Time series methods for fault detection and identification in vibrating structures

Transcription

1 Time series methods for fault detection and identification in vibrating structures By Spilios D. Fassois and John S. Sakellariou Stochastic Mechanical Systems (SMS) Group Department of Mechanical & Aeronautical Engineering University of Patras, GR Patras, Greece sms Revised Version February 6, 2006 Abstract 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 non-linear simulated structure. Keywords: Structural fault diagnosis; vibration based methods; fault detection and identification; fault estimation; vibrating structures; time series methods; statistical methods. 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 t = 1, 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 x N 1 = {x[i], i = 1, 2,..., N}. A hat designates estimator/estimate of the indicated quantity; for instance ˆθ is an estimator/estimate of θ. 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. Copyright c by S.D. Fassois and J.S. Sakellariou. All rights reserved. PLEASE ADDRESS ALL CORRESPONDENCE TO: Prof. S.D. Fassois, Department of Mechanical and Aeronautical Engineering, University of Patras, GR Patras, Greece. Tel/Fax (direct): (++ 30) ; Tel/Fax (central): fassois@mech.upatras.gr Philosophical Transactions of the Royal Society: Mathematical, Physical and Engineering Sciences, Vol. 365, pp , doi: /rsta

2 2 S.D. Fassois and J.S. Sakellariou 1. Introduction Fault detection, identification (localization), and (magnitude) estimation (collectively referred to 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 aging infrastructure. For these reasons, a number of nondestructive fault detection techniques have been developed over the past several years (for instance see Braun 1986, Doherty 1987, Doebling et al. 1996, Salawu 1997, Natke & Cempel 1997, Doebling et al. 1998, 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, and are, furthermore, 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, Salawu 1997, Doebling et al. 1998, 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 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 behavioral 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 vibration based 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 analyzing their observed and future behavior. 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: (a) No requirement for physical or finite element models. (b) 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. (c) Inherent accounting for (measurement, environmental, and so on) uncertainty through statistical tools. (d) Statistical decision making with specified performance characteristics.

3 Time series methods for fault detection and identification 3 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 be also combined with complete structural models; this subject will not be, however, treated in the present paper. 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 non-linear simulated structure. The rest of the paper is organized as follows: The problem statement is presented in section 2, and the structure of time series methods in section 3. An overview of time series model representations is given in section 4. Non-parametric methods are presented in section 5, and parametric methods in section 6. Three case studies are presented in Section 7, and concluding remarks are summarized in Section Problem statement Let S o designate the vibrating structure under study in its nominal (healthy) state. Also 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 SA k, with the subscript designating fault type and the superscript fault magnitude. The symbol FA k 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 on line 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, behavioral data, that is force excitation x u [t] and/or vibration response y u [t] (t = 1,2,...,N) signals, are obtained from the structure in its current (unknown) state (notice that t refers to discrete time, with the corresponding analog time being (t 1) T s and T s 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 (for instance see Lütkepohl 1991),

4 4 S.D. Fassois and J.S. Sakellariou while also requiring multivariate statistical decision making procedures. As a consequence this extension is not necessarily a straightforward task (see Basseville & Nikiforov 1993 and Gertler 1998). Let the complete signal records be obtained during inspection designated as (x u ) N 1 and (y u ) N 1, and let the excitation response signals be collected into the vector z u [t] = [x u [t] y u [t]] T (t = 1,2,...,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 (z u ) N 1. Notice that all collected signals need to be suitably pre-processed (Fassois 2001, Doebling et al. 1998). 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 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 crosscorrelation among them. Given the data (z u ) N 1, collected during the inspection phase, the characterization of the current state of the structure, which is the problem generally referred to as fault detection and identification (fdi), may be thought of as consisting of three conceptually distinct subproblems: (a) 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 u = S o (the current structure is healthy) or S u S o (the current structure is faulty). (b) Fault identification (also referred to 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 ). (c) Fault estimation is the subproblem concerning estimation of the exact fault magnitude (damage level). 3. The structure of time series methods 3.1 The operational viewpoint From an operational viewpoint the tackling of the aforementioned subproblems via time series methods requires (in addition to (z u ) N 1 ) the availability of similar data records from the nominal (healthy) and each considered faulty state of the structure. That is, in present terms, z o [t] (t = 1,2,...,N) corresponding to S o, as well as z A [t],...,z D [t] (t = 1,2,...,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

5 Time series methods for fault detection and identification 5 models or mathematical like finite element models) capable of providing reasonably accurate representations of the actual structural dynamics under various conditions may be used. These data sets 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 been already referred to as the inspection phase. 3.2 The conceptual viewpoint From a conceptual viewpoint time series methods include analysis and statistical decision making. (a) The analysis part includes signal or system characterization and parametric or non-parametric modelling. The aim is the extraction, from each data set, of a characteristic quantity, designated as Q = Q(z N 1 ) (a function of z N 1 ), which is instrumental in the decision making part. (b) The statistical decision making is the part where decisions are made by comparing, via formal statistical hypothesis testing procedures, the current characteristic quantity Q u to its counterparts Q o, Q A,...,Q D pertaining to the various possible structural states (o,a,...,d, respectively). Fault detection is then formulated as a binary composite hypothesis testing problem which may be generally expressed as: H o : Q o Q u (null hypothesis - healthy structure) (3.1) H 1 : Q o Q u (alternative hypothesis - faulty structure) with 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: H A : Q A Q u (hypothesis A - fault type A).. H D : Q D Q u (hypothesis D - fault type D) (3.2) Augmenting this formulation with the original null hypothesis H o 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 α and also referred to as the type I risk is the probability of rejecting the null hypothesis H o although it is true (false alarm). The second probability designated as β and also referred to as type II risk is the probability of accepting the null hypothesis H o although it is not true (missed fault). The designs presented in this paper are based upon selected type I error occurrence probability (α), and utilize the probability density function of a relevant random quantity under the null (H o ) 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 α it should be born in mind that a decrease/increase in it results in a corresponding increase/decrease in β. The reader is referred

6 6 S.D. Fassois and J.S. Sakellariou Methods Advantages Disadvantages Non-parametric Simplicity Potentially reduced accuracy methods Computational efficiency Some user expertise required Parametric Improved parsimony Increased complexity methods Potentially increased accuracy Computationally involved Increased user expertise required Table 1. Comparison of the main characteristics of non-parametric and parametric time series methods. to references such as Basseville & Nikiforov (1993, subsection 4.2) and Montgomery (1991, subsection 3.3) for details on statistical hypothesis testing. 3.3 Types of time series methods The diagram of a general time series fdi method is depicted in Figure 1. Notice that the baseline phase is indicated by the dashed frame, while the inspection phase is in its exterior. Also notice 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 to as excitation-response methods, while those falling into the second are referred to 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 next section; 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 next section). A rough comparison of the main characteristics of non-parametric and parametric time series methods is presented in Table Overview of time series representations The objective of this section is to provide a concise overview of Gaussian time series representations used in fault detection and identification. As already indicated, for the sake of simplicity, the univariate case (scalar excitation and response signals) is considered. Let h[t] designate the impulse response function describing the excitationresponse 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: y[t] = h[τ] x[t τ] + n[t] (convolution summation + noise) (4.1) τ=0 Using the backshift operator B (B i y[t] = y[t i]) the above may be re-written as: y[t] = H(B) x[t] + n[t] H(B) = h[τ] B τ (4.2) with H(B) designating the structure s discrete-time transfer function. τ=0

