## 2 Simplified DI-SMC Design of DC-AC Buck Converter
### Equivalent Control Law  
According to [3], if the hysteresis-modulation-based SMC method is utilized with the power converters, a variable switching frequency is generated, which complicates the design of the filters and worsens the EMI issues [3]. Hence, the equivalent control method should be applied to the PWM-based SMC design to maintain a constant switching frequency. To derive the equivalent control law, the hitting condition must be satisfied [3]. This condition can be achieved by choosing a suitable switching control law, which can be defined as  
\[u= \begin{cases}1&\text{for }\psi>0\\ 0&\text{for }\psi<0\end{cases}  \tag{6}\]  
where \(\psi\) is the sliding surface, which is given by  
\[\psi=\alpha_{1}x_{1}+\alpha_{2}x_{2}+\alpha_{3}x_{3}+\alpha_{4}x_{4}. \tag{7}\]  
The parameters \(\alpha_{1}\), \(\alpha_{2}\), \(\alpha_{3}\), and \(\alpha_{4}\) in (7) are positive constants that represent the sliding coefficients. The direct implementation of the switching control law, \(u\), in (6) with a pulse-width modulator is not possible [3]. Instead, an equivalent control law, \(u_{e}\), should be derived and mapped onto a duty cycle, \(d\). Now, if (7) is derived with respect to time, then one obtains the following sliding surface dynamics:  
\[\dot{\psi}=\alpha_{1}\dot{x}_{1}+\alpha_{2}\dot{x}_{2}+\alpha_{3}\dot{x}_{3}+ \alpha_{4}\dot{x}_{4}=0. \tag{8}\]It should be noticed that (8) is equated to zero based on the invariance conditions presented in [4], which means that the dynamics of the control-state variables become zero at the sliding surface, \(\psi\), if a proper control input is applied to the power converter. Next, the averaged model in (5) is substituted into (8), yielding  
\[\alpha_{1}\bigg{(}-\frac{\beta}{C}\tilde{i}_{C}\bigg{)}+\alpha_{2}\bigg{(}\frac {\beta}{FC^{2}}\tilde{i}_{C}-\frac{\beta\overline{v}_{l}}{LC}u_{e}+\frac{ \beta\overline{v}_{O}}{LC}\bigg{)}+\alpha_{3}(V_{r}-\beta\overline{v}_{O})+ \alpha_{4}\bigg{(}\int(V_{r}-\beta\overline{v}_{O})dt\bigg{)}=0. \tag{9}\]  
It is known that the averaged capacitor current, \(\tilde{i}_{C}\), at a steady state is zero [4]. Hence, if the terms \(\tilde{i}_{C}\) in (9) are neglected, then one obtains  
\[\alpha_{2}\bigg{(}-\frac{\beta\overline{v}_{l}}{LC}u_{e}+\frac{\beta \overline{v}_{O}}{LC}\bigg{)}+\alpha_{3}(V_{r}-\beta\overline{v}_{O})+\alpha_ {4}\bigg{(}\int(V_{r}-\beta\overline{v}_{O})dt\bigg{)}=0. \tag{10}\]  
Now, solving for the equivalent control law, \(u_{e}\), we obtain  
\[u_{e}=K_{p}\bigg{(}\frac{V_{r}-\beta\overline{v}_{O}}{\beta\overline{v}_{I}} \bigg{)}+K_{i}\bigg{[}\frac{\int(V_{r}-\beta\overline{v}_{O})dt}{\beta \overline{v}_{I}}\bigg{]}+\frac{\overline{v}_{O}}{\overline{v}_{I}}. \tag{11}\]  
The controller parameters \(K_{p}\) and \(K_{i}\) represent the proportional and integral gains, respectively, which are given by  
\[\big{[}K_{p}\quad K_{i}\big{]}^{T}=\Big{[}LC\frac{\alpha_{3}}{\alpha_{2}} \quad LC\frac{\alpha_{4}}{\alpha_{2}}\Big{]}^{T} \tag{12}\]  
In [3; 4], the equivalent control law, \(u_{e}\), is related to the duty cycle, \(d\), and the peak ramp voltage, \(V_{T}\), by the following relationship: \(0<d=\frac{\tilde{u}_{e}}{V_{T}}<1\). This maps (11) onto the following control equation:  
\[\left\{ \begin{array}{l}\tilde{u}_{e}=\gamma\big{[}K_{p}(V_{r}-\beta \overline{v}_{O})+K_{i}\int(V_{r}-\beta\overline{v}_{O})dt+\beta\overline{v}_{ O}\big{]}\\ V_{T}=\gamma\beta\overline{v}_{I}\end{array} \right. \tag{13}\]  
Note that the control voltage, \(\tilde{u}_{e}\), divided by the peak ramp voltage, \(V_{T}\), gives the duty cycle, \(d\), which varies between \(0\) and \(1\). In (13), parameter \(\gamma\) is a scaling factor (\(0<\gamma<1\)), which can be utilized to scale the parameters of the control equation down within the practical range of the analog control circuit. For instance, if \(V_{T}=\beta\overline{v}_{I}\) is \(10\) V and the maximum ramp voltage of the pulse-width modulator, IC, is \(5\) V, then \(V_{T}\) is multiplied by a scaling factor of \(0.5\).