Non-Stationary Random Vibration Parametric Modeling and its Application to Structural Health Monitoring

Transcription

1 Non-Stationary Random Vibration Parametric Modeling and its Application to Structural Health Monitoring Luis David Avendaño-Valencia and Spilios D. Fassois Stochastic Mechanical Systems and Automation (SMSA) Laboratory Department of Mechanical & Aeronautical Engineering University of Patras, 654 Patras, Greece s: Version of May, 14 Synonyms: Non-stationary random vibration, signal-based modeling (identification), time-frequency analysis, time-dependent ARMA modeling, Structural Health Monitoring, fault diagnosis. 1 Introduction Non-stationary random vibration is characterized by time-dependent (evolutionary) characteristics (Priestley 1988; Roberts and Spanos 199, ch. 7; Bendat and Piersol, ch. 1; Newland 1993, pp ; Preumont 1994, ch. 8; Hammond and White 1996; Kitagawa and Gersch 1996). Typical examples include earthquake ground motion and resulting structural vibration response, as well as the vibration of surface vehicles, flying aircraft, mechanisms, rotating machinery, cranes, bridges with passing vehicles, and so on. Non-stationary random vibration typically originates from either time-varying or non-linear dynamics. An example of a system exhibiting non-stationary random vibration is the mechanism of Figure 1(a). It is a -DOF pick-and-place mechanism, consisting of two coaxially aligned linear motors carrying prismatic links (arms) connected to their ends. The mechanism is clamped on an aluminium base and is excited by a zero-mean Gaussian stationary random excitation force, applied vertically with respect to the base by means of an electromechanical shaker, while the linear motors are following predetermined trajectories. The non-stationary nature of the resulting vibration (see Figure 1(b)) is due to the time-varying position of the linear motors controlling the position of the links, and is evident in the temporal evolution of the peaks seen in the Time-Varying Power Spectral Density (TV- PSD) estimate shown in Figure 1(c). Further details on this example may be found in (Spiridonakos and Fassois 13). From a mathematical point of view, non-stationary random vibration is characterized by time-dependent statistical moments. In the presently assumed Gaussian case this means that the mean is a function of time t and the AutoCovariance Function (ACF) a function of two time arguments t 1 and t. That is for a random vibration signal x(t) one has (E{ } designates statistical expectation): Mean : µ(t)=e{x(t)}, ACF : γ(t 1,t )=E{(x(t 1 ) µ(t 1 )) (x(t ) µ(t ))} (1) In many random vibration problems the mean is constant, and is thus easily estimated and subtracted from the signal (sample mean adjusted signal). The case of a time-dependent mean (also referred to as deterministic trend) may be treated in similar fashion using proper techniques, such as curve fitting, high pass filtering, or special parametric models (such as integrated models with a deterministic trend parameter (Box et al. 1994)). This article focuses on the following two subjects: (i) The signal based modeling (identification) of non stationary random vibration based on uniformly sampled (with sampling period T ) signal realization x[t], for t = 1,,...,N. Absolute time is t T, N is the signal length in samples, while the use of brackets indicates function of an integer variable. (ii) The use of an identified random vibration model (or a set of such models) for Structural Health Monitoring (SHM), where the objective is the detection of potential structural damage and its characterization (identification) (Fassois and Sakellariou 9). Other important uses of an identified model not discussed in 1

2 Motor A Exciter Motor B Base Guide (a) (c) Accelerometer Normalized Acceleration Conditioning and Acquisition Time [s] PC (b) Figure 1: Example of a laboratory pick-and-place mechanism exhibiting non-stationary random vibration: (a) Schematic diagram of the mechanism and the laboratory set-up; (b) measured vibration acceleration response signal (normalized); (c) estimate of the TV-PSD (time-frequency spectrum). this article include model based analysis (like the extraction of time-dependent Power Spectral Density (PSD) and time-dependent vibration modes (Newland 1993, p. 18; Preumont 1994, ch. 8; Poulimenos and Fassois 6)), prediction, classification, and control. Non-stationary random vibration signal-based modeling (identification) has received significant attention in recent years. The available methods may be broadly classified as parametric or non-parametric (Poulimenos and Fassois 6). Non-parametric methods have received most of the attention and are based upon non-parametrized representations of the vibration energy as a simultaneous function of time and frequency (time-frequency representations). They include the classical spectrogram (based upon the Short-Time Fourier Transform STFT) and its ramifications (Newland 1993, p. 18; Hammond and White 1996; Bendat and Piersol, p. 54), Mark s physical spectrum (Preumont 1994, sec. 8.3), the Cohen class of distributions (Hammond and White 1996), Priestley s evolutionary spectrum (Priestley 1988, sec. 6.3; Preumont 1994, sec. 8.4), as well as wavelet-based methods (Newland 1993, ch. 17). On the other hand, parametric methods are based upon parametrized representations, usually of the Time-dependent AutoRegressive Moving Average (TARMA) type, with apparently similar form to their conventional (stationary) counterparts, but with time-dependent parameters and innovations variance. Thus, a TARMA(n a,n c ) model, with n a and n c designating its AR and MA orders, respectively, is defined as follows (Poulimenos and Fassois 6) : TARMA(n a,n c ) model : x[t]= n a i=1 a i [t]x[t i]+ n c i=1 c i [t]w[t i]+w[t]; w[t] NID (,σ w[t] ) () where x[t] represents the non-stationary random signal modeled, w[t] an unobservable normally and independently distributed (NID) (thus white) non-stationary innovations sequence with zero mean and time-dependent variance σ w[t], n a, n c the AutoRegressive (AR) and Moving Average (MA) orders, respectively, and a i [t] and c i [t] the corresponding AR and MA time-dependent parameters. Parametric TARMA models are of three main types, according to the form of structure imposed upon the evolution of the time-dependent parameters and innovations variance (Poulimenos and Fassois 6): (a) Unstructured Parameter Evolution (UPE) models, in which no particular structure is imposed on parameter evolution. Prime models include Short-Time ARMA (ST-ARMA) and Recursive models (such as Recursive ARMA, or in short RARMA) models. (b) Stochastic Parameter Evolution (SPE) models, in which stochastic structure is imposed on parameter evolution via stochastic smoothness constraints. (c) Deterministic Parameter Evolution (DPE) models, in which deterministic structure is imposed on parameter

