3.1.1 Prognostics and SHM

Using suitable SHM sensors that can interrogate the system health state and assess in real time any change in fault severity are of paramount importance. Since damage predictions are sequentially updated from periodical measurements, the higher the accuracy expected from prognostics, the better the quality required for the information obtained from the sensing system. However, this information comes at the expense of more targeted sensing and significant computational requirements. Complex systems subjected to a variety of fault modes (cracks, voids, delamination, corrosion, etc.) often require dedicated sensors and sensor networks for detection as no sensor can provide sufficient information to cover all fault modes. The choice of the sensing method is typically guided by the feature or set of features to be monitored. For example, weight loss or power demand sensors on-board airspace systems lead to a different sensor choice than the one made for monitoring vibrations in buildings or corrosion in bridge structures [166].

Sensor locations are chosen such that the expected type of damage produces observable and statistically significant effects in features derived from the measurements at these locations, which is often determined through numerical simulations or physical tests. Low-level local response caused by damage (e.g. cracks onset and closing) must be separated from a large-amplitude global response, such as that caused by heavy traffic loads on pavements, by determining the required sensitivity and dynamic range through analysis or experimentation. There are well-known methods for optimal placement of sensors that consider the uncertainty in both measurements and model response (see for example [139, 141, 33]).

3.1.2 Damage response modelling

As mentioned above, underlying any prediction is a model that describes how the component or system behaves under nominal conditions and how it will evolve as it experiences wear or a fault condition. To that end, one needs to represent that process in a mathematical fashion. The model might be derived from laws of physics, encapsulated by empirical relations, or learned from data. Models can also be formed using a combination of these approaches. For example, parts of a component that are well-understood may be constructed using physics-based models, with unknown physical parameters learned from data using an appropriate inference technique, as pointed out in Chapter 1.

Modelling of damage or degradation can be accomplished at different levels of granularity; for example, micro and macro levels. At the micro level, models are embodied by a set of constitutive equations that define relationships about primary physical variables and the observed component damage under specific environmental and operational conditions. Since measurements of critical damage properties (such as micro-cracks, voids, dislocations, etc.) are rarely available, sensed system parameters have to be used to infer the damage properties. Micro-level models need to account for the assumptions and simplifications in the uncertainty management. In contrast, macro-level models are characterised by a somewhat abridged representation that makes simplifying assumptions, thus reducing the complexity of the model (typically at the expense of accuracy). Irrespectively, as pointed out in Chapter 1, a model is just an idealisation to the reality that suffices for an approximation, henceforth the disparity with respect to the reality (manifested through the data) should be accounted for via explicit uncertainty management [160].

3.1.3 Interpreting uncertainty for prognostics

Developing methods for uncertainty quantification in the context of prognostics is challenging because uncertainty methods are generally computationally expensive, whereas prognostics requires real time computation for anticipated decision-making. An important aspect of prognostics is the prediction of RUL and EOL, and several publications [63, 156] have discussed the importance of quantifying the uncertainty of these measures.

Typically, probabilistic methods have widely been used for uncertainty quantification and management in various engineering applications, although the interpretation of probability is, at times, not straightforward. As mentioned in Chapter 1, there are two main interpretations of probability: frequentist versus subjective. In the context of prognostics and health management, the frequentist interpretation of uncertainty is applicable only for testing-based methods, mostly when several nominally identical specimens are tested to understand the inherent (random) variability across them. On the other hand, in condition-based prognostics, the focus is typically on one particular unit (e.g. a bridge). At any specific time-instant, this particular unit is in a given state with no variability, basically because there is not any other identical individual to compare it with. Nonetheless, there exists an associated uncertainty regarding this state, which is simply reflective of our degree of belief about it. Bayesian tracking methods (Kalman filtering, particle methods, etc.) [9] used for sequential state estimation, which constitutes the previous step before EOL/RUL prediction, are called Bayesian not only because they use Bayes’ theorem for state estimation but also because they are based on probabilities that express the degree of plausibility of a proposition (damage state) conditioned on the given information (SHM data), which provides a rigorous foundation in a subjective uncertainty context [52]. For example, the future degradation of a component is subjected to epistemic (lack of knowledge) uncertainty; hence, the resulting RUL estimate that accounts for such future degradation needs to be interpreted probabilistically considering such uncertainty. Thus, it can be clearly seen that the subjective interpretation of uncertainty adopted across the entire book is also consistent in the domain of condition-based prognostics presented in this chapter [160].

