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!