2.1.1 Model‐Based Prognostic Approaches

Model‐based prognostic approaches include the modeling and use of expressions related to reliability, probability, and physics of failure (PoF) models. Such models are used to study and compare, for example, the relationships of materials, manufacturing, and utilization of the reliability, robustness, and strength of a product, often in structured, designed, controlled experiments and life tests. Such modeling offers potentially good accuracy, but it is difficult to apply and use in complex, fielded systems (Speaks 2005). Those models include distributions and probability models; fundamentals of reliability theory; and models based on reliability testing, such as acceleration factors (AFs), presented in Chapter 1. Other methods will be discussed later.

2.1.2 Data‐Driven Prognostic Approaches

Data‐driven prognostic approaches include statistical and machine learning (ML) methods; are generally simpler to apply, compared to model‐based prognostic approaches; but can produce less precision and less accuracy in prognostic estimations. Statistical methods include both parametric and nonparametric models, such as those shown in (Ross 1987) and (Hollander et al. 2014); and K‐nearest neighbor (KNN), a nonparametric method for classification or regression of an object with respect to its neighbors (Medjaher and Zerhouni 2013). Machine learning includes examples such as linear discriminant analysis (LDA) to characterize or separate multiple objects, hidden Markov modeling (HMM) to model a system having hidden states, and principal component analysis (PCA) to convert observations into linearly uncorrelated variables.

2.1.3 Hybrid Prognostic Approaches

Hybrid approaches employ both model‐driven and data‐driven approaches to further improve the accuracy of and/or to better understand the relationships of parameters and objects (Medjaher and Zerhouni 2013 ). Drawbacks are increased computational processing and complexity (see Figure 2.2). We will return to these and other methods later in this chapter.

2.1.4 Chapter Objectives

The objectives of this chapter are to present classical methodologies to support prognostics for PHM and then to present an approach to condition‐based maintenance (CBM) for PHM: an approach based on CBD signatures that lays the foundation for Chapter 3.

2.1.5 Chapter Organization

The remainder of this chapter is organized to present and discuss classical approaches to modeling to support prognostics for PHM and to introduce our approach to CBM:

• 2.2 Model‐Based Prognostics

  This section presents approaches to model‐based prognostics, including topics on analytical modeling, distribution modeling, PoF and reliability modeling, acceleration factors, complexity related to reliability modeling, failure distribution, failure rate and failures in time, and advantages and disadvantages of model‐based prognostics.

• 2.3 Data‐Driven Prognostics

  This section presents approaches to data‐driven prognostics including topics on statistical methods and machine learning – classification and clustering.

• 2.4 Hybrid‐Driven Prognostics

  This section presents approaches to hybrid‐driven prognostics: model‐based combined with data‐driven prognostics.

• 2.5 An Approach to Condition‐Based Maintenance (CBM)

  This section presents an approach to CBM, including topics on modeling CBD signatures, comparing life consumption and PoF, and CBD signature methodologies. An illustration of CBD‐signature modeling is included.

• 2.6 Approaches to PHM: Summary

  This section summarizes the material presented in this chapter.

2.2.1 Analytical Modeling

Analytical models, also referred to as physical models, employ load parameters such as those shown in Table 2.1 to estimate how a particular prognostic target in a system changes from a state of 100% healthy (not damaged) to zero health (failed) as damage accumulates (Pecht 2008 ; Hofmeister et al. 2016, 2017 ; Vichare 2006; Vichare et al. 2007). The PHM system performs health monitoring, detects an unhealthy condition, and uses, for example, fault‐tree or state‐diagram analysis to identify and determine the location(s) of the most likely prognostic target(s) causing the fault. Analytical approaches can be divided into two major groups: inductive and deductive approach.

2.2.2 Distribution Modeling

When many physical causes and/or complex reactions result in failure, a distribution model is often used instead of a physical model. Engineering experience can relate special distribution types to given failure types. Examples of distribution models and associated applications include those shown in Table 2.2 (Medjaher and Zerhouni 2013 ; Viswanadham and Singh 1998 ; Hofmeister et al. 2006, 2013; Hofmeister and Vohnout 2011; Silverman and Hofmeister 2012).

