## 4 Modified Non-Linear Quasi OBE Algorithm  
Having a neural network structure it is possible to obtain its parameters \(\boldsymbol{p}\) in a way that it is possible to minimize the mismatch between its response and the one of the system being modeled. Such a process can be realized with a wide spectrum of approaches among which the gradient-based, statistical, evolutionary and genetic algorithms can be distinguished [3]. Such algorithms enable the effective training of the MLP, however, they do not let to obtain the mathematical description of its uncertainty which is required in the RFD. The objective of this section is to propose a novel algorithm which may be used for training of MLP with description of its uncertainty. Subsequently, it allows for gradual decrease of modeling uncertainty of MLP by the application of the SDED approach developed in the Section 5. First, let us start with the development of a suitable parameter estimation algorithm. For that purpose, Modified Quasi OBE Algorithm (MQOA) [17] is utilized and suitably extended to cope with the nonlinearities presented in (3). It goes without saying that the parameter estimate \(\hat{\boldsymbol{p}}\) should be selected as follows:  
\[\varepsilon_{k+1}=y_{k+1}-f(\hat{\boldsymbol{p}}_{k},\boldsymbol{u}_{k+1}), \tag{7}\]  
and should be as small as possible for the entire dataset samples enumerated by \(k=0,\ldots,n_{t}\). One way to settle the nonlinearity issue is to use the [PERSON] expansion-based linearization which leads to:  
\[f(\hat{\boldsymbol{p}}_{k},\boldsymbol{u}_{k+1})\approx\boldsymbol{r}_{k+1}^{ T}\hat{\boldsymbol{p}}_{k}. \tag{8}\]  
Figure 3: Application of the adaptive threshold for the RFD. where the so-called regressor is given by:  
\[\mathbf{r}_{k+1}=\left.\frac{\partial f(\mathbf{p},\mathbf{u}_{k+1})}{\partial\mathbf{p}}\right|_ {\mathbf{p}=\hat{\mathbf{p}}_{k}}. \tag{9}\]  
To avoid the above approximation the following exact equality is introduced:  
\[f(\hat{\mathbf{p}}_{k},\mathbf{u}_{k+1})=\mathbf{r}_{k+1}^{\mathrm{T}}\hat{\mathbf{p}}_{k}+\bm {\omega}_{k}, \tag{10}\]  
where \(\mathbf{\omega}_{k}\) stands for the linearization error which is over-bounded as:  
\[-q\leq\mathbf{\omega}_{k}\leq q, \tag{11}\]  
\[\frac{1}{q^{2}}\mathbf{\omega}_{k}^{2}\leq 1,\quad q>0. \tag{12}\]  
Since the linearization error \(\mathbf{\omega}_{k}\) is unknown, the output error cannot theoretically attain its possible low value. Thus, following the general guidance of the MQOA, it is reasonable to set up a constraint on the magnitude of the estimation error (17), which can be expressed as:  
\[|\epsilon_{k+1}|\leq\gamma,\quad\gamma>0. \tag{13}\]  
Such a constraint leads to the feasible parameter set:  
\[\mathbf{P}_{m,k}=\bigcap_{k=1}^{n_{\mathrm{f}}}\{\mathbf{p}\in\mathbb{R}^{n_{p}}:\quad |y_{k}-f(\hat{\mathbf{p}}_{k-1},\mathbf{u}_{k})|\leq\gamma\}, \tag{14}\]  
rather then to a single parameter estimate \(\hat{\mathbf{p}}_{k}\). After the above-detailed deliberations, it is possible to propose Modified Nonlinear Quasi OBE Algorithm (MNQOA):  
**Initialization:**:  
\[\mathbf{P}_{0} =\delta\mathbf{I}_{n_{p}}, \tag{15}\] \[f(\hat{\boldsymbol{p}}_{k},\boldsymbol{u}_{k+1})\approx\boldsymbol{r}_{k+1}^{ T}\hat{\boldsymbol{p}}_{k}. \tag{8}\]0  
where \(\delta>0\) is a large constant and \(\mathbf{p}_{0}\) is initial parameter estimate.