Model-based failure detection for cylindrical shells from noisy vibration measurements J. V. Candy,a) K. A. Fisher, B. L. Guidry, and D. H. Chambers University of California, Lawrence Livermore National Laboratory, P.O. Box 808, L-151, Livermore, California 94551
(Received 29 July 2014; revised 23 September 2014; accepted 6 October 2014) Model-based processing is a theoretically sound methodology to address difficult objectives in complex physical problems involving multi-channel sensor measurement systems. It involves the incorporation of analytical models of both physical phenomenology (complex vibrating structures, noisy operating environment, etc.) and the measurement processes (sensor networks and including noise) into the processor to extract the desired information. In this paper, a model-based methodology is developed to accomplish the task of online failure monitoring of a vibrating cylindrical shell externally excited by controlled excitations. A model-based processor is formulated to monitor system performance and detect potential failure conditions. The objective of this paper is to develop a real-time, model-based monitoring scheme for online diagnostics in a representative structural C 2014 Acoustical Society of America. vibrational system based on controlled experimental data. V [http://dx.doi.org/10.1121/1.4898421] PACS number(s): 43.60.Uv, 43.60.Gk, 43.60.Cg, 43.30.Zk [ZHM]
I. INTRODUCTION
Structural vibration systems1 operating over long periods of time can be subjected to component and overall system failures due to fatigue caused by external excitations or by cracking and resonances during the aging process.1–6 In some cases the system can fail catastrophically, causing complete deterioration of the overall structure and the consequences that follow. It is essential to provide timely information about component failure and the conditions leading up to it as well as predicting the integrity of the total system in the future. A model-based approach7–11 to develop diagnostic techniques for structural failure prediction is based on a wellestablished theoretical formalism that ensures a careful and systematic analysis of the vibrational system or classes of systems under investigation and their associated failure mechanisms. Models can be developed from a firstprinciples mathematical representation, or by utilizing sensor measurements to “fit” the model parameters to the data—the approach we pursue in this paper. In cases where the failure mechanism is not well understood, the model fitting approach offers an effective solution. In fact, in contrast to many approaches of solving this problem, this model-based approach does not require an explicit representation of the failure mechanism at all to design an online monitor for initial failure detection. It is based on obtaining a representation of the vibrating structure during normal certification operations, or even more desirable, during quality assurance or acceptance testing. Once these normal operational characteristics of the system are known, the processor is developed based on the properties of the so-called residuals or innovations sequence. This sequence is the difference a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
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between the measured and predicted sensor outputs obtained from the underlying normal structural system model. Once this condition or “difference” monitor detects a failure, it is then classified according to a set of well-established failure mechanisms. Failure detection for mechanical systems has long been the subject of much research with most of the work being concentrated on single-channel, spectral or cepstral analysis techniques for diagnostic purposes.2 There have been several successful applications of the model-based approach8–17 to these type problems, but none devoted to solving the complete failure condition detection, failure isolation, and timeto-failure prediction problem for a real-time maintenance problem. The inclusion of a process model in any signal processing scheme provides a means of introducing information in a self-consistent manner.18–21 This is shown in Fig. 1 where we observe how vibrational measurements are input to a model-based processor (MBP) utilized to provide information to the failure monitor. As mentioned, structural vibration systems operating over long periods of time can be subjected to failures due to fatigue caused by various external forces or simply the aging process. When this happens it is possible for the structure to deteriorate causing a complete malfunction and possibly catastrophic failure. Thus, to attain reliable system performance it is necessary to monitor critical structural components, to provide timely information about current and/or imminent failures, and to predict its integrity. Standard approaches to detect failure mechanisms at the onset range from a simple accelerometer strategically placed, to observing the Fourier spectrum of known response, to using cepstral analysis to identify periodic responses, to sophisticated Bayesian processing schemes.22–27 Measures of failure can deteriorate significantly if noise is present—a common situation in an operational environment. A model-based signal processing approach8 to solve the structural system failure detection
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FIG. 1. (Color online) Conceptual model-based failure monitor.
problem is depicted in Fig. 2. Here we see that the first step is monitoring the current condition of the underlying structure. Should a failure be detected, a classifier processes the data to report the final condition and type of failure incurred (not addressed here). Most of the current monitoring approaches for failure detection and isolation lead to single-channel processing of measured sensor data.1 For instance, the so-called waterfall vibrational spectral plot, or the equivalent Campbell diagram,3 is limited to a stacking display of single-channel spectral information to identify failures. Multiple sensors (such as accelerometers for vibrations, microphones for acoustics, and thermocouples for temperature) in a structure provide enhanced information about the system. This implies a multi-channel (multi-input, multi-output) system representation, which is most easily handled in state-space form. Note that the state-space representation of a system is the transformation of an nth-order set of differential equations describing the system to a set of n first-order differential equations, without restrictions to single-channel spectral representations. Once this representation is accomplished, then the process model of the structural system can be incorporated into an effective signal processing scheme. This representation leads to the model-based approach, which can be stated succinctly as “incorporating mathematical models of both physical phenomenology and the measurement processes including noise into the processor to extract the desired information.” The incorporation of a mathematical model that represents the phenomenology under investigation can vastly improve the performance of any processor, provided such model is accurate.8,17 We summarize the development and application of such online “model-based processors” (MBP) in the context of failure