3.1.4 Prognostic performance metrics

Quantifying the prediction performance is a natural step for an efficient prognostics framework after a component or subsystem is being monitored using an appropriate sensor system. Decisions based on poor and/or late predictions may increase the risk of system failure, whereas wrong predictions of failure (false positives) trigger unnecessary maintenance actions with unavoidable cost increase. Saxena et al. [163] provided a detailed discussion about deriving prognostics requirements from top level system goals, so the reader is referred to this work for further insight. These requirements are generally specified in terms of a prediction performance that prognostics must satisfy for providing a desired level of safety or cost benefit. A variety of prognostics performance evaluation metrics have been defined in the literature, like the prediction horizon (PH), the α-λ accuracy measure, and other relative accuracy measures [162, 161]. As described by [164], prognostics performance can be evaluated using the following three main attributes, namely:

correctness, which is related to the prediction accuracy using the data as a benchmark;

timeliness, which accounts for how fast a prognostics method (or algorithm) produces the output as compared to the rate of upcoming outcomes from the system being monitored;

confidence, which deals with the uncertainty in a prognostics output, typically from a prognostics algorithm.

Among the metrics proposed by Saxena et al. [162, 161], the PH and the α-λ accuracy measures are widely used in the prognostics literature and are also adopted in the majority of works about prognostics. The PH serves to determine the maximum early warning capability that a prediction algorithm can provide with a user-defined confidence level denoted by α. Typically, a graphical representation using a straight line with a negative slope serves to illustrate the “true RUL”, that decreases linearly as time progresses. The predicted PDFs of RUL are plotted against time using error bars (e.g. by 5%–95% error bars), as depicted in Figure 3.1a. The median of the RUL predictions should stay within the accuracy regions specified by α, and ideally on the dotted line (RUL*) that represents the true RUL. By means of this representation, the PH can be directly determined as shown in Figure 3.1a. The PH metric can be further parameterised by a parameter β (thus denoted by PH_α,β) that specifies the minimum acceptable probability of overlap between the predicted RUL and the α accuracy bands delimited by the dashed lines in Figure 3.1a. Both α and β are scaling parameters for the prognostics which should be fixed considering the application scenario.

For the α-λ accuracy metric, a straight line with negative slope is also used to represent the true RUL. Predicted PDFs of RUL are plotted against time of prediction (which is termed as λ in the original paper by Saxena et al. [162]) using error bars. As in Figure 3.1a, accurate predictions should lie on this line as long as they are sequentially updated using SHM data. In this case, the accuracy region is determined by parameter α representing a percentage of the true RUL, in the sense that the accuracy of prediction becomes more critical as EOL approaches. See Figure 3.1b for illustration. In this case, two confidence regions are employed by adopting 0 < α₁ < α₂ < 1, so that each predicted RUL can be validated depending on whether or not it belongs to any of the α₁ or α₂ regions. The interested reader is referred to [45, 44, 42] for further insight about using prognostics metrics in the context of engineering case studies.

3.2.1 Linear Bayesian Processor: The Kalman Filter

The Kalman Filter (KF), proposed in 1960 by Rudolf E. Kálmán [107], triggered a huge revolution by that time in a wide range of disciplines such as signal processing, communication networks and modern control theory. It offers an analytic expression for the solution of the optimal filtering problem of linear Gaussian systems. Indeed, the structure of Equations (3.1) and (3.2) in this setting is

$x_k=A_{k-1}{x}_{k-1}+B_{k-1}{u}_{k-1}+{\omega }_{k-1}$ (3.5)

$y_k=C_k{x}_k+{\nu }_k$ (3.6)

with matrices A_k ∈ M_{nx ×nx} (ℝ), B_k ∈ M_{nx ×nx} (ℝ), C_k ∈ M_{ny ×nx} (ℝ) and D_k ∈ M_{ny ×nu} (ℝ). The random vectors ω_k ~ N(0, Q_k) and ν_k ~ N(0,R_k) are uncorrelated zero mean white process and measurement noises, respectively.

Provided the system is linear and Gaussian, if x₀ has a Gaussian distribution and considering that the sum of Gaussian distributions remains Gaussian, the linearity of the system guarantees x_k to be Gaussian for all k ≥ 0. In fact, since the state posterior distribution p(x_k|y_1:k) is always Gaussian, then we also know that the conditional expectation

$E_{p\left(x_k|y_{1:k}\right)}\left\{x_k\right\}={\displaystyle {\int }_{{ℝ}^{n_x}}{x}_{k}p\left(x_k|y_{1:k}\right)d{x}_k}$ (3.7)

and the conditional covariance matrix

$ℂo{v}_{p\left(x_k|y_{1:k}\right)}\left\{x_k\right\}={\displaystyle {\int }_{{ℝ}^{n_x}}\left(x_k-E_{p\left(x_k|y_{1:k}\right)}\right)}{\left(x_k-E_{p\left(x_k|y_{1:k}\right)}\right)}^{T}p\left(x_k|y_{1:k}\right)d{x}_k$ (3.8)

are the only elements that fully determine the state posterior distribution.

The KF is a recursive algorithm that provides an optimal unbiased estimate of the state of a system – under the aforementioned setting – in the sense of minimising the mean squared error. It recursively updates the conditional expectation and covariance matrix as new measurements are acquired sequentially with each sample time. In this setting, the minimum mean squared error estimate, x̂^MMSE_k, coincides with the conditional expectation, so it is directly obtained from the KF’s recursion.

Let us denote the conditional expectation and covariance matrix at time k given measurements until time k̄ by x̂_k|k̄ and P_k|k̄, respectively. The KF algorithm is summarised below.

