Identification of correlated damage parameters under noise and bias using Bayesian inference

Transcription

1 Article Identification of correlated daage paraeters under noise and bias using Bayesian inference Structural Health Monitoring 11(3) Ó The Author(s) 11 Reprints and perissions: sagepub.co.uk/journalsperissions.nav DOI: / sh.sagepub.co Dawn An 1, Joo-Ho Choi 1 and Na H. Ki Abstract This article presents statistical odel paraeter identification using Bayesian inference when paraeters are correlated and observed data have noise and bias. The ethod is explained using the Paris odel that describes crack growth in a plate under ode I loading. It is assued that the observed data are obtained through structural health onitoring systes, which ay have rando noise and deterinistic bias. It was found that a strong correlation exists (a) between two paraeters of the Paris odel, and (b) between initially easured crack size and bias. As the level of noise increases, the Bayesian inference was not able to identify the correlated paraeters. However, the reaining useful life was predicted accurately because the identification errors in correlated paraeters were copensated by each other. It was also found that the Bayesian identification process converges slowly when the level of noise is high. Keywords paraeters identification, daage growth paraeters, correlated paraeters, Bayesian inference Introduction Structural health onitoring (SHM) facilitates condition-based aintenance that provides a cost effective aintenance strategy by providing an accurate quantification of degradation and daage at an early stage without intrusive and tie-consuing inspections. 1 Most SHM systes utilize on-board sensors/actuators to detect daage, find the location of daage, and estiate the significance of daage. Since SHM systes can assess daage frequently, they can also be used to predict the future behavior of the syste, which is critically iportant for aintenance scheduling and fleet anageent. SHM systes can have a significant ipact on increasing safety by allowing predictions of the structure s health status and reaining useful life (RUL), which is called prognostics. In general, prognostics ethods can be categorized into data-driven, odel-based, 3 and hybrid 4 approaches, based on the usage of inforation. The data-driven ethod uses inforation fro collected data to predict the future status of the syste without using any particular physical odel. It includes leastsquare regression, Gaussian process regression, etc. The odel-based ethod assues that a physics odel describing the behavior of the syste is available. This ethod cobines the physics odel with easured data to identify odel paraeters and predict future behavior. Modeling the physical behavior can be accoplished at different levels, for exaple, icro- and acro-levels. Crack growth odel 5 or fatigue life odel 6 are often used for acro-level daage, and first principle odels 7 are used for icro-level daage. The hybrid ethod cobines the above-entioned two ethods, and includes particle filters 8 1 and Bayesian techniques Since the data-driven ethod identifies abnorality based on the trend of data, it is powerful in predicting near-future behaviors, while the odel-based ethod has advantages in predicting longter behaviors of the syste. It is noted that in the 1 Departent of Aerospace & Mechanical Engineering, Korea Aerospace University, 1 Hanggongdae-gil, Hwajeon-dong, Deokyang-gu, Goyang- City, Gyeonggi-do, , Korea. Departent of Mechanical & Aerospace Engineering, University of Florida, Gainesville, FL, 3611, USA. Corresponding author: Na H. Ki, Departent of Mechanical & Aerospace Engineering, University of Florida, Gainesville, FL, 3611, USA Eail: nki@ufl.edu

2 94 Structural Health Monitoring 11(3) odel-based ethod for fatigue applications, the history of load is required in addition to the easured crack data. In this article, a physics-based odel for structural degradation due to fatigue daage is applied for prognostics, since daage grows slowly and the physics governing its behavior is relatively well-known. The ain purpose of prognostics is to identify and repair those daages that threaten the syste safety (conditionbased aintenance) and to predict an appropriate aintenance schedule. Paris-faily odels are coonly used in describing the growth of cracks in aircraft panels under fatigue loading. 