ETH Chair of Structural Mechanics Structural Identiﬁcation ...

ETH Chair of Structural Mechanics

Structural Identification & Health Monitoring

Lecture 11: Nonparametric Identification B – correlation analysis

Dr. V.K. Dertimanis & Prof. Dr. E.N. Chatzi

Outline

Structural identificationStochastic modellingThree fundamental resultsImplementationFurther ReadingAppendix

ETH Chair of Structural Mechanics 25.03.2020 2

Structural identificationMain concepts

What do we know thus far

- A vibrating structure is completely characterized by its impulseresponse g(t) (time domain), its transfer function G(s) (Laplacedomain) and its frequency response function G(f ) (FRF,frequency domain)

- There’s a one-to-one correspondence among these quantities

- If the sampling theorem is satisfied, the digital counterparts ofthese quantities are sufficient approximations of the structure indiscrete-time



What do we know thus far

- In real life, data arrives always in digital form

- When we measure a structure, we gain availability of the excitations(optional) u[k ] and the responses x[k ] for k = 1, . . . ,N

- We have to infer decisions based exclusively on this data

- Decisions −→ natural frequencies, damping ratios and modeshapes (e.g. vibration modes)



The structural identification procedure

- Measure a structure

- Estimate a quantity (impulse response, transfer function, FRF)

- Decide on the number of vibration modes

- Calculate the vibration modes


Stochastic modellingRationale

Real life signals...

... cannot be expressed by any mathematical function!

... require a statistical framework for their characterization


Stochastic modellingRationale

Main goals of stochastic modelling:

1. The description/analysis of stochastic signals and systems.

2. The prediction of future values of stochastic signals, based onprevious and current data.

3. The diagnostic checking of systems for change detection and/orfault diagnosis.

4. The control of stochastic dynamical systems.


Stochastic modellingClassification1

1.3. Classification of digital signals 13

STOCHASTIC

SIGNALS

STATIONARY NON – STATIONARY

ERGODIC NON – ERGODIC

VARIOUS CLASSES OF

NONSTATIONARITY

Figure 1.11 Classification of stochastic signals.

given by

µx[tk] = limN→∞

N∑

j=1

xtk [j] (1.17)

γx[τ ] = limN→∞

N∑

j=1

(xtk [j] − µx[tk]

)·(xtk+τ [j] − µx[tk + τ ]

)(1.18)

Using the moments described by Equations 1.17–1.18, Figure 1.11 illustrates a classification of

stochastic signals.

1.3.2.1 Stationary stochastic signals

They are realizations of stochastic processes whose mean value and autocovariance are invariant

in time. In this case the stochastic process is called weakly stationary. If all the distributions of

its random variables are invariant in time, the process is called strongly stationary.

Usually, in real–world applications only one time–series are available (that is a single re-

alization of the process). In this case, the statistical moments are estimated on the basis of

the available data. For the kth time–series of a stochastic process, the mean value and the

autocovariance are given by the estimators

µk =1

N·

N∑

t=1

xk[t] (1.19)

γk[τ ] =1

N·

N∑

t=1

(xk[t] − µk

)·(xk[t + τ ] − µk

)(1.20)

where N the data length of the time–series. When the quantities estimated by Equations 1.19–

1.20 are invariant in k (that is, when they are estimated using another time–series of the same

stochastic process), then the process is called ergodic. Stationary ergodic time–series allow the

1Review the introductory presentation on probability, random variables and stochasticprocesses in Supplementary Material Table


Stochastic modellingNonparametric representations

Sample functions u[k ] and x [k ] for k = 1, 2, . . . ,N, and for samplingperiod Ts (sampling frequency Fs = 1/Ts).