3 evolution. Prime models are the so-called Functional Series TARMA (FS-TARMA) models in which the parameters are projected on properly selected functional subspaces. Parametric models, and their respective signal-based (identification) methods, are known to be characterized by a number of important advantages, such as representation parsimony, improved accuracy and resolution, improved tracking of the TV dynamics, flexibility in analysis, synthesis (simulation), prediction, diagnosis, and control. For instance, once a TARMA model is available, the corresponding frozen type TV-PSD may be readily obtained as: Frozen TV-PSD: S F (ω,t) = 1+ n c i=1 c i[t] e jωt si 1+ n a i=1 a i[t] e jωt si σw[t] (3) with ω representing frequency in rad/s, j the imaginary unit, and complex magnitude. Notice that this would be the PSD of the vibration signal if the system were frozen (made stationary) at the time instant t. This article focuses on parametric models and methods, and in particular on the SPE & DPE types. The problem of Structural Health Monitoring (SHM) for structures exhibiting non-stationary random vibration responses may be treated in a Statistical Time Series (STS) framework (Fassois and Sakellariou 9), using either non-parametric or parametric models and statistical decision making schemes. In the majority of applications thus far, non-parametric models (non-parametric time-frequency type representations) are used, and damage detection and identification are based on potential discrepancies observed in them between a baseline (healthy) phase and an inspection (current) phase in which the health state of the structure is under question (Feng et al. 13). Although the use of parametric models and methods may potentially lead to significant improvements, it has thus far received limited attention (Poulimenos and Fassois 4; Spiridonakos and Fassois 13). Brief Historical Notes. Among parametric models, the Unstructured Parameter Evolution (UPE) family of methods was initially developed. Prime methods in this area include the short-time ARMA (ST-ARMA) method (Niedzwieki, pp.79-8; Owen et al. 1) and the class of recursive (or adaptive) methods (Ljung 1999, ch. 11; Niedzwieki, ch. 4 & 5). The Stochastic Parameter Evolution (SPE) family of methods was developed primarily for the modeling of earthquake ground motion signals (Kitagawa and Gersch 1996; Gersch and Akaike 1988). The Deterministic Parameter Evolution (DPE) family was introduced in a broader context in (Rao 197) and later in (Kozin 1977). In (Kozin 1988) and (Fouskitakis and Fassois ) it was employed for earthquake ground motion modeling too. In (Poulimenos and Fassois 9b) it was applied to the vibration of a bridge-like structure with a moving mass. In a broader context the reader is also referred to (Niedzwieki, ch. 4). Article roadmap. This article is organized as follows: SPE and DPE non-stationary random vibration modeling is discussed in Section, where specific model forms and identification schemes are briefly reviewed. Structural Health Monitoring (SHM) based on non-stationary random vibration parametric modeling is presented in Section 3. The application of these concepts to random vibration modeling and SHM for the pick-and-place mechanism of Figure 1 is outlined in Section 4, while a summary is provided in Section 5. Parametric TARMA Modeling of Non-Stationary Random Vibration.1 Stochastic parameter evolution (SPE) TARMA modeling In the context of Stochastic Parameter Evolution (SPE) TARMA models the parameters are assumed to follow stochastic smoothness constraints in the form of linear Integrated AutoRegressive (IAR) models with integration order q (in this context referred to as smoothness priors order). A Smoothness Priors TARMA (SP-TARMA) model thus has parameters that obey the relations (Kitagawa and Gersch 1996): (1 B) q a i [t]=v ai [t], v ai [t] NID (,σv ) (1 B) q c i [t]=v ci [t], v ci [t] NID (,σv ) where the signal innovations w[t] from Equation (), and the parameter innovations v ai [t], and v ci [t] are mutually independent, Normally and Identically Distributed (NID) sequences, each being zero-mean and with variance σ w[t] and σ v, respectively. These smoothness constraints are characterized by unit roots that represent integrated stochastic models describing homogeneously non-stationary evolutions (Box et al. 1994, Ch. 4). Generalization of SP-TARMA models, in the form of Generalized Stochastic Constraint Time-dependent AutoRegressive Moving Average TARMA (GSC-TARMA) models, were recently introduced (Avendaño-Valencia and (4a) (4b) 3