15 In general, these odels have odel uncertainty as they are not exact in describing crack growth. However, the objective of this article is to present how odel paraeters can be identified for a given odel. Therefore, odel uncertainty is ignored. In this article, the original Paris odel 5 is used because it has the least nuber of paraeters. It is assued that the Paris odel exactly describes the daage growth behavior if the odel paraeters are also exact. The ain purpose of the article is to present the usage of Bayesian inference in identifying odel paraeters and predicting the RUL the reaining cycles before aintenance. The study focuses on crack growth in a fuselage panel under repeated pressurization loading, which can be considered regular loading cycles. In this type of application, the uncertainty in applied loading is sall copared to other uncertainties. Therefore, the crack growth behavior and the RUL can be predicted based on the identified odel paraeters before the crack becoes dangerous. The iproved accuracy in these odel paraeters allows ore accurate prediction of the RUL of the onitored structural coponent. Identifying the odel paraeters and predicting daage growth, however, is not a siple task due to the noise and bias of data fro SHM systes and the correlation between paraeters, which is prevalent in practical probles. The noise coes fro variability of rando environents, while the bias coes fro systeatic departures of easureent data, such as calibration error. However, research for identifying odel paraeters under noise and bias, without entioning correlated paraeters, is liited. 9,16 The ain objective of this article is to deonstrate how Bayesian inference can be used to identify odel paraeters and to predict RUL, especially when the odel paraeters are correlated. In order to find the effects of noise and bias on the identified paraeters, nuerical studies utilize synthetic data; that is, the easureent data are produced fro the assued odel of noise and bias. The key interest is how the Bayesian inference identifies the correlated paraeters under noise and bias in data. The article is organized as follows: in the second section, a siple daage growth based on Paris odel is presented in addition to the uncertainty odel of noise and bias; in the third section, paraeter identification and RUL prediction using Bayesian inference and MCMC siulation ethod 17 are presented with different levels of noise and bias; and conclusions are presented in the final section. Daage growth and easureent uncertainty odels Daage growth odel In this article, a siple daage growth odel is used to deonstrate how to characterize daage growth paraeters. Although soe experiental data on fatigue daage growth are available in the literature, 18 they are not easured using SHM systes. Therefore, the level of noise and bias is uch saller than the actual data that will be available in SHM systes. In this article, synthetic daage growth data are used in order to perfor statistical study on the effect of various levels of noise and bias. It is assued that a through-the-thickness center crack exists in an infinite plate under the ode I loading condition. In aircraft structure, this corresponds to a fuselage panel under repeated pressurization loading (Figure 1), which is the ain cause of fatigue in fuselage panels. Therefore, one flight corresponds to one cycle. In this approxiation, the effect of finite plate size and the curvature of the plate are ignored. When the stress range due to the pressure differential is, the rate of daage growth can be written using the Paris odel 5 as da dn ¼ CðK p Þ, K ¼ ffiffiffiffiffi a ð1þ where a is the half crack size, N is the nuber of cycles (one cycle corresponds to one flight), K is the range of stress intensity factor, and other paraeters are shown in Table 1 for 775-T651 aluinu alloy. Although the nuber of cycles, N, is an integer, it is treated as a real nuber in this odel. The above odel has two daage growth paraeters, C and, which are estiated to predict daage propagation and RUL. In Table 1, these two paraeters are assued to be uniforly distributed, whose lower and upper bounds were obtained fro the scatter of experiental data. 19 They can be considered as the daage growth paraeters of generic Al 775-T651 aterial. In general, it is wellknown that the two Paris paraeters are strongly correlated, but it is assued initially that they are

3 An et al. 95 s Bayesian inference Paris odel Daage growth paraeter RUL Sensor Daage size s Figure 1. Through-the-thickness crack in a fuselage panel. Table 1. Loading and fracture paraeters of 775-T651 Aluinu alloy Property Noinal stress s (MPa) Fracture toughness p K IC (MPa ffiffiffiffi Daage ) paraeter Daage paraeter log(c) Value/Distribution case 1: 86.5 Deterinistic 3 Unifor case : 78.6 (3.3, 4.3) case 3: 7.8 Unifor (log(5e-11), log(5e-1)) uncorrelated because there is no prior knowledge on the level of correlation. Using easured data of crack sizes, the Bayesian inference will show the correlation structure between these two paraeters. Since the scatter is so wide, the prediction of RUL using the initial distributions of paraeters is eaningless. The specific panel being onitored using SHM systes ay have uch narrower distributions