PSA processes are operated on repeated cycles of adsorption and regeneration. As shown in Figure 1, the bed pressure is raised during the adsorption step and the impurities are adsorbed by the adsorbent, providing the high-purity product gas. During the regeneration step, the bed pressure is lowered and the impurities are cleaned or purged from the adsorbent, allowing the adsorption-regeneration cycle to be repeated. Therefore, a PSA process is a continuous process but never operates at any single steady state. Instead, it repeats a sequence of operation steps over and over. This is usually termed a cyclic steady-state process, where cycles are very similar to each other, and a whole cycle is considered a “steady-state”.

To take full advantage of the feed pressure and to recover more product gas, multi-bed multi-step PSA systems have been widely applied in industrial applications. In terms of process monitoring, the bed pressure is always measured and is often the only variable constantly measured for PSA processes. The industrial data utilized in this work were collected from one of Linde’s 12-bed 15-step PSA systems. Figure 2 shows two common ways to visualize pressure trajectories in a multi-bed PSA process. Due to the sensitivity of the process’s actual operation and production data, all axis tick labels in this and other figures are omitted when real operation data are used. Figure 2A shows time-series pressure profiles of multiple beds (only three out of twelve beds are shown here to reduce clutter). This type of pressure time-series plot is useful for visualizing and observing between-bed variations. However, only severe faults that significantly deviate from the nominal trajectory can be detected by the naked eyes using this type of plot; in addition, it becomes very cluttered and difficult to read if all beds were plotted on the same figure. Another way to visualize the pressure profile within a bed over multiple cycles is to overlay cycles based on the start of each cycle, as illustrated in Figure 2B. This type of plot can be used to visualize within-bed variations. However, due to the variable duration of cycles, again, only severe faults that show significant deviations from the normal operation can be detected directly by the naked eyes from this type of plot.

In terms of process monitoring, PSA processes share more similarities with batch processes than with continuous processes. For example, PSA and batch processes can both have variable batch/cycle duration and step durations; they are often dynamic transient processes and do not have a steady state. The variable nature of the PSA cycle duration is demonstrated in Figure 3A, which plots the durations of different cycles from one PSA bed. For the PSA process studied in this work, each cycle consists of 15 steps, as illustrated in Figure 2. For the step durations, about half of the steps follow similar trends as the cycle duration, while the remaining steps have relatively constant durations. Figure 3B plots the variable step duration of the adsorption step across different cycles, and Figure 3C plots the relatively constant step duration of an equalization step across different cycles. For the PSA process studied in this work, the cycle duration is in the order of tens of minutes and the step durations vary from seconds to minutes.

Several observations can be made from these plots. First, the cycles are asynchronous across different beds; for the same bed, the cycles do not exactly overlap with each other either. Second, despite the overall highly nonlinear behavior for each cycle, the pressure profile for each individual step is usually much simpler and can be approximated by a simple linear or polynomial function. Finally, not only the cycle durations but also the step durations vary from cycle to cycle. It is important to note that the variations in cycle/step duration is not caused by unmeasured normal process variations, instead, it is a result of deliberate control of cycle and step durations to ensure product quality in response to dynamic scheduling and/or measured disturbances such as demand change and weather conditions. In addition, these characteristics are not unique to PSA processes but are rather common to other cyclic steady-state processes, such as heat exchanger networks under fouling with cleaning-in-place (CIP) operations (Georgiadis and Papageorgiou, 2000), and catalytic conversion processes where the catalyst undergoes periodic deactivation and activation (Jain and Grossmann, 1998).

As discussed above, normal PSA operations cover a wide range of pressure trajectories, due to the dynamically controlled step/cycle durations in response to external disturbances. It is clearly a highly challenging task to detect a subtle fault early from a wide range of normal cycle/step durations with the bed pressure as the only monitored variable. In addition, the characteristics of the PSA processes (and cyclic steady-state processes in general), including asynchronous trajectories, variable cycle/step durations, and nonlinear dynamics, present significant challenges to process monitoring. These challenges cannot be effectively addressed by commonly used multivariate statistical monitoring (MSPM) methods, including both conventional MSPM methods such as MPCA, trilinear decomposition (TLD), and parallel factor analysis (PARAFAC) (Wise et al., 1999), and more recent methods such as multi-way independent component analysis (MICA) (Yoo et al., 2004) and kernel PCA (KPCA) (Choi et al., 2005). These methods assume that the normal process data follow the same distribution and require the construction of a two-dimensional (2-D) data matrix (for data unfolding approaches) or a 3-D data array (for multi-way approaches). In other words, they require synchronization of all steps within a cycle to achieve equal step and cycle durations. Trajectory synchronization can be done through different ways, including simple cut, interpolation, dynamic time warping (DTW), etc. However, these preprocessing steps have their drawbacks, including trajectory distortion, information loss, etc (He and Wang, 2007; He and Wang, 2018). In particular, synchronization is undesirable for PSA processes because the step durations are dynamically controlled and may contain important information on the state of the process operation. Artificially changing the step/cycle durations may distort the contained information and negatively affect the fault detection and diagnosis performance.

