## 4 Theoretical Background for Stability Analysis of Limit Cycles
### Approximate Expression of the Monodromy Matrix and Stability Region in the Parametric Space  
The computational efforts with the previous approach is very high and it also precludes obtaining simple expressions for the stability boundary in the parametric space. In this section, some realistic assumptions will be made in order to obtain simple expressions for the stability borders. These expressions will be validated from numerical simulationsusing the exact semi-analytical approach detailed previously. A comparison will be performed between the results obtained from the exact approach and those obtained from the simplified approximate approach. The purpose of this comparison is to give a further support to the feasibility of the using the approximate expression of the monodromy matrix for design purposes. In practice, the switching period is selected much smaller than the time constants of the different converter configurations. Under this condition, it is possible to approximate the matrix exponential as follows \(e^{\mathbf{A}t}\approx I+\mathbf{A}t\). By using this simplified expression, we can obtain an approximate expression for the limit cycles, saltation matrices and finally monodromy matrix. The approximate expression of the limit cycle evaluated at the start of the switching period is given by  
\[\mathbf{x}(0) \approx \left( \begin{array}{c}\frac{k_{\mathrm{ref}}\mu_{\mathrm{ref}} \sigma_{\mathrm{g}}}{R+\kappa_{\mathrm{g}}\
u_{\mathrm{g}}}\\ \frac{\
u_{\mathrm{g}}}{\kappa}\\ \frac{2\
u_{\mathrm{g}}}{3}\end{array} \right) \tag{39}\]  
In order to make a validation of the simplified expression of the \(\mathbf{x}(0)\) let us compare the result from both the exact and the approximate approach. Namely, the validity of (39) is first demonstrated before obtaining the approximate expression of the monodromy matrix. Figure 8 shows the evolution of the voltage and the current coordinates of \(\mathbf{x}(0)\) as function of the feedback gain \(\kappa_{i}\) for two different values of the current reference \(i_{\mathrm{ref}}\) using the exact expression (9) and the approximate one (39). A remarkable agreement can be clearly observed which demonstrates that for design-purposes, the approximate expression (39) can be used without having to use the state exact transition matrices whose computation could be very time consuming in a repetitive simulation process. This also indicates that very little error expected if the stability analysis is performed using an approximate expression for the monodromy matrix. With the approximated vector \(\mathbf{x}(0)\) given in (39) and the simplified expressions of the state transition and the saltation matrices, the expressions of the monodromy matrix evaluated at the previous limit cycle can be explicitly expressed as follows  
\[\mathbf{M}\approx\left( \begin{array}{ccc}1-\frac{T\
u_{\mathrm{g}}\kappa_{i }}{L}-\frac{RT}{L}&\frac{T\
u_{\mathrm{g}}\kappa_{1}}{3L}&-\frac{T\
u_{ \mathrm{g}}\kappa_{2}}{3L}\\ 0&1-\frac{T\
u_{\mathrm{g}}i_{\mathrm{ref}}\mu_{\mathrm{ref}}\kappa_{1}}{C_{ 1}\left(\
u_{\mathrm{g}}\kappa_{i}+R\right)}&0\\ 0&0&1-\frac{T\
u_{\mathrm{g}}i_{\mathrm{ref}}\mu_{\mathrm{ref}}\kappa_{1}\kappa _{2}}{C_{2}\left(\
u_{\mathrm{g}}\kappa_{i}+R\right)}\end{array} \right) \tag{40}\]  
Figure 8: Evolution of the coordinates of the vector \(\mathbf{x}(0)\) according to the exact expression (9) and its approximate (39) with respect to the feedback gain \(\kappa_{i}\) for (**Left**): \(i_{\mathrm{ref}}=50\) A and (**Right**): \(i_{\mathrm{ref}}=80\) A. Table 3 shows a comparative analysis of the results from the exact and the approximate monodromy matrix. This table shows the limit cycle represented by their initial values \(\mathbf{x}(0)\), the corresponding Floquet multiplies, the exact and the approximate monodromy matrix \(\mathbf{M}\) and the stability of limit cycles. The values of the Floquet multipliers are obtained by solving the characteristic equation \(\det(\mathbf{M}-\lambda\mathbf{I})=0\) for both cases.