## 3 Adaptive TSK Fuzzy Self-Organizing Recurrent Cerebellar Model Articulation Controller Design
### Design of Improved Compensator
\begin{array}{ll}\dot{V}_{c2}&=s_{b}s_{b}-B_{n}[\frac{1}{\beta_{w2}}\widetilde {\mathbf{W}}_{a}^{T}\hat{\mathbf{W}}_{a}+\frac{1}{\beta_{w2}}\widetilde{ \mathbf{m}}_{a}^{T}\hat{\mathbf{m}}_{a}+\frac{1}{\beta_{v2}}\widetilde{ \mathbf{v}}_{a}^{T}\hat{\mathbf{v}}_{a}+\frac{1}{\beta_{v2}}\widetilde{ \mathbf{v}}_{a}^{T}\hat{\mathbf{v}}_{a}]\\ &=s_{b}B_{n}[\widehat{\mathbf{W}}_{a}^{T}\hat{\mathbf{M}}_{a}+\hat{\mathbf{W} }_{a}^{T}(\mathbf{C}^{T}\hat{\mathbf{m}}_{a}+\mathbf{G}^{T}\widetilde{\mathbf{ v}}_{a}^{T}+\mathbf{H}^{T}\widetilde{\mathbf{v}}_{a})+D_{2}-u_{c}]\\ &-B_{n}[\frac{1}{\beta_{w2}}\widetilde{\mathbf{W}}_{a}^{T}\hat{\mathbf{W}}_{ a}+\frac{1}{\beta_{w2}}\widetilde{\mathbf{m}}_{a}^{T}\hat{\mathbf{m}}_{a}+ \frac{1}{\beta_{v2}}\widetilde{\mathbf{v}}_{a}^{T}\hat{\mathbf{v}}_{a}+\frac{ 1}{\beta_{v2}}\widetilde{\mathbf{v}}_{a}^{T}\hat{\mathbf{v}}_{a}]\\ &=s_{b}B_{n}(D_{2}-u_{c})+s_{b}B_{n}\widetilde{\mathbf{W}}_{a}^{T}\hat{ \mathbf{M}}_{a}+s_{b}B_{n}\widetilde{\mathbf{W}}_{a}^{T}\mathbf{C}^{T}\hat{ \mathbf{m}}_{a}+s_{b}B_{n}\hat{\mathbf{W}}_{a}^{T}\mathbf{C}^{T}\widetilde{ \mathbf{v}}_{a}+s_{b}B_{n}\hat{\mathbf{W}}_{a}^{T}\mathbf{C}^{T}\widetilde{ \mathbf{v}}_{a}+s_{b}B_{n}\hat{\mathbf{W}}_{a}^{T}\mathbf{H}^{T}\widetilde{ \mathbf{v}}_{a}\\ &-\frac{B_{n}}{\beta_{w2}}\widetilde{\mathbf{W}}_{a}^{T}\hat{\mathbf{W}}_{a}- \frac{B_{n}}{\beta_{w2}}\widetilde{\mathbf{m}}_{a}^{T}\hat{\mathbf{m}}_{a}- \frac{B_{n}}{\beta_{w2}}\widetilde{\mathbf{v}}_{a}^{T}\hat{\mathbf{v}}_{a}- \frac{B_{n}}{\beta_{w2}}\widetilde{\mathbf{v}}_{a}^{T}\hat{\mathbf{v}}_{a}- \frac{B_{n}}{\beta_{r2}}\widetilde{\mathbf{v}}_{a}^{T}\hat{\mathbf{v}}_{a}\\ &=s_{b}B_{n}(D_{2}-u_{c})+\widetilde{\mathbf{W}}_{a}^{T}[s_{b}B_{n}\hat{ \mathbf{A}}_{a}-\frac{B_{n}}{\beta_{w2}}\hat{\mathbf{W}}_{a}]+\widetilde{ \mathbf{m}}_{a}[s_{b}B_{n}\hat{\mathbf{W}}_{a}^{T}\mathbf{C}^{T}-\frac{B_{n}} {\beta_{w2}}\hat{\mathbf{m}}_{a}]\\ &+\widetilde{\mathbf{v}}_{a}[s_{b}B_{n}\hat{\mathbf{W}}_{a}^{T}\mathbf{G}^{T}- \frac{B_{n}}{\beta_{v2}}\hat{\mathbf{v}}_{a}]+\widetilde{\mathbf{v}}_{a}[s_{b }B_{n}\hat{\mathbf{W}}_{a}^{T}\mathbf{H}^{T}-\frac{B_{n}}{\beta_{v2}}\hat{ \mathbf{v}}_{a}]\end{array}  \tag{117}\]  
As \(V_{c2}\geq 0\) and in order to achieve the goal of \(\dot{V}_{c2}<0\), we can assign  
\[\widetilde{\mathbf{W}}_{a}^{T}\bigg{[}s_{b}B_{n}\hat{\mathbf{\Lambda}}_{a}- \frac{B_{n}}{\beta_{w2}}\hat{\mathbf{W}}_{a}\bigg{]}=0, \tag{118}\]  
\[\bigg{[}s_{b}B_{n}\hat{\mathbf{W}}_{a}^{T}\mathbf{C}^{T}-\frac{B_{n}}{\beta_{ m2}}\hat{\mathbf{m}}_{a}\bigg{]}\widetilde{\mathbf{m}}_{a}=0, \tag{119}\]  
\[\bigg{[}s_{b}B_{n}\hat{\mathbf{W}}_{a}^{T}\mathbf{G}^{T}-\frac{B_{n}}{\beta_{ v2}}\hat{\mathbf{v}}_{a}\bigg{]}\widetilde{\mathbf{v}}_{a}=0, \tag{120}\]  
\[\bigg{[}s_{b}B_{n}\hat{\mathbf{W}}_{a}^{T}\mathbf{H}^{T}-\frac{B_{n}}{\beta_{ r2}}\hat{\mathbf{i}}_{a}\bigg{]}\widetilde{\mathbf{v}}_{a}=0. \tag{121}\]  
Substituting all formulas from Equation (111) to Equation (115) into Equation (117), we get  
\[ \begin{array}{ll}\dot{V}_{c2}&=s_{b}B_{n}(D_{2}-\delta_{2}\text{sgn}[s_{b}] )\\ &=s_{b}B_{n}D_{2}-|s_{b}|B_{n}\delta_{2}\leq-(\delta_{2}-|D_{2}|)B_{n}|s_{b}| \leq 0\end{array}  \tag{122}\]  
From Equation (71), as \(\dot{V}_{c2}\leq 0\) is a negative semi-definite function, there is no guarantee that the error will converge to zero. Using [PERSON]'s Lemma, we can prove that when time approaches infinity, \(\dot{V}_{c2}(t)\) approaches zero. The function \(\Xi_{2}(t)\) is defined as follows:  
\[\Xi_{2}(t)=(\delta_{2}-|D_{2}|)B_{n}|s_{b}|\leq-\dot{V}_{c2}(t). \tag{123}\]  
Integrating Equation (123) with respect to time, we get Equation (124), where \(V_{c2}(t)\) is as shown in Equation (116).