# Fisher Discriminative Sparse Representation Based on DBN for Fault Diagnosis of Complex System
## 4 Case Study  
The FDSR method based on a deep neural network was applied to the TE process. The flowchart of the TE process is shown in Figure 5, with a plantwide control structure proposed by [PERSON] and [PERSON]. The structure describes the nonlinear relationship among devices, materials, and energy. The system includes five major unit operations: an exothermic two-phase reactor, a vapor-liquid flash separator, a product condenser, a recycling compressor, and a reboiled product stripper. Four gaseous materials are involved, A, C, D, and E, with products G and H having a byproduct of F. In addition, inert gas B is added to the reactants. First, A, D, and E are fed into the reactor, then the products are sent into the condenser. The condensed product flows to a vapor-liquid flash separator, in which the products that have not yet been compressed are recycled by the compressor and sent to reactor, whereas the gases that has been compressed are sent to the stripper to remove the remaining reactants. Then, the byproduct F and inert gas B are removed by the purge valve to obtain products G and H.  
[PERSON] and [PERSON] [48] developed a simulator for the TE process, allowing 21 disturbances (faults), 12 manipulated variables (XMV1 to XMV12), and 41 measured process variables, of which 22 are continuously monitored process variables (XMEAS1 to XMEAS22) and 19 are composition measurements (Table 2). Detailed descriptions of this process can be found in [PERSON] et al. [44]. In this case study, the 22 continuously monitored variables (Table 2) and 11 manipulated variables were monitored for online fault diagnosis with the agitation speed set to 50%. Six kinds of operating modes were possible. We completed the simulation under the first mode, which is the basic mode. The normal datasets were sampled under normal operating conditions, and 21 fault datasets were generated under 21 types of faulty operating conditions, as shown in Table 3. The faults in bold were chosen to test the effectiveness of the proposed algorithm. Faults 16 to 20 were unknown types. Faults 1 to 7 were associated with step variation, faults 8 to 12 were the random causes of variations in the process variables, whereas slow drift caused fault 13, and the sticking leads were faults 14, 15, and 21. In this case study, we used the PI cascade control structure proposed by [PERSON] [48], and the data were sampled every 3 min. Each dataset contained 960 samples. The fault was introduced into the process after the 161 st sample. The task defined here was to diagnose different working conditions based on the process data. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable Name** & **Symbol** & **Variable Name** & **Symbol** \\ \hline D feed rate & XMV1 & Reactor feed rate & XMEAS6 \\ E feed rate & XMV2 & Reactor pressure & XMEAS7 \\ A feed rate & XMV3 & Reactor level & XMEAS8 \\ A and C feed rate & XMV4 & Reactor temperature & XMEAS9 \\ Recycle compressor & XMV5 & Purge rate & XMEAS10 \\ Purge valve & XMV6 & Separator temperature & XMEAS11 \\ Separator liquid flow & XMV7 & Separator level & XMEAS12 \\ Stripper liquid flow & XMV8 & Separator pressure & XMEAS13 \\ Steam valve of stripper & XMV9 & Separator underflow & XMEAS14 \\ Water flow in reactor & XMV10 & Stripper level & XMEAS15 \\ Water low in condenser & XMV11 & Stripper pressure & XMEAS16 \\ Agitation speed & XMV12 & Stripper underflow & XMEAS17 \\ D feed ratio & XMEAS1 & Stripper temperature & XMEAS18 \\ E feed ratio & XMEAS2 & Stripper steam flow & XMEAS19 \\ A feed ratio & XMEAS3 & Compressor power & XMEAS20 \\ A and C feed ratio & XMEAS4 & Reactor cooling water outlet temperature & XMEAS21 \\ Recycle flow & XMEAS5 & Condenser cooling water outlet temperature & XMEAS22 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Manipulated and continuously monitored variables. Figure 5: Tennessee Eastman (TE) process flowchart.