4 Fassois 13). In these, the model parameters are allowed to follow more general AutoRegressive (AR) models of the forms: a i [t]= c i [t]= q k=1 q k=1 µ k a i [t k]+v ai [t], v ai [t] NID (,σv ) (5a) µ k c i [t k]+v ci [t], v ci [t] NID (,σ v), (5b) where, as in the SP-TARMA case, w[t], v ai [t], and v ci [t] are mutually independent, Normally and Identically Distributed (NID) innovation sequences. The coefficients µ k are referred to as the stochastic constraint parameters. These are collected in the vector µ = [ ] T µ 1 µ q, which along with the covariance Σv = σv I na +n c (where I na +n c is the identity matrix with the indicated dimensions), and the innovations variance σw[t] define the model hyperparameters. Then the time-dependent AR/MA parameters θ[t], hyperparameters P, and structural parameters M of a GSC-TARMA model are: θ[t]= [ a 1 [t] a na [t]. c 1 [t] c nc [t]] T, P={µ,σ w[t], Σ v }, M={n a,n c,q} (6) It should be noted that the stochastic constraint parameters and parameter innovations variances may be more generally different for each AR/MA parameter, but the above simple form is adopted here for purposes of presentation simplicity. Comparing the GSC-TARMA and SP-TARMA model forms, one sees that in the latter case the stochastic constraint parameters are essentially pre-fixed, limiting the types of trajectories that each AR/MA parameter would be capable of describing. The identification of an SP-TARMA or GSC-TARMA model consists of the selection of the model structure M and the estimation of the time-dependent parameter vector θ[t] and hyperparameters P. The estimation of the model parameters and hyperparameters may be posed as the maximization of the joint a posteriori probability density function of θ N 1 ={θ[1],..., θ[n]} given the available observations x1 N ={x[1],...,x[n]}, namely p(θ N 1,P x1 N ). These, combined with the Gaussianity assumption for of w[t], v[t] lead to the following cost function: J(θ N 1,P)= N q (n a+ n c ) ln σ v + 1 N t=1 ( ) lnσw[t]+ w [t] σw[t] + vt [t] v[t] σv which must be minimized in order to provide the optimal Maximum A Posteriori (MAP) estimate of θ[t] and P. An estimate of θ[t] based on fixed values of P can be obtained recursively using the Kalman Filter (or a proper non-linear approximation filter in the full TARMA case) based on the following state space representation of the SP/GSC-TARMA model (Poulimenos and Fassois 6): (7) z[t]= F(µ) z[t 1]+ G v[t] x[t]= h T [t] z[t]+w[t] (8a) (8b) with: θ[t 1] θ[t ] z[t 1]=., θ[t q] µ 1 µ µ q 1 F(µ)= I na +n c, G=. I na +n c, x[t 1] w[t 1] h[t]=. where x[t 1]= [ x[t 1] x[t ] x[t n a ] ] T and w[t 1]= [ w[t 1] w[t ] w[t n c ] ] T. Notice that in the full TARMA case the innovations w[t] may be replaced by their respective a posteriori estimates ŵ[t] = x[t] h T [t] ẑz[t t], where ẑz[t t] is the a posteriori state estimate (Niedzwieki, p. 63). After initial estimation with the Kalman filter, refined parameter estimates may be obtained by using the Kalman smoother (Poulimenos and Fassois 6). The estimation of an SP-TARMA model is performed by computing θ N 1 as described above, using fixed values of σ w and σ v. Since the values of these variances are generally unavailable, a normalized form of the Kalman filter 4

5 may be used, where the prediction/update equations are divided by σ w. In this way, a single parameter λ = σ v/σ w is left as a design (user selected) parameter that adjusts the tracking speed versus smoothness of the estimates in the algorithm (Poulimenos and Fassois 6). In the GSC-TARMA case, an Expectation-Maximization scheme can be used as follows (Avendaño-Valencia and Fassois 13): (i) Expectation Step: obtain θ[t] using the Kalman Filter as previously described, based on fixed values of P. (ii) Maximization Step: estimate µ and σ w[t] based on the estimated values of θ[t]. The E and M steps are sequentially repeated until convergence of the cost functionj(θ N 1,P) is achieved. The estimation of σ v is avoided within the M step, as it destabilizes the algorithm. Thus, as in the SP-TARMA case, σ v is left as a design parameter to adjust the tracking speed and the parameter smoothness. For both SP and GSC-TARMA models the selection of the value of σ v may not be straightforward, since a very low value may over-smooth the estimated parameter trajectories, while a high value may lead to noisy trajectories. The selection of σ v may be guided by comparing the innovations (prediction residuals) with the parameter innovations (prediction error of the parameters) that is the Residual Sum of Squares (RSS) to the Parameter prediction Error Sum of Squares (PESS): RSS= N t=1 ŵ [t], PESS= N t=1 ˆv T [t] ˆv[t] (9) where ŵ[t]=x[t] h T [t] ẑz[t t] stands for the one-step ahead prediction errors (residuals), ˆv[t]= ˆθ[t t] ˆθ[t t 1] is an estimate of the parameter innovations with ˆθ[t t 1] being the a priori and ˆθ[t t] the a posteriori Kalman Filter estimates of θ[t]. Both RSS and PESS are computed from the Kalman Filter predictions of w[t] and v[t] obtained for a specific value of σ v. A high RSS indicates poor modeling accuracy, whereas high PESS indicates noisy parameter estimates. A curve displaying the PESS versus the RSS (parametrized in terms of σ v ) may be constructed and used for selecting a good compromise (and hence a proper σ v ). Remarks (i) In the definition of the GSC-TARMA model it is also possible to include a stochastically time-dependent innovations variance, as in (Kitagawa and Gersch 1996). (ii) Additional definitions are possible for the general SPE- TARMA model class. For instance non-gaussian signals or non-linear stochastic parameter evolution dynamics may be included, which may be appropriate for some rapidly evolving processes (Kitagawa and Gersch 1996).. Deterministic parameter evolution (DPE) TARMA modeling DPE-TARMA models are typically defined in terms of the Functional Series TARMA (FS-TARMA) model form, for which the temporal evolution of the parameters is expressed via projections in a proper functional subspace. Thus for an FS-TARMA(n a,n c ) [pa,p c,p s ] model, with p a, p c, p s designating its AR, MA and innovations variance functional subspace dimensionalities, the evolution of the parameters is as follows (Poulimenos and Fassois 6): a i [t]= p a k=1 a i,k G ba(k) [t], c i [t]= p c k=1 c i,k G bc(k) [t], σ w[t]= p s k=1 s k G bs(k) [t] { } { { } F AR = G ba(1) [t],...,g ba(pa) [t], F MA = G bc(1) [t],...,g bc(pc) [t] }, F σ w [t] = G bs(1) [t],...,g bs(ps) [t] (1a) (1b) with F designating the functional subspace of the indicated quantity, b a(k) (k = 1..., p a ), b c(k) (k = 1..., p c ) and b s(k) (k = 1..., p s ) indices indicating the specific basis functions included in each subspace, and a i,k, c i,k and s k the AR, MA, and innovations variance, respectively, coefficients of projection. Thus, for an FS-TARMA model the model parameter vector is ϑ =[ϑ T a ϑ T c ϑ T s ] T, while model structure is specified by the AR and MA orders n a,n c and the AR, MA and innovations variance basis function indices vectors b a =[b a(1)...b a(pa )] T, b c =[b c(1)...b c(pc )] T, and b s =[b s(1)...b s(ps )] T, that is: ϑ = [ ϑ T a ϑ T c ϑ T ] T [ ] T s = a1,1...a na,p a c 1,1...c nc,p c s 1...s ps (11a) M={n a,n c, b a, b c, b s } (11b) In a recent extension, Adaptable FS-TARMA (AFS-TARMA) models that employ functional subspaces parametrized by a parameter vector δ were introduced (Spiridonakos and Fassois 14a). The model definition is as in Equation (1), but the functional bases are parametrized,f AR = { G ba (1)[t, δ a ],...,G ba (p a )[t, δ a ] },F MA = { G bc (1)[t, δ c ],...,G bc (p c )[t, δ c ] }, 5

