Autogressive Model for Structural Condition

Assessment in Presence of Parametric

Uncertainty

Koushik Roy and Samit Ray-Chaudhuri

Abstract Long-term and often short-term vibration-based structural health

monitoring data show variations in response even though there is no visible change

in structural properties. Although these variations are attributed to environmental

factors causing change in structural parameters, such uncertainties in structural

parameters make the condition assessment of a structure difficult. In this study, the

autoregressive (AR) model, where the coefficients of the model are related to

structural model parameters and are considered as one of the efficient tools often

used in modal identification and damage detection, is used to investigate its

efficiency in damage detection and localization when parametric uncertainties are

present. A numerical study conducted with an eight-story shear building, where

uncertainties in stiffness are assumed in terms of known probability density

functions, shows that the AR model is highly efficient in damage detection and

localization even when significant parametric uncertainties are present. For this

purpose, damage is being induced in a particular story, and the response is analyzed

with the autoregressive model to gauge the efficiency of the model. To broaden the

practical applicability of the method when noise is present in the measurement data,

the Kalman filter approach has been adopted and successfully shown to handle the

noisy data.

Keywords Autoregressive model • Uncertainty • Damage detection • Sensitivity

analysis

K. Roy • S. Ray-Chaudhuri (*)

Indian Institute of Technology Kanpur, Kanpur, India

e-mail: koushik@iitk.ac.in; samitrc@iitk.ac.in

S. Chakraborty and G. Bhattacharya (eds.), Proceedings of the International Symposiumon Engineering under Uncertainty: Safety Assessment and Management (ISEUSAM - 2012),DOI 10.1007/978-81-322-0757-3_73, # Springer India 2013

1061

1 Introduction

Vibration-based structural damage detection has drawn considerable attention in

recent years due to its nondestructive nature. Since the last few decades, several

methodologies have been developed for this purpose. Brincker et al. [1] proposed afrequency domain method for structural modal identification of output-only

systems. Sohn et al. [15] applied statistical pattern recognition paradigm in struc-

tural health monitoring. Kiremidjian et al. [9] adopted an enhanced statistical

damage detection algorithm using time series analysis. Caicedo et al. [2]

demonstrated the efficacy of natural excitation technique (NExT) and eigensystem

realization algorithm (ERA) by applying on simulated data generated from the

popular IASC-ASCE benchmark problem. Lu and Gao [12] proposed an efficient

time-domain technique based on autoregressive prediction model for structural

damage localization. Later they extended their study to quantify damage with

noisy signal with the help of Kalman filter-based algorithm. Samuel da Silva

et al. [14] applied autoregressive AR-ARX models and statistical pattern recogni-

tion on damage detection of a structure. Xiaodong et al. [16] took a practical model

excited with simulated ambient signal to identify the modal parameters using

NExT-ERA. Cheung et al. [4] validated experimentally the statistical pattern

recognition methods for damage detection to field data. Gao and Lu [8] set up a

proposition with acceleration residual generation for structural damage identifica-

tion. Le and Tamura [10] used two frequency domain techniques—frequency

domain decomposition and wavelet transform for modal identification from ambi-

ent vibration data. Liu et al. [11] explained the usage of extended Kalman filter in

health monitoring of linear mechanical structures. Chiang et al. [5] put forwardERA with its modified form to identify modal parameters from ambient vibration

data. Chiang and Lin [6] stated NExT-ERA in complete time domain with correla-

tion technique. Caicedo [3] provided some practical guidelines for the NExT-ERA

approach of modal identification using ambient vibration data. There are many

other prediction models for structural damage detection [13] based on neural

network, fuzzy logic, genetic algorithm, etc. These models have been used success-

fully with mostly simulated data.

In addition to deterministic methodologies in damage detection, stochastic

damage detection procedures attract equal attention due to their capability in

dealing with structural parametric uncertainties and noise in measured data. Struc-

tural parametric uncertainty can be classified in two categories: epistemic and

aleatory. Epistemic uncertainty deals with the errors associated with measurement

noise, whereas aleatory uncertainty deals with inherent modeling errors, i.e., the

error due to parametric variation. From a different perspective, uncertainties in

structural response in terms of errors can be categorized as biased errors and

random errors. Random errors are evidently with zero mean and usually expected

to follow Gaussian distribution. Measurement noise is also assumed as a zero mean

random process. For uncertainty in structural parameters such as mass and stiffness,

the response of a structure is of biased uncertain nature, which can be dealt with

1062 K. Roy and S. Ray-Chaudhuri

probability density functions by applying the reverse procedures as proposed by Xu

et al. [17]. Zhang et al. [20] developed a probabilistic damage detection approach

for output-only structures with parametric uncertainties.