In this work, built upon our work in batch process monitoring that can naturally handle variable batch/step durations, we develop an FSM approach for PSA processes. We show that a balance between sensitivity and robustness of the FSM approach can be achieved through feature engineering and selection, which enables early detection of subtle faults with very low false alarm rate.

As shown in Figure 2, although a complete cycle of a PSA process is highly nonlinear, each step is much simpler and can be described by a simple linear or polynomial model. Therefore, in this work, we compute different features for each step separately. In addition to univariate statistics, we explore morphological features to better capture the characteristics of pressure profile in each step of the process. To handle the irregularities in industrial data, for each characteristic we consider multiple features that may exhibit different level of sensitivity to outliers. For example, to assess the dispersion of pressure measurement during a processing step, we compare features that use the mean of the pressure measurements as the reference with others that use the median as the reference. Based on the observation of the pressure profiles and discussions with process engineers, totally 12 features (as defined in the remaining section), both statistical and morphological, are examined in this work to determine if they would provide adequate process monitoring. All features are calculated for each step using raw pressure measurements without any preprocessing such as synchronization, centering, scaling or normalization.

In this work, we use a vector x i ∈ R N i to represent the N i pressure measurements in cycle i ( i = 1,2 , … , C ) , where C is the number of cycles from all beds and N i varies from cycle to cycle. A subset of x i : x i , j ∈ R N i , j represents the pressure measurements of step j ( j = 1,2 , … , S ) during cycle i ; and x i , j ( t ) ( t = 1,2 , … , N i , j ) represents an individual pressure measurement during step j of cycle i at time t ; N i , j is the duration of step j of cycle i . Note that total sample number in cycle i : N i = ∑ j = 1 S N i , j and total sample number across all C cycles: N = ∑ i = 1 C N i = ∑ i = 1 C ∑ j = 1 S N i , j . For each step of a given cycle, the definitions of different features are given below.

1. Mean ( μ i , j ), which captures the central tendency of pressure.

For the steps with relatively flat pressure profiles (e.g., adsorption, hold and purge steps), we first estimate the global mean of step j over all cycles under normal conditions (i.e., the training data). μ g l o b a l , j = 1 M ∑ j = 1 M μ i , j (12) where M is the total number of cycles in the training data. Then MAE can be calculated as the following M A E i , j = 1 N i , j ∑ t = 1 N i , j | x i , j ( t ) − μ g l o b a l , j | (13)

For the steps with sloped pressure profiles (e.g., equalization, provide purge, blowdown or evacuation, and pressurization steps), the predicted pressure measurements, x ^ i , j ( t )   ( t = 1,2 , … N i , j ) are computed by linear regression using the linear model estimated from all training data. M A E i , j = 1 N i , j ∑ t = 1 N i , j | x i , j ( t ) − x ^ i , j ( t ) | (14)

Finally, for each cycle, we have all the above-described features combined. f i = { μ i | s i | γ i | κ i | C V i | I R Q i | Q C D i | D m e a n , i | D m e d , i | S i | S L L , i | M A E i } (15) where μ i = { μ i , 1 , μ i , 2 , ⋯ , μ i , T } is a row vector of dimension ( 1 × S ) . S is the total number of steps in a cycle. The same concatenation convention applies to all types of features. Therefore, f i is a row vector of dimension ( 1 × ∑ j = 1 S Q j ) where Q j is the number of features used to characterize step j of the PSA process. For simplicity, we use F to denote the total number of features included in each cycle so that f i is a row vector containing F features. In this work, we utilize 12 different types of features, as defined in Eqs 1–14, for all the 15 steps, which would result in 180 features for each cycle (i.e., F = 180 ).

