## 3 Adaptive TSK Fuzzy Self-Organizing Recurrent Cerebellar Model Articulation Controller Design
### Design of TSK Fuzzy Self-Organizing Recurrent Cerebellar Model Articulation Controller
For convenience of derivation, assigning \(\widetilde{\mathbf{m}}_{a}=\mathbf{m}_{a}^{*}-\hat{\mathbf{m}}_{a}\), \(\widetilde{\mathbf{v}}_{a}=\mathbf{v}_{a}^{*}-\hat{\mathbf{v}}_{a}\), \(\widetilde{\mathbf{v}}_{a}=\mathbf{v}_{a}^{*}-\hat{\mathbf{v}}_{a}\) and  
\[\mathbf{C}=\left[\left(\frac{\partial b_{1}f^{T}}{\partial\mathbf{m}_{a}}\right),\;(\frac{\partial b_{2}f^{T}}{\partial\mathbf{m}_{a}}),\;\cdots,(\frac{\partial b _{p}f^{T}}{\partial\mathbf{m}_{a}})\right], \tag{53}\]  
\[\mathbf{G}=\left[\left(\frac{\partial b_{1}f^{T}}{\partial\mathbf{v}_{a}}\right),\;(\frac{\partial b_{2}f^{T}}{\partial\mathbf{v}_{a}}),\;\cdots,(\frac{\partial b _{p}f^{T}}{\partial\mathbf{v}_{a}})\right], \tag{54}\]  
\[\mathbf{H}=\left[\left(\frac{\partial b_{1}f^{T}}{\partial\mathbf{r}_{a}}\right),\;(\frac{\partial b_{2}f^{T}}{\partial\mathbf{r}_{a}}),\;\cdots,(\frac{\partial b _{p}f^{T}}{\partial\mathbf{r}_{a}})\right]. \tag{55}\]  
Then, Equation (52) can be rewritten as  
\[\widetilde{\mathbf{\Lambda}}_{a}=\mathbf{C}^{T}\widetilde{\mathbf{m}}_{a}+\mathbf{G}^ {T}\widetilde{\mathbf{v}}_{a}+\mathbf{H}^{T}\widetilde{\mathbf{r}}_{a}+\mathbf{O}_{t}. \tag{56}\]  
In addition, \(\widetilde{\mathbf{\Lambda}}_{a}=\mathbf{\Lambda}_{a}^{*}-\hat{\mathbf{\Lambda}}_{a}\), so Equation (55) can also be expressed as  
\[\mathbf{\Lambda}_{a}^{*}=\hat{\mathbf{\Lambda}}_{a}+\mathbf{C}^{T}\widetilde{\mathbf{m}}_ {a}+\mathbf{G}^{T}\widetilde{\mathbf{v}}_{a}+\mathbf{H}^{T}\widetilde{\mathbf{r}}_{a} +\mathbf{O}_{t}. \tag{57}\]  
Then, substituting Equations (56) and (57) into Equation (51), we get  
\[\begin{array}{rl}\dot{s}_{a}(t)&=B_{n}[\widetilde{\mathbf{W}}_{a}^{T}\mathbf{\Lambda }_{a}^{*}+\hat{\mathbf{W}}_{a}^{T}\widetilde{\mathbf{\Lambda}}_{a}+\mathbf{W}_{i}^{*T}\mathbf{ \Lambda}_{i}^{*}+\rho-u_{c}]\\ &=B_{n}[\widetilde{\mathbf{W}  
**Theorem 1**.: _Considering a chaotic synchronization control system as Equation (34), if the synchronization controller is designed to be Equation (49) and while the ATFSORC's adaptive laws and compensation controller output are as shown from Equation (60) to Equation (64), then the stability of such a controlled system can be guaranteed._  
\[\dot{\mathbf{W}}_{a}=s_{a}\beta_{w1}\mathbf{\hat{A}}_{a}, \tag{60}\]  
\[\dot{\mathbf{\hat{m}}}_{a}=s_{a}\beta_{m1}\mathbf{C\hat{W}}_{a}, \tag{61}\]  
\[\dot{\mathbf{\hat{v}}}_{a}=s_{a}\beta_{v1}\mathbf{G\hat{W}}_{a}, \tag{62}\]  
\[\dot{\mathbf{\hat{z}}}_{a}=s_{a}\beta_{r1}\mathbf{H\hat{W}}_{a}, \tag{63}\]  
\[u_{c}=\delta_{1}\text{sgn}[s_{a}]. \tag{64}\]  
**Proof of Theorem 1. ** Using the Lyapunov theorem to derive the adaptive laws, the Lyapunov function candidate \(V_{c1}\) is first defined as follows:  
\[ \begin{array}{ll}V_{c1}=\frac{1}{2}s_{a}^{2}+B_{n}[\frac{1}{2\beta_{w1}} \widetilde{\mathbf{W}}_{a}^{T}\widetilde{\mathbf{W}}_{a}+\frac{1}{2\beta_{w1} }\widetilde{\mathbf{m}}_{a}^{T}\widetilde{\mathbf{m}}_{a}\\ +\frac{1}{2\beta_{v1}}\widetilde{\mathbf{v}}_{a}^{T}\widetilde{\mathbf{v}}_{ a}+\frac{1}{2\beta_{r1}}\widetilde{\mathbf{v}}_{a}^{T}\widetilde{\mathbf{r}}_{a} ]\end{array}  \tag{65}\]  
where \(\beta_{w1}\), \(\beta_{m1}\), \(\beta_{v1}\), and \(\beta_{r1}\) are normal numbers greater than zero.