6 F σ w [t] ={ G bs (1)[t, δ s ],...,G bs (p s )[t, δ s ] }, where δ a, δ c and δ s indicate the AR, MA and innovations variance functional subspace parameter vector, respectively. The model structure is in this case defined by just the model orders and basis dimensionalities, while the complete parameter vector includes the functional subspace parameters as well: [ θ = ϑ T δ T] T [ ] T, δ = δ T a δ T c δ T s, M={na,n c, p a, p c, p s } (1) The advantage of the AFS-TARMA model structure is that the selection of the basis function which is a structural problem in conventional FS-TARMA models, becomes part of the parameter estimation problem and is thus significantly simplified. The adaptable models may thus better adapt to a given non-stationary signal, plus the modeling procedure is easier for the user. The estimation of AFS/FS-TARMA models is typically accomplished within a Maximum Likelihood (ML) framework. In the FS-TARMA case, and under Gaussian innovations, the log-likelihood function is (Spiridonakos and Fassois 14b): lnl(ϑ x N 1 )= N lnπ 1 N t=1 ( ) lnσw[t]+ w [t] σw[t] As the likelihood function in Equation (13) is non-quadratic in terms of the unknown parameter vector, the maximization is based on iterative schemes and rather accurate initial parameter estimates are required for avoiding potential local extrema. In the simpler case of FS-TAR models, initial parameter values may be obtained by estimating ϑ a via Ordinary Least Squares (OLS) and subsequently estimating ϑ s via an overdetermined set of equations after estimating σ w[t] via a sliding window approach (Poulimenos and Fassois 6). In the FS-TARMA case more elaborate techniques are necessary, as the prediction error is a non-linear function of the projection coefficient vector ϑ c. These include linear multi-stage or recursive methods (Poulimenos and Fassois 6). Linear multi-stage methods first obtain an initial estimate of the prediction error sequence based on high order TAR models, and subsequently employ the obtained values to estimate the FS-TARMA model coefficients of projection (the Two-Stage-Least Squares (SLS) approach). Recursive methods use the Recursive Extended Least Squares (RELS) or the Recursive Maximum Likelihood (RML) algorithms to obtain initial coefficient of projection coefficient estimates (Poulimenos and Fassois 6). Several runs over the data are typically recommended to avoid the influence of the unknown initial conditions and ensure convergence of the algorithm. For adaptable (AFS-TARMA) models, an estimation scheme based on Separable Non- Linear Least Squares (SNLS), in which the vectors ϑ and δ are estimated separately in an sequential fashion, has been suggested (Spiridonakos and Fassois 14a,b). The reader is referred to (Poulimenos and Fassois 6; Spiridonakos and Fassois 14b) for further details on AFS/FS-TARMA models and their estimation. Remarks: (i) For conventional FS-TARMA models the functional subspaces include linearly independent basis functions selected from an ordered set, such as Chebyshev, trigonometric, b-splines, wavelets, and other functions. For simplicity a functional subspace is often selected to include consecutive basis functions up to a maximum index. Yet, for purposes of model parsimony (economy) and effective estimation, some functions may not be necessary and may be dropped. (ii) An FS-TARMA(n a,n c ) [pa,p c,p s ] model of the form (1) is referred to as a fully parametric FS- TARMA model. The term semi-parametric FS-TARMA(n a,n c ) [pa,p c ] model implies that the innovations variance is not parametrized, that is it is not projected on basis functions..3 Model structure selection Model structure selection is the process by which the structural parameters of the model are obtained. This is typically an iterative and tedious procedure, in which models corresponding to various candidate structures are first estimated, and the one providing the best fitness is selected. This procedure may be facilitated via integer optimization schemes, or backward/forward regression schemes (Spiridonakos and Fassois 14b). Model fitness may be judged in terms of a number of criteria, which may include the Residual Sum of Squares (RSS) (often normalized by the Series Sum of Squares, SSS), the likelihood function, and the Akaike Information Criterion (AIC) or the Bayesian Information Criterion (BIC). The latter two are typically preferred as they maintain a balance between model fit (13) 6