After the extracted features are concatenated into a row vector for each cycle following Eq. 15, the features from multiple cycles are concatenated into a matrix as the following. F T R = [ f 1 f 2 ⋮ f C ] (16)

In this way, a training feature matrix F T R based on C normal cycles is obtained, which has a dimension of ( C × F ) , despite different step/cycle durations in the data. The features of test cycle(s) F T E are extracted in the same way except that some of the features are generated with reference to the training cycles (e.g., μ g l o b a l , j in Eq. 12).

It is worth noting that the statistical and morphological features are extracted for each step of each cycle based on the raw training data without any preprocessing, without synchronization, scaling, normalization nor alignment. Since the features are calculated using all measurements from each step of the cycle, they are all scalars regardless of the step/cycle durations. Therefore, FSM naturally handles unequal step/cycle durations and asynchronous step/cycle trajectories. In addition, the structure shown in Eq. 15 has the flexibility of allowing different number of features for different steps. In addition, cycle-based features can be conveniently added in a similar fashion.

Feature selection has been widely studied in supervised learning where it has been shown that including irrelevant and noisy features increases model complexity and can degrade model prediction performance (Lindgren et al., 1994; Andersen and Bro, 2010; Wang et al., 2015; Lee et al., 2020). Feature selection also has other benefits including reducing computational cost, improving interpretability of the model, etc. Since MSPM method based on PCA is a dimension reduction technique that is capable of handling collinearity in the process data, it may appear that feature selection is redundant and unnecessary. However, as shown in Ghosh et al. (2014), feature selection can have a significant impact on the monitoring performance of PCA-based MSPM. Specifically, experiments were conducted to show that including irrelevant and noisy features in a PCA-based MSPM model can degrade process monitoring performance (Ghosh et al., 2014).

In general, variable selection is more challenging for process monitoring as it is an unsupervised learning. Specifically, in process monitoring available data for model training are predominantly normal data. Even if fault data were available, they do not represent all possible fault scenarios. Therefore, for process monitoring, it is reasonable to assume that only normal operations data are available for feature selection, as features that are sensitive for detecting one type of fault may not be sensitive for detecting other (potentially unseen) faults. In this work we propose a new feature selection method for process monitoring that utilizes normal operation data only. We assume the true relevant features that are important for process monitoring should capture the key characteristics of the normal operation; consequently, if the true relevant features were used for process monitoring, the monitoring performance would be insensitive to the subsets of the training data used for model building. In other words, features extracted from a set of normal operation data (e.g., the training data) should show (highly) similar behavior as those extracted from another independent set of normal data (e.g., the validation data). In this work, we use false alarm rate (FAR) and false alarm magnitude (FAM) to quantify the difference between the training and validation performance, where FAM is defined as the difference between the monitoring statistic (e.g., T² or SPE) of the false alarm sample and the threshold of that statistic. In this work, 10-fold cross-validation is conducted using normal operation data to select features that result in similar FAR and FAM in the validation data, and feature selection is conducted through exhaustive search. A more systematic approach is under investigation. In the end, the following four features were selected: mean ( μ ), standard deviation ( s ), slope of linear regression line ( S L L ) and mean absolute error ( M A E ). It is worth noting that all median or quartile based robust features were not selected in this work. The possible reason is that since the PSA process is tightly controlled, these robust (hence less sensitive) features do not offer advantage over mean-based features that are more sensitive to changes in FAR and FAM.

After feature selection, a multivariate statistical model can be developed to extract the patterns of normal cycles by examining the correlations among all features. This model enables the determination of a boundary or threshold for process monitoring. In this work, we assume that under normal operations, the features form a multivariate normal distribution. Since all features are the averages of some functions of multiple measurements in a step/cycle, their distributions are asymptotically Gaussian (Pène, 2005). Similar to He and Wang (2011), here we choose PCA to capture the directions of maximum covariances among all the features. Other SPM methods, such as independent component analysis (ICA), can be applied as well.

Because the features in FSM are usually different types, it is reasonable to scale the training feature matrix F T R to zero mean and unit variance for correlation based PCA as follows. F T R = T P T + F ∼ T R = T P T + T ∼ P ∼ T (17) where P ∈ R F × P is the loading matrix with its columns containing the directions of the first P (i.e., the number of principal components) maximum correlations among all features in descending order. T ∈ R M × P is the score matrix with its columns containing the projections of F T R onto P . P ∼ and T ∼ are the residual loading and score matrices, respectively. The principal component subspace (PCS) is S P = s p a n { P } , which captures the systematic variations of the normal process operation, including measured and unmeasured disturbance, as well as set-point changes. The residual subspace (RS) is S R = s p a n { P ∼ } , which captures the remaining variations after subtracting the systematic variations. They are the random variations of the process, including measurement noise under normal operations.

