## 7 An Illustrative Example of the RFD  
### Experimental Study with the Three-Screw Spindle Oil Pump  
The primary goal of the illustrative part of the paper is to show the details of designing of the MLP and its application in the RFD of the oil pump [56]. The scheme of such system is presented in Figure 6.  
Such class systems are widely applied in many industries e.g., the chemical, petrochemical and automotive [57; 58; 59]. It is worth to emphasize that the appropriate monitoring of such systems being part of several industrial plants allows for an early RFD and avoidance of the economic losses. The relation describing the diagnosed pump is the following: \(y_{k}=f(\mathbf{u}_{k},\mathbf{p})\) where:  
Figure 5: The adaptive threshold in the RFD. * \(u_{k,1}\)--the differential pressure between inlet and outlet of the pump,
* \(u_{k,2}\)--the motor speed,
* \(y_{k}\)--the torque of the pump. To simulate the faults in the diagnosed system a simulator of the pump was implemented in the Simulink environment. On the basis of such simulator the training, validation and faults containing data sets required to the research were generated. All data were generated with sampling time \(t=0.01\) s. The training data were used to estimate the MLP parameters in the fault-free mode. The validation data were applied to check the generalization ability of the MLP. Moreover, it allows to confirm the increase of the modeling quality resulting from the application of the developed SDED approach. The faults containing data were used to show the effectiveness of the MLP-based RFD method. On the beginning of the system identification procedure the MLP consisting of neurons with the hyperbolic tangent transfer function in the hidden layer and one neuron with the linear transfer function in the output layer were assumed. Subsequently, the structure of MLP was optimized with the application of the Cascade Correlation Method (CCM) [45] to minimize the MLP uncertainty following from model over-parametrization. Finally, the neural model consisting of 8 neurons in hidden layer was obtained. After obtaining the MLP architecture, the initial training process with the LM algorithm was made. Next, the SDED algorithm developed in Section 5 was used in order to choose the training data samples which bring the most information in the training process. Figures 7 and 8 show the chosen data samples for inputs \(u_{k,1}\) and \(u_{k,2}\), where circles represent the measurements which taking into account during the training results in decreasing of the MLP uncertainty. Subsequently, in the Section 4 the developed modified non-linear quasi OBE algorithm was used to retrain the network and in the consequence to decrease MLP uncertainty. Figure 9 shows the comparison of the modeling quality with and without the application of the developed SDED approach. Simultaneously with the retraining of the MLP, the feasible parameter set was estimated with the application of the proposed MNQOA and adaptive threshold containing the outlet of the pump in the fault-free mode was calculated (c.f. Figure 10). Figure 6: Three-screw spindle oil pump. Figure 8: Additional dedicated data samples selected for input \(u_{2}\). Figure 7: Additional dedicated data samples selected for input \(u_{1}\). Figure 9: Real and modeled pump torque estimated with and without Sequential D-optimum Experimental Design (SDED). As it can be observed, the application of the developed MNQOA join with the SDED radically decrease the neural model uncertainty. Therefore, the narrower adaptive threshold is obtained and more sensitive the RFD system is achieved. Such designed FD system allows to detect even small or incipient faults. To validate such scenario the data containing the 20% performance degradation of an induction motor feeding the pump were simulated. Thus, in the case of such a fault, the system (1):  
\[y_{k}=f\left(\mathbf{p},\mathbf{u}_{k}+ \begin{bmatrix}0\\ 1\end{bmatrix} f_{k}\right)+\varepsilon_{k}, \tag{51}\]  
where  
\[f_{k}=\left\{