Uncertainty Assessment in Structural Damage Diagnosis

Uncertainty Assessment in Structural Damage Diagnosis

Shankar Sankararaman and Sankaran Mahadevan Department of Civil and Environmental Engineering Box 1831-B, Vanderbilt University, Nashville, TN – 37235 United States of America

ABSTRACT This paper develops methods for the quantification of uncertainty in each of the three steps of damage diagnosis (detection, localization and quantification), in the context of continuous online monitoring. A model-based approach is used for diagnosis. Sources of uncertainty include physical variability, measurement uncertainty and model errors. Damage detection is based on residuals between nominal and damaged system-level responses, using statistical hypothesis testing whose uncertainty can be captured easily. Localization is based on the comparison of damage signatures derived from the system model. Both classical statistics-based methods and Bayesian statistics-based methods are investigated to quantify the uncertainty in all the three steps of diagnosis, i.e. detection, localization, and quantification. While classical statistics-based methods use the concept of least squares-based optimization, Bayesian methods make use of likelihood function and Bayes theorem. The uncertainties in damage detection, isolation and quantification are combined to quantify the overall uncertainty in diagnosis. The proposed methods are illustrated using two types of example problems, a structural frame and a hydraulic actuation system.

1. INTRODUCTION Damage diagnosis consists of three major steps – damage detection, damage localization (or isolation), and damage quantification. Practical applications often consist of multi-level, multi-domain systems, where only a few response quantities are monitored but the number of prospective damage candidates is very high. In such cases, it is almost impossible to achieve precise damage diagnosis. Further, the model inputs, physical parameters, sensor measurements, the model form, etc. may all be uncertain and hence render the results of diagnosis uncertain. The quantification of uncertainty in damage diagnosis is an essential step to guide decision-making with respect to operations, maintenance, and risk management. The uncertainty in detection [1] has been previously addressed by non-destructive evaluation (NDE) techniques through the quantity probability of detection (POD). In such POD calculations, nominally identical damage is introduced in a number of nominally identical specimens, and the number of successful detections is used to calculate the probability of detection. However, such an approach is only applicable to offline testing and not directly applicable to real time diagnosis [2]. System identification techniques have been pursued by several researchers for the purpose of damage isolation and quantification. Fundamentally, these methods could be viewed as statistical calibration problems where the model parameters are calibrated and the uncertainty in calibration can be expressed through confidence bounds in the parameter estimation. Several studies have investigated classical statistics-based methods [e.g. 3] and Bayesian methods [4, 5] for uncertainty quantification in system identification and structural health monitoring. These methods involve estimating all the system parameters simultaneously; this makes the computation time-consuming and thus not suitable for online diagnosis. System identification-based methods also combine damage isolation and damage quantification into one procedure; the individual contributions from different stages of diagnosis (detection, localization, and quantification) to the overall uncertainty in diagnosis are not separated. This paper presents two different approaches – based on classical and Bayesian statistics - to quantify the uncertainty in the three steps of diagnosis, and thereby the overall uncertainty in diagnosis. The scope of the paper is currently limited to single faults and multiple fault scenarios will be considered in future.

