uncertainty quantification in structural damage diagnosis

STRUCTURAL CONTROL AND HEALTH MONITORING

Struct. Control Health Monit. 2011; 18:807–824Published online 28 May 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/stc.400

Uncertainty quantification in structural damage diagnosis

Shankar Sankararaman and Sankaran Mahadevan�,y

Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN 37235, U.S.A.

SUMMARY

This paper develops methods for the quantification of uncertainty in each of the three steps of damagediagnosis (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 anddamaged system-level responses, using statistical hypothesis testing whose uncertainty can be capturedeasily. Localization is based on the comparison of damage signatures derived from the system model.A metric based on least squares is proposed to assess the uncertainty in damage localization, when thedamage signatures fail to localize the damage uniquely. The uncertainty in damage quantification isevaluated through statistical non-linear regression, resulting in confidence bounds for the damageparameter. The uncertainties in damage detection, isolation and quantification are combined to quantifythe overall uncertainty in diagnosis. The proposed methods are illustrated using two types of exampleproblems, a structural frame and a hydraulic actuation system. Copyrightr 2010 John Wiley & Sons, Ltd.

Received 3 August 2009; Revised 22 January 2010; Accepted 6 April 2010

KEY WORDS: structural health monitoring; damage diagnosis; damage detection; confidence intervals;uncertainty quantification; bond graph; non-linear regression; F-statistic; inverse problems

1. INTRODUCTION

Damage diagnosis of an engineering system consists of three steps: damage detection, damagelocalization (isolation) and damage quantification. Model-based diagnosis involves the use of amathematical model which attempts to capture the behavior of the system. Diagnosis techniquesand system identification techniques form the components of a health monitoring system, aidingthe three steps of damage detection, isolation and quantification.

Diagnosis methods can be broadly classified into two groups, data-driven methods andmodel-based methods [1]. Data-driven approaches include neural networks [2], autoregressivemoving average models [3], fuzzy logic [4], similarity-based methods [5] and hybridmethodologies combining traditional knowledge-based methods and neural networks [6].Model-based techniques attempt to emulate nominal system behavior using physicallymeaningful system models, such as state-space models [7], finite element models [8], bondgraph models [9], etc. This paper uses a generic model-based diagnosis technique where thesystem measurements of quantities related to system-level performance are measuredcontinuously online and used in diagnosis in combination with model prediction. Damage

*Correspondence to: Sankaran Mahadevan, Department of Civil and Environmental Engineering, VanderbiltUniversity, Nashville, TN 37235, U.S.A.yE-mail: [email protected]

Contract/grant sponsor: US Air Force Research Laboratory; contract/grant number: USAF-0060-43-0001

Copyright r 2010 John Wiley & Sons, Ltd.

detection is based on statistical significance testing of the residuals (differences betweenmeasurements and model predictions). Damage isolation is done qualitatively based on thedamage signatures developed from cause–effect relationships between system measurements andsystem parameters. Damage quantification is based on the method of non-linear least squares.

The aim of this paper is to quantify the uncertainty in the results of this diagnosis technique,arising from each of the three stages. Uncertainty quantification in diagnosis helps to achieverobustness in decision making with respect to future operations, maintenance, risk management, etc.There can be several sources of uncertainty in the diagnosis procedure, and these can be broadlyclassified into three different types—measurement uncertainty, modeling errors and physicalvariability. System responses are measured through sensors and the data may contain noise.Further, the sensors themselves may be faulty. These different issues lead to measurementuncertainty. Any model-based approach to diagnosis would be prone to uncertainty, arising fromsolution approximation errors as well as model form errors. Uncertainty also arises from thephysical variability in the model parameters and inputs. All these sources lead to uncertainty indamage detection, isolation and quantification, and this paper develops methods to quantify theuncertainty in each of these three stages. The methods developed in this paper may be used with anygeneric model of the system. However, in the damage localization step, the cause–effect relationshipsare generated using the bond graph model [10] of the system. These signatures can also be generatedusing other procedures such as sensitivity analysis, finite differences, etc. This paper uses the bondgraph-based procedure to generate qualitative signatures that aid in rapid damage localization. Themethods for uncertainty quantification in diagnosis (detection, localization and quantification) donot depend on bond graph modeling and can be used with any model-based diagnosis procedure.

The uncertainty in detection [11] has been previously addressed through non-destructiveevaluation (NDE) techniques through the quantity probability of detection (POD). In such PODcalculations, nominally identical damage is introduced in a number of nominally identicalspecimens, and the number of successful detections gives the POD. However, such an approach isonly applicable to offline testing and not directly applicable to real-time diagnosis [12,13]. Systemidentification techniques estimate all the parameters of the system simultaneously and hencecombine damage isolation and damage quantification into one procedure; the individualcontributions of different sources of uncertainty to diagnosis uncertainty are not separated.Further, these methods are computationally expensive, time-consuming and hence, not suitablefor online diagnosis. This paper uses separate methods for detection and isolation, and quantifiesthe uncertainty in each of them through well-established statistical procedures. The uncertaintyin damage quantification is expressed through confidence bounds in the parameter estimation.