The following distributions form characteristic curves that are useful for modeling complex failure mechanisms (a more complete list can be found in Chapter 1):

In engineering applications, the Weibull, an extreme‐value type of distribution, and the lognormal distribution are frequently used for modeling because they can be fitted to data from a large number of applications: especially lifetime distributions where failures are bounded below zero (Xu et al. 2015). The versatility of the Weibull distribution is evidenced by the following example life‐time applications and by the example Weibull plots in Figure 2.8:

• Maximum flood levels of rivers; maximum wind gusts; steam boiler pressure
• Fatigue life of bearings; blending time of powder coating materials
• Tensile strength of wires; yield strength of beams
• Lifetimes of passive electronic components, such as resistors and capacitors
• Time‐dependent dielectric breakdown (TDDB) phenomena; count of detectable high‐resistance events in solder joints

2.2.3 Physics of Failure (PoF) and Reliability Modeling

PoF used in reliability modeling and simulation is significantly different from the constant‐failure rate (CFR) modeling (based on exponential distribution) used as the basis for the Military Handbook 217 series (MIL‐HDBK‐217C). The PoF approach has dominated since the 1980s: root causes of failure, such as fatigue, fracture, wear, and corrosion, are studied and corrected to achieve lifetime design requirements by designing out causes of wear‐out failures in components. Such modeling is used to study system performance and reduce failures. The following is a summary of a PoF approach (Weisstein 2015):

1. Identify potential failure modes.
2. Design and perform highly accelerated life tests (HALTs) or highly accelerated stress tests (HASTs) to verify and select dominant failure modes leading to failure.
3. Model the failure mode, and fit data to the model using statistical distributions.
4. Develop an equation for a PoF model to calculate mean time to failure (MTTF) and/or mean time before failure (MTBF).
5. Develop design, materials, and manufacturing improvements to increase MTTF to meet requirements.

The benefits of a PoF approach include the following:

• Reliability is designed in.
• Failures are reduced or eliminated prior to testing.
• Reliability of fielded systems is increased.
• Operational costs are decreased.

The reliability of a system can be obtained from the reliability of its building blocks.

• Assume first that the blocks have a series combination, meaning the system works if all blocks are in working condition. Let n be the number of blocks, and let R₁(t), …, R_n(t) be the reliability functions of the blocks. Then that of the entire system can be obtained as
  

  where X₁, …, X_n denote the failure times of the blocks. Assuming that the blocks are independent, then

  2.3

  showing that the reliability functions are multiplied. If a new block is added to the system, then a new factor is added to the product, which is less than one, so the system reliability decreases.

• Assume next that the connection is parallel, meaning the system works if at least one of the blocks works. In this case
  

If a new block is added to the system, then the second term decreases, so R(t) increases.

In many systems, the connection of the blocks is a combination of series and parallel combinations. Then Eqs. (2.3) and (2.4) are used repeatedly, as shown in the following example.

2.2.4 Acceleration Factor (AF)

Chapter 1 discussed how TTF distributions change when an item is subject to extreme stresses. These formulas can be used backward, when observations are made under extreme conditions, to estimate failure times: based on the data, distributions of TTFs can be estimated under normal conditions. This idea is known as accelerated testing (AT), ALT, and so on.

Reliability projections, such as MTTF based on accelerated testing (such as a HALT) when the object is subject to extreme stress, are projected estimates from test results to a future, slower rate of failure at lower levels of stress under normal use conditions. Such projections assume the use of a correct model for life distribution and the use of a correct acceleration model (Tobias 2003; Nelson 2004). In an acceleration model, the TTF (or t_F) at a stress level (s) is given by the following (Kentved and Schmidt 2012):

2.5

where A = constant, G(s) = stress function.

The acceleration factor (AF) is the ratio of the TFFs at two different levels of stress:

2.6

AFs used in reliability estimations (or usage modeling) include the following (White and Bernstein 2008).

Arrhenius Law for Temperature

AFs used in reliability estimations (or usage modeling) include the following (White and Bernstein 2008 ):

2.7

where

Tuse =	Product temperature in service use
Ttest =	Product temperature in laboratory test
Ea =	Activation energy for damage mechanism and material
k =	Bolzman's constant = 8.617 * 10−5eV/°K

Kemeny Law for Voltage

2.8

where in addition

	Material constants
Tj =	Junction temperature
Vcb =	Collector‐base voltage
Vcbmax =	Maximum collector‐base voltage before breakdown

Peck's Law for Temperature and Humidity

2.9

where in addition

Muse =	Moisture level in service use
Mtest =	Moisture level in test

This formula can be derived from the temperature‐humidity relationship discussed in Chapter 1.

