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An industrial process monitoring scheme with moving window slow feature analysis

With the development of the modern industries, the requirement for comprehensive and effective monitoring scheme of the industrial production process is growing significantly. Conventional monitoring methods treat the deviations as the abnormities and thus result in the invalid monitoring results, because the dynamic information cannot be extracted accurately, which may be caused by the transient process or new operation conditions, and real faults cannot be separated from the normal process changes. To cope with this limitation, a moving window slow feature analysis is proposed in this paper. First, the temporal dynamic features of the industrial production process are extracted to separate the temporal dynamics from the steady state. Second, an adaptive monitoring strategy is presented to accurately acquire the normal changes of the production process, including the normal shift of operation conditions and the slow time-varying behaviors, through updating model parameters and monitoring statistics when a query sample comes. In this way, the real dynamic anomalies can be distinguished from the normal dynamic behaviors and reduce the false alarms effectively. Finally, the effectiveness and practicality are demonstrated through an evaporation process.


INTRODUCTION
With the development of the instruments and measure technology, numerous sensors are positioned in various operating units to reveal the status of the industrial production process (Souza et al., 2016). Due to the huge amount of process data stored in the data bank, data-driven monitoring approaches can transform the historical data to process information and effectively detect faults and abnormal events in the process, where accurate mechanical models are hard to be constructed (Yan et al, 2016).
As the data-driven method, multivariate statistical process monitoring (MSPM) techniques, including principal component analysis, partial least squares and their derivative methods, have been studied and achieved the remarkable success in the recent decades (Dong et al., 2018, Jiang et al., 2016, Wang et al., 2018, but they may trigger false alarms in the practical application because of the time-varying behaviors of the production processes, such as equipment aging, load changes, catalyst deactivation and preventive maintenance. Traditionally, once the deviation between the actual monitoring data and the preset control limits of the monitoring model occurs, the fault alarms are raised. However, the deviations may be resulted from the transient process which could be well-compensated by the control strategy or the normal shifts of the operating conditions, so the operators and engineers have to distinguish these unnecessary false alarms based on their experience and knowledge in practice, which reduces the work efficiency and even ignores the real faults. To remove the false alarms, an alternative process monitoring method based on slow feature analysis (SFA) has been proposed recently (Shang et al., 2015, Zhang et al., 2017. Unlike the conventional MSPM methods, SFA-based monitoring method can better describe the temporal behaviors in the process through the extracting the slow latent variables, hence, the temporal dynamics could be isolated from the steady conditions and the reasons for the temporal dynamics could be analyzed by constructing different monitoring statistics. The meaningful information provided by SFA-based monitoring model can help operators and engineers separate the real faults from the nominal dynamics in the process. To maintain a good monitoring performance, adaptive schemes should be implemented to update the SFA-based monitoring model. Shang et al. (2018) proposed a recursive SFA method to update the monitoring model when new samples are collected. Though the false alarms can be reduced, the nonlinear variation features cannot be captured effectively because SFA is a linear method. Jiang, et al. (2019) presented a locally weighted SFA based on justin-time learning framework to deal with the missing data and predict octane number barrel values, but the continuity of time series production data could be violated. Yu, et al. (2019) proposed a recursive exponential SFA method to extract the nonlinear features and update the monitoring model. Nevertheless, as the new collected samples increase, the complexity and computational burden of the monitoring model will be a challenge. Therefore, how to effectively address the issues of nonlinear and slow time-varying characteristics and correctly update the monitoring model to precisely distinguish the normal process changes from the anomalies should be further studied.
To solve the aforementioned limitations, an adaptive monitoring scheme integrated moving window (MW) and SFA, named as MWSFA, is proposed. As slow and stable changes occur in the process, it is necessary to update MWSFA model from time to time so that the complex nonlinear features of the process data can be well captured and the different operating conditions can be well adapted. Besides, the temporal dynamics and steady state can be separated, and the real dynamic anomalies and the normal process dynamic behaviors can be distinguished based on the developed static and dynamic statistics for decreasing the false alarms.
The remainder of the paper is structured as follows: Section II the preliminaries of SFA are briefly revisited. The detailed procedures of the proposed MWSFA method are presented in Section III. Section IV discusses the effectiveness and practicability of the MWSFA method via an evaporation production process. Finally, the conclusions are drawn in Section V.