Several studies in system identification [14] have investigated methods for estimatingconfidence bounds in parameter estimation problems. Dalai et al. [15] proposed a method basedon higher order statistics to generate non-conservative confidence regions for parameters. Basedon the well-known concepts of poles and zeros in system identification, Vuerinckx et al. [16]quantify elliptic regions of uncertainty based on the covariance matrix of the estimatedparameters. The same question has been approached through a Bayesian perspective as well.Peterka [17] describes in detail the application of Bayesian inference to system identification.Ching et al. [18] use the extended Kalman filter to estimate the state of a dynamic system andthen quantify the uncertainty in estimation using particle filter and Bayesian techniques.

Most of the above-mentioned methods used for confidence interval calculation involveestimating all the system parameters simultaneously; this makes the computation time-consuming and thus not suitable for online diagnosis. In the current research work, aprospective set of damage parameters is isolated qualitatively using the damage signaturesderived from the bond graph model of the system, thus making it faster to compute theconfidence bounds for this set of isolated parameters only. In this paper, the damage signaturesare limited to the effects of single faults on system response. The single-fault assumption canalso be used to reduce damage quantification to a set of one-dimensional least-squaresoptimizations, instead of a multi-dimensional least-squares estimation of all parameters at once.These one-dimensional estimations do not use derivatives and can be easily parallelized, thusreducing computational time.

S. SANKARARAMAN AND S. MAHADEVAN808

Copyright r 2010 John Wiley & Sons, Ltd. Struct. Control Health Monit. 2011; 18:807–824

DOI: 10.1002/stc

The effects of concurrent multiple faults are not considered in this paper. This would requirea multivariate approach to damage diagnosis and will be considered in future work.

Section 2 discusses the damage diagnosis methodology where bond graph models have beenused for rapid damage localization thereby facilitating rapid diagnosis. The methods used fordamage detection, localization and quantification are also described in this section. (Note: Theterms ‘fault’ and ‘damage’ are used interchangeably in this paper. Although the former iscommonly used in electrical and electronic systems, the latter is used for mechanical andstructural components. Also, the terms ‘localization’ and ‘isolation’ are used interchangeably inthis study. Although localization implies narrowing down the location of damage, isolation is amore general term referring to a parameter space; some of the parameters need not be associatedwith a particular location in the system). Methods for uncertainty quantification in damagedetection, isolation and quantification are developed in Section 3. The uncertainties in detection,isolation and damage quantification are combined together to calculate the overall uncertaintyin diagnosis. The proposed methods are illustrated using two different kinds of systems, astructural frame (Section 4) and a hydraulic actuator (Section 5).

2. DIAGNOSIS METHODOLOGY

This paper uses a model-based approach for damage diagnosis. The state-space equations of thesystem are solved to calculate the nominal model output yðtÞ. Measurements y(t) are continuouslycollected from the system and the residuals r(t) are calculated by differencing the model output andmeasurements. In many cases, the model is not able to replicate the system accurately, because oferrors in the model and in the measurements. This problem is overcome by the use of an extendedKalman filter, which estimates the nominal state of the system, as a function of both modelprediction and system measurement, thereby accounting for model error and sensor noise. Damagedetection is based on a statistical hypothesis test on the residuals, explained later in Section 2.1.

Damage isolation is based on the method developed by Moustafa et al. [10] where the bondgraph modeling technique is used to qualitatively isolate damage parameters. Bond graphs areused to simulate the dynamics of complex systems by modeling model the flow of energy(expressed as the product of flow and effort) and provide a systematic framework for buildingmodels of systems spanning multiple domains (e.g. structural, electrical, mechanical, hydraulicand thermal) using energy as the connecting measure across domains. The bond graph theory iswell established, and is described in detail in several textbooks [19,20]. Bond graphs inherentlycontain qualitative cause–effect relationships between the parameters and outputs of the systemmodel. Each damage parameter is associated with a model parameter and each measurement isassociated with a model output. These qualitative cause–effect relationships are depictedgraphically in a Temporal Causal Graph (TCG) from which the damage signatures can bederived. The damage signatures describe the qualitative changes in the 0th and 1st derivatives ofa system measurement as a result of a change in a system parameter.

Consider the two-story frame shown in Figure 1. This structure has six parameters: damping(D1 and D2), stiffness (k1 and k2) and inertia (m1 and m2). The inputs to this model are theexcitations (F1(t) and F2(t)) at the two levels and the outputs are the displacements (u1(t) andu2(t)) at the two levels.

The bond graph model for this structure was constructed from first principles by Moustafaet al. [10], and is shown in Figure 1(c). This bond graph model is an exact representation of thesystem equations in a graphical form. The bond graph model in Figure 1(c) was validated byderiving the state-space equations and comparing them with the equations of motion of thesystem derived from classical structural dynamics. Note that the sensors for measuringdisplacements are also included as separate blocks in the bond graph model, thereby facilitatingdiagnosis of sensor faults. The details are given in Moustafa et al. [10].

The combined bond graph model of the system and sensors was used to construct a TCGfrom which the damage signatures were derived. These damage signatures are shown in Table I.

UNCERTAINTY QUANTIFICATION 809


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These signatures correspond to changes in the 0th derivative and the 1st derivative of thedisplacements due to changes in stiffness and damping parameters.

This entire procedure of generating damage signatures from bond graphs can be automated[21] and performed offline using the system model. Damage isolation is based on these damagesignatures and is explained in Section 2.2. Once the damage parameters are isolated, damagequantification is achieved using a least-squares approach.

