identification in vibrating structures Time-series ...

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Time-series methods for fault detection and identification in vibrating structures Spilios D Fassois and John S Sakellariou Phil. Trans. R. Soc. A 2007 365, doi: 10.1098/rsta.2006.1929, published 15 February 2007

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Phil. Trans. R. Soc. A (2007) 365, 411–448 doi:10.1098/rsta.2006.1929 Published online 13 December 2006

Time-series methods for fault detection and identification in vibrating structures B Y S PILIOS D. F ASSOIS *

AND

J OHN S. S AKELLARIOU

Stochastic Mechanical Systems (SMS) Group, Department of Mechanical & Aeronautical Engineering, University of Patras, GR 265 00 Patras, Greece An overview of the principles and techniques of time-series methods for fault detection, identification and estimation in vibrating structures is presented, and certain new methods are introduced. The methods are classified, and their features and operation are discussed. Their practicality and effectiveness are demonstrated through brief presentations of three case studies pertaining to fault detection, identification and estimation in an aircraft panel, a scale aircraft skeleton structure and a simple nonlinear simulated structure. Keywords: structural fault diagnosis; vibration-based methods; fault detection and identification; statistical methods; vibrating structures; time-series methods

1. Introduction Fault detection, identification (localization) and (magnitude) estimation (collectively referred as fault diagnosis or as fault detection and identification—fdi) in vibrating structures, such as aerospace and mechanical structures, marine structures, buildings, bridges and offshore platforms, are of paramount importance for reasons associated with proper operation, maintenance and safety. Furthermore, they become crucial for the assessment of ageing infrastructure. For these reasons, a number of non-destructive fault detection techniques have been developed over the past several years (see Braun 1986; Doherty 1987; Doebling et al. 1996, 1998; Natke & Cempel 1997; Salawu 1997; Zou et al. 2000; Balageas 2002; Boller & Staszewski 2004; Staszewski et al. 2004). Most are ‘local’, requiring access to the vicinity of the suspected fault location; furthermore, they are typically time consuming and costly. They are usually based upon radiography, eddy current, acoustic, ultrasound, magnetic and thermal field principles. In recent years, significant attention has been paid to fault detection via vibration-based methods (Doebling et al. 1996, 1998; Salawu 1997; Zou et al. 2000; Farrar et al. 2001). These appear as particularly promising and offer a number of potential advantages, such as no requirement for visual inspection, ‘automation’ * Author for correspondence ([email protected]). Copyright 2003–2006 by S. D. Fassois and J. S. Sakellariou. All rights reserved. One contribution of 15 to a Theme Issue ‘Structural health monitoring’.

411

q 2006 The Royal Society


412

S. D. Fassois and J. S. Sakellariou

Important conventions Bold-face upper/lower case symbols designate matrix/column-vector quantities, respectively. Matrix transposition is indicated by the superscript T. A functional argument in parentheses designates function of a real variable. A functional argument in brackets designates function of an integer variable. For instance, x[t] is a function of normalized discrete time tZ1, 2, . Time instants used as subscript and superscript to a function of an integer variable designate the set of values of the function from the subscript to the superscript; for instance, x1N bfx½i; iZ 1; 2; .; N g. A hat designates estimator/estimate of the indicated quantity; for instance, q^ is an estimator/ estimate of q. For simplicity of notation, no distinction is made between a random variable and its value(s). The subscripts ‘o’ and ‘u’ designate quantities associated with the nominal (healthy) and current (in unknown state) structure, respectively.

capability, ‘global’ coverage (in the sense of covering large areas of the structure) and capability of working at a ‘system level’. Furthermore, they tend to be time effective and less expensive than most alternatives. The fundamental principle upon which vibration-based methods are founded is that small changes (faults) in a structure cause behavioural discrepancies in its vibration response. The goal thus is the reliable detection of such discrepancies in a structure’s vibration response and their precise association with a specific cause (fault of specific type and magnitude). Time-series methods form an important category within the broader vibrationbased family of methods. The term time-series has originated in statistics (see Box et al. 1994) and refers to a time-ordered sequence of random (stochastic) scalar or vector observations (random signal(s)). Within the present context, these include the excitation and/or vibration response of a given structure. Time-series analysis uses statistical tools for developing (identifying and estimating) mathematical models describing one or more measured random signals and analysing their observed and future behaviour. By their own nature, time-series methods account for uncertainty, while fundamentally they are of the inverse type, in that the models developed are data based rather than physics based (although the former are obviously related to the latter and the underlying physics). Time-series methods for fault detection and identification in vibrating structures offer a number of potential advantages over alternatives. These include the following. (i) No requirement for physical or finite element models. (ii) No requirement for complete structural models; in fact, they may operate on partial models with a limited number of measurable excitation and/or response signals. (iii) Inherent accounting for (measurement, environmental and so on) uncertainty through statistical tools. (iv) Statistical decision making with specified performance characteristics. On the other hand, as complete structural models are not employed, time-series methods may identify (locate) a fault only to the extent allowed by the type of model used. Of course, the methods may also be combined with complete structural models; this subject will not be, however, treated in the present paper. Phil. Trans. R. Soc. A (2007)


Time-series methods for fault detection

413

The goal of this paper is a concise and tutorial overview of the principles and techniques of Gaussian time-series methods for fault detection, identification (localization) and estimation in vibrating structures. The paper does not attempt to provide an exhaustive survey of the literature, as such. Certain new results and methods are also briefly presented. The methods are classified, and their features and operation are discussed. Their practicality and effectiveness are demonstrated through brief presentations of three case studies pertaining to fault detection, identification and estimation in an aircraft panel, a scale aircraft skeleton structure and a simple nonlinear simulated structure. The rest of the paper is organized as follows: the problem statement is presented in §2 and the structure of time-series methods in §3; an overview of time-series model representations is given in §4; non-parametric methods are presented in §5 and parametric methods in §6; three case studies are presented in §7; and concluding remarks are summarized in §8.

2. Problem statement Let S o designate the vibrating structure under study in its nominal (healthy) state. In addition, let S A, ., S D designate the structure under fault types (fault modes) A, ., D, respectively. Each fault type includes a continuum of faults (each one of distinct magnitude), characterized by common nature (for instance, damage of various possible magnitudes to a specific structural element). The structure under each such fault is designated as S kA , with the subscript designating fault type and the superscript fault magnitude. The symbol FAk is also used for designating the fault itself. Over the course of its service life, the structure is assumed to be periodically inspected for faults (for instance, following a major loading or the completion of a certain operation cycle). It is at such times that fault diagnosis is sought. This is to say that the off-line problem (as opposed to the online problem in which a structure is continuously monitored in real time) is presently treated. During inspection, the structure is in a currently unknown (to be determined) state designated as S u. Suppose that during the inspection phase, behavioural data, i.e. force excitation x u[t] and/or vibration response yu[t] (tZ1, 2, ., N ) signals, are obtained from the structure in its current (unknown) state (note that t refers to discrete time, with the corresponding analogue time being (tK1)$Ts and Ts standing for the sampling period; the subscript ‘u’ designates the current/unknown state of the structure). For convenience and simplicity of presentation, it is assumed that the excitation and response signals are scalar (the univariate case). The extension to the more general multivariate (vector signal) case requires the establishment of pertinent vector statistics and the use of corresponding models. Despite their phenomenal resemblance to their univariate counterparts, multivariate models generally have a much richer structure and give rise to parametrization, identifiability and other important questions (see Lu ¨tkepohl 1991), while also requiring multivariate statistical decision-making procedures. As a consequence, this extension is not necessarily a straightforward task (see Basseville & Nikiforov 1993; Gertler 1998). Phil. Trans. R. Soc. A (2007)


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Let the complete signal records be obtained during inspection designated as N ðx u ÞN 1 and ðyu Þ1 , and let the excitation–response signals be collected into the vector z u ½t b½x u ½t yu ½tT (tZ1, 2, ., N ) (lower/upper case bold face symbols designate vector/matrix quantities, respectively; by convention, all vectors are column vectors), the complete record of which is designated as ðz u ÞN 1 . Note that all collected signals need to be suitably pre-processed (Doebling et al. 1998; Fassois 2001). This may include low or band pass filtering within the frequency range of interest, signal subsampling (in case the originally used sampling frequency is too high) as well as proper scaling. The latter is used not only for numerical reasons, but also for counteracting—to the extent possible— different operating (including excitation levels) and/or environmental conditions. In the common case of linear systems, scaling typically involves subtraction of each signal’s sample (estimated) mean and normalization by its sample (estimated) standard deviation. In case of multiple excitations, care should be exercised in order to ensure minimal cross-correlation among them. Given the data ðz u ÞN 1 , collected during the inspection phase, the characterization of the current state of the structure, which is the problem generally referred as fault detection and identification (fdi), may be thought of as consisting of three conceptually distinct subproblems. (i) Fault detection is the binary decision-making subproblem pertaining to the presence or not of a fault in the current structure. The two available possibilities thus are either S uZS o (the current structure is healthy) or S usS o (the current structure is faulty). (ii) Fault identification (also referred as fault localization or fault classification) follows fault detection and is the multiple decision-making subproblem pertaining to the identification of the particular type of fault incurred. In our present notation, the available possibilities include fault type (mode) A (S A) through fault type (mode) D (S D). (iii) Fault estimation is the subproblem concerning estimation of the exact fault magnitude (damage level). 3. The structure of time-series methods (a ) The operational viewpoint From an operational viewpoint, the tackling of the aforementioned subproblems via time-series methods requires (in addition to ðz u ÞN 1 ) the availability of similar data records from the nominal (healthy) and each considered faulty state of the structure. In other words, in present terms, zo[t] (tZ1, 2, ., N) corresponding to S o, as well as zA[t], ., zD[t] (tZ1, 2, ., N) corresponding to S A, ., S D, respectively (in reality, data records corresponding to various possible fault magnitudes within each fault type are required). Note that, due to obvious reasons, this may not be necessarily possible on a real structure. In such a case, either data coming from similar (if available) structures under the given conditions, or more realistically, structural models (laboratory scale models or mathematical—like finite element—models) capable of providing reasonably accurate representations of the actual structural dynamics under various conditions may be used. Phil. Trans. R. Soc. A (2007)


