## 2 Higher-Order Sliding Mode Regulation for Boost Converter Output
### Stability Analysis of the Sliding Mode  
When the designed sliding mode happens, i.e., \(\sigma(t)=z_{2}(t)+c_{1}z_{1}(t)=0\)\(\forall t\geq t_{s}\), the sliding mode dynamics are described by the following first-order system:  
\[\dot{z}_{1}=-\frac{z_{2}+\pi_{2}(\omega)}{L}v+\frac{E}{L}-\frac{\partial\pi_{1} (\omega)}{\partial\omega}S\omega\bigg{|}_{ \begin{array}{c}z_{2}=-c_{1}z_{1} \\ v=v_{eq}(z,\omega)\end{array} }, \tag{27}\]  
where the equivalent control \(v_{eq}\) is calculated from \(\dot{\sigma}=0\) as follows:  
\[\dot{\sigma}=\eta(z,\omega)+\delta(z,\omega)v_{eq}(z,\omega)=0, \tag{28}\]  
\[v_{eq}(z,\omega)=-\frac{\eta(z,\omega)}{\delta(z,\omega)}. \tag{29}\]  
This results in  
\[\dot{z}_{1}=f_{0}(z_{1},\omega), \tag{30}\]  
where  
\[f_{0}(z_{1},\omega)=\left(\frac{c_{1}z_{1}+\pi_{2}(\omega)}{L}\right)\left( \frac{c_{1}\frac{E}{L}-c_{1}\frac{\partial\pi_{1}(\omega)}{\partial\omega}S \omega-\frac{c_{1}z_{1}+\pi_{2}(\omega)}{RC}-\frac{\partial\pi_{2}(\omega)}{ \partial\omega}S\omega}{\frac{z_{1}+\pi_{1}(\omega)}{C}-c_{1}\frac{-c_{1}z_{1 }+\pi_{2}(\omega)}{L}}\right)+\frac{E}{L}-\frac{\partial\pi_{1}(\omega)}{ \partial\omega}S\omega. \tag{31}\]  
**Lemma 1. **_Under the conditions in Equations (13) and (14), origin \(z_{1}=0\) is an equilibrium point of the sliding mode dynamics in Equation (30)._  
**Proof. ** Assume that \(z_{1}=0\); then, the sliding mode dynamics in Equation (30) reduces to the following expression:  
\[f_{0}(0,\omega)=\left(\frac{\pi_{2}(\omega)}{L}\right)\left(\frac{c_{1}\frac{ E}{L}-c_{1}\frac{\partial\pi_{1}(\omega)}{\partial\omega}S\omega-\frac{\pi_{2}( \omega)}{RC}-\frac{\partial\pi_{2}(\omega)}{\partial\omega}S\omega}{\frac{\pi _{1}(\omega)}{C}-c_{1}\frac{\pi_{2}(\omega)}{L}}\right)+\frac{E}{L}-\frac{ \partial\pi_{1}(\omega)}{\partial\omega}S\omega. \tag{32}\]  
By substituting Equations (13) and (14) into Equation (32), after simple manipulations, one obtains that  
\[f_{0}(0,\omega)=-\frac{\pi_{2}(\omega)}{L}c(\omega)+\frac{\pi_{2}(\omega)}{L} c(\omega)=0. \tag{33}\]  
\(\square\)  
**Lemma 2. **_The sliding mode dynamics equilibrium point \(z_{1}=0\) (30) is locally attractive if \(c_{1}\) is selected in a way that_  
\[\overline{a}_{11}-\overline{a}_{12}c_{1}=\lambda_{1}<0, \tag{34}\]  
_where_  
\[\overline{a}_{11}=a_{11}-\frac{b_{1}(a_{21}+c_{1}a_{11})}{(b_{2}+c_{1}b_{1})},\ \ \overline{a}_{12}=a_{12}-\frac{b_{1}(a_{22}+c_{1}a_{12})}{(b_{2}+c_{1}b_{1})}, \tag{35}\]  
_with_  
\[a_{1,j}=\left.\frac{\partial f_{i}}{\partial x_{j}}\right|_{(b^{2}/(RE),b)}, \ \ b_{i}=\left.\frac{\partial f_{i}}{\partial v}\right|_{(b^{2}/(RE),b)}, \tag{36}\]  
_where \(f_{1}\) and \(f_{2}\) are the right-hand sides of Equations (8)-(9), respectively._  
**Proof. ** The corresponding linear approximation of Equations (13) and (14) is as follows:  
\[\Pi_{1}S\omega=R_{1}\omega+b_{1}\Gamma\omega, \tag{37}\]\[v_{eq}(z,\omega)=-\frac{\eta(z,\omega)}{\delta(z,\omega)}. \tag{29}\]0  
with  
\[v_{eq}(z,\omega)=-\frac{\eta(z,\omega)}{\delta(z,\omega)}. \tag{29}\]1  
\[v_{eq}(z,\omega)=-\frac{\eta(z,\omega)}{\delta(z,\omega)}. \tag{29}\]2  
where  
\[v_{eq}(z,\omega)=-\frac{\eta(z,\omega)}{\delta(z,\omega)}. \tag{29}\]3  
Let us define the steady-state errors as \(z_{1}=x_{1}-\Pi_{1}w\) and \(z_{2}=x_{2}-\Pi_{2}w\), where the corresponding dynamic equations are as follows:  
\[v_{eq}(z,\omega)=-\frac{\eta(z,\omega)}{\delta(z,\omega)}. \tag{29}\]4  
\[v_{eq}(z,\omega)=-\frac{\eta(z,\omega)}{\delta(z,\omega)}. \tag{29}\]5  
The sliding function dynamics (the function defined in Equations (20)-(22)) take the following form:  
\[ \begin{array}{c}\dot{\sigma}=a_{21}z_{1}+a_{22}z_{2}+b_{2}v+R_{2}\omega-\Pi_ {2}S\omega,\\ +c_{1}a_{11}z_{1}+c_{1}a_{12}z_{2}+c_{1}b_{1}v+c_{1}R_{1}\omega-c_{1}\Pi_{1}S \omega+\phi_{\sigma}(\cdot),\end{array}  \tag{44}\]  
with \(\phi_{\sigma}(z,w)=\phi_{2}(z,w)+c_{1}\phi_{1}(z,w)\). The equivalent control is calculated as a solution of \(\dot{\sigma}=0\):  
\[