3.2.2 Unscented Transformation and Sigma Points: The Unscented Kalman Filter

When linearity and Gaussianity conditions are not fulfilled, the KF cannot ensure optimal state estimates in the mean square error sense. Two other alternative Bayesian tracking methods inspired on KF have been developed to offer sub-optimal solutions: the Extended Kalman Filter (EKF) and the Unscented Kalman Filter (UKF). On the one hand, the EKF proposes a first order approximation of the system model, to allow a setup where the standard KF could still be used. On the other hand, the UKF focuses on approximating the statistics of the random state vectors after undergoing non-linear transformations.

Indeed, the Unscented Transform (UT) [105, 106] is a method designed for approximating nonlinear transformations of random variables. Although some nonlinear functions can be approximated by truncating Taylor expansions, UT focuses on preserving some of the statistics related to the nonlinear transformation, using sets of deterministic samples called sigma-points. For illustrative purposes, let us suppose x ∈ ℝ^n_x is a random variable with mean x̄ and covariance matrix P_x, and g(·) is a nonlinear real-valued function. The objective is to approximate y = g(x). The UT method states the following: Firstly, compute a set of 2n_x + 1 deterministically chosen weighted samples, namely the sigma-points S_i = {X⁽ⁱ⁾, W⁽ⁱ⁾}, i = 0,..., 2n_x, such that

$X^{\left(i\right)}=\overline{x}i=0$ (3.9)

$X^{\left(i\right)}=\overline{x}+{\left(\sqrt{\left(n_x+\kappa \right)P_x}\right)}_{i}i=1,\dots ,n_x$ (3.10)

$X^{\left(i\right)}=\overline{x}-{\left(\sqrt{\left(n_x+\kappa \right)P_x}\right)}_{i}i=n_x+1,\dots ,2n_x$ (3.11)

where ${\left(\sqrt{\left(n_x+\kappa \right)P_x}\right)}_i$ is the i^th column of the matrix square root of (n_x + κ)P_x. On the other hand, there are weights defined as

$W^{\left(i\right)}=\frac{\kappa }{n_x+\kappa }i=0$ (3.12)

$W^{\left(i\right)}=\frac{1}{2\left(n_x+\kappa \right)}i=1,\dots ,2n_x$ (3.13)

The variable i indexes the deterministic weighted samples {X(i), W(i)}2nxi=0 that approximate the first two moments of x as

$\begin{array}{l}\overline{x}={\displaystyle \sum _{i=0}^{2n_x}{W}^{\left(i\right)}{X}^{\left(i\right)}}\\ P_x={\displaystyle \sum _{i=0}^{2n_x}{W}^{\left(i\right)}}\left(X^{\left(i\right)}-\overline{x}\right){\left(X^{\left(i\right)}-\overline{x}\right)}^T\end{array}$

Weights satisfy ${\displaystyle {\sum }_{i=0}^{2n_x}{W}^{\left(i\right)}=1.}$ On the other hand, κ is a parameter (commonly used to adjust higher order moments) that determines how far the sigma-points set are from the mean.

Applying the nonlinear transformation on each sigma-point

$y^{\left(i\right)}=g\left(X^{\left(i\right)}\right)i=0,\dots ,2n_x,$ (3.14)

then the statistics of y can be approximated as

$\overline{y}\approx {\displaystyle \sum _{i=0}^{2n_x}{W}^{\left(i\right)}{y}^{\left(i\right)},}$ (3.15)

$P_y\approx {\displaystyle \sum _{i=0}^{2n_x}{W}^{\left(i\right)}\left(y^{\left(i\right)}-\overline{y}\right){\left(y^{\left(i\right)}-\overline{y}\right)}^T}$ (3.16)

$P_{xy}\approx {\displaystyle \sum _{i=0}^{2n_x}{W}^{\left(i\right)}\left(X^{\left(i\right)}-\overline{x}\right){\left(y^{\left(i\right)}-\overline{y}\right)}^T}$ (3.17)

As it can be seen in the expressions above, the variable i now indexes the transformed weighted samples that define the mean, covariance, and cross-covariance matrices of the random variable y, respectively.

To avoid non-positive definite covariance matrices, weights in Equations (3.12) and (3.13) can be optionally modified as follows:

$W_m^{\left(i\right)}=\frac{\lambda }{n_x+\lambda }i=0$ (3.18)

$W_c^{\left(i\right)}=\frac{\lambda }{n_x+\lambda }+\left(1-{\alpha }^2+\beta \right)i=0$ (3.19)

$W_m^{\left(i\right)}=W_c^{\left(i\right)}=\frac{1}{2\left(n_x+\lambda \right)}i=1,\dots ,2n_x$ (3.20)

so that the set of weights ${\left\{W_m^{\left(i\right)}\right\}}_{i=0}^{2n_x}$ are used to compute mean vectors, whereas the set of weights ${\left\{W_c^{\left(i\right)}\right\}}_{i=0}^{2n_x}$ are used to compute covariance (or cross-covariance) matrices instead. Note that, in comparison to Equations (3.12) and (3.13), we have replaced the tuning parameter κ → λ = α²(n_x + κ) − n_x, leading to a scaled form of the UT. The parameter α regulates the speediness of sigma-points around the mean vector and according to [31], typical values are set to 0.01 ≤ α ≤ 1. Similarly, κ is commonly chosen as 0 or 3 − n_x, whereas β provides and additional degree of freedom for further adjustments if there is prior knowledge about the distribution of the random vector to be non-linearly transformed.

