# Self-Evolving Fuzzy Controller Composed of Univariate Fuzzy Control Rules
## 4 Results
### Results' Analysis  
The results, along time, are presented in Figures 4-6, where:  
* Figure 4 presents the global results of the direct FLC controller online designed by the proposed self-evolving methodology, in which Figure 4a presents the evolution of the tracking performance for the unknown regions of operation, Figure 4b presents the evolution of control signal, \(u(k)\), Figure 4c shows the evolution of the number of fuzzy control rules for each input variable, \(x_{1}\) and \(x_{2}\), and Figure 4d shows the evolution of \(\Delta e_{c}(k)\), which is associated with Criterion 1 to add new fuzzy control rules. Figure 4a-b presents also the results of the SEDFLC evolving method proposed in [31]. * Figures 5 and 6 present the evolution of the antecedent and consequent parameters, i.e., the structural changes along time and the consequent adaptation, and the final membership functions for \(x_{1}(k)=r(k)\) and \(x_{2}(k)=y(k)\), respectively. Figure 4: _Cont._Figure 4: Results of the proposed self-evolving FLC design methodology on the CSTR plant. (**a**) Results of the proposed methodology, and of the SEDFLC method [31]. (**b**) Command signal of the proposed method and of the SEDFLC method [31]. (**c**) Evolution of the number of fuzzy control rules. (**d**) Evolution of \(\Delta e_{c}(k)\). Figure 5: Cont. Figure 5: Evolution of the antecedent and consequent parameters of \(x_{1}=r(t)\). (**a**) Evolution of the antecedent parameters of \(x_{1}=r(t)\). (**b**) Evolution of the consequent parameters of \(x_{1}=r(t)\). (**c**) Final membership functions of \(x_{1}=r(t)\). Figure 6: _Cont._  
From the results, analyzing the performance of the direct FLC, online designed by the proposed self-evolving methodology, it can be seen that:  
* Since, initially, the controller is offline designed using only the variables range values, using two fuzzy control rules per input variable, where the membership functions were defined as presented in Figure 2, thus, the controller is initialized without any control knowledge or previous data of the process under control, i.e., without knowledge of any region of operation. It can be seen from the results that the tracking performance increases during the time of operation and that the proposed evolving methodology adds new fuzzy control rules (see Figure 4c) when unknown regions of operation are reached. * When the reference has the value \(r(k)=0.065\,(1500\leq k<1800)\), that \(r(k)\) region of operation has not been learned (reached) previously, and new control rules were added in both input variables. Afterwards, as that region of operation was never reached again, the control rules that were previously added for that region were deleted, due to the fact that Criteria 3 and 4 considered these rules obsolete (\"less active\" and \"less informative\"). * Additionally, it can be seen that for \(k<1800\), there are large changes in the control structures, namely in the antecedent (Figures 5a and 6a) and consequent parts (Figures 5b and 6b) because, until then (\(k<1800\)), most of the regions of operation were unknown to the controller, and the proposed design methodology online evolved the control structure (i.e., the fuzzy control rules); afterwards, for \(k\geqslant 1800\), with the exception of the process of deleting the control rules, only small changes were made in the antecedent and the consequent parts of the rules, since in that time interval, the regions of operation were similar to the ones already learned before. Figure 6: Evolution of the antecedent and consequent parameters of \(x_{2}=y(t)\). (**a**) Evolution of the antecedent parameters of \(x_{2}=y(t)\). (**b**) Evolution of the consequent parameters of \(x_{2}=y(t)\). (**c**) Final membership functions of \(x_{2}=y(t)\).