7 Time series methods for fault detection and identification 7 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: µ y = E{y[t]} γ yy [τ] = E{y[t] y[t τ]} S yy (ω) = γ yy [τ] e jωτts (4.3) τ= with E{ } designating statistical expectation, j the imaginary unit, τ time lag, ω frequency in rad/sec, and T s the sampling period. Notice that γ yy [0] is the variance (σ 2 y) of the response y[t], and S yy (ω) its auto spectral density defined as the Fourier transform of the autocovariance (Box et al pp , Kay 1988 p. 3). These quantities may be related to those of x[t] and n[t] as follows: µ y = H(jω) ω=0 µ x S xy (jω) = H(jω) S xx (ω) S yy (ω) = H (jω) S xy (jω) + S nn (ω) } S yy (ω) = H(jω) 2 S xx (ω) + S nn (ω) (4.4) where H(jω) is the structure s frequency response function (frf) obtained by setting B = e jωts in the second of Equations (4.2), designates complex conjugate, complex magnitude, S nn (ω) the noise auto spectral density, while the (complex) cross spectral density S xy (jω) is defined as the Fourier transform of the cross covariance function γ xy [τ] = E{x[t] y[t τ]}. Notice that, by convention, complex functions are designated by incorporating the imaginary unit in their argument. 4.1 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] w[t] (zero-mean uncorrelated (white) sequence) and n[t] 0. Non-parametric representations. The response may be non-parametrically described via its mean µ y and autocovariance γ yy [τ] or the auto spectral density S yy (ω) = H(jω) 2 σw, 2 with σw 2 designating the excitation s variance (from the rightmost of Equations (4.4)). Time-frequency representations or polyspectra or wavelet analysis (Staszewski 1998, Peng & Chu 2004) may be used in the non-stationary or non-linear cases. Parametric representations. The parametrization of Equation (4.1) leads to an AutoRegressive Moving Average (ARMA) representation of the form (Box et al pp ): ( ) ( ) na nc na nc y[t] + a i y[t i] = w[t] + c i w[t i] 1 + a i B i y[t] = 1 + c i B i w[t] i=1 i=1 i=1 i=1 A(B) y[t] = C(B) w[t] w[t] iid N(0,σ 2 w) (4.5) with a i, c i, A(B), 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. n a, n c are the model s AR and MA orders, respectively. The model parameter vector is θ = [coef A, coef B] T. It should be noted that the excitation w[t] may be shown (Box et al p. 134, Ljung 1999 p. 70) to coincide with the model-based one-step-ahead prediction error, and is also referred to as the residual or innovations. Other parametrizations are generally possible (especially in the multivariate case for instance

8 8 S.D. Fassois and J.S. Sakellariou see Söderström & 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 subsection 6.1 and subsection 6.2.4). Alternatively, Equation (4.1) may be set into state space form (Söderström & Stoica 1989 p. 157, Box et al pp ): ψ[t + 1] = A ψ[t] + v[t] y[t] = C ψ[t] v[t] iid N(0,Σ v ) (4.6) with ψ[t] designating the model s state vector and v[t] a zero-mean uncorrelated vector sequence with covariance Σ v. The model parameter vector may be selected as θ = [(vec A) T,(vec C) T ] 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 subsection for a selection that is invariant under changes in the state space basis). Time-varying versions of the above (characterized by time-dependent parameters) may be used in the nonstationary case (Petsounis & Fassois 2000, Poulimenos & Fassois 2006), while various non-linear models, like Nonlinear ARMA (NARMA) models, may be used in the non-linear case (Leontaritis & Billings 1985, Sakellariou & Fassois 2002). 4.2 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 µ x, µ y and the auto and cross covariances γ xx [τ], γ yy [τ], γ xy [τ] (equivalently the auto and cross spectral densities S xx (ω), S yy (ω), S xy (jω)). 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): γ 2 (ω) = S xy(jω) 2 S xx (ω) S yy (ω) = S nn(ω) H(jω) 2 S xx(ω) γ 2 (ω) [0,1] (4.7) with S nn (ω) 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 non-linear cases. Parametric representations. The parametrization of Equation (4.1) may lead to an AutoRegressive Moving Average with exogenous excitation (ARMAX) representation of the form (Söderström & Stoica 1989 p. 149, Ljung 1999 p. 83, Fassois 2001): na nb nc y[t] + a i y[t i] = b i x[t i] + w[t] + c i w[t i] i=1 A(B) y[t] = B(B) x[t] + C(B) w[t] y[t] = B(B) A(B) i=1 x[t] + C(B) A(B) w[t] }{{} n[t] i=1 w[t] iid N(0,σ 2 w) (4.8) with parameter vector θ = [coef A, coef B, coef C] T. A(B), B(B), C(B) are the AR, X, and MA polynomials, respectively, and na, nb, 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 to as residual or innovations.

9 Time series methods for fault detection and identification 9 Alternatively, it may lead to the Box-Jenkins representation (Söderström & Stoica 1989 pp , Ljung 1999 p. 87, Fassois 2001): y[t] = B(B) A(B) x[t] + C(B) D(B) w[t] }{{} n[t] w[t] iid N(0,σ 2 w) (4.9) with parameter vector θ = [coef A, coef B, coef C, coef D] T. On the other hand, the Output Error (OE) representation models the excitation-response dynamics (Ljung p. 85, Fassois 2001): with parameter vector θ = [coef A, coef B] T. y[t] = B(B) x[t] + n[t] n[t] : auto correlated zero-mean (4.10) A(B) A state space representation may be also 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 for additional versions): ψ[t + 1] = A ψ[t] + B x[t] + v[t] y[t] = C ψ[t] + D x[t] v[t] iid N(0,Σ v ) (4.11) with ψ[t] designating the model s state vector and v[t] a zero-mean uncorrelated vector sequence with covariance Σ v. The model parameter vector may be selected as θ = [(vec A) T,(vec B) T,(vec C) T,(vec D) T ] 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 non-linear models, like Nonlinear ARMAX (NARMAX) models (Leontaritis & Billings 1985, Chen & Billings 1989, Sakellariou & Fassois 2002; also see the case study of subsection 7.3), neural network models (Masri et al. 2000), or frequency-domain ARX models (Adams 2002) may be used in the non-linear case. 5. Non-parametric methods Non-parametric methods are those in which the characteristic quantity Q is constructed based upon nonparametric time series representations (see section 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 sequel: A spectral density function based method (Sakellariou et al. 2001), a frequency response function (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), Staszewski (1998), Worden et al. (2000), Worden & Manson (2003), Peng & Chu (2004), and references therein. 5.1 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 Equations (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