Damage detection is based on statistical hypothesis testing of the residuals, i.e. the difference between the model predictions and system measurements. In classical hypothesis testing [6], the choice of the significance level of testing is used to calculate the uncertainty in detection. In the Bayesian hypothesis testing-based approach, the likelihoods of two scenarios, i.e. “no fault” and “fault” are calculated and the uncertainty in the detection is quantified. Damage isolation is first done qualitatively based on cause-effect relationships between model parameters and measurements, and this helps to isolate a prospective set of candidate damage parameters. In the classical statistics-based approach, a heuristic measure based on the least squares corresponding to each candidate damage parameter is used to quantify the uncertainty in damage isolation. In the Bayesian statistics-based approach, the probability of a particular parameter being faulty is calculated directly. Damage quantification is based on least squares approach or likelihood maximization. The uncertainty in damage quantification in calculated using statistical confidence intervals in the classical statistics-based approach, and using Bayesian inference in the Bayesian approach. Note that the latter approach directly calculates the probability distribution of the damage parameter while the former approach only gives a measure of uncertainty. The overall uncertainty in diagnosis is calculated by combining the uncertainties in the three stages of diagnosis using the principle of total probability, and the overall probability distribution of the damage parameter is calculated. The methods for uncertainty quantification in damage detection, localization and quantification are developed in the Sections 2-4. The proposed methods are illustrated using two different kinds of systems, a structural frame (Section 5) and a hydraulic actuator (Section 6). 2. UNCERTAINTY IN DAMAGE DETECTION Detection is based on comparison between healthy system response (predicted by the model) and the observed system response. ‘Significant’ deviation of y from the expected performance indicates the presence of damage. The difference, i.e the residuals r(t) are averaged over a moving window and the mean (μ) is subjected to a hypothesis test at a chosen significance level α, as follows: H0: μ = 0 (No damage)

(1)

H1: μ ≠ 0 (Damage)

(2)

The null hypothesis (H0) in Eq. 1 states that the residuals are not significantly different from zero (no damage) while the alternate hypothesis (H1) in Eq. 2 states that the residuals are significantly different from zero (damage). Classical Statistics-based Approach The uncertainty in detection is calculated as the probability that the damage is true after having detected a fault. This is established using the choice of the significance level α in hypothesis test. The probabilities of different types of errors in hypothesis testing are well established in literature. The error of rejecting a correct null hypothesis is known as Type-I error while the error of not rejecting a false null hypothesis is known as Type-II error. The probability of Type-I error is equal to α and the probability of Type-II error is denoted by β. Suppose that the residuals calculated from the system have been subjected to statistical hypothesis testing at significance level α and damage has been detected. Hence, the hypothesis H0 has been rejected. The confidence in damage detection can be calculated as the probability that the actual state of the system is H1, subject to having rejected the hypothesis H0. This probability is equal to (1 – α) and is denoted as Pd. Bayesian Statistics-based Approach In this method, residuals are assumed to be normally distributed, i.e. r ~ N (μ, σ2), and the probability distribution of the mean μ is continuously updated with measurements. Let the prior distribution be N (ρ, τ2). If no additional information is available, Migon and Gamerman [7] suggest ρ = 0 and τ2 = σ2. Then the posterior distribution of μ can be calculated to be normal with mean and variance as follows: … 1

3

1

4

Damage detection can be achieved through the use of Bayes factor which is defined as the ratio of likelihood of the two scenarios: “damage” and “no damage” as follows: | |

5

In Eq. 5, D refers to the data on the residuals obtained during health monitoring. Jiang and Mahadevan [5] derived the expression for the (natural) logarithm of Bayes factor.

log

log

1

6

In Eq. 6, R stands for the mean of the observed residuals, i.e. R = (r1+ r2+ r3 …+ rn) / n. If the Bayes factor B is greater than 1, it implies that the data favors the hypothesis H1, and hence suggests that there is damage. The confidence in damage detection, i.e. the probability that the fault is true can be calculated as P(H1|D) and is equal to B/B+1 if the prior probabilities for P(H0) and P(H1) are chosen to be equal. This metric gives a direct measure of the probability of damage and hence is easy to compute and interpret in comparison with the classical statistics-based metric. Further, the probabilities P(H0) and P(H1) can be updated continuously with measurements and thereby helps in tracking the uncertainty with acquisition of more measurements. 3. UNCERTAINTY IN DAMAGE LOCALIZATION Damage localization is based on the comparison between damage signatures calculated from the system model and the symbols calculated from the system measurements. First, the system model is used to derive cause-effect relationships between the model parameters (that correspond to damage parameters) and model outputs (that correspond to system measurements). These cause-effect relationships, collectively called as the damage signatures, describe the qualitative changes in the 0th and 1st derivatives of a system output as a result of a change in a system parameter. There are several ways to obtain these signatures; for example, by using finite difference with the system model to calculate approximate derivatives. This paper uses a bond graph-based methodology to derive the damage signatures; this method is qualitative and hence, can aid in rapid online damage localization of damage parameters [6, 8]. However, practical engineering systems usually have a large number of parameters that could become faulty, but only a small set of measurements. It may be difficult to exclusively isolate one particular damage parameter if several candidates have the same set of damage signatures. Instead, a set of prospective candidate damage parameters, θi (i = 1 to m) may be suggested by the isolation procedure. This section associates a measure of uncertainty to each of the candidate damage parameters. Classical Statistics-based Approach The uncertainty in damage localization is calculated using a heuristic least squares-based metric. Consider a particular damage parameter θ and let and y denote the model prediction and sensor measurements respectively. Suppose that the fault was detected at tf and consider N further measurements. Then a least square estimate of the damage parameter θ can be calculated by minimizing Eq. 7.