detection. One of the major advantages of a model-based processor is that it is capable of monitoring its own performance by testing the residual error (or innovations) between the measurement and its prediction. The model-based approach for failure detection is based on the concept that the process or vibrational system under consideration is modeled either from first principles or using system identification/parameter estimation techniques15,16,19 to “fit” models to the data. Once the system models are developed, then the sensor suite (or measurement system) models are developed. Usually the bandwidth of each sensor is much wider than the dominant dynamics or resonances of the system, and therefore each sensor is represented by a gain. However, if sensor dynamics must be included in the system model, the model-based approach easily accommodates them. Sensor dynamics models can be obtained from manufacturer specifications, independent experiments, or transfer function estimation to name a few (see Sec. II for some details). After these models are completed, then it remains to model the noise. If noise data are unavailable, then a reasonable approach is to model the noise as additive and random, leading to a Gauss–Markov model.19 Once a representation of the overall system (structure, sensors, and noise) is developed, then a failure monitor/detector can be developed to monitor the status of the vibrational system. Should a failure (that is, a deviation from the normal operation) be detected, then this failure must be classified using information from known or measured failures. Then it is possible to predict the time-to-failure for the overall system, and decide on the appropriate action, including simply reporting the results. Here we concentrate on the model development and online monitoring function. In Sec. II, we discuss the basic models of the cylindrical shell system under investigation along with both distributed and lumped governing dynamic systems as well as the embedded novel optical strain measurement sensor employed. Next we present the fundamental model-based approach in Sec. III including the underlying failure detection problem. The model-based processor is developed in Sec. IV and applied to simulated and experimental data in Sec. V. We summarize our results in the final section. II. DYNAMIC MODELING OF A VIBRATING CYLINDRICAL SHELL AND STRAIN SENSOR NETWORK
In this section we discuss the basic dynamic models used to represent the vibrating cylindrical shell under study for failure detection along with the novel optical strain sensor network. We start with the fundamental wave dynamics and corresponding numerical model using a finite element approach. Next we discuss a coupled lumped multi-variable dynamic model and corresponding transfer function matrices that provide the structure and justification of the MBP employed in the failure detector. The fundamental governing equation for the vibrating elastic cylindrical shell is5 FIG. 2. (Color online) Model-based failure monitor and classifier structure. J. Acoust. Soc. Am., Vol. 136, No. 6, December 2014
q
@2u lr2 u ðk þ lÞrðr uÞ ¼ qf; @t2 Candy et al.: Model-based failure detection
(1) 3115
where u(x, t) is the vector displacement field, f(x, t) is the forcing function, q is the material density, and k and l are the material elastic moduli. The displacement field is a superposition of longitudinal and shear displacement fields. Mathematically these can be separated, however, our measurement system is sensitive only to the total surface displacement and the individual contributions of the longitudinal and shear displacement fields are not necessary for failure detection. This equation, along with the boundary conditions, describes the dynamic response of the cylindrical shell to a given set of applied forces. We can convert this to a lumped model using a set of orthonormal vector basis functions un(x) such that ð um ðxÞ un ðxÞ; dx ¼ dmn : The displacement field and forcing excitation can be expanded in terms of these basis functions as uðx; tÞ ¼
Nd X
dn ðtÞun ðxÞ;
fðx; tÞ ¼
n¼1
Nd X
pn ðtÞun ðxÞ;
n¼1
which leads to the following matrix equation for the coefficients dn(t) (Ref. 6): € þ CdðtÞ _ þ KdðtÞ ¼ pðtÞ; MdðtÞ
(2)
where d is the Nd 1 displacement vector, p is the Np 1 excitation force, and M, C, K, are the Nd Nd lumped mass, damping function, and spring constant matrices characterizing the structural process model, respectively, 2 3 M1 0 0 0 0 6 0 M2 0 0 0 7 6 7 6 .. .. . . .. .. 7 M¼6 . ; C ¼ ½cij . . . . 7 6 7 4 0 5 0 0 0 MNd 1 0 0 0 0 MNd and 2 6 6 6 6 K ¼6 6 6 4
ðK1 þK2 Þ K2 0 0 0
3 0 0 0 7 .. 7 ðK2 þK3 Þ . 0 0 7 7 .. .. 7: . . KNd 1 0 7 7 0 KNd 1 ðKNd 1 þKNd Þ KNd 5 0 0 KNd KNd K2
Though the damping function is not explicitly part of the original governing equation, it is standard practice to include damping in the lumped model. This model is in the form of a linear dynamical system, which can be converted to a state-space representation for signal processing purposes. That is, if we define the 2Nd-state vector as _ 0 , then the state-space representation of this xðtÞ :¼ ½dðtÞjdðtÞ process can be expressed as 3116
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x_ ðtÞ ¼ AxðtÞ þ BuðtÞ; " # 0 I 0 x_ ðtÞ ¼ pðtÞ; x ðt Þ þ M1 K M1 C M1 and yðtÞ ¼ M1 j0 xðtÞ;
(3)
where we have assumed that a sensor directly proportional to displacement is used, y 2 RNd 1 . For our measurement system we actually use a Bragg fiber optic strain sensor that is small ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ a R ðiÞ=N R ðiÞ=N for i ¼ 1; …; Ny :
(8)
Therefore, the zero-mean test,8 on each component innovation i is given by Candy et al.: Model-based failure detection
^ ðiÞ l
>Reject H0 s for i ¼ 1; ;Nd : 30: Iq ¼ q N
(14)
^ ði; kÞ values Hence, under the null hypothesis, 95% of the q must lie within this confidence interval to accept H0 . That is, for each component innovation sequence to be considered statistically white. The whiteness test of Eq. (14) is very useful to detect modeling inaccuracies from individual component innovations. However, for complex systems with a large number of measurement channels, it becomes computationally burdensome to investigate each innovation componentwise especially under the limiting ergodicity assumptions. C. Vector whiteness detection
A statistic containing all of the innovation information is the weighted sum-squared residual (WSSR). It aggregates all J. Acoust. Soc. Am., Vol. 136, No. 6, December 2014
N 1 0 ðkÞR1 : ðkÞðtk Þ for ‘ M; 2 R
k¼‘Mþ1
q ð‘Þ
^ ði; kÞ ¼ q
‘ X
q ð‘Þ :¼
>H1 s: 30, q(‘) is approximately Gaussian N (NM, 2NM) (see Refs. 8–11 for more details). At the a-significance level, the probability of rejecting the null hypothesis is given by ! jqð‘Þj N M s N M (17) Pr pffiffiffiffiffiffiffiffiffiffiffiffi > pffiffiffiffiffiffiffiffiffiffiffiffi ¼ a: 2N M 2N M For a level of significance of a ¼ 0.05, we have pffiffiffiffiffiffiffiffiffiffiffiffi s ¼ N M þ 1:96 2N M:
(18)
Thus, we see that the WSSR can be considered a “whiteness test” of the innovations vector over a finite window of length N. Note that since [{(tk)}, {R(tk)}] are obtained from the state-space MBP algorithm directly, they can be used for both stationary as well as nonstationary processes. In fact, in practice for a large number of measurement components, the WSSR is used to “tune” the filter and then the component innovations are individually extracted and checked for zeromean/whiteness to detect failures. Note also that the adjustable parameter of the WSSR statistic is the window length M, which essentially controls the width of the window sliding through the innovations sequence. Therefore, even if we cannot implement the optimal detector, we can still perform a set of pragmatic statistical hypothesis tests to investigate the condition of the underlying structure and classify its failure.
IV. FAILURE DETECTION
In this section we extend the basic principles of modelbased detection and apply them to our particular problem of cylindrical shell failure detection. P First we start with the development of a “normal” model ð N Þ providing a standard for monitoring and comparison. Here we assume that we have measured the vibrational response of the structural system (cylindrical shell) and developed a model to capture its performance, that is, the normal (state-space) system, Candy et al.: Model-based failure detection
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P
N :¼ fAN ; BN ; CN g is characterized by the structural matrices of Eq. (3). We measure an ensemble of vibrational responses from the system using a controlled excitation over the frequency range of interest and then use them to obtain a representative model by averaging over the estimated parameters for each realization, y‘(tk); ‘ ¼ 1,…, L. That is, we “fit” a model to the ‘th realization X ðH‘ Þ :¼ fAN ðH‘ Þ; BN ðH‘ Þ; CN ðH‘ Þg (19) N
measurement is given by y(tk; r) where r is the sensor position vector on the structure. We will (tacitly) assume that the positions are weakly coupled at first and ignore the crosscoupling terms simplifying the models and measurements. Therefore, we separate the analysis of each subsystem defined by: A(H) ¼ diag [A1(H) A2(H)]. The subsystem model parameters will be “fit” to each subsystem individually and the measurements and detections processed accordingly. With this in mind we proceed to process the measured vibrational response data from the cylinder.