10 10 S.D. Fassois and J.S. Sakellariou Q = S yy (ω) = S(ω). Let the vibration response data record y N 1 be segmented into K non-overlapping segments (each of length L). Then the Welch auto spectral density estimator (sample spectrum) is defined as (Kay 1988 p. 76): Ŝ(ω) = 1 K K I (i) (ω) with I (i) (ω) = 1 L 2 a[t] y (i) [t] e jωtts L i=1 with the superscript i designating the i-th segment, and a[t] a proper data window (notice that estimators/estimates are designated by a hat). This estimator, multiplied by 2K and normalized by the true auto spectral density S(ω), may be shown to follow a (central) chi-square distribution with 2K degrees of freedom (see Appendix for the distribution s definition), or explicitly (Kay 1988 p. 76, Bendat & Piersol 2000 p. 309): 2KŜ(ω) S(ω) t=1 (5.1) χ 2 (2K) (5.2) Fault detection. Fault detection is based upon confirmation of statistically significant deviations (from the nominal/healthy) in the current structure s auto spectral density function at one or more frequencies through the hypothesis testing problem (for each ω): H o : S u (ω) = S o (ω) (null hypothesis - healthy structure) H 1 : S u (ω) S o (ω) (alternative hypothesis - faulty structure) (5.3) Toward this end, owing to Equation (5.2), the following quantity follows (for each frequency ω) F distribution with 2K,2K degrees of freedom (ratio of normalized chi-square distributions; see Appendix): F = Ŝo(ω)/S o (ω) Ŝ u (ω)/s u (ω) F(2K,2K) with Ŝo(ω) and Ŝu(ω) 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 (H o ) hypothesis (current structure is healthy), the true auto spectral densities coincide (S o (ω) = S u (ω)) and the above quantity becomes: Under H o : F = Ŝo(ω) Ŝ u (ω) F(2K,2K) (5.4) Equality of the true auto spectral densities S o (ω) and S u (ω) is then examined at the α risk level (type I error probability of α) via the statistical test: f α 2 (2K,2K) F f 1 α 2 (2K,2K) ( ω) = H 0 is accepted (healthy structure) Else = H 1 is accepted (faulty structure) (5.5) with f α 2, f 1 α 2 designating the F distribution s α 2 and 1 α 2 critical points (f α is defined such that Prob(F f α ) = α) (Figure 3). 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.

11 Time series methods for fault detection and identification 11 Remark. Notice that signal scaling is particularly important in order to properly account for different excitation levels, while the environmental conditions should be kept constant. 5.2 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 utilizes the frequency response function (frf) magnitude as its characteristic quantity (Q = H(jω) ) (see the first central of Equations (4.4)). Yet, it may be also used in case the excitation is unavailable, but more than one responses 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 S xx (ω) and S xy (jω), respectively, using the first central of Equations (4.4): Ŝxy(jω) Ĥ(jω) = Ŝ xx (ω) This estimator may be shown to follow a distribution approximated as Gaussian (for instance see Bendat & Piersol 2000 p. 338; alternatively see Koopmans 1974 p. 284 for a discussion that is based upon the F distribution): (5.6) Ĥ(jω) N( H(jω),σ2 (ω)) with σ 2 (ω) 1 γ2 (ω) γ 2 (ω) 2K H(jω) 2 (5.7) with the mean coinciding with the true frf magnitude and the indicated variance (γ 2 (ω) designates the coherence function see Equation (4.7)). 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 ω): H o : δ H(jω) = H o (jω) H u (jω) = 0 (null hypothesis - healthy structure) H 1 : δ H(jω) = H o (jω) H u (jω) 0 (alternative hypothesis - faulty structure) (5.8) Toward this end the difference of the frf magnitude estimates in the nominal and current states is considered ( Ĥo(jω) is obtained in the baseline phase, Ĥu(jω) in the inspection phase) which, due to Equation (5.7) and mutual independence of Ĥo(jω), Ĥu(jω) (owing to the fact that the estimates corresponding to the nominal and current states are based upon distinct data records), follows Gaussian distribution with mean δ H(jω) = H o (jω) H u (jω) (the true difference) and variance δσ 2 (ω) = σo(ω) 2 + σu(ω), 2 that is: δ Ĥ(jω) = Ĥ o (jω) Ĥu(jω) N ( δ H(jω),δσ 2 (ω) ) (5.9) Under the null (H o ) hypothesis (current structure is healthy), the true frf magnitudes coincide ( H o (jω) = H u (jω) ) and so do the respective estimator variances (σo(ω) 2 = σu(ω)). 2 Hence δ Ĥ(jω) follows the Gaussian distribution: Under H o : δ Ĥ(jω) = Ĥo(jω) Ĥu(jω) N(0,2σo(ω)) 2 (5.10) The variance σo(ω) 2 is generally unknown, but may be estimated in the baseline phase through Equation (5.7). Treating this estimate as a fixed quantity, that is a quantity characterized by negligible variability (which is

12 12 S.D. Fassois and J.S. Sakellariou reasonable for estimation based upon a large number of samples), the equality of H o (jω) and H u (jω) is examined at the α (type I) risk level through the statistical test (Figure 4): δ Ĥ(jω) Z 1 α 2 2 σ 2 o (ω) ( ω) = H 0 is accepted (healthy structure) Else = H 1 is accepted (faulty structure) (5.11) with Z 1 α 2 designating the standard normal distribution s 1 α 2 critical point (typical values are α = 0.05, Z 1 α 2 = 1.96). Alternatively (treating σ2 o(ω) as a random variable) the t distribution may be used (see Appendix; also subsection 6.3; notice 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. 5.3 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) 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 excitationresponse case. The coherence estimator γ 2 (ω) is obtained by replacing the auto and cross spectral 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 ): E{ γ 2 (ω)} γ 2 (ω) + 1 K [1 γ2 (ω)] 2 with γ 2 (ω) designating the true coherence value. σ 2 (ω) 2γ2 (ω) K [1 γ2 (ω)] 2 (5.12) Next consider a frequency discretization ω i (i = 1,2,...,n) (frequency resolution δω), and define the coherence measure: n Γ = δω γ 2 (ω i ) (5.13) i=1 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 ) the coherence measure estimator approximately follows, thanks to the central limit theorem (see Appendix), Gaussian distribution, that is: n Γ N(Γ,σΓ) 2 with σγ 2 = δω 2 σ 2 (ω i ) (5.14) with Γ designating the measure s true value. Notice 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 coherence being close to unity), while the variance of Γ is equal to the sum of the individual variances due to the fact that the coherence estimates γ 2 (ω i ) (i = 1,2,...,n) are mutually independent random variables (Brillinger 1981 p. 204). i=1

