## 4 Interval Type-2 Fuzzy Logic Controller for Constant Switching Frequency  
Any acceptable constant or flexible value can be used to set the hysteresis bandwidth. The switching frequency should be considered when choosing the hysteresis band value since there is a strong association between them. To figure out the relationship between the switching frequency (\(f_{s}\)) and the hysteresis bandwidth (\(\Delta\)), Figure 2 can be utilized. From the figure, it is required to calculate the derivative of \(S\) in order to find the relationship between \(f_{s}\) and \(\Delta\). Note that \(\dot{S}^{+}\) and \(\dot{S}^{-}\) are considered the positive slope and negative slope of the time derivatives of the proposed sliding surface. According to the geometry of Figure 2 and with the help of (2), (5), and (6), the ON and OFF periods of the switch can be expressed as below by assuming that the error variable \(x_{1}\) is negligible and the desired inductor current is \(I_{Ldesird}=\frac{\left(V_{ref}\right)^{2}}{\text{RE}}\) in the steady state. \[T_{on}=\frac{-2\Delta}{\dot{S}^{-}}=\frac{-2\Delta}{\lambda\frac{V_{0}}{\text {RC}}-\gamma\frac{E}{L}} \tag{9}\]  
\[T_{off}=\frac{2\Delta}{\dot{S}^{+}}=\frac{2\Delta}{\lambda\left(\frac{V_{0}}{ \text{RC}}-\frac{V_{ref}\right^{2}}{\text{REC}}\right)-\gamma\left(\frac{E}{ L}-\frac{V_{0}}{L}\right)} \tag{10}\]  
With the help of (9) and (10), the expression for the switching frequency can be obtained as  
\[f_{s}=\frac{1}{T_{on}+T_{off}}=\frac{1}{2\Delta\left(\frac{1}{a_{1}}-\frac{1}{ a_{2}}\right)} \tag{11}\]  
where \(a_{1}=\lambda\left(\frac{V_{0}}{\text{RC}}-\frac{V_{ref}\right^{2}}{\text{REC }}\right)-\gamma\left(\frac{E-V_{0}}{L}\right)\) and \(a_{2}=\lambda\frac{V_{0}}{\text{RC}}-\gamma\frac{E}{L}\). By choosing an optimal hysteresis bandwidth, one can obtain a desirable switching frequency based on (11). However, this method raises a concern that the switching frequency varies over time as it depends on the parameters of the circuit and also coefficients of the designed surface. This is not desirable, as discussed in the previous section. To remove all  
Figure 2: Hysteresis-modulated sliding function. of the problems caused by a time-varying switching frequency and to prevent running the converter at a different switching frequency than that desired, a basic controller should be employed. With the hypothesis of piecewise linear behavior of the sliding function [4; 15], the time-varying switching frequency issue can be resolved. Unfortunately, as it was noted, this strategy results in an excessive amount of mathematical complexity. For this reason, type-2 fuzzy logic control is proposed in this study. Block diagrams of a type-1 fuzzy logic system (T1 FS) and a type-2 fuzzy logic system (T2 FS) are presented in Figure 3. Both systems operate on the same fundamental principle. Nevertheless, there are some significant distinctions between them. While both T1 FS and T2 FS have the fuzzification, fuzzy inference engine, knowledge base, and defuzzification, T2 FS also has a type-reducer stage that sits between the inference engine and the defuzzifier, as demonstrated in Figure 3. Another difference is that the membership functions of the T1 FS are certain, whereas T2 FS employs type-2 fuzzy sets, which are described by fuzzy membership functions. By doing so, T2 FS are able to simulate and govern measurement uncertainty (converter structure, sensors) as well as any rule uncertainty, resulting in more satisfied dynamic responses [23; 24]. In contrast to T1 FS, several investigations have also supported this circumstance, namely the robustness of T2 FS [23; 24; 25]. One of the most significant steps in the T2 FS design is defining the input and output variables, as well as the controller's membership functions. Because the switching frequency of the converter is controlled, the input variables of the proposed IT2 FS are the switching frequency error (e) and the change in switching frequency error (ce).