and then average over the entire ensemble X N
A. Cylindrical shell vibrational response (normal) model
X X L X ðH‘ Þ :¼ EH ðH‘ Þ ¼ ðH‘ Þ N N
‘¼1
¼ fAN ðHÞ; BN ðHÞ; CN ðHÞg:
(20)
A model order (dimension of the AN ) is selected to guarantee a zero-mean/white innovations sequence using the ensemble as ðtk ; H‘ Þ ¼ y‘ ðtk Þ y^ðtk ; H‘ Þ for ‘ ¼ 1; …; L:
(21)
The model extraction (parameter estimation) and validation method is shown in Fig. 4. Here we see that an initial normal model is estimated from part of the ensemble data (training data) to generate a set of parameters producing an average model (above). Next the parameterized model is applied to test data where the innovations sequence is generated and tested for acceptability. If acceptable, the parameter ðHN Þ is adjusted (averaged) and the model is updated yielding an acceptable system model. In this way the normal model is trained, tested and validated over an ensemble of both training and test data to ensure its validity. Once we have performed the parameter estimation,8 we construct the state-space model from its parameters. Since we use a prediction error parameter estimation technique, the resulting state-space model is the discrete (sampled) innovations model parameterized by the following x ðtk jtk1 Þ þ Kðtk ;HN Þðtk ;HN Þ; x^ðtk jtk Þ ¼ AðHN Þ^ ðtk ;HN Þ ¼ yðtk Þ y^ðtk ; HN Þ; x ðtk jtk1 Þ: y^ðtk Þ ¼ CðHN Þ^
(22)
In our case, since we have two sensors available and we choose two positions to locate them, then our vector
FIG. 4. Normal vibrational model extraction and validation: Ensemble parameter estimation/prediction, prediction error whiteness, model parameter updating. 3120
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The normal model is developed using the parameter estimation techniques resulting in the innovations model of Eq. (22). A typical set of cylindrical shell response data is shown in Fig. 5 where we see the noisy time series measurements and corresponding power spectra. The cylindrical shell excitation was a chirp (swept sinusoid) that was varied from 100 Hz to 5 kHz. This range was selected from numerical model studies5,6 of the cylinder. Next the models were fit as discussed above with the validation over a test ensemble (30 members) of vibrational response measurements. A typical fit and validation are shown in Fig. 6 where we observe the innovations sequence in Fig. 6(a) with the associated zero-mean/whiteness and WSSR statistic in Figs. 6(b) and 6(c). In Fig. 6(b), the zero-mean test is passed as well as the component “whiteness” test (95% of the correlation samples must lay within the bounds (dashed lines). In Fig. 6(c) the WSSR statistic must lay below the threshold (dashed lines) and it does. Thus, this realization passes the “whiteness” criteria tests and is acceptable for model updating (averaging) parameters. With the average normal (pristine) model validated in terms of the ensemble zero-mean/whiteness testing, a set of data were generated to test the performance of the estimated prediction model. The results for these tests are shown in Fig. 7. In Fig. 7(a), we see the results for both sensors. Here we observe that from an ensemble of 30 (2 sensors) that 100% of the innovations (correlation) lie within 10% out while 75% were “statistically white” lying within the 5% out bounds for sensor 1 (circles). The sensor 2 (solid circles) results were above the 5% bound with 93% of the ensemble below the 10% bound. These results indicate that the estimated average prediction model (Fig. 5) is adequate for failure detection processing. Next we create failure conditions by attaching masses at the end of the cylinder: one of 300 g and one of 150 g The resulting spectra for both sensors are shown in Fig. 8 where we also indicate the number of modes or resonant peaks residing in the 1000–2000 Hz band. We note that for sensor 1 the number of modes or resonant peaks under normal conditions are 5, while for the 150 and 300 g masses the number of modes increase to (5 ! 10,5 ! 12). For sensor 2 normal conditions of 8, the number of modes increases (5 ! 12,5 ! 11). In conclusion, we see that for both masses the changes from normal (increased modes) enable a detection indicating the robustness of the process. Candy et al.: Model-based failure detection
FIG. 5. (Color online) Cylindrical shell vibrational response data: Time series, average power spectra and ensembles for each sensor with the number of modes (resonant peaks) (5,8) for sensors 1 (5) and 2 (8).
V. RESULTS
As mentioned, failure conditions were created by attaching the specified mass for each experiment and generating a corresponding ensemble (30 members) of resonance data for performance testing. The failure detections using the average
normal (pristine) model was applied after validation tests (above) were performed indicating the robustness of the embedded processor model. The results are shown in Fig. 9 for both sensors and both cases (150 g/300 g). For the 300 g mass induced failure conditions all of the members of the test ensemble were detected successfully
FIG. 6. (Color online) Ensemble zeromean/whiteness and validation testing for normal cylindrical shell vibrational models. (a) Innovations sequence. (b) Zero-mean/whiteness test. (c) WSSR test.
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FIG. 7. (Color online) Self-validation ensemble whiteness test results for vibrational models: (a) Normal (pristine) model self-validation (sensor No. 2 solid circles). (b) Failure model for 300 g mass self-validation (sensor No. 2 solid circles). (c) Failure model for 150 g mass self-validation (sensor No. 2 solid circles).
with over >60% bound (non-white), that is, recall to be declared normal each test should lie within the 5% whiteness bound—these failures exceed >60% for sensor 1 and >55% for sensor 2. The results for the 150 g mass induced failure conditions are slightly better with failures exceeding >60% in each run for sensor 1 and >58% for sensor 2. From the figure we see that the failure detector performs quite well.
As a final part of the analysis, we estimate the residual sequences comparing the ensembles of normal spectra to that of the failure (induced) spectra and the results are shown in Fig. 10. The normal spectra are flat with no resonant peaks illustrating “white” spectra. Clearly there are spectral differences induced by the attached masses as illustrated in the figure. The differences are quite
FIG. 8. (Color online) Change detection: Sensor average and ensemble spectra for failure conditions: (a) 150 g mass spectra from sensor 1 (5 ! 10 modes or resonant peaks). (b) 150 g mass spectra from sensor 2 (8 ! 12 modes or resonant peaks). (a) 300 g mass spectra from sensor 1 (5 ! 12 modes or resonant peaks). (a) 300 g mass spectra from sensor 2 (8 ! 11 modes or resonant peaks). 3122
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VI. CONCLUSIONS
FIG. 9. (Color online) Change detection: Failure detector performance on vibrational test data (sensor No. 2 solid circles): (a) 300 g mass induced failure condition for sensors 1 and 2 (>60% out, >55% out). (b) 150 g mass induced failure condition for sensors 1 and 2 (>60% out, >58% out).
interesting for this particular location of the mass, sensor 1 shows a large spectral difference from the normal compared to that of sensor 2 indicating that is more sensitive to this vibrational response. This completes the discussion of the results, next we conclude and discuss future effort.
We have demonstrated the effectiveness of a modelbased approach to failure detection by performing parameter estimation and model-based detection on sets of controlled vibrational response experimental data. By investigating the response of data and fitting parametric models we were able to use whiteness testing of the innovations (residual) sequence to both validate and detect two sets of failure conditions induced by attaching masses to a cylindrical test object. The processor detects a departure from the normal structural condition. The results indicate that this is a viable approach. Future efforts will be aimed at location the failure itself by utilizing the residuals that can be extracted from the normal spectra when compared to the abnormal. Here the residual sequence is obtained using the model-based approach.8 The required spectral are estimated using the average normal model and a failure mode model to create the residual spectra. These spectra could then be backpropagated using a propagation model and like time-reversal “focuses” back to the failure position performing the desired localization. Preliminary results appear to be quite promising.
