dynamicdynamicdynamic responseresponseresponse (OMA)(OMA)(OMA)CatchingCatching the the mechanicalmechanical parameters parameters evolutionevolution and and changeschanges (on(on--line SHMline SHM--OMA)OMA)
ExperimentalExperimental knowledgeknowledge forfor residualresidualsafetysafety evaluationevaluation after severe after severe externalexternaleventsevents likelike earthquakesearthquakes ((nonnon--linearlinearidentificationidentification strategiesstrategies, no OMA) , no OMA)
Line
arLi
near
Line
arLi
near
Non
linea
rN
onlin
ear
SHM in CIVIL ENGINEERING:SHM in CIVIL ENGINEERING:
lightweightlightweight modernmodernstructuresstructures, new , new highhigh--riserise towerstowers, new , new longlong--spanspan bridgesbridges•• FlexibleFlexible, mostly low , mostly low
damped structures;damped structures;•• WellWell characterizad and characterizad and
reliable reliable materialsmaterials;;•• WellWell knownknown geometrygeometry•• nearly hyperelastic nearly hyperelastic
serviceservice behaviourbehaviour;;•• SHM looking SHM looking forfor
localizedlocalized single or single or multiple multiple defectsdefects..
RigidRigid bridgesbridges, , existingexistingand and damageddamagedconstructionsconstructions, , ancientancientheritageheritage•• stiffstiff, , mostlymostly highhigh--
dampeddamped structuresstructures;;•• poorpoor, , degradedegrade, and non , and non
reliablereliable base base materialsmaterials, , withwith largelylargely scatteredscatteredmachanicalmachanical propertiesproperties;;
•• uncertainuncertain geometrygeometry;;•• nonnon--linearlinear behaviourbehaviour;;
•• SHM SHM ??
Class 1 Class 2Class 1 Class 2
Class 2, SHM FIRST STEP:Class 2, SHM FIRST STEP:EXPLORING THE ACTUAL CONDITIONS:EXPLORING THE ACTUAL CONDITIONS:
uncertainties in uncertainties in geometric and geometric and mechanical propertiesmechanical properties
•local variability of geometric properties and masonry internal organization;
lack of material continuity, hidden empty volumes
•local variability of the material strength and stiffness, due to original defects or electro-chemical degradation;
distribution of cracks, subject to thermal path (seasonal width oscillation with basic trend to increase continuously, due to cumulated debris inside the crack);
effects of past, non documented, damages and repairs, architectural changes, local manipulations.
SECOND STEP:SECOND STEP:DESIGNING AND ASSESSING THE MONITORING DESIGNING AND ASSESSING THE MONITORING
DEVICES AND PROCEDURESDEVICES AND PROCEDURES1)Preliminary analyses:1)Preliminary analyses:•• risk analysis; risk analysis; •• optimisation of the measurement network; optimisation of the measurement network; •• design of the signal acquisition procedure and sensor choice; design of the signal acquisition procedure and sensor choice; •• dislocation of the permanent sensors in order to obtain the autodislocation of the permanent sensors in order to obtain the automatic monitoring matic monitoring
system be sensitive to possible damage and defects.system be sensitive to possible damage and defects.
2) Realization and assessment:2) Realization and assessment:•• testing of permanent measurement networks on laboratory samples testing of permanent measurement networks on laboratory samples and real and real
existing structural components; existing structural components; •• positioning supports for temporary or periodic observations.positioning supports for temporary or periodic observations.
3) Global on3) Global on--line testing:line testing:•• dynamic dynamic vibrationalvibrational response measurement acquisition and elaboration;response measurement acquisition and elaboration;•• Mechanical identification;Mechanical identification;•• symptom based diagnosis;symptom based diagnosis;
4) Modelling and Model Updating:4) Modelling and Model Updating:•• FE model based diagnosis;FE model based diagnosis;•• safety evaluations. safety evaluations. •• Model aided identification of faults detectable by monitoring anModel aided identification of faults detectable by monitoring and those that do d those that do
not show apparent or perceptible forewarnings.not show apparent or perceptible forewarnings.
““RobustRobust”” monitoring.monitoring.