7

Let the least square estimate be and the corresponding least squares value. Then the probability that θ is the true damage parameter can be heuristically said to be inversely proportional to and the constant of proportionality by considering all prospective damage parameters and equating the total probability to unity.

Bayesian Statistics-based Approach The Bayesian approach directly calculates the probability that a given damage parameter is faulty and provides an exact measure of uncertainty as against a heuristic measure provided by the classical statistics-based approach. Let A denote the event that θi is the true damage parameter. The proposed metric calculates the probability of event A after observing the data (D). A prior probability Pi' is first assumed (a suitable initial assumption would be 1/m as any of the candidate damage parameters is equally likely to be the damage parameter) and then updated to calculate a posterior probability Pi'’ based on the following likelihood function. P (D|A) = P (D | θi is the true damage parameter) = L(θi)

(8)

This probability is the likelihood of event A, denoted by L(θi). The calculation of this probability is explained in the following paragraphs. Let represents the model prediction. Let B denote the event that qi is the value of the damage parameter θi. Thus, for a given x and t, depends on the value of θi. , ;

(9)

Assume that damage has been detected at time ‘tf’, when the value qi of the parameter θi changes to an unknown value. Let measurements continue to be available after damage detection and y denote the measurements available (for example, pressure, velocity, etc.) at any time instant from tf to tf + N. The standard deviation of these measurements can be monitored until damage is detected and the covariance matrix (Σ) of the residuals can be calculated. Then the likelihood function corresponding to event (i.e. the probability of observing the data conditioned on (i) event A, i.e. θi is the true damage parameter and (ii) event B, i.e. the true damaged value is qi can be calculated by assuming an appropriate distribution (normal, below) for the residuals: |

exp

,

10

Note that y and are column vectors of length n, and the covariance matrix Σ is of size n, where n is the number of measured quantities. Hence, the right hand side of Eq. 10 is a scalar. The joint likelihood L(θi, qi) in Eq. 10 can be used in Bayesian inference to update Pi. Let f'(qi |θi) denote the probability density function (PDF) of the damage value qi conditioned on event A. The posterior probabilities Pi'’ and f''(qi |θi) can be calculated based on Bayes theorem. While the former indicates the uncertainty in damage localization, the latter indicates the uncertainty in damage quantification. Section 4 focuses on the latter topic and the choice of prior and the estimation of the posterior f''(qi |θi) is discussed in detail in Section 4. This section deals only with the estimation of Pi’’. It can be proved that |

,

|

11

Note that there is proportionality constant in Eq. 11. This constant can be evaluated by imposing the constraint that the sum of all Pis should be equal to unity. These probabilities can be continuously updated with the acquisition of measurements. 4. UNCERTAINTY IN DAMAGE QUANTIFICATION This section calculates the uncertainty in the damage quantification stage. The classical statistics-based approach calculates statistical confidence intervals on the damage parameter while the Bayesian statistics-based approach directly calculates the probability distribution of the damage parameter. Classical Statistics-based Approach A least squares estimate for θ was obtained in Section 3, and the least squares error was estimated as by minimizing Eq. 7. To calculate the confidence bounds on θ, the first step is to assign bounds on the squared error S(θ) as shown in Eq. 12. The values of θ that correspond to this error define the confidence bounds on θ: 1