PCA based fault detection is well established for monitoring multivariate processes at steady state, where Hotelling’s T² can be employed for monitoring variations in the PCS while SPE or Q statistic can be employed for monitoring variations in the RS. By monitoring features of individual cycles, we can straightforwardly extend PCA for monitoring cyclic processes which are inherently non-steady state. Similar to PCA-based monitoring of a continuous process, T² can capture faults that shift away from the normal operation region without violating the covariance among measured/monitored process variables. These faults are usually large operational changes such as a change of feedstock or raw material. On the other hand, SPE are sensitive to the process faults that violate the collinear relationships among the monitored features. The control limits of T² and SPE can be defined theoretically based on the Gaussian assumption of the features. They can also be determined empirically, e.g., by kernel density estimation (KDE). The latter is used in this work.

By design, PSA processes are tightly controlled to operate in a targeted optimal region. Therefore, we expect there are few process changes that could violate the threshold in PCS and are detectable by T². In addition, as multiple features included in the FS are closely related to each other, such as the same features from different steps, there could be significant collinearities among features. Therefore, we expect SPE to be sensitive to the process faults with small magnitude but violating the collinearities among features, and enable early detection of potentially catastrophic faults. As shown in Section 5.2, both the traditional multi-way PCA (MPCA) and the proposed FSM detected the faults largely through SPE, as expected.

Once a fault is detected by SPE statistic, the contribution plot can be used for fault diagnosis. In this work, we propose a hierarchical fault diagnosis using SPE to first determine in which step the fault occurred based on the step-wise contribution plot, then postulate what type of fault occurred based on the feature-wise contribution plot.

At the step level (i.e., step-wise diagnosis), SPE statistic is broken down by step: S P E = ∑ j = 1 T S P E j = ∑ j = 1 T ‖ f ∼ j ‖ 2 (18) where S P E j = ‖ f ∼ j ‖ 2 = ∑ k = 1 Q j f ∼ k 2 is the contribution from the j t h step. f ∼ k 2 is the residual (row) vector of features extracted from step j . The bar chart of S P E j S P E × 100 %   ( j = 1 ⋯ T ) provides information for the step-wise diagnosis for the faulty cycle. The step(s) with the highest contribution(s) are identified as the root cause(s). Note that if different number of features are used for different steps, a normalization (e.g., dividing by number of features) can be applied. Once the faulty step is identified, a feature-wise diagnosis is performed to identify the nature of the fault. The bar chart of f ∼ k 2 S P E j × 100 %   ( k = 1 , ⋯ Q j ) provides information for the feature-wise diagnose for step j. For example, if mean contributed significantly to S P E j , then more likely there were step change(s) in the pressure profile during step j . Similar diagnosis can be made for other features. It is worth noting that the features proposed in this work specifically target the PSA process. For other processes, other features, including auto- and cross-correlation coefficients and higher-order statistics (HOS), could and have been utilized for quantifying process dynamics and nonlinearity (J. Wang and He, 2010), (He and Wang, 2011).

In this section, we use an industrial PSA case study to demonstrate the performance of the proposed FSM method, and compare it to the traditional MPCA method. Note that the method proposed by Wang et al. (2017) is not applicable because pressure measurements are the only available measurements for process monitoring in this study. The patented method by Arslan et al. (2014) cannot be implemented either, because there are no details as how the peaks are defined or classified as “relevant”, and the criteria used for peak selection and control limits determination are unknown. The state space modeling approach by Pan et al. (2004) is not applicable due to the absence of cycle-to-cycle dynamics of the PSA processes and the lack of quality-relevant measurement for the PSA process studied in this work. Because MPCA requires that each step across all cycles has the same duration, two different data preprocessing techniques are studied: one with simple cut denoted as MPCA_SC and the other with dynamic time warping (DTW) denoted as MPCA_DTW. For MPCA_SC, the shortest step durations across all cycles are used as the reference while the last few measurements of any cycle with longer step duration are simply removed to match the shortest, which resulted in 438 variables for the whole cycle. For MPCA_DTW, the number of variables after unfolding is 705. The significant difference in the number of variables for MPCA_SC and MPCA_DTW reflects the significant variation of step durations across different cycles. For FSM, four features are used for each of the 15 steps, which resulted in 60 variables. To ensure that enough number of samples are available for all methods, 2,070 cycles under normal operations are used as the training set, which is about three times the number of variables for MPCA_DTW. The first 1,449 cycles (70% of 2,070 cycles) are used for model training and the remaining 621 cycles for model validation.

