# Self-Evolving Fuzzy Controller Composed of Univariate Fuzzy Control Rules
## 3 Proposed Self-Evolving FLC Design Method
### Online Stage
#### 3.3.5 Delete Fuzzy Control Rule  
Taking into account the complex characteristics of industrial processes, such as unknown and time-varying dynamics, and disturbances, some control rules created earlier may become obsolete, and the presence of outliers (besides the criteria to avoid that) can create the wrong rules. However, in order to not remove a fuzzy control rule with significant information, the deletion of a control rule must be done carefully. In this way, two criteria to delete \"less active\" or \"less informative\" fuzzy control rules are used [36].  
**Criterion 3**.: \(\frac{N_{ij}^{total}(k)}{k-k_{ji}^{min}}<\epsilon\)_, for \(i_{j}=1,\ldots,N_{j}\) and \(j=1,\ldots,n\), where \(N_{i_{j}}^{total}(k)=N_{i_{j}}^{total}(k-1)+\omega_{j}^{i_{j}}[\mathbf{x}(k)]\) is the indicator of the total activation degrees (antecedent values) associated with the \(i_{j}\)-th fuzzy rule, being \(N_{i_{(}0)}^{total}=0\)[37; 38]. \(k\) is the current instant of time, and \(k_{i_{j}}^{ini}\) is the instant of time at which the fuzzy control rule \(R_{j}^{i_{j}}\) was created. \(\epsilon\in[0,1]\) is a positive constant._  
A usual choice has been \(\epsilon=0.1\)[36; 39].  
**Criterion 4**.: \(1-\frac{N_{ij}^{min}(k)}{k-k_{ij}^{min}}<\epsilon/2\)_, for \(i_{j}=1,\ldots,N_{j}\) and \(j=1,\ldots,n\), where \(N_{i_{j}}^{min}(k)\) is given by (18), which is the total number of times (during the interval of time \([k_{i_{j}}^{ini},k]\)) in which the antecedent value of the \(i_{j}\)-th fuzzy rule, \(\omega_{j}^{i_{j}}[\mathbf{x}(k)]\)(3), is the minimum among all the antecedent values of all fuzzy rules, and \(k_{i_{j}}^{ini}\) is the instant of time at which fuzzy control rule \(R_{j}^{i_{j}}\) was created. \(\epsilon\in[0,1]\) is the same positive constant as in Criterion (3)._  
\[N_{ij}^{min}(k)=\sum_{k_{j}^{ini}}^{k}\left[\omega_{j}^{i_{j}}[\mathbf{x}(k)] =\min_{\begin{subarray}{c}i_{j}=1,\ldots,N_{j}\\ j=1,\ldots,n\end{subarray}}(\omega_{j}^{i_{j}}[\mathbf{x}(k)])\right]\,, \tag{18}\]  
where \([\ldots]\) are the Iverson brackets and \([P]\) is defined to be one if \(P\) is true, and otherwise zero.  
Criterion 3 gives an indication of the use of a given fuzzy rule, by its antecedent value, \(\omega_{j}^{i_{j}}[\mathbf{x}(k)]\)(3), during an interval of time (\([k_{i_{j}}^{ini},k]\)). In this way, Criterion 3 is defined in order to remove control rules that are less active. On the other hand, Criterion (4) gives an indication of how long a given rule had the lowest antecedent value \(\omega_{j}^{i_{j}}[\mathbf{x}(k)]\) (3), during an interval of time (\([k_{i_{j}}^{ini},k]\)). In this way, Criterion (4) is defined in order to remove control rules that are non-informative.  
For all instants of time \(k\), Criteria 3 and 4 are verified for all control rules, and when simultaneously they are met, then fuzzy control rule \(R_{j-}^{del}\) will be deleted from the current knowledge base, as well as the associated membership function \(A_{j-}^{del}\). After removing membership function \(A_{j-}^{del}\), the nearest left (\(A_{j-}^{left}\)) and right (\(A_{j-}^{right}\)) membership functions (which are updated due the elimination of \(A_{j-}^{del}\)) are given by:  
* For \(A_{j-}^{left}\): 1) \(c_{j-left}=b_{j-right}\);
* For \(A_{j-}^{right}\): 1) \(a_{j-right}=b_{j-left}\).  
Since complementary triangular membership functions are used (Figure 1), only one of the parameters is updated for each of \(A_{j-}^{left}\) and \(A_{j-}^{right}\).