Coffin‐Manson Law of Fatigue

2.10

where

ΔT =	Difference between the high and low temperatures for the product in service use and in the laboratory test.

Notice that (2.10) is derived from the inverse power law.

Eyring Formula

2.11

Similar formulas can be derived from any other known rules dealing with extreme stress levels such as the generalized Eyring, temperature non‐thermal relations, or the general log‐linear law discussed in Chapter 1.

2.2.5 Complexity Related to Reliability Modeling

In addition to requiring the use of a correct distribution model and a correct acceleration model, reliability modeling is further complicated by an almost infinite number of methods, like those in Table 2.3. Each has advantages and disadvantages in comparison to the others for a given application.

Reliability modeling is even more complicated because of the variability in the fitting of values used within the models of each version of a procedure. This variance is evidenced by the modeling examples of the AFs for temperature as shown in Table 2.4.

In addition to differences in acceleration factors and parameter values used in modeling, there are differing versions of distribution models. Examples include three different Coffin‐Manson models for calculating a probable test cycle (N_f) that a solder joint fails when subjected to cyclic loading of temperature during accelerated testing (Viswanadham and Singh 1998 ):

2.12

2.13

2.14

where subscript u refers to a use value, subscript t refers to a test value, parameter T is in °K, and f_u is a model parameter.

Now, suppose a prognostic target, such as an FPGA attached to a fiber‐resin (FR‐4) printed wire board (PWB), which is formerly and sometimes still referred to as printed circuit board (PCB), is operated such that the temperature varies in any given 24‐hour period of time from less than −40 °C to over 100 °C, with different temperature ramp‐up rates, different dwell times at high temperature, different ramp‐down rates, and different dwell times at low temperature. Also suppose that during any given 24‐hour period of time, the PWB is subjected to different rates of different magnitudes of vibration and shock (such as might be experienced from being mounted in an engine compartment of a vehicle). Further, suppose that the PWB comprises over a dozen different FPGAs, some of which use standard PbSn solder, some of which use lead‐free solder balls, some of which use plastic grid array (PGA) die packages with and without staking, and some of which use ceramic‐column grid array (CCGA); all are mounted at different distances from centers of maximum stress‐strain. Estimating with a high degree of accuracy when the primary clock‐input pin of a specific FPGA will fail and thereby cause the PWB to fail becomes a daunting task (Javed 2014).

2.2.6 Failure Distribution

As the independent variables change in distribution models, the rate of change of a given curve varies, which creates a family of failure curves having a failure distribution with a TTF as illustrated in Figure 2.11. The failure distribution is a probability density function (PDF), and the TTF is the expectation or the 0.50 value of the cumulative distribution function (CDF) of that PDF: note that TTF is not the same entity as MTTF – more on this topic is presented in Chapter 7.

2.2.7 Multiple Modes of Failure: Failure Rate and FIT

When prognostic targets are prone to more than one dominant failure mode, multiple distribution models and/or multiple‐parameter models must be used in a PoF approach. For example, a transistor device could fail because of temperature cycling and also because of failure related to high levels of voltage. Two distribution models might then apply: Arrhenius temperature, Eq. (2.7); and Kemeny distribution, Eq. (2.8). A simplifying approach is to assume that all failures are random and all failure modes are equally dominant. Overall MTTF and failure in time (FIT) values can be calculated by applying a sum‐of‐failure‐rate model and an improved AF to account for two different temperatures, use and test, as shown in Eq. (2.9). One FIT is equal to one failure in 1 billion part hours (White and Bernstein 2008 ):

2.15

2.2.8 Advantages and Disadvantages of Model‐Based Prognostics

Advantages of model‐based prognostics are many and include the following: (i) such modeling leads to a better understanding of how prognostic targets, especially devices, fail because of defects and weaknesses in manufacturing processes and materials, and how and why they fail because of loading and environmental stresses and strain; (ii) manufacturing processes, materials, electrical and physical designs, and control of operational loading and environmental conditions can be improved to increase reliability; and (iii) simple estimates of state of health (SoH) and remaining useful life (RUL) are possible.

Disadvantages of model‐based prognostics are also many and include the following: (i) modeling for other than single‐mode failures is complex; (ii) simple models for non‐steady state and multiple and variable environment loading generally do not exist; (iii) modeling of large, complex systems of hundreds or thousands of different parts becomes extremely difficult, if not computationally intractable; and, perhaps most important, (iv) model‐based approaches are not applicable to a specific prognostic target in a system, as exemplified by Figure 2.11 : MTTF, for example, applies to a population of like prognostic targets rather than a fielded, specific prognostic target in a specific system in an operational, non‐test environment.