In this chapter, the behavior of an autoregressive model is investigated in

presence of structural parametric uncertainties. An autoregressive model is one of

the most popular models for safety assessment of a structure. The model is used to

predict future possible data of a signal based on its previous data. The coefficient of

this model is directly related to the modal properties of the structure. The following

section describes the details of the autoregressive model.

2 Model Description

For a system with n degrees-of-freedom (DOF) having mass, stiffness, and damping

matrices M0, K0, and C0, respectively, the equations of motion can be expressed as

follows:

M0€xðtÞ þ C0

_xðtÞ þ K0xðtÞ ¼ L0uðtÞ€xðtÞ þ J _xðtÞ þ KxðtÞ ¼ LuðtÞ

Where J ¼ M�10 C0;K ¼ M�1

0 K0 ¼ FLFT ; and L ¼ M�10 L0 with L0 being the input

coefficient or influence matrix with dimensions n by m. The defining equation of theautoregressive model thus takes the following form:

yðtÞ ¼Xp

i¼1

fiyðt� iÞ þ eðtÞ (2)

where fi ¼ ith coefficient of the model, y(t) ¼ signal at time step t to be predicted,e(t) ¼ model residual error, and p ¼ order of the model.

A modified form of this prediction model, called autoregressive model with

exogenous (ARX) input, is employed here mainly for the purpose of damage

localization. The model expression relating among input data u(t), output data y(t), and residual error e(t) in the time domain can be written as

AðqÞyðtÞ ¼ BðqÞuðtÞ þ CðqÞeðtÞ (3)

where A, B, and C are polynomials in the delay operation q�1 and can be expressed

as follows:

AðqÞ ¼ 1þ a1 q�1 þ a2 q

�2 þ � � � þ a1na q�na

BðqÞ ¼ 1þ b1 q�1 þ b2 q

�2 þ � � � þ bnb q�nbþ1

CðqÞ ¼ 1þ c1 q�1 þ c2 q

�2 þ � � � þ anc q�nc

Autogressive Model for Structural Condition Assessment. . . 1063

In Eq. (4), the numbers na, nb, and nc are the order of the respective

polynomials. Based on these orders, the autoregressive model can be categorized

as AR model, ARMA model, ARX model, and ARMAX model.

3 Theory

Due to lack of space, a detailed mathematical derivation is not presented here.

Rather, a very brief theoretical background of the new approach in time domain

to find out modal properties is discussed here. An autoregressive model with order

[2, 3, 0] is expressed as follows:

yðkÞ ¼X2i¼1

Aiyðk � iÞ þX3i¼1

Biuðk � iÞ (5)

According to the proposed method, only the first AR coefficient A1 is the only

parameter of interest, and A1 is given by

A1 ¼ 2f cos L12Dt

� �fT (6)

where

L ¼o2

1 � � �... . .

. ...

� � � o2n

264

375; cos L

12Dt

� �¼

cosðo1DtÞ � � �... . .

. ...

� � � cosðonDtÞ

264

375 (7)

Thus, A1 can be expressed as

A1 ¼ 2f

cosðo1DtÞ � � �... . .

. ...

� � � cosðonDtÞ

264

375fT (8)

4 Parametric Uncertainty

Due to variation in different environmental factors, model error, measurement error

etc., uncertainty is indispensible resulting in variation in structural responses. To

deal with uncertainty in structural parameters, various approaches are commonly

utilized. These are probability theory, fuzzy set theory, stochastic finite element

1064 K. Roy and S. Ray-Chaudhuri

method (FEM), interval analysis, control methodology, and meta-modeling.

