## 3 Model-Based RFD  
Let us assume that the following relation is used to describe the diagnosed system:  
\[y_{k}=f(\mathbf{p},\mathbf{u}_{k})+\varepsilon_{k}, \tag{1}\]  
where variables \(\mathbf{u}_{k}=[u_{k,1},u_{k,2},\ldots,u_{k,n_{k}}]^{T}\) and \(y_{k}\) denote the diagnosed system inputs and the corresponding output. The expression \(f(\cdot,\cdot)\) denotes the general relation describing the diagnosed non-linear system behavior, \(\varepsilon_{k}\) is a noise/disturbances and vector \(\mathbf{p}\) represents model parameters. In practice such values represent the real parameters of systems which are manufactured during the process of production. Unfortunately, as the result of the influence of the disturbances during the manufacturing process the obtained parameters can be biased and in consequence, the faulty systemcan be obtained. Similarly, the changes of the parameters during the system operation may denote that the system is faulty. One of the most popular FD method depends on the application of a diagnosed system model. Such model usually is obtained involving the known physical laws specifying the diagnosed system or it can be created in the process of system identification [3]. During the system identification, an analytical model of the real system is achieved. It imitates the nominal system behavior despite that its structure does not reflect the system structure. Therefore, the quality of the model is a significant factor which determines how efficiently the model-based FD system operates. The model-based FD relies on the calculation of the spread between the system and model outputs which is called residuum. Such a signal may contain information about faults and may be used for the fault detection, isolation and identification [9; 11]. Usually, the fault detection stage of the FD comes down to check out whether the absolute value of the residuum is smaller than the assumed constant threshold \(\gamma\). \[|y_{k}-y_{m,k}|\leq\gamma. \tag{2}\]  
If such a condition is not fulfilled for any value of \(k\) the diagnosed system is faulty. In the literature several examples of using constant threshold obtained with the use of the MLP can be found, e.g., in the power systems [41], nuclear plant [42] and valve actuators [43]. It should be noticed that such relatively simple FD method is usually safer in real applications. It may arise from the fact that such technique is not robust against uncertainty. Among the sources of uncertainty the variance of the system parameters, time-varying properties of the system, disturbances and measurements noise, model uncertainty resulting from its inappropriate structure and the parameters estimation errors can be distinguished [8]. In such a case the variation of the residuum caused by the noise and/or uncertainty does not enable to perform the reliable fault detection process and leads to undetected faults or signalization of the false alarms. To avoid such situations wider thresholds for the fault detection are assumed. Unfortunately, such solution reduces the sensitivity of the system for fault detection. To overcome this problem a new RFD scheme is proposed where the MLP [3; 44] with its uncertainty description during the nominal/non-faulty system identification process is obtained. It is assumed that the parameters of the MLP (c.f. Figure 1) are approximated by the feasible parameter set \(P\). Its size mainly depends on the inaccuracy of parameters estimates, neural model structure errors and represents neural model uncertainty [8]. Figure 1: The architecture of the Multi-Layer Perceptron (MLP) used for the system identification. During the research the following structure of the neural model is applied:  
\[y_{m,k}=\mathbf{P}^{(l)}\mathcal{F}(\mathbf{P}^{(n)},\mathbf{u}_{k})=f(\mathbf{p},\mathbf{u}_{k}), \tag{3}\]  
where vector of neural model inputs is defined as \(\mathbf{u}_{k}\in\mathbb{R}^{n_{u}+1}\), i.e.