415


These datasets are all obtained and processed in an initial baseline phase. On the other hand, the current data acquisition, processing and decision making are taking place in a second operational phase that has already been referred as the inspection phase. (b ) The conceptual viewpoint From a conceptual viewpoint, time-series methods include analysis and statistical decision making. (i) The analysis part includes signal or system characterization and parametric or non-parametric modelling. The aim is the extraction, from each dataset, of a characteristic quantity designated as N QZ Qðz N 1 Þ (a function of z 1 ), which is instrumental in the decision making part. (ii) The statistical decision making is the part where decisions are made by ‘comparing’, via formal statistical hypothesis testing procedures, the current characteristic quantity Qu to its counterparts Qo, QA, ., QD pertaining to the various possible structural states (o, A, D, ., D, respectively). Fault detection is then formulated as a binary composite hypothesis testing problem which may be generally expressed as: ) H0 : Qo wQu ðnull hypothesis-healthy structureÞ; ð3:1Þ H1 : Qo §Qu ðalternative hypothesis-faulty structureÞ; with w designating a proper relationship (such as equality, inequality and so on). Fault identification, on the other hand, is formulated as a multiple hypothesis testing problem which may be generally expressed as: HA :

QA wQu

«

«

HD :

QD wQu

ðhypothesis A-fault type AÞ :

ð3:2Þ

ðhypothesis D-fault type DÞ

Augmenting this formulation with the original null hypothesis H0 leads to combined treatment of fault detection and identification. Fault estimation is a generally more complicated issue. It requires proper formulation and the use of interval estimation techniques. The design of a binary statistical hypothesis test (such as that of equation (3.1)) may be based upon the probabilities of type I and type II error occurrence. The first probability—designated as a and also referred as the type I risk—is the probability of rejecting the null hypothesis H0 although it is true (false alarm). The second probability—designated as b and also referred as type II risk—is the probability of accepting the null hypothesis H0 although it is not true (missed fault). The designs presented in this paper are based upon selected type I error occurrence probability (a) and use the probability density function of a relevant random quantity under the null (H0) hypothesis of a healthy current structure. The computation of the type II risk depends upon a number of factors and is, in general, more complicated. In selecting a, it should be born in mind that a decrease/increase in it results in a corresponding increase/decrease in b. The reader is referred to references such as Basseville & Nikiforov (1993, subsection 4.2) and Montgomery (1991, subsection 3.3) for details on statistical hypothesis testing. Phil. Trans. R. Soc. A (2007)


416


excitation xu[t]

current structure Su

response yu[t]

estimation of characteristic quantity Q

Qˆ u

statistical decision making

… Qˆ o Qˆ A … Qˆ D healthy fault fault A D baseline phase Figure 1. General structure of time-series-based fault detection and identification methods (the inspection phase is depicted outside the dashed box). Table 1. Comparison of the main characteristics of non-parametric and parametric time-series methods. methods

advantages

disadvantages

non-parametric methods

simplicity computational efficiency some user expertise required improved parsimony potentially increased accuracy

potentially reduced accuracy

parametric methods

increased complexity computationally involved increased user expertise required

(c ) Types of time-series methods The diagram of a ‘general’ time-series fdi method is depicted in figure 1. Note that the baseline phase is indicated by the dashed frame, while the inspection phase is in its exterior. In addition, note that although the characteristic quantity Q appears as a function of the response, it may or may not be a function of the excitation as well. Methods falling into the first category are referred as excitation–response methods, while those falling into the second are referred as response-only methods. Depending upon the way the characteristic quantity Q is constructed, time-series methods may be also classified as parametric or non-parametric. Parametric are those methods in which the statistic is constructed through parametric time-series representations, such as the autoregressive moving average (ARMA) representation (see §4; also Fassois 2001). Non-parametric are those methods in which the statistic is constructed via non-parametric time-series representations, such as spectral models (see §4). A rough comparison of the main characteristics of non-parametric and parametric time-series methods is presented in table 1. 4. Overview of time-series representations The objective of this section is to provide a concise overview of Gaussian timeseries representations used in fault detection and identification. As already indicated, for the sake of simplicity, the univariate case (scalar excitation and Phil. Trans. R. Soc. A (2007)


417

Time-series methods for fault detection n[t]

x[t]

structure h[t]

y[t]

+

Figure 2. Representation of a linear time-invariant system with additive response noise.

response signals) is considered. Let h[t] designate the impulse response function describing the excitation–response causality relationship between an excitation and a response location on a linear time-invariant structure (figure 2). Let n[t] designate zero-mean stationary Gaussian corrupting noise that is of unknown autocovariance structure and mutually uncorrelated with the excitation x[t]. Then y½t Z

N X

h½t$x½tKt C n½t

ðconvolution summation C noiseÞ:

ð4:1Þ

tZ0

Using the backshift operator B ðBi $y½tZ y½tKiÞ, the above may be rewritten as y½t Z H ðBÞ$x½t C n½t

H ðBÞ Z

N X

h½t$Bt ;

ð4:2Þ

tZ0

with H(B) designating the structure’s discrete-time transfer function. Assuming x[t] to be a random stationary excitation, y[t] will also be stationary in the steady state. In addition, y[t] will be Gaussian if x[t] and n[t] are jointly Gaussian. In such a case, each signal is fully characterized by its first two moments (mean and autocovariance). For y[t], these are my Z Efy½tg

gyy ½t Z Efy½t$y½tKtg

Syy ðuÞ Z

N X

gyy ½t$eKjutTs ;

tZKN

ð4:3Þ with E{$} designating statistical expectation; j is the imaginary unit; t is the time lag; u is the frequency in rad sK1; and Ts is the sampling period. Note that gyy[0] is the variance ðs2y Þ of the response y[t], and Syy(u) its auto-spectral density defined as the Fourier transform of the autocovariance (Kay 1988 p. 3; Box et al. 1994 pp. 39–40). These quantities may be related to those of x[t] and n[t] as follows: ) Sxy ðjuÞ Z H ðjuÞ$Sxx ðuÞ my Z H ðjuÞjuZ0 $mx Syy ðuÞ Z H ðjuÞ$Sxy ðjuÞ C Snn ðuÞ

ð4:4Þ

0Syy ðuÞ Z jH ðjuÞj2 $Sxx ðuÞ C Snn ðuÞ; where H(ju) is the structure’s frequency response function (frf) obtained by setting BZ eKjuTs in the second of equations (4.2), designates complex conjugate; j$j is the complex magnitude; and Snn(u) is the noise auto-spectral density, while the (complex) cross-spectral density Sxy(u) is defined as the Fourier transform of Phil. Trans. R. Soc. A (2007)


418


the cross-covariance function gxy ½tZ Efx½t$y½t Ktg. Note that, by convention, complex functions are designated by incorporating the imaginary unit in their argument. (a ) Response-only representations In this case, the properties of the excitation x[t] (and also of the noise n[t], if it is present) are assumed known. Typically, both signals are zero mean, serially and mutually uncorrelated. Without loss of generality, it is presently assumed that x[t]hw[t] (zero-mean uncorrelated (white) sequence) and n[t]h0. Non-parametric representations. The response may be non-parametrically described via its mean my and autocovariance gyy[t] or the auto-spectral density Syy ðuÞZ jH ðjuÞj2 $s2w , with s2w designating the excitation’s variance (from the rightmost of equation (4.4)). Time-frequency representations or polyspectra or wavelet analysis (Staszeswski 1998; Peng & Chu 2004) may be used in the nonstationary or nonlinear cases. Parametric representations. The parametrization of equation (4.1) leads to an ARMA representation of the form (Box et al. 1994, pp. 52–53) na nc X X y½t C ai $y½tKi Z w½t C ci $w½tKi iZ1

5 1C

na X

!

iZ1

ai Bi $y½t Z 1 C

iZ1

5 AðBÞ$y½t Z C ðBÞ$w½t

nc X

! ci Bi $w½t

ð4:5Þ

iZ1

w½t wiid N 0; s2w ;

with ai, ci, A(B) and C(B) designating the AR and MA parameters and polynomials, respectively. iid stands for identically independently distributed, while N ($, $) designates normal distribution with the indicated mean and variance. na and nc are the model’s AR and MA orders, respectively. The model parameter vector is qZ ½coef A; coef BT . It should be noted that the excitation w[t] may be shown (Box et al. 1994, p. 134; Ljung 1999, p. 70) to coincide with the model-based one-step-ahead prediction error and is also referred as the residual or innovations. Other parametrizations are generally possible (especially in the multivariate case—see So¨derstro ¨m & Stoica 1989). In the scalar case, the model poles (eigenvalues) may be alternatively used (this is commonly done through the use of modal models; also see the remark in §6a and §6b(iv)). Alternatively, equation (4.1) may be set into state space form (So ¨derstro ¨m & Stoica 1989, p. 157; Box et al. 1994, pp. 163–164) j½t C 1 Z A$j½t C v½t y½t Z C$j½t

v½t wiid N ð0; Sv Þ;

ð4:6Þ

with j[t] designating the model’s state vector and v[t] a zero-mean uncorrelated vector sequence with covariance Sv. The model parameter vector may be selected as qZ ½ðvec AÞT ; ðvec CÞT T , with vec($) designating the column vector produced by stacking the columns of the indicated matrix on top of each other (the previous comment on different parametrizations applies; also see §6b(iv) for a selection that is invariant under changes in the state space basis). Phil. Trans. R. Soc. A (2007)



419

Time-varying versions of the above (characterized by time-dependent parameters) may be used in the non-stationary case (Petsounis & Fassois 2000; Poulimenos & Fassois 2006), while various nonlinear models, like nonlinear ARMA (NARMA) models, may be used in the nonlinear case (Leontaritis & Billings 1985; Sakellariou & Fassois 2002). (b ) Excitation–response representations In this case, n[t] (figure 2) is assumed to be stationary, zero mean but of unknown autocovariance structure, and mutually Gaussian and uncorrelated with the excitation x[t]. Non-parametric representations. A complete non-parametric representation includes the mean values mx and my and the auto- and cross-covariances gx x[t], gyy[t] and gxy[t] (equivalently the auto- and cross-spectral densities Sx x(u), Syy(u) and Sxy(ju)). These quantities are interrelated through the relationships of equations (4.4). In addition, the (squared) coherence function is defined as (Bendat & Piersol 2000, p. 196) g2 ðuÞ Z

jSxy ðjuÞj2 Z Sxx ðuÞ$Syy ðuÞ 1 C

1 Snn ðuÞ jH ðjuÞj2

$Sxx ðuÞ

g2 ðuÞ 2½0; 1;

ð4:7Þ

with Snn(u) designating the noise auto-spectral density. Time-frequency representations or polyspectra or wavelet analysis (Peng & Chu 2004) may be used in the non-stationary or nonlinear cases. Parametric representations. The parametrization of equation (4.1) may lead to an AutoRegressive Moving Average with eXogenous excitation (ARMAX) representation of the form (So ¨derstro ¨m & Stoica 1989, p. 149; Ljung 1999, p. 83; Fassois 2001) na nb nc X X X y½t C ai $y½t Ki Z bi $x½tKi C w½t C ci $w½tKi5 AðBÞ$y½t iZ1

iZ0

iZ1

Z BðBÞ$x½t C C ðBÞ$w½t5 y½t Z

BðBÞ CðBÞ $x½t C $w½t AðBÞ AðBÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

w½t wiid N 0; s2w ;

