# Fault Detection, Diagnosis, and Prognosis of a Process Operating under Time-Varying Conditions
## 4 Case Study  
The study presented in this paper can be defined as the continuation of research proposed in a previous publication [5]. In the first stages of the prior research, a methodology for managing the non-linearities of a process was proposed to allow its monitoring by using a well-known multivariate statistics technique, i.e., CVA. As discussed in the introductory  
Figure 1: Methodology for the proposed model. part of the paper, countless monitoring challenges arise in the processes that operate under time-varying conditions, and, for this reason, in-depth research is necessary. Considering the level of detail with which the fault detection application is described in the previous contribution, the authors refer to that paper for further description of the implementation [5]. As previously mentioned, fault detection has a double role in this research:  
* Input for the prognostic and decision-making step. * Continuous process control to compensate for errors and inaccuracies in the prognostic model. The production process considered in this contribution is a dataset from a real-world industrial plant. This ensures that the case study is relevant for research. As asserted before, the distinguishing features of the considered production process are the management and the structure of the production process itself. The process does not operate continuously, but it only runs in specific time windows, defined by managerial choices or necessary maintenance interventions. The production intervals taken under analysis are:  
* 2-12 January (A);
* 11-21 February (B);
* 4-15 May (C);
* 1-22 June (D);
* 29 July-17 September (E). Another feature that exacerbates the process monitoring complexity is its division into seven sub-phases, each having specific process characteristics and, consequently, specific anomaly situations:  
* \(p_{1}\) = Initializing;
* \(p_{2}\) = Conditioning;
* \(p_{3}\) = 1 st Spray;
* \(p_{4}\) = Heating;
* \(p_{5}\) = 2 nd Spray;
* \(p_{6}\) = Drying;
* \(p_{7}\) = Unloading. There are 13 monitored parameters, and every data point \(d\) = \(\{x_{1}\),..., \(x_{13}\}\) represents a set of 13 measures, with a time rate of 1 min:  
* \(x_{1}\) = Spray Percentage;
* \(x_{2}\) = Air IN Flow;
* \(x_{3}\) = Spray Flow;
* \(x_{4}\) = Air Pressure Spray;
* \(x_{5}\) = Microclimate Pressure;
* \(x_{6}\) = Cleaning Pressure;
* \(x_{7}\) = Air IN Temperature;
* \(x_{8}\) = Washing Air Temperature;
* \(x_{9}\) = Air OUT Temperature;
* \(x_{10}\) = Product Temperature;
* \(x_{11}\) = Cooling Temperature;
* \(x_{12}\) = Absolute Air IN Humidity;
* \(x_{13}\) = Relative Air IN Humidity. The results obtained in the first phase of the project paved the way for the inclusion of an additional section, with the aim to create a complete methodology that adds more value to the implemented maintenance process. Currently, the previous research focuses only on fault detection, i.e., real-time monitoring of the process to identify trends that are not consistent with what is expected, and that are consequently identified as faults. However, as mentioned at the beginning of the paper, in this study greater attention was paid to the decision-making process. To achieve this goal, the methodology was divided into two subphases:* A phase of process structure analysis in a state of failure to circumscribe its most indicative and representative variables;
* Prognostic analysis of the failure based on information obtained in the previous phase to predict the machinery's useful life by monitoring only a subset of variables. This allows for streamlining of the analysis process and to eliminate any noise in the model caused by variables that are not representative of the failure. Contribution plots were used to analyze the role of the variables related to failure. The role of a contribution plot is to show the contribution of each process variable to the calculated statistic, in this case, \(T^{2}\) and \(Q\). In this way, when the statistic is out of control, it is possible to identify which variable caused the anomaly.