appliedsciences  
DC-DC Zeta Power Converter: Ramp Compensation Control Design and Stability Analysis [PERSON] 1,*c, [PERSON] 2 and [PERSON] 3  
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The common feature in most of the controllers analyzed so far in the literature for the ZETA converter is that they are based on the state-space averaging technique (SSA) (one remarkable exception can be found in [26]). In SSA, the nonlinear dynamics of the converter inherited from the switching behavior is replaced by an averaged linear model which is easy to analyze and allows the application of widely known linear-control theory techniques [27]. However, average linear models only avoid slow-scale phenomena [28], and the nonlinearity induced by the switch is known to produce fast-scale instabilities in power converters, such as quasi-periodicity and chaos [29], which decrease the efficiency and saturate the controllers. Therefore, to capture these non-trivial dynamics in switched systems, one needs to employ [PERSON]'s formalism to model the behavior of the ZETA converter [30]. Within this more realistic framework, the desired operation in a power converter is a periodic orbit swirling nearby the operation point as the switch turns ON and OFF according to a control law. The problem is then to asses the stability of such solutions in terms of suitable dynamical indicators such as Floquet multipliers and Lyapunov exponents. On the one hand, Floquet multipliers are used to describe the stability of periodic orbits and are therefore useful in the analysis of power converters [31]. With the aim of calculating the Floquet multipliers, it is necessary to compute the fundamental matrix associated to an equivalent discrete map [32]. The analytical procedure implies the use of a saltation matrix [33; 34; 35], in order to infer some implicit time derivatives related to the switching times. This tool has been used in several power converters such as boost converter [36], buck converter [32; 33; 37], boost-flyback [34; 35], Cuk [38], and buck-boost [39], among others. On the other hand, Lyapunov exponents are used to characterize the stability of generic solutions (not only periodic), and they are especially useful to characterize chaotic behavior. Lyapunov exponents have also been used to describe the dynamics of power converters (see, e.g., [40; 41; 42]), but not for the ZETA converter. For this reason, Lyapunov exponents and Floquet multipliers are complementary indicators. While Lyapunov exponents are unable to capture the transitions from a periodic dynamics to another (for example, period doubling transitions), Floquet multipliers capture precisely these changes in the periodic solutions. Meanwhile, Floquet multipliers can only capture the stability of periodic orbits and do not provide any information of other types of dynamics such as quasi-periodic motion and chaotic behavior. With these ideas in mind, in this paper, we propose to control and theoretically analyze the behavior of the synchronous ZETA (sZETA) converter. The control includes a proportional-integral control action (a PI controller) to avoid steady state error, handle the transient dynamics, and add robustness to the system [18; 19; 20; 21]. Hence, the values of this controller are tuned to satisfy these two requirements. Next, a sawtooth signal of period \(T\) is subtracted from the PI output and compared with the current flowing through one of the coils of the converter. This is a strategy that has been widely used in power converters and is known as peak-current control with ramp compensation [2]. Two remarkable features of this control are: (i) its relatively easy implementation; and (ii) a fixed switching frequency.