## 3 Adaptive TSK Fuzzy Self-Organizing Recurrent Cerebellar Model Articulation Controller Design
### Design of TSK Fuzzy Self-Organizing Recurrent Cerebellar Model Articulation Controller
Differentiating Equation (65) and substituting Equation (58) into the equation can give  
\[\begin{array}{ll}\dot{V}_{c1}&=s_{a}\dot{s}_{a}-B_{n}[\frac{1}{2\beta_{w1}} \widetilde{\mathbf{W}}_{a}^{T}\dot{\mathbf{W}}_{a}+\frac{1}{2\beta_{m1}} \widetilde{\mathbf{m}}_{a}^{T}\dot{\mathbf{m}}_{a}+\frac{1}{\beta_{r1}} \widetilde{\mathbf{v}}_{a}^{T}\dot{\mathbf{v}}_{a}+\frac{1}{\beta_{r1}} \widetilde{\mathbf{v}}_{a}^{T}\dot{\mathbf{z}}_{a}]\\ &=s_{a}B_{n}[\widetilde{\mathbf{W}}_{a}^{T}\dot{\mathbf{A}}_{a}+\dot{\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})+D_{1}-u_{ c}]\\ &-B_{n}[\frac{1}{2\beta_{w1}}\widetilde{\mathbf{W}}_{a}^{T}\dot{\mathbf{W}}_{ a}^{T}\dot{\mathbf{w}}_{a}+\frac{1}{2\beta_{w1}}\widetilde{\mathbf{v}}_{a}^{T} \dot{\mathbf{z}}_{a}+\frac{1}{2\beta_{w1}}\widetilde{\mathbf{v}}_{a}^{T} \dot{\mathbf{z}}_{a}+\frac{1}{2\beta_{w1}}\widetilde{\mathbf{v}}_{a}^{T} \dot{\mathbf{z}}_{a}]\\ &=s_{a}B_{n}(D_{1}-u_{c})+s_{a}B_{n}\widetilde{\mathbf{W}}_{a}^{T}\dot{ \mathbf{A}}_{a}+s_{a}B_{n}\dot{\mathbf{W}}_{a}^{T}\dot{\mathbf{C}}^{T} \widetilde{\mathbf{m}}_{a}+s_{a}B_{n}\dot{\mathbf{W}}_{a}^{T}\mathbf{G}^{T} \widetilde{\mathbf{v}}_{a}+s_{a}B_{n}\dot{\mathbf{W}}_{a}^{T}\mathbf{H}^{T} \widetilde{\mathbf{r}}_{a}\\ &-\frac{B_{n}}{\beta_{v1}}\widetilde{\mathbf{W}}_{a}^{T}\dot{\mathbf{W}}_{a}- \frac{B_{n}}{\beta_{v1}}\widetilde{\mathbf{mFrom Equation (71), as \(\dot{V}_{c1}\leq 0\) is a negative semi-definite function, there is no guarantee that the error will converge to zero. By using [PERSON]'s lemma, we can prove that when time approaches infinity, \(\dot{V}_{c1}(t)\) will approach zero.  
The function of \(\Xi_{1}(t)\) is defined as follows:  
\[\Xi_{1}(t)=(\delta_{1}-|D_{1}|)B_{n}|s_{a}|\leq-\dot{V}_{c1}(t). \tag{72}\]  
Integrating Equation (72) with respect to time can give Equation (73), where \(V_{c1}(t)\) is as shown in Equation (65). \[\int_{0}^{t}\Xi_{1}(\tau)d\tau\leq V_{c1}(0)\,-V_{c1}(t) \tag{73}\]  
Since \(V_{c1}(0)\) is bounded, while \(V_{c1}(t)\) is bounded and a non-incremental function, it can be determined from Equation (73) that \(\Xi_{1}(t)\) is bounded by time, i.e.,  
\[\lim_{t\to\infty}\int_{0}^{t}\Xi_{1}(\tau)d\tau<\infty. \tag{74}\]  
Next, since \(V_{c1}(t)>0\) and \(\dot{V}_{c1}(t)\leq 0\), Equation (65) shows that \(\widetilde{\textbf{W}}_{a}\), \(\widetilde{\textbf{m}}_{a}\), \(\widetilde{\textbf{v}}_{a}\), and \(\widetilde{\textbf{r}}_{a}\) are bounded. As inferred from Equation (71), when \(s_{a}\
eq 0\), Equation (58) implies that \(\dot{s}_{a}\) is bounded, and \(s_{a}\) is uniformly continuous; therefore, from Equation (72), it is learnt that \(\tilde{\Xi}_{1}(t)\) is bounded, thus proving that \(\dot{V}_{c1}(t)\) is uniformly continuous and \(\ddot{V}_{c1}(t)\) is bounded. From [PERSON]'s Lemma, we can further obtain the following:  
\[\lim_{t\to\infty}\dot{V}_{c1}(t)=0,\,\forall s_{a}\
eq 0. \tag{75}\]  
In addition, from Equation (71), when \(s_{a}=0\), \(\dot{V}_{c1}(t)=0\). The above-mentioned inference indicates that for arbitrary \(s_{a}\), when the time \(t\) approaches infinity, \(\dot{V}_{c1}(t)\) will converge to zero, and \(s_{a}\) at this point will also converge to zero. Equation (41) shows that if an appropriate \(k_{2}\) is selected to conform to the Hurwitz polynomial, the errors \(e\) and \(v\) will also converge to zero when the time approaches infinity. Through the above derivations, we can ensure that the designed adaptive TSK fuzzy self-organizing recurrent cerebellar model articulation controller system is stable. \(\square\)