## 3 Adaptive TSK Fuzzy Self-Organizing Recurrent Cerebellar Model Articulation Controller Design
### Design of TSK Fuzzy Self-Organizing Recurrent Cerebellar Model Articulation Controller
It takes the signals of integrated error function as the input, as shown in Figure 4:8 Here, \(y_{d}\) is the desired output; \(y\) is the actual output; \(e\) is the difference value between the desired output and actual output; \(\beta_{w}\), \(\beta_{m}\), \(\beta_{v}\), and \(\beta_{r}\) are the learning rates of weight value for weight memory, midpoint, and standard deviation for Gaussian function, and recurrent weight value, respectively; and \(\gamma\) is the compensation coefficient. The total control amount \(u\) of ATFSORC is as shown in Equation (25), where \(u_{TFSORC}\) is the control amount of the TSK fuzzy self-organizing cerebellar model articulation controller; and \(u_{c}\) is the control amount of the improved compensation controller. When there is an ideal control amount \(u^{*}\), the error between the \(u_{TFSORC}\) of the TSK fuzzy self-organizing cerebellar model articulation controller and the ideal control amount \(u^{*}\) will be compensated by the improved compensation controller \(u_{c}\). Figure 4:9In this study, the proposed ATFSORC is used as a synchronization controller for chaotic systems. As shown in Figure 6, the controlled body in Figure 5 is converted into a slave system; \(y_{d}\) can be regarded as a master system output \((x_{1},y_{1});y\) can be regarded as a slave system output \((x_{2},y_{2})\); \(e_{x}\) and \(e_{y}\) are the error values of \(x_{1}\), \(x_{2}\) and \(y_{1}\), \(y_{2}\), respectively. Moreover, in order to reduce the architectural complexity of ATFSORC, a method of signed distance is introduced to convert the two errors \(e_{x}\) and \(e_{y}\) in Figure 6 into an error \(v\). Figure 7 shows the \(e_{x}-e_{y}\) plane and the switch line \(L_{S}\), which is as shown in Equation (26). Figure 3:0  
where \(\lambda\) is a normal number; and \(v\) is the signed distance between \(M(e_{x},e_{y})\) and \(L_{S}\), as shown in Equation (27). Figure 3:1  
Figure 3:2 Adaptive TSK fuzzy self-organizing cerebellar model articulation controller architecture. Figure 3:3 Adaptive Takagi–Sugeno–Kang (TSK) fuzzy self-organizing recurrent cerebellar model articulation controller (ATFSORC) architecture diagram for chaotic system synchronization. In a neuronal system, a gap junction is an electrical synapse. Through the gap junction, neurons can communicate with each other, and the synaptic current is directly proportional to the membrane potentials between the two neurons. In the previous literature [26], the effect of gap junction on the synchronization of two coupled FitzHugh-Nagumo (FHN) neurons was studied. A circuit diagram of the neuron model is shown in Figure 8, where:  
\(V_{1}\), \(V_{2}\): Membrane voltages of two gap-junction-coupled neurons,  
\(f(V_{1}),f(V_{2})\), \(L_{m}\), \(C_{m}\): Electrical parameters of neurons,  
\(g\): Coupling strength between neurons,  
\(I_{o}\): External electrical stimulation. The two gap-junction-coupled FHN neuron model is expressed in Equations (28) and (29). Master system:  
\[ \begin{array}{l}\dot{x}_{1}=x_{1}(x_{1}-1)(1-rx_{1})-y_{1}-g(x_{1}-x_{2})+I\\ \dot{y}_{1}=bx_{1}\end{array}  \tag{28}\]  
Slave system:  
\[ \begin{array}{l}\dot{x}_{2}=x_{2}(x_{2}-1)(1-rx_{2})-y_{2}-g(x_{2}-x_{1})+I+u \\ \dot{y}_{2}=bx_{2}\end{array}  \tag{29}\]  
First, the errors of two gap-junction-coupled FHN neurons are defined as follows:  
\[ \begin{array}{l}e_{x}=x_{1}-x_{2}\\ e_{y}=y_{1}-y_{2}\end{array}  \tag{30}\]  
Next, the error dynamic system can be expressed as Equation (31). \[ \begin{array}{l}\dot{e}_{x} =\dot{x}_{1}-\dot{x}_{2}\\ =-rx_{1}{}^{3}+(r+1)x_{1}{}^{2}+rx_{2}{}^{3}-(r+1)x_{2}{}^{2}\\ -2g(x_{1}-x_{2})-(x_{1}-x_{2})-(y_{1}-y_{2})-u\\ \dot{e}_{y} =\dot{y}_{1}-\dot{y}_{2}\\ =b(x_{1}-x_{2})\end{array}  \tag{31}\]  
Merging the two errors of Equation (30) through signed distance into one error \(v\), as shown in Equation (32), we get  
\[v=(\lambda_{1}e_{x}+e_{y})/\sqrt{1+\lambda_{1}{}^{2}}. \tag{32}\]  
Differentiating Equation (32) and substituting in Equation (31) gives Equation (33).