ð4:8Þ

n½t

with parameter vector qZ ½coef A; coef B; coef C T . A(B), A(B) and C(B) are the AR, X and MA polynomials, respectively, and na, nb and nc the corresponding orders. Like before, w[t] is a zero-mean uncorrelated (white) signal, which coincides with the model-based one-step-ahead prediction error and is also referred as residual or innovations. Alternatively, it may lead to the Box–Jenkins representation (So ¨derstro ¨m & Stoica 1989, pp. 148–154; Ljung 1999, p. 87; Fassois 2001) BðBÞ C ðBÞ y½t Z w½t wiid N 0; s2w ; ð4:9Þ $x½t C $w½t AðBÞ DðBÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} n½t

with parameter vector qZ ½coef A; coef B; coef C ; coef DT . Phil. Trans. R. Soc. A (2007)


420


On the other hand, the output error (OE) representation models the excitation–response dynamics (Ljung p. 85; Fassois 2001) y½t Z

BðBÞ $x½t C n½t n½t : auto correlated zero-mean; AðBÞ

ð4:10Þ

with parameter vector qZ ½coef A; coef BT . A state space representation may also be used. This may be of the form of equation (4.6), but augmented by terms that include the excitation x[t] (see Ljung 1999, pp. 97–101 for additional versions) j½t C 1 Z A$j½t C B$x½t C v½t y½t Z C$j½t C D$x½t

v½t wiid N ð0; Sv Þ;

ð4:11Þ

with j[t] designating the model’s state vector and v[t] a zero-mean uncorrelated vector sequence with covariance Sv. The model parameter vector may be selected as qZ ½ðvec AÞT ; ðvec BÞT ; ðvec CÞT ; ðvec DÞT T (the comment of the previous subsection on different parametrizations applies). Time-varying versions of the above may be used in the non-stationary case (Poulimenos & Fassois 2004), while a variety of nonlinear models, like nonlinear ARMAX (NARMAX) models (Leontaritis & Billings 1985; Chen & Billings 1989; Sakellariou & Fassois 2002; also see the case study of §7c), neural network models (Masri et al. 2000) or frequency-domain ARX models (Adams 2002) may be used in the nonlinear case.

5. Non-parametric methods Non-parametric methods are those in which the characteristic quantity Q is constructed based upon non-parametric time-series representations (see §4). The generic structure of these methods follows that of the general time-series fdi method (figure 1). Three non-parametric methods are briefly presented in the following: a spectral density function-based method (Sakellariou et al. 2001), a frf-based method (Rizos et al. 2001) and a coherence measure-based method (Rizos et al. 2002). For additional non-parametric methods, including time-frequency- and wavelet analysis-based methods, the interested reader is referred to Farrar & Doebling (1997), Staszeswski (1998), Worden et al. (2000), Worden & Manson (2003), Peng & Chu (2004) and references therein. (a ) Spectral density function-based method This method attempts fault detection, identification and magnitude estimation via characteristic changes in the auto-spectral density function (defined via the third of equation (4.3)) of the measured vibration y[t] when the excitation x[t] is not available (response-only case). The method’s characteristic quantity thus is QZSyy(u)ZS(u). Let the vibration response data record y1N be segmented into K non-overlapping segments (each of length L). Then the Welch auto-spectral Phil. Trans. R. Soc. A (2007)



density estimator (sample spectrum) is defined as (Kay 1988, p. 76) 2 X K L X 1 1 ^ I ðiÞ ðuÞ with I ðiÞ ðuÞ Z a½t$y ðiÞ ½t$eKjutTs ; SðuÞ Z K iZ1 L tZ1

421

ð5:1Þ

with the superscript i designating the ith segment and a[t] a proper data window (note that estimators/estimates are designated by a hat). This estimator, multiplied by 2K and normalized by the true auto-spectral density S(u), may be shown to follow a (central) chi-square distribution with 2K degrees of freedom (see appendix A for the distribution’s definition) or explicitly (Kay 1988, p. 76; Bendat & Piersol 2000, p. 309) ^ 2K SðuÞ wc2 ð2KÞ: SðuÞ

ð5:2Þ

Fault detection. Fault detection is based upon confirmation of statistically significant deviations (from the nominal/healthy) in the current structure’s autospectral density function at one or more frequencies through the hypothesis testing problem (for each u) ) H0 : Su ðuÞ Z So ðuÞ ðnull hypothesis-healthy structureÞ; ð5:3Þ H1 : Su ðuÞ sSo ðuÞ ðalternative hypothesis-faulty structureÞ: Towards this end, owing to equation (5.2), the following quantity follows (for each frequency u) F distribution with 2K, 2K degrees of freedom (ratio of normalized chi-square distributions; see appendix A): Fb

Sô ðuÞ=So ðuÞ wFð2K; 2KÞ; Sû ðuÞ=Su ðuÞ

with Sô ðuÞ and Sû ðuÞ designating the sample (that is estimated) auto-spectral density for the healthy and current structure, respectively (the first obtained in the baseline phase and the second in the inspection phase). Under the null (H0) hypothesis (current structure is healthy), the true auto-spectral densities coincide (So(u)ZSu(u)) and the above quantity becomes Under H0 :

Fb

Sô ðuÞ wFð2K; 2KÞ: Sû ðuÞ

ð5:4Þ

Equality of the true auto-spectral densities So(u) and Su(u) is then examined at the a risk level (type I error probability of a) via the statistical test fða=2Þ ð2K;2KÞ%F %f1Kða=2Þ ð2K;2KÞ ðcuÞ 0 H0 is accepted ðhealthy structureÞ Else

; 0 H1 is accepted ðfaulty structureÞ ð5:5Þ

with f(a/2), f1K(a/2) designating the F distribution’s a/2 and 1K(a/2) critical points (fa is defined such that ProbðF % fa ÞZ a; figure 3). Phil. Trans. R. Soc. A (2007)


422


fF

a/2 a/2

0

fa /2

H1 accepted (fault)

f1–a /2 H0 accepted (no fault)

H1 accepted (fault)

Figure 3. Statistical hypothesis testing based upon an F-distributed statistic (two-tailed test).

Fault identification. Fault identification may be, in principle, achieved by performing hypotheses testing similar to the above separately for faults from each potential fault mode. Fault estimation. Fault estimation may be achieved by possibly associating specific quantitative changes in the auto-spectral density (at one or more frequencies) with specific fault magnitudes. Remark. Note that signal scaling is particularly important in order to properly account for different excitation levels, while the environmental conditions should be kept constant. (b ) Frequency response function (frf)-based method This method is similar to the previous one, except that it requires the availability of both the excitation and response signals and uses the frf magnitude as its characteristic quantity (QZjH(ju)j) (see the first central of equations (4.4)). Yet, it may be also used in case the excitation is unavailable, but more than one response are available (Sakellariou & Fassois 2000). The frf magnitude is estimated via Welch estimates (K non-overlapping data segments, each of length L; see previous subsection) of the auto- and cross-spectral density functions Sx x(u) and Sx y(ju), respectively, using the first central of equations (4.4) ^ ^ ðjuÞj Z jS x y ðjuÞj : jH ð5:6Þ S^x x ðuÞ This estimator may be shown to follow a distribution approximated as Gaussian (see Bendat & Piersol 2000, p. 338; alternatively see Koopmans 1974, p. 284 for a discussion that is based upon the F distribution) ^ ðjuÞj wN ðjH ðjuÞj; s2 ðuÞÞ jH

with

s2 ðuÞ z

1Kg2 ðuÞ jH ðjuÞj2 ; g2 ðuÞ$2K

ð5:7Þ

with the mean coinciding with the true frf magnitude and the indicated variance (g2(u) designates the coherence function, see equation (4.7)). Phil. Trans. R. Soc. A (2007)



423

Fault detection. Fault detection is based upon confirmation of statistically significant deviations (from the nominal/healthy) in the current structure’s frf at one or more frequencies through the hypothesis testing problem (for each u) ) H0 : djH ðjuÞjbjHo ðjuÞjKjHu ðjuÞj Z0 ðnull hypothesis-healthy structureÞ; H1 : djH ðjuÞjbjHo ðjuÞjKjHu ðjuÞjs0 ðalternative hypothesis-faulty structureÞ: ð5:8Þ Towards this end, the difference of the frf magnitude estimates in the nominal ^ o ðjuÞj is obtained in the baseline phase and and current states is considered (jH ^ jH u ðjuÞj in the inspection phase) which, due to equation (5.7) and mutual ^ o ðjuÞj, jH ^ u ðjuÞj (owing to the fact that the estimates independence of jH corresponding to the nominal and current states are based upon distinct data records), follows Gaussian distribution with mean djH ðjuÞj bjHo ðjuÞjKjHu ðjuÞj (the true difference) and variance ds2 ðuÞZ s2o ðuÞC s2u ðuÞ, i.e. ^ ðjuÞj bjH ^ o ðjuÞjKjH ^ u ðjuÞj wN ðdjH ðjuÞj; ds2 ðuÞÞ: djH ð5:9Þ Under the null (H0) hypothesis (current structure is healthy), the true frf magnitudes coincide (jH0(ju)jZjHu(ju)j) and so do the respective estimator b ðjuÞj follows the Gaussian distribution variances ðs2o ðuÞZ s2u ðuÞÞ. Hence, djH Under H0 :

^ ðjuÞj Z jH ^ o ðjuÞjKjH ^ u ðjuÞj wN ð0; 2s2o ðuÞÞ: djH

ð5:10Þ

The variance s2o ðuÞ is generally unknown, but may be estimated in the baseline phase through equation (5.7). Treating this estimate as a fixed quantity, i.e. a quantity characterized by negligible variability (which is reasonable for estimation based upon a large number of samples), the equality of jHo(ju)j and jHu(ju)j is examined at the a (type I) risk level through the statistical test (figure 4) ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ ðjuÞjj% Z1Kða=2Þ 2^ s2o ðuÞ ðcuÞ 0 H0 is accepted ðhealthy structureÞ jdjH ; Else 0 H1 is accepted ðfaulty structureÞ ð5:11Þ with Z1K(a/2) designating the standard normal distribution’s 1K(a/2) critical point (typical values are aZ0.05, Z1K(a/2)Z1.96). Alternatively (treating s^2o ðuÞ as a random variable), the t distribution may be used (see appendix A; also §6c; note that the t distribution approaches normality for long data records). Fault identification. Fault identification may be, in principle, achieved by performing hypotheses testing similar to the above separately for each potential fault from each fault mode. Fault estimation. Fault estimation may be achieved by possibly associating specific quantitative changes in the frf magnitude (at one or more frequencies) with specific fault magnitudes. (c ) Coherence measure-based method This method is founded upon the heuristic premise that under constant experimental and environmental conditions, the overall coherence (coherence over the complete frequency range; see equation (4.7) for the definition of coherence) Phil. Trans. R. Soc. A (2007)