of the paraeters, or even deterinistic values. The half crack size a i, after N i cycles of fatigue loading, can be obtained by integrating Equation (1) and solving for a i as h a i ¼ N i C 1 p ffiffiffi i þa 1 ðþ where a is the initial half crack size. In SHM, the initial crack size does not have to be the icro-crack in the panel before applying any fatigue loading. This can be the crack size that is detected by SHM systes the first tie. In such a case, N i should be interpreted as the nuber of cycles since the crack is detected. It is assued that the panel fails when a i reaches a critical half crack size, a C. Here, we assue that this critical crack size is when the stress intensity factor exceeds the fracture toughness K IC. This leads to the following expression for the critical crack size: K IC a C ¼ p ffiffiffi ð3þ Even if the crack growth odel in Equation (1) is the siplest for, it requires identifying various paraeters. First, the daage growth paraeters, C and, need to be identified, which can be estiated fro specien-level fatigue tests. 18 However, due to aterial variability, these paraeters show different values for different batches of panels. In addition, the initial crack size, a, needs to be found. Liu and Mahadevan 1 used an equivalent initial flaw size, but in this case, it corresponds to error in the first data in SHM easureent. In addition, the fracture toughness, K IC, also shows randoness due to variability in anufacturing. Measureent uncertainty odel In SHM-based inspection, the sensors installed on the panel are used to detect the location and size of daage. Even if the on-line inspection can be perfored continuously, it would not be significantly different fro on-ground inspection because the structural daage will not grow quickly. In addition, the

4 96 Structural Health Monitoring 11(3) on-ground, copared to on-line, inspection will have uch saller levels of noise. The on-ground inspection ay provide a significant weight advantage because only sensors, not easureent equipent, are on-board. Our preliinary study showed that there is no need to inspect at every flight because daage growth at each flight is extreely sall. A crack in the fuselage panel grows according to the applied load, pressurizations in this case. Then, the SHM systes detect the crack. In general, the SHM syste cannot detect a crack when it is too sall. Many SHM systes can detect a crack between the size of 5 1. Therefore, the necessity of identifying the initial crack size becoes uniportant by setting a to be the initially detected crack size. However, a ay still include noise and bias fro the easureent. In addition, the fracture toughness, K IC, is also uniportant because airliners ay want to send the airplane for aintenance before the crack becoes critical. The ain objective of this article is to show that the easured data can be used to identify crack growth paraeters, and then, to predict the future behavior of the cracks. Since no airplanes are equipped with SHM systes yet, we siulate the easured crack sizes fro SHM. In general, the easured daage includes the effect of bias and noise. The forer is deterinistic and represents a calibration error, while the latter is rando and represents noise in the easureent environent. The synthetic easureent data are useful for paraeter study, that is, the different noise and bias levels show how the identification process is affected. In this context, bias is considered as two different levels,, and noise is uniforly distributed between u and þu. Four different levels of u are considered:,.1, 1, and 5. The varying levels of noise represent the quality of SHM systes. The synthetic easureent data are generated by (a) assuing that the true paraeters, true and C true, and the initial half crack size, a, are known; (b) calculating the true crack sizes according to Equation () for a given N i and ; and (c) adding a deterinistic bias and rando noise to the true crack size data including the initial crack size. Once the synthetic data are obtained, the true values of crack sizes as well as the true values of paraeters are not used in the prognostics process. In this article, the following true values of paraeters are used for all nuerical exaples: true ¼ 3:8, C true ¼ 1:5 1 1, and a ¼ 1. Table 1 shows three different levels of loading; the first two ( ¼ 86.5 and 78.6 MPa) are used for estiating odel paraeters, while the last ( ¼ 7.8) is used for validation purposes. The reason for using two sets of data to estiate daage growth paraeters is to utilize ore data having daage propagation inforation at an early stage. Theoretically, the true values of paraeters can be identified using a single set of data because the Paris odel is a nonlinear function of paraeters. However, rando noise can ake the identification process