Figure 2 summarizes the diagnosis technique, explaining the three stages of diagnosis. Notethat the bond graph model has been used only for qualitative isolation and neither in damagedetection nor in damage quantification. Furthermore, the bond graph model is not required toquantify the uncertainty in diagnosis, as seen in Section 3.

The following subsections describe the diagnosis methodology in three steps—detection,localization and quantification.

2.1. Damage detection

Detection is based on comparison between healthy system response y(predicted by the model)and the observed system response y. ‘Significant’ deviation of y from the expected performanceindicates the presence of damage. This significance is evaluated through statistical hypothesistesting [22]. The residuals are averaged over a moving window and subjected to a hypothesis testat a chosen significance level a, as follows:

H0 : rðtÞ ¼ 0 ð1Þ

H1: rðtÞ 6¼ 0 ð2Þ

1 0 1

3

2 4

1 7

8

10

9 5

6

11

C: 1/k1 R: D1

C: 1/k2 R: D2

Se: F2(t)

I: m2I: m1First floor Second floor

Se: F1(t) 1112 14

13

Se: B1

MSf: f3 12

C: 1

First floor sensor

112 17

16

Se: B2

MSf: f11 15

Second floor sensorf3

f11

C: 1

F2(t) m2u1(t)

m1 m2

k1

D1 D2

u2(t)

F1(t) m1

k1,D1

k2,D2F1(t) k2 F2(t)

(b)(a)

(c)

Figure 1. (a) Frame structure; (b) Mechanical model; and (c) Bond graph model for the frame structureand the sensors.

Table I. Damage signatures for the two-story frame.

Actual damage First floor u1(t) Second floor u2(t)

Decrease in k1 01 01Decrease in k2 01 0�Decrease in D1 01 01Decrease in D2 01 0�First floor sensor (1 bias) 1 0Second floor sensor (1 bias) 0 1



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The null hypothesis (H0) in Equation (1) states that the residuals are not significantlydifferent from zero, while the alternate hypothesis (H1) in Equation (2) states that the residualsare significantly different from zero.

If the measured deviation (from the nominal behavior) is found to be significant, thendamage is assumed to have occurred. Furthermore, this deviation is also used to assign the firstsymbol used in damage localization in Section 2.2. If the measurement of the damaged structureat a given point of time is above normal, a (1) symbol is assigned, and if below normal, a (�)symbol is assigned.

If there are several measurements (e.g. multiple sensors), a hypothesis test needs to beconducted for each separately. Although some of these measurements may be correlated, thedeviation from healthy behavior alone is of interest. The change in any model parameter willaffect at least one of the measurements. Hence, several independent hypothesis tests (one foreach measurement) are used and the sufficient condition for the presence of damage is ‘rejectionof null hypothesis for any of the sensors’.

2.2. Damage localization

Once the presence of damage is detected, the residuals r(t) are measured and two sets of symbolsare generated for each measurement. The first symbol can be (0) or (1) or (�) depending onthree different scenarios: no discontinuity, discontinuity with a positive residual anddiscontinuity with a negative residual, respectively. The second symbol refers to the slope ofthe residual. It can be either (0), (1) or (�).

During continuous monitoring, several measurements are collected (for example, pressure,accelerations, etc.) and symbols are generated for each of these measurements. Starting from alist of possible damage scenarios, a particular damage parameter is dropped when the observedsymbols do not correspond to the damage signatures. By repeating this process for everydamage parameter, the incorrect parameters are all dropped and it is possible to arrive at thecorrect damage parameter. As this isolation procedure is qualitative in nature, itsimplementation is inexpensive and hence aids in rapid online damage localization.

However, if two (or more) candidate damage parameters share the same set of damagesignatures, then this qualitative localization procedure cannot isolate further between them. Inthis case, further isolation is possible only through quantitative analysis, to be described inSection 3.2. An alternative solution is to make an additional system measurement that producesa difference in the set of damage signatures between the two candidates.

2.3. Damage quantification

Damage quantification is achieved through a least-squares procedure, similar to that used insystem identification techniques. However, only those parameters isolated in Section 2.2 areestimated. Further, because of the single-fault assumption made in Section 1, the parameterestimation reduces to one-dimensional optimization for each isolated candidate damage

Input

y(t)

y(t)

r(t) DAMAGELOCALIZAZATION

DAMAGEDETECTION

DAMAGEQUANTIFICATION

Symbols Generated

For Comparison

With Symbols

BOND GRAPH MODEL

TEMPORAL CAUSAL GRAPH

DAMAGE SIGNATURES

PLANT

OBSERVER^

Figure 2. Diagnosis scheme.



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parameter. This reduces the computational effort from a multi-dimensional optimizationproblem to a few one-dimensional optimization problems. Further, these optimizations can beexecuted in parallel, aiding in rapid damage quantification and online decision making.

Concurrent multiple faults would require a multivariate approach and are not considered inthis paper; they will be considered in future work.