7 (model accuracy) and model size (thus discouraging overfitting). The RSS/SSS and BIC criteria are defined as: RSS/SSS = N t=1 ŵ [t]/ N t=1 x [t], BIC= lnl( )+ lnn d (14) where ŵ[t] is the one-step-ahead prediction error, L( ) is the likelihood of the respective model class, and d the number of estimated parameters in the model (FS-TARMA: d = dim ϑ, AFS-TARMA: d = dim θ, SPE-TARMA: d = dim z[t] = (n a + n c ) q) (Poulimenos and Fassois 6; Kitagawa and Gersch 1996, Ch. ). Notice that in the DPE-TARMA case the likelihood is defined by Equation (13), while in the SPE-TARMA case the log-likelihood function (w.r.t the hyperparameters) is equivalent to J( ˆθ N 1, ˆP) (Avendaño-Valencia and Fassois 13). It should be noted that model estimation always requires attention on part of the user in order to detect numerical problems (for instance those due to inverting an ill-conditioned matrix) or estimating a number of parameters that is not commensurate with the signal length. As a rough guide the number of signal Samples Per estimated Parameter (SPP) should be at least 15 (Spiridonakos and Fassois 14b). Finally formal model validation which examines the validity of the model assumptions (such as innovations whiteness and Gaussianity) should be performed before final model acceptance (Box et al. 1994, Ch. 8; Poulimenos and Fassois 6; Spiridonakos and Fassois 14b). 3 Structural Health Monitoring (SHM) Based on Non-Stationary Random Vibration Let s v designate a given structure in one of several potential health states. v = o designates the healthy state, while any other v from the set V ={a, b,...} designates the structural in a damaged (faulty) state of a distinct type a, b,... and so forth (for instance damage in a particular region, or of a particular nature). In general, each damage can include a continuum of damages, each being characterized by its own damage magnitude. The SHM problem may be then posed as follows: Given the structure in a currently unknown state u, first determine whether or not the structure is currently damaged (u = o or u o) (the damage detection sub-problem). In case the structure is found to be damaged, determine which one of {a,b,...} is the current damage type (the damage identification sub-problem). The damage magnitude estimation sub-problem, which focuses on estimating the magnitude of the current damage will not be treated in this article. When the main information available for solving this problem is in the form of measured structural vibration, then it is classified as a vibration based SHM problem. If additional information such as an analytical structural dynamics model is used for its solution, then the method is classified as analytical model based, otherwise as a data based. This article focuses in the latter case where analytical models are not available or are hard to obtain the reader is referred to (Fassois and Sakellariou 9) and (Farrar and Worden 13) for a broader overview and details. In the context of this article the important but obviously more difficult and scarcely studied case of non-stationary random vibration is considered. From an operational viewpoint, vibration based SHM is organized into two distinct phases: First, an initial baseline phase in which a set of vibration response signals x v [t] (t = 1,...,N), possibly for all v {o,a,b,...}, are obtained and properly processed (this phase is carried out only once). Second, an inspection phase in which (typically) a single vibration response signal x u [t] is obtained and decisions on the presence and type of damage need to be made (the aforementioned damage detection and identification sub-problems). This phase is typically carried out continuously or periodically, each time using a fresh vibration signal. In this article two SHM approaches are presented within a non-stationary random vibration context: a parameter based and a residual based. Both are based on the modeling of the non-stationary random vibration signals via the earlier discussed FS-TARMA representations. For purposes of simplicity the damage identification sub-problem is treated via successive binary hypothesis testing (instead of a single multiple hypothesis testing). In this (former) context, once the presence of damage is detected, its type is determined via successive (pairwise) comparisons with each potential damage type. An implicit assumption behind both approaches is that the operating and vibration signal measurement conditions in the baseline and inspection phases are identical. Hence the various signals correspond to each other in a proper way in particular in their time duration they describe the exact same motion or operational cycle. 3.1 A parameter based approach 7

