## 3 Adaptive TSK Fuzzy Self-Organizing Recurrent Cerebellar Model Articulation Controller Design
### Design of Improved Compensator
Next, the integrated error function \(s_{b}\) is defined as in Equation (91):  
\[s_{b}(t)=e(t)+k_{3}{\int_{0}}^{t}e(\tau)d\tau. \tag{91}\]  
Differentiating Equation (91) with respect to time \(t\), we get  
\[\dot{s}_{b}=\dot{e}+k_{3}e. \tag{92}\]  
When the control amount is an ideal control amount \(u^{*}\), error \(e\) can become asymptotically stable. However, in fact, such an ideal control amount cannot be obtained, because system parameters and the unknown dynamic model will vary along with many uncertain factors, and these cannot be known precisely in advance. Therefore, this study proposes ATFSORC that can use the good approximation capability of TFSORC to learn the ideal control amount \(u^{*}\), and use the compensation controller to compensate for the error between TFSORC and \(u^{*}\). Figure 9 is the ATFSORC architectural diagram for chaotic system synchronization; its output is  
\[u=u_{TFSORC}+u_{c}, \tag{93}\]  
where \(u\) is the total control amount; \(u_{TFSORC}\) is the output amount of TFSORC; and \(u_{c}\) is the output amount of the compensation controller. Next, the association memory space is partitioned so that the optimal number of stage layers is \(p^{*}\), which is divided into an activated part as \(p\) and an inactivated part as \(p^{*}-p\). Therefore, the optimal parameters \(\mathbf{\Lambda}^{*}\), \(\mathbf{W}^{*}\), \(\mathbf{m}^{*}\), \(\mathbf{v}^{*}\), and \(\mathbf{r}^{*}\) are also divided into two parts. \[\mathbf{\Lambda}^{*}=\left[ \begin{array}{c}\mathbf{\Lambda}_{a}^{*}\\ \mathbf{\Lambda}_{i}^{*}\end{array} \right], \tag{94}\]  
\[\mathbf{W}^{*}=\left[ \begin{array}{c}\mathbf{W}_{a}^{*}\\ \mathbf{W}_{i}^{*}\end{array} \right], \tag{95}\]  
\[\mathbf{m}^{*}=\left[ \begin{array}{c}\mathbf{\mathrm{m}}_{a}^{*}\\ \mathbf{\mathrm{m}}_{i}^{*}\end{array} \right], \tag{96}\]  
\[\mathbf{v}^{*}=\left[ \begin{array}{c}\mathbf{v}_{a}^{*}\\ \mathbf{\mathrm{v}}_{i}^{*}\end{array} \right], \tag{97}\]  
\[\mathbf{r}^{*}=\left[ \begin{array}{c}\mathbf{\mathrm{r}}_{a}^{*}\\ \mathbf{\mathrm{r}}_{i}^{*}\end{array} \right], \tag{98}\]  
where \(\mathbf{\Lambda}^{*}\), \(\mathbf{W}^{*}\), \(\mathbf{m}^{*}\), \(\mathbf{v}^{*}\), and \(\mathbf{r}^{*}\) are the optimal parameters corresponding to \(\mathbf{\Lambda}\), \(\mathbf{W}\), \(\mathbf{m}\), \(\mathbf{v}\), and \(\mathbf{r}\), respectively; \(\mathbf{\Lambda}_{a}\), \(\mathbf{W}_{a}\), \(\mathbf{m}_{a}\), \(\mathbf{\mathrm{v}}_{a}\), and \(\mathbf{r}_{a}\) represent the activated parts, whereas \(\mathbf{\Lambda}_{i}\), \(\mathbf{W}_{i}\), \(\mathbf{\mathrm{m}}_{i}\), \(\mathbf{\mathrm{v}}_{i}\), and \(\mathbf{r}_{i}\) represent the inactivated parts. Based on the universal approximation theorem, an optimal output control amount for a TSK fuzzy self-organizing recurrent cerebellar model articulation controller, \(u_{TFSORC}^{*}\), is assumed, and the ideal control amount \(u^{*}\) is expressed as Equation (99). \[ \begin{array}{ll}u^{*}&=u_{TFSORC}^{*}(x\,,\,\textbf{W}^{*},\,\textbf{m}^{*},\, \textbf{v}^{*},\textbf{r}^{*})+\rho\cong\textbf{W}^{*T}\boldsymbol{\Lambda}^{* }+\rho\\ &=\textbf{W}_{a}^{*T}\boldsymbol{\Lambda}_{a}^{*}+\textbf{W}_{i}^{*T} \boldsymbol{\Lambda}_{i}^{*}+\rho\end{array}  \tag{99}\]  
where \(\rho\) represents the error between the two control amounts. Since the optimal output control amount \(u_{TFSORC}^{*}\) cannot be obtained easily, \(u_{TFSORC}^{*}\) is estimated in an iterative manner. That is, we assign  
\[u=u_{TFSORC}(x\,,\,\hat{\textbf{W}}_{a},\,\hat{\textbf{m}}_{a},\,\hat{\textbf{ v}}_{a},\hat{\textbf{r}}_{a})+u_{c}\cong\hat{\textbf{W}}_{a}^{T}\hat{ \boldsymbol{\Lambda}}_{a}+u_{c}, \tag{100}\]  
where \(\hat{\boldsymbol{\Lambda}}\,,\,\hat{\textbf{W}}\,,\,\hat{\textbf{m}}\,,\, \hat{\textbf{v}}\,,\,\hat{\textbf{r}}\) are estimated values corresponding to \(\boldsymbol{\Lambda},\textbf{W},\textbf{m},\textbf{v}\), and \(\textbf{r}\), respectively.