Consider an engineering system which is modeled by Equation (3). In this equation, yrepresents the model prediction and y represents the damaged model parameter. Thus, for agiven x and t, y depends on y

y ¼ f ðx; t; yÞ ð3Þ

Assume that damage has been detected at time ‘tf’, when the value of the parameter y changesto an unknown value y�. The task is to estimate this value for y with a small number ofmeasurements, each observation y being made at a time step. Assume that N is the minimumnumber of observations (after damage has been detected) to estimate y to a predefined level oftolerance. An error function S(y) can be associated with the set of observations made over thistime period. In Equation (4), y is a vector of length equal to the number of measurementsavailable (for example, pressure, velocity, etc.) at any time instant.

SðyÞ ¼Xtf 1N

t¼tf

ðyðyÞ � yÞTðyðyÞ � yÞ ð4Þ

A least-squares estimate for y can be obtained by minimizing S(y) with respect to y. Let thisleast-square estimate be denoted by y. If the different response quantities are of different ordersof magnitudes and different units, then a weighted least-squares approach may need to beemployed. Using an optimization procedure, a least-squares estimate can be obtained almostimmediately, having made N observations after damage has been detected. Typically, the valueof N depends on the dynamic behavior of the system. A least-squares estimate for y� can bedetermined for every additional measurement made and can be tracked along with the numberof measurements. It can be proved that y converges to y� asymptotically [23]. But it may not beaffordable to wait for the convergence of y. An alternative solution is to estimate confidencebounds for y�. Acceptable confidence bounds estimated with a minimal N would be an efficientalternative to a convergent y. The estimation of confidence bounds is discussed in Section 3.3.

3. UNCERTAINTY IN DIAGNOSIS

Section 2 presented the proposed methodology for online diagnosis of engineering systems. Thissection quantifies the uncertainty in each of these procedures: damage detection, isolation andquantification.

3.1. Uncertainty in damage detection

As described earlier in Section 2.1, the damage detection procedure is based on statisticalhypothesis testing. The residuals r(t) (as defined in Section 2) are calculated with the acquisitionof measurements. These residuals are subjected to a hypothesis test as explained in Section 2.1.

The choice of the significance level a in the hypothesis test is a critical issue. A higher a wouldlead to delayed detection, whereas a lower a would lead to a lower confidence in detection.The probabilities of different types of errors in hypothesis testing are well established inliterature [24]. The error of rejecting a correct null hypothesis is known as Type-I error, whereasthe error of not rejecting a false null hypothesis is known as Type-II error. The probability ofType-I error is equal to a and the probability of Type-II error is denoted by b. This is tabulatedin Table II.

In the context of damage diagnosis, the null hypothesis is that there is no damage in thesystem, and the alternative hypothesis is that there is damage in the system. Therefore, a Type-Ierror refers to detecting damage when there is no damage in the system (false alarm). A Type-IIerror refers to concluding that there is no damage when there is actually damage in the system



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(missed detection). NDE techniques use the term POD (defined in Section 1) to quantify theprobability of detecting damage when there is one [11]. Therefore, the probability of misseddetection (Type-II error) is b5 1–POD.

Suppose that the residuals calculated from the system have been subjected to statisticalhypothesis testing at significance level a and damage has been detected. From Table II, thismeans that the hypothesis H0 has been rejected. The confidence in damage detection can becalculated as the probability that the actual state of the system is H1, subject to having rejectedthe hypothesis H0. This probability is equal to (1–a) and is denoted as Pd in Section 3.5 (This isnot to be confused with POD which is equal to (1–b)).

3.2. Uncertainty in damage localization

The procedure for damage localization is based on damage signatures and was described inSection 2. Practical engineering systems usually have a large number of parameters that couldbecome faulty, but only a small set of measurements. It may be difficult to exclusively isolateone particular damage parameter if several candidates have the same set of damage signatures.Instead, a set of prospective candidate damage parameters, yi (i5 1–m) may be suggested by theisolation procedure. Recalling the single-fault assumption made in Section 1, only one of thesecandidates corresponds to the actual damage parameter. In addition, because of measurementnoise and modeling errors, none of these prospective candidate damage parameters can beisolated with a 100% confidence.

In case of ambiguity in damage localization, the parameter estimation scheme in Section 2.3is executed assuming that one of the m parameters (yJ) is the true damage parameter and that allother prospective candidate damage parameters yi (i5 1 to j�1, j11 to m) retain their nominalvalues. The least-squares error, SðyJ Þ corresponding to the convergent �y is estimated as inEquation (5). This least-squares error is calculated for every prospective damage parameter yiover the same set of observation points (N).

SðyJ Þ ¼Xtf 1N

t¼tf

ðyðyJ Þ � yÞTðyðyJ Þ � yÞ ðJ ¼ 1 tomÞ ð5Þ

The confidence that a particular parameter yJ is the damage parameter can be assumed to beinversely proportional to the corresponding least-squares error SðyJ Þ. Hence, a total of 100%confidence is split between each of the prospective candidates, depending on the correspondingleast-squares error. The confidence in yi being the actual damage parameter, denoted by Pi, canbe calculated as

Pi ¼ðSðyiÞÞ

�1

PðSðyJ ÞÞ

�1ðJ ¼ 1 tomÞ ð6Þ

This is a heuristic approach connected to the goodness of the statistical fit. Higher thesquared error, lower the fit and confidence.

3.3. Uncertainty in damage quantification

As mentioned earlier in Section 2.3, the damage quantification procedure is analogous to aparameter estimation problem in non-linear regression. Consider the model in Equation (3).A least-squares estimate for y was obtained in Section 2.3, and the least-squares error wasestimated as SðyÞ in Equation (4). To calculate the confidence bounds on y, the first step is to

Table II. Probabilities of making decisions.

