## 5 Enhancing Performance of Neural Networks with SDED  
The algorithm presented in the preceding section aims at guaranteed convergence iterative improvements of the MLP parameter estimates based on the sequentially provided input and output measurements. It is well known fact that the quality of such an iterative process can be measured using \(\mathbf{P}_{k+1}\)[22; 23]. Having a parameter updating algorithm along with the description of parameter uncertainty it is desirable to develop an algorithm capable of determining additional input values leading to uncertainty minimization. Indeed, while analyzing (20) it can be observed that parameters uncertainty is strongly correlated with input selection policy. In the MQOA developed in Section 4, the ellipsoidal shape of the parameter uncertainty region is employed. Thus, this section aims at developing an algorithm capable of determining a sequence of D-optimal input values [22] minimizing the determinant of the ellipsoidal uncertainty region which is proportional to the volume of such an ellipsoid. The developed algorithm will be called Sequential D-optimum Experimental Design. It should be underlined that the selection of determinant criterion is not accidental as it is a common practice to do it while assessing the parameter estimation quality [53; 54]. The developed iterative improvement algorithm guarantees that the resulting neural network will make it possible to design more reliable fault diagnosis scheme capable of making appropriate decisions about the current state of the system. To proceed with further deliberations, let us start with identifying the determinant of \(\mathbf{P}_{k}\) by exploiting (20):  
\[\det(\mathbf{P}_{k+1})= \det\left(\frac{1}{l_{k}}\mathbf{P}_{k}\right)\det\left(I_{n_{p}}- \frac{\mathbf{P}_{k}\mathbf{r}_{k+1}\mathbf{r}_{k+1}^{T}}{l_{k}+g_{k+1}}\right), \tag{43}\]  
Taking into account the following identity \(\det(\mathbf{I}+\mathbf{a}\mathbf{b}^{T})=1+\mathbf{b}^{T}\mathbf{a}\) yields:  
\[\det\left(\mathbf{I}_{n_{p}}-\frac{\mathbf{P}_{k}\mathbf{r}_{k+1}\mathbf{r}_{k+1}^{T}}{l_{k}+g _{k+1}}\right)=\frac{1}{1+l_{k}^{-1}g_{k+1}}, \tag{44}\]  
which results in more condensed form of (43):  
\[\det(\mathbf{P}_{k+1})=\frac{l_{k}^{-n_{p}}}{1+l_{k}^{-1}g_{k+1}}\det(\mathbf{P}_{k}), \tag{45}\]  
While analyzing (45) and noticing that the regressor \(\mathbf{r}_{k}\) depends on \(\mathbf{u}_{k^{\prime}}\) it is clear that an appropriate input selection may lead to a gradual decrease of \(\mathbf{P}_{k}\) resulting in an enhanced model quality. Therefore, bearing in mind the fact that \(g_{k+1}=\mathbf{r}_{k+1}^{T}\mathbf{P}_{k}\mathbf{r}_{k+1}\), \(\mathbf{u}_{k+1}\) minimizing \(\det(\mathbf{P}_{k+1})\) is:  
\[\mathbf{u}_{k+1}^{*}=\arg\max_{\mathbf{u}_{k+1}\in\mathcal{U}}\mathbf{r}_{k+1}^{T}\mathbf{P}_{ k}\mathbf{r}_{k+1}, \tag{46}\]where \(\mathcal{U}\) denotes an admissible input set, which can be expressed by various constraints, e.g., commonly used linear ones. The above optimization problem can be solved by the application of various global optimization algorithms, e.g., an evolutionary or genetic algorithms or an adaptive random search [55], which is employed within the framework of this paper. To summarize, the structure of the proposed iterative algorithm (cf. Figure 4) is:  
**Step 1:**: Set \(k=0\), determine an initial parameter \(\hat{p}_{0}\) by exploiting the collected input-output data measurement set of the diagnosed system in the nominal state, set \(\boldsymbol{P}_{0}=\delta I\) where \(\delta\) denotes sufficiently large constant. Select \(n_{t}\), the number of gradual improvement iterations. **Step 2:**: Obtain \(\boldsymbol{u}_{k}\) by D-optimal input value, and then fed it into the modeled system to get \(y_{k}\). **Step 3:**: Update \(\hat{p}_{k+1}\) the parameter estimate. If \(k=n_{t}\) then STOP the algorithm, else set \(k=k+1\) and go to Step 2.