## 2 Higher-Order Sliding Mode Regulation for Boost Converter Output
### Load Resistor Estimation  
Here, we remove the assumption of a known load resistor as stated in Section 2.2, by designing a sliding mode observer. The estimate of the load resistor is proposed by means of a super-twisting sliding mode observer for the voltage equation in Equations (1)-(3).  
\[\dot{\bar{x}}_{2}=\frac{vx_{1}}{C}+l_{1}\sqrt{|x_{2}-\hat{x}_{2}|}\text{sign}( x_{2}-\hat{x}_{2})-\bar{\xi}_{1}. \tag{59}\]  
\[\dot{\bar{\xi}}_{1}=-l_{2}\text{sign}(x_{2}-\hat{x}_{2}). \tag{60}\]Then, by means of the equivalent control method, one can retrieve the load resistance information. For that end, the estimation error is defined as \(\tilde{x}_{2}=x_{2}-\hat{x}_{2}\) where its dynamics are as follows:  
\[\dot{\tilde{x}}_{2}=-\frac{x_{2}}{RC}-l_{1}\sqrt{|\tilde{x}_{2}|}\text{sign}( \tilde{x}_{2})+\xi_{1}, \tag{61}\]  
\[\dot{\xi}_{1}=-l_{2}\text{sign}(\tilde{x}_{2}), \tag{62}\]  
where \(l_{1}\) and \(l_{2}\) are constant design parameters, and \(\xi_{1}\) is an integral action. The observer is ensured to converge uniformly when the following bound is satisfied for some constants \(\mu>0\):  
\[\left|\frac{x_{2}}{RC}\right|\leq\mu. \tag{63}\]  
The corresponding sufficient condition for the finite time convergence to the sliding manifold \(\tilde{x}_{2}=0\) is  
\[l_{2}>\frac{\mu}{\Gamma_{2,m}}, \tag{64}\]  
where \(0<\Gamma_{2,m}\leq 1\leq\Gamma_{2,M}\), such that the estimation error \(\tilde{x}_{2}\) will tend to zero in finite time [33]. Using the equivalent method, i.e., setting \(\dot{\tilde{x}}\equiv 0\) in Equations (61) and (62), one can determine  
\[\tilde{\xi}_{1}=\frac{x_{2}}{RC}. \tag{65}\]  
If \(\xi_{1}\
eq 0\), the resistance estimate is  
\[\hat{R}=\frac{x_{2}}{\tilde{\xi}_{1}C}. \tag{66}\]  
If \(\tilde{x}_{2}\equiv 0\), it is true for \(t\geq T_{S}\), \(T_{S}>0\), a finite time instant; then, \(\hat{R}=R\) for \(t\geq T_{S}\). On the other hand, for \(t<T_{S}\), the difference \(\hat{R}-R\) remains bounded since the convergence of \(\tilde{x}_{2}\) to zero is in finite time. Lastly, the control action to be implemented is as follows:  
\[\hat{\sigma}=z_{2}+c_{1}\hat{z}_{1}, \tag{67}\]  
\[\hat{\sigma}=\left(-k_{1}\sqrt{|\hat{\sigma}|}\text{sign}(\hat{\sigma})+\hat{ \sigma}_{1}\right)/\delta(z,\omega), \tag{68}\]  
\[\dot{\hat{\sigma}}_{1}=-k_{2}\text{sign}(\hat{\sigma}), \tag{69}\]  
where \(\hat{z}_{1}=x_{1}-\hat{\sigma}_{1}(w)\), and \(\hat{\sigma}_{1}(w)\) is the updated version of \(\pi_{1}(w)\) in Equation (57) with the resistor estimate \(\hat{R}\).  
**Remark 3**.: _The resistance is estimated through a sliding mode observer. Nevertheless, the control law utilizes the inductance and capacitance. In contrast to the load (which can suffer variations during the operation), the inductance and capacitance are chosen during the design, and their variations during the operation are small (e.g., smaller than 5%) in a well-designed converter, mainly due to the quality (tolerance) of components and to the change in the current in the case of the inductance. Therefore, considering substantial variations of \(L\) and \(C\) is out of the scope of this work. This can be evaluated in future work. Nevertheless, it is worth mentioning that any mismatch between nominal values and real values for \(L\) and \(C\) will appear in a perturbation term that belongs to the control subspace, which the sliding mode controller will reject due to its robustness property. However, in particular, variations in \(C\) will slightly affect the steady-state solution for \(\pi_{1}\) as appreciated in coefficients \(a_{6}\) and \(a_{15}\), as can be noted in Equation (53)._