Actual state of system

Hypothesis testing H0 (no damage) H1 (damage)

Decision H0 (no damage) 1�b (no error) b (Type-II error)H1 (damage) a (Type-I error) 1�a (no error)



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assign bounds on the squared error S(y) as shown in Equation (7). The values of e thatcorrespond to this error define the confidence bounds on y:

SðyÞpSð~yÞ 11p

n� pF a

p;N�p

� �ð7Þ

In Equation (7), N denotes the number of observations over which the least-squares error wasestimated and p denotes the length of the vector y. In this case of single-fault assumption, p5 1.F a

p;N�p denotes the F-statistic, with estimation (numerator) degrees of freedom p and residual(denominator) degrees of freedom N�p at significance level a. The bounds (at a-significancelevel) for y� are schematically shown in Figure 3.

The bounds on y, i.e. [ymin, ymax], are calculated using an optimization procedure as follows.For a given significance level a, a target error value Starget is defined in Equation (8) as:

StargetðaÞ ¼ SðyÞ 11p

n� pF a

p;N�p

� �ð8Þ

The upper and lower bounds for y satisfying Equation (8) are estimated through theoptimization in Equation (9).

Minimize=Minimize y

s:t: SðyÞ ¼ Starget

ð9Þ

These optimization problems are functionally constrained and hence might becomputationally expensive for complicated systems. To simplify the computation, these arerewritten as shown in Equations (10) and (11).

Minimize ðSðyÞ � StargetÞ2

s:t: yoyðto solve for yminÞ ð10Þ

Minimize ðSðyÞ � StargetÞ2

s:t: yoyðto solve for ymaxÞ ð11Þ

These optimization problems are functionally unconstrained, one-dimensional and boundedin one direction. Hence, techniques such as golden section search and parabolic interpolationcan be used for quicker solutions. These methods don’t use derivatives and hence arecomputationally efficient as well. The following section extends this concept to obtain the entireprobability distribution of the damage parameter.

3.4. Probability distribution of damage parameter

The method explained in Section 3.3 can be used to obtain confidence bounds for the damageparameter. For example, a5 0.05 would yield a 95% confidence interval for y� whose boundswould correspond to cumulative probabilities of 0.025 and 0.975 of the ‘true’ value, y�. In general,

S ( )

S ( )

Starget ( )

*minmin

^

^

Figure 3. Estimation of confidence bounds.



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a value of a would lead to a (1�a) % confidence interval, whose bounds would correspond to thea/2 and 1�a/2 cumulative probabilities of the ‘true’ value y� (Hereon, this would be referred to asthe damage parameter y). Thus, by repeating the procedure in Section 3.3 for different values of a,it is possible to obtain the entire cumulative probability distribution of the damage parameter y.

However, this probability distribution is conditioned on two events. The first event is thedetection of damage, whose uncertainty was quantified in Section 3.1. The second event is thelocalization of damage, whose uncertainty was quantified in Section 3.2. Thus, this conditionalprobability distribution of y given that the damage is ‘true’ and given that y is the ‘true’ damageparameter is denoted as Fy (y|Damage is true, Damage is in y). Using the uncertaintiesquantified in Section 3.2 and Section 3.3, the unconditional distribution of the damageparameter can be obtained. This is discussed in the next section.

3.5. Overall uncertainty in diagnosis

This section develops the method to obtain the unconditional distribution of y, i.e. overalluncertainty in diagnosis.

Let Pd denote the confidence in damage detection, as calculated in Section 3.1. This is equalto the probability that there is damage. Let Pi denote the confidence in isolation, as calculated inSection 3.2. This is equal to the probability that the damage is in y, given that there is damage.To calculate the unconditional distribution of y, the various possible scenarios that lead todamage detection and isolation need to be identified.

(i) The damage detection system might have triggered a false alarm. The probability forthis event to happen is equal to (1 � Pd). In this case, the distribution of the damageparameter y remains the same as the nominal (healthy) distribution, Fy (yhealthy).

(ii) The damage detection might be correct but the isolation might be wrong, and ymight notbe the real candidate. This event has a probability to Pd

�(1 � Pi). In this case also, thedistribution of y� remains the same as the nominal distribution, because y is not thedamage parameter.

(iii) Both damage detection and isolation might be correct. This event has a probability Pd�

Pi. In this case, the distribution of the damage parameter y is what was calculated inSection 3.4. This was denoted by Fy (y | Damage is true, Damage is in y).

Using the theorem of total probability [25], the unconditional probability of y� can beevaluated as in Equation (12), where F(.) represents the cumulative probability distribution.

FyðyÞ ¼ ð1� Pd Þ � FyðyhealthyÞ1Pd � ð1� PiÞ � FyðyhealthyÞ

1Pd � Pi � FyðyjDamage is true;Damage is in yÞ ð12Þ

The expression in Equation (12) quantifies the overall uncertainty in the diagnosis procedure,by calculating the unconditional cumulative distribution of the damage parameter. Further, thecorresponding probability density function can also be calculated by differentiating theexpression in Equation (12).

The following sections illustrate the proposed methods through two numerical examples, astructural frame and a hydraulic actuation system.