slow, especially when paraeters are correlated; that is, any different cobinations of correlated paraeters can achieve the sae crack size. This property delays the convergence of Bayesian process such that eaningful paraeters can only be obtained toward the end of RUL. Based on preliinary study, two sets of data at different loadings can help the Bayesian process converge quickly. This situation corresponds to two fuselage panel with different thickness. Figure shows the true crack growth curves for three different levels of loading (solid curves) and synthetic easureent data a eas i (triangles) with noise and bias. It is noted that the positive bias shifts the data above the true crack growth. On the other hand, the noises are randoly distributed between easureent cycles. It is assued that the easureents are perfored at every 1 cycles. Let there be n easureent data. Then, the easured crack sizes and corresponding cycles are represented by a eas ¼fa eas, a eas 1, a eas,..., a eas n g N ¼fN ¼, N 1 ¼ 1, N ¼,..., N n g ð4þ It is assued that after N n, the crack size becoes larger than the threshold and the crack is repaired. Bayesian inference for characterization of daage properties Daage growth paraeters estiation Once the synthetic data (daage sizes vs. cycles) are generated, they can be used to identify unknown daage growth paraeters. As entioned before,, C, and a can be considered as unknown daage growth paraeters. In addition, the bias and noise are also unknown because they are only assued to be known in generating crack size data. In the case of noise, the standard deviation,, of the noise and deterinistic bias, b, are considered as an unknown paraeters. The identification of will be iportant as the Bayesian process depends on it. Therefore, the objective is to identify (or, iprove) these five paraeters using the easured crack size data. The vector of unknown paraeters is defined by y ¼f, C, a, b, g. Paraeter identification can be done in various ways. The least-squares ethod is a traditional way of identifying deterinistic paraeters. For crack propagation, Coppe et al. used the least-square ethod to identify unknown daage growth paraeters along with bias. However, in the least-squares

5 An et al. 97 (a) 6 5 Synthetic data data set : Δσ=78.6 (b) 6 5 Synthetic data data set : Δσ= data set 1: Δσ= data set 1: Δσ= Δσ=7.8 1 Δσ= Bias = + and noise = Bias = + and noise = 5 Figure. Crack growth of three different loading conditions and two sets of synthetic data. ethod, it is non-trivial to estiate the uncertainty in the identified paraeters. In this paper, Bayesian inference is used to identify the unknown paraeters as well as the level of noise and bias. Coppe et al. 3 used Bayesian inference in identifying daage growth paraeters, C or. They used the grid ethod to calculate the posterior distribution of one variable and discussed that updating ulti-diensional variables can be coputationally expensive. The grid ethod coputes the values of PDF at a grid of points after identifying the effective range, and calculates the value of the posterior distribution at each grid point. This ethod, however, has several drawbacks such as the difficulty in finding correct location and scale of the grid points, spacing of the grid, and so on. In addition, it becoes coputationally expensive when the nuber of updating paraeters increases. Markov Chain Monte Carlo (MCMC) siulation is a coputationally efficient alternative to obtain the probability density function (PDF) by generating a chain of saples. 17 In Baye s theore, 4 the knowledge of a syste can be iproved with additional observation of the syste. More specifically, the joint PDF of y will be iproved using the easured crack sizes a eas. The joint posterior PDF is obtained by ultiplying the prior PDF with the likelihood as p Y ðyja eas Þ¼ 1 K p Aða eas jy ¼ yþ p Y ðyþ ð5þ where p Y ðyþ is the prior PDF of paraeters, p A ða eas jy ¼ yþ is the likelihood or the PDF values of crack size at a eas given paraeter value of y, and K is a noralizing constant. It is noted that the likelihood is constructed using n easured crack size data. For prior distribution, the unifor distribution is used for the daage growth paraeters, and C, as described in Table 1. For other paraeters, no prior distribution is used; that is, noninforative. Therefore, the prior PDF becoes p Y ðyþ ¼p Y ðþ p Y ðcþ. The likelihood is the probability of obtaining the observed crack sizes a eas given values of paraeters. For the likelihood, it is assued to be a noral distribution for the given five paraeters including the standard deviation of the easured size, :! n p A a eas 1 ð jy ¼ yþ / pffiffiffiffiffi y5 " exp 1 X n a eas # i a i ðy 1:4 Þ, y ¼f, C, a, b, g y i¼1 5 ð6þ where h a i ðy 1:4 Þ¼ N i C 1 p ffiffiffi i þa 1 þb ð7þ is the crack size fro the Paris odel with bias and a eas i is the easureent crack size at cycle N i. In general, it is possible that the noral distribution in Equation (6) ay have a negative crack size, which is physically ipossible; therefore, the noral distribution is truncated at zero. A priitive way of coputing the posterior PDF is to evaluate Equation (5) at a grid of points after identifying the effective range. This ethod, however, has several drawbacks such as the difficulty in finding correct location and scale of the grid points, the spacing of

