## 2 Simplified DI-SMC Design of DC-AC Buck Converter
### Control-Oriented Model  
The control-oriented model should be first developed to reflect the error dynamics of the power converter, which will be used to derive the simplified control equation based on the equivalent control method. The time-variant control-state variables \(x_{1}\), \(x_{2}\), \(x_{3}\), and \(x_{4}\) have been chosen as an output voltage error, a time derivative of output voltage error, an integral of output voltage error, and a double integral of output voltage error, respectively.  
\[\left\{\begin{array}{l}x_{1}=V_{r}-\beta v_{O}\\ x_{2}=\dot{x}_{1}\\ x_{3}=\int x_{1}dt\\ x_{4}=\int(\int x_{1}dt)dt\end{array}\right. \tag{1}\]  
where \(V_{r}\), \(\beta\), and \(v_{O}\) represent the reference voltage, output voltage sensor gain, and instantaneous output voltage, respectively. The fourth state, \(x_{4}\), in (1) contains a double integral term, which is added to improve the tracking performance and eliminate the steady-state error at the output voltage.  
On the other hand, the ideal model of the buck converter in CCM is given by  
\[\left\{\begin{array}{l}\frac{di_{L}}{dt}=-\frac{1}{L}v_{O}+\frac{1}{L}v_{I}u \\ \frac{dv_{O}}{dt}=\frac{1}{C}i_{L}-\frac{1}{rC}v_{O}\end{array}\right. \tag{2}\]  
where \(r\), \(v_{I}\), and \(i_{L}\) represent the instantaneous load resistance, input voltage, and inductor current, respectively. The capacitor and inductor of the power converter are \(C\) and \(L\), respectively. The switching control input, \(u\), takes a value of 1 or 0. As reported in [31], if the ideal model of the buck converter in (2) is reflected by the control-state variables in (1), then one obtains  
Figure 1: The block diagram of the simplified DI-SMC scheme of PWM dc-ac buck converter.  
\[\left\{\begin{array}{l}x_{1}=V_{r}-\beta v_{O}\\ x_{2}=-\frac{\beta}{C}i_{C}\\ x_{3}=\int(V_{r}-\beta v_{O})dt\\ x_{4}=\int[\int(V_{r}-\beta v_{O})dt]dt\end{array}\right. \tag{3}\]  
where \(i_{C}\) represents the instantaneous capacitor current of the power converter. Thus, the time derivatives of the control-state variables in (3) yield the following dynamics:  
\[\left\{\begin{array}{l}\dot{x}_{1}=-\frac{\beta}{C}i_{C}\\ \dot{x}_{2}=\frac{\beta}{rC^{2}}i_{C}-\frac{\beta v_{I}}{LC}u+\frac{\beta v_{ O}}{LC}\\ \dot{x}_{3}=V_{r}-\beta v_{O}\\ \dot{x}_{4}=\int(V_{r}-\beta v_{O})dt\end{array}\right. \tag{4}\]  
Hence, the averaged control-oriented model of the power converter is obtained via averaging (4), according to [4], and gives  
\[\left\{\begin{array}{l}\dot{\overline{x}}_{1}=-\frac{\beta}{C}\dot{\overline {i}}c\\ \dot{\overline{x}}_{2}=\frac{\beta}{rC^{2}}i_{C}-\frac{\beta\overline{v}_{I}}{ LC}u_{e}+\frac{\beta\overline{v}_{O}}{LC}\\ \dot{\overline{x}}_{3}=V_{r}-\beta\overline{v}_{O}\\ \dot{\overline{x}}_{4}=\int(V_{r}-\beta\overline{v}_{O})dt.\end{array}\right. \tag{5}\]  
The terms \(\dot{\overline{i}}_{C}\), \(\overline{r}\), \(u_{e}\), and \(\overline{v}_{O}\) represent the averaged quantities of the capacitor current, load resistance, input voltage, control input, and output voltage, respectively. The averaged control-oriented model in (5) is the key to simplifying the equivalent control equation and the corresponding analog control circuit.