4. ILLUSTRATION USING A TWO-STORY FRAME

Consider the two-story frame discussed in Section 2. This system has six parameters, m1, m2, k1,k2, D1 and D2. The masses of the first and second floors are assumed not to change. Hence, thereare only four parameters that could be affected by damage. A typical degradation of the systemwould be a decrease in the stiffness and/or damping values. The bond graph model of thissystem and the TCG were derived by Moustafa et al. [10]. The damage signatures derived fromthis TCG were given earlier in Table I.



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It is evident that some candidate damage parameters have unique signatures and somecandidates don’t. For example, a bias in the first floor sensor has unique damage signatures andhence can be isolated with a 100% confidence. However, the stiffness and damping at a particularfloor share the same set of damage signatures and hence, it is not possible to isolate between themqualitatively. However, it is possible to localize the damage to a particular floor, either the firstfloor (k1, D1) or the second floor (k2, D2). These observations are presented in Table III.

Thus, using the damage signatures, it is possible to identify the floor that is damaged. Hence,it is sufficient to consider the equations governing the motion of that particular floor alone.Using the dynamics of motion of this floor, Equation (3) can be rewritten as a differentialequation. The damage quantification scheme and uncertainty quantification procedures arecarried out as explained in Section 2 and Section 3. These are illustrated in this section bysimulating damage in the structural frame.

The various parameters used in the system are listed in Table IV. The stiffness parameters ofeach floor are represented by k1 and k2, the damping parameters are represented by D1 and D2,the mass of each floor is represented by m1 and m2 and the loading at each level is represented byF1(t) and F2(t). The system is measured at every 0.01 s, hence making 100 observations persecond. Damage is triggered at 5 s.

4.1. Damage diagnosis—an illustration

The value of k1 is reduced by 10% at 5 s. This means a change from 30 700N/m to 27 630N/m.The story displacements are observed using sensors. Sensor noise is introduced by addingGaussian white noise (ranging from 1–5%) to the simulated signal. Damage is detected with aconfidence of 95%, hence Pd 5 0.95. The results of damage localization, as given in Table III,suggest that either k1 or D1 could be the true damage parameter. The next step would be to carryout parameter estimation as in Section 2.3.

Then, the confidence in isolation is determined as explained in Section 3.2. The sameparameter estimation scheme is run, assuming D1 is the damage parameter and the final estimateof D1 5 0. It is observed that the least-squares approach performed on the wrong damageparameter usually gives a physically meaningless result. Further, the level of confidence can beassociated with k1 being the damage parameter and D1 being the damage parameter arecalculated as in Section 3.2. These values are 99.94 and 0.06%, respectively. This corresponds toPi 5 0.9994.

Table III. Qualitative damage isolation results.

Actual damage Qualitative isolation results

Decrease in k1 Decrease in k1 or decrease in D1

Decrease in k2 Decrease in k2 or decrease in D2

Decrease in D1 Decrease in k1 or decrease in D1

Decrease in D2 Decrease in k2 or decrease in D2

First floor sensor (1 bias) First floor sensor (1 bias)Second floor sensor (1 bias) Second floor sensor (1 bias)

Table IV. Two-story frame: parameter values.

Parameter Value

k1 30 700N/mk2 44 300N/mD1 307Ns/mD2 443Ns/mm1 136Ns2/mm2 66Ns2/mF1(t) 75 sin(9t) NF2(t) 100 sin(9t) N



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The 90% confidence bounds are estimated to be 27 232N/m and 27 876N/m. Approximately20 time steps of measurements were required for convergence. This is shown in Figure 4.

Having obtained the least-squares estimate, the method in Section 3.4 is used to calculate theentire distribution of k1. The cumulative distribution function of k1 is numerically differentiatedto calculate the probability density function, as shown in Figure 5.

The least-squares estimate and the true (simulated) damage are also shown in Figure 7. Theleast-squares estimate (27 554N/m) corresponds to the mode of the distribution. Due to variouserrors, this least-squares result underestimates the true value (276 30N/m) by 0.28%. Further, atriangular-like distribution is obtained for the damage parameter k1. This may be due to the factthat the system (two-story frame) is a linear system and hence the confidence bounds increaselinearly with the confidence level a.

The next step is to calculate the unconditional probability distribution of k1 (overalluncertainty in diagnosis) using the method in Section 3.5. For the sake of simplicity, the healthyvalue of k1 is considered deterministic and hence, the expression Fy(yhealthy) in Equation (12) isequal to one. The resulting probability distribution is shown in Figure 6.

While differentiating Equation (12) to calculate the probability density function of k1, the firsttwo terms are zero and the last term alone needs to be evaluated. This is equivalent to multiplying

Figure 4. Convergence study on stiffness (k1) at 5% noise level.

Least Squares Estimate True Fault

Figure 5. Conditional probability density of stiffness (k1).



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the conditional cumulative probability distribution by Pd� Pi and therefore, the unconditional

cumulative probability at k1 5 28 000N/m is less than unity. Hence, the unconditionalprobability density function k1 5 28 000N/m is finite and not equal to zero. However, thecumulative probability at the end of the distribution should be equal to one and hence, theprobability that k1428 000N/m (approximately) is equal to zero. Hence the probability densityat k1 5 28 000N/m shows a jump from a finite density to zero, as shown in Figure 6.