““robustrobust”” applies to every algorithm, applies to every algorithm, process, method or technique able to process, method or technique able to reduce the sensitivity of analysis reduce the sensitivity of analysis results against input data and results against input data and measure errors, noise or measure errors, noise or uncertainties.uncertainties.In case of largeIn case of large distributed distributed uncertainties uncertainties ““non robustnon robust”” means means often often ““unable to give a responseunable to give a response””
Some basic principlesSome basic principlesMethods able to solve by complex Methods able to solve by complex algorithms only simple problems are of algorithms only simple problems are of none interestnone interestReliable methods can treat complex Reliable methods can treat complex problems with reasonable simplicity problems with reasonable simplicity (method complexity shall not increase (method complexity shall not increase with the complexity of the problem)with the complexity of the problem)Methods shall survive real experimental Methods shall survive real experimental validationsvalidationsUnfit models cannot be Unfit models cannot be updatedupdated
Global (dynamic) testing in detailGlobal (dynamic) testing in detail
RobustRobust identificationidentification algorythmalgorythm
RobustRobust model model updatingupdating techniquetechnique
IIdentifying the modal parameters of dentifying the modal parameters of a a linearlinear systemsystem
SMART STRUCTURES ON-LINE MONITORING
OUTPUT ONLY VIBRATION MEASURESOUTPUT ONLY IDENTIFICATION PROCEDURES
Linear OMA identification Linear OMA identification
Time domain methods Time domain methods (linear time invariant)(linear time invariant)
Spectral methodsSpectral methods•• Frequency domain methodsFrequency domain methods (linear time invariant)(linear time invariant)
•• TimeTime--frequencyfrequency or or timetime--scalescale domain domain methods methods (linear time variant)(linear time variant)
Linear OMA identification Linear OMA identification
Modal analysis by time domain Modal analysis by time domain techniques techniques
There are several Time Domain There are several Time Domain techniques but their conceptual contents techniques but their conceptual contents are strictly mutually related. The starting are strictly mutually related. The starting point is the Lagrange solution of the point is the Lagrange solution of the dynamic response of linear MIMO dynamic response of linear MIMO mechanical systems.mechanical systems.
Response of a MIMO damped linear systemResponse of a MIMO damped linear system
{ } [ ]{ } [ ] { })(0 tFbZAZ ⊗+={ }
{ }
actions.forcingexternaltheanddecay)(freeresponseimpulsetheofnconvolutiothe2.
0Zstateinitialtheafterdecayfreethe1.:onscontributitwo
ofsumtheisspacestatetheinZresponseThe
{ } [ ]{ } [ ]{ })(tFZZ Θ+Φ=& { } { }{ } { } { }
{ }
[ ] [ ] [ ] [ ] [ ][ ] [ ]
[ ] [ ] [ ][ ] [ ]⎥⎦
⎤⎢⎣
⎡=Θ
⎥⎦
⎤⎢⎣
⎡ −−=Φ
⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧
=
−
−−
000
;0
;0
)()(;
1
11
M
IKMCM
tftF
xx
Z&
The general Lagrange solutionThe general Lagrange solution
{ } [ ]{ } [ ]{ } { }{ } [ ]{ } [ ]{ }
{ } [ ] [ ]{ } [ ]{ }( ) [ ]{ } .......::..
;
1121
1
00
kkkkkkkk
kkkkk
k
FBFBZAAZprocedurerecursiveaInFBZAZge
steptimepreviousanythetorelatedbecanZFBZAZ
++=
+=
+=−
−−−−
−
:domaintimediscretetheIn
The part of the recursive process related to the past The part of the recursive process related to the past responses is called responses is called ““autoregressiveautoregressive””If the input F is not known, it is necessary to assume If the input F is not known, it is necessary to assume that its statistic nature is given.that its statistic nature is given.Generally an assumption of stationary white noise Generally an assumption of stationary white noise input is required.input is required.When each state depends only from the previousWhen each state depends only from the previous--one one the process is a recursive Markov process (ERA, the process is a recursive Markov process (ERA, SubspaceSubspace…………..)...).If each state depends from more previous states, the If each state depends from more previous states, the recursive process is nonrecursive process is non--Markov or semiMarkov or semi--Markov ; Markov ; autoauto--regressive coefficient are overlapped ( ARMAV or regressive coefficient are overlapped ( ARMAV or DSPI approach, DSPI approach, much more robust than ERA or much more robust than ERA or SubspaceSubspace))
SpectralSpectral AnalysisAnalysis
StationaryStationary signalssignals::(FREQUENCY ANALYSIS)(FREQUENCY ANALYSIS)
Non Non StationaryStationary signalssignals::(TIME(TIME--FREQUENCY ANALYSIS)FREQUENCY ANALYSIS)
SLOW VARIATIONSSLOW VARIATIONS
RAPID VARIATIONSRAPID VARIATIONS
•Fourier Transform,•Periodogram,•AR modelling
•STFT,•Spectrogram
•Wigner-Ville &T-F distributions of Coehn’s class ,
•Evolutionary spectra
•Wavelets
Time-frequency methods+ high accuracy in parameter extimation
+ possibility of managing effectively non-stationary signals
+ ability in handling modarate non-linearities
+ high robustness against noise (even in high-frequency range)
- heavy computational cost
- higher theoretical and algorythmic complexity
MOREOVER:
Time-Frequency methods supply the spectral content evolution in time. Therefore:
intrinsecally fit to continuous on-line monitoringmonitoring needingsneedings
TT--ff bibi--linearlinear transformstransforms: : autospectraautospectra
ModalModal responsesresponses are are notnot allall excitedexcited simultaneouslysimultaneously! A time ! A time window window cleverlycleverly chosenchosen can help can help addessingaddessing closeclose modesmodes
It is possible to define the estimator for the phasedifference between two signals as:
( ) ( ){ }( ){ }
0
,,
arctg,,
,,
ffyx
yxyx ftXWV
ftXWVft
=⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
ℜ
ℑ=Δϕ
consistent for any distribution belonging to theCohen class.