424

S. D. Fassois and J. S. Sakellariou fN

a/2

a /2

H1 accepted (fault)

Z1–a / 2

0

Za/ 2

H0 accepted (no fault)

H1 accepted (fault)

Figure 4. Statistical hypothesis testing based upon a Gaussian-distributed statistic (twotailed test).

decreases with fault occurrence. This is due to the fact that nonlinear effects are introduced, or strengthened, with fault occurrence. Like the previous method, this also pertains to the excitation–response case. The coherence estimator g^2 ðuÞ is obtained by replacing the auto- and crossspectral densities in the first of equations (4.7) by their respective Welch estimates (obtained via K non-overlapping data segments). The mean and variance of this estimator may be shown to be (for large K; see Carter et al. 1983; Bendat & Piersol 2000, pp. 333–335) Efg^2 ðuÞg zg2 ðuÞ C

1 ½1Kg2 ðuÞ2 K

s2 ðuÞ z

2g2 ðuÞ ½1Kg2 ðuÞ2 ; K

ð5:12Þ

with g2(u) designating the true coherence value. Next, consider a frequency discretization ui (iZ1, 2, ., n) (frequency resolution du) and define the coherence measure G bdu$

n X

g2 ðui Þ;

ð5:13Þ

iZ1

which consists of the sum of individual coherence values and constitutes the method’s characteristic quantity (Q). For a large number of discrete frequencies n (n/N), the coherence measure estimator approximately follows, thanks to the central limit theorem (see appendix A), Gaussian distribution, i.e. ^ wN ðG; s2G Þ with s2G Z du2 G

n X

s2 ðui Þ;

ð5:14Þ

iZ1

with G designating the measure’s true value. Note that in obtaining this expression, the bias terms (second term on the right-hand side of the leftmost of equations (5.12)) are neglected (which is justified for either large K or for true ^ is equal to the sum of the coherence being close to unity), while the variance of G individual variances due to the fact that the coherence estimates g^2 ðui Þ (iZ1, 2, ., n) are mutually independent random variables (Brillinger 1981, p. 204). Phil. Trans. R. Soc. A (2007)



425

Fault detection. Fault detection is based upon confirmation of a statistically significant reduction in the coherence measure Gu of the current structure (compared to Go of the healthy structure) through the statistical hypothesis testing problem ) H0 : dG bGo KGu % 0 ðnull hypothesis-healthy structureÞ; ð5:15Þ H1 : dG bGo KGu O 0 ðalternative hypothesis-faulty structureÞ: Towards this end, the difference of the coherence measure estimates in the nominal and current states of the structure is considered ^ bG ô KG û wN ðdG; ds2G Þ with dG Z Go KGu ds2G Z s2G C s2G ; ð5:16Þ dG o u which, due to equation (5.14), follows normal distribution with the indicated mean ô is obtained in the baseline phase and (true G difference) and variance. Note that G ^ Gu in the inspection phase and are mutually independent. Under the null (H0) hypothesis (current structure is healthy), dGZ Go KGu % 0 and ds2G Z 2ðs2G Þo , thus ^bG ô KG û wN dG; 2 s2G Under H0 : dG ðdG% 0Þ: ð5:17Þ o The variance ðs2G Þo is unknown, but may be estimated in the baseline phase based upon equations (5.12) and (5.14) by replacing the true coherence by its estimate. The presence or not of a statistically significant reduction in the coherence measure Gu of the current structure (compared to Go of the healthy) may be then examined through a proper statistical test. Since the null hypothesis is composite, the establishment of such a test at any selected (type I) risk level a requires knowledge of the unavailable mean dG%0. To overcome this difficulty, a conservative procedure is adopted by allowing for a risk level a in the worst possible (under H0) case of dGZ0. By treating the estimated variance ð^ s2G Þo as a fixed quantity (quantity with negligible variability), this leads to the following test characterized by a maximum type I (false alarm) risk of a: 9 ^ dG > = qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 H is accepted ðhealthy structureÞ ! Z 1Ka 0 2 ; ð5:18Þ 2 s^G o > ; Else 0 H1 is accepted ðfaulty structureÞ with Z1Ka designating the standard normal distribution’s 1Ka critical point (typical values are aZ0.05, Z1KaZ1.645). Fault identification. The method is not appropriate for fault identification because different types of faults, or faults of the same type but of different magnitudes, may cause the same reduction in the coherence measure. Fault estimation. Assuming that only one type of fault is possible, fault estimation may be achieved by possibly associating specific quantitative reductions in the coherence measure with specific fault magnitudes. 6. Parametric methods Parametric methods are those in which the characteristic quantity Q is constructed based upon parametric time-series representations (see §4; also Fassois 2001). They are applicable to both the response-only and excitation–response cases, as each Phil. Trans. R. Soc. A (2007)


426


situation may be dealt with through the use of proper representations (also see Basseville & Nikiforov 1993; Natke & Cempel 1997; Gertler 1998). Parametric methods may be classified into the following categories. (i) Model parameter-based methods. These attempt fault detection and identification using a characteristic quantity Q which is the function of the parameter vector q, QZf(q), of a parametric time-series representation (model). In these methods, the model has to be re-estimated during the inspection phase based upon the current signal(s) ðz u ÞN 1 . (ii) Residual-based methods. These attempt fault detection and identification using characteristic quantities, Q, that are functions of model residuals generated by driving the current signal(s) ðz u ÞN 1 through pre-determined representations (models) corresponding to the various states of the structure (healthy structure and structure under specific magnitude faults of types A, ., D). In this case, QZ f ððeXu ÞN 1 Þ, with eXu[t] designating the residual generated by driving z u[t] through the model corresponding to the X structural state (XZo, A, ., D). No model re-estimation is needed in the inspection phase. (iii) The functional model-based method. Conceptually, this may be thought of as a sort of combined method that uses characteristic quantities constructed based upon a model parameter k representing the fault magnitude on one hand and model residuals eXu[t] on the other. A prime advantage of the method is that it allows for effective identification of the fault type in the important case that the continuum of possible fault magnitudes is considered. In addition, it allows for accurate fault magnitude estimation. It is noted that model estimation is required in the inspection phase. The three families of parametric methods are briefly presented in the following. (a ) Model parameter-based methods This category of methods attempts fault detection and identification via changes in the parameter vector q of a suitable parametric representation (see Isermann 1993; Natke & Cempel 1997 is more focused on structures; Gertler 1998). In the typical case, QZq is the methods’ characteristic quantity (sometimes a transformed version of the parameter vector may be used). Let q^ designate a proper (say maximum likelihood; see So¨derstro¨m & Stoica 1989, pp. 198–199; Ljung 1999, pp. 212–213; Fassois 2001) estimator of the parameter vector q based upon excitation and/or response measurements z N 1 . The particular model structure employed (see §4) depends upon the specific problem at hand. For sufficiently long signals (N large), the estimator q^ is (under mild assumptions) Gaussian distributed with mean equal to its true value q and covariance Pq (see So¨derstro ¨m & Stoica 1989, pp. 205–207; Ljung 1999, p. 303 on issues relating to the numerical computation/estimation of Pq) q^ wN ðq; P q Þ

ð6:1Þ

Fault detection. Fault detection is based upon testing for statistically significant changes in the parameter vector q between the nominal and current structures Phil. Trans. R. Soc. A (2007)


427


through the hypothesis testing problem H0 : dq Z qo Kqu Z 0

ðnull hypothesis-healthy structureÞ;

H1 : dq Z qo Kqu s0

ðalternative hypothesis-faulty structureÞ:

) ð6:2Þ

Towards this end, observe that the difference of the parameter vector estimators corresponding to the nominal and current states of the structure (the first obtained in the baseline phase using ðz o ÞN 1 and the second in the inspection phase using ðz u ÞN 1 ) follows Gaussian distribution dq^ b qô Kqû wN ðdq; dPÞ with dq Z qo Kqu

dP Z P o C P u ;

ð6:3Þ

with Po and Pu designating the estimator covariance matrices for the nominal and current structure, respectively. ^ qô Kqû wN ð0; 2P o Þ and the quantity Q, Under the null (H0) hypothesis dqZ below, follows chi-square distribution with d (parameter vector dimensionality) degrees of freedom (as it may be shown to be the sum of squares of independent standardized Gaussian variables; see appendix A; also Ljung 1999, p. 558) T Q b dq^ $dPK1 $dq^ wc2 ðdÞ; ð6:4Þ with dPZ2Po. Since the covariance Po corresponding to the healthy structure is ^ o is used in practice. Treating this unavailable, its sample (estimated) version P sample covariance as a deterministic quantity, i.e. a quantity characterized by negligible variability (which is reasonable for large N ) leads to the following test constructed at the a (type I) risk level (figure 5; also see Ljung 1999, p. 559 for an alternative approach based upon the F distribution) Q! c21Ka ðdÞ 0 H0 is accepted ðhealthy structureÞ ; ð6:5Þ Else 0 H1 is accepted ðfaulty structureÞ with c21Ka ðdÞ designating the chi-square distribution’s, with d degrees of freedom, 1Ka critical point. Fault identification. Fault identification could, in principle, be based upon the multiple hypotheses testing problem of equations (3.2), comparing the current parameter vector qû to those corresponding to different fault types, say qÂ ; .; q^D . Nevertheless, such an approach will work only for faults of specific magnitudes, but will generally fail to account for the continuum of fault magnitudes possible within each fault type. A geometric approach, aiming at circumventing this difficulty, has been introduced by Sadeghi & Fassois (1997, 1998). This uses point and covariance estimates of q collected into the vector q b½q1 .qr T . For each fault type (the continuum of faults of all possible magnitudes), a suitable geometrical representation (typically linear, in the form of a hyperplane) is constructed within the q (r-dimensional) space (figure 6) Z q1 C ui1 $q2 C/C uirK1 $qr Kuir Z 0 ði-th fault mode hyperplaneÞ; ð6:6Þ g i ðqÞ with uij ’s designating the i th hyperplane’s coefficients. These hyperplanes are constructed during the baseline phase using linear regression and vibration datasets obtained from the structure under faults of various magnitudes from each fault type. Phil. Trans. R. Soc. A (2007)


428


Figure 5. Statistical hypothesis testing based upon a c2-distributed statistic (one-tailed test).

Figure 6. Principle of the geometric method for fault identification.

