3.2 State space models

If it is assumed that the data matrix X contains all the dynamic information of the system, then the use of predictive models can be viewed as an attempt to remove all the dynamic information from the system to yield Gaussian residuals that can be monitored in the normal way. State space models offer a principled approach for the identification of the subspaces containing the data. This can be summarized as follows

$$\mathbf{x}\_{\mathbf{k}+1} = f(\mathbf{x}\_{\mathbf{k}}) + \mathbf{w}\_{\mathbf{k}} \tag{6}$$

x<sup>k</sup> and y<sup>k</sup> are the respective state and measurement vectors of the system, and w<sup>k</sup> and v<sup>k</sup> are the plant disturbances and measurement errors, respectively, at time k. State space models and their variants have been considered by several authors [65–75].

In principle, machine learning models are better able to deal with complex nonlinear systems than linear models, and some authors have considered the use of

perceptron neural network to remove the nonlinear and dynamic characteristics of processes to generate residuals that could be used as input to a PCA model for the construction of simple monitoring charts. Guh and Shiue [63] have used a decision tree to detect shifts in the multivariate means of process data. Auret and Aldrich [48] have considered the use of random forests in the detection of change points in process systems. In addition, Aldrich and Auret [2] have compared the use of random forests with autoassociative neural networks and singular spectrum analy-

The application of deep learning in process monitoring is an emerging area of

Finally, as an example of the application of a process monitoring scheme incorporating feature extraction from time series data in a moving window, the following study can be considered. It is based on the Tennessee Eastman benchmark process widely used in these types of studies. The feature extraction process considered here is an extension of recurrent quantitative analysis discussed in Section 2.5.2. Instead of using thresholded recurrence plots, unthresholded or global recurrence plots are

The Tennessee Eastman (TE) process as proposed by Downs and Vogel [97] and has been used as a benchmark in numerous process control and monitoring studies [98]. It captures the dynamic behavior of an actual chemical process, the layout of

The plant consists of 5 units, namely a reactor, condenser, compressor, stripper and separator, as well as eight components (four gaseous reactants A, C, D, E, one inert reactant B, and three liquid products F, G, H) [97]. In this instance, the plantwide control structure suggested by Lyman and Georgakis [99] was used to simulate the process and to generate data related to varying operating conditions. The

A total of four data sets were used, that is, one data set associated with NOC and the remaining three associated with three different faults conditions. The TE process comprises 52 variables, of which 22 are continuous process measurements, 19 are composition measurements, and the remaining 11 are manipulated variables. These variables are presented in Table 3. Each data set consisted of 960 measure-

research that shows particular promising. This includes the use of stacked autoencoders [76], deep long short term memory (LSTM) neural networks [77], and convolutional neural networks. Table 2 gives an overview of the feature extraction methods that have been investigated over the last few decades.

these approaches. For example, Chen and Liao [62] have used a multilayer

Process Fault Diagnosis for Continuous Dynamic Systems Over Multivariate Time Series

sis in a conventional process monitoring framework.

4. Case study: Tennessee Eastman process

considered, as explained in more detail in below.

data set is available at http://web.mit.edu/braatzgroup.

4.1 Tennessee Eastman process data

which is shown in Figure 3.

ments sampled at 3 min intervals.

11

3.3 Machine learning models

DOI: http://dx.doi.org/10.5772/intechopen.85456

$$\mathbf{y}\_{\mathbf{k}} = \mathbf{g}(\mathbf{x}\_{\mathbf{k}}) + \mathbf{v}\_{\mathbf{k}} \tag{7}$$


#### Table 2.

Approaches to the monitoring of continuous dynamic process systems.

x<sup>k</sup> and y<sup>k</sup> are the respective state and measurement vectors of the system, and w<sup>k</sup> and v<sup>k</sup> are the plant disturbances and measurement errors, respectively, at time k. State space models and their variants have been considered by several authors [65–75].