TimeTime--frequencyfrequency estimatorsestimators methodmethod::modalmodal frequencyfrequency
( )0
,, ffyx ft =Δϕ is expected to be nearly constant versus time if 0f is a modal frequency.
TimeTime--frequencyfrequency estimatorsestimators methodmethod::modalmodal frequencyfrequency
EstimatingEstimating the the modalmodal frequencyfrequency
Strategy: We explore, along the frequency axis, the ( ) ),(. , fyxt ftDevSt ϕΔ , i.e., for each f value, we compute the standard deviation along the t axis of the function :
( ) fyx ft ,,ϕΔ
If it approaches zero, then: f is a modal frequency
EstimatingEstimating the the modalmodal shapeshape
( ) ( ) ( ) ( )ftDftDftDftDkjkiji ssssss ,,, ,, ,,
( ) ( )( )ftD
ftDftAR
j
i
s
sji ,
,,, =
( ) ( )( )ftD
ftDftAR
kj
ki
ss
ssji ,
,,, =
GivenGiven the the signalssignals::
)(
)(,
0ffwithsinusoidalpurevirtuals
signalsrycontemporarecordedrealss
k
ji
=
IndicatingIndicating by:by:
coherentcoherent auto and cross auto and cross CohenCohen--classclass transformstransforms of the of the describeddescribed signalssignals,,thenthen wewe can estimate the can estimate the modalmodal shapesshapes asas amplitudeamplitude ratiosratios, , asas followsfollows::
Or, Or, betterbetter::
The work done by our groupThe work done by our group
A Nonlinear System Identification Method Based A Nonlinear System Identification Method Based upon the Timeupon the Time--varying Trend of the Instantaneous varying Trend of the Instantaneous Vibration Frequency and AmplitudeVibration Frequency and Amplitude
System Identification of Bilinear Systemsthrough Separating Responses and Application to the Detection of Breathing Crack
Nonlinear damage detection using chaotic metricssuch as Lyapunov Exponent, Correlation dimension
VolterraVolterra series applied to series applied to tt--ff distribution core distribution core
Non OMA identification for nonNon OMA identification for non--linear linear responseresponse
Review on Nonlinear System IdentificationReview on Nonlinear System Identification
M odel-free methods
M odel-dependent methods
Methods for identifying the time- domain input-output relationship
Methods for identifying the frequency-domain input-output relationship
Methods based on an extremely general model
Methods based on a polynomial model
Methods for time-varying systems
Methods based on an exact model
M ethods based on an equivalent linearilization model
Symptomatic and Symptomatic and timetime--frequency techniques frequency techniques
to nonto non--linear structural linear structural identificationidentification
NonNon--linear behavior typeslinear behavior types•• Hysteretic and plastic nonHysteretic and plastic non--linearity linearity Evolving Evolving
properties, energy dissipation, strength and properties, energy dissipation, strength and stiffness increasing degradation: parametric stiffness increasing degradation: parametric identification: e.g. based on the inverse identification: e.g. based on the inverse BoucBouc--WenWen model assessment (The model assessment (The BoucBouc--WenWen model model does not respect fully some basic physical does not respect fully some basic physical energy principles, so it should be used very energy principles, so it should be used very carefully)carefully)
•• Elastic nonElastic non--linearity,linearity, not evolving, not evolving, often generated by local damages, often generated by local damages, cracks, soft contact problems, cracks, soft contact problems, anolonomousanolonomous restrains. restrains.