During the filtering stage, uncertainty arises from i) initial state condition, ii) state transition model (process noise), and iii) observation model (measurement noise). Hence, in order to incorporate all these uncertainty sources, the implementation of the UKF for a general nonlinear system (see Equations (3.1) and (3.2)) includes the usage of an augmented vector:

$x_k^a={\left[x_k^T{\omega }_k^T{\nu }_k^T\right]}^T$ (3.21)

In this sense, uncertainty is jointly propagated trough nonlinear transformations. The computational cost may be lowered by assuming that both the process and measurement noises are purely additive, which is very convenient in the context of online monitoring. The resulting system would be described as

$x_k=f\left(x_{k-1},u_{k-1}\right)+{\omega }_{k-1}$ (3.22)

$y_k=g\left(x_k,u_k\right)+{\nu }_k$ (3.23)

with ω_k and ν_k zero mean process and measurement noises; with second moments denoted as Q_k and R_k, respectively. The UKF version for systems with additive sources of uncertainty is summarised in Algorithm 7.

3.2.3 Sequential Monte Carlo methods: Particle Filters

Except from a reduced number of cases, it is generally not possible to obtain an analytic solution for the filtering problem. Nonlinear systems may either be approximated by linearised versions within bounded regions of the state space (e.g. EKF) or even characterised in terms statistical approximations of state transitions (e.g. the UKF). However, KF-based implementations can only offer a sub-optimal solution and, moreover, underlying uncertainties may even distribute differently from Gaussian probability density functions (PDFs). In this regard, Sequential Monte Carlo methods, also known as Particle Filters (PFs) [9, 58], provide an interesting alternative for the solution of the filtering problem in the context of nonlinear non-Gaussian systems. PFs approximate the underlying PDF by a set of weighted samples, recursively updated, providing an empirical distribution from where statistical inference is performed.

PFs efficiently simulate complex stochastic systems to approximate the posterior PDF by a collection of N_p weighted samples or particles ${\left\{x_{0:k}^{\left(i\right)},w_k^{\left(i\right)}\right\}}_{i=1}^{N_p},{\displaystyle {\sum }_{i=1}^{N_p}{w}_k^{\left(i\right)}=1,}$ such that

$p\left(x_{0:k}|y_{1:k}\right)\approx {\widehat{p}}_{N_p}\left(x_{0:k}|y_{1:k}\right)={\displaystyle \sum _{i=1}^{N_p}{w}_k^{\left(i\right)}{\delta }_{x_{0:k}^{\left(i\right)}}\left(x_{0:k}\right)}$ (3.47)

where ${\delta }_{x_{0:k}^{\left(i\right)}}\left(x_{0:k}\right)$ is a Dirac delta distribution located at the i^th state trajectory x⁽ⁱ⁾_0:k. The weighting process is performed by applying sequential importance resampling (SIR) algorithms in two stages: sequential importance sampling and resampling. The resulting algorithm is commonly known as “bootstrap filter”, after the celebrated paper by Gordon et al. [79]. A pseudo-code implementation for the PF is given as Algorithm 8, which includes a systematic resampling step [9] to limit the well-known degeneracy problem¹.

3.3.1 The failure prognosis problem

Given that both the future conditions in which the systems operate as well as the dynamic models that seek to characterise them have various sources of uncertainty, the EOL of a system should therefore be expressed in terms of a random variable, which is denoted here as τ_ε. If E_k is a binary random variable that indicates at each future time instant whether the event ε = “System failure” occurs or not (which in the last case would be equivalent to saying that the complement event occurs, i.e. ε^c = “System without failure”), then τ_ε is defined as [3]:

${\tau }_{\epsilon }\left(k_p\right):=\inf \left\{k\in \mathbb{N}:\left\{k>k_p\right\}\wedge \left\{E_k=\epsilon \right\}\right\}$ (3.52)

where k_p is the current prediction time. In other words, τ_ε corresponds to the first future time instant k (since it is greater than k_p) in which there is a system failure. This definition is that of a random variable and, consequently, induces an associated probability distribution. According to [3], this probability distribution (applicable to any type of system) is equivalent to the probability that the system is in a faulty condition at time k and that before it has necessarily been operating under a healthy condition:

$\mathbb{P}\left({\tau }_{\epsilon }=k\right):=\mathbb{P}\left(\underset{Faultyatk}{\underbrace{\left\{E_k=\epsilon \right\}}},\underset{\begin{array}{l}Healthyfrom\\ k_{p}uptok-1\end{array}}{\underbrace{{\left\{E_j={\epsilon }^c\right\}}_{j=k_p}^{k-1}}}\right)$ (3.53)