In the inspection phase, following fault detection, fault identification is based upon computation of the distance of the estimated current point qû in the r-dimensional space from each hyperplane (figure 6). The current fault mode is identified as that associated with the hyperplane to which the distance is minimal. This involves the solution of a constrained minimization problem (the distance may be of the Euclidean or other proper type). Fault estimation. Within the context of the geometric approach, fault estimation may be achieved via computation of the distance of qû from the point qo corresponding to the healthy structure (figure 6; distances must have been suitably graded in the baseline phase; also see Sakellariou & Fassois 2000). Remark. Obviously, the model parameter-based methods may also be used with alternative representations, such as modal models. In such a case, fault detection Phil. Trans. R. Soc. A (2007)


429


excitation current response structure xu[t] yu[t] S u

Qˆ ou residual estimation of model Mo (healthy structure) eou[t] characteristic quantity Q model MA (fault A) . . . model MD (fault D)

Qˆ Au residual estimation of eAu [t] characteristic quantity Q . . . . . . Qˆ Du residual estimation of eDu [t] characteristic quantity Q


baseline phase … Qˆ oo Qˆ AA… Qˆ DD fault D

healthy fault A

baseline phase

Figure 7. Schematic of residual-based methods (the inspection phase is depicted outside the dashed boxes).

and identification are attempted based upon changes incurred in the system’s modal parameters (Hearn & Testa 1991; Doebling et al. 1996; Farrar & Doebling 1997; Salawu 1997; Rizos et al. 2001; also see §6b(iv)). (b ) Residual-based methods These attempt fault detection, identification and estimation using characteristic quantities that are functions of residual sequences obtained by driving the current signal(s) ðz u ÞN 1 through suitable pre-determined (estimated in the baseline phase) models Mo, MA, ., MD, each one corresponding to a particular state of the structure (healthy structure and structure under specific magnitude faults of types A, ., D; figure 7). The methods have a relatively long history—mainly within the general context of engineering systems (see Mehra & Peschon 1971; Willsky 1976; Basseville 1988; Frank 1990; Basseville & Nikiforov 1993; Natke & Cempel 1997 is more focused on structures; Gertler 1998). Let MX designate the model representing the structure in its X state (XZo or XZA, ., D under specific fault magnitudes). In addition, let the residual series generated by driving zu[t] through each one of the above models be designated as eou[t], eAu[t], ., eDu[t ] and be characterized by respective variances s2ou , s2Au , ., s2Du (the first subscript designating the model and the second the excitation and/or response signal(s) used). Fault detection, identification and estimation may then be based upon the fact that under the hypothesis HX (i.e. the structure is in its X state, for XZo or XZA, ., D under specific fault magnitudes), the residual series generated by driving the current signal(s) ðz u ÞN 1 through the model MX possesses the property UnderHX : eXu ½twiid N ð0;s2Xu Þ with s2Xu !s2Yu for any state Y sX;

ð6:7Þ

where iid designates identically independently distributed random variables. It is tacitly assumed that UnderHX : s2Xu Zs2XX ; ð6:8Þ implying that the excitation and environmental conditions are the same in the baseline and inspection phases. Phil. Trans. R. Soc. A (2007)


430


A first method within this category is based upon examination of the residual series obtained by driving the current signal(s) ðz u ÞN 1 through the aforementioned bank of models (estimated in the baseline phase). The model matching the current state of the structure should generate a residual sequence characterized by minimal variance. A second method is based upon the likelihood function evaluated for the current signal(s) ðz u ÞN 1 under each one of the considered hypotheses (Ho, HA, ., HD). The hypothesis corresponding to the largest likelihood is selected as corresponding to the current state. A third method is based upon examination of the residual series obtained by driving the current signal(s) ðz u ÞN 1 through the aforementioned bank of models (just as in the first approach above). The model matching the current state of the structure should generate a white (uncorrelated) residual sequence. These methods use classical tests on the residuals and offer simplicity and no need for model estimation in the inspection phase. Yet, they are subject to performance limitations, as certain subtle faults may go undetected (see Willsky 1976; Basseville & Benveniste 1983; Basseville & Nikiforov 1993). A fourth method is based upon the examination of residuals associated with subspace identification. Concise descriptions of the four methods follow. (i) Method A: using the residual variance In this method, the characteristic quantity is the residual variance (also see Sohn & Farrar 2001; Sohn et al. 2001 for a related scheme). Fault detection. According to the preceding discussion, fault detection is based upon the fact that the residual series eou[t], obtained by driving the current signal(s) ðz u ÞN 1 through the model Mo corresponding to the nominal (healthy) structure, should be characterized by variance s2ou which becomes minimal (specifically equal to s2oo ; see equation (6.8)) if and only if the current structure is healthy (S uZS o). The hypothesis testing problem of equation (3.1) may be then expressed as ðQZ s2Xu Þ H0 : s2oo R s2ou

ðnull hypothesis-healthy structureÞ;

H1 : s2oo ! s2ou

ðalternative hypothesis-faulty structureÞ:

ð6:9Þ

Under the null (H0) hypothesis, the residuals eou[t] are (just like the residuals eoo[t]) identically independently distributed (iid) zero-mean Gaussian with variance s2oo (equations (6.7) and (6.8)). Hence, the quantities ðNu K1Þ^ s2ou =s2oo 2 2 and ðNo Kd K1Þ^ soo =soo follow (central) chi-square distributions with NuK1 and NoKdK1 degrees of freedom, respectively (sum of squares of independent standardized Gaussian random variables; see appendix A). Note that No and Nu designate the number of samples used in estimating the residual variance in the healthy and current cases, respectively (typically NoZNuZN ), whereas d designates the dimensionality of the model parameter vector. Consequently, the following statistic follows F distribution with NuK1 and NoKdK1 degrees of freedom (ratio of two independent and normalized c2 random variables; see appendix A): Under Ho :

Fb

Phil. Trans. R. Soc. A (2007)

ðNuK1Þ^ s2ou s2oo ðNuK1Þ ðNoKdK1Þ^ s2oo s2oo ðNoKdK1Þ

Z

s^2ou wFðNu K1; No Kd K1Þ: s^2oo

ð6:10Þ


431


The following test is then constructed at the a (type I) risk level: F % f1Ka ðNu K1; No Kd K1Þ 0 H0 is accepted ðhealthy structureÞ Else

0

H1 is accepted ðfaulty structureÞ

) ; ð6:11Þ

with f1Ka(NuK1,NoKdK1) designating the F distribution’s 1Ka critical point. Fault identification. Fault identification may be similarly achieved through pairwise tests of the form H0 : s2XX R s2Xu for X Z A; B; .; D: ð6:12Þ H1 : s2XX ! s2Xu An alternative possibility may be based upon obtaining the residual series eAu[t], ., eDu[t], estimating their variances, and declaring as current fault that corresponding to minimal variance. Note that by including eou[t], fault detection may also be treated. The advantage of this approach is that only the signal(s) ðz u ÞN 1 are used in the inspection phase. As with the model parameter-based methods, however, these approaches may work only for faults of specific magnitudes, but not for the continuum of magnitudes possible within each fault mode. Fault estimation. Fault estimation may be achieved in the limited case in which only one type of faults is possible. In that case, specific values of the residual variance may be potentially associated with specific fault magnitudes. (ii) Method B: using the likelihood function Fault detection. In this method, fault detection is based upon the likelihood function under the null (Ho) hypothesis of a healthy structure (see Gertler 1998, pp. 119–120). The hypothesis testing problem considered is ) H0 : qo Z qu ðnull hypothesis-healthy structureÞ; ð6:13Þ H1 : qo squ ðalternative hypothesis-faulty structureÞ; with qo and qu designating the healthy and current structure’s parameter vectors, respectively. Assuming independence of the residual sequence, the Gaussianlikelihood function for the data y1N , given x1N , is (Box et al. 1994, p. 226) 2 N N Y N Y 1 e ½t; q N pffiffiffiffiffiffiffiffiffiffi $exp K Ly y1 ; q=x1 Z fw ðe½t; qÞ Z 2s2 2ps2 tZ1 tZ1 ( ) N 1 1 X Z pffiffiffiffiffiffiffiffiffiffi N $exp K 2 e2 ½t; q ; ð6:14Þ 2s 2 tZ1 2ps with e[t,q] designating the model’s residual (one-step-ahead prediction error) characterized by zero mean and variance s2, and fw($) its probability density function. Under the null (H0) hypothesis, the residual series eou[t] generated by driving the current signal(s) through the nominal model Mo is (just like eoo[t]) identically independently distributed (iid) Gaussian with zero mean and variance Phil. Trans. R. Soc. A (2007)


432


s2oo (equations (6.7) and (6.8)). Decision making may be then based upon the likelihood function under H0 (qZqo) evaluated for the current data, by requiring it to be larger or equal to a threshold l (which is to be selected) in order for the null (H0) hypothesis to be accepted, i.e. ) N Ly ððyu ÞN 1 ; qo =ðx u Þ1 ÞR l 0 H0 is accepted ðhealthy structureÞ : ð6:15Þ Else 0 H1 is accepted ðfaulty structureÞ N The evaluation of the likelihood Ly ððyu ÞN 1 ; qo =ðx u Þ1 Þ requires knowledge of the 2 true innovations variance soo . If this quantity is known, or may be estimated with very good accuracy in the baseline phase (which is reasonable for estimation based upon a large number of samples N ) so that it may be treated as a fixed quantity (negligible variability), the above decision-making rule may (following some algebra) be re-expressed as 9 N 2 X > eou ½t N s^2ou QN b Z 2 % l + 0 H0 is accepted ðhealthy structureÞ = 2 ^ ; sôo tZ1 soo > ; Else 0 H1 is accepted ðfaulty structureÞ ð6:16Þ where l+ designates the resulting (to be selected) threshold. The statistic QN follows (under the H0 hypothesis) c2 distribution with N degrees of freedom (sum of squares of mutually independent standardized Gaussian variables; see appendix A), and this leads to the following test at the a risk level (figure 5): 9 N 2 X > eou ½t N s^2ou 2 QN Z Z 2 % c1Ka ðN Þ 0 H0 is accepted ðhealthy structureÞ = 2 ^ sôo ; tZ1 soo > ; Else 0 H1 is accepted ðfaulty structureÞ ð6:17Þ with

c21Ka ðN Þ

designating the chi-square distribution’s 1Ka critical point.

