## 3 Adaptive TSK Fuzzy Self-Organizing Recurrent Cerebellar Model Articulation Controller Design
### Design of Improved Compensator
\begin{array}{ll}\dot{s}_{b}(t)&=B_{n}[\widetilde{\mathbf{W}}_{a}^{T} \mathbf{\Lambda}_{a}^{*}+\hat{\mathbf{W}}_{a}^{T}\mathbf{\Lambda}_{a}^{*}+ \mathbf{W}_{i}^{*T}\mathbf{\Lambda}_{i}^{*}+\rho-u_{c}]\\ &=B_{n}[\widetilde{\mathbf{W}}_{a}^{T}(\hat{\mathbf{\Lambda}}_{a}+\mathbf{C}^ {T}\widetilde{\mathbf{m}}_{a}+\mathbf{G}^{T}\widetilde{\mathbf{v}}_{a}+ \mathbf{H}^{T}\widetilde{\mathbf{v}}_{a}+\mathbf{O}_{t})\\ &+\hat{\mathbf{W}}_{a}^{*}(\mathbf{C}^{T}\widetilde{\mathbf{m}}_{a}+\mathbf{G }^{T}\widetilde{\mathbf{v}}_{a}+\mathbf{H}^{T}\widetilde{\mathbf{r}}_{a}+ \mathbf{O}_{t})+\mathbf{W}_{i}^{*T}\mathbf{\Lambda}_{i}^{*}+\rho-u_{c}]\\ &=B_{n}[\widetilde{\mathbf{W}}_{a}^{T}\hat{\mathbf{\Lambda}}_{a}+(\widetilde {\mathbf{W}}_{a}^{T}+\widetilde{\mathbf{W}}_{a}^{T})(\mathbf{C}^{T}\widetilde {\mathbf{m}}_{a}+\mathbf{G}^{T}\widetilde{\mathbf{v}}_{a}+\mathbf{H}^{T} \widetilde{\mathbf{r}}_{a}+\mathbf{O}_{t})+\mathbf{W}_{i}^{*T}\mathbf{r}_{i} ^{*}+\rho-u_{c}]\\ &=B_{n}[\widetilde{\mathbf{W}}_{a}^{T}\hat{\mathbf{\Lambda}}_{a}+\hat{ \mathbf{W}}_{a}^{T}(\mathbf{C}^{T}\widetilde{\mathbf{m}}_{a}+\mathbf{G}^{T} \widetilde{\mathbf{v}}_{a}+\mathbf{H}^{T}\widetilde{\mathbf{r}}_{a})\\ &+(\widetilde{\mathbf{W}}_{a}^{T}\mathbf{C}^{T}\widetilde{\mathbf{m}}_{a}+ \widetilde{\mathbf{W}}_{a}^{T}\mathbf{G}^{T}\widetilde{\mathbf{v}}_{a}+ \widetilde{\mathbf{W}}_{a}^{T}\mathbf{H}^{T}\widetilde{\mathbf{r}}_{a}+ \mathbf{H}^{*T}\widetilde{\mathbf{r}}_{a})+\mathbf{W}_{i}^{*T}\mathbf{\Lambda }_{i}^{*}+\rho)-u_{c}]\\ &=B_{n}[\widetilde{\mathbf{W}}_{a}^{T}\hat{\mathbf{\Lambda}}_{a}+\hat{ \mathbf{W}}_{a}^{T}(\mathbf{C}^{T}\widetilde{\mathbf{m}}_{a}+\mathbf{G}^{T} \widetilde{\mathbf{v}}_{a}+\mathbf{H}^{T}\widetilde{\mathbf{v}}_{a})+D_{2} -u_{c}]\end{array}  \tag{109}\]  
where \(D_{2}=\widetilde{\mathbf{W}}_{a}^{T}\mathbf{C}^{T}\widetilde{\mathbf{m}}_{a} +\widetilde{\mathbf{W}}_{a}^{T}\mathbf{G}^{T}\widetilde{\mathbf{v}}_{a}+ \widetilde{\mathbf{W}}_{a}^{T}\mathbf{H}^{T}\widetilde{\mathbf{r}}_{a}+ \mathbf{W}_{a}^{*T}\mathbf{O}_{t}+\mathbf{W}_{i}^{*T}\mathbf{\Lambda}_{i}^{*}+\rho\). Assuming that the upper limit of \(D_{2}\) is a normal number \(\delta_{2}\)  
\[|D_{2}|<\delta_{2}. \tag{110}\]  
**Theorem 2. **_Considering a chaotic stability control system as Equation (84), if the stability controller is designed to be Equation (100) and the ATFSORC's adaptive laws and compensation controller output are as shown from Equation (111) to Equation (115), then the stability of such a controlled system can be guaranteed._  
\[\dot{\mathbf{\mathbf{W}}}_{a}=s_{b}\beta_{w2}\hat{\mathbf{\Lambda}}_{a}, \tag{111}\]  
\[\dot{\hat{\mathbf{m}}}_{a}=s_{b}\beta_{m2}\mathbf{C}\hat{\mathbf{W}}_{a}, \tag{112}\]  
\[\dot{\hat{\mathbf{v}}}_{a}=s_{b}\beta_{v2}\mathbf{G}\hat{\mathbf{W}}_{a}, \tag{113}\]  
\[\dot{\hat{r}}_{a}=s_{b}\beta_{r2}\mathbf{H}\hat{\mathbf{W}}_{a}, \tag{114}\]  
\[u_{c}=\delta_{2}\text{sgn}[s_{b}]. \tag{115}\]  
**Proof of Theorem 2. ** Using the Lyapunov theorem to derive the adaptive laws, the Lyapunov function candidate \(V_{c2}\) is first defined as follows:  
\[ \begin{array}{ll}V_{c2}&=\frac{1}{2}{s_{b}}^{2}+B_{n}[\frac{1}{2B_{n2}} \widetilde{\mathbf{W}}_{a}^{T}\widetilde{\mathbf{W}}_{a}+\frac{1}{2B_{n2}} \widetilde{\mathbf{m}}_{a}^{T}\widetilde{\mathbf{m}}_{a}\\ &+\frac{1}{2B_{n2}}\widetilde{\mathbf{v}}_{a}^{T}\widetilde{\mathbf{v}}_{a}+ \frac{1}{2B_{n2}}\widetilde{\mathbf{v}}_{a}^{T}\widetilde{\mathbf{u}}_{a}] \end{array}  \tag{116}\]where \(\beta_{w2}\), \(\beta_{m2}\), \(\beta_{v2}\), and \(\beta_{r2}\) are normal numbers greater than zero. By differentiating Equation (116) and substituting Equation (109) into the equation, we get  
\[