## 2 System Description
### Discrete-Time Controller  
Power electronic converters are generally used to provide a desired voltage/current for the load. This can be accomplished through suitable feedback loops. For flying capacitor multi-cell converters, the controller task it to regulate the voltage/current of the load and also to balance the flying capacitor voltages to effectively reduce the switches voltage stresses. It can be demonstrated that if the duty cycles for all the cells is the same, a natural balancing phenomenon takes place without using the flying capacitor voltages in the feedback loop [34]. However, the transient response speed in this case depends on the circuit parameters and could not undesirably very slow. To avoid this problem, the voltages of the cell are also used in the feedback loop. The controller must be able to regulate the current \(i_{L}\) and balance the input voltage between the different cells. The expression of the duty cycles in terms of the state variables error can be fixed according to the control objectives. A purely proportional control will be used here with the aim to minimize the error between the current and voltages and their references values. As the system contains three controlled switches, three different duty cycles must be generated and their corresponding driving signals must be applied for the three switches. The error between the controlled outputs and their references are appropriately combined and used in a discrete time modulation strategy. In particular, the inductor current (which is also the load current) \(i_{L}\) is controlled to a desired current reference \(i_{\text{ref}}\) and the flying capacitor voltages \(v_{1}\) and \(v_{2}\) are controlled to \(2/3\) and \(1/3\) of the input voltage respectively. The \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Configurations** & **A Matrices** & **B Vectors** & **C\({}_{\mathbf{1}}\) Charge** & **C\({}_{\mathbf{2}}\) Charge** & **L Charge** \\ \hline \(\mathcal{C}_{101}\) & \(\mathbf{A}_{101}=\left(\begin{array}{ccc}-\dfrac{R}{L}&\dfrac{1}{L}-\dfrac{1}{L} \\ -\dfrac{1}{C_{1}}&0&0\\ \dfrac{1}{C_{2}}&0&0\end{array}\right)\) & \(\mathbf{B}_{101}=\left(\begin{array}{ccc}v_{g}\\ \dfrac{1}{L}\\ 0\end{array}\right)\) & discharged & charged & charged \\ \(\mathcal{C}_{100}\) & \(\mathbf{A}_{100}=\left(\begin{array}{ccc}-\dfrac{R}{L}&\dfrac{1}{L}\\ -\dfrac{1}{C_{1}}&0&0\\ 0&0&0\end{array}\right)\) & \(\mathbf{B}_{100}=\left(\begin{array}{ccc}0\\ 0\end{array}\right)\) & discharged & maintained & discharged \\ \(\mathcal{C}_{110}\) & \(\mathbf{A}_{110}=\left(\begin{array}{ccc}-\dfrac{R}{L}&0\dfrac{1}{L}\\ 0&0&0\\ 1&-\dfrac{1}{C_{2}}&0&0\end{array}\right)\) & \(\mathbf{B}_{110}=\left(\begin{array}{ccc}0\\ 0\end{array}\right)\) & maintained & discharged \\ \(\mathcal{C}_{010}\) & \(\mathbf{A}_{010}=\left(\begin{array}{ccc}-\dfrac{R}{L}&1&1\\ \dfrac{1}{C_{1}}&0&0\\ -\dfrac{1}{C_{2}}&0&0\end{array}\right)\) & \(\mathbf{B}_{010}=\left(\begin{array}{ccc}0\\ 0\end{array}\right)\) & charged & discharged & discharged \\ \(\mathcal{C}_{011}\) & \(\mathbf{A}_{011}=\left(\begin{array}{ccc}-\dfrac{R}{L}&1&1\\ \dfrac{1}{C_{1}}&0&0\\ 0&0&0\end{array}\right)\) & \(\mathbf{B}_{011}=\left(\begin{array}{ccc}\dfrac{v_{g}}{L}\\ 0\\ 0\end{array}\right)\) & charged & maintained & charged \\ \(\mathcal{C}_{001}\) & \(\mathbf{A}_{001}=\left(\begin{array}{ccc}-\dfrac{R}{L}&0-\dfrac{1}{L}\\ 0&0&0\\ \dfrac{1}{C_{2}}&0&0\end{array}\right)\) & \(\mathbf{B}_{001}=\left(\begin{array}{ccc}\dfrac{v_{g}}{L}\\ 0\\ 0\end{array}\right)\) & maintained & charged & discharged \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Matrices for the eight different configurations. charge and discharge of the capacitors depend on the flow direction of the current across it.