ACKNOWLEDGMENTS
This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. APPENDIX
To formally pose the sequential failure detection problem, we appeal to classical detection theory.20 We are to test the binary hypothesis that the innovations sequence is zeromean and white H0 : ðtk Þ N ½0; R ðtk Þ H1 : ðtk Þ N ½ l ; R ðtk Þ
½WHITE; ½NON WHITE;
which is a statistical test for the zero-mean and whiteness of the innovations sequence. Note that we assume that we know ðtk Þ and R ðtk Þ a the model error and how to calculate l priori. The optimal solution to this binary detection problem is based on applying the Neyman–Pearson theorem leading to the likelihood ratio23 and is given by the ratio of probabilities for the sequential innovations detector (SID), that is
H1 Pr E tN j H1 > s; L½E tN :¼
Pr E tN j H0 < H0
FIG. 10. (Color online) Estimated spectral residuals with normal (flat) spectrum: (a) Mass induced failure condition for sensor 1. (b) Mass induced failure condition for sensor 2. J. Acoust. Soc. Am., Vol. 136, No. 6, December 2014
where E tN :¼ fðt0 Þ; ðt1 Þ; …; ðtN Þg or expanding Pr ðt0 Þ; ðt1 Þ; …; ðtN ÞjH1 L½E tN ¼ Pr ðt0 Þ; ðt1 Þ; …; ðtN ÞjH0 but from the chain rule of probability,8 we have that Candy et al.: Model-based failure detection
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PrðE tk j Hi Þ ¼ Pr½ðtk ÞjE tk1 ; Hi
which yields the new test
Pr½ðt1 Þjðt0 Þ; Hi Pr½ðt0 Þj Hi ;
K ðtk Þ !1
which can be expressed succinctly using Bayes’ rule as
!0 K ðtk Þ !1
PrðE tk j Hi Þ ¼ Pr½ðtk Þ; E tk1 jHi ¼ Pr½ðtk ÞjE tk1 ; Hi PrðE tk1 j Hi Þ:
K ðtk Þ !0
Taking logarithms we obtain the relation for the sequential log-likelihood k ðtk Þ :¼ lnL½E tk lnPr½ðtk ÞjE tk1 H0 : The corresponding Wald or sequential probability ratio test8 is then given by Accept H1 ;
s0 k ðtk Þ s1 k ðtk Þ s0
Continue;
Accept H0
and the underlying conditional Gaussian distributions are given by N=2 jR ðtk Þj1=2 Pr ðtk ÞjE tk1 ; H0 ¼ ð2pÞ 1 exp 0 ðtk ÞR1 t t ð Þ ð Þ k k 2 and N=2 jR ðtk Þj1=2 Pr ðtk ÞjE tk1 ; H1 ¼ ð2pÞ 1 ðt k Þ 0 exp ½ðtk Þ l 2 1 t l R t t ð k Þ : ð k Þ½ ð k Þ If we include the determinants in the thresholds, we have the modified decision function 1 K ðtk Þ ¼ K ðtk1 Þ þ 0 ðtk ÞR1 ðtk Þðtk Þ 2 1 1 ðtk Þ½ðtk Þ l ðt k Þ 0 R ðt k Þ ; ½ðtk Þ l 2 3124
Accept H0 ;
1 1 1=2 !1 ¼ s1 þ lnjR ðtk Þj1=2 jR ; ðtk Þj 2 2 1 1 1=2 : !0 ¼ s0 þ lnjR ðtk Þj1=2 jR ðtk Þj 2 2 The implementation of this “optimal” failure detector/ monitor presents a basic problem, such as how to obtain estimates of the innovations (abnormal) mean and covariance required, but does illustrate a potential optimal solution to the model monitoring problem. As mentioned, the SID requires a priori knowledge of the actual model “mismatch” and structurally how it enters the process model to obtain ðtk Þ for the monitor.16–18 ½ l ðtk Þ; R 1
¼ k ðtk1 Þ þ lnPr½ðtk ÞjE tk1 ; H1
k ðtk Þ s1
Continue;
where
Substituting these expressions (replacing tN by tk) into the equation and grouping we obtain "
# Pr ðtk ÞjE tk1 ; H1 Pr E tk1 j H1
; L½E tk ¼ Pr ðtk ÞjE tk1 ; H0 Pr E tk1 j H0 and therefore we have the recursion or equivalently sequential likelihood as Pr ðtk ÞjE tk1 ; H1 : L½E tk :¼ L½E tk1 Pr ðtk ÞjE tk1 ; H0
Accept H1 ;
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V. Wowk, Machinery Vibration: Measurement and Analysis (McGrawHill, New York, 1991), 358 pp. 2 H. C. Pusey and S. C. Pusey, Focus on Mechanical Failures: Mechanisms and Detection (Vibration Institute, Willowbrook, IL, 1991), 362 pp. 3 A. D. Dimarogonas and S. Haddad, Vibration for Engineers (PrenticeHall, Englewood Cliffs, NJ, 1992), 825 pp. 4 C. J. Li, T. Kim, and G. W. Nickerson, “Screw compressor fault detection and diagnosis based on modeling and estimation,” in Proc. 45th Meeting Mech. Failures Prevention Group, Annapolis, MD (1991), pp. 35–45. 5 J. D. Achenbach, Wave Propagation in Elastic Solids (Elsevier-NorthHolland, New York, 1980), 440 pp. 6 O. C. Zienkiewicz, The Finite Element Method (McGraw-Hill, New York, 1983), 648 pp. 7 M. Basseveille and I. Nikiforov, Detection of Abrupt Changes: Theory and Application (Prentice-Hall, Englewood Cliffs, NJ, 1993), 586 pp. 8 J. V. Candy, Model-Based Signal Processing (Wiley/IEEE, Hoboken, NJ, 2006), 677 pp. 9 F. Gustafsson, Adaptive Filtering and Change Detection (Wiley/IEEE, Hoboken, NJ, 2000), 500 pp. 10 A. Willsky, “A survey of design methods for failure detection in dynamic systems,” Automatica 12, 601–611 (1976). 11 R. Mehra and J. Peshon, “An innovations approach to fault detection and diagnosis in dynamical systems,” Automatica 7, 637–640 (1971). 12 S. G. Azevedo, J. V. Candy, and D. L. Lager, “On-line failure detection of vibrating structures,” in Proc. ASME Conf. on Mechanical Vibrations and Noise, Hartford, CT (1980), pp. 37–48. 13 D. L. Lager, S. G. Azevedo, and J. V. Candy, “The application of largescale state estimation to vibrating structures,” in Proc. 6th IFAC Symp. Identification System Param. Est., Washington, DC (1983), pp. 163–172. 14 J. V. Candy and H. E. Jones, “Processing of prosthetic heart valve sounds for single leg separation classification,” J. Acoust. Soc. Am. 97, 3663–3673 (1995). 15 J. V. Candy and H. E. Jones, “Classification of prosthetic heart valve sounds: A parametric approach,” J. Acoust. Soc. Am. 97, 3675–3687 (1995). 16 J. V. Candy and R. B. Rozsa, “Safeguards design for a plutonium concentrator—An applied estimation approach,” Automatica 16, 615–627 (1980). 17 J. V. Candy and E. J. Sullivan, “Ocean acoustic signal processing: A model-based approach,” J. Acoust. Soc. Am. 92(6), 3185–3201 (1992). 18 J. R. Roman and D. W. Davis, “State-space models for multichannel detection,” Final Report for Contract No. F30602-92-C-0081, RL-TR-93146, Rome Laboratory, Griffiss AFB, NY, (1993), 129 pp. 19 R. Luppold, J. R. Roman, and S. Roth, “Sensor diagnostics using the detection filter algorithm,” in Proc. 6th Digital Avionics Systems Conf., Baltimore MD (1984), pp. 213–219. Candy et al.: Model-based failure detection
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A. Jazwinski, Stochastic Processes and Filtering Theory (Academic, New York, 1970), 376 pp. 21 L. Ljung, System Identification: Theory for the User (Prentice-Hall, Englewood Cliffs, NJ, 1987), 519 pp. 22 P. Maybeck, Stochastic Models, Estimation, and Control (Academic, New York, 1979), 423 pp. 23 H. Van Trees, Detection, Estimation and Modulation Theory, Pt.1, (Wiley, New York, 1968), 694 pp. 24 D. G. Childers, D. P. Skinner, and R. C. Kemerait, “The cepstrum: a guide to processing,” Proc. IEEE, 65(10), 1428–1443 (1981). 25 R. O. Duda and P. E. Hart, Pattern Classification and Scene Analysis, (Wiley, New York, 2001) 657 pp. 26 D. F. Specht, “Probabilistic neural networks,” Neural Networks 3, 109–118 (1990).