13 Time series methods for fault detection and identification 13 Fault detection. Fault detection is based upon confirmation of a statistically significant reduction in the coherence measure Γ u of the current structure (compared to Γ o of the healthy structure) through the statistical hypothesis testing problem: H o : δγ = Γ o Γ u 0 (null hypothesis - healthy structure) H 1 : δγ = Γ o Γ u > 0 (alternative hypothesis - faulty structure) (5.15) Toward this end the difference of the coherence measure estimates in the nominal and current states of the structure is considered: δ Γ = Γ o Γ u N(δΓ,δσΓ) 2 with δγ = Γ o Γ u δσγ 2 = (σγ) 2 o + (σγ) 2 u (5.16) which, due to Equation (5.14), follows normal distribution with the indicated mean (true Γ difference) and variance. Notice that Γ o is obtained in the baseline phase and Γ u in the inspection phase and are mutually independent. Under the null (H o ) hypothesis (current structure is healthy) δγ = Γ o Γ u 0 and δσγ 2 = 2(σ2 Γ ) o, thus: Under H o : δ Γ = Γ o Γ u N(δΓ,2(σΓ) 2 o ) (δγ 0) (5.17) The variance (σγ 2) o is unknown, but may be estimated in the baseline phase based upon Equations (5.12), (5.14) by replacing the true coherence by its estimate. The presence or not of a statistically significant reduction in the coherence measure Γ u of the current structure (compared to Γ o 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 α requires knowledge of the unavailable mean δγ 0. To overcome this difficulty, a conservative procedure is adopted by allowing for a risk level α in the worst possible (under H o ) case of δγ = 0. By treating the estimated variance ( σ Γ 2) o as a fixed quantity (quantity with negligible variability), this leads to the following test characterized by a maximum type I (false alarm) risk of α: δ Γ < Z 2( σ 2 1 α = H 0 is accepted (healthy structure) Γ ) o Else = H 1 is accepted (faulty structure) (5.18) with Z 1 α designating the standard normal distribution s 1 α critical point (typical values are α = 0.05, Z 1 α = 1.645). 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 section 4; also Fassois 2001). They are applicable to both the response-only and excitation-response cases, as each situation may be dealt with through the use of proper representations (also

14 14 S.D. Fassois and J.S. Sakellariou see Basseville & Nikiforov 1993, Natke & Cempel 1997, Gertler 1998). Parametric methods may be classified into the following categories: (a) Model parameter based methods. These attempt fault detection and identification using a characteristic quantity Q that is function of the parameter vector θ, Q = f(θ), 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) (z u ) N 1. (b) Residual based methods. These attempt fault detection and identification by using characteristic quantities Q that are functions of model residuals generated by driving the current signal(s) (z u ) N 1 through predetermined 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 Q = f((e Xu ) N 1 ), with e Xu [t] designating the residual generated by driving z u [t] through the model corresponding to the X structural state (X=o,A.,...,D). No model re-estimation is needed in the inspection phase. (c) 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 e Xu [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 sequel. 6.1 Model parameter based methods This category of methods attempts fault detection and identification via changes in the parameter vector θ of a suitable parametric representation (see Isermann 1993, Gertler 1998; Natke & Cempel 1997 is more focused on structures). In the typical case Q = θ is the methods characteristic quantity (sometimes a transformed version of the parameter vector may be used). Let θ designate a proper (say Maximum Likelihood; see for instance Söderström & Stoica 1989 pp , Ljung 1999 pp , Fassois 2001) estimator of the parameter vector θ based upon excitation and/or response measurements z N 1. The particular model structure employed (see section 4) depends upon the specific problem at hand. For sufficiently long signals (N large), the estimator θ is (under mild assumptions) Gaussian distributed with mean equal to its true value θ and covariance P θ (see Söderström & Stoica 1989 pp , Ljung 1999 p. 303 on issues relating to the numerical computation / estimation of P θ ): θ N(θ,P θ ) (6.1) Fault detection. Fault detection is based upon testing for statistically significant changes in the parameter vector θ between the nominal and current structures through the hypothesis testing problem: H o : δθ = θ o θ u = 0 (null hypothesis - healthy structure) (6.2) H 1 : δθ = θ o θ u 0 (alternative hypothesis - faulty structure)

15 Time series methods for fault detection and identification 15 Toward 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 (z o ) N 1 and the second in the inspection phase using (z u ) N 1 ) follows Gaussian distribution: δ θ = θ o θ u N(δθ,δP) with δθ = θ o θ u δp = P o + P u (6.3) with P o, P u designating the estimator covariance matrices for the nominal and current structure, respectively. Under the null (H o ) hypothesis δ θ = θ o θ u N(0,2P o ) and the quantity Q, 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; also Ljung 1999 p. 558): Q = δ θ T δp 1 δ θ χ 2 (d) (6.4) with δp = 2P o. Since the covariance P o corresponding to the healthy structure is unavailable, its sample (estimated) version P o is used in practice. Treating this sample covariance as a deterministic quantity, that is a quantity characterized by negligible variability (which is reasonable for large N) leads to the following test constructed at the α (type I) risk level (Figure 5) (also see Ljung 1999 p. 559 for an alternative approach based upon the F distribution): Q < χ 2 1 α(d) = H 0 is accepted (healthy structure) Else = H 1 is accepted (faulty structure) (6.5) with χ 2 1 α(d) designating the chi-square distribution s with d degrees of freedom 1 α 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 θ u to those corresponding to different fault types, say θ A,..., θ 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 and Fassois (1997, 1998). This utilizes point and covariance estimates of θ collected into the vector θ = [ θ 1... θ ρ ] 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 θ (ρ-dimensional) space (Figure 6): g i ( θ) = θ 1 + ω i 1 θ ω i ρ 1 θ ρ ω i ρ = 0 (i-th fault mode hyperplane) (6.6) with ωj i s designating the i-th hyperplane s coefficients. These hyperplanes are constructed during the baseline phase using linear regression and vibration data sets obtained from the structure under faults of various magnitudes from each fault type. In the inspection phase, following fault detection, fault identification is based upon computation of the distance of the estimated current point θ u in the ρ-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).

16 16 S.D. Fassois and J.S. Sakellariou Fault estimation. Within the context of the geometric approach, fault estimation may be achieved via computation of the distance of θu from the point θ o 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 be also used with alternative representations, such as modal models. In such a case fault detection 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 subsection 6.2.4). 6.2 Residual based methods These attempt fault detection, identification, and estimation by using characteristic quantities that are functions of residual sequences obtained by driving the current signal(s) (z u ) N 1 through suitable pre-determined (estimated in the baseline phase) models M o, M A,...,M D, 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; for instance see Mehra & Peschon 1971, Willsky 1976, Basseville 1988, Frank 1990, Basseville & Nikiforov 1993, Gertler 1998; Natke & Cempel 1997 is more focused on structures. Let M X designate the model representing the structure in its X state (X = o or X = A,...,D under specific fault magnitudes). Also let the residual series generated by driving z u [t] through each one of the above models be designated as e ou [t], e Au [t],..., e Du [t] and be characterized by respective variances σou, 2 σau 2,..., σ 2 Du (the first subscript designating the model and the second the excitation and/or response signal(s) used). Fault detection, identification, and estimation may be then based upon the fact that under the hypothesis H X (that is the structure is in its X state, for X = o or X = A,...,D under specific fault magnitudes) the residual series generated by driving the current signal(s) (z u ) N 1 through the model M X possesses the property: Under H X : e Xu [t] iid N(0,σXu) 2 with σxu 2 < σy 2 u for any state Y X (6.7) where iid designates identically independently distributed random variables. It is tacitly assumed that: Under H X : σxu 2 = σxx 2 (6.8) implying that the excitation and environmental conditions are the same in the baseline and inspection phases. A first method within this category is based upon examination of the residual series obtained by driving the current signal(s) (z u ) N 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) (z u ) N 1 under each one of the considered hypotheses (H o,h A,...,H D ). 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) (z u ) N 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

