## 1 _Article_  
Nonlinear Dynamics and Stability Analysis of a Three-Cell  
Flying Capacitor DC-DC Converter  
[PERSON] 1,*,+\({}^{\text{(c)}}\), [PERSON] 2,+ [PERSON] 3,+ and [PERSON] 4,+\({}^{\text{(d)}}\)  
In power electronic converters, these kinds of nonlinear behaviors should be avoided and suppressed and therefore a deep understanding of the mechanism of their occurrence is necessary [22]. To analyze these behaviors, an appropriate and accurate model is required. The conventional approach used for modeling of this kind of switching systems is based on the averaging technique [1; 23]. It is true that the average model reduces system complexity and can predict the low frequency behavior, but it was demonstrated that it fails to predict the nonlinear behavior of the system that could taking place at the frequency ranges close to one half switching frequency. This behavior is known in the literature as period doubling bifurcation or fast-scale instability in the nonlinear dynamics field and subharmonic oscillation in the power electronics field. With the averaging procedure, it is assumed that the steady-state behavior of the system is an equilibrium point while in reality it is a limit cycle whose stability analysis is mathematically more involved. To predict the actual behavior of the system, the exact switched model or the related discrete-time model must be used. There are many works dealing with the nonlinear dynamics of switching converters most of them use a discrete-time model. While obtaining such a model for elementary switching converters is a simple task, it is not the case for advanced converter topologies such as multi-level flying capacitor converters mainly because the number of circuit configurations among which the circuit toggles is high [4]. To perform the stability analysis of the desired limit cycles of the switched model, Floquet theory can be applied without passing through the discrete-time model [24; 25]. Therefore, it offers an efficient and accurate tool to perform stability analysis leading to the same results as the discrete-time modeling approach but more straightforwardly. It also separately reveals the effects of each switching on the system stability [25]. Floquet approach is based on the theory originally developed by [PERSON] [26] to study the stability of periodic orbits in dynamical systems with time varying periodic coefficients. These approach has been extended by [PERSON] [27], and later by [PERSON] [28] to analyze impacting motion and stick-slip oscillations in mechanical switching systems. A good tutorial on how to use this theory is in [24]. On the other hand, flying capacitor multi-level multi-cell converters have been proposed for higher power industrial applications. These converters are based on adding and subtracting voltage of storage elements in the form of capacitors creating multi-level voltages [2]. A special characteristic of flying capacitor converters are their commutation cells that allow to operate with high supply voltages making them good candidate for high voltage applications [2]. They can be also used to reduce the stress in the switching devices and the state variables ripples by interleaving operation. Their control objective is two-fold. First, the controller must ensure a regulated output voltage/current. Second, the control circuit must ensure a voltage balancing between the different cells of the converter. This is accomplished just as in conventional switching converters by applying appropriate signals to control the time durations of the on/off state of the switches. This is accomplished by using suitable error signals that represents the difference between the reference and the actual value of the state variable to be controlled. Selecting the right value of the feedback gain of these controller is crucial for ensuring stability of the system and suppressing underscored nonlinear behavior. The feedback control may be designed in the continuous time or in the discrete time domain.