## 2 Application Example  
This paper focuses on the application example of a boost converter, although the idea can be adapted to other topologies that face non-minimum phase issues. The proposed method is necessary for boost converters because conventional FCS-MPC strategies struggle to manage the inherent non-minimum phase dynamics, which cause initial voltage undershoots when increasing the output voltage. This topology is also useful for understanding the non-minimum phase phenomena in power converters and FCS-MPC due to its simplicity. In a boost converter, the main objective is to have a higher output voltage than in the input; this means that there is an increment of the DC voltage. Between the elements composing this circuit, there is a DC input voltage source (_Vin_), an inductor to store energy, a power electronic switch (represented by the variable \(Q\)), a diode (\(D\)) to transfer the energy stored in the inductor to the load, and a capacitor to smooth the output voltage. While different types of power transistors, such as MOSFETs or IGBTs, can be used as the switch, MOSFETs are more suitable for the mid-high switching frequencies used in the experimental results. IGBTs, on the other hand, may limit the maximum attainable frequency. For this specific design, a MOSFET is used and shown in the figures and schematics. When the converter works in Continuous Conduction Mode (CCM), there are two equivalent sub-circuits, shown in Figure 1: when the MOSFET is turned _on_ and when it is turned _off_. When the MOSFET is _on_ (Figure 1a), it creates a low-impedance path for the current, allowing the inductor to store energy from the input voltage. During this state, the diode is reverse-biased, preventing current from flowing to the output. When the MOSFET is turned _off_ (Figure 1b), the inductor's stored energy is released. The inductor generates a high voltage, which forward-biases the diode, allowing current to flow to the output and simultaneously charging the output capacitor. This process boosts the output voltage to a level higher than the input voltage, hence the name \"boost converter\". The rapid switching between these two states ensures a continuous transfer of energy, maintaining a higher steady-state output voltage. In Discontinuous Conduction Mode (DCM), illustrated in Figure 1c, the inductor current falls to zero during the _off_ state of the MOSFET. This introduces a third sub-circuit where the diode is _off_, and both the inductor and capacitor are disconnected from the input voltage. During this state, the inductor does not transfer energy, and the output voltage is maintained solely by the charge stored in the capacitor. This mode typically occurs under light load conditions when the energy demand is low, resulting in periods where no current flows through the inductor. The discrete differential equations of the boost converter, as represented by Equations (1) and (2), are essential for accurately modeling the system's dynamics and are crucial for the prediction block in FCS-MPC. These equations enable the controller to predict the future inductor current and capacitor voltage, allowing for precise switching decisions. \[v_{\text{C}}(k+1)\ =\ v_{\text{C}}(k)\ +\ \frac{\Delta t}{\text{C}}\cdot i_{ \text{C}}(k) \tag{1}\]  
\[i_{L}(k+1)\ =\ i_{L}(k)\ +\ \frac{\Delta t}{L}\cdot v_{L}(k) \tag{2}\]  
For the sake of clarity, \(i_{L}\)(\(k\)) and \(v_{\text{C}}\)(\(k\)) represent the current values of the inductor current and output voltage, respectively, while \(i_{L}(k+1)\) and \(v_{\text{C}}(k+1)\) denote their future values. These future values will be the predicted ones in the next section. There are two primary states, as shown in Table 1, representing the possible decisions that the controller can make: switching the MOSFET _on_ (situation I) or _off_ (situation II). Table 1 also provides the inductor voltage (\(v_{L}\)) and the capacitor current (\(i_{\text{C}}\)) for these two states, which are needed for the prediction block Equations (1) and (2).