## 3 Adaptive TSK Fuzzy Self-Organizing Recurrent Cerebellar Model Articulation Controller Design
### Design of Improved Compensator
 \tag{79}\]  
The errors are defined as  
\[ \begin{array}{ll}e_{x}(t)=x(t)-x(t-\tau)\\ e_{y}(t)=y(t)-y(t-\tau)\end{array}  \tag{80}\]  
Figure 9: Architectural diagram of ATFSORC stability controller for chaotic system. Next, the error dynamic system can be expressed as Equation (81). \[ \begin{array}{ll}\dot{e}_{x}(t)&=\dot{x}(t)-\dot{x}(t-\tau)\\ &=x(t)(x(t)-1)(1-rx(t))-y(t)+I(t)+f(t)\\ &-[x(t-\tau)(x(t-\tau)-1)(1-rx(t-\tau))\\ &-y(t-\tau)+I(t-\tau)+f(t-\tau)]\\ \dot{e}_{y}(t)&=\dot{y}(t)-\dot{y}(t-\tau)\\ &=b(x(t)-x(t-\tau))\end{array}  \tag{81}\]  
Merging the two errors of Equation (80) through the signed distance into an error \(v\), we get Equation (82). \[ \begin{array}{ll}v(t)&=(\lambda_{2}e_{x}(t)+e_{y}(t))/\sqrt{1+\lambda_{2}{} ^{2}}\\ &=[\lambda_{2}(x(t)-x(t-\tau))+(y(t)-y(t-\tau))]/\sqrt{1+\lambda_{2}{}^{2}}] \end{array}  \tag{82}\]  
Differentiating Equation (82) and substituting Equation (81) into it, we get Equation (83). \[ \begin{array}{ll}\dot{v}(t)&=(\lambda_{2}\dot{e}_{x}(t)+\dot{e}_{y}(t))/ \sqrt{1+\lambda_{2}{}^{2}}\\ &=\big{\{}\lambda_{2}[x(t)(x(t)-1)(1-rx(t))-y(t)+I(t)+f(t)-x(t-\tau)(x(t-\tau) -1)(1-rx(t-\tau))\\ &+y(t-\tau)-I(t-\tau)-f(t-\tau)]+b(x(t)-x(t-\tau))\big{\}}/\sqrt{1+\lambda_{2} {}^{2}}\\ &=-\lambda_{2}v(t)+(f(t)-f(t-\tau))\lambda_{2}/\sqrt{1+\lambda_{2}{}^{2}}+ \big{\{}\lambda_{2}[x(t)(x(t)-1)(1-rx(t))+I(t)\\ &-x(t-\tau)(x(t-\tau)-1)(1-rx(t-\tau))-I(t-\tau)]+b(x(t)-x(t-\tau))+\lambda_{2 }{}^{2}x(t)\\ &-\lambda_{2}{}^{2}x(t-\tau)\big{\}}/\sqrt{1+\lambda_{2}{}^{2}}\\ &=Av(t)+B(f(t)-f(t-\tau))+d\end{array}  \tag{83}\]  
where  
\[ \begin{array}{ll}A=-\lambda_{2},\\ B=\frac{\lambda_{2}}{\sqrt{1+\lambda_{2}{}^{2}}},\\ d=\big{\{}\lambda_{2}[x(t)(x(t)-1)(1-rx(t))+I(t)-x(t-\tau)(x(t-\tau)-1)\\ (1-rx(t-\tau))-I(t-\tau)]+b(x(t)-x(t-\tau))+\lambda_{2}{}^{2}x(t)-\lambda_{2 }{}^{2}x(t-\tau)\big{\}}/\sqrt{1+\lambda_{2}{}^{2}}\\ \end{array} .\]  
In an actual chaotic system, considering the dynamics of the chaotic system, the parameters will change accordingly under different conditions. Therefore, Equation (83) can be rewritten as  
\[ \begin{array}{ll}\dot{v}(t)&=(A_{n}+\Delta A)v(t)+(B_{n}+\Delta B)(f(t)-f(t -\tau))+(d+\Delta d)\\ &=A_{n}v(t)+B_{n}(f(t)-f(t-\tau))+\Delta A\dot{v}(t)+\Delta B(f(t)-f(t-\tau)) +d_{1}\\ &=A_{n}v(t)+B_{n}(f(t)-f(t-\tau))+d_{2}\end{array}  \tag{84}\]  
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), where \(d_{1}=d+\Delta d\) and \(d_{2}=\Delta Av+\Delta B(f(t)-f(t-\tau)+d_{1}\). By transposing and arranging Equation (84), we get  
\[f(t)-f(t-\tau)=\frac{1}{B_{n}}[-A_{n}v(t)-d_{2}+\dot{v}(t)]. \tag{85}\]  
From Equation (85), the proposed ATFSORC output control amount \(u\) can be assigned as \(u=f(t)-f(t-\tau)\); rearranging this can give  
\[u=\frac{1}{B_{n}}[-A_{n}v(t)-d_{2}+\dot{v}(t)]. \tag{86}\]  
Ideally, if all the parameters of the chaotic system are known, an ideal control quantity \(u^{*}\) can be constructed as  
\[u^{*}=\frac{1}{B_{n}}[-A_{n}v(t)-d_{2}+\dot{v}^{*}(t)-k_{3}e(t)], \tag{87}\]where \(v^{*}\) is the ideal error amount and \(k_{3}\) is a normal number. Error \(e\) is defined as Equation (88). \[e=v-v^{*}. \tag{88}\]  
If the control amount \(u\) approaches the ideal control amount \(u^{*}\), Equation (89) will be obtained based on Equations (86) and (87), and Equation (90) will be obtained from Equations (88) and (89). \[\dot{v}=\dot{v}^{*}-k_{3}e, \tag{89}\]  
\[\dot{e}+k_{3}e=0. \tag{90}\]  
Since \(k_{3}\) is a positive real number, and Equation (90) is a strictly Hurwitz polynomial, the error will converge to zero as time passes to infinity, i.e., \(\underset{t\rightarrow\infty}{\lim}e=0\).