# Fault Detection, Diagnosis, and Prognosis of a Process Operating under Time-Varying Conditions
## 2 Literature Review  
The complexity of modern engineering systems and manufacturing processes is rapidly increasing, and their reliability management becomes more and more challenging in modern dynamic operational settings. In this context, the research area of CBM data management is still immature, with many open challenges at different levels, due to the fact that several CBM approaches are data driven [6] ([PERSON] et al. 2008). Fault detection and diagnosis have always been the focus of analysis to maintain adequate production standards, with multivariate statistical process control proving to be an excellent tool to support these analyses [7] ([PERSON] et al. 2021). In recent years, the processes operating under time-varying conditions, which are at the center of this research proposal, became of great interest in the field of fault detection and maintenance [8; 9; 10; 11]. In these processes, the assumption of the linearity and static nature of the underlying processes represents a risk in the application of multivariate statistical techniques. For this reason, there are currently some limitations for CVA in such processes, with the consequent need, for example, to implement changes to the traditional CVA. If properly managed or modified, CVA proves to be high performing in this context [12; 13]. It should be noted that CVA has achieved excellent results in many maintenance applications, both in its original or modified forms [14; 15; 16]. The fundamental characteristic of CVA is that it calculates linear combinations of the past and future values of the system to maximize the correlation between them [17]. It is crucial to go beyond the assumption that process variables are time independent. The dependence of process variables on time is particularly relevant in processes that show a significant correlation between past and future instances, such as the ones used in the chemical sector. The idea of the variables' time-independence is at the heart of many widely applied multivariate statistics techniques for fault detection, such as principal component analysis. Consequently, unlike other multivariate statistical techniques, CVA allows consideration of the time dependence of variables during process monitoring, thus making this analysis extremely suitable for fault detection in many contexts where the assumption of independence of processes from time is not valid [18]. From the available literature, it was noted that topics concerned with residual useful life and time to failure prediction are largely covered in research areas that involve aspects of the degradation processes affecting a system. Similarly, the issue of predicting the remaining useful life of equipment was found to be associated with research on prioritizing maintenance activities [19]. Since different degradation processes require different approaches [20; 21; 22; 23], a comprehensive review of these processes is fundamental to this research, and future research should aim at developing more accurate models. In this context, the use of machine learning is still an open topic, especially as far as residual useful life and time to failure prediction are concerned. Examples are represented by the application of support vector machine [24; 25], random forests [26] or neural networks [27; 28], and incremental learning [29]. In the literature, the use of CVA is described in combination with other techniques to predict the trend of degradation and the behavior of a system after the onset of a failure [30; 31]. However, in practice, prognosis analysis is more often applied after a failure to monitor its progress. Considering the prediction of residual useful life in support of the decision-making process, it can be said that the understanding of the complex interactions between operating conditions and component capability is crucial to estimate the degradation of a component and predict its remaining useful life, thus allowing decision making that directly impact the results [32], even financially [33].