Remark. Note that under the assumption of a fixed s^2oo , this method leads to a test statistic which is similar to that of equation (6.10) (method A). The difference is that in method A, the variability of the estimate s^2oo is accounted for, thus leading to an F distribution of the pertinent statistic. Of course, the two tests coincide for No/N in equation (6.10), as the F distribution then approaches the chi-square distribution (see appendix A). Fault identification. Fault identification may be achieved by computing the likelihood function for the current signal(s) for the various values of q (qA, ., qD) and selecting that hypothesis HX- which corresponds to the maximum value of the likelihood, i.e. N N N Ly ððyu ÞN 1 ; qX- =ðx u Þ1 Þ Z max Ly ððyu Þ1 ; qX =ðx u Þ1 Þ X

0 The HX- hypothesis is accepted;

ð6:18Þ

with XZA, ., D (faults of specific magnitudes). Obviously, also by including the nominal model, the scheme may be used for combined fault detection and identification. Phil. Trans. R. Soc. A (2007)



433

Fault estimation. Fault estimation may be achieved in the limited case in which only one type of faults is possible. In that case, specific values of the likelihood may be possibly associated with specific fault magnitudes. (iii) Method C: using the residual uncorrelatedness In this method, the characteristic quantity is a function of the residual series autocovariance sequence. Fault detection. Fault detection may be based upon assessment of the uncorrelatedness (whiteness) of the residual series ðeou ÞN 1 obtained by driving the through the nominal (healthy) model Mo. In this case, current signal(s) ðz u ÞN 1 the hypothesis testing problem of equation (3.1) may be expressed as ) H0 : r½i Z 0 i Z 1; 2; .; r ðnull hypothesis-healthy structureÞ; ð6:19Þ H1 : r½i s0 for some i ðalternative hypothesis-faulty structureÞ; with r½i bg½i=g½0 designating the normalized autocovariance (correlation coefficient; Box et al. 1994, p. 26) of the eou[t] residual sequence (see equation (4.3) for the definition of g[i]). The method’s characteristic quantity thus is QZ ½r½1r½2.r½rT (r being a design variable). Under the null (H0) hypothesis, eou[t ] is (just like eoo[t ]) identically independently distributed (iid) Gaussian with zero mean, and the statistic Q below follows c2 distribution with rKd (rKdK1 in case the mean is also estimated) degrees of freedom (dZdim(q); Box et al. 1994, p. 314) Under H0 :

Q b N $ðN C 2Þ$

r X

ðN KiÞK1 $^ r½i2 wc2 ðr KdÞ;

ð6:20Þ

iZ1

with r^½i designating the sample (estimated) r[i] and N the number of signal samples ( estimation based upon the autocovariance e stimator P g^½iZ ð1=N Þ N ðe ½tK^ meou Þðeou ½tKiK^ meou Þ, with mêou designating the tZiC1 ou residual sample mean). The hypothesis testing problem of equation (6.19) then leads to the following test at the a (type I) risk level (figure 5): ) Q! c21Ka ðr KdÞ 0 H0 is accepted ðhealthy structureÞ ; ð6:21Þ Else 0 H1 is accepted ðfaulty structureÞ with c21Ka ðr KdÞ designating the chi-square distribution’s 1Ka critical point. Fault identification. Fault identification may be achieved by similarly examining which one of the eXu[t] (XZA, B, ., D) residual series is uncorrelated. As with the previous methods, only faults of specific magnitudes (not the continuum of fault magnitudes) may be considered. Fault estimation. The approach, as such, is not suitable for fault estimation. (iv) Method D: using residuals associated with subspace identification This method is motivated by stochastic subspace identification (the responseonly case is presently treated; see Basseville et al. 2000 for details). It attempts Phil. Trans. R. Soc. A (2007)


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detection of changes in the modal space by considering the parameter vector " # l qb ; ð6:22Þ vec F where l designates the vector containing the eigenvalues of a discrete-time system representation (the eigenvalues of the system matrix A in the state space representation of equation (4.6)), F designates the matrix with columns being the vectors fl where fl bC4l (with 4l being the corresponding eigenvectors and C the output matrix in equation (4.6)). Fault detection. Fault detection is based upon forming (for a given q) the system’s (pC1)th order (p large enough) observability matrix as follows: 1 0 F C B B FL C C; B ð6:23Þ OpC1 ðqÞ b B C @ « A FLp with Lbdiag(l) (diagonal matrix with the system eigenvalues), and also the system’s block Hankel matrix (for the case of equation (4.6)) 1 0 gyy ð1Þ . gyy ðqK1Þ gyy ð0Þ C B gyy ð2Þ . gyy ðqÞ C B gyy ð1Þ C bHankðgyy Þ; B HpC1;q b B ð6:24Þ C « « 1 « A @ gyy ðpÞ gyy ðp C 1Þ .

gyy ðp C q K1Þ

with qRpC1 and gyy(t) designating the response’s theoretical autocovariance (see equation (4.3)). The main idea for fault detection lies in the fact that under the hypothesis of the observability matrix and the block Hankel matrix corresponding to the same system (common q), OpC1(q) and HpC1,q have the same left kernel space. Whether this is the case or not may be checked through the following procedure. Use an available parameter vector q in order to form the observability matrix OpC1(q). Then pick an orthonormal basis of the left kernel space of the matrix W1OpC1(q) (W1 is a selectable invertible weighting matrix), in terms of the columns of a matrix S of co-rank n (the system order), such that S T $S Z I ;

ð6:25Þ

S T $W 1 $OpC1 ðqÞ Z 0;

ð6:26Þ

with sZ ðpC 1Þr Kn (r designating the output dimensionality). Note that the matrix S is not unique. It may be, for instance, obtained through the singular value decomposition of W 1 OpC1 ðqÞ, and implicitly depends upon the parameter vector q; hence, it may be designated as S(q). The block Hankel matrix corresponding to the same system should then satisfy the property S T ðqÞ$W 1 $H HpC1;q $W T 2 Z 0: (W2 being an additional selectable invertible weighting matrix). Phil. Trans. R. Soc. A (2007)

ð6:27Þ



435

Now assume that the parameter vector qo corresponding to the healthy system is available from the baseline phase. For checking whether the current data ðyu ÞN 1 actually correspond to the healthy system, the following residual is defined (compare with equation (6.27)): pffiffiffiffiffi ^ pC1;q $W T ð6:28Þ zN ðqo Þ b N vec S T ðqo Þ$W 1 $H 2 ; ^ where HpC1;q is the sample block Hankel matrix obtained from the current data ðyu ÞN 1 . This residual, which should be zero for the theoretical block Hankel matrix under H0 (quZqo) (see equation (6.27)), has zero mean under H0 (quZqo) and nonzero mean under H1 (qusqo). Testing whether the current system (with parameter vector qu and associated with the sample block Hankel matrix) coincides with the healthy system (with parameter vector qo) is based upon a ‘statistical local approach’, according to which the following ‘close’ hypotheses are considered: 9 H0 : q u Z q o ðnull hypothesis-healthy structureÞ; > = 1 ð6:29Þ H1 : qu Z qo C pffiffiffiffiffi dq ðalternative hypothesis-faulty structureÞ; > ; N with dq designating an unknown but fixed error vector. Let M(qo) be the Jacobian matrix designating the mean sensitivity of zN and Sðqo Þ blimN/NEqo fzN zT N g, with Eqo f$g designating the expectation operator when the actual system parameter is qo. Note that these matrices do not depend upon the sample size N and may be estimated from the healthy structure during the baseline phase. Provided that Sðqo Þ is positive definite, the residual zN(qo) of equation (6.28) asymptotically follows Gaussian distribution, i.e. ( N ð0; Sðqo ÞÞ under H0 ðN/NÞ zN ðqo Þ $$% : ð6:30Þ N ðM ðqo Þdq; Sðqo ÞÞ under H1 This indicates that a deviation in the system parameters is reflected into a change in the mean of zN. ^ and S ^ be consistent estimates of M(qo) (assumed to be full column rank) Let M and Sðqo Þ, respectively, obtained in the baseline phase (details in Basseville et al. 2000). Under the null (H0) hypothesis, the statistic QN, below, follows c2 distribution with rankM degrees of freedom, i.e. 2 ^K1 ^ ^ T ^K1 ^ K1 ^ T ^K1 Under H0 : QN bzT N ðqo ÞS M ðM S M Þ M S zN ðqo Þ wc ðrankM Þ: ð6:31Þ Based upon this, a suitable statistical test may be constructed for deciding (at a certain risk level) whether the residual sequence has mean that is significantly different from zero (for instance as in §6a). Fault identification. Fault identification could be similarly based upon consecutive tests of the above form, each one corresponding to each one of the parameter vectors qA, ., qD. Nevertheless, as with other methods, this may only work with faults of specific magnitudes but not for the continuum of fault magnitudes possible within each fault type. Fault estimation. The method, as such, is not immediately suitable for fault estimation. Phil. Trans. R. Soc. A (2007)


436

S. D. Fassois and J. S. Sakellariou kÂ

estimation of Qˆ A estimation residual characteristic quantity Q eAu[t]

functional model MA(k) (fault mode A)


current excitation structure response xu[t] yu[t] Su

kˆD

functional model MD(k) (fault mode D)

estimation residual eDu[t]

estimation of Qˆ D characteristic quantity Q

baseline phase

Figure 8. Schematic of the functional model-based method (the inspection phase is depicted outside the dashed box).

(c ) The functional model-based method (fmbm) The functional model-based method (fmbm) has recently been introduced (Sakellariou & Fassois 2002; Sakellariou et al. 2002a,b) in an attempt to (i) effectively tackle the problem of identifying faults of any type in the important case that the continuum of possible fault magnitudes is considered, (ii) accurately estimate fault magnitude, and (iii) provide a ‘combined’ solution to the subproblems of fault detection, identification and magnitude estimation. The method’s schematic is given in figure 8. Its cornerstone is its unique ability to accurately represent a structure in a given fault mode (fault type) for the mode’s continuum of fault magnitudes. This is achieved using a single representation (model) that is directly parametrized in terms of fault magnitude. This representation is referred as a stochastic functional model (FM ), and depending upon the problem treated, it may assume various forms (Sakellariou & Fassois 2002; Sakellariou et al. 2002a). Letting k represent fault magnitude (within a particular fault mode X), a simple linear form for the structural dynamics under fault mode X is the Functional AutoRegressive with eXogenous excitation (FARX) representation na X 2 MX ðaij ; bij ; sw ðkÞÞ : yk ½t C ai ðkÞ$yk ½tKi iZ1

Z

nb X

bi ðkÞ$xk ½tKi C wk ½t

wk ½t wiid

ð6:32Þ N ð0; s2w ðkÞÞ;

iZ0

with : ai ðkÞ b

p X

aij $Gj ðkÞ;

jZ1

bi ðkÞ b

p X jZ1

bij $Gj ðkÞ fault magnitude : k 2R: ð6:33Þ

This model form may be viewed as the union of a continuum of conventional ARX models, each one corresponding to a particular fault magnitude k. xk[t], yk[t] and wk[t] are the excitation, response and innovations (prediction error) signals, respectively, corresponding to the particular k (note that kZ0 corresponds to the healthy structure). It should be noted that the above model resembles the conventional ARX representation (equation (4.8) with C(B)h1), but explicitly accounts for the continuum of fault magnitudes within the fault mode it represents. This is Phil. Trans. R. Soc. A (2007)