,

12

In Eq. 12, N denotes the number of observations over which the least squares error was estimated and p denotes the length of denotes the F-statistic, with estimation (numerator) degrees the vector θ. In this case of single fault assumption, p = 1. , of freedom p and residual (denominator) degrees of freedom N - p at significance level α. For example, α = 0.05 would yield a 95% confidence interval for the damage parameter whose bounds would be approximate estimates of values that correspond to cumulative probabilities of 0.025 and 0.975 of the ‘true’ value. By repeating this procedure for different values of α, it is possible to obtain the entire an approximation to the cumulative probability distribution of the damage parameter. Bayesian Statistics-based Approach This section calculates the uncertainty in qi, the value of the damage parameter θi. A maximum likelihood estimate of the damage value can be obtained easily by maximizing the RHS of Eq. 10. The likelihood function calculated in Eq. 10 can be used in Bayes theorem and the entire probability distribution of the value qi of the damage parameter θi can be calculated as: |

, ,

| |

13

In Eq. 23, f'(qi| θi) is the prior density function and represents the knowledge about qi. As there is no “prior” knowledge about the damage value before the damage quantification, a uniform distribution may be assumed for the prior distribution to begin with. Then, this distribution can be updated continuously with acquisition of more measurements. This paper uses advanced numerical integration [9] to calculate the denominator in Eq. 13. The probability distribution of the value qi of the damage parameter θi can be calculated, and updated continuously with acquisition of more measurements. 5. OVERALL UNCERTAITNY IN DIAGNOSIS The probability distribution calculated in the previous section is conditioned on two events: damage detection and localization being correct. This section develops the method to obtain the unconditional distribution of the damage parameter, i.e. overall uncertainty in diagnosis. Let Pd denote the confidence in damage detection, as calculated in Section 2. This is equal to the probability that there is damage. Let Pi denote the confidence in isolation, as calculated in Section 3. This is equal to the probability that the damage is in θ, given that there is damage. The unconditional distribution of θ can be calculated as: Fθ(θ) = (1-Pd)* Fθ(θhealthy)+Pd*(1-Pi)*Fθ(θhealthy)+Pd*Pi*Fθ(θ|Damage is true,Damage is in θ)

(14)

The expression in Eq. 14 quantifies the overall uncertainty in the diagnosis procedure, by calculating the unconditional cumulative distribution of the damage parameter. Further, the corresponding probability density function can also be calculated by differentiating the expression in Eq. 14. The following sections illustrate the proposed methods through two numerical examples, a structural frame and a hydraulic actuation system. 6. ILLUSTRATION USING STRUCTURAL FRAME Consider the two story frame discussed in Moustafa et al. [8]. This system has 6 parameters, m1, m2, k1, k2, D1, D2. The inputs to the system are forces at the two levels and the measured outputs are the accelerations at the two levels. The masses of the first and second floors are assumed not to change. Hence, there are only 4 parameters that could be affected by damage. Using the damage signatures (cause-effect relationships between the model parameters and measurements), it is possible to identify the floor that is damaged, and hence the prospective damage parameters are the stiffness and damping of that particular floor. The value of k1 is reduced to 80% at 5 seconds. This means a change from 30700 N/m to 24560 N/m. Sensor noise is simulated by adding Gaussian white noise to the measurements. Damage detection is done through hypothesis testing, as explained in Section 2. A moving window is used to calculate the mean of the residuals and this mean is tested to detect damage. While classical statistics-based hypothesis testing detects

damage with 95% confidence, the Bayesian method facilitates easy quantification and updating of uncertainty from a 61% confidence at 5 seconds to 99.98% confidence at 5.05 seconds. Following damage detection, three consecutive sets of measurements are assumed to be available. The qualitative damage localization results suggest that either k1 or D1 could be faulty. The heuristic measures of uncertainty using classical statisticsbased methods yield 98% and 2% probabilities for k1 and D1 to be faulty, considering the 3 sets of measurements. The uncertainty in localization of k1, using the Bayes method, is calculated for the I set as 99% , and updated using Bayes theorem for the following two sets to 100% and 100%.