6 98 Structural Health Monitoring 11(3) the grid, and so on. Especially when a ulti-variable joint PDF is required, which is the case in this article, the coputational cost is proportional to M 5, where M is the nuber of grids in one-diension. On the other hand, the MCMC siulation can be an effective solution as it is less sensitive to the nuber of variables. 17 Using the expression of posterior PDF in Equation (5), 5 saples of paraeters are drawn by using the Metropolis-Hastings (M-H) algorith, which is a typical ethod of MCMC. The effect of correlation between paraeters Since the original data of crack sizes are generated fro the assued true values of paraeters, the objective of Bayesian inference is to ake the posterior joint PDF converge to the true values. Therefore, it is expected that the PDF becoes narrower as n increases; that is, ore data are used. This process sees straightforward, but the preliinary study shows that the posterior joint PDF ay converge to values different fro the true ones. It is found that this phenoenon is related to the correlation between paraeters. For exaple, let the initially detected crack size be a eas when the easureent environent has no noise. This easured size is the outcoe of the initial crack size and bias: a eas ¼ a þ b ð8þ Therefore, there exist infinite possible cobinations of a and b to obtain the easured crack size. It is generally infeasible to identify the initial crack size and bias with a single easureent when the easured data is linearly dependent on ultiple paraeters. It was also well known that the two Paris odel paraeters, and C, are strongly correlated. 5 This can be viewed fro the crack growth rate curve, as illustrated in Figure 3. In this graph, the paraeter is the slope of the curve, while C corresponds to the y-intercept at K ¼ 1. If a specific value of crack growth rate da=dn is observed, this can be achieved by different cobinations of these two paraeters. However, in the case of Paris odel paraeters, it is feasible to identify the because the stress intensity factor gradually increases as the crack grows. However, the ebedded noise can ake it difficult to identify the two odel paraeters because the crack growth rate ay not be consistent with the noisy data. In addition, this can slow down the convergence in the posterior distribution, because when the crack is sall, there is no significant crack growth rate. The effect of noise is relatively diinished as the crack growth rate increases, which occurs toward the end of life. log (da/d/v) C C 1 Measured crack growth rate 1 In order to overcoe the above-entioned difficulty in identifying correlated paraeters, two different strategies are used in this article. There are any correlations between unknown paraeters, but only the strongest correlated relationships are considered: the two Paris odel paraeters and a and b. First, the two Paris odel paraeters are kept because they can be identified as the crack grows. Second, the relationship between a and b has different characteristics fro the odel paraeters (, C). a and b are tie independent, and the su of the two paraeters are constant. Therefore, the bias is reoved fro the Bayesian identification process using Equation (8), assuing that the bias and initial crack size are perfectly correlated. This process sees straightforward, but the difficulty exists fro the fact that the constant a eas in Equation (8) is unknown. Below is the procedure of estiating a eas. log (DK) Figure 3. Illustration of showing the sae crack growth rate with different cobinations of paraeters. a. Assue that the easured initial crack size is a ¼ a eas b. With given a, use the Bayesian ethod to update the posterior joint PDF of, C, b, c. Calculate the axiu likelihood value b of b fro the posterior joint PDF d. Estiate a eas ¼ a þ b e. Eliinate b using b ¼ a eas a and update the posterior joint PDF of, C, a, Figure 4 shows the posterior PDFs for the case of true bias of (a) when n ¼ 13 (N 1 ¼ 1 cycles) and (b) when n ¼ 17 (N 16 ¼ 16 cycles). The posterior joint PDFs are plotted separately into three groups for the plotting purpose. In this case, it is assued that there is no noise in the data. The true values of paraeters are arked using a star sybol. Siilar results were also obtained in the case with bias ¼.

