## 3 Stability Analysis  
### Floquet Multipliers  
As shown in Figure 2, the ideal operation of the sZETA converter consists of a periodic orbit with period \(T\), which commutes the position of the switch between 1 and 0 and assures that the orbit oscillates around the desired operation point. We now turn our attention to describe the stability of this periodic orbit, by making use of Floquet theory. Let us assume a periodic solution \(\mathbf{x}_{p}(t)=\mathbf{x}_{p}(t+T)\). According to Floquet theory, the stability of an infinitesimally perturbation around the reference orbit \(\delta\mathbf{x}_{p}\) evolves from one period to the other as:  
\[\delta\mathbf{x}_{p}(t+T)=M\delta\mathbf{x}_{p}(t). \tag{8}\]  
Here, \(M\) is the monodromy matrix and is equal to the state transition matrix \(\Phi(\mathbf{x}(t_{0}),t_{0},t)\) evaluated at \(t=t_{0}+T\), namely:  
\[M=\Phi(\mathbf{x}(t_{0}),t_{0},t_{0}+T) \tag{9}\]  
The eigenvalues of the monodromy matrix \(\mu_{i}\) are the so-called Floquet multipliers and indicate the stability of the periodic orbit as follows:  
* If \(|\mu_{i}|<1\) for all \(i\), then the orbit is stable. * If at least one \(|\mu_{i}|>1\), the orbit is unstable. * The definition of the monodromy matrix in Equation (9) is valid for smooth systems. However, consider that the sZETA converter switches its dynamics from topology \(u=1\) to topology \(u=0\) whenever condition (7) is met, which requires special treatment for \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{3}{c}{**Parameter Values**} \\ \hline \(V_{in}=10\) V & \(V_{ref}=15\) V \\ \(C_{1}=100\)\(\mu\)F & \(C_{1}=220\)\(\mu\)F \\ \(L_{1}=100\)\(\mu\)H & \(L_{2}=55\)\(\mu\)H \\ \(R=100\)\(\Omega\) & \(T=50\)\(\mu\)s \\ \(k_{V}=1\) & \(k_{int}=500\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values used throughout the paper. Figure 2: (**A**) Dynamics of the ramp compensation \(I_{ramp}\) and current \(i_{1}\) showing the transitions of the switch; (**B**) Position of the switch as a function time; (**C**) Evolution of the voltages \(v_{1}\) and \(v_{2}\) with the reference voltage in dashed lines. Inset: Steady state evolution of the voltages across the capacitors showing a good agreement of \(v_{2}\) with \(V_{ref}\). To solve this system, a fourth-order Runge–[PERSON] method with variable step method was used with an event detection algorithm. the calculation of the monodromy matrix. Let us call \(\mathbf{x}^{*}\) the state of the system where \(h_{r}(\mathbf{x}^{*},t^{*})=0\), and the corresponding switching time is \(t^{*}\). It is easy to see that \(t^{*}\) can be expressed as \(t^{*}=t_{0}+dT\) if we set \(t_{0}\) at the beginning of the ramp cycle. Without loss of generality, we can make \(t_{0}=0\), therefore \(\mathbf{x}^{*}=\mathbf{x}(dT)\). Then, for a non-smooth system that switches its dynamics from a vector field \(\mathbf{f}_{ON}\) to \(\mathbf{f}_{OFF}\), the monodromy matrix can be constructed as follows (see [31] for a detailed explanation):  
\[M=\Phi_{OFF}(\mathbf{x}(dT),dT,T)\cdot\mathcal{S}\cdot\Phi_{ON}(\mathbf{x}(0),0,dT), \tag{10}\]  
where \(\Phi_{ON}\) (\(\Phi_{OFF}\)) are the state transition matrices associated with \(\mathbf{f}_{ON}\) (\(\mathbf{f}_{OFF}\)). Thanks to the linear nature of each individual vector field, the state transition matrices can be straightforwardly calculated as:  
\[\Phi_{ON} = e^{A_{ON}Td} \tag{11}\] \[\Phi_{OFF} = e^{A_{OFF}T(1-d)} \tag{12}\]  
with  
\[A_{ON} = \frac{\partial\mathbf{f}_{ON}}{\partial\mathbf{x}}= \begin{bmatrix} 0&0&0&0\\ 0&0&1/L_{2}&-1/L_{2}&0\\ 0&-1/C_{1}&0&0&0\\ 0&1/C_{2}&0&-1/(RC_{2})&0\\ 0&0&0&-1&0\end{bmatrix}  \tag{13}\] \[A_{OFF} = \frac{\partial\mathbf{f}_{OFF}}{\partial\mathbf{x}}= \begin{bmatrix} 0&0&-1/L_{1}&0&0\\ 0&0&0&-1/L_{2}&0\\ 1/C_{1}&0&0&0&0\\ 0&1/C_{2}&0&-1/(RC_{2})&0\\ 0&0&0&-1&0\end{bmatrix}