2.3.1 Statistical Methods

Statistical methods can be divided into parametric and nonparametric methods (Pecht 2008 ), including those shown in Table 2.5.

Maximum Likelihood Method

The maximum likelihood method (Ross 1987 ) is a common procedure to estimate unknown parameters of probability distributions. As in Section 1.4, let f(t|θ) denote the PDF where θ is unknown. Assume we have a random sample t₁, t₂, …, t_N from this distribution. The likelihood function is defined as

2.16

which represents the probability of the sampling event that actually occurred. Since the logarithmic function strictly increases, instead of L(θ), its logarithm is maximized:

2.17

and the optional θ value is accepted as the estimate of the unknown parameter.

Likelihood Ratio Test

The likelihood ratio test (Casella and Berger 2002) is usually used to determine the validity of an estimate. For example, let θ₀ be an estimate of an unknown parameter of a PDF f(t|θ). The likelihood ratio test is based on the likelihood ratio

2.19

where the likelihood function is denoted by L(θ) and the denominator is the maximal value of L(θ). In other cases, two estimates are compared. Let θ₁ and θ₂ be two estimates for θ; then

2.20

If the value of r is small, then in the first case θ₀ is unacceptable, and in the second case θ₂ is a much better estimate.

Minimum Mean Square Error

Minimum mean square error is mainly used in fitting function forms with unknown parameters. For example, a density histogram is obtained from a sample with points (t₁, f₁), (t₂, f₂), …, (t_N, f_N), and it is known that the corresponding PDF is f(t|θ). The least square estimate of θ is obtained by minimizing the overall squared error:

2.22

Maximum a Posteriori Estimation

The maximum a posteriori estimation (Stein et al. 2002) is based on Bayesian principles. Consider again a PDF f(t|θ) depending on the unknown parameter θ. The likelihood function is given as L(θ), which is maximized in order to get the best estimate for θ. In practical cases, usually its logarithm is maximized. Assume now that a prior distribution is known for θ, with PDF g(θ). By the Theorem of Bayes

2.24

where the integration domain is the domain of all possible values of θ. Since the denominator is independent of θ, we need to optimize only L(θ)g(θ).

K‐Nearest Neighbor Classifier

The KNN classifier (Cover and Hart 1967) is based on the following simple procedure. Assume we have N vectors; each of them is attached with a class label. We need to put a given vector into the most appropriate class. We select a positive integer k ≧ 1 and determine the k closest vectors from the given N vectors by using any distance measure, such as the Euclidean distance. Then the selected k closest vectors “vote” about the class by selecting the class that appears the most times among the k closest vectors. In this way, we can order any set of vectors into given classes based on a given sample.

In a two‐dimensional case, the Euclidean distance of vectors (x_c, y_c) and (x_i, y_i) is the following:

2.26

The Euclidean distance is based on the Pythagorean theorem. If the dimension is larger than two, say n, then similarly

2.27

where the components of vector x_c are and those of vector x_i are .

Kernel Density Estimation

This method is based on the following equation:

2.28

which can be interpreted as follows (Wand and Jones 1995). Let x₁, x₂, …, x_N be the sample elements. A kernel function K(x) is selected that is nonnegative and that has an integral of one (like the properties of any PDF). A parameter h > 0 is also chosen, which is called the bandwith. Then Eq. (2.28) gives an estimate of the PDF, from which the sample is generated, at point x. Table 2.6 gives a collection of commonly used kernel functions.

Kernel	K(x)	Domain
Uniform		(−1, 1)
Triangle	1 −  ∣x∣	(−1, 1)
Epanechnikov		(−1, 1)
Quartic		(−1, 1)
Twiweight		(−1, 1)
Gaussian		(−∞, ∞)
Cosinus		(−1, 1)

Chi‐Square Test

The chi‐square test (Ross 1987 ) is used to test whether a given sample comes from a population with a specific distribution, and therefore it does not provide the distribution; it can only be used to check whether a user‐selected distribution is appropriate. The data is divided into K bins, with y_k and y_k + 1 being the lower and upper limits of class k. It is assumed that the bins are defined by subintervals between the consecutive nodes y₀ < y₁ < … < y_K. Let O_k be the observed frequency for bin k, and E_k; then the expected frequency is defined as

where N is the total number of sample elements and F(y) is a CDF. The chi‐square test computes the value of χ² as

2.29

and the distribution F(y) is rejected if

where α is a user‐selected significance level, and c is the number of the unknown parameters. The threshold can be found in the chi‐square test tables.