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J. Beck and S-K. Au, “Bayesian updating of structural model and reliability using Markov chain Monte Carlo simulation,” J. Eng. Mech. 128(4), 380–391 (2002). 28 M. Wippich and K. Dessau, “Tunable lasers and fiber-Bragg-grating sensors,” Web, AIP, The Industrial Physicist, http://www.aip.org/tip/ INPHFA/vol-9/iss-3/p24.html (2003). 29 D. Betz, B. Thursby, and W. Staszewski, “Acousto-ultrasonic sensing using fiber Bragg gratings,” Smart Mater. Struct. 12, 122–128 (2003). 30 P. Formitchov and S. Krishnaswamy, “Response of a fiber Bragg grating sensor,” Opt. Eng. 42(4), 956–963 (2003). 31 E. Udd, W. Shultz, J. Seim, M. Morrell, T. Weaver, J. Bush, and G. Adamovsky, “Fiber optic distributed sensing systems for harsh aerospace environments,” SPIE 3674, 136–144 (1999).
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(Received 29 July 2014; revised 23 September 2014; accepted 6 October 2014) Model-based processing is a theoretically sound methodology to address difficult objectives in complex physical problems involving multi-channel sensor measurement systems. It involves the incorporation of analytical models of both physical phenomenology (complex vibrating structures, noisy operating environment, etc.) and the measurement processes (sensor networks and including noise) into the processor to extract the desired information. In this paper, a model-based methodology is developed to accomplish the task of online failure monitoring of a vibrating cylindrical shell externally excited by controlled excitations. A model-based processor is formulated to monitor system performance and detect potential failure conditions. The objective of this paper is to develop a real-time, model-based monitoring scheme for online diagnostics in a representative structural C 2014 Acoustical Society of America. vibrational system based on controlled experimental data. V [http://dx.doi.org/10.1121/1.4898421] PACS number(s): 43.60.Uv, 43.60.Gk, 43.60.Cg, 43.30.Zk [ZHM]
I. INTRODUCTION
Structural vibration systems1 operating over long periods of time can be subjected to component and overall system failures due to fatigue caused by external excitations or by cracking and resonances during the aging process.1–6 In some cases the system can fail catastrophically, causing complete deterioration of the overall structure and the consequences that follow. It is essential to provide timely information about component failure and the conditions leading up to it as well as predicting the integrity of the total system in the future. A model-based approach7–11 to develop diagnostic techniques for structural failure prediction is based on a wellestablished theoretical formalism that ensures a careful and systematic analysis of the vibrational system or classes of systems under investigation and their associated failure mechanisms. Models can be developed from a firstprinciples mathematical representation, or by utilizing sensor measurements to “fit” the model parameters to the data—the approach we pursue in this paper. In cases where the failure mechanism is not well understood, the model fitting approach offers an effective solution. In fact, in contrast to many approaches of solving this problem, this model-based approach does not require an explicit representation of the failure mechanism at all to design an online monitor for initial failure detection. It is based on obtaining a representation of the vibrating structure during normal certification operations, or even more desirable, during quality assurance or acceptance testing. Once these normal operational characteristics of the system are known, the processor is developed based on the properties of the so-called residuals or innovations sequence. This sequence is the difference a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
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between the measured and predicted sensor outputs obtained from the underlying normal structural system model. Once this condition or “difference” monitor detects a failure, it is then classified according to a set of well-established failure mechanisms. Failure detection for mechanical systems has long been the subject of much research with most of the work being concentrated on single-channel, spectral or cepstral analysis techniques for diagnostic purposes.2 There have been several successful applications of the model-based approach8–17 to these type problems, but none devoted to solving the complete failure condition detection, failure isolation, and timeto-failure prediction problem for a real-time maintenance problem. The inclusion of a process model in any signal processing scheme provides a means of introducing information in a self-consistent manner.18–21 This is shown in Fig. 1 where we observe how vibrational measurements are input to a model-based processor (MBP) utilized to provide information to the failure monitor. As mentioned, structural vibration systems operating over long periods of time can be subjected to failures due to fatigue caused by various external forces or simply the aging process. When this happens it is possible for the structure to deteriorate causing a complete malfunction and possibly catastrophic failure. Thus, to attain reliable system performance it is necessary to monitor critical structural components, to provide timely information about current and/or imminent failures, and to predict its integrity. Standard approaches to detect failure mechanisms at the onset range from a simple accelerometer strategically placed, to observing the Fourier spectrum of known response, to using cepstral analysis to identify periodic responses, to sophisticated Bayesian processing schemes.22–27 Measures of failure can deteriorate significantly if noise is present—a common situation in an operational environment. A model-based signal processing approach8 to solve the structural system failure detection
0001-4966/2014/136(6)/3114/12/$30.00
C 2014 Acoustical Society of America V
FIG. 1. (Color online) Conceptual model-based failure monitor.