Time Time domaindomain HilbertHilberttransformtransform
[ ] ∫+∞
∞− −== τ
ττ
πd
tytytyH )(1)(~)(
)(~)()( tyjtytY +=
Detection of Detection of elasticelastic nonnon--linearitylinearity: : euristiceuristic approachapproach
AnalyticalAnalytical signalsignal
Free decay analytical signal for linear behaviourFreeFree decaydecay analyticalanalytical signalsignal forfor linearlinear behaviourbehaviour
Force
displacement
K1
K1
-100-50
050
100
-100-50
050
1000
0.5
1
1.5
2
2.5
3
Time (s)
y(t) y(t)_ -150 -100
-150
-100
-50
0
50
100
Im[Y
(t)]
ekr
y
ekn
k1
ykl
ykjwir wji
yk1
ykj
ykl
xkl
xki
xkp
ek1 ek1
ekr
ekn
NL1 Force
displacement
K1
aK1
K1aK1
NL2 Force
displacement
aK1
aK1
K1
K1
NL3
Force
displacement
aK1
K1
NL4 Force
displacement
aK1
K1
NL5
Force
displacement
aK1
K1
FreeFree decaydecay analyticalanalytical signalsignalforfor nonnon--linearlinear behaviourbehaviour
Each kind of elastic non-linear behavior has a different “signature” on the complex planeEach kind of elastic non-linear behavior has a different “signature” on the complex plane
Invariant moments of free decay plotsand correct prediction probabilityInvariantInvariant momentsmoments of of freefree decaydecay plotsplotsand and correctcorrect predictionprediction probabilityprobability
∫ ∫+∞
∞−
+∞
∞−−−= dxdyyxfyxqpU q
yp
x ),()()(),( μμ
)0,0()1,0(
MM
y =μ)0,0()0,1(
MM
x =μ
Two 3-layers NNs for two stages: invariant moments fill the input nodesnetwork 1; 26+10+6 nodes:identification of the presence and type of non-linearity network 2;52+15+3 nodes: quantification of non-linearity
NNNN--1 1 outcomesoutcomes: : examplesexamples fromfrom simulatedsimulatedexperimentsexperiments
a)V
11
V02
V12
V30
V31
V32
V33
-5.00E-02
0.00E+00
5.00E-02
1.00E-01
1.50E-01
LIN
NL1
NL2
NL3
NL4
NL5
b)
LIN
NL2
NL4
LIN
NL2
NL4
0
0.2
0.4
0.6
0.8
1
The probability of correct predictionby a trained Neural Network for asimple simulated case
The invariant moment distributionfor different non-linearity tipes
a) NL4 V
11
V02
V12
V30
V31
V32
V33
LIV1LIV2
LIV3-0.02
0
0.02
0.04
0.06
0.08
0.1
LIV1
LIV2
LIV3
b) NL4
LIV1 LIV2 LIV3
LIV1
LIV2
LIV3
0
0.2
0.4
0.6
0.8
1
c) NL5
V11
V02
V12
V30
V31
V32
V33
LIV1LIV2
LIV3-0.10
0.10.20.30.40.50.6
LIV1
LIV2
LIV3
d) NL5
LIV1 LIV2 LIV3
LIV1
LIV2
LIV3
0
0.2
0.4
0.6
0.8
1
NNNN--2 2 outcomesoutcomes: : examplesexamples fromfrom simulatedsimulatedexperimentsexperiments
Detection and Detection and simulationsimulation of of anan elasticelastic nonnon--linearitylinearity: : approachapproach byby Volterra Volterra expansionsexpansions
Time-Frequency domain identification holds in case of non-linear identification too.Time-Frequency domain identification holds in case of non-linear identification too.
Key-fact:
Time-localized (“instantaneous”) frequency domain analysis
Key-fact:
Time-localized (“instantaneous”) frequency domain analysis
Linear IdentificationLinear IdentificationFrequency domain Frequency domain Time domainTime domainTimeTime--Frequency domainFrequency domain
mm is the mass, is the nonlinear is the mass, is the nonlinear damping term and is the damping term and is the nonlinear elastic restoring forcenonlinear elastic restoring force
LetLet’’s assume and as: s assume and as: •• polynomial functions of polynomial functions of yy•• Depending from few unknown Depending from few unknown
parametersparameters
yy expandable in expandable in VolterraVolterra seriesseries
( )yfd
( ) ( ) ( )txyfyfym sd =++&&
( )yfd
D’Alembert equation for a SDOF systemD’Alembert equation for a SDOF system
( )yfs
( )yfs
Volterra Volterra seriesseries representationrepresentation in time in time domaindomain( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
..........................