Among them, the probability theory is the most popular and thus used in this

work to deal with parametric uncertainty. According to Xu et al. [17] and Zhang

et al. [20], uncertainty in structural parameters can be expressed in the form of a

certain distribution function. For the convenience of the numerical execution, in

this study, only normal distribution has been considered to model uncertainty in

structure parameters. It is also assumed that the structure is undamped or slightly

damped and of the linear time invariant (LTI) nature. Among various structural

parameters, only stiffness is taken as uncertain. The stiffness properties are

modeled using a truncated normal distribution (no negative side) with mean m as

deterministic stiffness value and a reasonable value of standard deviation s as

suggested in previous studies.

In this work, 1% of the mean value is taken as the standard deviation in the

numerical calculation (coefficient of variation is assumed as 0.01). The reason

behind choosing such a small standard deviation to represent uncertainty is the

assumption that the level of environmental perturbation is much less compared

to the intensity of damage. Figure 1 provides the probability density function of

particular story stiffness.

5 Detailed Procedure

Lu and Gao [12] made a new proposition to localize and quantify damage with the

help of Kalman filter using AR model. In their study, the major goal was to study

the influence of uncertainty in stiffness in detecting damage. In addition, a new

Fig. 1 Probability density function of story stiffness

Autogressive Model for Structural Condition Assessment. . . 1065

absolutely time-domain method [19] has also been proposed by them to determine

the modal properties from the response signal of an output-only system with the

help of autoregressive model coefficients. Further, the performance of AR model

under different damping level was also studied by them.

In this work, to localize damage, all the structural response is collected from

structure’s undamaged (reference) and damaged (unknown) states. Further, any one

of the output (structural response) is considered as an input as the ARX

(autoregressive model with exogenous input) model is employed to investigate

the position of damage. The residual error s of the damaged state with respect to the

reference state estimated from the model represents the deviation of the system

from its original condition. Hence, the standard deviation of this residual error s(e)works as a damage-sensitive feature (DSF), i.e., s(e) is more at a damaged spot in

this feature-based technique. Now, once uncertainty is present in an output signal

due to uncertainty in stiffness, the DSF varies and shows its competence to

undertake such circumstances.

Now, to quantify damage, an algorithm with Kalman filter as proposed by Gao

and Lu [7] is followed, where another DSF is assigned to be the ratio of the

aforementioned DSF for damage localization to the standard deviation of the output

response. That is,

DSF ¼ sðeÞsðyÞ � 100% (9)

The step-by-step algorithm to determine the modal properties of an MDOF

undamped system using the proposed time-domain technique is given as follows:

1. Collect the ambient vibration acceleration response data from each DOF.

2. Pick any one of these responses and consider it as base excitation of the

structure.

3. Then the base excitation and all the output acceleration responses are used for

the autoregressive-exogenous-input (ARX) model to calculate the model

coefficients for prediction.

4. All the coefficients contain the modal properties of the structure. The first

coefficient is taken to calculate the modal properties.

5. A singular value decomposition (SVD) is applied to calculate the matrix with

singular values as well as vectors.

6. The singular values are directly associated with the modal frequencies, and the

associated vectors are nothing but the mode shapes.

6 Numerical Results

Representative models of a six-story (Fig. 2) and two-story shear buildings

are considered for numerical investigation. The two-story model is employed

only when the Kalman filter [18] is used to denoise data and quantify damage.

1066 K. Roy and S. Ray-Chaudhuri

The six-story model is however used to show the efficacy of the autoregressive

model in damage localization in the presence of parametric uncertainty. For both

the buildings, the story stiffness is taken as 56.7 kN/m, and the mass of each floor is

taken as 400 kg (lumped mass).

For the two-story building, to show the proficiency as well as the performance of

ARMAX model (Kalman filter) in data denoising [7], several levels of noise

scenarios (10 and 20% in the input and output signals of models as well as state

conditions) are explored. Before considering the moving average part with

autoregressive model, the DSF for no damage condition of the structure is evaluated

and presented in Table 1a. Table 1b provides the DSF values after applying Kalman

filter. It can be observed from this table that the DSF values are significant and

increase with an increase in intensity of noise even though there is no damage.

However, after applying the Kalman filter, the DSF values become less than unity

highlighting the importance of Kalman filter in tackling noisy data.

