## 1 _Article_
Discrete-time controllers have been proposed recently as substitutes of continuous-time ones due to ease in programming, insensibility to noise and the possibility of implementing complex algorithms. The flying capacitor multi-cell mult-level converters have attracted a lot of attention from the power electronics and control communities [29; 30; 31; 32; 33; 34; 35; 36; 37; 38] since their appearance in [2; 3]. The reasons behind the interest for these topologies are modularity, applicability to high voltage fields and ease of balancing the flying capacitor voltage and possibility to use small reactive components [36] which could reduce the size and increase the power density. Several studies have been conducted to solve the flying capacitor balancing issues [35]. A sensorless stabilization technique is presented in [37] where a discrete-time digital platform based on FPGA was used. The use of discrete-time control techniques in the field of power electronics converters is more and more increasing. A discrete-time digital predictive control for a three-level flying capacitor buck converter with voltage balancing is presented in [38]. Discrete-time PWM control has been applied to a two-cell DC-DC converters [29; 30; 31; 32; 39]. This procedure can be adapted to any multi-cell converter with a generic number of cells [40]. Floquet theory has been successfully applied in the past to perform stability analysis of simple power electronics converters such as the conventional buck converter [25], cascaded converters [33] and interleaved converter [41] among others. The application of this approach to multi-level flying capacitor converters has not been reported for the best knowledge of the authors. It is the aim of this paper to apply this theory to predict the nonlinear dynamics and to perform stability analysis of a three-cell flying capacitor DC-DC buck converter under discrete-time controller. First, the system modeling is addressed and the nonlinear switched model is derived. Second, the different operating modes are determined according the value of the operating steady-state duty cycle. Then, a discrete time controller is proposed which will be in charge of regulating the output current and balancing the cell voltages. Subsequently, the limit cycles of the converter will be computed and the result will be compared with numerical simulations using the exact switched model. Floquet theory is applied which leading to an accurate prediction of the instability phenomena exhibited by the system. However with the exact expression of the monodromy matrix, the computational efforts are high. Therefore, some realistic approximations are used to obtain the limit cycles and the monodromy matrix analytically predicting the onset of instability in the three-cell DC-DC buck converter with a much lower computational effort. The rest of the paper is organized as follows: in Section 2, a system description and the operating modes are presented. The mathematical modeling of the power stage and the discrete-time controller are also be presented in the same section. The switched model and the switching conditions are derived. The system nonlinear dynamics is explored in Section 3 using bifurcation diagrams showing that the system may exhibit period doubling bifurcation, subharmonics and chaotic regimes when suitable system parameters are varied. Floquet theory for stability of limit cycles is presented in Section 4 and adapted for the system under study. A procedure to compute limit cycles and to analyze their stability is presented. The monodromy matrix and Floquet multipliers explaining the observed period doubling bifurcation phenomena in Section 3. A simplified approach is proposed to derive a simplified expression for the monodromy matrix and Floquet multipliers in the same section where also a validation of the theoretical predictions is provided. Finally, in Section 5, the concluding remarks of this study are given.