17 Time series methods for fault detection and identification 17 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 Method A: Using the residual variance In this method the characteristic quantity is the residual variance (also see Sohn & Farrar 2001 and Sohn et al for a related scheme). Fault detection. According to the preceding discussion, fault detection is based upon the fact that the residual series e ou [t], obtained by driving the current signal(s) (z u ) N 1 through the model M o corresponding to the nominal (healthy) structure, should be characterized by variance σou 2 which becomes minimal (specifically equal to σoo; 2 see Equation (6.8)) if and only if the current structure is healthy (S u = S o ). The hypothesis testing problem of Equation (3.1) may be then expressed as (Q = σxu 2 ): H o : σ 2 oo σ 2 ou (null hypothesis - healthy structure) H 1 : σ 2 oo < σ 2 ou (alternative hypothesis - faulty structure) (6.9) Under the null (H o ) hypothesis the residuals e ou [t] are (just like the residuals e oo [t]) identically independently distributed (iid) zero mean Gaussian with variance σoo 2 (Equations (6.7) and (6.8)). Hence the quantities (N u 1) σ ou/σ 2 oo 2 and (N o d 1) σ oo/σ 2 oo 2 follow (central) chi-square distributions with N u 1 and N o d 1 degrees of freedom, respectively (sum of squares of independent standardized Gaussian random variables; see Appendix). Notice that N o, N u designate the number of samples used in estimating the residual variance in the healthy and current cases, respectively (typically N o = N u = N), whereas d designates the dimensionality of the model parameter vector. Consequently, the following statistic follows F distribution with N u 1 and N o d 1 degrees of freedom (ratio of two independent and normalized χ 2 random variables; see Appendix): Under H o : F = (N u 1) σ 2 ou σoo 2 (Nu 1) (N o d 1) σ oo 2 ou σ oo 2 σ 2 oo (No d 1) = σ2 F(N u 1,N o d 1) (6.10) The following test is then constructed at the α (type I) risk level: F f 1 α (N u 1,N o d 1) = H 0 is accepted (healthy structure) Else = H 1 is accepted (faulty structure) (6.11) with f 1 α (N u 1,N o d 1) designating the F distribution s 1 α critical point. Fault identification. Fault identification may be similarly achieved through pairwise tests of the form: H o : H 1 : σ 2 XX σ2 Xu σ 2 XX < σ2 Xu for X = A,B,...,D (6.12) An alternative possibility may be based upon obtaining the residual series e Au [t],..., e Du [t], estimating their variances, and declaring as current fault that corresponding to minimal variance. Notice that by including e ou [t], fault detection may be also treated. The advantage of this approach is that only the signal(s) (z u ) N 1 are used in the inspection phase.

18 18 S.D. Fassois and J.S. Sakellariou 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 Method B: Using the likelihood function Fault detection. In this method fault detection is based upon the likelihood function under the null (H o ) hypothesis of a healthy structure (see Gertler 1998 pp ). The hypothesis testing problem considered is: H o : θ o = θ u (null hypothesis - healthy structure) H 1 : θ o θ u (alternative hypothesis - faulty structure) (6.13) with θ o, θ u designating the healthy and current structure s parameter vectors, respectively. Assuming independence of the residual sequence, the Gaussian likelihood function for the data y N 1 given x N 1 is (Box et al p. 226): N N L y (y1 N,θ/x N 1 ) = f w (e[t,θ]) = t=1 t=1 { } 1 exp e2 [t,θ] 2πσ 2 2σ 2 = { 1 ( 2πσ 2 ) exp 1 N 2σ 2 } N e 2 [t,θ] t=1 (6.14) with e[t, θ] designating the model s residual (one-step-ahead prediction error) characterized by zero mean and variance σ 2, and f w ( ) its probability density function. Under the null (H o ) hypothesis, the residual series e ou [t] generated by driving the current signal(s) through the nominal model M o is (just like e oo [t]) identically independently distributed (iid) Gaussian with zero mean and variance σ 2 oo (Equations (6.7) and (6.8)). Decision making may be then based upon the likelihood function under H o (θ = θ o ) 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 (H o ) hypothesis to be accepted, that is: L y ((y u ) N 1,θ o /(x u ) N 1 ) l = H 0 is accepted (healthy structure) Else = H 1 is accepted (faulty structure) (6.15) The evaluation of the likelihood L y ((y u ) N 1,θ o /(x u ) N 1 ) requires knowledge of the true innovations variance σoo. 2 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: e 2 ou [t] σ 2 oo N Q N = t=1 = N σ2 ou σ l = H oo 2 0 is accepted (healthy structure) Else = H 1 is accepted (faulty structure) (6.16) where l designates the resulting (to be selected) threshold. The statistic Q N follows (under the H o hypothesis) χ 2 distribution with N degrees of freedom (sum of squares of mutually independent standardized Gaussian variables; see Appendix), and this leads to the following test at the α risk level (Figure 5): Q N = N e 2 ou [t] t=1 σ = N σ2 oo 2 ou σ χ 2 1 α(n) = H oo 2 0 is accepted (healthy structure) Else = H 1 is accepted (faulty structure) (6.17)

19 Time series methods for fault detection and identification 19 with χ 2 1 α(n) designating the chi-square distribution s 1 α critical point. Remark. Notice that under the assumption of a fixed σ 2 oo this method leads to a test statistic that is similar to that of Equation (6.10) (method A). The difference is that in method A the variability of the estimate σ 2 oo is accounted for, thus leading to an F distribution of the pertinent statistic. Of course, the two tests coincide for N o in Equation (6.10), as the F distribution then approaches the chi-square distribution (see Appendix). Fault identification. Fault identification may be achieved by computing the likelihood function for the current signal(s) for the various values of θ (θ A,...,θ D ) and selecting that hypothesis H X that corresponds to the maximum value of the likelihood, that is: L y ((y u ) N 1,θ X /(x u ) N 1 ) = max X L y((y u ) N 1,θ X /(x u ) N 1 ) = The H X hypothesis is accepted (6.18) with X = A,...,D (faults of specific magnitudes). Obviously, by also including the nominal model, the scheme may be used for combined fault detection and identification. 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 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 (e ou ) N 1 obtained by driving the current signal(s) (z u ) N 1 through the nominal (healthy) model M o. In this case the hypothesis testing problem of Equation (3.1) may be expressed as: H o : ρ[i] = 0 i = 1,2,...,r (null hypothesis - healthy structure) H 1 : ρ[i] 0 for some i (alternative hypothesis - faulty structure) (6.19) with ρ[i] = γ[i]/γ[0] designating the normalized autocovariance (correlation coefficient; Box et al p. 26) of the e ou [t] residual sequence (see Equation (4.3) for the definition of γ[i]). The method s characteristic quantity thus is Q = [ρ[1]ρ[2]... ρ[r]] T (r being a design variable). Under the null (H o ) hypothesis e ou [t] is (just like e oo [t] identically independently distributed (iid) Gaussian with zero mean, and the statistic Q below follows χ 2 distribution with r d (r d 1 in case the mean is also estimated) degrees of freedom (d = dim(θ); Box et al p. 314): r Under H o : Q = N (N + 2) (N i) 1 ρ[i] 2 χ 2 (r d) (6.20) with ρ[i] designating the sample (estimated) ρ[i] and N the number of signal samples (estimation based upon the autocovariance estimator γ[i] = 1 N N t=i+1 (e ou[t] ˆµ ) (e eou ou[t i] ˆµ ), with ˆµ eou e ou designating the residual sample mean). The hypothesis testing problem of Equation (6.19) then leads to the following test at the α (type I) risk level (Figure 5): i=1 Q < χ 2 1 α(r d) = H 0 is accepted (healthy structure) Else = H 1 is accepted (faulty structure) (6.21) with χ 2 1 α(r d) designating the chi-square distribution s 1 α critical point.