437

accomplished by allowing its parameters and innovations variance s2w to be functions of the fault magnitude k. As indicated by equations (6.33), the model parameters are specifically assumed to belong to a p-dimensional functional subspace spanned by the mutually independent functions G1(k), ., Gp(k) (functional basis). The constants aij and bij designate the model’s AR and X, respectively, coefficients of projection. A suitable functional model, corresponding to each particular fault mode, is estimated in the baseline phase using data obtained from the structure under various (sufficiently large number of) fault magnitudes (details in Sakellariou et al. 2002a). Fault detection. Fault detection may be based upon the functional model (obtained in the baseline phase) corresponding to any specific fault mode (say X ). This model is now re-parametrized in terms of the currently unknown fault magnitude k and the innovations variance s2e (the coefficients of projection being available from the baseline phase; e[t] represents the re-parametrized model’s innovations—corresponding to w[t] in equation (6.32)) MX ðk; s2e Þ :

y½t C

na X

ai ðkÞ$y½tKi Z

iZ1

nb X

bi ðkÞ$x½tKi C e½t:

ð6:34Þ

iZ0

Estimates of k and s2e are obtained based upon the currently available signal(s) ðz u ÞN 1 and the nonlinear least squares estimator k^ b arg min RSSðkÞ b arg min k

k

N X tZ1

e2 ½t

s^2e Z

N 1 X e^2 ½t; N tZ1

ð6:35Þ

with RSS standing for residual sum of squares. Owing to the non-quadratic nature of the RSS criterion with respect to k, the estimator is nonlinear and thus obtained using the Gauss–Newton and Levenberg–Marquardt schemes (Ljung 1999, pp. 326–329). Assuming that the structure is indeed under a fault belonging to the specific fault mode (or else healthy), the estimator may be shown to be asymptotically (N/N) Gaussian distributed, with mean equal to its true k value and variance s2k ðk^ wN ðk; s2k ÞÞ, the latter corresponding to the Cramer–Rao lower bound (Sakellariou et al. 2002a). This variance is also estimated, with its estimate ^ s^2k shown to be asymptotically (N/N) uncorrelated with k. Since the healthy structure corresponds to kZ0 (zero fault magnitude), fault detection may be based upon the hypothesis testing problem ) H0 : k Z 0 ðnull hypothesis-healthy structureÞ; ð6:36Þ H1 : k s0 ðalternative hypothesis-faulty structureÞ: Under the hypothesis that the structure is indeed under a fault belonging to the specific fault mode (or else healthy), the random variable t, below, follows a t distribution with NK2 degrees of freedom (as it may be shown to be the ratio of a standardized normal random variable over the square root of an independent and normalized chi-square random variable with NK2 degrees of freedom; see appendix A), i.e. k^ t b wtðN K2Þ; ð6:37Þ s^k Phil. Trans. R. Soc. A (2007)


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S. D. Fassois and J. S. Sakellariou ft

a /2

a /2

0

ta /2 H1 accepted (fault)

t1–a /2

H0 accepted (no fault)

H1 accepted (fault)

Figure 9. Statistical hypothesis testing based upon a t-distributed statistic (two-tailed test).

which leads to the following test at the a (type I) risk level (figure 9): ta=2 ðN K2Þ% t% t1Kða=2Þ ðN K2Þ 0 H0 is accepted ðhealthy structureÞ Else

0

H1 is accepted ðfaulty structureÞ

) ;

ð6:38Þ with ta and t1K(a/2) designating the t distribution’s (with the indicated degrees of freedom) a and 1K(a/2) critical points, respectively. Fault identification. Once a fault is detected, fault identification is achieved through successive estimation (using the current data ðz u ÞN 1 ) and validation of the re-parametrized FARX models MX ðk; s2e Þ (XZA, ., D) (equation (6.34)) corresponding to the various fault modes. The procedure stops as soon as a particular model is successfully validated; the corresponding fault mode then being identified as current. Model validation may be based upon statistical tests examining the hypothesis of excitation and residual sequence uncrosscorrelatedness and/or residual uncorrelatedness (see §6b(iii)). Fault estimation. Once the current fault mode has been identified, an interval ^ s^2k estimates estimate of the fault magnitude is constructed based upon the k, obtained from the corresponding re-parametrized FARX model. Thus, using equation (6.37), the interval estimate of k at the a risk level is h i k interval estimate : k^ C ta=2 ðN K2Þ$^ sk ; k^ C t1Kða=2Þ ðN K2Þ$^ sk : ð6:39Þ 7. Case studies (a ) Case study A: fault detection in an aircraft stiffened panel In this case study, fault detection for the skin of an aircraft stiffened panel is considered via the coherence measure-based method (§5c). For a detailed presentation, the reader is referred to Rizos et al. (2002). The panel and the experimental set-up. The structure used in the study (figure 10a) is a lightweight (887 g) trapezoidal 2024-T3-bare aluminium stiffened (via ribs) panel of dimensions 450!320!320 mm (large base–small base–height) Phil. Trans. R. Soc. A (2007)


Time-series methods for fault detection (a)

(b)

439

(c)

B C

A

F

skin fault

Figure 10. Aircraft stiffened panel: (a) the panel with the measurement positions (front view), (b) the panel with the skin fault area (rear view) and (c) part of the experimental set-up (Rizos et al. 2002).

obtained from a Vought A-7 Corsair aircraft. The fault considered is a 25 mm long sawcut (figure 10b) on the skin (skin thicknessw2 mm). The panel is suspended through strings and tested under free–free boundary conditions (figure 10c). The excitation is a horizontal random Gaussian force applied at point F (figure 10a) through an electromechanical shaker equipped with a stinger. The exerted force is measured via an impedance head, and the resulting acceleration (at points A, B and C; figure 10a) is measured via miniature accelerometers (sampling frequency fsZ6 kHz). A series of experiments are performed for both the healthy and faulty states of the panel. The coherence measure is estimated within the [0.38, 2.00] kHz frequency range using NZ492! 103 samples per signal (frequency resolution 0.3662 Hz, number of non-overlapped segments KZ30, Hanning windowing). Baseline phase. Interval estimates of the coherence measure corresponding to the three (points A, B and C) excitation–response pairs are obtained from experimental vibration test data with the healthy panel. Inspection phase. Interval estimates of the coherence measure (three excitation–response pairs) are re-obtained for the panel in its current (faulty) state. Figure 11a depicts the coherence measure interval estimates (at the aZ0.05 risk level) for each panel state (healthy–faulty) and for each one of the vibration response measurement positions (points A, B and C). As expected, the coherence measure decreases with fault occurrence. The test statistic (left-hand side of equation (5.18)) is depicted in figure 11b and confirms (for positions A and C) the statistically significant reduction in the coherence measure, and, hence, fault presence by being greater than the critical point corresponding to the aZ0.05 risk level. (b ) Case study B: fault detection and estimation in an aircraft skeleton structure In this case study, fault detection and estimation in a scale aircraft skeleton structure is considered via the functional model-based method. For a detailed presentation, the reader is referred to Sakellariou et al. (2002a). The structure and the experimental set-up. The scale aircraft skeleton structure used (figure 12) has been constructed at the University of Patras based upon the Garteur SM-AG19 specifications (see Balmes & Wright 1997). It consists of six solid beams representing the fuselage, the wings, the horizontal and vertical Phil. Trans. R. Soc. A (2007)


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coherence measure

1630

(b)

12

test statistic

(a)

4

healthy:

damaged:

1615 maximum possible coherence measure 1600

–4

healthy versus damaged damage

damage

damage

no damage

no damage

no damage

point A

point B

point C

Figure 11. Aircraft stiffened panel: coherence-based measure method applied to three measurement positions (A, B and C): (a) interval estimates of the coherence measure and (b) fault detection results (!, test statistic; ., critical value).

point B

point A

Figure 12. Aircraft scale skeleton structure: experimental set-up indicating the force/vibration measurement position (point A) and the fault position (point B; Sakellariou et al. 2002a).

stabilizers, and the right and left wing tips. All parts of the skeleton are constructed from standard aluminium and are jointed together via two steel plates and screws (total skeleton massw50 kg). The faults considered correspond to the placement of a variable number of small masses (simulating local elasticity reductions), of approximately 6.5 g each, at point B of the structure (figure 12). Each such fault is designated as F k, with k representing the fault magnitude (grams of added mass). The corresponding state of the structure is designated as S k. F 0 designates a zero magnitude fault, thus referring to the healthy state of the structure (S o). Fault detection and estimation are based upon vibration testing of the structure under free–free boundary conditions (figure 12). Excitation (in the form of a random Gaussian force) is applied vertically, at the right wing tip (point A), through an electromechanical shaker equipped with a stinger. The exerted force, Phil. Trans. R. Soc. A (2007)


441

Time-series methods for fault detection F37.7

F0 (b) 0.02

0.02

0.5

0

RSS/SSS (%)

RSS/SSS (%)

(a) 0.04

0.5

nominal

0

65 parameter k (g)

130

0.01

37 37.5 38 38.5

fault F37.7

0

65 parameter k (g)

130

Figure 13. Aircraft scale skeleton structure: fault detection/estimation results—residual sum of squares normalized by the series sum of squares versus k: (a) healthy structure and (b) fault F 37.7 (the dashed vertical lines indicate the true k; the shaded strips in the zooms indicate corresponding interval estimates).

along with the resulting vertical vibration accelerations, are measured via an impedance head and lightweight accelerometers, respectively (sampling frequency fsZ128 Hz, signal length NZ1000 samples). Baseline phase. AutoRegressive with eXogenous excitation (ARX) modelling of the healthy structure based upon 4 s (NZ1000 sample) long excitation and single response signals leads to an ARX(18,18) representation. Fault mode modelling (for the single fault mode characterized by mass placement at point B) is based upon signals obtained from a series of 21 experiments, one corresponding to the healthy structure (kZ0 g of added mass) and the rest corresponding to various fault magnitudes (faults F k with k2[6.5, 130] g; increment dkz6.5 g). The functional ARX (FARX) modelling procedure leads to a FARX(18,18) model (equations (6.32) and (6.33)) with functional basis consisting of the first pZ7 Chebyshev type II polynomials (Abramowitz & Stegun 1970). Inspection phase. Two test cases, the first corresponding to the healthy structure (F 0) and the second to a fault characterized by added mass of 37.7 g (F 37.7), are considered. Fault detection and estimation results are, for each test case, pictorially presented in figure 13. A normalized version of the cost function of equation (6.35) is, for each case, shown as a function of the fault magnitude k. The corresponding zooms depict the true value of the added mass (dashed line), along with the corresponding interval estimate (shaded strip; the middle line indicates the point estimate and the left and right lines the lower and upper confidence bounds at the aZ0.05 risk level). (i) Test case I (healthy structure). As it would have been rightly expected, no fault is detected in the first case, as the fault magnitude’s interval estimate ðk^Z 4:8356 !10K5 G0:1571Þ includes the nominal kZ0 value (figure 13a). The proximity of the k point estimate to the true, kZ0, value is quite remarkable. (ii) Test case II (F 37.7 fault; added mass of 37.7 g). A fault is clearly detected in this case, as the fault magnitude’s interval estimate ðk^Z 37:7530G0:5253Þ does not include the kZ0 value (figure 13b). The accuracy attained in estimating the fault magnitude is, again, excellent. Phil. Trans. R. Soc. A (2007)


442


x1

k1

faults F kk2

k2 m1 c1

F : input

x2 : output k3 m2 c3

Figure 14. Two-DOF system with cubic stiffness.