REVISIT OF SFA
SFA method can extract features based on the temporal slowness, which contain more relevant information about the process. It aims at finding a set of function   g x that maps the time series data   t x to their features   t s (Zhang et al., 2012). SFA method can be described as an optimization problem (Wiskott et al., 2002), where s  indicates the velocity of the feature, which is , and t s represents the time averaging of s , which can be formulated as For the linear SFA, the mapping functions are When the time series data x is zero mean, the optimization problem of SFA can be transformed into (Yu et al., 2019), Consequently, the Lagrangian multiplier method can be applied to solve this generalized eigenvalue decomposition (GED) problem, which is, , and  is a diagonal matrix, which consists of the singular values of 1 Finally, the features s can be calculated as, According to the variation speed of features, s can be further divided into where d s indicates the slow variation of data which tend to catch essential process variations, and e s means the residual features with fast variations which is regarded as noise. The number k of slow features can be determined when the cumulative percentage (CP) of the derivative values reaches a preset threshold value, which is defined as, 3. METHODOLOGY OF MWSFA

The derivation of MWSFA
The covariance of where std X  is the first-order derivative of std X .
From the perspective of moving window, the new data block is added while the old one is dropped out, which means the model data set is changed from , and its corresponding mean vectors X m and standard deviation X Σ are also updated, which can be calculated as, Hence, the normalized updated data block is, According to the Eq.(11), the relationship between ˆs td  and std  can be represented as, Therefore, the updated matrix ˆs td A and ˆs td B can be iterated by the previous data information instead of recalculation, which reduces the calculation loadings. Then, according to Eq.(6), the singular value decomposition (SVD) is applied to solve the GED problem for obtaining the corresponding eigenvector matrix Ŵ , and the updated features ŝ could be obtained by Eq.(7).

Adaptive monitoring scheme based on MWSFA
The static deviation and dynamic fluctuation of the process are analyzed according to two pairs of monitoring statistics indices (static and dynamic indices), respectively.
The static indices are utilized for evaluating the systematic static variation and residual information, which are where 2 d T and 2 e T represent the systematic tendency and residual information of the production process, respectively. Because s obeys independently Gaussian distribution, 2 d T follows a χ² distribution with k degrees of freedom, and 2 e T follows a χ² distribution with M k  degrees of freedom.
The dynamic indices are developed for considering the temporal fluctuation in the process. To be specific, the distribution of dynamic variations can be measured by

X
, which is also normalized by X m and X  .
5. Compare with the control limits and analyze the query data based on the above 4-case monitoring scheme to make the corresponding actions.

CASE STUDY
In this section, the closed-loop evaporation process is applied to validate the effectiveness and practicality of the proposed MWSFA method. The feed materials will be evaporated by the steam to produce the product in the evaporation process. In the evaporator, the material level should be in the safe range, in case of evacuation and overflow, so the level controller should be used to adjust the level. Besides, the steam provides the heat for evaporating the feed materials to obtain the required product, and the quantity of heat will have an effect on the quality of the product, hence the steam flow should also be controlled by the controller for adjusting the quantity of heat. The evaporation process control diagram is shown in Fig 2.     S are calculated by the extracted slow features, which reflects the essential process variations, it reveals the real fault occurred and the fault alarm should be triggered. According to the process analysis, the valve for the feed materials was out of control and the corresponding maintenance actions should be taken in time.

CONCLUSION
In this paper, an adaptive monitoring scheme integrated SFA and moving window is developed for distinguishing the real dynamic faults and reducing the nuisance alarms in the process. The proposed MWSFA method cannot only extract slow features to better capture the general tendency of the process state, but also can update the monitoring model with the new arriving samples for adapting the slow time-varying behaviors in the process. Based on the proposed MWSFA method, the temporal dynamic information of the process can be separated from the steady state completely for analyzing the temporal behaviors of the process. Besides, the real anomalies can be distinguished effectively from the normal behaviors, so the invalid alarm can be removed and help operators focus on the real process faults. The effectiveness of the MWSFA-based adaptive monitoring scheme is validated by a practical production process and the results demonstrated the MWSFA method can accurately detect the real process faults and adapt the normal process variations.