2.3.2 Machine Learning (ML): Classification and Clustering

Machine learning (ML), a form of artificial intelligence, predicts future behavior by learning from the past: classification and clustering are forms of ML divided into supervised and unsupervised techniques, which are further divided into discriminative and generative approaches. Certain ML approaches, such as regression and ranking, are less useful compared to classification and clustering, which use computational and statistical methods to extract information from data (Pecht 2008 ). Table 2.7 is a summary list of some of the ML techniques.

These techniques are well presented in the literature; therefore we do not discuss them in detail. Instead, we will select one method from each category. The other methods are briefly described as examples.

Neural Networks

Neural networks are database input‐output relations (Faussett 1994). They are “connectionist” computer systems. Let the input vector be denoted as x = (x₁, x₂, …, x_m) and the output vector as y = (y₁, y₂, …, x_n). The transformation x → y is performed in several stages. The initial nodes of the network are the input variables, the final (terminal) nodes are the output variables, and the different stages of the transformation are represented by hidden layers including the hidden nodes. Figure 2.13 shows a neural network structure with three input, three output, and two hidden nodes.

The first step is to transform the input and output variables into the same order of magnitude. The hidden variables are linear combinations of the transformed inputs as

2.30

where f() represents the transform function of the variables x. In many applications

2.31

is selected, since it is strictly increasing, f(−∞) = 0, and f(∞) = 1; that is, the input values are transformed into the unit interval (0, 1). The transformed output variables are also linear combinations of the hidden variables:

2.32

In Eqs. (2.30) and (2.32) the coefficients w_ij and are the unknowns, and their values are determined so that the resulting input/output relation

has the best fit to the measured input and output data.

Assume that we have N input‐output data sets:

then similar to the least squares method, the overall fit is minimal as measured by

2.33

where the unknowns are the w_ij and values. The optimization can be done by using software packages or by using special neural network algorithms like back propagation.

In practical cases, the structure (number of hidden layers and number of nodes on them) of the neural network is selected, and the weights w_ij and are determined.

For the optimization, usually only half of the data set is used; the other half is then used for validation, when Q is computed based on data that was not used in determining the weights. If Q is sufficiently small, then the structure and weights of the network are accepted; otherwise, a new structure (with new added hidden layers and/or added nodes) is chosen, and the procedure is repeated. The optimal choice of the weights is usually called the training of the network.

Support Vector Machines

Support vector machines (Cortes and Vapnik 1995) can be described for the case of two groups of vectors. Assume there are N training data points (x₁, y₁), …, (x_N, y_N), where y_i =  + 1 or −1, indicating the class the vectors x_i belong to. The objective is to find a maximum‐margin hyperplane that divides the set of data points (x_i, y_i) into two groups. In one, y_i =  + 1; and in the other, y_i =  − 1. The hyperplane is selected so the distance between the hyperplane and the closest point from either group is maximized. If the training vectors are heavily separable, then there is a vector w such that the hyperplane w^Tx + b = 0 satisfies the following property:

These relations can be summarized as

2.34

In order to maximize the distance between the hyperplanes

we have to minimize the length of w = (w_i):

2.35

subject to the constraints y_i(w^Tx_i + b) ≥ 1. The vector w and scale b define the classifier as

The vectors closest to the separating hyperplane are called support vectors.

Naive Bayesian Classifier

The naive Bayesian classifier technique (Webb et al. 2005) is also based on m classes C₁, …, C_m. An attribute vector x belongs to class C_i if and only if for the conditional probabilities,

2.36

for all j ≠ i. By the Bayesian theorem

And since P(x) is independent of the classes, optimal class C_i is selected by maximizing

2.37

The value of P(C_i) is usually taken as the relative frequency of the sample vectors belonging to class C_i. The naive Bayesian classifier assumes the class conditional independence as

where the components of vector x are the x_k values.

Independent Component Analysis

Independent component analysis (Stone 2004) is a procedure that finds underlying factors or components from multivariate or multidimensional data. Let the observation of random variables be denoted by x₁(t), x₂(t), …, x_N(t). The method finds a matrix M and variables y_j(t) such that

2.38

where the components of y and x are y_j and x_i, respectively.