20 20 S.D. Fassois and J.S. Sakellariou Fault identification. Fault identification may be achieved by similarly examining which one of the e Xu [t] (X = A,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 Method D: Using residuals associated with subspace identification This method is motivated by stochastic subspace identification (the response-only case is presently treated; see Basseville et al for details). It attempts detection of changes in the modal space by considering the parameter vector: [ θ = λ vec Φ ] (6.22) where λ 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)), Φ designates the matrix with columns being the vectors φ λ where φ λ = Cϕλ (with ϕ λ being the corresponding eigenvectors, and C the output matrix in Equation (4.6)). Fault detection. Fault detection is based upon forming (for a given θ) the system s (p + 1)-th order (p large enough) observability matrix as follows: O p+1 (θ) = Φ ΦΛ. ΦΛ p (6.23) with Λ = diag(λ) (diagonal matrix with the system eigenvalues), and also the system s block Hankel matrix (for the case of Equation (4.6)): γ yy (0) γ yy (1)... γ yy (q 1) γ yy (1) γ yy (2)... γ yy (q) H p+1,q = = Hank(γ yy ) (6.24) γ yy (p) γ yy (p + 1)... γ yy (p + q 1) with q p + 1, and γ yy (τ) 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 θ), O p+1 (θ) and H p+1,q have the same left kernel space. Whether this is the case or not may checked through the following procedure. Use an available parameter vector θ in order to form the observability matrix O p+1 (θ). Then pick an orthonormal basis of the left kernel space of the matrix W 1 O p+1 (θ) (W 1 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 = I (6.25) S T W 1 O p+1 (θ) = 0 (6.26) with s = (p+1)r n (r designating the output dimensionality). Notice that the matrix S is not unique. It may be, for instance, obtained through the Singular Value Decomposition (SVD) of W 1 O p+1 (θ), and implicitly depends

21 Time series methods for fault detection and identification 21 upon the parameter vector θ; hence it may be designated as S(θ). The block Hankel matrix corresponding to the same system should then satisfy the property: S T (θ) W 1 H p+1,q W T 2 = 0 (6.27) (W 2 being an additional selectable invertible weighting matrix). Now assume that the parameter vector θ o corresponding to the healthy system is available from the baseline phase. For checking whether the current data (y u ) N 1 actually correspond to the healthy system, the following residual is defined (compare with Equation (6.27)): ζ N (θ o ) = ( ) N vec S T (θ o ) W 1 Ĥp+1,q W T 2 (6.28) where Ĥp+1,q is the sample block Hankel matrix obtained from the current data (y u ) N 1. This residual, which should be zero for the theoretical block Hankel matrix under H o (θ u = θ o ) (see Equation (6.27)), has zero mean under H o (θ u = θ o ) and non-zero mean under H 1 (θ u θ o ). Testing whether the current system (with parameter vector θ u and associated with the sample block Hankel matrix) coincides with the healthy system (with parameter vector θ o ) is based upon a statistical local approach, according to which the following close hypotheses are considered: H o : θ u = θ o (null hypothesis - healthy structure) H 1 : θ u = θ o + 1 N δθ (alternative hypothesis - faulty structure) (6.29) with δθ designating an unknown but fixed error vector. Let M(θ o ) be the Jacobian matrix designating the mean sensitivity of ζ N, and Σ(θ o ) = lim N E θo {ζ N ζ T N }, with E θo { } designating the expectation operator when the actual system parameter is θ o. Notice 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 Σ(θ o ) is positive definite, the residual ζ N (θ o ) of Equation (6.28) asymptotically follows Gaussian distribution, that is: ζ N (θ o ) (N ) { N (0,Σ(θo )) under H o N (M(θ o )δθ,σ(θ o )) under H 1 (6.30) This indicates that a deviation in the system parameters is reflected into a change in the mean of ζ N. Let M and Σ be consistent estimates of M(θ o ) (assumed to be full column rank) and Σ(θ o ), respectively, obtained in the baseline phase (details in Basseville et al. 2000). Under the null (H o ) hypothesis the statistic Q N, below, follows χ 2 distribution with rankm degrees of freedom, that is: Under H o : Q N = ζ T N (θ o ) Σ 1 M( MT Σ 1 M) 1 MT Σ 1 ζn (θ o ) χ 2 (rankm) (6.31) 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 subsection 6.1). 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 θ A,...,θ D. 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.

22 22 S.D. Fassois and J.S. Sakellariou Fault estimation. The method, as such, is not immediately suitable for fault estimation. 6.3 The functional model based method (fmbm) The functional model based method (fmbm) has been recently introduced (Sakellariou et al. 2002a, Sakellariou et al. 2002b, Sakellariou & Fassois 2002) in an attempt to (a) effectively tackle the problem of identifying faults of any type in the important case that the continuum of possible fault magnitudes is considered, (b) accurately estimate fault magnitude, and (c) provide a combined solution to the subproblems of fault detection, identification, and magnitude estimation. The method s schematic representation 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 by using a single representation (model) that is directly parametrized in terms of fault magnitude. This representation is referred to as a stochastic Functional Model (FM), and, depending upon the problem treated, it may assume various forms (Sakellariou et al. 2002a, Sakellariou & Fassois 2002). 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 nb M X (a ij,b ij,σw(k)) 2 : y k [t] + a i (k) y k [t i] = b i (k) x k [t i] + w k [t] w k [t] iid N(0,σw(k)) 2 i=1 i=0 j=1 j=1 (6.32) p p with: a i (k) = a ij G j (k), b i (k) = b ij G j (k) fault magnitude: k R (6.33) 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. x k [t], y k [t] and w k [t] are the excitation, response, and innovations (prediction error) signals, respectively, corresponding to the particular k (notice the k = 0 corresponds to the healthy structure). It should be noted that the above model resembles the conventional ARX representation (Equation (4.8) with C(B) 1), but explicitly accounts for the continuum of fault magnitudes within the fault mode it represents. This is accomplished by allowing its parameters and innovations variance σw 2 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 G 1 (k),...,g p (k) (functional basis). The constants a ij, b ij 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 by 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 σe 2 (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

