## 4 Training of NN Weights  
As the true nominal NN weights \(W^{*}\) remain unknown, in this section, we devise a composite-learning OBE-based identification algorithm to estimate \(\hat{W}\) in real time. To lay the groundwork, we present the following definition of an ellipsoid:  
**Definition 1**. : _An \(N\)-dimensional real ellipsoid set \(\Omega\), centered at point \(z^{*}\), is mathematically represented as:_  
\[\Omega(z^{*},\digamma):=\{z\in\mathbb{R}^{N}:(z-z^{*})^{T}\digamma(z-z^{*}) \leq 1\}, \tag{33}\]  
_where \(\digamma\in\mathbb{R}^{N\times N}\) denotes a positive-definite symmetric matrix._  
To guarantee the boundedness of the control signal and the stability of the closed-loop system, the estimation result \(\hat{W}\) is expected to be bounded by preset compact sets \(\Omega_{\xi_{wi}}\) as  
\[\Omega_{\xi_{wi}}:=\{\hat{W}_{i}\ |\ \|\hat{W}_{i}\|<c_{wi}\}. \tag{34}\]  
To facilitate the learning law design, a set of filtered variables, \(x_{2f}\), \(\bar{H}_{f}\), \(\Phi_{f}\), \(\bar{\tau}_{1f}\) and \(\zeta_{f}\) are first defined by using a first-order filter \(F_{f}(s)=\frac{1}{ks+1}(k>0)\) as  
\[k\dot{x}_{2f}+x_{2f} =x_{2} \tag{35a}\] \[k\hat{H}_{f}+\bar{H}_{f} =-M^{-1}H_{r}\] (35b) \[k\dot{\Phi}_{f}+\Phi_{f} =\Phi\] (35c) \[k\dot{\tau}_{1f}+\bar{\tau}_{1f} =M^{-1}\tau_{1}\] (35d) \[k\dot{\xi}_{f}+\zeta_{f} =\zeta \tag{35e}\]  
Applying the filter \(F_{f}(s)\) to both sides of (7b) and subsequently rearranging, we obtain  
\[\psi_{f}(t)=\dot{x}_{2f}-\hat{H}_{f}-\bar{\tau}_{1f}=W^{*\bar{T}}\Phi_{f}+\zeta _{f}. \tag{36}\]  
Since \(\zeta\) is bounded, then the filtered optimal NN approximation error \(\zeta_{f}\) is also bounded; that is, \(|\zeta_{fil}|\leq\zeta_{fi}\), where \(\zeta_{fi}\) is the \(i\)th element of \(\zeta_{f}\), \(\zeta_{fi}\in\mathbb{R}^{+}\) is a constant [40]. Employing the \(i\)th column of the matrix \(W^{*}\), denoted \(W^{*}_{i}\), from (35), we derive  
\[\psi_{fi}(t)=W^{*\bar{T}}_{i}\Phi_{f}+\zeta_{fi}, \tag{37}\]  
where \(\psi_{fi}\) is the \(i\)th elements of \(\psi_{f}\),\(i\)=1 to \(n\). Define the identification error as  
\[\tilde{\psi}_{fi}=\psi_{fi}-\hat{\psi}_{fi}, \tag{38}\]  
where \(\hat{\psi}_{fi}(t)=\hat{W}^{T}_{i}\[\Omega_{\xi_{wi}}:=\{\hat{W}_{i}\ |\ \|\hat{W}_{i}\|<c_{wi}\}. \tag{34}\]0 \[\Omega_{\xi_{wi}}:=\{\hat{W}_{i}\ |\ \|\hat{W}_{i}\|<c_{wi}\}. \tag{34}\]1with  
\[\Omega_{\xi_{wi}}:=\{\hat{W}_{i}\ |\ \|\hat{W}_{i}\|<c_{wi}\}. \tag{34}\]2 \[\Omega_{\xi_{wi}}:=\{\hat{W}_{i}\ |\ \|\hat{W}_{i}\|<c_{wi}\}. \tag{34}\]3 (40b) \[\Omega_{\xi_{wi}}:=\{\hat{W}_{i}\ |\ \|\hat{W}_{i}\|<c_{wi}\}. \tag{34}\]4 (40c) \[\Omega_{\xi_{wi}}:=\{\hat{W}_{i}\ |\ \|\hat{W}_{i}\|<c_{wi}\}. \tag{34}\]5 (40d) \[b_{i}= \begin{cases}1,&\text{if }|\bar{\psi}_{fi}|>\overline{\xi}_{fi} \ \&\[PERSON]_{f}^{T}\digamma_{i}\[PERSON]_{f}>\overline{\xi}_{fi}^{2}\\ 0,&\text{if }|\bar{\psi}_{fi}|\leq\overline{\xi}_{fi}\text{ or }\Phi_{f}^{T}\digamma_{i}\Phi_{f}\leq \overline{\xi}_{fi}^{2}\end{cases}  \tag{40e}\]  
for \(i=1\cdots n\), let \(g_{is}\) be the \(i\)th element of \(g_{s}\), \(\lambda_{i}\in\mathbb{R}^{+}\) be a design parameter satisfying \(\lambda_{i}<1/\overline{\xi}_{fi}^{2}\), and \(o_{io}\in R^{+}\) be a forgetting factor satisfying \(0<o_{io}<1\). \(\digamma_{il}\in R^{N\times N}\) is a positive definite diagonal matrix, and \(\digamma_{iu}\in R^{+}\) is a positive constant that satisfies \(\digamma_{il}<\digamma_{iu}I\), where \(I\in R^{N\times N}\) is the identity matrix. The projection operator, denoted by \(\mathbb{N}(\bullet)\), is defined as follows:  
\[\mathbb{N}(\bullet)= \begin{cases}\bullet,\text{if }||\hat{W}_{i}||<c_{wi} \text{ or }||\hat{W}_{i}||=c_{wi}\ \&\ \hat{W}_{i}^{T}\bullet\leq 0\\ \bullet-\hat{W}_{i}\hat{W}_{i}^{T}\bullet/||\hat{W}_{i}||^{2},\text{ otherwise}\end{cases}  \tag{41}\]  
and \(b_{p}(\bullet)\) is given by  
\[b_{p}(\bullet)= \begin{cases}1,\text{ if }||\hat{W}_{i}||<c_{wi}\text{ or }||\hat{W}_{i}||=c_{wi}\ \&\ \hat{W}_{i}^{T}\bullet\leq 0\\ 0,\text{ otherwise}\end{cases}