,,
,
...
32132132133
21212122
11111
321
∫ ∫ ∫
∫ ∫
∫
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
⋅⋅⋅−⋅−⋅−⋅=
⋅⋅−⋅−⋅=
⋅−⋅=
+++=
τττττττττ
ττττττ
τττ
dddtxtxtxhty
ddtxtxhty
dtxhty
tytytyty
For a Volterra system there must be convergence conditions to guarantee that the description is meaningful, involving generally bounds on the time interval and the input
InstantaneousInstantaneous identificationidentificationApplying the definition of the STFT to the Applying the definition of the STFT to the VolterraVolterra series expansion:series expansion:
( ) ( ) ( )
( ) ( ) ( )( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ...,,
,
...
,
23213213213
22121212
21111
2321
2
+−⎟⎟⎠
⎞⎜⎜⎝
⎛−−−+
+−⎟⎟⎠
⎞⎜⎜⎝
⎛−−+−⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
=−+++=
=−=
∫ ∫ ∫ ∫
∫ ∫ ∫∫ ∫
∫
∫
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
+∞
∞−
ττττττττττττττ
ττττττττττττττττ
τττττ
τττ
τπ
τπτπ
τπ
τπ
detwdddxxxh
detwddxxhdetwdxh
detwyyy
detwyftD
fi
fifi
fi
fi
If an If an analytical analytical form for form for
functions h1, functions h1, h2, h3h2, h3…… can can
be be hypothesized, hypothesized,
then it is then it is possible to possible to resort to a resort to a parametric parametric
method.method.
W(τ-t) is a time-domain
window function
IdentificationIdentification of Volterra of Volterra systemssystems
In structural engineering applications it is not possible to obtain a number of experimental measurements as large as necessary to estimate the statistical quantities of interest especially when the dynamic tests are conducted in situ
The availability of a limited number of experimentalmeasurements can be obviated by taking into account the “localisation” in time of the frequency content introducing a timetime--frequencyfrequency descriptiondescription of the signals themeselves
Kernels are estimated through statistical moments of spectra obtained directly from experimental time series
InstantaneousInstantaneous identificationidentificationBASIC ASSUMPTIONS
•System’s response measured in N instants •A vector p of unknown parameters governs damping and elastic restoring forces
( ) ( ) ( )∑−
=
⎟⎠⎞⎜
⎝⎛ −=
1
0
2*2** ,,,N
msob mnDmnDnF p
Fob describes the difference between the instantaneous energy of the signal at time t*=n*Dt and the instantaneous energy of the signal corresponding to a given configuration of the unknown parameters p, given by a Volterra series approximation
The parameters pcan be identified by convenient optimization procedures
Objective function to minimize
Case studyCase study
Instantaneous identification of Instantaneous identification of rigid bodies on nonlinear support rigid bodies on nonlinear support
based on based on VolterraVolterra series series representationrepresentation
ApplicationApplication domaindomain limitslimitsThe following procedure applies:to the case, frequent in practice, of relatively
soft and dissipating contact between upper block and lower non-linear elastic support. In such case the contact surface can be reduced during motion, but dynamic rebound effect are not present and the static push-over definition of the constitutive is well fit to the dynamic case too.
to the case, frequent in practice, of relatively soft and dissipating contact between upper block and lower non-linear elastic support. In such case the contact surface can be reduced during motion, but dynamic rebound effect are not present and the static push-over definition of the constitutive is well fit to the dynamic case too.