Six fault scenarios of a PSA process are studied in this work, which are listed in Table 1. The first four are simulated faults while the last two are from real industrial data. For the simulated faults, similar faulty process behaviours have been observed in actual operations. They are reproduced based on historical plots of those faults because the historical data are no longer available. All faults are related to valve malfunctions. For example, an internally leaky valve could result in higher or lower pressure in one vessel depending on whether the vessel serves as a pressure provider or receiver, such as the fault scenarios 1, 3 and 5. A sticky valve could result in higher pressure variation, such as the fault scenario 2, sudden pressure increase or drop, such as the fault scenario 4, or a non-smooth (e.g., zig-zag) pressure profile, such as the fault scenario 6. As discussed in the Introduction section, due to frequent open and close operations of valves, it has been found that the process faults are most often caused by valve-related problems. For each fault scenario, totally 16 cycles are used as the test set and among which 3 (deliberately arranged as cycle 4, 9 and 14 for better comparison across all scenarios) are faulty cycles. In these cases, the simulated faults are introduced by modifying a normal cycle randomly selected from the industrial data set. The pressure trajectories of the test set for fault scenarios 1 (simulated fault) and 5 (real fault) are shown in Figure 4.

For all fault detection methods, the number of principal component (PCs) is selected to cover 90% of the variance of their corresponding full feature space. The control limits on Hoteling’s T² and squared prediction error (SPE) are calculated empirically based on the kernel density estimation of the corresponding statistics (i.e., T² or SPE) of the training dataset at confidence level 99%. The number of PCs and other information discussed above are listed in Table 2.

By considering faults detected in both residual subspace using SPE and principal subspace using T², the overall fault detection results are shown in Table 3. Specifically, the table lists faulty cycles detected by either SPE, or T², or both. The fault detection rate (FDR) and false alarm rate (FAR) of each method are also summarized in Table 3. These results show that FSM detects all faulty cycles under all fault scenarios without generating false alarms. In comparison, MPCA_SC has missed detection under fault scenario 1, while MPCA_DTW has missed detection under fault scenario 5. In addition, both MPCA_SC and MPCA_DTW have false alarms. More details of the fault detection results by T² and SPE are shown in Table 4. It can be seen that SPE statistic in general is more effective in detecting faults, although it also generates false alarms in the cases of MPCA_SC and MPCA_DTW. In comparison, T² does not generate false alarms for all methods. However, it misses several faults in MPCA_SC and performs even worse in MPCA_DTW. Overall, FSM performs robustly with both T² and SPE and is significantly better than MPCA_SC and MPCA_DTW.

To provide additional details on the performance of different methods, we visualized some fault detection results. Due to limited space, only the detection results of fault scenarios one and five in the residual subspace (i.e., using SPE statistic) are visualized in Figures 5, 6 and discussed in detail below. Figure 5 shows that for fault scenario 1, MPCA_SC has difficulty in detecting Fault 1: missing two out of three faulty cycles. MPCA_DCW detects all three faulty cycles but also generates a false alarm. Only FSM detects all three faulty cycles without generating false alarms. Figure 6 shows that for fault scenario 5, MPCA_SC detects all three faulty cycles while generating a false alarm. MPCA_DCW failed to detect two out of three faulty cycles while generating a false alarm. Again, only FSM successfully detects all faulty cycles without generating false alarms. It is worth noting that FSM results in linear models, which have low risk of overfitting. This is demonstrated in Figure 5C and Figure 6C where the normal test samples have similar SPE values as those of the normal training samples.

We further investigated the false alarms (i.e., cycle 10 for MPCA_SC and cycle 15 for MPCA_DTW) to understand why those two cycles generate false alarms. Figure 7 plots the trajectory of all test cycles for the fault scenario 5. All true faulty cycles are plotted in red dashed lines, and all normal test cycles are plotted in black solid lines—except cycle 10 in the cyan dash-dotted line, and cycle 15 in the green dotted line. As described in Table 1, the pressure profiles of the faulty cycles do not follow the normal cycle pressure trajectory during the re-pressurization step (highlighted in the blue dashed-line box). It can be seen that cycles 10 and 15 both behave normally during the re-pressurization step. However, if we zoom in to visualize the pressure profile in other steps (e.g., the insert in Figure 7, which is the zoom-in view of the adsorption step), it can be seen that cycles 10 and 15 are at the lower boundary of all normal cycles, suggesting that the MPCA based approaches may be more sensitive to mean shift. Although the mean shift in this case is within the normal operation range, the cumulative effect (i.e., the persistent small shift that lasts for a period of time) would likely be captured by MPCA as a fault.

