## 4 Results
### Disturbance Rejection  
We now proceed to evaluate the performance of the ramp controlled sZETA converter under different types of perturbations. First, we focus on disturbances at the load \(R\). This simulation is depicted in Figure 7. In this figure, \(R\) is set to the nominal value of 100 \(\Omega\), where one can see that the converter rapidly converges to the reference voltage in around 8 ms, starting from \(\mathbf{x}(0)=\mathbf{0}\) (see Figure 7A), while the error rapidly decreases below 1% (Figure 7B). At \(t=15\) ms, the load is abruptly changed to \(R=200\)\(\Omega\), where it again rapidly responds to the perturbation by transiently changing the duty cycle, until it relaxes again to the expected value \(d=(V_{ref}/V_{in})/(1+V_{ref}/V_{in})\), which for the chosen values of the reference and input voltage corresponds to \(d=0.6\) (see Figure 7C). Finally, another perturbation is presented to the system at \(t=30\) ms, where \(R\) is again changed to 50 \(\Omega\). It can be seen here that the controller handles the perturbation in a satisfactory manner. In fact, the last two disturbances are practically negligible. This means that, once the system reaches the steady state, disturbances in the load are unimportant for the system's performance. Figure 6: (**A**) Bifurcation diagram of the duty cycle; (**B**) maximal Lyapunov exponent near the disappearance of the Period 1 orbit in the interval \(V_{ref}\in[42\;V,50\;V]\). Inset: Bifurcation diagram of the error. For each point in this diagram, a transient evolution of 2000 cycles is allowed before calculating the tangent evolution during 3000 cycles. Renormalization of the tangent vector is performed every 10 cycles. Similarly, we performed a perturbation analysis in a more interesting set-up, i.e., by introducing disturbances in \(V_{ref}\) in such a way that the converter switches its behavior from step-up to step-down regime. This situation is depicted in Figure 8. From the time trace of the output voltage, we can see that the system reaches the three desired values of the reference despite the rather large deviations from one value to the other. Notice that, in the second transition, \(V_{ref}\) changes from \(V_{ref,2}=5\) V to \(V_{ref,3}=20\) V, which corresponds to a percent change of 300%. Even in this extreme case, the system reaches steady operation in approximately 5 ms, as seen by the small error values in Figure 8B a few milliseconds after the presentation of the disturbance. As expected, the transitions to different reference voltages are accompanied by variations in the steady state value of the duty cycle, which meet the expected value of the gain \(\mathcal{G}\) (Figure 8C). Figure 8: Time trace of the ramp controlled sZETA converter under perturbations of the reference voltage: (**A**) voltage trace; (**B**) error percentage; (**C**) duty cycle. In all panels, \(V_{ref,1}=12\) V for \(t<15\) ms, \(V_{ref,2}=5\) V for \(15\) ms \(\leq t<30\) ms, and \(V_{ref,3}=20\) V for \(t\geq 30\) ms  
Figure 7: Time trace of the ramp controlled sZETA converter under perturbations of the load: (**A**) voltage trace; (**B**) error percentage; and (**C**) duty cycle. In all panels, \(R=100\)\(\Omega\) for \(t<15\) ms, \(R=200\)\(\Omega\) for \(15\) ms \(\leq t<30\) ms, and \(R=50\)\(\Omega\) for \(t\geq 30\) ms.  
We performed the same perturbation analysis with \(V_{in}\) as a perturbation parameter. Following the same procedure as for \(V_{ref}\), \(V_{in}\) was varied in such a way that the sZETA converter alternates between step-up and step-down behavior. In particular, it transits from step-up to step-down at \(t=15\) ms and returns to a step-up regime at \(t=30\) ms, as shown in Figure 9. As reported with the disturbances in \(V_{ref}\), the converter rapidly reaches steady evolution in all the considered cases, despite the rather large variations of \(V_{in}\).