4.2. Studies on noise levels

This section investigates the effect of noise level in measurements on the amount of timerequired for damage quantification and the level of confidence in the diagnosis results.

However, it is essential to observe that the results below only pertain to the current system ofstudy, i.e. a two-story frame. With the increase in complexity of the system and with increase incharacteristics of dynamics involved, these results may vary from system to system.

Damage in k1 is simulated at 5 s. The value reduces from 30700N/m to 27630N/m, a reductionof 10%. The effect of noise on the accuracy of the parameter estimation result is shown in Table V.

A similar analysis has been carried out for each of the damage scenarios and the results suggestthat as the noise increases, (1) more measurements are required for desired accuracy, (2) theaccuracy (measured by the deviation from the true amount of damage) of the least-squares estimatedecreases, (3) the confidence in isolation decreases and (4) the confidence bounds become wider.

4.3. Extension to multi-story frames

As explained in Section 2, the bond graph for a two-story frame can be easily extended to multi-story frames. For example, a five-story frame has 10 parameters, two for each story—stiffness

Figure 7. Actuation system.

True Fault Least Squares Estimate

Figure 6. Unconditional probability density of stiffness k1.



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and damping co-efficient. The measurements made are the displacements at the five differentfloors. The TCG is able to localize the damage to the respective story once damage is detected.Further, a quantitative isolation needs to be performed to identify if the damage parameter isthe stiffness or the damping factor.

Different damage scenarios were considered for the five-story frame, and these were detectedand isolated (localized) satisfactorily with the proposed method. The damage was quantifiedwith high accuracy. Uncertainty quantification associated with each of these three processes wasalso successful and revealed narrow bounds in the estimated parameters, thus providing highconfidence in the diagnosis. The easy extension to multi-story frames clearly indicates that thisdiagnosis methodology is applicable to practical structures.

5. HYDRAULIC ACTUATION SYSTEM

The developed approach for structural health monitoring is now illustrated for a hydraulicactuation system. The Pump-Actuator-Load system shown in Figure 7 is modeled using bondgraphs to aid in damage isolation. The system is studied in detail by Sankararaman et al. [26]and only the data and results pertaining to diagnosis uncertainty quantification are provided inthis section.

A pump (powered by a voltage source V) drives a hydraulic actuator. The pump fills the firstchamber of the actuator with fluid, and the increase in pressure causes the piston to move. As aresult, the fluid in the second chamber is also compressed and the excess fluid flows out into thereservoir. The piston is connected to the load through a rigid rod and hence, the movement ofthe piston guides the load. The bond graph model of this system is constructed and is shown inFigure 8. This is used to derive the cause–effect relationships between the various parameters ofthe entire model and hence, the TCG of the pump actuation system.

The model shown in Figure 7 consists of three components: a pump, an actuator chamberand a load (control surface). Individual bond graph models were constructed for each of thesecomponents, viz. pump [27], actuator and control surface. As bond graphs model the flow ofenergy, the three models are easily connected to represent the entire system. The varioussymbols in the bond graph model are explained in Table VI.

Six different fault candidates are considered: control surface damping (1), control surfacestiffness (�), valve resistance (1), mass of motor assembly (�), area of pump vanes (�) andresistance of pump (1). Both abrupt faults (the value of the parameter changes abruptly) andincipient faults (the value of the parameter increases/decreases slowly and steadily) can bediagnosed using the proposed approach. For example, the area of the vanes could eitherdecrease suddenly by chipping (abrupt damage) or could degrade over time due to corrosion(incipient damage). In the case of abrupt faults, the fault parameter is estimated directly; in thecase of incipient faults, the parameters (such as the rate of material removal) that govern theevolution of the damage with time are estimated.

Five measurements are made from the system. These are pressure in the first actuation chamber,pressure in the second actuation chamber, speed of mass M, pump output pressure and speed ofpump vanes.

The damage signatures corresponding to these set of fault candidates and measurements areshown in Table VII.

Table V. Damage quantification for a 10% decrease in k1 (30 700N/m-27 630N/m).

Noise level inmeasurements (%)

No. of measurementsafter detectionof damage

Least squaresestimate(N/m)

Confidencein isolation

(%)

90% lowerbound(N/m)

90%upper bound

(N/m)

1 12 27 645 99.8 27 568 27 7052 16 27 690 99.3 27 435 27 7805 20 27 554 91.2 27 232 27 876



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Several faults were simulated in the system at t5 150 s and were diagnosed successfully. Theuncertainties in damage detection, isolation and quantification were also quantified.

Table VIII contains the results of detection and isolation. The first column states the damageparameter and the second indicates the amount of damage (ratio of increase) simulated. Thetime taken for detection, the symbols generated and the time of isolation are also indicated.

1 01 1 0 Se

Rv C1

C2

TF 1

M

B

0 C=1/k

f

Vel. Disp.

0

R2

Actuation Subsystem

Load Subsystem

f

1 G1 1 0V

m1 R m2

f

Pump connected to the Actuator

Pump Subsystem

Figure 8. Bond graph model of the pump actuation system.

Table VI. Parameters of the pump actuation system.