8 The essence of this approach is on the use of the projection coefficient vector ϑ of an FS-TARMA model of the non-stationary random vibration as the characteristic quantity (or feature) in the decision making mechanism. The underlying thesis is that each distinct health state is characterized by its own projection coefficient vector, thus comparing that of the current state u to that of the healthy state o leads to damage detection. In the positive (damage) case, comparing that of the current state u sequentially (pairwise) to that of each damage state {a,b,...} leads to damage identification. Hence, in the (initial) baseline phase FS-TARMA models corresponding to the healthy o and each damage state {a, b,...} are estimated. Then, in the (current) inspection phase, a (similar in structure) FS-TARMA model corresponding to the current structural state u is estimated based on the currently available vibration signal. Then damage detection may be treated via a hypothesis test of the form: H : ϑ o ϑ u = H 1 : ϑ o ϑ u Null hypothesis healthy state Alternative hypothesis damaged state As the true coefficient of projection vectors are not available, decision making is based on corresponding estimates ˆϑ o and ˆϑ u, and since these are random quantities, on their distributions. In this context, under mild assumptions, the estimators are shown to be asymptotically (for long data records, that is N ) Gaussian distributed, with mean equal to the true coefficient of projection vector and covariance Σ that may be estimated (Poulimenos and Fassois 9a, Spiridonakos and Fassois 13). Under these conditions, the quantity dm, below, follows (assuming negligible variability for the covariance estimator) chi-square distribution with d degrees of freedom (d = dim ϑ). Hence decision making may be made as follows at the α risk level (that is false alarm probability equal to α): d M =(ˆϑ u ˆϑ o ) ˆΣ 1 o ( ˆϑ u ˆϑ o ) T χ d (1 α) Accept H o (15) otherwise Accept H 1 with χ d (1 α) designating the 1 α critical point of the chi-square distribution (d degrees of freedom) and ˆΣ o the estimator covariance in the healthy case. As already indicated, damage identification may be treated via similar pairwise tests in which the current true coefficient of projection vector ϑ u is compared to the true vector ϑ v corresponding to each damage state (for v {a, b,...}). Of course, ramifications of the method are possible, for instance by only including specific elements of the coefficient of projection vector, or a properly transformed (for instance via Principal Component Analysis) version. An obvious disadvantage of the general parameter based approach is that model estimation needs to be carried out in the inspection phase as well. 3. A residual based approach The essence of this approach is on the use of the FS-TARMA model residual signal ŵ[t], and more specifically its time-dependent variance σ w[t], as the characteristic quantity (or feature) in the decision making mechanism. The underlying thesis is that for each distinct health state (for instance the healthy state) the model residual signal (then ŵ o [t]) is characterized by its own residual variance (then σ w o [t]) this may be obtained in the baseline phase. Then, under the hypothesis that the structure is still in the same health state during inspection, the new residual signal (ŵ u [t]) obtained by driving the current (fresh) vibration signal (x u [t]) through the same baseline model (no model re-estimation involved) should be characterized by the same time-dependent variance (then σ w o [t]) if and only if the hypothesis of the structure being in the same health state (for instance healthy state) is correct (as a change would result in increased variance). Then, a decision on the health state of the structure may be made based on comparing the current time-dependent residual variance (σ w u [t]) to the baseline time-dependent variance (then σ w o [t]) at each time instant. Of course, this procedure may be repeated for any other health state of the structure in the baseline phase for binary damage identification. Then damage detection may be treated via a hypothesis test of the form (Poulimenos and Fassois 4): H : σ w u [t] σ w o [t] H 1 : σ w u [t]>σ w o [t] Null hypothesis healthy state Alternative hypothesis damaged state As the theoretical variances σ w u [t] and σ w o [t] are unavailable, they need to be estimated from the obtained 8

9 residual series ŵ u [t] and ŵ o [t] using a moving average filter (sliding window) as follows: ˆσ w u [t]= 1 l t τ=t l+1 τ=t (l o 1)/ ŵ u[τ], ˆσ w o [t]= 1 t+(l o 1)/ ŵ l o[τ] (16) o withl,l o designating the corresponding window lengths. Under the null hypothesis (H ), given the residual normality and uncorrelatedness, the statistic defined as the ratio of the two variance estimators follows an F distribution with (l 1,l o 1) degrees of freedom, that is: F[t]= ˆσ w u [t] ˆσ w o [t] F l 1,lo 1 (17) This leads to the following sequential F-test (at the α risk level, that is false alarm probability equal to α): F[t] F 1 α Accept H (18) otherwise Accept H 1 with F 1 α = F l 1,lo 1(1 α) indicating the distribution s (1 α) critical point. An obvious advantage of this approach is that no model re-estimation is required in the inspection phase. 4 Illustrative Example: Non-Stationary Random Vibration Modeling and SHM for a Pick-And-Place Mechanism 4.1 The structure and its non-stationary random vibration response The system studied in this example is the -DOF pick-and-place mechanism mentioned earlier (Figure 1(a)). The random vibration response is measured in the same direction as the excitation, using lightweight piezoelectric accelerometers. During a single experiment a single cycle is performed, in which the linear motors move from their rightmost to their leftmost position and back. The measured vibration response is conditioned and driven into a data acquisition module, which digitizes the signal with a sampling frequency f s = 51 Hz signal length 1 s (N = 51 samples). Each signal is subsequently sample mean corrected and normalized (scaled). The frequency range of interest is 5- Hz, with the lower limit set in order to avoid instrument dynamics and rigid body modes. For the SHM problem six damage scenarios are considered, which correspond to the loosening or removal of various bolts at different points of the mechanism (damages A to C and E), loosening the slider of motor B (damage D), and adding a mass at the free end of the slider of motor A (damage F). For each health state, a set of 4 random vibration responses are recorded see (Spiridonakos and Fassois 13) for details. 4. Non-stationary random vibration parametric modeling (healthy structure) The non-stationary random vibration response (see Figure 1(b)) of the healthy structure is now modeled via the Stochastic Parameter Evolution (both SP-TAR & GSC-TAR models) and the Deterministic Parameter Evolution (FS- TAR models) methods using a single data record obtained from the healthy structure. The details for each method are summarized as follows: Stochastic Parameter Evolution modeling: SP-TAR models are estimated via the Kalman Filter smoother method (Poulimenos and Fassois 6) and GSC-TAR models via the Expectation-Maximization method (Avendaño-Valencia and Fassois 13). The innovations variance is in each case estimated via a moving rectangular (6 sample long) window. The following structural parameters are considered: AR order n a {3,...,3}, with {q = 1,σv = exp( 11)} and {q=,σv = exp( 4)}. The value of σv is further optimized by estimating SP/GSC-TAR models with the selected model order and σv = exp(ν),ν ={ 1, 11,..., 3}, q={1,}. The optimal σv is determined by considering the RSS and PESS. Fully parametrized FS-TAR models with trigonometric basis func- Deterministic Parameter Evolution modeling: tions of the form: G [t]=1, G k 1 [t]=sin(πkt/n), G k [t]=cos(πkt/n) k=1,,... (19) 9