problem is depicted in Fig. 2. Here we see that the first step is monitoring the current condition of the underlying structure. Should a failure be detected, a classifier processes the data to report the final condition and type of failure incurred (not addressed here). Most of the current monitoring approaches for failure detection and isolation lead to single-channel processing of measured sensor data.1 For instance, the so-called waterfall vibrational spectral plot, or the equivalent Campbell diagram,3 is limited to a stacking display of single-channel spectral information to identify failures. Multiple sensors (such as accelerometers for vibrations, microphones for acoustics, and thermocouples for temperature) in a structure provide enhanced information about the system. This implies a multi-channel (multi-input, multi-output) system representation, which is most easily handled in state-space form. Note that the state-space representation of a system is the transformation of an nth-order set of differential equations describing the system to a set of n first-order differential equations, without restrictions to single-channel spectral representations. Once this representation is accomplished, then the process model of the structural system can be incorporated into an effective signal processing scheme. This representation leads to the model-based approach, which can be stated succinctly as “incorporating mathematical models of both physical phenomenology and the measurement processes including noise into the processor to extract the desired information.” The incorporation of a mathematical model that represents the phenomenology under investigation can vastly improve the performance of any processor, provided such model is accurate.8,17 We summarize the development and application of such online “model-based processors” (MBP) in the context of failure
detection. One of the major advantages of a model-based processor is that it is capable of monitoring its own performance by testing the residual error (or innovations) between the measurement and its prediction. The model-based approach for failure detection is based on the concept that the process or vibrational system under consideration is modeled either from first principles or using system identification/parameter estimation techniques15,16,19 to “fit” models to the data. Once the system models are developed, then the sensor suite (or measurement system) models are developed. Usually the bandwidth of each sensor is much wider than the dominant dynamics or resonances of the system, and therefore each sensor is represented by a gain. However, if sensor dynamics must be included in the system model, the model-based approach easily accommodates them. Sensor dynamics models can be obtained from manufacturer specifications, independent experiments, or transfer function estimation to name a few (see Sec. II for some details). After these models are completed, then it remains to model the noise. If noise data are unavailable, then a reasonable approach is to model the noise as additive and random, leading to a Gauss–Markov model.19 Once a representation of the overall system (structure, sensors, and noise) is developed, then a failure monitor/detector can be developed to monitor the status of the vibrational system. Should a failure (that is, a deviation from the normal operation) be detected, then this failure must be classified using information from known or measured failures. Then it is possible to predict the time-to-failure for the overall system, and decide on the appropriate action, including simply reporting the results. Here we concentrate on the model development and online monitoring function. In Sec. II, we discuss the basic models of the cylindrical shell system under investigation along with both distributed and lumped governing dynamic systems as well as the embedded novel optical strain measurement sensor employed. Next we present the fundamental model-based approach in Sec. III including the underlying failure detection problem. The model-based processor is developed in Sec. IV and applied to simulated and experimental data in Sec. V. We summarize our results in the final section. II. DYNAMIC MODELING OF A VIBRATING CYLINDRICAL SHELL AND STRAIN SENSOR NETWORK
In this section we discuss the basic dynamic models used to represent the vibrating cylindrical shell under study for failure detection along with the novel optical strain sensor network. We start with the fundamental wave dynamics and corresponding numerical model using a finite element approach. Next we discuss a coupled lumped multi-variable dynamic model and corresponding transfer function matrices that provide the structure and justification of the MBP employed in the failure detector. The fundamental governing equation for the vibrating elastic cylindrical shell is5 FIG. 2. (Color online) Model-based failure monitor and classifier structure. J. Acoust. Soc. Am., Vol. 136, No. 6, December 2014
q
@2u lr2 u ðk þ lÞrðr uÞ ¼ qf; @t2 Candy et al.: Model-based failure detection
(1) 3115
where u(x, t) is the vector displacement field, f(x, t) is the forcing function, q is the material density, and k and l are the material elastic moduli. The displacement field is a superposition of longitudinal and shear displacement fields. Mathematically these can be separated, however, our measurement system is sensitive only to the total surface displacement and the individual contributions of the longitudinal and shear displacement fields are not necessary for failure detection. This equation, along with the boundary conditions, describes the dynamic response of the cylindrical shell to a given set of applied forces. We can convert this to a lumped model using a set of orthonormal vector basis functions un(x) such that ð um ðxÞ un ðxÞ; dx ¼ dmn : The displacement field and forcing excitation can be expanded in terms of these basis functions as uðx; tÞ ¼
Nd X
dn ðtÞun ðxÞ;
fðx; tÞ ¼
n¼1
Nd X
pn ðtÞun ðxÞ;
n¼1
which leads to the following matrix equation for the coefficients dn(t) (Ref. 6): € þ CdðtÞ _ þ KdðtÞ ¼ pðtÞ; MdðtÞ
(2)
where d is the Nd 1 displacement vector, p is the Np 1 excitation force, and M, C, K, are the Nd Nd lumped mass, damping function, and spring constant matrices characterizing the structural process model, respectively, 2 3 M1 0 0 0 0 6 0 M2 0 0 0 7 6 7 6 .. .. . . .. .. 7 M¼6 . ; C ¼ ½cij . . . . 7 6 7 4 0 5 0 0 0 MNd 1 0 0 0 0 MNd and 2 6 6 6 6 K ¼6 6 6 4
ðK1 þK2 Þ K2 0 0 0
3 0 0 0 7 .. 7 ðK2 þK3 Þ . 0 0 7 7 .. .. 7: . . KNd 1 0 7 7 0 KNd 1 ðKNd 1 þKNd Þ KNd 5 0 0 KNd KNd K2
Though the damping function is not explicitly part of the original governing equation, it is standard practice to include damping in the lumped model. This model is in the form of a linear dynamical system, which can be converted to a state-space representation for signal processing purposes. That is, if we define the 2Nd-state vector as _ 0 , then the state-space representation of this xðtÞ :¼ ½dðtÞjdðtÞ process can be expressed as 3116
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x_ ðtÞ ¼ AxðtÞ þ BuðtÞ; " # 0 I 0 x_ ðtÞ ¼ pðtÞ; x ðt Þ þ M1 K M1 C M1 and yðtÞ ¼ M1 j0 xðtÞ;
(3)
where we have assumed that a sensor directly proportional to displacement is used, y 2 RNd 1 . For our measurement system we actually use a Bragg fiber optic strain sensor that is small ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ a R ðiÞ=N R ðiÞ=N for i ¼ 1; …; Ny :
(8)
Therefore, the zero-mean test,8 on each component innovation i is given by Candy et al.: Model-based failure detection
^ ðiÞ l
>Reject H0 s for i ¼ 1; ;Nd : 30: Iq ¼ q N
(14)
^ ði; kÞ values Hence, under the null hypothesis, 95% of the q must lie within this confidence interval to accept H0 . That is, for each component innovation sequence to be considered statistically white. The whiteness test of Eq. (14) is very useful to detect modeling inaccuracies from individual component innovations. However, for complex systems with a large number of measurement channels, it becomes computationally burdensome to investigate each innovation componentwise especially under the limiting ergodicity assumptions. C. Vector whiteness detection
A statistic containing all of the innovation information is the weighted sum-squared residual (WSSR). It aggregates all J. Acoust. Soc. Am., Vol. 136, No. 6, December 2014
N 1 0 ðkÞR1 : ðkÞðtk Þ for ‘ M; 2 R
k¼‘Mþ1
q ð‘Þ
^ ði; kÞ ¼ q
‘ X
q ð‘Þ :¼
>H1 s: 30, q(‘) is approximately Gaussian N (NM, 2NM) (see Refs. 8–11 for more details). At the a-significance level, the probability of rejecting the null hypothesis is given by ! jqð‘Þj N M s N M (17) Pr pffiffiffiffiffiffiffiffiffiffiffiffi > pffiffiffiffiffiffiffiffiffiffiffiffi ¼ a: 2N M 2N M For a level of significance of a ¼ 0.05, we have pffiffiffiffiffiffiffiffiffiffiffiffi s ¼ N M þ 1:96 2N M:
(18)
Thus, we see that the WSSR can be considered a “whiteness test” of the innovations vector over a finite window of length N. Note that since [{(tk)}, {R(tk)}] are obtained from the state-space MBP algorithm directly, they can be used for both stationary as well as nonstationary processes. In fact, in practice for a large number of measurement components, the WSSR is used to “tune” the filter and then the component innovations are individually extracted and checked for zeromean/whiteness to detect failures. Note also that the adjustable parameter of the WSSR statistic is the window length M, which essentially controls the width of the window sliding through the innovations sequence. Therefore, even if we cannot implement the optimal detector, we can still perform a set of pragmatic statistical hypothesis tests to investigate the condition of the underlying structure and classify its failure.
