## 3 Model-Based RFD
\(\mathbf{u}_{k}=[u_{1},\ldots,u_{n_{u}-1},1]\), where 1 stands for the so-called bias [3] and the values \(y_{m,k}\) represents a neural model output. The vector \(f(\cdot)\) consists of \(n_{h}\) non-linear activation functions \([f_{1}(\cdot),\ldots,f_{n_{h}}(\cdot)]^{T}\) of neurons in the hidden layer, where \(n_{h}\) represents its number. It should be underlined that the non-linear activation functions is depicted as \(f(\cdot)=\tanh(\cdot)\). This assumption is not restrictive and other types of non-linear activation functions can be chosen [3]. Moreover, \(f_{l}(\cdot)\) denotes the linear function of the output neuron of the MLP. Furthermore, both linear and non-linear parameters describing the properties of the identified system may be described as \(\mathbf{p}=[\mathbf{P}^{(l)},\mathbf{P}^{(n)}]^{T}=[p_{1}^{(l)},\ldots,p_{n_{h}}^{(l)^{T} },p_{1,1}^{(n)},\ldots,p_{n_{u},n_{h}}^{(n)}]^{T}\in\mathbb{R}^{n_{p}}\) whereas \(n_{p}=n_{u}n_{h}+n_{h}\). The performance of the fault detection scheme shall be conditional upon the quality of the MLP which is obtained at the system identification stage. Despite the efforts such model can be uncertain. One of the main sources of the MLP uncertainty follows from the inappropriate structure of the MLP. The structure of the network is usually chosen in an arbitrary way which relies on the subjective indication of the number of neurons in each layer what may result in an inadequate network architecture. Such a problem can be reduced by various well-known selection methods of the neural models structure [45; 46; 47; 48; 49]. The description of the MLP uncertainty, defined as \(\mathbf{P}\), is required to design so called adaptive threshold [50]. On the contrary to the constant threshold, the adaptive one over-bounds the diagnosed system output. Its width depends on neural model uncertainty (c.f. Figure 2) described by the feasible parameter set \(\mathbf{P}\) and hence, it provides a much more reliable FD. The adaptive threshold, which includes the response of the diagnosed system \(y_{k}\) in the fault-free case, is defined by the following general form:  
\[\hat{y}_{k}^{m}\leq y_{k}\leq\hat{y}_{k}^{M}, \tag{4}\]  
where \(\hat{y}_{k}^{m}\) and \(\hat{y}_{k}^{M}\) denote the bounds of the adaptive threshold. Such values can be calculated as follows:  
\[\hat{y}_{k}^{m}=\min_{p\in\mathbf{P}_{mk}}y_{m,k}-\gamma, \tag{5}\]  
Figure 2: The diagram of the neural model-based Robust Fault Detection (RFD). \[\hat{y}_{k}^{M}=\max_{p\in\mathcal{P}_{m,k}}y_{m,k}+\gamma, \tag{6}\]  
whereas \(y_{m,k}\) is the estimate of the diagnosed system, \(\boldsymbol{P}_{m,k}\) represents the neural network uncertainty and \(\gamma\) is the noise. Thus, the spread of the threshold (4) depends on the values of the noise and the neural model uncertainty expressed as the feasible parameter set which should be minimized in order to enhance the RFD sensitivity. To achieve such goal a new approach of MLP model improving with the SDED approach is developed. Such approach allows for appropriate training data selection what results in smaller neural model uncertainty and more narrow adaptive threshold. An exemplary application of the proposed threshold is depicted in Figure 3. The values of the intervals (4) vary with the changes of the values of the system input \(\boldsymbol{u}_{k}\). The application of the proposed SDED method enables to obtain the narrow internal. The fault may be detected when the diagnosed system response \(y_{k}\) crosses the upper bound of the designed adaptive threshold. Furthermore, it can be seen that such small faults can not be detected by the wider adaptive threshold obtained for classical MLP without the application of the proposed approach.