10 are considered. The functional subspace dimensionality p a corresponds to the number of sine and cosine functions used by the FS-TAR model plus one (the constant component), t = 1,,...,N is the normalized discrete time, and N the signal length. Parameter estimation is based on Ordinary Least Squares, while the innovations variance is estimated via the instantaneous method (Poulimenos and Fassois 6). The following structural parameters are considered: AR order n a {3,...,3}, p a = p s {3,5,...,1}. (a) RSS/SSS [%] FS-TAR SP-TAR q=1 SP-TAR q= GSC-TAR q=1 GSC-TAR q= n a = 4 (b) BIC (x 1 4 ) p a = Functional subspace dimensionality (c) Normalized BIC PESS n a = n -1 v = e -1 v = e SP-TAR q=1 SP-TAR q= GSC-TAR q=1 GSC-TAR q= RSS/SSS [%] (d) BIC (x 1 4 ) RSS/SSS [%] Proj. coeff a i,k Basis function index C D C C D D H B A E F H A B F H A B F E E FS-TAR SP-TAR GSC-TAR H A D E F B C H A C D B E F C H A D F B E FS-TAR SP-TAR GSC-TAR Figure : Model structure selection for SP-TAR, GSC-TAR and FS-TAR models: (a) Model order selection: RSS/SSS and BIC (the latter normalized between and 1) curves for SP/GSC/FS-TAR models with orders n {1,...,3}. (b) Selection of the basis dimensionality of FS-TAR models: Top BIC of FS-TAR(4) models with p a = p s {3,5,...,1}; Bottom estimated projection coefficients for each basis dimensionality (continuous lines connect the point estimates and bars indicate±one standard deviation). (c) Optimization of σ v for SP/GSC-TAR models PESS versus RSS/SSS for SP-TAR(4) and GSC-TAR(4) models. (d) Sample distribution of RSS/SSS and BIC for the selected model structure re-estimated for each healthy (H) and damage (A... F) data record (4 models per health state). Modeling results: The results of the model structure selection procedure are depicted in Figure (a),(b),(c). Figure (a) shows the RSS/SSS and normalized BIC for models from each considered class (FS/SP/GSC-TAR), with the selected order (n a = 4) being indicated by an arrow. Figure (b) shows the functional basis dimensionality selection for FS models. The top plot shows the BIC of FS-TAR(4) models versus functional basis dimensionality; the selected dimensionality (p a = 5) is indicated by an arrow. The bottom plot shows the absolute value of the estimated coefficients of projection with their corresponding ±1 standard deviation interval indicated by the bars. It is evident that the standard deviation interval of the estimated coefficients of projection for the basis indexes and 4, indicated by the arrows in the plot, consistently contain zero, and may be thus removed from the model. An FS-TAR(4) [3,5] model with functional basis indices b a =[,1,3] and b s =[,1,,3,4] is finally selected. 1

11 Figure (c) shows the PESS versus RSS/SSS plot for the selection of σ v for SP-TAR and GSC-TAR models. An arrow indicates the model selected, which corresponds to a good compromise between low PESS and RSS/SSS. The selected model structures are: SP-TAR(4) with q=1, σ v = e 1 and GSC-TAR(4) model with q=1, σ v = e 1. Models of the selected (under the healthy condition) structure are subsequently fitted (estimated) for each data record corresponding to the healthy and each damaged state of the structure (4 models per health state, each one based on a distinct data record). In Figure (d) the sample distribution of RSS/SSS and BIC of the above selected models are presented for the various health states. The FS-TAR model structure uniformly (for all health states and data records) achieves the lowest BIC; although its RSS/SSS values are not minimal. The frozen type TV-PSDs of the healthy structure, as obtained by the aforementioned three model types and a single data record, are presented in Figure 3. For purposes of comparison, the non-parametric spectrogram (Gaussian window σ = 8, N f ft = 14 samples, 51 samples advance ( 5% of N f ft )) estimate is also shown. While all TV-PSDs are in rough overall agreement, it is obvious that the parametric model-based ones are much cleaner and informative than their non-parametric (spectrogram) counterpart. This is an important feature of parametric methods. (a) (b) (c) (d) Figure 3: Spectrogram and frozen TV-PSD estimates obtained from estimated TAR models using a single vibration response of the healthy structure: (a) Spectrogram; (b) SP-TAR(4) (q = 1); (c) GSC-TAR(4) (q = 1); (d) FS- TAR(4) [3,5]. 4.3 Non-stationary vibration response based SHM As already mentioned the healthy (H) and six damage scenarios (A to F) are considered in SHM, with 4 data records used in each health state. Two versions of the FS-TAR model parameter based approach are used: (a) The original version in which the coefficient of projection covariance matrix Σ o is estimated based on a single data record, and (b) an alternative version 11

