## 2 Simplified DI-SMC Design of DC-AC Buck Converter
### Existence and Stability Conditions  
The existence and stability conditions must be derived to ensure a proper selection for the controller gains that satisfy the SMC operation [3]. The local reachability condition is applied to derive the existence condition \(\lim\limits_{\psi\to 0}\psi\dot{\psi}<0\), which can be expressed as  
\[\left\{\begin{array}{l}K_{p}x_{1(max)}+K_{i}x_{3(max)}<\beta\bigg{(}v_{I(min )}-v_{O}\bigg{)}\\ -K_{p}x_{1(min)}-K_{i}x_{3(min)}<\beta v_{O}.\end{array}\right. \tag{14}\]  
The minimum and maximum quantities of \(v_{I}\), \(x_{1}\), and \(x_{3}\) are included in (14) to take the full-load operating conditions into account. This condition ensures that all the trajectories remain within the vicinity of the sliding surface [31].  
In order to analyze the stability of the closed-loop dynamics of the power converter around the desired equilibrium point, the linearization of the nonlinear dc-ac buck converter model is required. The averaged closed-loop power converter dynamics are \[\left\{\begin{array}{l}\frac{d\tilde{i}_{L}}{dt}=-\frac{1}{L}\overline{v}_{O}+ \frac{1}{L}\overline{v}_{I}u_{e}\\ \frac{d\overline{v}_{O}}{dt}=\frac{1}{C}\tilde{i}_{L}-\frac{1}{\tilde{r}C} \overline{v}_{O}\end{array}\right. \tag{15}\]  
where \(u_{e}\) is the equivalent control law given in (11). The linearized model of the power converter is obtained by perturbing (15) around the equilibrium point \(I_{L}=V_{O}/R\), which results in  
\[\begin{bmatrix}\tilde{\tilde{x}}_{1}\\ \tilde{\tilde{x}}_{2}\\ \tilde{\tilde{x}}_{3}\end{bmatrix}=\begin{bmatrix}0&j_{12}&j_{13}\\ j_{21}&j_{22}&0\\ 0&1&0\end{bmatrix}\begin{bmatrix}\tilde{\tilde{x}}_{1}\\ \tilde{\tilde{x}}_{2}\\ \tilde{\tilde{x}}_{3}\end{bmatrix}. \tag{16}\]  
The states of the linearized model \(\tilde{\tilde{x}}_{1}\), \(\tilde{\tilde{x}}_{2}\), and \(\tilde{\tilde{x}}_{3}\) are defined as \(\widetilde{i}_{L}\), \(\widetilde{v}_{O}\), and \(\int\widetilde{v}_{O}dt\), respectively. The Jacobian Matrix \(J\) in (16) is defined by  
\[J=\begin{bmatrix}0&-\frac{K_{p}}{L}&-\frac{K_{i}}{L}\\ \frac{1}{C}&-\frac{K_{i}}{RC}&0\\ 0&1&0\end{bmatrix}. \tag{17}\]  
It should be noted that the linearized model in (16) is obtained considering that \(v_{I}=V_{I}\), \(r=R\), \(V_{r}-\beta V_{O}=0\), \(I_{L}>\widetilde{i}_{L}\), and \(V_{O}>\widetilde{v}_{O}\). The characteristic equation of the linearized model is given by  
\[\begin{vmatrix}\lambda&-j_{12}&-j_{13}\\ -j_{21}&\lambda-j_{22}&0\\ 0&-1&\lambda\end{vmatrix}=\lambda^{3}+P_{1}\lambda^{2}+P_{2}\lambda+P_{3}=0 \tag{18}\]  
where the parameters \(P_{1}\), \(P_{2}\), and \(P_{3}\) are defined as  
\[\begin{bmatrix}P_{1}&P_{2}&P_{3}\end{bmatrix}^{T}=\begin{bmatrix}-j_{22}&-j_ {12}j_{21}&-j_{13}j_{21}\end{bmatrix}^{T} \tag{19}\]  
The linearized closed-loop power converter model is stable if all the Eigen values of the Jacobian Matrix have a negative real part. Using the Routh-Hurwitz stability criterion, one obtains the following stability conditions \(P_{1}>0\), \(P_{3}>0\), and \(P_{2}>\frac{P_{3}}{P_{1}}\).  
The existence and stability conditions ensure that the controller gains \(K_{p}\) and \(K_{i}\) in the equivalent control equation, maintain the control-state variables within the vicinity of the sliding surface, and drive them toward the desired equilibrium point.