7 An et al. 99 (a) 5 (b) 15 Joint PDF Joint PDF log(c) log(c) log(c) b () a () log(c) b () a () σ ().5.1 σ () n = 13 n = Figure 4. Posterior distributions of paraeters with zero noise and true bias of. Firstly, it is clear that the two Paris odel paraeters are strongly correlated. The sae is true for the initial crack size and bias; in fact, the PDF of bias is calculated fro Equation (8) with the initial crack size. Secondly, it can be observed that the PDFs at n ¼ 17 is narrower than that of n ¼ 13, although the PDFs at n ¼ 13 is quite narrow copared to the prior distribution. As the uncertainty in odel paraeters is reduced, the shape of distribution approaches a Gaussian distribution, regardless of the nonlinearity of the syste. In addition, the central liit theore 6 can be considered as a part of the reason why the joint distribution of (, log C) appears Gaussian. Lastly, the identified results look different fro the true values due to the scale, but the errors between the true values and the edian of identified results are at a axiu of around 5%, except for bias. The error in bias looks large, but that is because the true value of bias is sall. The error in bias is about.5. The sae agnitude of error exists for the initial crack size due to the perfect correlation between the. Table lists all six cases considered in this paper, and all of the show a siilar level of errors. It is noted that the identified standard deviation of noise,, does not converge to its true value of zero. This occurred because the original data did not include any noise. Zero noise can cause a proble in the likelihood calculation, as the denoinator becoes zero in Equation (6). However, this would not happen for practical cases in which noise always exists. The next exaple is to investigate the effect of noise on the posterior PDFs of paraeters. The results of identified posterior distributions with different levels of noise were shown in Figure 5 when the true bias was. Siilar results were obtained when bias was. The black, blue, and red colors represent noise levels of.1, 1, and 5, respectively. The edian location is denoted by a sybol (a circle for.1 noise, a square for 1 noise, and a star for 5 noise). Each vertical line represents a 9% confidence interval (CI) of posterior PDF. The solid horizontal line is the true value of the paraeter. In the case

8 3 Structural Health Monitoring 11(3) Table. The edian of identified paraeters and the errors with the true values n ¼ 13 n ¼ 15 n ¼ 17 log(c) a b log(c) a b log(c) a b True values b ¼ error (%) b ¼ error (%) a () n =13 n =15 n =17 5 n =13 n =15 n =17 1 log (C) bias () n =13 n =15 n =17 1 n =13 n =15 n =17 Figure 5. Posterior distributions with three different levels of noise (bias ¼ ). of noise level ¼.1, all paraeters were identified accurately with very narrow CIs. In the case of noise level ¼ 1, the initial crack size and bias were identified accurately as the nuber of data increased, whereas the CIs of two Paris paraeters were not reduced. In addition, the edian values were soewhat different fro the true paraeter values. Increasingly inaccurate results were observed as the level of noise increased to 5. Therefore, it is concluded that the level of noise plays an iportant role in identifying correlated paraeters using Bayesian inference. However, this does not ean that it is not able to predict RUL. Even if these paraeters were not accurately identified because of correlation, the predicted RUL was relatively accurate, which will be discussed in detail next subsection. Daage propagation and RUL prediction Once the paraeters are identified, they can be used to predict the crack growth and estiate RUL. Since the joint PDF of paraeters are available in the for of 5 saple, the crack growth and RUL will also be estiated using the sae nuber of saple. First, using 5 sets of paraeters obtained fro the MCMC ethod, Equation () is utilized to calculate 5 nubers of crack size a i after N i cycles. Then, rando easureent errors are added to the predicted crack sizes. For that purpose, 5 saples of easureent errors are generated fro noral distribution with zero ean and identified 5 saples of. Then, the quality of prediction can be evaluated in ters of how close the edian is to the true crack growth and how large the prediction interval (PI) is. The results of crack growth are shown in Figure 6 when the true bias is. Different colors represent the three different loading conditions. The solid curves are true crack growth, while the dashed curves are edians of predicted crack growth distribution. The results are obtained as a distribution due to the uncertainty in paraeters, but the edians of predicted crack growth are only shown in the figures for visibility. In addition, the critical crack sizes with different loadings use horizontal lines. Since the posterior distributions of

