*Appendix A.2. Proof of Theorem 1*

**Proof.** From Theorem 8 in [38], consider the common Lyapunov candidate function *V* = *Cv*<sup>2</sup> *<sup>L</sup>* <sup>+</sup> *<sup>i</sup>* 2 2 ; hence, *<sup>V</sup>*(0) = <sup>0</sup> and *<sup>V</sup>* ∈ C∞. For *<sup>γ</sup>* <sup>=</sup> <sup>1</sup> and replacing Equation (25) on Equations (14) and (15), the time derivative along the system's trajectory is sought to comply with:

$$\begin{aligned} \dot{V} &= -\frac{2CPv^3 - \mathsf{C}\Delta Eiv^3 - \Delta Ek\_3i^2v + Pk\_4iv^3 - \Delta Ek\_4i^2v^3}{Lv(ik\_3 + \mathsf{C}v^2 + ik\_4v^2 + Pk\_3iv)} \\ &- \frac{4LPfk\_3i^2 - \mathsf{C}Pk\_1iv^3 + \mathsf{C}Pk\_2iv^4 + 4LPfk\_4i^2v^2}{Lv(ik\_3 + \mathsf{C}v^2 + ik\_4v^2)} \end{aligned} \tag{A3}$$

$$-\frac{4CLPf\dot{v}^2}{Lv(ik\_3 + Cv^2 + ik\_4v^2)} < 0\tag{A4}$$

where *E* ≤ ∆*E* = (1 + 2*u*<sup>1</sup> − 2*u* 2 1 )*<sup>E</sup>* <sup>≤</sup> <sup>3</sup>*<sup>E</sup>* 2 . Since *v*, *i* > 0 the denominator is greater than zero, and using *k*1, *k*<sup>4</sup> from (26) and (27) one can look equivalently for:

$$-\quad \text{CP}k\_2iv^4 + \frac{9Ek\_3}{16LPf}i^2v^3 - 2CPv^3 - 4\text{CLP}fiv^2 - k\_3Piv$$

$$-\quad 4\text{LPf}k\_3i^2 - \frac{3k\_3}{2}i^2v^2 + \frac{3k\_3E}{2}i^2v < 0. \tag{A5}$$

From the last two terms of the previous inequality, with *k*<sup>3</sup> > 0, and from expressions (26) and (27):

$$\begin{aligned} -\frac{3k\_3}{2}i^2v^2 + \frac{3k\_3E}{2}i^2v &< \ 0\\ \frac{3k\_3E}{2}i^2v &< \ \frac{3k\_3}{2}i^2v^2\\ \ E^2v &< \ \ i^2v^2\\ E &< \ v \end{aligned} \tag{A6}$$

which is true for *P* > 0; that is, in the Boost converter this condition is feasible under regular parameterization, and one looks for stability under this condition. Hence, inequality (A5) turns equivalently in:

$$-\mathbb{C}k\_2iv^4 + \frac{9\mathbb{E}k\_3}{16\mathbb{E}Pf}i^2v^3 - 2\mathbb{C}Pv^3 - 4\mathbb{C}LPfiv^2 - k\_3Piv - 4\mathbb{L}Pfk\_3i^2 < 0. \tag{A7}$$

Taking the *v* 3 terms, one can find that their sum is negative if:

$$\frac{9Ek\_3}{16LPf}i^2v^3 - 2\text{CP}v^3 \quad < \quad 0$$

$$i^2 \quad < \quad \frac{32\text{CL}fP^2}{9Ek\_3}.\tag{A8}$$

Note that *k*<sup>3</sup> > 0 can be arbitrarily small to allow any finite *i*, therefore inequality (A7) turns in:

$$-\mathcal{C}k\_2iv^4 - 4\mathcal{C}LPfiv^2 - k\_3Piv - 4LPfk\_3i^2 < 0\tag{A9}$$

which is always negative.

On the other hand, for *γ* = 0 and replacing Equation (25) on Equations (14) and (15), the time derivative along the system's trajectory is sought to comply with:

$$
\dot{V}\_{\perp} = -Ei - 2P + Pk\_1 i - Pk\_2 i v < 0. \tag{A10}
$$

Separating into two inequalities, one has:

$$\begin{array}{cccc} \text{Ei} - Pk\_2iv & < & 0, \\ -2P + Pk\_1i & < & 0. \end{array} \tag{A11}$$

For the first one:

$$\begin{aligned} \text{iEi} &< Pk\_2 \text{iE} < Pk\_2 \text{iv}\_{\prime} \\ &\frac{1}{P} < k\_{2\prime} \end{aligned} \tag{A12}$$

and for the second one:

$$\begin{array}{ccccc} \text{P}k\_1i & < & \text{2P}\_1 \\ & i & < & \frac{2}{k\_1} \end{array} \tag{A13}$$

Since *V* is a common Lyapunov function, the proof is complete.
