## 4 Results
### Stability Regions  
Now that we have chosen a suitable value of \(A\), we aim to characterize the stability regions in the ramp compensation controlled sZETA converter at variations of different parameters and to characterize the mechanisms by which the stability is lost. To do this, we calculate the stability of the Period 1 solution taking \(R\), \(V_{in}\), and \(V_{ref}\) as bifurcation parameters (recall that the nominal parameters are \(R=100\) O, \(V_{ref}=15\) V, and \(V_{in}=10\) V). The results are depicted in Figure 4. It is possible to see in Figure 4A that the load has little effect on the stability of the controller, as testified by \(|\mu|_{max}<1\) in the whole range of resistances \(R\in[1\ \Omega,\ 1000\ \Omega]\) that we considered. It is also not surprising that the value of the duty cycle remains unchanged over the whole range and very close to the expected gain \(\mathcal{G}\) defined in Equation (4), as the gain only depends on the values \(V_{in}\) and \(V_{ref}\). Next, we focus on the evolution of \(|\mu|_{max}\) as a function of \(V_{ref}\) in Figure 4B. Here, we can observe that the orbit is stable in an ample range of \(V_{ref}\), and it only loses stability at either very small references (\(V_{ref}<1\) V) or very large ones (\(V_{ref}>42.9\) V). This of course corresponds to very low or very high values of the duty cycle where power converters do not perform adequately. As above, the duty cycle solution obtained by solving (4) is in good agreement with the expected gain. Finally, we found a similar behavior choosing \(V_{in}\) as the bifurcation parameter (see Figure 4C). Namely, the system loses stability only at very large values of \(V_{in}\) corresponding to extreme values of the duty cycle. This suggests that the peak-current with ramp compensation is a quite stable strategy, except for cases in which the gain is either very small of very large. Figure 3: Norm of the maximal Floquet multiplier as a function of the input voltage \(V_{in}\) and the ramp slope \(A\) at a fixed value of \(V_{ref}\), indicated on the title of each panel. **(A)**\(V_{ref}=5\) V; **(B)**\(V_{ref}=15\) V; **(C)**\(V_{ref}=25\) V; **(D)**\(V_{ref}=35\) V. In all panels, the magenta dashed line denotes the selected value of \(A=10\) that guarantees stable operation for most of the parameters selected. To test this hypothesis, we proceeded to calculate \(|\mu|_{max}\) in the two-dimensional parameter space \((V_{ref},V_{in})\) and the value of the duty cycle consistent with the Period 1 solution in Equation (18). The results are shown in Figure 5, where it can be seen that, indeed, the regions where the Period 1 orbit loses stability are those when the ratio \(V_{in}/V_{ref}\) is either too large or too small (bright red regions in Figure 5A). These are indeed parameter regions where the duty cycle has very large or very low values (see Figure 5B), hence supporting the initial hypothesis. A rather interesting exception appears for cases in which \(V_{in}\approx V_{ref}<5\) V. This is not surprising, because, as shown in Figure 3A, for the chosen value of the ramp slope \(A\), there is a region of small and comparable values of \(V_{ref}\) and \(V_{in}\) where the controller is unstable. This result indicates that the proposed controller handles much better medium to high values of voltages, at least for the parameters chosen here. Figure 4: Norm of the maximal Floquet multiplier as a function of different bifurcation parameters: (**A**) \(R_{i}\) (**B**) \(V_{ref}\); (**C**) \(V_{in}\). In all panels, the inset shows the solution of the duty cycle which solves the self-consistent Equation (18) (symbols) and the corresponding steady state expected solution (4) (red). Figure 5: (**A**) Heat map of the maximal Floquet multiplier in the parameter space \((V_{ref},V_{in})\); (**B**) Heat map of the duty cycle solution (18) as in (**A**).