## 3 Adaptive TSK Fuzzy Self-Organizing Recurrent Cerebellar Model Articulation Controller Design
### Design of TSK Fuzzy Self-Organizing Recurrent Cerebellar Model Articulation Controller
In this study, the physical memory space is the product of the receptive field space and the TSK output; each recurrent input state \(x_{j}\) corresponds to a TSK fuzzy system. Wherein, the fuzzy control rules for the \(j\)-th fuzzy system are designed as follows:  
\(R^{(1)}\): _if \(x_{j}\) is NB, then \(t_{j1}=c_{j0}^{1}+c_{j1}^{1}x_{j}\),_  
\(R^{(2)}\): _if \(x_{j}\) is NM, then \(t_{j2}=c_{j0}^{2}+c_{j1}^{2}x_{j}\),_  
\(R^{(3)}\): _if \(x_{j}\) is NS, then \(t_{j3}=c_{j0}^{3}+c_{j1}^{3}x_{j}\),_  
\(R^{(4)}\): _if \(x_{j}\) is \(Z\), then \(t_{j4}=c_{j0}^{4}+c_{j1}^{4}x_{j}\),_  
\(R^{(5)}\): _if \(x_{j}\) is PS, then \(t_{j5}=c_{j0}^{5}+c_{j1}^{5}x_{j}\),_  
\(R^{(6)}\): _if \(x_{j}\) is PM, then \(t_{j6}=c_{j0}^{6}+c_{j1}^{6}x_{j}\),_  
\(R^{(7)}\): _if \(x_{j}\) is PB, then \(t_{j7}=c_{j0}^{7}+c_{j1}^{7}x_{j}\),_  
where \(t_{j\eta}\) is the linear combination of the posterior parts in TSK, as shown in Equation (13). \[t_{j\eta}=c_{j0}^{4}+c_{j1}^{4}x_{j}=\mathbf{c}_{1}^{T}\mathbf{f},\,\,q\,\, =\,\,1,\,2,\,...,\,7, \tag{13}\]  
where \(\mathbf{c}_{j}=\left[c_{j0}^{4},c_{j1}^{q}\right]^{T},\mathbf{f}=\left[1,\,x_ {j}^{T}\right]^{T}\), for \(j=1,\,2,\,\ldots,\,p\). The function value in receptive field vector \(\mathbf{b}\) is defined as firing strength \(\alpha_{j}\), which is used for determining the self-organizing layer decision-making mechanism. \[\alpha_{j}=b_{j},\,\text{for}\,j=1,\,2,\,\ldots,\,p. \tag{14}\]  
Next, it is to determine the maximum value \(\alpha_{max}\) from these firing strengths. The maximum firing strength \(\alpha_{max}\) is compared with a preset threshold value \(\alpha_{th}\). If \(\alpha_{max}\) is less than or equal to \(\alpha_{th}\), one new layer of association memory will be added. The threshold value \(\alpha_{th}\) is in between 0 and 1. \[\alpha_{max}=\max_{j=1}^{p(k)}\alpha_{j}, \tag{15}\]  
\[\alpha_{max}\leq\alpha_{th}. \tag{16}\]The determinant of Equation (16) will cause the number of layers to increase by at most one layer at a time, as shown in Equation (17). Newly added association memory is equally implanted with one Gaussian function. For the new Gaussian function, the midpoint \(m_{i}^{new}\), standard deviation \(v_{i}^{new}\), weight \(w^{new}\), and recurrent weight value \(r_{l}^{new}\) are defined as below:  
Figure 4:0  
Figure 4:1  
Figure 4:2  
Figure 4:3  
Figure 4:4  
Here, \(x_{j}\) is the current input state; \(v_{o}\) and \(r_{o}\) are the constants given in advance; and \(w_{o}\) is the TSK parameter given in advance. If \(\alpha_{j}\) is less than or equal to the threshold value \(\beta_{th}\), the contribution of the layer will not be large, so deleting that layer of association memory will not affect the control performance; thus, the computation amount can be reduced. Figure 4:5  
The physical memory space is shown by Equation (23). Figure 4:6  
where \(j=1,\,2,\,\ldots,\,p\) and \(q=1,\,2,\,\ldots,\,7\). After summation, the resulting total output control amount is as shown in Equation (24). Figure 4:7  
where \(\mathbf{\Lambda}=[b_{1}\textbf{f}^{T},\,b_{2}\textbf{f}^{T},\,...,\,b_{p}\textbf{ f}^{T}]^{T}\); \(\textbf{W}=[\textbf{c}_{1}^{T},\,\textbf{c}_{2}^{T},\,\ldots,\,\textbf{c}_{ j}^{T}]^{T}\). The proposed TSORC can enable the CMAC with originally static and fixed architecture to possess dynamic memory performance, and the number of association memory layers can increasingly or decreasingly change along with the self-organizing layer decision-making mechanism in order to improve the controller output precision and reduce the computation amount, thereby providing better control performance than the traditional CMAC. Design of Adaptive TSK Fuzzy Self-Organizing Recurrent Cerebellar Model Articulation Controller for Chaotic Synchronization Control System  
In this study, TSORC combines the adaptive theorem with an integrated error function along with an improved compensation controller to form the adaptive TSK fuzzy self-organizing recurrent cerebellar model articulation controller (ATFSORC).