## 3 Adaptive TSK Fuzzy Self-Organizing Recurrent Cerebellar Model Articulation Controller Design
### Design of Improved Compensator  
The adaptive TSK fuzzy self-organizing cerebellar model articulation controller proposed in this study contains an improved compensation controller. Equation (64) indicates that the output of compensation controller \(u_{c}\) is composed of a normal number \(\delta_{1}\) and a sign function of \(s_{a}\), wherein the function of \(\text{sgn}[s_{a}]\) is defined as in Equation (76). Since \(\delta_{1}\) is a normal number, the output of the compensation controller depends on the integrated error function \(s_{a}\) value. In fact, there are only three types of states as \(\delta_{1}\), \(0\), and \(-\delta_{1}\), which cause the shortcoming of having insufficient transient-state compensation or excessive steady-state compensation. \[\text{sgn}[s_{a}]=\left\{ \begin{array}{ll}1&,s_{a}>0\\ 0&,s_{a}=0\\ -1&,s_{a}<0\end{array} \right. \tag{76}\]  
Therefore, this study proposes an improved compensator to ameliorate the original compensation controller as presented arithmetically in Equation (64). The design of an 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 \(s_{a}\) can be used as the basis to determine whether the ATFSORC has completed learning and entered the steady state. The above-mentioned design concept can be implemented from Equation (77). \[u_{c}=\left\{ \begin{array}{ll}\delta_{1}\,\gamma_{1}\,s_{a},&|s_{a}|>\zeta_{1} \\ \delta_{1}\text{sgn}[s_{a}],&0<|s_{a}|<\zeta_{1}\end{array} \right., \tag{77}\]  
where \(\zeta_{1}\) is the threshold value for switching the compensation control amount; \(\gamma_{1}\) is a compensation coefficient; and \(\gamma_{1}>1\), which is used to elastically adjust the magnitude of compensation amount in the transient state. When \(|s_{a}|>\zeta_{1}\), it is a transient state, whereas when \(0<|s_{a}|<\zeta_{1}\), it is a steady state. The output amount of the improved compensation controller at the transient state is much larger than that at the steady state, which can effectively ameliorate the shortcoming of poor response of the self-organizing recurrent cerebellar model articulation controller at the transient state. From Equation (77), we see that if the selected \(\delta_{1}\) still meets \(|D_{1}|<\delta_{1}\) and an appropriate \(\gamma_{1}\) is selected, the stability of the overall system will not change. Introducing ATFSORC with an improved compensation controller still can guarantee the convergence in errors of the system and maintain the stability of the system. Design of Adaptive TSK Fuzzy Self-Organizing Recurrent Cerebellar Model Articulation Controller for Chaotic Stability Control System  
In this study, the proposed adaptive TSK fuzzy self-organizing recurrent cerebellar model articulation controller is used as a stability controller for chaotic systems. Therefore, by converting the controlled body in Figure 9 into a chaotic system, \(y_{d}\) can be regarded as the chaotic system outputs of \(x(k)\) and \(y(k)\); \(y\) can be regarded as the chaotic system delayed feedback outputs of \(x(k-1)\) and \(y(k-1)\); \(\varepsilon_{x}\) and \(e_{y}\) are errors of \(x(k)-x(k-1)\) and \(y(k)-y(k-1)\), respectively. Moreover, in order to reduce the architectural complexity of ATFSORC, the two errors \(\varepsilon_{x}\) and \(\varepsilon_{y}\) are merged into one error \(v\) through the signed distance. First, the chaotic system is expressed as Equation (78)  
\[ \begin{array}{ll}\dot{x}(t)=x(t)(x(t)-1)(1-rx(t))-y(t)+I(t)+f(t)\\ \dot{y}(t)=bx(t)\end{array}  \tag{78}\]  
After Equation (78) goes through the time delay, Equation (79) can be obtained. \[ \begin{array}{ll}\dot{x}(t-\tau)=x(t-\tau)(x(t-\tau)-1)(1-rx(t-\tau))-y(t- \tau)+I(t-\tau)+f(t-\tau)\\ \dot{y}(t-\tau)=bx(t-\tau)\end{array}