16

First Set of Measurements Second Set of Measurements Third Set of Measurements

Probability Density Function

14

MLE

12 10 8

True Damaged Stiffness

6 4 2 0 0.6

0.65

0.7

0.75 0.8 0.85 x 100 (30700) N/m Stiffness : First Storey

0.9

0.95

1

Fig. 1. PDF of k1 Damage quantification using the classical statistics procedure yields a 90% confidence interval of (24509 N/m, 24612 N/m). This can be repeated for multiple confidence intervals and the approximation to the cumulative distribution function can be calculated. The Bayesian approach directly calculates the cumulative distribution function as explained in Section 4. The overall uncertainty in diagnosis can be calculated as in Section 5 and the corresponding PDF is shown in Fig. 1. From Fig. 1, it can be seen that (i) the maximum likelihood estimate (MLE) of k1 matches closely with the true damage value, and (ii) the uncertainty in the estimate of k1 decreases with acquisition of more measurements. The above diagnosis procedure was also extended to five-story frames, where qualitative localization is able to isolate to the floor of damage accurately. Different damage scenarios were considered for the five-story frame, and these were detected and isolated (localized) satisfactorily with the proposed method. The damage was quantified with high accuracy. Uncertainty quantification associated with each of these three processes was also successful and both classical statistics-based and Bayesian methods were investigated. The easy extension to multi-story frames clearly indicates that this diagnosis methodology is applicable to practical structures. 7. HYDRAULIC ACTUATION SYSTEM The developed approach for structural health monitoring is now illustrated for a hydraulic actuation system. The PumpActuator-Load system shown in Fig. 2 is modeled using bond graphs to aid in damage isolation. The system is described and studied in detail by Sankararaman et al. [10] and only the data and results pertaining to diagnosis uncertainty quantification are provided in this section. Six different fault candidates are considered and five measurements are made from the system. Several faults were introduced in the system at t = 150 seconds and were diagnosed successfully. Measurements are collected at the rate of 2/sec. The uncertainties in damage detection, isolation and quantification were also quantified. One sample fault diagnosis example is discussed below.

Fig. 2 Actuation System The value of C, the load compliance, is suddenly doubled at 150 seconds. Similar to the previous section, measurements with noise are simulated by adding Gaussian white noise to the signals. Damage detection is done through hypothesis testing, as explained in Section 2. A moving window is used to calculate the mean of the residuals and this mean is tested to detect damage. While classical statistics-based hypothesis testing detects damage with 95% confidence, the Bayesian method facilitates easy quantification and updating of uncertainty from a 52% confidence at 150 seconds to 99.1% confidence at 152 seconds. 8

First Set of Measurements Second Set of Measurements Third Set of Measurements

Probability Densiy Function

7 6 5

MLE 4 3

True Damage

2 1 0 0

1

2

3 4 5 6 Ratio of Change in Value of Compliance C

7

8

Fig. 3. PDF of Ratio of Increase in C Following damage detection, three consecutive sets of measurements are assumed to be available. The qualitative damage localization results suggest that either C or B could be faulty. The heuristic measures of uncertainty using classical statisticsbased methods yield 99.9% and 0.1% probabilities for C and B to be faulty, considering the 3 sets of measurements. The uncertainty in localization of C, using the Bayes method, is calculated for the I set as 99.9% , and updated using Bayes theorem for the following two sets to 100% and 100%. Damage quantification using the classical statistics procedure yields a 90% confidence interval of (1.93, 2.06 N/m) for the ratio of increase. This can be repeated for multiple confidence intervals and the approximation to the cumulative distribution function can be calculated. The Bayesian approach directly calculates the cumulative distribution function as explained in Section 4. The overall uncertainty in diagnosis can be calculated as in Section 5 and the corresponding PDF is shown in Fig. 3. From Fig. 3, it can be observed that (i) the maximum likelihood estimate (MLE) of damage is reasonably equal to the true damage value, and (ii) the uncertainty associated with the PDF decreases with acquisition of more measurements. 8. SUMMARY This paper developed methods to quantify the uncertainty in the three steps of damage diagnosis, i.e. damage detection, damage localization, and damage quantification. Classical statistics-based methods and Bayesian methods were investigated