Nonparametric estimators:1. Mean values2. Probability density functions3. Auto/cross–correlation functions4. Auto/cross spectral densities ETH Chair of Structural Mechanics 25.03.2020 9


Mean values

µu =1N

N∑i=1

u[i ]

µx =1N

N∑i=1

x [i ]

- Unbiased estimator

- Upon availability of a sample function, we always subtract the meanbefore further processing



Probability density function



Probability density functionData range [α, β]Data intervals KInterval width W = β−α

KInterval limits di = α + iW , i = 0, 1, . . . ,KInterval data counts Ni (e.g. histogram)PDF

p[i ] =Ni

NW=

Ni

NK

β − α



Probability density function


Stochastic modellingCorrelation

u[k ], x [k ] −→ discrete-time stochastic processes

Auto/cross-correlation functions

γuu[h] = E{u[k + h]u[k ]}γxx [h] = E{x [k + h]x [k ]}γxu[h] = E{x [k + h]u[k ]}

h −→ time lag


Stochastic modellingCorrelation


Three fundamental resultsRationale

deterministic excitation −→ closed-form solution for the response

stochastic excitation −→ no closed-form solution for the response

stochastic excitation −→ stochastic response

stochastic response −→ fully characterized by µx and γxx [h]

zero-mean stochastic excitation −→ zero-mean stochastic response

question −→ closed-form expression for γxx [h]?


Three fundamental resultsResult 1

The result

γxx [h] is related to γux [h] by the same discrete-time convolution, asthe one between the response and the excitation

The equationγxx [h] = g[h] ∗ γux [h]

The proof −→ Appendix



The result

γux [h] is related to γuu[h] by a “similar” discrete-time convolution,as the one between the response and the excitation

The equationγux [h] = g[−h] ∗ γuu[h]




The result

γxx [h] is related to γuu[h] by a discrete-time convolution that isexclusively based on the impulse response between the responseand the excitation

The equationγxx [h] = gx [h] ∗ γuu[h]



Three fundamental resultsAn important special case

Zero-mean Gaussian white noise excitation

µu = 0

γuu[h] = σ2uuδ[h] =

σ2uu, h = 0

0 , h 6= 0

What happens when we “pass” white noise through the structure?


Three fundamental resultsAn important special case

Modal decomposition of the digital impulse response

g[k ] =2n∑`=1

R`pk` , k = 0, 1, . . .

R` → digital residues, p` → digital poles

Then→ modal decomposition of the response’s autocorrelation function2

γxx [h] =2n∑`=1

d`ph`

2Proof in AppendixETH Chair of Structural Mechanics 25.03.2020 21

Three fundamental resultsExample

Digital impulse response of the underdamped SDOF system

g[k ] =2∑

m=1

Rmpkm ≡ (TsR1)(eλ1Ts )k + (TsR2)(eλ2Ts )k

=Ts

mωde−ζnωnkTs sin(ωdkTs)

Continuous-time residues/poles (Lecture 5)

R1,2 = ∓ 12mωd

j and λ1,2 = −ωnζn ± ωd j



Autocorrelation function of the response

γxx [h] =2∑

q=1

dqphq ≡ d1(eλ1Ts )h + d2(eλ2Ts )h

for

d1 = σ2uu

{T 2

s R21

1− p21

+T 2

s R1R2

1− p1p2

}, d2 = σ2

uu

{T 2

s R22

1− p22

+T 2

s R1R2

1− p1p2

}Doing the algebra

γxx [h] = 2|d |e−ζnωnhTs cos(ωdhTs + θ)



Impulse response

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

0

1

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

0

1ETH Chair of Structural Mechanics 25.03.2020 24


Autocorrelation function0 1 2 3 4 5 6 7 8 9 10-1

0

1

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

0

1


ImplementationEstimation

Sample functions u[k ] and x [k ] for k = 1, 2, . . . ,N, and for samplingperiod Ts (sampling frequency Fs = 1/Ts)

γ̂uu[h] =1N

N−h∑i=1

u[i + h]u[i ]

γ̂xx [h] =1N

N−h∑i=1

x [i + h]x [i ]

γ̂xu[h] =1N

N−h∑i=1

x [i + h]u[i ]


ImplementationWhiteness test

White noise −→ sample autocorrelation within the ±1.96/√

N zone

.[h]

.[0]

h


Further Reading

1. Papoulis, A. (1991), Probability, Random Variables & Stochastic Processes, 3thEd., McGraw–Hill, New York, USA.

2. Bendat, J.S. and Piersol, A.G. (2010), Random Data: Analysis and MeasurementProcedures, 4th Ed., John Wiley & Sons Ltd., Chichester, UK.