12 in which the covariance matrix is estimated based on several (presently 35) data records (using the sample covariance estimator). Damage detection results using version (a) are depicted in Figure 4(a). All 4 cases corresponding to the healthy structure provide d M values lying below the selected detection threshold (dashed horizontal line: α = 1 14 ), thus correctly detecting the current healthy state. Also, in all cases corresponding to damages A...D, the obtained d M values are above the detection threshold, thus correctly detecting damage. Only certain cases corresponding to damages E and F are not properly detected. As indicated by the ROC curves (Receiver Operating Characteristic curves which depict the True Positives, TPs, versus False Positives, FPs, as the threshold varies) of Figure 4(b), the two versions of the parameter based approach perform quite adequately and similarly (the performance is almost ideal if damages E and F are excluded). Next, the FS-TAR residual based approach is employed using variance estimates obtained via a sliding window of length l=l o =. Figure 4(c) provides a comparison of the obtained F[t] statistic for a healthy and a damaged (damage D) case. F[t] is, at all times, under the threshold F u (α = 1 4 ) indicating healthy structure, or above it (for at least some times) indicating damaged structure. As indicated by the ROC curves of Figure 4(d), the residual based approach provides, in this application, inferior performance, which is somewhat improved when only two types of damage, namely C and D, are considered. (a) log 1 ( d M ( θ u, θ o )) (c) F[t] χ (1- α), α=1-14 d O A Damaged F, α -4 1-α =1 B C D Healthy E Damaged Healthy 1 Healthy Time [s] F Fault D (b) True Positive Rate (d) True Positive Rate α=1-14 FS-TAR all (th) FS-TAR reduced (th) FS-TAR all (smp) FS-TAR reduced (smp) False Positive Rate. FS-TAR all FS-TAR reduced False Positive Rate Figure 4: Summary of damage detection results: (a) Boxplots of dm for the parameter based approach (original version; 4 experiments per health state) a damage is detected if dm exceeds the threshold (dashed horizontal line; α = 1 14 ); (b) ROC curves for the original ( th ) and alternative ( smp ) versions of the approach when all damage types are included ( all ) and when damage types E and F are excluded ( reduced ); (c) Performance of the residual based approach F[t] versus the threshold (l = l o =, α = 1 4 ) for a single healthy and a damaged state; (d) ROC curves for the residual based approach when all damage types are included ( all ) and when only damage types C and D are included ( reduced ). 5 Summary 1

13 Models and methods for non-stationary random vibration parametric modeling have been presented, with focus on the Stochastic Parameter Evolution (SP-TAR and GSC-TAR models) and Deterministic Parameter Evolution (FS- TAR models) methods. Approaches for non-stationary random vibration based Structural Health Monitoring (SHM) employing these model types have been also discussed. An illustrative example of the application of the methods to the vibration response modeling and SHM for a laboratory pick-and-place mechanism has been also presented. Some key points are summarized below: Parametric SPE and DPE modeling of non-stationary random vibration are more involved than their nonparametric counterparts, but offer unique opportunities for more accurate and compact representations, improved time-frequency resolution, analysis, and SHM. Other areas such as prediction and control may significantly benefit as well. SPE modeling is better suited to slow or medium variations in the non-stationary dynamics, whereas DPE modeling is also suited (with proper functional subspace selection) for fast variations. The problem of model structure selection is important for both SPE and DPE modeling. Yet, in the latter case it may be significantly alleviated via the new class of adaptable models (Spiridonakos and Fassois 14a). The problem of local extrema in the (non-convex) estimation criterion is important for both SPE and DPE modeling. Good initial values and careful use of model validation and diagnostic tools are thus necessary (Poulimenos and Fassois 9b). SHM in non-stationary random vibration environments is a rather recent, but important area with many potential applications. The presented approaches should be viewed as initial attempts to address the problem. Main Cross-references Model Class Selection for Prediction Error Estimation Operational modal analysis in civil engineering: an overview Stochastic structural identification from vibrational and environmental data System and damage identification of civil structures References Avendaño-Valencia, L. and Fassois, S. (13). Generalized stochastic constraint TARMA models for in-operation identification of wind turbine non-stationary dynamics. In Basu, B., editor, Key Engineering Materials (Volumes ) - Damage Assessment of Structures X, pages Trans Tech Publications, Switzerland. Bendat, J. and Piersol, A. (). Random data analysis and measurement procedures. John Wiley and Sons. Box, G., Jenkins, G., and Reinsel, G. (1994). Time Series Analysis Forecasting and Control. Prentice-Hall. Farrar, C. and Worden, K. (13). Structural Health Monitoring a machine learning perspective. John Wiley. Fassois, S. and Sakellariou, J. (9). Statistical time series methods for structural health monitoring. In Encyclopedia of Structural Health Monitoring, pages John Wiley & Sons Ltd. Feng, Z., Liang, M., and Chu, F. (13). Recent advances in time-frequency analysis methods for machinery fault diagnosis: a review with application examples. Mechanical Systems and Signal Processing, 38(1): Fouskitakis, G. and Fassois, S. (). Functional series TARMA modeling and simulation of earthquake ground motion. Earthquake Engineering and Structural Dynamics, pages Gersch, W. and Akaike, H. (1988). Smoothness Priors in Time Series. In Spall, J., editor, Bayesian Analysis of Time Series and Dynamic Models, pages Marcel Dekker. 13

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