## 3 Design of Sliding Mode Controller  
The SMC is designed in two steps. One of the two steps is determining a suitable sliding manifold [7]. The designed sliding surface is given below [1]  
\[S=\lambda x_{1}+\beta\int\limits_{0}^{t}x_{1}(t)dt-\gamma I_{L} \tag{2}\]  
where \(\lambda,\beta,\gamma\) are the positive parameters, \(x_{1}\) is the output voltage error, which is defined as \(x_{1}=V_{ref}-V_{0}\), and \(V_{ref}\) is the reference value of output voltage. Figure 1: Boost converter (blue line: the switch is ON, red line: the switch is OFF). This converter exhibits non-minimum phase characteristics, as seen by the space-state equations in (1). In a more intelligible expression, the control signal is linked to the inductor current as well as the output voltage. This characteristic of the converter precludes direct voltage control [1; 18; 19]. To address this issue, the inductor variable must be located within the specified surface. The surfaces chosen for non-minimum phase converters in HM-SMC investigations are commonly classified into two major classes in the literature: Firstly, surfaces that depend just on inductor current error [1; 11; 18; 20], and secondly, surfaces that depend on both inductor current error and output voltage error [10; 12; 13; 14; 17; 19]. Both categories of surface topologies have their own share of advantages and disadvantages. In the surface utilized in this study, there is no extra value specified for the reference value of the inductance current, and it is generated by using the output voltage error, which makes it the best option. For the second stage of SMC design, which is the determination of the control law, the following equation is provided from the study in [8]. \[u=\left\{ \begin{array}{ll}0&\qquad\quad\text{when}\quad S<-\Delta\\ 1&\qquad\quad\text{when}\quad S>\Delta\end{array} \right. \tag{3}\]  
where \(\Delta\) is a hysteresis bandwidth. The expression formulated in (3) is derived from the formulation \(u=(1+sign(S))/2\) because, as highlighted in the introduction section, the formula in (3) enables the practical application and eliminates several additional issues. By the way, it should be noticed that, in some studies, a PWM-based SMC has taken the place of the HM-SMC [3; 21; 22]. However, the control becomes sensitive to the parameters. In this case, SMC robustness should be re-established by introducing additional terms to compensate for parameter dependence [5]. In addition, with this approach, the control signal is created by comparing it with a trigger signal rather than going directly to the switch. All of these disadvantages result in more complicated control structures. The stability of sliding mode control is assured if the following criterion is met. \[S\frac{dS}{dt}<0 \tag{4}\]  
where \(\frac{dS}{dt}=\lambda\left(\frac{V_{0}}{RC}-\frac{(1-u)I_{t}}{C}\right)+\beta x _{1}-\gamma\left(\frac{E}{L}-\frac{(1-u)V_{0}}{L}\right)\). Existence conditions, such as the hitting condition mentioned in (3), must be observed for the stable functioning of SMC. With the help of the conditions given in (3) and (4), one can generate the existence conditions shown below [3]. When \(S<0\Rightarrow u=0\), then the following inequality can be written for \(\dot{S}>0\)  
\[\lambda\left(\frac{V_{0}}{RC}-\frac{I_{t}}{C}\right)+\beta x_{1}-\gamma\left( \frac{E}{L}-\frac{V_{0}}{L}\right)>0 \tag{5}\]  
When \(S>0\Rightarrow u=1\), then the following inequality can be written for \(\dot{S}<0\)  
\[\lambda\left(\frac{V_{0}}{RC}\right)+\beta x_{1}-\gamma\left(\frac{E}{L} \right)<0 \tag{6}\]  
The coefficients \(\lambda,\beta\) and \(\gamma\) should be selected to satisfy (5) and (6). The equivalent control [1; 4; 8], represented by \(u_{eq}\), is obtained by applying the invariance criterion provided by the following equation. \[ \begin{array}{l}\frac{dS}{dt}_{S=0}=0\\ u=u_{eq}\\ u_{eq}=1-\frac{\frac{\frac{\lambda V_{0}}{RC}-\beta x_{1}+\frac{E}{L}}{-\frac{ \lambda I_{t}}{C}}}{-\frac{\lambda I_{t}}{C}+\frac{V_{0}}{C}}\end{array}