The following procedure applies:The following procedure applies:
The following procedure
does not generally apply
The following procedure
does not generally apply
to the case of hard and low dissipating contact between upper block and lower support. In such case the dynamic rebound and sudden change in rotation centre requires a more accurate cinematic model
Hor. acceleration
time
Masonry on mortar
or ground
Stone on stone
InstantaneousInstantaneous identificationidentification of a of a block block structurestructure
A non-linear structure made of twoblocks is considered
The blocks interacts one to eachother via a contact surfaces withoutstrength in traction
Boundary conditions make possibleonly the relative rotation of the upper block (non-linear inverse pendulum)
InstantaneousInstantaneous identificationidentification of a of a block block structurestructure
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
x 10-3
-6000
-4000
-2000
0
2000
4000
6000
Displacement - X direction (m)
Forc
e - X
dire
ctio
n (N
)
As loads increase opening can happen and the structure exhibit a non-linear behaviour in the force-displacement law
InstantaneousInstantaneous identificationidentification of a of a block block structurestructure
To check the results of identification, the force-displacement law hasbeen approximated via a cubic polynom
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
x 10-3
-6000
-4000
-2000
0
2000
4000
6000
Displacement - x direction (m)
Forc
e - x
dire
ctio
n (N
)
ADINAP3
( ) 3126331 10673,310193,8 xxxkxkxk ⋅−⋅=+=
InstantaneousInstantaneous identificationidentification of a of a block block structurestructure
0 5 10 15 20-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
Time (s)
Ext
erna
l loa
d (N
)
0 5 10 15 20-3
-2
-1
0
1
2
Time (s)S
yste
m a
ccel
erat
ion
(m/s
2 )
System’s responseExternal load
InstantaneousInstantaneous identificationidentification of a of a block block structurestructure
0 5 10 15 20-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Time (s)
Gro
und
acce
lera
tion
(m/s
2 )
0 5 10 15 20-3
-2
-1
0
1
2
3
Time (s)S
yste
m a
ccel
erat
ion
(m/s
2 )
Seismic ground acceleration System’s response
InstantaneousInstantaneous identificationidentification of a of a block block structurestructure
The instantaneousestimators of ζ and fare charcterised by a certain stability over the exact value, whileestimator of k3 is muchmore unsettled
The mean value of k3is quite different fromthe exact value, but itis sufficient to reach a better model of the system’s dynamic
10 10.2 10.4 10.6 10.8 11-3
-2
-1
0
1
2
3
Time (s)
Acc
eler
atio
n (m
/s2 )
System responseidentified responselinear response
Reliability of a structure Reliability of a structure RR((tt) ) •• probability that the time to reach a probability that the time to reach a
reference limit state, reference limit state, ttbb , is greater than a , is greater than a given time t:given time t:
Hazard function h(t) • instantaneous rate of reliability
deterioration
)()( bttPtR ≤=
( );lim)(
0 ttttttP
th bb
t Δ≥Δ+<
=→Δ
Reliability and hazard as stochastic process
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛∫−=t
dxh(x)tR0
exp)(
.
.
JumpingJumping into the symptom spaceinto the symptom space
∫∞
==≤=S
dSfSbSSPSR S) valuesuitable()(
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛∫−=S
dxxhSR0
)(exp)(
This formulation includes continuous time (slow degradation) or/and discrete time processes (earthquakes, storms etc.), given that time and symptom evolution can be correlated by suitable lows.
Condition Monitoring is essentially a Condition Monitoring is essentially a search for structural or material disease search for structural or material disease
symptomssymptoms. . Symptoms can be regarded as Symptoms can be regarded as evolutionary and sudden changes in evolutionary and sudden changes in observable qualitative properties and/or observable qualitative properties and/or measurable responses. measurable responses. Symptoms search can require a Symptoms search can require a knowledge based direct search or model knowledge based direct search or model based predictive assessment. In both based predictive assessment. In both cases a stochastic procedure is needed.cases a stochastic procedure is needed.In some applications direct search and In some applications direct search and model based simulations can provide an model based simulations can provide an integrated procedure. integrated procedure.
The HolyThe Holy--Shroud Chapel caseShroud Chapel caseDesigned by Designed by GuarinoGuarinoGuariniGuarini, built from , built from 1667 to 1694 1667 to 1694 Heavily damaged by fire in Heavily damaged by fire in 19971997
The The numericalnumerical modelmodel
ThermalThermal AnalysisAnalysis ((firefire effecteffectsimulationsimulation))
Deformed state after fireStress concentrations
(confirmed by direct observation)
Damage scenarioDamage scenarioDamage scenario=state of structure Damage scenario=state of structure caused by an expected damage caused by an expected damage configurationconfiguration
