## 5 Discussion and Conclusions  
In this study, we theoretically analyzed the stability of the sZETA converter controlled via ramp compensation using non-smooth systems tools. We found that the proposed control technique is stable over a large parameter range including operation in step-up and step-down regimes. We demonstrated that the system loses the stability of the Period 1 orbit at extremely low values of the duty cycle, which are in general unrealistic in most applications. Nevertheless, we found that the sZETA converter also loses stability for high--but still realistic--values of the gain (i.e., \(d>0.8\)) where a period doubling occurs and eventually the system transits to chaos by means of a skipped-cycle mechanism. In this sense, it is known that the sZETA converter is unsuited for high gains applications due to decreased efficiency of the converter [52]. To deal with this, some improvements have been proposed for the sZETA converter to deal with extreme conditions [52; 53] using switched capacitors, but this requires the addition of several non-linear elements to the circuit, and the cheapest solution is still to choose a more suited topology for the application in hand (for a complete review on new step-up topologies, the reader is referred to [54]). It should be noticed that we used two different (although very much related) tools in the analysis of our system, namely Floquet multipliers and Lyapunov exponents. The former characterizes the stability of a particular target periodic orbit, which is often the requirement in the design of power converters, and the latter describes the nature of the attractor in which the system evolves (fixed point, periodic, quasi-periodic, or chaotic). Unfortunately, it is uncommon in the literature to see the evolution of these two indicators, and some authors prefer one tool over the other. We showed that, using both indicators, one can have a much deeper understanding of the system and the way in which a certain dynamics loses stability. Both Floquet multipliers and Lyapunov exponents, however, are based on linear stability, therefore they only characterize the attractors in a neighborhood. This means that we can only guarantee local stability of the solutions, and more refined formalisms are required to capture the global convergence, especially in switched systems [55; 56]. Despite this drawback, the disturbance rejection analysis showed that indeed the stable solutions attract a large perturbation. This feature is particularly important in applications where disturbances are important factors to consider, for instance in renewable energies applications, where the unsteady nature of the renewable sources may produce variations in the input power. As a matter of fact, the ZETA converter has been proven to be suitable on small scale (<350 W) and medium scale (<10,000 W) applications [57], where the converter supports a high efficiency over 90% [58]. Such is the case of the single-switch ZETA converter [59], using basic cells to step up the input voltage and get high gain and efficiency. To the authors' knowledge, this is the first time that a complete dynamical description using non-linear systems tools was performed for the ZETA converter, with the only exception being the work of [PERSON] and [PERSON] [60], which was made in a different configuration of the converter. Despite this, we propose as a future direction of research applying our non-smooth methodology to the analysis of other previously reported controllers for the ZETA converter which used the SSA approximation, with the aim of revealing possible nonlinearities that are not explained by the averaging technique. This also opens a wide range of possibilities that allow us to apply non-smooth systems tools and analyses other types of controllers that have been shown to perform adequately in power converters [34; 35; 61], including MPPT controller for photovoltaic applications [62], as well as the design of slope compensation to avoid overcompensation on light loads [43].