IV. FAILURE DETECTION
In this section we extend the basic principles of modelbased detection and apply them to our particular problem of cylindrical shell failure detection. P First we start with the development of a “normal” model ð N Þ providing a standard for monitoring and comparison. Here we assume that we have measured the vibrational response of the structural system (cylindrical shell) and developed a model to capture its performance, that is, the normal (state-space) system, Candy et al.: Model-based failure detection
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P
N :¼ fAN ; BN ; CN g is characterized by the structural matrices of Eq. (3). We measure an ensemble of vibrational responses from the system using a controlled excitation over the frequency range of interest and then use them to obtain a representative model by averaging over the estimated parameters for each realization, y‘(tk); ‘ ¼ 1,…, L. That is, we “fit” a model to the ‘th realization X ðH‘ Þ :¼ fAN ðH‘ Þ; BN ðH‘ Þ; CN ðH‘ Þg (19) N
measurement is given by y(tk; r) where r is the sensor position vector on the structure. We will (tacitly) assume that the positions are weakly coupled at first and ignore the crosscoupling terms simplifying the models and measurements. Therefore, we separate the analysis of each subsystem defined by: A(H) ¼ diag [A1(H) A2(H)]. The subsystem model parameters will be “fit” to each subsystem individually and the measurements and detections processed accordingly. With this in mind we proceed to process the measured vibrational response data from the cylinder.
and then average over the entire ensemble X N
A. Cylindrical shell vibrational response (normal) model
X X L X ðH‘ Þ :¼ EH ðH‘ Þ ¼ ðH‘ Þ N N
‘¼1
¼ fAN ðHÞ; BN ðHÞ; CN ðHÞg:
(20)
A model order (dimension of the AN ) is selected to guarantee a zero-mean/white innovations sequence using the ensemble as ðtk ; H‘ Þ ¼ y‘ ðtk Þ y^ðtk ; H‘ Þ for ‘ ¼ 1; …; L:
(21)
The model extraction (parameter estimation) and validation method is shown in Fig. 4. Here we see that an initial normal model is estimated from part of the ensemble data (training data) to generate a set of parameters producing an average model (above). Next the parameterized model is applied to test data where the innovations sequence is generated and tested for acceptability. If acceptable, the parameter ðHN Þ is adjusted (averaged) and the model is updated yielding an acceptable system model. In this way the normal model is trained, tested and validated over an ensemble of both training and test data to ensure its validity. Once we have performed the parameter estimation,8 we construct the state-space model from its parameters. Since we use a prediction error parameter estimation technique, the resulting state-space model is the discrete (sampled) innovations model parameterized by the following x ðtk jtk1 Þ þ Kðtk ;HN Þðtk ;HN Þ; x^ðtk jtk Þ ¼ AðHN Þ^ ðtk ;HN Þ ¼ yðtk Þ y^ðtk ; HN Þ; x ðtk jtk1 Þ: y^ðtk Þ ¼ CðHN Þ^
(22)
In our case, since we have two sensors available and we choose two positions to locate them, then our vector
FIG. 4. Normal vibrational model extraction and validation: Ensemble parameter estimation/prediction, prediction error whiteness, model parameter updating. 3120
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The normal model is developed using the parameter estimation techniques resulting in the innovations model of Eq. (22). A typical set of cylindrical shell response data is shown in Fig. 5 where we see the noisy time series measurements and corresponding power spectra. The cylindrical shell excitation was a chirp (swept sinusoid) that was varied from 100 Hz to 5 kHz. This range was selected from numerical model studies5,6 of the cylinder. Next the models were fit as discussed above with the validation over a test ensemble (30 members) of vibrational response measurements. A typical fit and validation are shown in Fig. 6 where we observe the innovations sequence in Fig. 6(a) with the associated zero-mean/whiteness and WSSR statistic in Figs. 6(b) and 6(c). In Fig. 6(b), the zero-mean test is passed as well as the component “whiteness” test (95% of the correlation samples must lay within the bounds (dashed lines). In Fig. 6(c) the WSSR statistic must lay below the threshold (dashed lines) and it does. Thus, this realization passes the “whiteness” criteria tests and is acceptable for model updating (averaging) parameters. With the average normal (pristine) model validated in terms of the ensemble zero-mean/whiteness testing, a set of data were generated to test the performance of the estimated prediction model. The results for these tests are shown in Fig. 7. In Fig. 7(a), we see the results for both sensors. Here we observe that from an ensemble of 30 (2 sensors) that 100% of the innovations (correlation) lie within 10% out while 75% were “statistically white” lying within the 5% out bounds for sensor 1 (circles). The sensor 2 (solid circles) results were above the 5% bound with 93% of the ensemble below the 10% bound. These results indicate that the estimated average prediction model (Fig. 5) is adequate for failure detection processing. Next we create failure conditions by attaching masses at the end of the cylinder: one of 300 g and one of 150 g The resulting spectra for both sensors are shown in Fig. 8 where we also indicate the number of modes or resonant peaks residing in the 1000–2000 Hz band. We note that for sensor 1 the number of modes or resonant peaks under normal conditions are 5, while for the 150 and 300 g masses the number of modes increase to (5 ! 10,5 ! 12). For sensor 2 normal conditions of 8, the number of modes increases (5 ! 12,5 ! 11). In conclusion, we see that for both masses the changes from normal (increased modes) enable a detection indicating the robustness of the process. Candy et al.: Model-based failure detection
FIG. 5. (Color online) Cylindrical shell vibrational response data: Time series, average power spectra and ensembles for each sensor with the number of modes (resonant peaks) (5,8) for sensors 1 (5) and 2 (8).
V. RESULTS
As mentioned, failure conditions were created by attaching the specified mass for each experiment and generating a corresponding ensemble (30 members) of resonance data for performance testing. The failure detections using the average
normal (pristine) model was applied after validation tests (above) were performed indicating the robustness of the embedded processor model. The results are shown in Fig. 9 for both sensors and both cases (150 g/300 g). For the 300 g mass induced failure conditions all of the members of the test ensemble were detected successfully
FIG. 6. (Color online) Ensemble zeromean/whiteness and validation testing for normal cylindrical shell vibrational models. (a) Innovations sequence. (b) Zero-mean/whiteness test. (c) WSSR test.
