## 4 Theoretical Background for Stability Analysis of Limit Cycles
### Approximate Expression of the Monodromy Matrix and Stability Region in the Parametric Space
The approximate expressions for the Floquet multipliers in terms of system parameters are given by  
\[\mu_{1} \approx 1-\frac{Tv_{g}\kappa_{i}}{L}-\frac{RT}{L} \tag{41}\] \[\mu_{2} \approx 1-\frac{Tv_{g}i_{\text{ref}}i_{\text{ref}}i_{\text{ref}}\kappa_{i} \kappa_{1}}{C_{1}\big{(}v_{g}\kappa_{i}+R\big{)}}\] (42) \[\mu_{3} \approx 1-\frac{Tv_{g}i_{\text{ref}}i_{\text{ref}}\kappa_{i}\kappa_{2}}{C_ {2}\big{(}v_{g}\kappa_{i}+R\big{)}} \tag{43}\]  
The approximate closed-form expressions of the Floquet multipliers can be used to determine the stability region of the desired limit cycle in the parameter space. A sufficient condition for stability is that all the Floquet multipliers \(\mu_{i}\) lie inside the unitary circle. For practical design reasons, the feedback coefficients are restricted within \(\mathbb{R}^{+}\). These limit cycles will be stable whenever the following conditions are fulfilled for the feedback gains \(\kappa_{i}\), \(\kappa_{1}\) and \(\kappa_{2}\)  
\[0 < \kappa_{i}<\frac{2L-RT}{Tv_{g}} \tag{44}\] \[0 < \kappa_{1}<\frac{2C_{1}(R+\kappa_{i}v_{g})}{Ti_{\text{ref}}i_{ \text{ref}}i_{\text{ref}}\kappa_{i}v_{g}}\] (45) \[0 < \kappa_{2}<\frac{2C_{2}(R+\kappa_{i}v_{g})}{Ti_{\text{ref}}i_{ \text{ref}}i_{\text{ref}}\kappa_{i}v_{g}} \tag{46}\]  
The first constraint (44) when evaluated for the set of parameter values of Table 2 gives a critical value of the feedback gain \(\kappa_{i}\approx 0.058\) A\({}^{-1}\) which is in perfect agreement with the numerical simulations in Section 3. Expressions (44)-(46) can be used to get 2-dimensional diagrams that would help in setting the design limits for the parameters since as they show the safe range of the parameters that would lead to a stable system. The stability boundaries can be plotted in a suitable projection of the parametric space. The parameters can be selected in this space for obtaining a desired response for the system under step type change of a certain reference. For instance, based on a small-signal analysis, the feedback gains that leads to a deadbeat response [42], characterized by the fact that all Floquet multipliers are zero, are given by the following expression  
\[\kappa_{i} = \frac{L-RT}{Tv_{g}} \tag{47}\] \[\kappa_{1} = \frac{C_{1}(R+\kappa_{i}v_{g})}{Ti_{\text{ref}}i_{\text{ref}}i_{ \text{ref}}\kappa_{i}v_{g}}\] (48) Figure 9:0  
The intersection of the different curves representing (47)-(49) would lead to deadbeat response. Figure 10 shows the constraints (44), (47) and (48) in the \((\kappa_{i},\kappa_{1})\) parameter space. The stable and the unstable region as well as the set of parameter values leading to a deadbeat response are shown in the same figure. The system parameters are selected on the intersection point, and since the system is of order three, the system will settle within three switching periods after a disturbance takes place. The stable and the unstable region as well as the set of parameter values leading to a deadbeat response are shown in the same figure. The left panel of Figure 11 shows the Floquet multipliers locus when the parameters are selected at the deadbeat response point in Figure 9:1 These multipliers are obtained from both the exact and the approximate monodromy matrix. As can predicted by the theoretical analysis, all the multipliers are located at the origin of the complex plane. The matching between the results in this particular case is also remarkable. The right panel of Figure 11 shows the corresponding system response to a 5% step change in the input voltage \(v_{g}\) with the feedback. It can be observed that the duration of the transient after a perturbation in the input voltage takes place is three periods agreed with the theoretical prediction.