## 3 Adaptive TSK Fuzzy Self-Organizing Recurrent Cerebellar Model Articulation Controller Design
\[\int_{0}^{t}\Xi_{2}(\tau)d\tau\leq V_{c2}(0)\,-V_{c2}(t) \tag{124}\]  
Since \(V_{c2}(0)\) is bounded, while \(V_{c2}(t)\) is bounded and a non-incremental function, it can be determined from Equation (124) that \(\Xi_{2}(t)\) is bounded by time, i.e.,  
\[\lim_{t\to\infty}\!\!\int_{0}^{t}\Xi_{2}(\tau)d\tau<\infty. \tag{125}\]  
Next, since \(V_{c2}(t)>0\) and \(\dot{V}_{c1}(t)\leq 0\), Equation (116) shows that \(\widetilde{\mathbf{W}}_{a}\), \(\widetilde{\mathbf{m}}_{a}\), \(\widetilde{\mathbf{v}}_{a}\), and \(\widetilde{\mathbf{r}}_{a}\) are bounded. As inferred from Equation (122) that when \(s_{b}\
eq 0\), from Equation (109), we see that \(\dot{s}_{b}\) is bounded and \(s_{b}\) is uniformly continuous; Equation (122) shows that \(\dot{\Xi}_{2}(t)\) is bounded, thus proving that \(\dot{V}_{\mathcal{L}2}(t)\) is uniformly continuous and \(\ddot{V}_{\mathcal{L}2}(t)\) is bounded. From [PERSON]'s Lemma, we can further obtain the following. \[\lim_{t\to\infty}\dot{V}_{\mathcal{L}2}(t)=0,\,\forall s_{b}\
eq 0. \tag{126}\]  
From Equation (122), we see that when \(s_{b}=0\), \(\dot{V}_{\mathcal{L}2}(t)=0\). From the above-mentioned inference, it can be learnt that for arbitrary \(s_{b}\), when the time \(t\) approaches infinity, \(\dot{V}_{\mathcal{L}2}(t)\) will converge to zero, and \(s_{b}\) at this point of time will also converge to zero. Equation (92) shows that if an appropriate \(k_{3}\) is selected to conform to the Hurwitz polynomial, the error \(e\) 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. \(\square\)  
### Design of Improved Compensator  
The adaptive TSK fuzzy self-organizing cerebellar model articulation controller proposed in this study contains an improved compensation controller. From Equation (115), the output of compensation controller \(u_{c}\) is composed of a normal number \(\delta_{2}\) and a sign function of \(s_{b}\), wherein the function of \(\text{sgn}[s_{b}]\) is defined as in Equation (127). Since \(\delta_{2}\) is a normal number, the output of the compensation controller depends on the integrated error function \(s_{b}\) value. In fact, there are only three types of states as \(\delta_{2}\), \(0\), and \(-\delta_{2}\), which cause the shortcoming of insufficient transient-state compensation or excessive steady-state compensation. \[\text{sgn}[s_{b}]=\left\{ \begin{array}{ll}1&,s_{b}>0\\ 0&,s_{b}=0\\ -1&,s_{b}<0\end{array} \right. \tag{127}\]  
Therefore, this study proposes an improved compensator to ameliorate the original compensation controller as presented arithmetically in Equation (115). Design of the improved compensation controller focuses on allowing the compensation controller to provide a larger compensation amount in ATFSORC at the transient state and a smaller compensation amount after ATFSORC enters the steady state. Errors will increase when the compensation amount provided by the compensation controller is insufficient; therefore, the integrated error function \(s_{b}\) can be used as the basis to determine whether ATFSORC has completed learning and entered the steady state. The above-mentioned design concept can be implemented from Equation (128). \[u_{c}=\left\{ \begin{array}{ll}\delta_{2}\,\gamma_{2}\,s_{b},\,\,\,|s_{b}|> \zeta_{2}\\ \delta_{2}\text{sgn}[s_{b}],\,\,\,0<|s_{b}|<\zeta_{2}\end{array} \right., \tag{128}\]  
where \(\zeta_{2}\) is the threshold value for switching compensation control quantity; \(\gamma_{2}\) is a compensation coefficient; and \(\gamma_{2}>1\), which is used to elastically adjust the magnitude of compensation amount at the transient state. When \(|s_{b}|>\zeta_{2}\), it is a transient state, while when \(0<|s_{b}|<\zeta_{2}\), it is a steady state.