SOME KNOWLEDGE BASED DAMAGE SCENARIOS ARE DEFINED IN SOME KNOWLEDGE BASED DAMAGE SCENARIOS ARE DEFINED IN DETERMINISTIC WAY;DETERMINISTIC WAY;
MANY OTHERS ARE THEN RANDOMLY GENERATED TROUGH STOCHASTIC MANY OTHERS ARE THEN RANDOMLY GENERATED TROUGH STOCHASTIC CHANGES IN MECHANICAL PARAMETERSCHANGES IN MECHANICAL PARAMETERS
EACH DAMAGE SCENARIO EACH DAMAGE SCENARIO DSkDSk PRODUCES A SET PRODUCES A SET
OF SENSITIVE SYMPTOMSOF SENSITIVE SYMPTOMS
Symptom vectors are generatedSymptom vectors are generated
From Damage Scenarios to Symptom vectorsFrom Damage Scenarios to Symptom vectors
DSDSkk S(S(k,1k,1)) S(S(k,2k,2)) S(S(k,3k,3)) S(S(k,Nk,N))
Damage Damage Indexes VectorIndexes Vector Symptom Observation MatrixSymptom Observation MatrixSymptom Observation MatrixSymptom Observation Matrix
INVERSE APPROACH (ILL CONDITIONED)INVERSE APPROACH (ILL CONDITIONED)
•• SensitivitySensitivity--oriented problemoriented problem--size reduction size reduction neededneeded
From Symptom vectors to Damage ScenariosFrom Symptom vectors to Damage Scenarios
Principal Principal Component searchComponent search
Proper Orthogonal Proper Orthogonal DecompositionDecomposition
Equivalent Equivalent actions based actions based
on SVDon SVD
INVERSE APPROACH (ILL CONDITIONED)INVERSE APPROACH (ILL CONDITIONED)
•• SensitivitySensitivity--oriented problemoriented problem--size reduction size reduction needed (needed (CempelCempel’’ss approach)approach)
S(S(k,1k,1)) S(S(k,2k,2)) S(S(k,3k,3)) S(S(k,Nk,N)) MMkk
Symptom Observation MatrixSymptom Observation Matrix MODELSMODELS
DIRECT APPROACH DIRECT APPROACH •• Many models are compared to select the Many models are compared to select the bestbest--fittingfitting--one (Ian Smith)one (Ian Smith)
Modal Identification after the main vibration test campaignModal Identification after the main vibration test campaign(TFIE approach)(TFIE approach)
modal frequencies are marked by the downward peaksmodal frequencies are marked by the downward peaks
X direction Y direction
First two modal shapesFirst two modal shapes
First mode
Second mode
The stochastic Model Updating process at a glanceThe stochastic Model Updating process at a glance
SeeSee, , e.ge.g.: .: I.F.C. Smith, I.F.C. Smith, MultiMulti--Model Interpretation of Measurement Data Model Interpretation of Measurement Data
with Errorswith Errors CANSMART 2005, RMC, Canada, 2005, pp 13CANSMART 2005, RMC, Canada, 2005, pp 13--2222
Objective (Target) function:Objective (Target) function:•• As usually, a (quadratic) function based on As usually, a (quadratic) function based on
average error between the computed and average error between the computed and measured output parameters.measured output parameters.
Process organized in two phasesProcess organized in two phases1.1. MultiMulti--model generation, premodel generation, pre--selection selection
and grouping;and grouping;2.2. Best fitting model final selection .Best fitting model final selection .
Model Updating in progress. Variable selected in a preliminary tentative process:•3D Livello02-2 (Variable S19)•Tamburo Esterno (Variable S11)•Timpani (Variable S6)•3D Livello 03 (Variable S18)•Tamburo Interno (Variable S12)
13
5
S1 S3 S5 S7 S9 S11 S13 S1
5S
17 S19
S21 S2
3S
25
0
0.05
0.1
0.15
0.2
0.25
Sensitivity
ModesParameters
Sensitivity Analysis - Modes 1-5
PHASE 1PHASE 1
Probabilistic Global Search LausanneProbabilistic Global Search Lausanne (PGSL)(PGSL)IITERATIVE PROCESS INCLUDING FOUR ENCASED LOOPSTERATIVE PROCESS INCLUDING FOUR ENCASED LOOPS..
STARTING UPSTARTING UPEach model is a point in Each model is a point in RRNN Input Variables Space Input Variables Space (N floating variable input parameters);(N floating variable input parameters);A set of M points are randomly generatedA set of M points are randomly generatedEach point has the same joint probability densityEach point has the same joint probability densityThe range of each coordinate of each point is The range of each coordinate of each point is divided into divided into p p equal intervals with the same equal intervals with the same probabilityprobability
SECOND STEPSECOND STEPAdaptive probabilistic new models generationAdaptive probabilistic new models generation
Conceptual core: Conceptual core: SEARCH FOR A BETTER MODEL NEAR TO A GOODSEARCH FOR A BETTER MODEL NEAR TO A GOOD--ONEONE
Extract the best model (target function minimum) Extract the best model (target function minimum) Increase Increase the probability density in the range the probability density in the range interval of each coordinate about the point interval of each coordinate about the point representing the best model; representing the best model; decreasedecrease the the probability density in the other intervals, as much probability density in the other intervals, as much as the distance from the best point is large.as the distance from the best point is large.