For localization of damage, the six-story shear building is considered. Damage is

induced by reducing the stiffness of spring k3 by 20%. Here, DSF is chosen to be the

ratio of standard deviation of AR model residual error of damage state (ed) to the

Fig. 2 Shear building

Autogressive Model for Structural Condition Assessment. . . 1067

reference state (eu) for every DOF. Now, plot of DSF with DOF in Fig. 3 shows the

plot of DSF versus DOF, which can be used for the localization of damage in this

deterministic model.

Now a confidence interval of 95% for the probability density function of the

story stiffness of any floor k is considered, i.e., m�1.96s � k � m + 1.96s, wheres ¼ 0.01 m and m ¼ 56.7 kN/m (see Fig. 1). Following the same procedure as

explained in case of the deterministic model, DSF versus DOF plot is generated and

presented in Fig. 4. It can be observed from this figure that the damage can be

localized even when there is uncertainty in stiffness parameters.

To see if the ARX model can localize damage in presence of damping in a

structure, the same approach is applied. Figure 5 presents the DSF for different

Table 1 Comparison

between the DSF before and

after applying Kalman filter

(a) DSF before applying Kalman filter

Noise at reference model

Noise at state model 0% 10% 20%

0% 0.0030 2.7354 10.8861

10% 35.4070 34.3964 35.6530

20% 70.8712 69.8263 68.6789

(b) DSF after applying Kalman filter

Noise at reference model

Noise at state model 0% 10% 20%

0% 0.2132 0.1852 0.7252

10% 0.3236 0.2516 0.8720

20% 0.1295 0.3625 0.2404

Fig. 3 Localization with

DSF

1068 K. Roy and S. Ray-Chaudhuri

values of critical damping. Note that the damping is assumed to be same for all

modes. It can be observed from Fig. 5 that the DSF drops down reducing the

difference of DSF of DOFs adjacent to the damaged spring in comparison to that

of the undamaged one.

Now, applying the time-domain approach described previously, the results of the

numerical study are presented in Table 2, which shows the calculated frequencies

Fig. 4 Localization in

presence of uncertainty

Fig. 5 Performance of the model with increase damping

Autogressive Model for Structural Condition Assessment. . . 1069

along with the error. It can be observed from Table 2 that, in general, this error is

negligibly small for all damping values. However, for the fundamental mode, this

error is not that insignificant, especially in case of undamped system. Figure 6

provides a comparison of the actual mode shapes and the mode shapes estimated

using the proposed algorithm for the first three modes. It can be observed from this

figure that the mode shapes are very close to the actual values.

7 Conclusion

Uncertainty in structural parameters may result in variability of structural response.

In this study, the autoregressive (AR) model, where the coefficients of the model

are related to structural model parameters, is used to investigate its efficiency in

Table 2 Actual and calculated frequencies using the proposed algorithm

Actual (Hz) Calculated (Hz) Error (%) Calculated (Hz) Error (%)

Undamped Damping ¼ 1%

0.4570 0.4614 0.9456 0.6807 0.3285

1.3445 1.3443 �0.019 1.5008 0.1041

2.1539 2.1538 �0.005 2.3054 0.0657

2.8381 2.8381 0.00 2.9912 0.0512

3.3573 3.3575 0.0024 3.5124 0.0441

3.6815 3.6816 0.003 3.8351 0.0400

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

Mode Shape, Φ

DO

F

Mode Shape#1

-1.5 -1 -0.5 0 0.5 10

1

2

3

4

5

6Mode Shape#2

Mode Shape, Φ-1.5 -1 -0.5 0 0.5 1 1.50

1

2

3

4

5

6Mode Shape#3

Mode Shape, Φ

theoreticalcalculated (ξ=0%)calculated (ξ=1%)

Fig. 6 Comparison of estimated and actual mode shapes

1070 K. Roy and S. Ray-Chaudhuri

damage detection and localization when parametric uncertainties are present.

A numerical study is conducted with a two-story and eight-story shear buildings.

Uncertainties in stiffness are assumed in terms of known probability density

functions. It is found that the proposed ARXmodel can localize damage, especially,

if the structure has negligible damping. It is also found that the ARMAX model

with Kalman filter can tackle noisy data, which is demonstrated through a two-story

shear building.

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