for this purpose. Damage detection was based on statistical hypothesis testing. While the classical hypothesis testing procedure calculated the uncertainty in detection based on the significance level of testing, the Bayesian hypothesis testing procedure calculated the likelihood of “no damage” scenario to “damage” scenario, thereby providing a robust Bayesian metric to assess the confidence in detection. The uncertainty in localization was quantified using heuristic measures in the classical statistics-based approach while the Bayesian method directly quantified the probability that a given damage parameter is faulty. The uncertainty in damage quantification was expressed in statistical confidence intervals in the classical statistics-based approach while the Bayesian method gives the entire probability distribution of the damage parameter. Finally, the uncertainties in the three different steps were combined to determine the PDF of the damage parameter as a measure of the overall uncertainty in diagnosis. While the two methods are fundamentally different and hence, it may not be correct to compare the results from the two methods, it is evidently seen that the Bayesian method is easy to implement and interpret. Further, the Bayesian method also facilitates easy updating of uncertainty with acquisition of measurements. The scope of this paper was limited to single faults only; in realistic problems, several faults may occur in the system. This will be considered in future work. 9. ACKNOWLEDGEMENT This study was partly supported by funds from the U. S. Air Force Research Laboratory through subcontract to General Dynamics Information Technology (Contract No. USAF-0060-43-0001, Project Monitor: Mark Derriso). The support is gratefully acknowledged. 10. REFERENCES [1] Achenbach, J.D. Quantitative nondestructive evaluation, International Journal of Solids and Structures, Volume 37, Issues 1-2, January 2000, Pages 13-27, ISSN 0020-7683, DOI: 10.1016/S0020-7683(99)00074-8. [2] Guratzsch, R. Sensor placement optimization under uncertainty for structural health monitoring systems of hot aerospace structures. Ph.D. Dissertation, Vanderbilt University, Nashville, TN. 2007. [3] Dalai, M., Weyer, E., and Campi M.C. Parametric Identification of Nonlinear Systems : Guaranteed Confidence Regions. Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15, 2005. [4] Beck J.L, Au S.K, and Vanik M.V. Monitoring structural health using a probabilistic measure. Computer-Aided Civil and Infrastructure Engineering 2001; 16:1–11. [5] Jiang, X., and Mahadevan, S. Bayesian Wavelet Methodology for Structural Damage Detection. Struct. Control Health Monit. 2008; 15:974–991. [6] Sankararaman, S., and Mahadevan, S. Uncertainty Quantification in Structural Damage Diagnosis. doi: 10.1002/stc.400 Published Online May, 2010. [7] Migon H.S, and Gamerman D. Statistical Inference: An Integrated Approach. Arnold: London, 1999. [8] Moustafa, A., Mahadevan, S., Daigle, M., Biswas, G. Structural and Sensor Damage Identification using the Bond Graph Approach. J. International Association of Structural Control and Health Monitoring. Published Online Jul. 2008. [9] McKeeman, M.W. Algorithm 145: Adaptive numerical integration by Simpson's rule. Commun. ACM, Vol. 5, No. 12, Aug. 1962. [10] Sankararaman, S., Bartram, G., Biswas, G., and Mahadevan, S. 2009. Bond Graph Method for Hierarchical MultiDomain Systems Diagnosis. In the Proceedings of 7th International Workshop on Structural Health Monitoring, Stanford, CA. Sept. 9-11, 2009.