# Compact Fuzzy Systems Based on Boolean Relations
## 4 Compact Fuzzy System Architectures Based on Boolean Relations
### Compact System with Architecture II  
In this case, the coding of Table 16 is employed to find a fuzzy system where the activation functions depend on a single input set. Figure 8: Representation of a compact system with architecture I.  
Figure 9: Configuration of the neuro-fuzzy systems for identification and control. From Table 16 is obtained \(Y_{1,1}=A_{1,1}(x_{1})\). Generally increasing the columns, the activation outputs are written as:  
\[Y_{1,i}=A_{1,i}(x_{1}) \tag{42}\]  
Having for each set \(A_{1,i}\) a membership function \(\mu_{1,i}\), and the respective virtual actuator \(v_{1,i}\), then it is obtained the partial output \(y_{1}\) shown in Figure 10.  
In this way, each activation function \(Y_{j,i}\) directly depends on a set \(A_{j,i}\). The resulting expression is:  
\[Y_{j,i}=A_{j,i}(x_{j}) \tag{43}\]  
Using the respective membership functions \(\mu_{j,i}\) associated with sets \(A_{j,i}\). Figure 11 shows the general scheme obtained. The output of the inference process can be calculated as:  
\[f=\sum_{j=1}^{M}\sum_{i=1}^{N}v_{j,i}\mu_{j,i}(x_{j}) \tag{44}\]  
As seen in Figure 11, the compact fuzzy system scheme can be interpreted as a neuro-fuzzy network since input and output relationships are obtained through membership functions and virtual actuators. The membership value for each input is obtained in the first layer. Then, this value is multiplied using respective virtual actuator \(v_{j,i}\) in the second layer. In the third layer, the functions \(f_{j}\) associated to each input \(x_{j}\) are calculated. Finally, in this same layer performing the summation of all \(f_{j}\) the inference output \(f\) is determined. In this way, the adaptation parameters correspond to the virtual actuators and the configuration values for the membership functions. This architecture can be used in identification and control since it allows for the representation of nonlinear relationships between the input and the output that can be associated with a dynamic system. \begin{table}
\begin{tabular}{c c c c c} \hline \(A_{1,1}(x_{1})\) & \(A_{1,2}(x_{1})\) & \(\cdots\) & \(A_{1,i}(x_{1})\) & \(Y_{1,1}\) \\ \hline
1 & \(X\) & \(\cdots\) & \(X\) & 1 \\
0 & \(X\) & \(\cdots\) & \(X\) & 0 \\ \hline \end{tabular}
\end{table} 
Table 16: Partial coding for a compact system with architecture II. \begin{table}
\begin{tabular}{c c c c c c c} \hline \(A_{j,1}(x_{j})\) & \(A_{j,2}(x_{j})\) & \(\cdots\) & \(A_{j,i}(x_{j})\) & \(Y_{j,1}\) & \(Y_{j,2}\) & \(\cdots\) & \(Y_{j,i}\) \\ \hline
1 & \(X\) & \(\cdots\) & \(X\) & 1 & 0 & \(\cdots\) & 0 \\ \(X\) & 1 & \(\cdots\Application for Dynamic Systems  
In this case, it is observed the direct relationship that a compact FIS-BBR has to represent a nonlinear discrete time dynamic system. This equivalence is observed when modifying a linear system of discrete time using fuzzy sets to represent the relationships of the system. The model is obtained by replacing the linear relationships present for each of the delays of both the input and the output feedback. Figure 12 shows the input and output of the linear and fuzzy systems. In the first place, a linear system in discrete time is considered where its transfer function is:  
\[C(z)=\frac{U(z)}{E(z)}=\frac{b_{0}+b_{1}z^{-1}+b_{2}z^{-2}+\cdots+b_{N_{\rm L}} z^{-N_{\rm c}}}{1+a_{1}z^{-1}+a_{2}z^{-2}+\cdots+b_{N_{\rm L}}z^{-N_{\rm L}}} \tag{45}\]  
Performing the respective operations:  
\[U(z)+a_{1}z^{-1}U(z)+a_{2}z^{-2}U(z)+\cdots+b_{N_{\rm L}}z^{-N_{ \rm L}}U(z)=\\ b_{0}E(z)+b_{1}z^{-1}E(z)+b_{2}z^{-2}E(z)+\cdots+b_{N_{\rm L}}z^{-N _{\rm L}}E(z) \tag{46}\]  
The discrete time equation of this system is:  
\[u[n]=b_{0}e[n]+b_{1}e[n-1]+b_{2}e[n-2]+\cdots+b_{p}e[n-p]+\cdots+ b_{N_{\rm L}}e[n-N_{\rm L}]\\ -a_{1}u[n-1]-a_{2}u[n-2]-\cdots-a_{q}u[n-q]-\cdots-a_{N_{\rm L}}u [n-N_{\rm L}] \tag{47}\]  
Figure 11: General diagram of a compact system with architecture II. Figure 12: Linear system and fuzzy system. where the respective coefficients \(a_{q}\), \(b_{p}\) are constant.