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FIG. 7. (Color online) Self-validation ensemble whiteness test results for vibrational models: (a) Normal (pristine) model self-validation (sensor No. 2 solid circles). (b) Failure model for 300 g mass self-validation (sensor No. 2 solid circles). (c) Failure model for 150 g mass self-validation (sensor No. 2 solid circles).
with over >60% bound (non-white), that is, recall to be declared normal each test should lie within the 5% whiteness bound—these failures exceed >60% for sensor 1 and >55% for sensor 2. The results for the 150 g mass induced failure conditions are slightly better with failures exceeding >60% in each run for sensor 1 and >58% for sensor 2. From the figure we see that the failure detector performs quite well.
As a final part of the analysis, we estimate the residual sequences comparing the ensembles of normal spectra to that of the failure (induced) spectra and the results are shown in Fig. 10. The normal spectra are flat with no resonant peaks illustrating “white” spectra. Clearly there are spectral differences induced by the attached masses as illustrated in the figure. The differences are quite
FIG. 8. (Color online) Change detection: Sensor average and ensemble spectra for failure conditions: (a) 150 g mass spectra from sensor 1 (5 ! 10 modes or resonant peaks). (b) 150 g mass spectra from sensor 2 (8 ! 12 modes or resonant peaks). (a) 300 g mass spectra from sensor 1 (5 ! 12 modes or resonant peaks). (a) 300 g mass spectra from sensor 2 (8 ! 11 modes or resonant peaks). 3122
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VI. CONCLUSIONS
FIG. 9. (Color online) Change detection: Failure detector performance on vibrational test data (sensor No. 2 solid circles): (a) 300 g mass induced failure condition for sensors 1 and 2 (>60% out, >55% out). (b) 150 g mass induced failure condition for sensors 1 and 2 (>60% out, >58% out).
interesting for this particular location of the mass, sensor 1 shows a large spectral difference from the normal compared to that of sensor 2 indicating that is more sensitive to this vibrational response. This completes the discussion of the results, next we conclude and discuss future effort.
We have demonstrated the effectiveness of a modelbased approach to failure detection by performing parameter estimation and model-based detection on sets of controlled vibrational response experimental data. By investigating the response of data and fitting parametric models we were able to use whiteness testing of the innovations (residual) sequence to both validate and detect two sets of failure conditions induced by attaching masses to a cylindrical test object. The processor detects a departure from the normal structural condition. The results indicate that this is a viable approach. Future efforts will be aimed at location the failure itself by utilizing the residuals that can be extracted from the normal spectra when compared to the abnormal. Here the residual sequence is obtained using the model-based approach.8 The required spectral are estimated using the average normal model and a failure mode model to create the residual spectra. These spectra could then be backpropagated using a propagation model and like time-reversal “focuses” back to the failure position performing the desired localization. Preliminary results appear to be quite promising.
ACKNOWLEDGMENTS
This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. APPENDIX
To formally pose the sequential failure detection problem, we appeal to classical detection theory.20 We are to test the binary hypothesis that the innovations sequence is zeromean and white H0 : ðtk Þ N ½0; R ðtk Þ H1 : ðtk Þ N ½ l ; R ðtk Þ
½WHITE; ½NON WHITE;
which is a statistical test for the zero-mean and whiteness of the innovations sequence. Note that we assume that we know ðtk Þ and R ðtk Þ a the model error and how to calculate l priori. The optimal solution to this binary detection problem is based on applying the Neyman–Pearson theorem leading to the likelihood ratio23 and is given by the ratio of probabilities for the sequential innovations detector (SID), that is
H1 Pr E tN j H1 > s; L½E tN :¼
Pr E tN j H0 < H0
FIG. 10. (Color online) Estimated spectral residuals with normal (flat) spectrum: (a) Mass induced failure condition for sensor 1. (b) Mass induced failure condition for sensor 2. J. Acoust. Soc. Am., Vol. 136, No. 6, December 2014
where E tN :¼ fðt0 Þ; ðt1 Þ; …; ðtN Þg or expanding Pr ðt0 Þ; ðt1 Þ; …; ðtN ÞjH1 L½E tN ¼ Pr ðt0 Þ; ðt1 Þ; …; ðtN ÞjH0 but from the chain rule of probability,8 we have that Candy et al.: Model-based failure detection
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PrðE tk j Hi Þ ¼ Pr½ðtk ÞjE tk1 ; Hi
which yields the new test
Pr½ðt1 Þjðt0 Þ; Hi Pr½ðt0 Þj Hi ;
K ðtk Þ !1
which can be expressed succinctly using Bayes’ rule as
!0 K ðtk Þ !1
PrðE tk j Hi Þ ¼ Pr½ðtk Þ; E tk1 jHi ¼ Pr½ðtk ÞjE tk1 ; Hi PrðE tk1 j Hi Þ:
K ðtk Þ !0
Taking logarithms we obtain the relation for the sequential log-likelihood k ðtk Þ :¼ lnL½E tk lnPr½ðtk ÞjE tk1 H0 : The corresponding Wald or sequential probability ratio test8 is then given by Accept H1 ;
s0 k ðtk Þ s1 k ðtk Þ s0
Continue;
Accept H0
and the underlying conditional Gaussian distributions are given by N=2 jR ðtk Þj1=2 Pr ðtk ÞjE tk1 ; H0 ¼ ð2pÞ 1 exp 0 ðtk ÞR1 t t ð Þ ð Þ k k 2 and N=2 jR ðtk Þj1=2 Pr ðtk ÞjE tk1 ; H1 ¼ ð2pÞ 1 ðt k Þ 0 exp ½ðtk Þ l 2 1 t l R t t ð k Þ : ð k Þ½ ð k Þ If we include the determinants in the thresholds, we have the modified decision function 1 K ðtk Þ ¼ K ðtk1 Þ þ 0 ðtk ÞR1 ðtk Þðtk Þ 2 1 1 ðtk Þ½ðtk Þ l ðt k Þ 0 R ðt k Þ ; ½ðtk Þ l 2 3124
Accept H0 ;
1 1 1=2 !1 ¼ s1 þ lnjR ðtk Þj1=2 jR ; ðtk Þj 2 2 1 1 1=2 : !0 ¼ s0 þ lnjR ðtk Þj1=2 jR ðtk Þj 2 2 The implementation of this “optimal” failure detector/ monitor presents a basic problem, such as how to obtain estimates of the innovations (abnormal) mean and covariance required, but does illustrate a potential optimal solution to the model monitoring problem. As mentioned, the SID requires a priori knowledge of the actual model “mismatch” and structurally how it enters the process model to obtain ðtk Þ for the monitor.16–18 ½ l ðtk Þ; R 1
¼ k ðtk1 Þ þ lnPr½ðtk ÞjE tk1 ; H1
k ðtk Þ s1
Continue;
where
Substituting these expressions (replacing tN by tk) into the equation and grouping we obtain "
# Pr ðtk ÞjE tk1 ; H1 Pr E tk1 j H1
; L½E tk ¼ Pr ðtk ÞjE tk1 ; H0 Pr E tk1 j H0 and therefore we have the recursion or equivalently sequential likelihood as Pr ðtk ÞjE tk1 ; H1 : L½E tk :¼ L½E tk1 Pr ðtk ÞjE tk1 ; H0
Accept H1 ;
J. Acoust. Soc. Am., Vol. 136, No. 6, December 2014
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