9 An et al. 31 (a) Measureent data Critical crack size (b) Measureent data Critical crack size Noise = 1, n = 13 Noise = 1, n = 17 (c) 6 (d) 6 Measureent data Measureent data 5 5 Critical crack size 4 3 Critical crack size Noise = 5, n = 13 Noise = 5, n = 17 Figure 6. Prediction of crack growth with bias ¼ +. paraeters are syetric, there was no difference between the ean, ode, and axiu likelihood estiates. Figure 6 shows that the results closely predicted the true crack growth when noise is less than 1. Even if the level of noise is 5, the results of predicted crack growth becoe close to the true one as the nuber of data increases. This eans that if there are any data (uch inforation about crack growth), the future crack growth can be predicted accurately, even if there is uch noise. However, when the level of noise is large, the convergence is slow such that the accurate prediction happened alost at the end of life. As can be seen fro Figure 6, crack growth and RUL can be predicted with reasonable accuracy even though the true values of the paraeters are not accurately identified. The reason is that the correlated paraeters and C work together to predict crack growth in Equation (). For exaple, if is underestiated, then the Bayesian process overestiates C to copensate for it. In addition, if there is large noise in the data, the distribution of estiated paraeters becoes wider, which can cover the risk that coes fro the inaccuracy of the identified paraeters. Therefore it is possible to safely predict crack growth and RUL. In order to see the effect of the noise level on the uncertainty of predicted RUL, Figure 7 plots the edian and 9% prediction interval (PI) of the RUL and copared the with the true RUL. The RUL can be calculated by solving Equation () for N when the crack size becoes critical: N f ¼ a1 = C a 1 = i Cð1 Þð p ffiffiffi Þ ð9þ The RUL is also expressed as a distribution due to the uncertainty of the paraeters, which is obtained by replacing a i and odel paraeters, and C in

10 3 Structural Health Monitoring 11(3) (a) 4 (b) 4 (c) 4 RUL True RUL Δσ=7.8 Δσ=78.6 Δσ=86.5 RUL True RUL Δσ=7.8 Δσ=78.6 Δσ=86.5 RUL Δσ=7.8 Δσ=78.6 Δσ=86.5 True RUL Noise = Noise = Noise = 5 Figure 7. and 9% of prediction interval of the predicted RUL (bias ¼ ). Equation (9) with 5 predicted crack growth and identified odel paraeters. In Figure 7, the solid diagonal lines are the true RULs at different loading conditions ð ¼ 86:5, 78:6, 7:8Þ. The precision and accuracy are fairly good when the noise is less than 1, which is consistent with the crack growth results. In the case of a large noise, 5, the edians are close to the true RUL, and the wide intervals are gradually reduced as ore data are used. Therefore, it is concluded that the RULs are predicted reasonably in spite of large nose and bias in data. Conclusions In this article, Bayesian inference and the Markov Chain Monte Carlo (MCMC) ethod are used for identifying the Paris odel paraeters that govern the crack growth in an aircraft panel using SHM systes that easure crack sizes with noise and bias. Focuses have been given to the effect of correlated paraeters and the effect of noise and bias. The correlation between the initial crack size and bias was explicitly iposed using analytical expression, while the correlation between two Paris paraeters was identified through Bayesian inference. It is observed that the correlated paraeter identification is sensitive to the level of noise, while predicting the reaining useful life is relatively insensitive to the level of noise. It is found that greater nubers of data are required to narrow the distribution of paraeters when the level of noise is high. When paraeters are correlated, it is difficult to identify the true values of the paraeters, but the correlated paraeters work together to predict accurate crack growth and RUL. Acknowledgent This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea governent (MEST) (No ). References 1. Giurgiutiu V. Structural health onitoring with piezoelectric wafer active sensors. Burlington, MA: Acadeic Press (an Iprint of Elsevier), 8.. Schwabacher MA. A survey of data-driven prognostics. In: AIAA Infotech@Aerospace Conference, Reston, VA, Luo J, Pattipati KR, Qiao L and Chigusa S. Model-based prognostic techniques applied to a suspension syste. IEEE Trans Syst Man Cyber 8; 38(5): Yan J and Lee J. A hybrid ethod for on-line perforance assessent and life prediction in drilling operations. In: IEEE International Conference on Autoation and Logistics, Jinan, Shandong, China, August 18 1, Paris PC and Erdogan F. A critical analysis of crack propagation laws. ASME J Basic Eng 1963; 85: Yu WK and Harris TA. A new stress-based fatigue life odel for ball bearings. Tribol Trans 1; 44: Jaw LC, Inc SM and Tepe AZ. Neural networks for odel-based prognostics. In: IEEE Aerospace Conference, Snowass at Aspen, Colorado, March 6 13, Orchard M and Vachtsevanos G. A particle filtering approach for on-line failure prognosis in a planetary carrier plate. Int J Fuzzy Logic Intell Syst 7; 7(4): Orchard, M., Kacprzynski, G., Goebel, K., Saha, B. and Vachtsevanos, G. (8). Advances in uncertainty representation and anageent for particle filtering applied to prognostics. In: International Conference on Prognostics and Health Manageent, Denver, CO. 1. Zio E and Peloni G. Particle filtering prognostic estiation of the reaining useful life of nonlinear coponents. Reliab Eng Syst Safety 11; 96(3): Sheppard JW, Kaufan MA, Inc A and Annapolis MD. Bayesian diagnosis and prognosis using instruent uncertainty. IEEE Autotestcon, 5 Conference Record, Orlando, Florida, Septeber 5 9, 5, pp Saha B and Goebel K. Uncertainty anageent for diagnostics and prognostics of batteries using Bayesian techniques. In: IEEE Aerospace Conference, Big Sky, Montana, March 1 8, 8.

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