If we further develop this expression by applying conditional probabilities, then we get

$\mathbb{P}\left({\tau }_{\epsilon }=k\right)=\mathbb{P}\left(E_k=\epsilon |{\left\{E_j={\epsilon }^c\right\}}_{j=k_p}^{k-1}\right)\mathbb{P}\left({\left\{E_j={\epsilon }^c\right\}}_{j=k_p}^{-1}\right)$ (3.54)

$\begin{array}{l}⋮\\ \mathbb{P}\left(E_k=\epsilon |{\left\{E_j={\epsilon }^c\right\}}_{j=k_p}^{k-1}\right){\displaystyle \prod _{j=k_p+1}^{k-1}\mathbb{P}\left(E_j={\epsilon }^c|{\left\{E_j={\epsilon }^c\right\}}_{j=k_p}^{-1}\right)}\cancel{\mathbb{P}{\left(E_{k_p}={\epsilon }^c\right)}^1}\end{array}$ (3.55)

$=\mathbb{P}\left(E_k=\epsilon |{\tau }_{\epsilon }\ge k\right){\displaystyle \prod _{j=k_p+1}^{k-1}\mathbb{P}\left(E_j={\epsilon }^c|{\tau }_{\epsilon }\ge j\right)}$ (3.56)

$=\mathbb{P}\left(E_k=\epsilon |{\tau }_{\epsilon }\ge k\right){\displaystyle \prod _{j=k_p+1}^{k-1}\left(1-\mathbb{P}\left(E_j=\epsilon |{\tau }_{\epsilon }\ge j\right)\right)}$ (3.57)

However, since the binary random process {E_k}_k∈ℕ depends on the system condition, the final expression is given by

$\mathbb{P}\left({\tau }_{\epsilon }=k\right){\displaystyle {\int }_{X_{k_p+1:k}}\mathbb{P}\left(E_k=\epsilon |x_k\right)}{\displaystyle \prod _{j=k_p+1}^{k-1}\left(1-\mathbb{P}\left(E_j=\epsilon |x_j\right)\right)p\left(x_{k_p+1:k}|y_{1:k_p}\right)d{x}_{k_p+1:k}}$ (3.58)

where p(x_kp+1:k |y_1:kp) corresponds to the joint probability density of the possible future trajectories of the system up to an instant k, with k > k_p, given the evidence in terms of measurements. The expression ℙ (E_k = ε|x_k) refers to the probability, at the time instant k, with which the event ε = “System failure” occurs given a system condition denoted by x_k.

There is a version of this probability distribution for continuous-time systems as well, for which the reader can refer to [3] .

Since it is generally assumed that system failures are triggered when the condition indicators of such systems cross certain thresholds, then in those cases the probability P (E_k = ε|x_k) is defined as

$\mathbb{P}\left({{\rm E}}_k=\epsilon |x_k\right)=1_{\left\{x\in {ℝ}^{n_x}:x>\overline{x}\right\}}\left(x_k\right)=\{\begin{array}{cc}1,& if{x}_k\ge \overline{x}\\ 0,& ~\end{array}$ (3.59)

That is, if x_k exceeds an upper threshold x̄ (it can also be set as a lower threshold depending on the context), the event ε occurs with probability 1, which is the same as simply saying that it occurs with certainty.

Therefore, if failures are triggered by a threshold, as in most cases, then the statistics of the random variable τ_ε associated with the EOL time are determined by

$\mathbb{P}\left({\tau }_{\epsilon }=k\right){\displaystyle {\int }_{{\mathbb{X}}_{k_p+1:k}}=1_{\left\{x\in {ℝ}^{n_x}:x>\overline{x}\right\}}\left(x_k\right){\displaystyle \prod _{j=k_p+1}^{k-1}\left(1-1_{\left\{x\in {ℝ}^{n_x}:x>\overline{x}\right\}}\left(x_j\right)\right)p\left(x_{k_p+1:k}|y_{1:k_p}\right)d{x}_{k_p+1:k}}}$ (3.60)

Once the probability distribution of τ_ε has been characterised, then decision-making becomes possible considering the risk of the future operation of the system. Each possible decision leads to consequences of varying severity, which must be analysed in proportion to the risk associated with incurring in each one of them.

There are even more general cases in which the failure condition is not determined by the crossing of thresholds. In this regard, the reader can refer to [3] for a detailed explanation.

3.3.2 Future state prediction

As we saw earlier, to calculate the probability that a system will fail in the future we must use Equation (3.60). However, this in turn requires:

• □ To define the time k_p from which the prediction is made onward.
• □ To define a failure threshold x̄.
• □ To know the joint probability distribution p(x_kp+1:k|y_1:kp).