Functionally similar to simulated annealingFunctionally similar to simulated annealing
THIRD STEPTHIRD STEPRefining the search and generating new modelsRefining the search and generating new models
Intervals with the highest probability Intervals with the highest probability density parted into equal subdensity parted into equal sub--intervalsintervalsProbability density distribution Probability density distribution refinedrefinedNew models generation driven by the New models generation driven by the updated probability distributionupdated probability distributionIterations to fulfill preIterations to fulfill pre--defined defined criteriacriteria
PHASE 2PHASE 2Evaluating modelsEvaluating models
STEP 1: preliminary filteringSTEP 1: preliminary filtering•• Discarding all models exceeding a Discarding all models exceeding a
““penaltypenalty”” thresholdthreshold defined on base of defined on base of target and additional constraints. target and additional constraints.
STEP 2: Reducing the size of STEP 2: Reducing the size of models and problem :models and problem :•• Proper Orthogonal Decomposition (or Proper Orthogonal Decomposition (or
SVD)SVD) on linear combinations of the input on linear combinations of the input parameters allows to reduce the size of the parameters allows to reduce the size of the space in which the models are described. space in which the models are described.
•• The influence of the input parameters on the The influence of the input parameters on the Target function can be assessed by few Target function can be assessed by few principal coordinates principal coordinates
STEP 3: Reducing the number of STEP 3: Reducing the number of modelsmodels•• Clustering (KClustering (K--means technique) makes it possible means technique) makes it possible
to collect the surviving models into groups. to collect the surviving models into groups. •• Models are associated to the cluster having the Models are associated to the cluster having the
centroidcentroid at the minimum Euclidean distance.at the minimum Euclidean distance.•• Due to the association of a new model, the Due to the association of a new model, the
centroidcentroid location evolves graduallylocation evolves gradually..
ClusteringClustering INITIALIZING: defining k initial centroids
ASSIGNING: each representative point is assigned to the cluster of the nearest centroid
RELOCATING CENTROIDS: new centroidsgenerated
CHECKING: |cnew-cold|<tolerance OR maximum number of iterations reached?
YESNO
STOP
The output parameters of the 5 The output parameters of the 5 centroidscentroids
Now we can proceed to finally select the best fitting model (minimum target value)
Or we can associate to each centroid a probability level related to the target value (bayesian estimator)
E_6
E_1
1
E_1
2
E_18
E_19
M_1
M_40,00E+005,00E+08
1,00E+09
1,50E+09
2,00E+09
2,50E+09
3,00E+09
M_1
M_2
M_3
M_4
M_5
ConclusionsConclusionsThe multiThe multi--model approach (Smith, EFPL) is a direct approach to model approach (Smith, EFPL) is a direct approach to model updating through a generationmodel updating through a generation--selection of models. It selection of models. It makes it dual of the inverse model updating and somehow makes it dual of the inverse model updating and somehow comparable to a GA approach.comparable to a GA approach.
++Not ill Not ill conditionnedconditionnedStochastically oriented, potentially usable as a Bayesian Stochastically oriented, potentially usable as a Bayesian approach.approach.Potentially robustPotentially robust
==No reliable updates if initial basic models not fitNo reliable updates if initial basic models not fit
--Reliability of results strongly influenced by the choice of the Reliability of results strongly influenced by the choice of the objective function. Wrong choice of the OF can cause objective function. Wrong choice of the OF can cause ambiguous outcomes and numerical noise, ambiguous outcomes and numerical noise, These problems can be avoided building the OF initially with These problems can be avoided building the OF initially with few highly sensitive parameters and then assessing the few highly sensitive parameters and then assessing the effect of the other parameters in a hierarchical iterative effect of the other parameters in a hierarchical iterative procedure.procedure.
ConclusionsConclusionsLargely uncertain data require distributed Largely uncertain data require distributed sensing. Onsensing. On--line dynamic testing is mainly line dynamic testing is mainly ambient testing, requiring robust identification ambient testing, requiring robust identification techniques and redundant information. techniques and redundant information. Redundant information requires lowRedundant information requires low--cost sensing cost sensing systems, with reduced needing of energy supply systems, with reduced needing of energy supply and protection from electric spikes and magnetic and protection from electric spikes and magnetic fields. optical sensors and MEMS fulfill such fields. optical sensors and MEMS fulfill such needingneedingAmong others, Among others, pecialpecial sensors are under study as sensors are under study as part of hierarchical sensor subpart of hierarchical sensor sub--systems to be systems to be placed in noisy locations.placed in noisy locations.
THEN, THANK YOU FOR YOUR PATIENCE!THEN, THANK YOU FOR YOUR PATIENCE!