23 Time series methods for fault detection and identification 23 (6.32)): na nb M X (k,σe) 2 : y[t] + a i (k) y[t i] = b i (k) x[t i] + e[t] (6.34) i=1 i=0 Estimates of k, σe 2 are obtained based upon the currently available signal(s) (z u ) N 1 and the NonLinear Least Squares (NLLS) estimator: k = arg min k RSS(k) = arg min k N e 2 [t] t=1 σ 2 e = 1 N N ê 2 [t] (6.35) t=1 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 is thus obtained by using the Gauss Newton and Levenberg- Marqurdt schemes (Ljung 1999 pp ). 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 ) Gaussian distributed, with mean equal to its true k value and variance σ 2 k ( k N(k,σk 2 )), the latter corresponding to the Cramer-Rao lower bound (Sakellariou et al. 2002a). This variance is also estimated, with its estimate σ k 2 shown to be asymptotically (N ) uncorrelated with k. Since the healthy structure corresponds to k = 0 (zero fault magnitude), fault detection may be based upon the hypothesis testing problem: H o : k = 0 (null hypothesis - healthy structure) H 1 : k 0 (alternative hypothesis - faulty structure) (6.36) 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 t distribution with N 2 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 N 2 degrees of freedom; see Appendix), that is: which leads to the following test at the α (type I) risk level (Figure 9): t = k σ k t(n 2) (6.37) t α 2 (N 2) t t 1 α 2 (N 2) = H 0 is accepted (healthy structure) Else = H 1 is accepted (faulty structure) (6.38) with t α and t 1 α 2 designating the t distribution s (with the indicated degrees of freedom) α and 1 α 2 critical points, respectively. Fault identification. Once a fault is detected, fault identification is achieved through successive estimation (using the current data (z u ) N 1 ) and validation of the re-parametrized FARX models M X (k,σ 2 e) (X = A,...,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 being then 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 subsection 6.2.3). Fault estimation. Once the current fault mode has been identified, an interval estimate of the fault magnitude is constructed based upon the k, σ k 2 estimates obtained from the corresponding re-parametrized FARX model.

24 24 S.D. Fassois and J.S. Sakellariou Thus, using Equation (6.37), the interval estimate of k at the α risk level is: k interval estimate: [ k + t α 2 (N 2) σ k, k + t1 α 2 (N 2) σ k ] (6.39) 7. Case studies 7.1 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 (subsection 5.3). 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 gr) trapezoidal 2024-T3-bare aluminium stiffened (via ribs) panel of dimensions mm (large base small base height) obtained from a Vought A-7 Corsair aircraft. The fault considered is a 25 mm long sawcut (Figure 10b) on the skin (skin thickness 2 mm). The panel is suspended through strings and tested under free-free boundary conditions (Figure 10c). The excitation is an 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, C; Figure 10a) is measured via miniature accelerometers (sampling frequency f s = 6 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 N = samples per signal (frequency resolution Hz, number of non-overlapped segments K = 30, Hanning windowing). Baseline phase. Interval estimates of the coherence measure corresponding to the three (Points A, B, 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 reobtained for the panel in its current (faulty) state. Figure 11a depicts the coherence measure interval estimates (at the α = 0.05 risk level) for each panel state (healthy faulty) and for each one of the vibration response measurement positions (Points A, B, 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 α = 0.05 risk level. 7.2 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 stabilizers,

25 Time series methods for fault detection and identification 25 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 mass 50 kg). The faults considered correspond to the placement of a variable number of small masses (simulating local elasticity reductions), of 6.5 gr each, at Point B of the structure (Figure 12). Each such fault is designated as F k, with k representing the fault magnitude (gr 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 is 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, along with the resulting vertical vibration accelerations, are measured via an impedance head and lightweight accelerometers, respectively (sampling frequency f s = 128 Hz, signal length N = 1000 samples). Baseline phase. AutoRegressive with exogenous excitation (ARX) modelling of the healthy structure based upon 4 sec (N = 1, 000 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 (k = 0 gr of added mass) and the rest corresponding to various fault magnitudes (faults F k with k [6.5,130] gr; increment δk 6.5 gr). The Functional ARX (FARX) modelling procedure leads to a FARX(18,18) model (Equations (6.32), (6.33)) with functional basis consisting of the first p = 7 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 gr (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 α = 0.05 risk level). a) 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 ( k = ± ) includes the nominal k = 0 value (Figure 13a). The proximity of the k point estimate to the true, k = 0, value is quite remarkable. b) Test case II (F 37.7 fault; added mass of 37.7 gr): A fault is clearly detected in this case, as the fault magnitude s interval estimate ( k = ± ) does not include the k = 0 value (Figure 13b). The accuracy attained in estimating the fault magnitude is, again, excellent. 7.3 Case study C: Fault detection and identification in a 2 DOF system with cubic stiffness In this case study fault detection, identification and estimation in a simple two degree-of-freedom non-linear

26 26 S.D. Fassois and J.S. Sakellariou system characterized by cubic stiffness (spring k 3 ; 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 T s = sec (sampling frequency f s = 1,500 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 Fk k 2, 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 y[t]+ a i y[t i]+a 5 y[t 4] y[t 3]+a 6 y 2 [t 4]+a 7 y 2 [t 4] y[t 3] + i=1 +a 8 y 3 [t 4] + a 9 y 2 [t 4]y[t 2] = b 1 x[t 4] + e[t] Fault mode modelling is (for the F k k 2 fault mode) based upon a series of 17 experiments, one corresponding to the healthy system (k = 0% variation in k 2 ) and the rest corresponding to various fault magnitudes (faults F k k 2 with k [ 32,32%]; increment δk = 4% change in k 2 ). The signals used are, in all cases, N = 2,000 samples long. A Functional NARX (FNARX) model with functional basis consisting of the first two (0th and 1st degree, thus p = 2) Chebyshev Type II polynomials (Abramowitz & Stegun 1970) is selected for representing the Fk k 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 α = 0.05). a) Test case I (healthy structure): In this case the fault magnitude interval estimate includes the k = 0 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. b) Test case II (Fk 3 2 fault; 3% reduction in k 2 ): In this case a small magnitude fault is injected. Yet, its detection is again remarkably accurate ( k = 3.02 ± ) for all 10 runs (Figure 15b). In addition, the Q statistic (Equation (6.20)) corresponding to the Fk k 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

27 Time series methods for fault detection and identification 27 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. Parametric methods were further classified as: a) Model parameter based, in which the characteristic quantity is a function of the model parameter vector, b) residual based, in which the characteristic quantity is a function of model residuals, and, c) the functional model based method, 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 non-linear 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 behavioral data sets corresponding to various fault conditions. This may not be necessarily possible (also see the discussion in subsection 3.1), 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 are expected to be sought, so that information from more measurement locations may be included in the decision making. Furthermore, methods suitable for non-stationary structures (structures with time-varying properties) as well as non-linear 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. Acknowledgement The authors are indebted to three anonymous referees whose comments helped in improving the manuscript.