The first point is associated with the detection of an anomalous behaviour in the system that suggests the incursion of an imminent failure. The second point, on the other hand, will depend on the context and the specific system that is being monitored; the failure threshold can be defined either by an expert or through post-mortem analysis, which would require historical data. But what about the third point? Frequently, degradation processes can be represented through Markovian models as shown in Equations (3.1)–(3.2). In such cases we have

$p\left(x_{k_p+1:k}|y_{1:k_p}\right)={\displaystyle {\int }_{{\mathbb{X}}_{0:k_p}}p\left(x_{0:k}|y_{1:k_p}\right)d{x}_{0:k_p}}$ (3.61)

$={\displaystyle {\int }_{{\mathbb{X}}_{0:k_p}}p\left(x_{k_p+1:k}|x_{0:k_p,}{y}_{1:k_p}\right)p\left(x_{0:k_p}|y_{1:k_p}\right)d{x}_{0:k_p}}$ (3.62)

$={\displaystyle {\int }_{{\mathbb{X}}_{0:k_p}}\underset{Future}{\underbrace{\left({\displaystyle \prod _{j=k_p+1}^{k}p\left(x_j|x_{j-1}\right)}\right)}}\underset{\mathrm{Pr}esent\&Past}{\underbrace{p\left(x_{0:k_p}|y_{1:k_p}\right)}}d{x}_{0:k_p}}$ (3.63)

Thus, the calculation of p(x_kp+1:k|y_1:kp) has a part related to the prediction itself, and another part that accounts for the current state of the system and how this state was reached. The latter is usually obtained using a Bayesian processor like those presented in Section 3.2. In this regard, the PF provides an expression for the joint probability density p(x_0:kp|y_1:kp), but the KF and its variants (such as the EKF and the UKF) provide the marginal probability density p(x_kp|y_1:kp). Therefore, p(x_kp+1:k|y_1:kp) can be calculated in either of the following two ways:

$p\left(x_{k_p+1:k}|y_{1:k_p}\right)=\{\begin{array}{cc}{\displaystyle {\int }_{{\mathbb{X}}_{0:k_p}}\left({\displaystyle \prod _{j=k_p+1}^{k}p\left(x_j|x_{j-1}\right)}\right)p\left(x_{0:k_p}|y_{1:k_p}\right)d{x}_{0:k_p}}& ,iffilteringwithaPF.\\ {\displaystyle {\int }_{{\mathbb{X}}_{0:k_p}}\left({\displaystyle \prod _{j=k_p+1}^{k}p\left(x_j|x_{j-1}\right)}\right)p\left(x_{0:k_p}|y_{1:k_p}\right)d{x}_{k_p}}& ,iffilteringwithaKF\mathrm{var}iant.\end{array}$ (3.64)

The two best-known methods for numerically solving the above integrals are random walks (sometimes referred to as “Monte Carlo simulations”) and particle-filtering-based prognostics, which we present below.

1. Random walks (Monte Carlo simulations)

This is the most popular method for predicting future events. In simple words, the method consists of simulating many (where “many” depends on the problem to be solved) possible future state trajectories of the system. Each trajectory crosses the failure threshold for the first time at a specific point in time. Then, with the information of all these instants of time, a histogram can be constructed, which corresponds to an approximation of the probability distribution for the failure time. An illustration of this process is provided in Figure 3.3. By the Law of Large Numbers, this approximation converges to the true probability distribution when the number of simulated state trajectories tends to infinity. The disadvantage of this method is that the computational cost it requires prevents its implementation in real-time monitoring systems.

The method consists of the following. Suppose that the state of the system at time k_p is x⁽ⁱ⁾_kp and is obtained by making one of the following approximations to p(x_0:kp|y_1:kp) or p(x_kp|y_1:kp), depending on the Bayesian tracking method used:

• $p\left(x_{0:k_p}|y_{1:k_p}\right)\approx {\displaystyle {\sum }_{i=1}^N{w}_{k_p}^{\left(i\right)}{\delta }_{x_{0:k_p}^{\left(i\right)}}\left(x_{0:k_p}\right),iffilteringwithaPF.}$ if filtering with a PF.
• $p\left(x_{k_p}|y_{1:k_p}\right)\approx {\displaystyle {\sum }_{i=1}^N{w}_{k_p}^{\left(i\right)}{\delta }_{x_{k_p}^{\left(i\right)}}\left(x_{k_p}\right),iffilteringwithaKF\mathrm{var}iant.}$ if filtering with a KF variant.

These approximations can be made by sampling either the probability distribution p(x_0:kp|y_1:kp) or p(x_kp|y_1:kp). If statistically independent samples are generated, using inverse transform sampling, for example, then in both cases $w_{k_p}^{\left(i\right)}=\frac{1}{N},i\in \left\{1,\mathrm{...},N\right\}.$ Under other sampling approaches such as importance sampling, the weights {w⁽ⁱ⁾_kp}^N_i=1 might not be equal.