The normal samples (i.e., cycles) in the test data are industrial data collected from the same PSA process and used for all the fault scenarios. For each faulty cycle, the fault occurred during different steps for different fault scenarios as defined in Table 1. As a result, the false alarms for a specific method across different fault scenarios are the same (e.g., cycle 15 for MPCA_SC and cycle 10 for MPCA_DTW) since they are the same normal cycles used in all fault scenarios. Clearly, the value of FAR in Table 1 would be affected by the normal cycles included in the test data. In this study, the normal testing samples were selected randomly to avoid any potential bias.

Further investigation is conducted to understand the reason for MPCA_DCW’s failure in detecting true faulty cycles under fault scenario 5 (a real fault), for which only one out of three faulty cycles is detected. Since MPCA_SC is able to detect all faulty cycles, we suspect that the failure is related to data preprocessing by DTW. Therefore, we plot the original pressure profiles of the 16 test cycles, which are shown in Figure 8A and compare them to the pressure profiles after DTW as shown in Figure 8B. A zoom-in view of the faulty step is included for both figures. The comparison clearly indicates that the irregular discrepancies of the faulty cycles shown in the original pressure profiles diminished after trajectory synchronization by DTW. This case suggests that DTW could cause severe information loss or distortion, which in turns affects the fault detection performance. This is consistent with our previous findings that data manipulations during preprocessing, including DTW, could cause information loss or distortion and should be avoided if possible (He and Wang, 2011). This example further raises the alarm that the widely used DTW for batch trajectory warping or alignment in process monitoring applications could potentially be a problematic practice that may lead to missed detections of process faults.

After fault detection, fault diagnosis is performed for the detected faulty cycles. For FSM, the procedure outlined in Sec. 4 is followed to construct step-wise and feature-wise contribution plots. For MPCA_SC and MPCA_DTW, since pressure is the only measured process variable, only step-wise contribution plot is applicable. Again, we use fault scenarios 1 and 5 as examples. For fault scenario 1, test cycle four is used for illustration as the fault is detected by all methods. For the same reason, test cycle nine is used for fault scenario 5. Figures 9A–C show the step-wise contribution plots for fault diagnosis of test cycle four in fault scenario 1. From Table 1, we know that this is a fault occurring during the adsorption step (i.e., step 1), and the faulty cycles have lower pressure than normal cycles. Figure 9 shows that MPCA_SC incorrectly attributes this fault to step 10, while MPCA_DTW and FSM correctly identify the faulty step, with FSM providing the strongest conviction. For FSM, once the faulty step is identified, the faulty step contribution is further broken down to the feature level following the procedure outlined in Sec. 4, as shown in Figure 9D. In this figure, MAE and the mean ( μ ) are identified as the most significant contributor to the fault, indicating that there might be a mean shift in pressure profile during step 1 of the faulty cycles. It is worth noting that MAE usually contributes the most to the fault detection indices as it captures the absolute deviation of a faulty trajectory from the nominal behavior, which includes the effects of both mean and spread (e.g., standard deviation) of the samples.

The fault diagnosis for cycle nine of fault scenario five is conducted similarly and the results are shown in Figure 10. Here the true fault is a deviation of the pressure profile from the normal trajectories during the last step of the cycle. In this care, both MPCA_SC and FSM correctly identify the faulty step, while MPCA_DTW misidentifies the step 10 as the faulty step. FSM also shows the clearest diagnosis among all methods. Feature-wise diagnosis from FSM identifies MAE and the standard deviation (s) as the most significant contributors to the fault, indicating that there could be an increased variation in pressure profile during the last step of the faulty cycle. The misdiagnosis of this fault by MPCA_DTW can be at least partially attributed to the pressure profile distortion during the DTW preprocessing step as discussed earlier and illustrated in Figure 8.

The overall fault diagnosis results are shown in Table 5. It can be seen that FSM correctly diagnoses all fault scenarios (i.e., correctly identifies all faulty steps). MPCA_SC has two misdiagnoses (fault scenarios three and 6) and one inconsistent diagnosis from T² and SPE (fault scenario 1). MPCA_DTW has two misdiagnoses (fault scenarios five and 6). In addition, FSM provides meaningful diagnoses at the feature level for all fault scenarios.