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31 Time series methods for fault detection and identification 31 Appendix: Central limit theorem and statistical distributions associated with the normal The Central Limit Theorem (CLT). (Montgomery 1991 p. 46, Stuart & Ord 1987 p. 273, Nguyen & Rogers 1989 Vol. I p. 420). Let Z 1,Z 2,...Z n designate mutually independent random variables each with mean µ k and (finite) variance σ 2 k. Then, for n the distribution of the random variable X = n k=1 Z k approaches the Gaussian distribution with mean E{X} = n k=1 µ k and variance var(x) = n k=1 σ2 k. The chi-square distribution. Let Z 1,Z 2,...Z n designate mutually independent, normally distributed, random variables, each with mean µ k and standard deviation σ k. Then the sum: X = n ( ) 2 Zk µ k (A.1) k=1 is said to follow a (central) chi-square distribution with n degrees of freedom (X χ 2 (n)). Its mean and variance are E{X} = n and var(x) = 2n, respectively. Notice that imposing p equality constraints among the random variables Z 1,Z 2,...Z n reduces the set s effective dimensionality, and thus the number of degrees of freedom, by p (Stuart & Ord 1987 pp ). For n the χ 2 (n) distribution tends to normality (Stuart & Ord 1987 p. 523). ( Zk The sum X = ) 2 n k=1 σ k is said to follow non-central chi-square distribution with n degrees of freedom ( ) and non-centrality parameter λ = µk 2. σ k This distribution is designated as χ 2 (n;λ) (Nguyen & Rogers 1989 Vol. II p. 33). Let x R n follow n-variate normal distribution with zero mean and covariance Σ (x N(0,Σ)). Then the quantity x T Σ 1 x follows (central) chi-square distribution with n degrees of freedom (Stuart & Ord 1987 pp , Gertler 1998 p. 120, Söderström & Stoica 1989 p. 557). σ k The 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: T = Z X/n (A.2) 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 } = 0 (n > 1) and var(t) = n n 2 (n > 2), respectively (Stuart & Ord 1987 p. 513). The (central) t distribution approaches the standard normal distribution N(0,1) as n (Stuart & Ord 1987 p. 523). The Fisher s F distribution. Let X 1, X 2 be mutually independent random variables following (central) chisquare distributions with n 1, n 2 degrees of freedom, respectively. Then the ratio: F = X 1/n 1 X 2 /n 2 (A.3)

32 32 S.D. Fassois and J.S. Sakellariou is said to follow a (central) F distribution with n 1,n 2 degrees of freedom (F F(n 1,n 2 )) (central because it is based on central chi-square distributions; Nguyen & Rogers 1989 Vol. II p. 34). Its mean and variance are E{F } = n2 n 2 2 (n 2 > 2) and var(f) = 2n2 2 (n1+n2 2) n 1(n 2 2) 2 (n 2 4) (n 2 > 4), respectively (Stuart & Ord 1987 p. 518). Note that for the distribution s 1 α critical point f 1 α (n 1,n 2 ) = 1/f α (n 2,n 1 ). The (central) F distribution approaches normality as n 1,n 2. For n 2 n 1 F approaches a (central) chi-square distribution with n 1 degrees of freedom (Stuart & Ord 1987 p. 523).

33 List of figures Time series methods for fault detection and identification 33 1 General structure of time series based fault detection and identification methods (the inspection phase is depicted outside the dashed box) Representation of a linear time-invariant system with additive response noise Statistical hypothesis testing based upon an F distributed statistic (two-tailed test) Statistical hypothesis testing based upon a Gaussian distributed statistic (two-tailed test) Statistical hypothesis testing based upon a χ 2 distributed statistic (one-tailed test) Principle of the geometric method for fault identification Schematic representation of residual based methods (the inspection phase is depicted outside the dashed boxes) Schematic representation of the functional model based method (the inspection phase is depicted outside the dashed box) Statistical hypothesis testing based upon a t distributed statistic (two-tailed test) Aircraft stiffened panel: (a) The panel with the measurement positions (front view); (b) the panel with the skin fault area (rear view); (c) part of the experimental setup (Rizos et al. 2002) Aircraft stiffened panel: Coherence based measure method applied to three measurement positions (A,B,C): (a) Interval estimates of the coherence measure, (b) fault detection results ( : test statistic, : critical value) 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) Aircraft scale skeleton structure: Fault detection/estimation results Residual Sum of Squares normalized by the Series Sum of Squares versus k: (a) Healthy structure; (b) fault F 37.7 (the dashed vertical lines indicate the true k; the shaded strips in the zooms indicate corresponding interval estimates) Two DOF system with cubic stiffness Two DOF system with cubic stiffness: Fault detection/estimation results: (a) Test case I (healthy system); (b) test case II (fault Fk 3 2 ) (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 α = 0.05 risk level) Two DOF system with cubic stiffness: Q statistic (bars) and the critical point (- - -) at the a = 0.05 risk level for test case II (values below the critical point indicate F k2 fault mode identification 10 Monte Carlo runs) Classification of time series fdi methods

34 34 S.D. Fassois and J.S. Sakellariou Excitation x u [t] Current structure S u Response y u [t] Estimation of characteristic quantity Q Qˆ u Statistical Decision Making Qˆ o Qˆ A healthy fault A Baseline Phase Qˆ D fault D Figure 1. General structure of time series based fault detection and identification methods (the inspection phase is depicted outside the dashed box). n[t] x[t] Structure h[t] + y[t] Figure 2. Representation of a linear time-invariant system with additive response noise. f F α/2 α/2 0 f α/2 f 1 α/2 H 1 accepted (fault) H o accepted (no fault) H 1 accepted (fault) Figure 3. Statistical hypothesis testing based upon an F distributed statistic (two-tailed test). f N α/2 Z α/2 0 Z 1 α/2 α/2 H 1 accepted (fault) H o accepted (no fault) H 1 accepted (fault) Figure 4. Statistical hypothesis testing based upon a Gaussian distributed statistic (two-tailed test).

35 ... Time series methods for fault detection and identification 35 f χ 2 0 H o accepted (no fault) 2 χ 1 α α H 1 accepted (fault) Figure 5. Statistical hypothesis testing based upon a χ 2 distributed statistic (one-tailed test). θ 2 θ u Current structure g i (θ) = 0 i-th fault mode θ 3 θ o healthy structure.. θ 1 θ g j (θ) = 0 j-th fault mode Figure 6. Principle of the geometric method for fault identification.

36 36 S.D. Fassois and J.S. Sakellariou Excitation x u [t] Current structure S u Response y u [t] Model M o (healthy structure) Model M A (fault A). Residual e ou [t] Residual e Au [t]. Estimation of char. quantity Q Estimation of char. quantity Q Qˆou Qˆ Au. Statistical Decision Making Model M (fault D) D Residual e Du [t] Estimation of char. quantity Q Qˆ Du Baseline Phase Qˆ oo Qˆ AA healthy fault A Baseline Phase Qˆ DD fault D Figure 7. Schematic representation of residual based methods (the inspection phase is depicted outside the dashed boxes). Excitation x u [t] Current structure S u Response y u [t] Functional Model M A (k ) (fault mode A). Estimation. kˆa Residual e Au [t] Estimation of characteristic quantity Q. Qˆ A Statistical Decision Making Functional Model M D (k) (fault mode D) Estimation kˆd Residual e Du [t] Estimation of characteristic quantity Q Qˆ D Baseline Phase Figure 8. Schematic representation of the functional model based method (the inspection phase is depicted outside the dashed box). f t α/2 α/2 t 0 t α/2 1 α/2 H 1 accepted (fault) H o accepted (no fault) H 1 accepted (fault) Figure 9. Statistical hypothesis testing based upon a t distributed statistic (two-tailed test).

37 Time series methods for fault detection and identification 37 Figure 10. Aircraft stiffened panel: (a) The panel with the measurement positions (front view); (b) the panel with the skin fault area (rear view); (c) part of the experimental setup (Rizos et al. 2002). Coherence Measure Healthy: Damaged: maximum possible Coherence Measure (a) 12 Healthy versus Damaged Test Statistic 4 4 Damage Damage Damage No damage No damage No damage Point A Point B Point C (b) Figure 11. Aircraft stiffened panel: Coherence based measure method applied to three measurement positions (A,B,C): (a) Interval estimates of the coherence measure, (b) fault detection results ( : test statistic, : critical value). 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).