If we knew the future inputs of the system u_k, k > k_p, and, at each future time instant, we take a realisation of the process noise w⁽ⁱ⁾_k ~ N(0, σ_w), then we can simulate random future state trajectories with the state transition Equation (3.1) as

$x_{k+1}^{\left(i\right)}=f\left(x_k^{\left(i\right)},u_k,w_k^{\left(i\right)}\right),$ (3.65)

which is equivalent to sample x⁽ⁱ⁾_k+1 ~ p(x_k+1|x⁽ⁱ⁾k). If we start with x⁽ⁱ⁾_kp, we can generate the future state trajectory

$x_{k_p:k}^{\left(i\right)}=\left\{x_{k_p}^{\left(i\right)},x_{k_p+1}^{\left(i\right)},\mathrm{...}{x}_k^{\left(i\right)}\right\}$ (3.66)

Consequently, if this procedure is repeated N times, the probability distribution p(x_kp+1:k|y_1:kp) that contains information about the past, present, and future of the state of the system (see Equation (3.64)), is approximated as

$p\left(x_{k_p+1:k}|y_{1:k_p}\right)\approx \{\begin{array}{cc}{\displaystyle {\sum }_{i=1}^N{w}_{k_p}^{\left(i\right)}{\delta }_{x_{0:k}^{\left(i\right)}}\left(x_{0:k}\right)}& ,iffilteringwithaPF.\\ {\displaystyle {\sum }_{i=1}^N{w}_{k_p}^{\left(i\right)}{\delta }_{x_{k_p:k}^{\left(i\right)}}\left(x_{k_p:k}\right)}& ,iffilteringwithaKF\mathrm{var}iant.\end{array}$ (3.67)

Finally, the approximation of the probability distribution of τ_ε yields

$\mathbb{P}\left({\tau }_{\epsilon }=k\right)={\displaystyle \underset{{\mathbb{X}}_{k_p+1:k}}{\int }{1}_{\left\{x\in {ℝ}^{n_x}:x>\overline{x}\right\}}\left(x_k\right){\displaystyle \prod _{j=k_p+1}^{k-1}\left(1-1_{\left\{x\in {ℝ}^{n_x}:x>\overline{x}\right\}}\left(x_j\right)\right)=p\left(x_{k_p+1:k}|y_{1:k_p}\right)d{x}_{k_p+1:k}}}$ (3.68)

$\approx {\displaystyle \sum _{i=1}^N{w}_{k_p}^{\left(i\right)}{1}_{\left\{x\in {ℝ}^{n_x}:x>\overline{x}\right\}}\left(x_k^{\left(i\right)}\right)}{\displaystyle \prod _{j=k_p+1}^{k-1}\left(1-1_{\left\{x\in {ℝ}^{n_x}:x>\overline{x}\right\}}\left(x_j^{\left(i\right)}\right)\right).}$ (3.69)

This last equation can be interpreted as: the probability of system failure at time τ_ε = k can be approximated by the fraction of future state trajectories that crossed the threshold x̄ for the first time at time k.

Note that in the common case where the future inputs of the system u_kp+1:k, k > k_p, were uncertain (i.e. u_k can be described as a stochastic process) then the generation of N future system state trajectories should consider one different realisation of the future system inputs per each future system state trajectory. In other words, in order to get the i^th future system state trajectory x_kp+1:k⁽ⁱ⁾, with i = 1, ... , N, we require an i^th simulation of the future system inputs u_kp+1:k⁽ⁱ⁾.

2. Particle-filtering-based prognostics

The computational cost associated with the previous method is humongous and impractical for real-time decision-making processes (such as the ones found in condition-based maintenance schemes). This fact forces the implementation of simplified methods and algorithms for the characterisation of future evolution of the uncertainty related to system states.

A particle-filtering-based prognostic algorithm was originally presented in [136] and became the de facto failure prognostic algorithm in the literature. This algorithm is very similar to the one we presented above, but it differs in that, in this case, the transitions of future states are regularised with Epanechnikov kernels.

The pseudo-code for this prognostic algorithm is as follows:

0. Resample either p(x_0:kp|y_1:kp) or p(x_kp|y_1:kp) as appropriate, and get a set of N_θ equally weighted particles. This is,

• – $p\left(x_{0:k_p}|y_{1:k_p}\right)\approx \frac{1}{N_{\theta }}{\displaystyle {\sum }_{i=1}^{N_p}{\delta }_{x_{0:k_p}^{\left(i\right)}}\left(x_{0:k_p}\right)},$ if filtering with a PF.
• – $p\left(x_{k_p}|y_{1:k_p}\right)\approx \frac{1}{N_{\theta }}{\displaystyle {\sum }_{i=1}^{N_p}{\delta }_{x_{k_p}^{\left(i\right)}}\left(x_{k_p}\right)},$, if filtering with a KF variant.