Symbol Physical parameter

V Voltage sourcem1 Mass of motorm2 Mass of vanes (Pump)A Area of vanes (Pump)R Resistance of pumpRv Valve resistanceC1 Capacity of first chamberC2 Capacity of second chamberM Mass (control surface)B Damping (control surface)K Stiffness (control surface)

Table VII. List of damage signatures for the actuation system.

FaultPressure in first

cylinderPressure in second

cylinderSpeed ofload mass

Pump outputressure Vane speed

B (�) 0� 01 01 0� 0�C5 1/k (1) 0� 01 01 0� 0�Rv (1) 0� 0� 0� 01 01m1 (�) 01 01 01 1� 01R (1) 01 01 01 01 0�A (�) 01 01 �1 �1 01



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Damage detection employed a statistical hypothesis test at 5% significance level. Hence, theconfidence in detection is Pd 5 0.95. The confidence in isolation is shown in parentheses in thelast column of Table VIII.

Table IX describes the results of damage quantification and the associated uncertainty. Forthe sake of completeness, the first three columns are also given from Table VIII. Further, thetime taken for damage quantification is also mentioned.

In each case, the entire probability distribution of the damage parameter was calculated usingthe least-squares approach and the overall uncertainty in diagnosis was also quantified. For thesake of illustration, these results are presented for only one fault scenario, increase in resistance(R) of the pump. The conditional and unconditional probability distributions of the damageparameter are plotted in Figures 9 and 10, respectively.

Furthermore, a study on noise levels, similar to that on structural frames, was alsoconducted. It was verified that the confidence interval for parameter estimation was narrower atlower noise levels. As observed in the case of frames, the confidence bounds grew in range withthe amount of noise in measurements.

6. CONCLUDING REMARKS

This study developed methods to quantify the uncertainty involved in damage detection,damage isolation and damage quantification for model-based online diagnosis with continuousmonitoring. In this study, bond graph models were used to characterize these cause–effect

Table VIII. Detection and isolation results.

Faultparameter

Faultmagnitudesimulated(ratio ofincrease)

Detectedafter (s)

Isolation

Symbols Time (s) C andidate list

R 6 1 Vane speed (�) 2 Pump resistance (1) (99%)Pressure in first cylinder (1) Valve resistance (�) (1%)

Rv 5 0.5 Pressure in first cylinder (�) 3.5 Pump resistance (�) (1%)Pressure in second cylinder (�) Valve resistance (1) (99%)Pressure in first cylinder (�)

C5 1/k 3 0.5 Pressure in second cylinder (1) 3.5 Load stiffness (1) (99.9%)Pressure in first cylinder (�) Load damping (�) (0.1%)Pressure in second cylinder (1)

A �0.20 0.5 Load velocity (�) 2 Area of Vanes (�) (100%)Load velocity (1)

Table IX. Parameter estimation results.

Parameter estimation

Faultparameter

Fault magnitudesimulated

(ratio of increase)Detectedafter (s)

Completeat (s)

LS estimate 90% bounds

(ratio of increase)

R 6 1 10 6.08 f4:75; 7:28g

Rv 5 0.5 10.5 5.26 f4:35; 6:53g

C5 1/k 3 0.5 10.5 2.99 f2:93; 3:06g

A �0.20 0.5 10.5 �20.05 f�0:19; 0:20g



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relationships. However, the methods for uncertainty quantification developed in this study areapplicable to any model-based technique that can assist in qualitative isolation of the damageparameter. Damage isolation is qualitative and rapid, based on damage signatures derived usingthe cause–effect relationships between the model parameters and system outputs. Damagequantification is performed through a least-squares technique. However, the computationalexpense is significantly reduced by estimating only those parameters that were isolatedqualitatively. The assumption of single faults reduces damage quantification to a set of one-dimensional least-squares estimation problems, which can be parallelized, resulting in furtherreduction of computational time.

Nayeri et al. [28] explain that practical engineering applications have several systemparameters that may drift due to changes in environmental conditions, temperature, etc., and amultivariate approach would be required to apply the proposed uncertainty quantificationmethods to such problems. Daigle [29] developed methods for damage detection and qualitativedamage isolation of multiple simultaneous faults and the uncertainty quantification methodsproposed in this study will be extended to such multiple fault scenarios in future. The scope ofthis study is limited to the quantification of uncertainty in the detection, isolation andquantification of single faults only.

As detection is based on statistical hypothesis testing, and the uncertainty in detection isquantified based on the significance level of testing. When several candidate damage parameters

True Fault Least Square Estimate

Figure 10. Unconditional probability density of valve resistance (R).

Least Square Estimate True Fault

Figure 9. Conditional probability density of valve resistance (R).



DOI: 10.1002/stc

have the same signature, causing ambiguity in damage isolation, the proportion of sum ofsquares error is used to quantify the uncertainty in isolation. The uncertainty in damagequantification is estimated through the F-statistic used in regression analysis. Finally, the overalluncertainty in diagnosis is calculated through the use of conditional probabilities and thetheorem of total probability. Such estimation of uncertainty helps to measure the confidence indiagnosis, and provides valuable information for decision making and risk management.

ACKNOWLEDGEMENTS

This study is partly supported by funds from the US Air Force Research Laboratory through subcontractto General Dynamics Information Technology (Contract No. USAF-0060-43-0001, Project Monitor:Mark Derriso). The support is gratefully acknowledged.

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uncertainty quantification in structural damage diagnosis

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