## 3 Controller Design and Analysis
\begin{array}{l}\widetilde{f}=\underline{\hat{W}_{5}^{T}\underline{\[PERSON]}_{W_{4 }}\widetilde{\underline{W}}_{4}}+\underline{\hat{W}_{5}^{T}\underline{\[PERSON]}_{W_ {4}}\widetilde{\underline{W}}_{4}}\underline{\hat{W}}_{4c}+\underline{\hat{W}_ {5}^{T}\underline{\[PERSON]}_{B_{4}}\widetilde{\underline{B}}_{4}}\underline{\hat{B }}_{4}+\underline{\hat{W}_{5}^{T}\underline{\[PERSON]}_{W_{3}}\underline{\hat{W}}_ {3}}+\underline{\hat{W}_{5}^{T}\underline{\[PERSON]}_{C_{2}}\widetilde{\underline{ C}}_{2}}+\underline{\hat{W}_{5}^{T}\underline{\[PERSON]}_{B_{2}}\widetilde{\underline{B}}_{2}}+ \underline{\hat{W}_{5}^{T}\underline{\Phi}}\underline{\hat{B}}_{4}\\ +\underline{\hat{W}_{5}^{T}\overline{\[PERSON]}_{W_{4}}\widetilde{\underline{W}}_{4 }}+\underline{\hat{W}_{5}^{T}\overline{\[PERSON]}_{W_{4}}\widetilde{\underline{W}}_ {4c}}\widetilde{\underline{W}}_{4c}+\underline{\hat{W}_{5}^{T}\overline{\Phi} _{\overline{W}_{3}}\widetilde{\underline{W}}_{3}}+\underline{\hat{W}_{5}^{T} \overline{\Phi}_{\overline{C}_{2}}\widetilde{\underline{C}}_{2}}+\underline{ \hat{W}_{5}^{T}\overline{\Phi}_{\overline{B}_{2}}\widetilde{\underline{B}}_{2 }}+\overline{\hat{W}_{5}^{T}\overline{\Phi}}\underline{\hat{B}}+\Delta_{0} \end{array}  \tag{29}\]  
where the high-order comprehensive approximation error is expressed as follows:  
\[\Delta_{0}=\underline{\hat{W}_{5}\underline{O}_{h}}+\hat{\overline{W}_{5} \overline{O}_{h}}+\epsilon_{0} \tag{30}\]  
**Theorem 1**. : _For the given APF system in (1), when the controller is designed according to (21) and the parameter adaptive rule is appropriately formulated as follows, the stability of the proposed IT2 FNN-SFR STSMC strategy is ensured:_  
\[ \begin{array}{l}\underline{\hat{W}_{4}}^{T}=\eta_{1}\text{Csgn}(s)\underline {\hat{W}_{5}^{T}\underline{\Phi}_{\underline{W}_{4}}};\dot{\overline{W}_{4}}^ {T}=\eta_{2}\text{Csgn}(s)\underline{\hat{W}_{5}^{T}\overline{\Phi}_{\overline {W}_{4}}}\end{array}  \tag{31}\]  
\[ \begin{array}{l}\underline{\hat{W}_{4c}}^{T}=\eta_{3}\text{Csgn}(s)\underline {\hat{W}_{5}^{T}\underline{\Phi}_{\underline{W}_{4}}};\dot{\overline{W}_{4c}}^ {T}=\eta_{4}\text{Csgn}(s)\underline{\hat{W}_{5}^{T}\overline{\Phi}_{\overline {W}_{4}}}\end{array}  \tag{32}\]  
\[ \begin{array}{l}\underline{\hat{B}_{4}}^{T}=\eta_{5}\text{Csgn}(s)\underline {\hat{W}_{5}^{T}\underline{\Phi}_{\underline{B}_{4}}};\dot{\overline{B}_{4}}^ {T}=\eta_{6}\text{Csgn}(s)\underline{\hat{W}_{5}^{T}\underline{\Phi}_{\overline {B}_{4}}}\end{array}  \tag{33}\]  
\[ \begin{array}{l}\underline{\hat{W}_{3}}^{T}=\eta_{7}\text{Csgn}(s)\underline {\hat{W}_{5}^{T}\underline{\Phi}_{\underline{W}_{3}}};\dot{\overline{W}_{3}}^ {T}=\eta_{8}\text{Csgn}(s)\underline{\hat{W}_{5}^{T}\overline{\Phi}_{\overline {W}_{3}}}\end{array}  \tag{34}\]  
\[ \begin{array}{l}\underline{\hat{\underline{C}}_{2}}^{T}=\eta_{9}\text{Csgn}(s )\underline{\hat{W}_{5}^{T}\underline{\Phi}_{\underline{C}_{2}}};\dot{\overline{ C}_{2}}^{T}=\eta_{10}\text{Csgn}(s)\underline{\hat{W}_{5}^{T}\overline{\Phi}_{ \overline{C}_{2}}}\end{array}  \tag{35}\]  
\[ \begin{array}{l}\underline{\hat{\underline{B}}_{2}}^{T}=\eta_{11}\text{Csgn}(s )\underline{\hat{W}_{5}^{T}\underline{\Phi}_{\underline{B}_{2}}};\dot{\overline{ B}_{2}}^{T}=\eta_{12}\text{Csgn}(s)\underline{\hat{W}_{5}^{T}\underline{\Phi}_{ \overline{B}_{2}}}\end{array}  \tag{36}\]  
\[ \begin{array}{l}\underline{\hat{W}_{5}}=\eta_{13}\text{Csgn}(s)\underline {\hat{\Phi}};\dot{\overline{W}_{5}}=\eta_{14}\text{Csgn}(s)\dot{\overline{ \hat{\Phi}}}\end{array}  \tag{37}\]  
\[ \begin{array}{l}\dot{\hat{k}}_{1}=-\dot{\hat{k}}_{1}=\eta_{15}\text{C}b|s|^{ \frac{1}{2}}\end{array}  \tag{38}\]  
_where \(\eta_{1}\sim\eta_{15}\) are the adaptive learning rate gains of the corresponding parameters, all of which are positive constants._  
Proof. : When \(s\
eq 0\), the Lyapunov function is designed as follows:  
\[