Then, for each future time instant k, k > k_p, perform the following steps:

1. Compute the expected state transitions x*_k⁽ⁱ⁾ = E{f(x_k−1⁽ⁱ⁾, u_k−1, w_k−1)}, ∀i ∈ {1, ... , N_θ}, and calculate the empirical covariance matrix

${\widehat{S}}_k=\frac{1}{N_{\theta }-1}{{\displaystyle \sum _{i=1}^{N_{\theta }}\left[x_k^{*}{}^{\left(i\right)}-{\overline{x}}_k^{*}\right]\left[x_k^{*}{}^{\left(i\right)}-{\overline{x}}_k^{*}\right]}}^T$

with ${\overline{x}}_k^{*}=\frac{1}{N_{\theta }}{\displaystyle {\sum }_{i=1}^{N_{\theta }}{x}_k^{*}{}^{\left(i\right)}}$

2. Compute D̂_k such that D̂_kD̂^T_k = Ŝ_k.
3. (Regularisation step) Update the samples as

$x_k^{\left(i\right)}=x_k^{*}{}^{\left(i\right)}+h_{\theta }{\widehat{D}}_k{\epsilon }_k^{\left(i\right)},{\epsilon }_k^{\left(i\right)}~\epsilon$

where ε is the Epanechnikov kernel and h_θ corresponds to its bandwidth.

The hyper-parameters vector of the prognostic algorithm is defined as θ^T= [N_θ h_θ], and has to be adjusted. A formal methodology for algorithm design in this regard can be found in [2, 3]. Finally, the approximation of the probability distribution of τ_ε yields

$\mathbb{P}\left({\tau }_{\epsilon }=k\right){\displaystyle {\int }_{{\mathbb{X}}_{k_p+1:k}}=1_{\left\{x\in {ℝ}^{n_x}:x>\overline{x}\right\}}\left(x_k\right){\displaystyle \prod _{j=k_p+1}^{k-1}\left(1-1_{\left\{x\in {ℝ}^{n_x}:x>\overline{x}\right\}}\left(x_j\right)\right)p\left(x_{k_p+1:k}|y_{1:k_p}\right)d{x}_{k_p+1:k}}}$ (3.70)

$\approx \frac{1}{N_{\theta }}{\displaystyle \sum _{i=1}^{N_p}{1}_{\left\{x\in {ℝ}^{n_x}:x>\overline{x}\right\}}\left(x_k^{\left(i\right)}\right)}{\displaystyle \prod _{j=k_p+1}^{k-1}\left(1-1_{\left\{x\in {ℝ}^{n_x}:x>\overline{x}\right\}}\left(x_j^{\left(i\right)}\right)\right)}$ (3.71)

Similar to the previous algorithm, this last equation can be interpreted as: the probability of system failure at time τ_ε = k can be approximated by the fraction of future state trajectories that crossed the threshold x̄ for the first time at time k.

Example 6 Consider an exponential degradation process described by the following discrete statetransition equation:

$x_{k+1}=e^{-1\zeta k}{x}_k+w_k$ (3.72)

where x_k ∈ ℝ are discrete system states for k ∈ ℕ, ζ ∈ ℕ is the decay parameter, a scaling constant that controls the degradation velocity, and w_k ∈ ℝ is the model error term which is assumed to be modelled as a zero-mean Gaussian distribution, that is w_k ~ N(0, σ_w). Let us now assume that the degradation process can be measured over time and that, at a certain time k, the measured degradation can be expressed as a function of the latent damage state x_k, as follows:

$y_k=x_k+{\nu }_k$ (3.73)

where v_k ∈ ℝ is the measurement error, which is also assumed to be modelled as a zero mean Gaussian PDF with standard deviation σ_v. The initial system condition is assumed to be uncertain and characterised as x₀ ~ N(0.9, 0.05), expressed in arbitrary units. We use synthetic data for y_k by generating it from Equations (3.72) and (3.73) considering σ_w = 0.02 and σ_v = 0.01. Additionally, the decay parameter is assumed to be ζ = 0.0004.

The prognostics results at prediction time k_p = 20 are presented in Figure 3.4. A particle filter (PF) with N_p = 1000 particles is used for the filtering stage (i.e. when k ≤ k_p). The probability distribution approximation $p\left(x_{0:k_p}|y_{1:k_p}\right)\approx {\displaystyle {\sum }_{i=1}^{N_p}{w}_{k_p}^{\left(i\right)}{\delta }_{x_{0:k_p}^{\left(i\right)}}\left(x_{0:k_p}\right)}$ yielded from the PF at time k_p is used as initial condition for the prognostics stage. In this regard, prognostics are performed via random walks (i.e. when k > k_p), for which N = 10 particles are resampled from the original PF approximation at k_p. Thus,

$p\left(x_{0:k_p}|y_{1:k_p}\right)\approx \frac{1}{N}{\displaystyle \sum _{i=1}^N{\delta }_{x_{0:k_p}^{\left(i\right)}}\left(x_{0:k_p}\right)}$

The failure region is defined by fixing the threshold value x̄ = 0.3 in arbitrary units.

According to the results, we can expect a failure time at about 55 seconds (which can be visually obtained just by looking at the area of higher probability content in the PDF at Figure 3.4), which approximately coincides with the time where the real system gets into the failure domain. This simple example reveals the potential of the methodology presented in this chapter for important engineering applications such as failure time estimation.