(c ) Case study C: fault detection and identification in a two-DOF system with cubic stiffness In this case study, fault detection, identification and estimation in a simple two degree-of-freedom nonlinear system characterized by cubic stiffness (spring k3; figure 14) is considered through the functional model-based method. For a detailed presentation, the reader is referred to Sakellariou & Fassois (2002). The system and the faults. System simulation is based upon discretization of the equations of motion with time-step TsZ6.666!10K4 s (sampling frequency fsZ 1500 Hz). Fault detection and identification is based upon measurement of the force excitation F (subsequently designated as x) and the vibration displacement response x 2 (subsequently designated as y). The faults considered correspond to stiffness changes in k 2 (k 2 fault mode; figure 14). Each individual fault is represented as Fkk2 , with the subscript k 2 indicating the fault mode and the superscript k the exact fault magnitude. Baseline phase. A nonlinear ARX (NARX) model of orders (4,4) is used for representing the healthy system (details in Sakellariou & Fassois 2002) 4 X y½t C ai $y½tKi C a5 $y½tK4$y½tK3 C a 6 $y 2 ½tK4 C a 7 $y2 ½tK4$y½tK3 iZ1

C a8 $y 3 ½tK4 C a 9 $y2 ½tK4y½tK2 Z b1 $x½tK4 C e½t:

Fault mode modelling (for the Fkk 2 fault mode) is based upon a series of 17 experiments, one corresponding to the healthy system (kZ0% variation in k 2) and the rest corresponding to various fault magnitudes (faults Fkk 2 with k2[K32,32%]; increment dkZ4% change in k 2). The signals used are, in all cases, NZ2000 samples long. A functional NARX (FNARX) model with functional basis consisting of the first two (zeroth and first degree, thus pZ2) Chebyshev type II polynomials (Abramowitz & Stegun 1970) is selected for representing the Fkk 2 fault mode. Inspection phase. Two test cases are considered via Monte Carlo experiments (10 runs per case). Monte Carlo fault detection and estimation results are pictorially presented in figure 15 and fault identification results in figure 16 (type I risk aZ0.05). (i) Test case I (healthy structure). In this case, the fault magnitude interval estimate includes the kZ0 value in each one of the 10 runs (figure 15a), thus no fault is (rightly) detected. The excellent accuracy achieved in estimating the correct k value in all 10 runs is remarkable. Phil. Trans. R. Soc. A (2007)



443

fault magnitude k (%)

(a) 0.2 0.0 – 0.2

fault magnitude k (%)

(b) 3.5 3.0 2.5

Q statistic

Figure 15. Two-DOF system with cubic stiffness—fault detection/estimation results: (a) test case I (healthy system) and (b) test case II (fault Fk32 ) (10 Monte Carlo runs per case; the solid horizontal lines designate true fault magnitude, the circles corresponding point estimates and the boxes interval estimates at the aZ0.05 risk level).

60 40 20 0

Figure 16. Two-DOF system with cubic stiffness: Q statistic (bars) and the critical point (– – –) at the aZ0.05 risk level for test case II (values below the critical point indicate Fk 2 fault mode identification—10 Monte Carlo runs).

(ii) Test case II (Fk32 fault; 3% reduction in k 2). In this case, a small magnitude fault is injected. Yet, its detection is again remarkably accurate ðk^Z 3:02G7:12 !10K2 Þ for all 10 runs (figure 15b). In addition, the Q statistic (equation (6.20)) corresponding to the Fkk 2 fault mode is, for all 10 runs, below the critical point (figure 16), thus correctly identifying (isolating) the current fault in the k 2 stiffness. 8. Concluding remarks In this paper, the principles and techniques of time-series methods for fault detection, identification and estimation in vibrating structures were presented, and certain new methods were introduced. As demonstrated, time-series methods offer inherent accounting of uncertainty, statistical decision making and the relaxation of the requirement for complete structural models, including physical or finite element models. The methods were classified as non-parametric or parametric, reflecting upon the way each method’s characteristic quantity is constructed (see figure 17 for a pictorial classification of time-series methods). Non-parametric methods are generally simpler, but mainly focus on the fault detection subproblem. Parametric methods, on the other hand, are more elaborate, but offer the possibility of increased accuracy along with more effective tackling of the fault identification and estimation subproblems. Phil. Trans. R. Soc. A (2007)


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S. D. Fassois and J. S. Sakellariou time-series fdi methods

non-parametric

PSD-based method [Q = S(w)]

FRF-based method [Q = |H( jw)|]

residual variance [Q = s 2Xu ]

parametric

coherence measurebased method [Q = G ]

parameter-based methods [Q = f (q)]

residual uncorrelatedness [Q = [r1,r2, ..., rr]t ]

residual-based methods [Q = f ((eXu)1N)]

likelihood function [Q = L(e)]

the functional model-based method [Q = k, Q = f ((eXu)1N)]

'residuals' associated with subspace identification [Q = zN(q0)]

Figure 17. Classification of time-series fdi methods.

Parametric methods were further classified as: (i) model parameter based, in which the characteristic quantity is a function of the model parameter vector, (ii) residual based, in which the characteristic quantity is a function of model residuals, and (iii) the functional model based, in which a characteristic quantity is selected as a model parameter and another as a function of the model residuals. The practicality and effectiveness of the methods were demonstrated through brief presentations of three case studies pertaining to fault detection, identification and estimation in an aircraft panel, a scale aircraft skeleton structure and a simple nonlinear simulated structure. It is evident that due to their stated advantages, time-series methods will attract increasing attention in the future. The fault detection subproblem has received most of the attention thus far. It is, however, clear that significant work has to be devoted to the fault identification and magnitude estimation subproblems as well. On the other hand, one of the strengths of time-series methods, which is their reliance upon relatively simple (typically partial) mathematical models identified from operating data records (as opposed to detailed physical or finite element models), may generally constitute a limiting factor with regard to the latter two subproblems. It seems that the limits of the methods in this respect have not been sufficiently explored. An additional issue that merits attention is the need for behavioural datasets corresponding to various fault conditions. This may not be necessarily possible (also see the discussion in §3a), and although the problem may be handled via data obtained from either laboratory scale models or mathematical (like finite element) models, it seems to set a practical limitation for (at least) certain cases. The fact that this primarily concerns fault identification and estimation (but not necessarily fault detection) is certainly encouraging, but it appears practically important to explore potential approaches for circumventing it. In addition, further development of non-parametric and parametric methods suitable for the multivariate case is expected to be sought, so that information from more measurement locations may be included in the decision making. Furthermore, Phil. Trans. R. Soc. A (2007)



445

methods suitable for non-stationary structures (structures with time-varying properties) as well as nonlinear structures are expected to be further developed. Other important issues that need to be addressed include effective fault detection and identification under varying operating and/or environmental conditions; a difficult but practically important problem. Also significant is the transition from the current, mainly Gaussian, to a broader non-Gaussian time-series framework that may be more appropriate for certain applications. The authors are indebted to three anonymous referees whose comments helped in improving the manuscript.

Appendix A. Central limit theorem and statistical distributions associated with the normal The central limit theorem (CLT) (Stuart & Ord 1987, p. 273; Nguyen & Rogers 1989, vol. I, p. 420; Montgomery 1991, p. 46). Let Z1,Z2, ., Zn designate mutually independent random variables each with mean mk and (finite) variance P s2k . Then, for n/N, the distribution of the random variable X Z nkZ1 Zk Pn approaches Pthe Gaussian distribution with mean EfXgZ kZ1 mk and variance varðXÞZ nkZ1 s2k . The chi-square distribution. Let Z1,Z2, ., Zn designate mutually independent, normally distributed, random variables, each with mean mk and standard deviation sk. Then the sum n

X Zk Kmk 2 XZ ; ðA 1Þ sk kZ1 is said to follow a (central) chi-square distribution with n degrees of freedom (Xwc2(n)). Its mean and variance are E(X )Zn and var(X )Z2n, respectively. Note that imposing p equality constraints among the random variables Z1,Z2, ., Zn reduces the set’s effective dimensionality, and thus the number of degrees of freedom, by p (Stuart & Ord 1987, pp. 506–507). For n/N, theP c2(n) distribution tends to normality (Stuart & Ord 1987, p. 523). The sum X Z nkZ1 ðZk =sk Þ2 is said to follow non-central chi-square distribution with n degrees of freedom and non-centrality parameter lZ(mk/sk)2. This distribution is designated as c2(n;l) (Nguyen & Rogers 1989, vol. II, p. 33). Let x2Rn follow n-variate normal distribution with zero mean and covariance S (xwN (0,S)). Then the quantity xTSK1x follows (central) chi-square distribution with n degrees of freedom (Stuart & Ord 1987, pp. 486–487; So¨derstro ¨m & Stoica 1989, p. 557; Gertler 1998, p. 120). Student’s t-distribution. Let Z be the standard (zero mean and unit variance) normal variable. Let X follow a (central) chi-square distribution with n degrees of freedom and be independent of Z. Then the ratio Z T Z pffiffiffiffiffiffiffiffiffiffi ; ðA 2Þ X=n is said to follow a Student or t (central) distribution with n degrees of freedom (central, because it is based on a central chi-square distribution; Nguyen & Rogers 1989, vol. II, p. 34). Its mean and variance are E(T )Z0 (nO1) and var(T )Z n/(nK2) (nO2), respectively (Stuart & Ord 1987, p. 513). Phil. Trans. R. Soc. A (2007)


446


The (central) t distribution approaches the standard normal distribution N (0,1) as n/N (Stuart & Ord 1987, p. 523). Fisher’s F-distribution. Let X1 and X2 be mutually independent random variables following (central) chi-square distributions with n1 and n2 degrees of freedom, respectively. Then the ratio FZ

X1 =n1 ; X2 =n2

ðA 3Þ

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