The objective is to find the minimal number of independent components y_j.

Fuzzy C‐Means Classifier

The fuzzy C‐means classifier approach (Bezdek et al. 1999) allows each piece of data to belong to two or more clusters. Let x₁, …, x_N be the data set, and let u_ij denote the degree of membership of data vector x_i in cluster j (j = 1, 2, …, K). The membership values are determined by using an iterative procedure as follows:

• Step 1: Select an initial set of u_ij values, u_ij(0).
• Step 2: At each later step k, compute the center vectors for each cluster as
  

  where m > 1 is a real number selected by the user and u_ij(k) is the current degree of membership of x.

• Step 3: Update the u_ij values as
  
• Step 4: Stop if for all i and j,
  

  where ε is a threshold.

The final u_ij values are accepted as degrees of membership of the data vectors in the clusters. In this approach, the number of clusters is assumed to be given. If some of the resulting cluster centers become close to each other, then we can reduce the number of clusters by merging and repeating the process.

2.5.1 Modeling of Condition‐Based Data (CBD) Signatures

An example of an alternative approach to CBM based on CBD signatures is shown in Figure 2.16. A sensor framework senses, collects, and transmits sensor output data to a feature‐vector framework that performs data processing such as data conditioning, data fusing, and data transforming to transform CBD into failure‐progression signatures: fault‐to‐failure progression (FFP) signature data, degradation progression signature (DPS) data, and functional failure signature (FFS) data.

Any of these signatures can be used as input to a prediction framework to produce prognostic information such as estimates of RUL, prognostic horizon (PH), and SoH for processing by a health‐management framework to make intelligent decisions about the health of the system and initiate, manage, and complete service, maintenance, and logistics activities to maintain health and ensure the system operates within functional specifications.

2.5.2 Comparison of Methodologies: Life Consumption and CBD Signature

A common model‐based approach to PHM is a life‐consumption methodology defined by the Center for Advanced Life Cycle Engineering (CALCE), University of Maryland at College Park, Maryland. As seen in the simplified diagrams in Figure 2.17, that model‐based approach (Pecht 2008 ) is similar to, but different from, using CBD signature models (Hofmeister et al. 2013 ). The primary differences (see Table 2.9) are related to the difference in modeling and data: modeling using (i) physical, reliability, and/or statistical modeling or (ii) modeling based on empirical data – CBD signatures; and data using (i) environmental, usage, and operational data such as voltage, current, and temperature or (ii) CBD signatures at nodes with environmental, usage, and operational data used to condition signature data.

2.5.3 CBD‐Signature Modeling: An Illustration

A switch mode power supply (SMPS) such as that shown in Figure 2.18 is used to illustrate modeling of CBD signatures with an analysis of a circuit or assembly, which in this case is the output filter of the SMPS. The output filter has also been simplified: for example, to exclude such components and subcircuits as a feedback loop, diodes, and high‐frequency noise filters. Additionally, the filter has been further simplified to lump inductance, capacitance, and resistance into three passive components (L1, C1, and RL).

Example Backup and Setup

Suppose you are asked to prognostic enable a SMPS, and it is known that the supply has a high failure rate caused by failure of tantalum oxide capacitors used in the output filter. Failure and repair information reveals that the capacitors fail short and then burn open. The fail‐short condition causes high current, resulting in an overload condition that turns off the SMPS – but, at the same time, damage continues and the capacitor burns open. Subsequently, maintenance personnel declare a no fault found (NFF)/cannot duplicate (CND) repair condition. The SMPS tests okay – the DC output voltage, ripple voltage, noise levels, and regulation of the supply under loading conditions are all within specifications. Upon further inspection after disassembling the circuit card assembly (CCA) – an enclosed metal frame – physical evidence in the form of color and smell reveals 1 of 22 parallel‐connected capacitors has failed. Removal (desoldering) and testing of the capacitor confirms the capacitor has burned open, which reduces the effective total capacitance in the output filter.

Also suppose the customer specifications for prognostic enabling of the SMPS includes the following requirements: (i) the prognostic system shall detect degradation at least 200 operational days prior to functional failure; and (ii) the prognostic system shall provide estimates of RUL that shall converge to within 5% accuracy at least 72 operational days, ±2 days, before functional failure occurs. Functional failure is defined as a level of damage at which an object, such as a power supply, no longer operates within specifications.

