## 3 Adaptive TSK Fuzzy Self-Organizing Recurrent Cerebellar Model Articulation Controller Design
### Design of TSK Fuzzy Self-Organizing Recurrent Cerebellar Model Articulation Controller
\[\begin{array}{l}\dot{v} =(\lambda_{1}\dot{e}_{x}+\dot{e}_{y})/\sqrt{1+\lambda_{1}{}^{2}} \\ =-2g[\lambda_{1}(x_{1}-x_{2})+\left(y_{1}-y_{2})\right]/\sqrt{1+\lambda_{1}{}^ {2}}-\lambda_{1}u/\sqrt{1+\lambda_{1}{}^{  
\(A=-2g\),  
\(B=\frac{-\lambda_{1}}{\sqrt{1+\lambda_{1}}^{2}}\),  
\(d=\left\{\lambda_{1}[-r{x_{1}}^{3}+(r+1){x_{1}}^{2}+r{x_{2}}^{3}-(r+1){x_{2}}^{2} ]+(2g-\lambda_{1})(y_{1}-y_{2})+(b-\lambda_{1})(x_{1}-x_{2})\right\}/\sqrt{1+{ \lambda_{1}}^{2}}\). In the actual chaos, considering the dynamics of a chaotic system, the parameters will change accordingly under different conditions. Therefore, Equation (32) can be rewritten as  
\[ \begin{array}{ll}\dot{v}&=(A_{n}+\Delta A)v+(B_{n}+\Delta B)u+(d+\Delta d)\\ &=A_{n}v+B_{n}u+\Delta Av+\Delta Bu+d_{1}\\ &=A_{n}v+B_{n}u+d_{2}\end{array}  \tag{34}\]  
where \(A_{n}\) and \(B_{n}\) are the nominal values of \(A\) and \(B\), respectively; \(\Delta A\) and \(\Delta B\) are the variation values of \(A\) and \(B\), respectively; and \(d_{1}\) and \(d_{2}\) are the uncertainty dynamic model (system uncertainty), in which \(d_{1}=d+\Delta d\), \(d_{2}=\Delta A_{v}+\Delta B_{u}+d_{1}\). Then, transposing and arranging the equation, we can obtain  
\[u=\frac{1}{B_{n}}[-A_{n}v-d_{2}+\dot{v}]. \tag{35}\]  
Ideally, if all the parameters of a chaotic system are known, an ideal control amount \(u^{\star}\) can be constructed. \[u^{\star}=\frac{1}{B_{n}}[-A_{n}v-d_{2}+\dot{v}^{\star}-k_{2}e], \tag{36}\]  
where \(v^{\star}\) is the ideal error amount; and \(k_{2}\) is a normal number. Error \(e\) is defined as Equation (37). \[e=v-v^{\star}. \tag{37}\]  
If the control amount \(u\) approaches the ideal control amount \(u^{\star}\), Equation (38) can be obtained based on Equations (35) and (36); Equation (39) can be obtained from Equations (37) and (38). \[\dot{v}=\dot{v}^{\star}-k_{2}e, \tag{38}\]  
\[\dot{e}+k_{2}e=0. \tag{39}\]  
Since \(k_{2}\) is a positive real number and Equation (39) is a strictly Hurwitz polynomial, the error will converge to zero as time passes to infinity, i.e., \(\lim\limits_{t\rightarrow\infty}e=0\). Next, the integrated error function \(s_{a}\) is defined as in Equation (40). \[s_{a}(t)=e(t)+k_{2}{\int_{0}}^{t}e(\tau)d\tau. \tag{40}\]  
Differentiating Equation (40) with respect to time \(t\), we get  
\[\dot{s}_{a}=\dot{e}+k_{2}e. \tag{41}\]  
When the control amount is the ideal control amount \(u^{\star}\), the error \(e\) can become asymptotically stable. However, in fact, such an ideal control amount cannot be obtained because the system parameters and an unknown dynamic model will vary along with many uncertain factors, and these cannot be known precisely in advance. Therefore, this study proposes ATFSORC, which can use the good approximation capability of a Takagi-Sugeno-Kang (TSK) fuzzy self-organizing recurrent cerebellar model articulation controller (TFSORC) to learn the ideal control amount \(u^{\star}\) and use the compensation controller to compensate for the error between TFSORC and \(u^{\star}\). Figure 6 shows the ATFSORC architecture diagram for chaotic system synchronization. Its output amount is  
\[u=u_{TFSORC}+u_{c}, \tag{42}\]where \(u\) is the total control amount; \(u_{TTSORC}\) is the output amount of TFSORC; and \(u_{c}\) is the output amount of compensation controller. Next, the association memory space is partitioned such 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. \[\mathbf{\Lambda^{*}}=\left[ \begin{array}{c}\mathbf{\Lambda^{*}_{a}}\\ \mathbf{\Lambda^{*}_{i}}\end{array} \right], \tag{43}\]  
\[\mathbf{W^{*}}=\left[ \begin{array}{c}\mathbf{W^{*}_{a}}\\ \mathbf{W^{*}_{i}}\end{array} \right], \tag{44}\]  
\[\mathbf{m^{*}}=\left[ \begin{array}{c}\mathbf{m^{*}_{a}}\\ \mathbf{m^{*}_{i}}\end{array} \right], \tag{45}\]  
\[\mathbf{v^{*}}=\left[