AppendixProof of Result 1

From Lecture 8 (slide 8):

x [k ] =∞∑

i=−∞g[i ]u[k − i ], k = 0, 1, . . .

u[k ] −→ weakly stationary and weakly ergodic process



Shift h time steps forward, multiply by x [k ] and apply the expectationoperator

E{x [k + h]x [k ]} = E{ ∞∑

i=−∞g[i ]u[k + h − i ]x [k ]

}

E{x [k + h]x [k ]} −→ γxx [h]



Right-hand side3

E{ ∞∑

i=−∞g[i ]u[k + h − i ]x [k ]

}=∞∑

i=−∞g[i ]E{u[k + h − i ]x [k ]}

=∞∑

i=−∞g[i ]γux [h − i ]

3Observe that (i) the impulse response g[i ] is a deterministic function, so it can jumpout of the expectation operator; and (ii) both the expectation and the sum operators arelinear, so they can be interchanged



Thus

γxx [h] =∞∑

i=−∞g[i ]γux [h − i ]

orγxx [h] = g[h] ∗ γux [h]



From Lecture 8 (slide 8):

x [k ] =∞∑

i=−∞g[i ]u[k − i ], k = 0, 1, . . .

u[k ] −→ weakly stationary and weakly ergodic process



Multiply u[k + h] and apply the expectation operator

E{

u[k + h]x [k ]}

= E{

u[k + h]∞∑

i=−∞g[i ]u[k − i ]

}

E{u[k + h]x [k ]} −→ γux [h]



Right-hand side

E{

u[k + h]∞∑

i=−∞g[i ]u[k − i ]

}= E

{ ∞∑i=−∞

g[i ]u[k + h]u[k − i ]}

=∞∑

i=−∞g[i ]E{u[k + h]u[k − i ]}

=∞∑

i=−∞g[i ]γuu[h + i ]



Then

γux [h] =∞∑

i=−∞g[i ]γuu[h + i ]

=∞∑

m=−∞g[−m]γuu[h −m]

= g[−h] ∗ γuu[h]

(Note: the i = −m change of variable has been applied)



Result 1 −→ γxx [h] = g[h] ∗ γux [h]

Result 2 −→ γux [h] = g[−h] ∗ γuu[h]

Then

γxx [h] = g[h] ∗(g[−h] ∗ γuu[h]

)=(g[h] ∗ g[−h]

)∗ γuu[h]

= gx [h] ∗ γuu[h]


AppendixModal decomposition of the response’s autocorrelation function

Result 3 −→ γxx [h] = gx [h] ∗ γuu[h]

White noise excitation −→ γuu[h] = σ2uu[h]

Plug into Result 3

γxx [h] = σ2uu(gx [h] ∗ δ[h]

)= σ2

uu

∞∑i=−∞

gx [i ]δ[h − i ]

δ[h] −→ all terms are cancelled, except the i = h one



Then

γxx [h] = σ2uugγ [h] = σ2

uu

∞∑i=−∞

g[i ]g[i + h]

Impulse response

g[k ] =2n∑

m=1

Rmpkm

Substitute to γxx [h] (recall that g[k ] = 0 for k < 0)



Then

γxx [h] = σ2uu

∞∑i=0

2n∑m=1

Rmpim

2n∑`=1

R`pi+h` = σ2

uu

2n∑m=1

2n∑`=1

RmR`ph`

∞∑i=0

(pmp`)i

= σ2uu

2n∑m=1

2n∑`=1

RmR`ph`

∞∑i=0

(pmp`)i =2n∑`=1

( 2n∑m=1

RmR`σ2uu

1− pmp`

)ph`

=2n∑`=1

d`ph`


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