Mahalanobis Distance Modeling of Failure of Capacitors

One data‐driven approach to modeling the failure of capacitors uses the Mahalanobis distance (MD) method, which reduces multiple parameters to a single parameter using correlation techniques after normalizing data. MD values are calculated between incoming data and a baseline representative of a healthy state to detect anomalies in the following way (Alan et al. 2011 ).

Assume we have n vectors with m components, and each component represents a parameter. Let (x_i1, x_i2, …, x_im) denote the ith vector. The mean of the kth parameter is given as

2.40

The standard deviation of the parameter is similarly

2.41

And so the standardized value of x_ik becomes

2.42

Define

Then the correlation matrix of the standardized values is given as

And finally, the MD value can be obtained as

2.43

where T denotes the transpose.

MD values are not normally distributed and need to be further transformed by using a Box Cox transformation and maximizing the logarithm of the likelihood function. However, in a highly accelerated test experiment, the MD method only predicted failure in 14 out of 26 samples (Alan et al. 2011 ). Even if the degree of accuracy were acceptable, that still leaves, for example, the problem of predicting when the power supply will fail because of the loss of sufficient capacitance for any specific failure mode. To address those kinds of problems, heuristic‐based modeling of CBD can be used rather than traditional models. We conclude that neither model‐based nor data‐driven approaches is likely to meet the requirements for resolution, precision, and prognostic accuracy.

There are other techniques as well, to reduce multiple measurements into a single health indicator. One example, for instance, is the multivariate state estimation technique (Cheng and Pecht 2007), which is based on the best healthy estimates of the sample vectors obtained by the use of least squares. However, other metrics can be also selected for the distance of the sample vectors and their estimates. These distances are indicative of SoH, since smaller distances indicate better health. Other examples include autoassociative kernel regression (Guo and Bai 2011), a method based on kernel smoothing (Wand and Jones 1995 ) discussed in Section 2.3.1. After the sequence of single health indicators is determined, the resulting time series can be used as input data for all procedures being discussed in this book.

CBD: Noise and Features

The output of an SMPS, as seen at the top of Figure 2.20, is very noisy. It consists of DC voltage (the desired output) plus all manner of noise in the form of signal variations that include the following: thermal noise; ripple‐voltage variations; switching noise; harmonic distortion; and responses to load variations such as damped‐ringing responses, an example of which is shown at the bottom of Figure 2.20 (Hofmeister et al. 2017 ).

Noise contains information, and when noise in CBD is appropriately isolated and conditioned, leading indicators of failure can be extracted as feature data (FD). Useful FD at the output node of an SMPS includes, but is not limited to, ripple voltage (amplitude and frequency), switching amplitude and frequency, and other features found in a damped‐ringing response. An important first step in prognostic‐enabling an object such as a component, circuit, or assembly in an electrical system is to perform analyses, such as PoF and FMEA, to identify FD of interest associated with failure modes of interest.

CBD: Feature Data and Degradation Function

From the results of deriving Eq. ( 2.49 ), we use the inductive approach to analytical modeling to state the following:

• Given a feature, FD, that has a nominal value FD₀ in the absence of degradation, then
• Given a degradation of a parameter that has a value of P₀ in the absence of degradation, and
• Let dP represent the amount of change in the value of P₀ caused by degradation,
• Then the value of a given feature is the non‐degraded value of the feature times a degradation function of a parameter that changes.

In Eq. ( 2.49 ), if we let FD = ω, FD₀ = ω₀, P₀ = C₀ and dP = ΔC we have

and, in general,

2.50

The model for a given degradation function, f(dP, P₀), depends on, for example, the component that is degrading, the failure mode, and the PoF. This is an important result, and more information is presented in following chapters of this book.

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Further Reading

1. Filliben, J. and Heckert, A. (2003). Probability distributions. In: Engineering Statistics Handbook. National Institute of Standards and Technology http://www.itl.nist.gov/div898/handbook/eda/section3/eda36.htm.
2. Saxena, A., Celaya, J., Saha, B. et al. (2009). On applying the prognostic performance metrics. Annual Conference of the Prognostics and Health Management Society (PHM09), San Diego, California, US, 27 Sep.–1 Oct.
3. Tobias, P. (2003). Extreme value distributions. In: Engineering Statistics Handbook. National Institute of Standards and Technology https://www.itl.nist.gov/div898/handbook/apr/section1/apr163.htm.