# Fault Detection, Diagnosis, and Prognosis of a Process Operating under Time-Varying Conditions
## 1 Introduction  
The increasing complexity of production systems triggers the need for condition monitoring (CM) and condition-based maintenance (CBM) techniques that can be adapted to suit specific requirements. This paper discusses a project aimed at optimizing the application of CBM in industrial settings that operate under time-varying conditions, i.e., processes whose standards change over time. Such processes are widespread in the current industrial scenario, e.g., inconsistent characteristics of the processed product, unfixed quantities of processed products throughout different production cycles, multiphase processes where different operations are carried out by a single machine without a clear shift between phases. Existing CM and CBM techniques are often limited to fixed operating conditions, and their implementation in real settings is very challenging. Consequently, changes that are inherent in the production process can often be mistaken for fault situations, thus causing false alarms. Today, companies and researchers are ever more frequently focusing on machine prognosis [1] in order to obtain a sufficiently large forecasting period before a fault onset, so as to allow the execution of maintenance tasks aimed at preventing its occurrence. This research investigates the application of multivariate statistical techniques, focusing on canonical variate analysis (CVA) alongside with machine learning algorithms for fault detection, diagnosis, and prognosis in time-varying conditions processes. Ultimately, this research aims to define a maintenance model that incorporates the main maintenance steps and areas, i.e., fault detection, diagnosis, and prognosis. The integrated and synergistic management of all these areas not only enhances performance but also ensures a simpler and more streamlined management. Furthermore, a highly debated aspect in the maintenance field is that of the decision-making process. The importanceof decision making and its correlation with fault detection and prognosis is noticeably clear, and, for this reason, it should be always taken into account [2; 3]. However, while it certainly plays an increasingly important role in achieving effective maintenance systems, the definition of a structured decision-making process is particularly challenging. In this regard, the authors believe that the contribution plots that increase the quality of the prognosis analysis can be of invaluable support to the decision-making process for maintenance activities. The role of contribution plots is particularly relevant, as it allows obtaining effective and highly robust models. The decision-making process has significant safety implications as well, especially in processes that are highly hazardous both for the plant and the operator [1; 4]. The research questions to be investigated, as the focus of this paper, are:  
**RQ1:** is it possible to define a comprehensive methodology for the joint management of fault detection, diagnosis, and prognosis, setting the focus of the analysis on the information that can be extracted from the application of CVA? **RQ2:** is it possible to implement an effective selection of the process variables to be considered for the prognosis analysis, by analyzing the relationship between these variables and the CVA health indexes? A fault detection analysis on data gathered during different production intervals is not effective, as varying conditions could change the variables that provide prognosis information. Nevertheless, it is possible to carry out the selection of variables for prognosis analyzing only one production interval. This is because the correlation logic between process variables and degradation of the machinery remains unchanged between various production intervals. Obviously, for subsequent prognosis models, it remains necessary, currently, to train a model for each subinterval.