## 2 Sliding Mode Control Design
### Fourth Step: Validation of Existence Condition  
This condition ensures the existence of a sliding mode around \(S=0\), such that SBBC must remain within the sliding surface when \(S\to 0\) (see (15)), guaranteeing not only \(S=0\) but also \(x_{1}=x_{2}=0\). Under these conditions, the system remains controlled around the equilibrium point. \[\lim_{S=0^{+}}\frac{dS}{dt}|_{u=1}<0\quad\text{and}\quad\lim_{S=0^{-}}\frac{dS }{dt}|_{u=0}>0 \tag{15}\]  
Equation (16) presents the first existence condition when the system operates over the sliding surface and the control signal (\(u=1\)) is applied for following the dynamic trajectories towards \(S=0\). \[\lim_{S=0^{+}}\frac{dS}{dt}|_{u=1}=\bigg{[}\frac{\alpha_{1}}{\alpha_{2}}\frac {1}{RC}+\gamma_{2}-\frac{\gamma_{1}}{RC}\bigg{]}(x_{1}+1)-\frac{\alpha_{3}}{ \alpha_{2}}x_{2}-\frac{v_{s}}{L}<0 \tag{16}\]  
Figure 2: Graphical representation of transversality condition. Figure 3a is the representation of (16), while Figure 3b is the zoom in at the extremes (\(wt=0\) and \(wt=\pi\)) which correspond to zero crossings. Note that \(dS/dt\) is lower than zero for almost the entire operating range, while \(dS/dt\) is greater than zero around zero crossings. Thus, for the entire operating range, regions near zero crossings correspond to unstable points and the SBBC cannot be controlled. In a strict sense of sliding mode theory, this condition is not fulfilled and sliding mode control should not be implemented. However, in practical terms, the instability of the system is produced in a short period of time, and basically this is the reason AC current deforms at zero crossing. Zero crossing deformation is not exclusive to the theory of sliding modes; this phenomenon also occurs when other control techniques are implemented [9; 11; 14]. In general terms, sliding mode theory allows explaining the phenomenon of zero crossing deformation. To mitigate the deformation at zero crossings, an adaptive hysteresis band based on current geometry is proposed, as described in Section 3. Equation (17) provides the second condition for \(\alpha_{1}/\alpha_{2}\) which was obtained by evaluating (16) in system boundaries and in steady state. This condition is more restrictive than the transversality one. \[\frac{\alpha_{1}}{\alpha_{2}}<\frac{2 wCV_{ref}^{2}}{V_{s}} \tag{17}\]  
Equation (18) presents the second existence condition when the system operates under the sliding surface, and the open control signal (\(u=0\)) is applied to follow the dynamic trajectories towards \(S=0\). Figure 4 shows that \(dS/dt\) is greater than zero for the entire operating range. This condition is obtained considering (17). \[ \begin{split}\lim_{S=0^{-}}\frac{dS}{dt}|_{u=0}&= \left(\frac{\alpha_{1}}{\alpha_{2}}-\gamma_{1}\right)\left[\frac{(x_{1}+1)}{ RC}-\frac{(x_{2}+i_{ref})}{CV_{ref}}\right]+\\ &\left(\frac{V_{ref}}{L}+\gamma_{2}\right)(x_{1}+1)-\frac{V_{s} }{L}-\frac{\alpha_{3}}{\alpha_{2}}x_{2}>0\end{split}  \tag{18}\]  
Another condition that is usually verified in SMC is the hitting condition. This one is related to the existence condition and can be satisfied if the switching function is appropriately chosen [33; 34]. Equation (19) corresponds to the switching function proposed in this paper. Satisfying this condition assures that, regardless of the initial condition, dynamic trajectories of the system are always towards the sliding surface. The suitable selection of the logic states of the switching function and its application in the SM control law allow changing the dynamic behavior of the system. As a consequence, for the design of the hitting condition, it is sufficient to consider state variables \(x_{1}\) and \(x_{2}\), during the reaching phase. If the measured magnitude of \(i_{s}\) is less than \(i_{ref}\) and \(v_{o}\) is less than \(v_{ref}\), then \(S\) is positive. In this condition, the necessary switching action required to force the  
Figure 3: \(dS/dt\) for existence condition when \(u=1\). (**a**) Representation in the range of \(wt=[0,\pi]\); (**b**) Zoom in at \(wt=0\) and \(wt=\pi\).