## 4 Theoretical Background for Stability Analysis of Limit Cycles  
### Limit Cycles Computation  
In steady state, for \(\frac{1}{3}<D<\frac{2}{3}\), there will be six time instants at which a switching from a configuration \(\mathcal{C}_{u_{1}u_{2}u_{3}}\) to another will take place (see Figure 3). These are \(t_{0}=nT\), \(t_{2}=(n+\frac{1}{3})T\) and \(t_{4}=(n+\frac{2}{3})T\). The other remaining switching instants are \(t_{1}=n+(D-\frac{2}{3})T\), \(t_{3}=(n+D)T\) and \(t_{5}=n+(D-\frac{1}{3})T\). The steady-state duty cycle \(D\) can be obtained by imposing volt-second balance on the inductor current [1]. This leads to \(D=Ri_{\text{ref}}/v_{g}\). If \(\frac{1}{3}<D<\frac{2}{3}\), during one switching cycle, the three-cell buck converter changes its structure among the six different configurations shown in Figure 4 and the the system will have a limit cycle using six different affine models. In steady-state, the limit cycle \(\mathbf{x}(t)\) starts  
Figure 6: Time-domain waveforms for different values of \(\kappa_{i}\) that are indicated in each panel title. \(i_{\text{ref}}=50\) A.  
with initial conditions \(\mathbf{x}(0)\), at the start of the switching cycle and evolves according to the switching sequence in (2) as shown below  
\[ \begin{array}{rcll}\mathbf{x}(t)&=&\mathbf{\Phi}_{101}(t)\mathbf{x}(0)+\mathbf{ \Psi}_{101}(t)&\text{for}&t\in(0,t_{1})\\ \mathbf{x}(t)&=&\mathbf{\Phi}_{100}(t-t_{1})\mathbf{x}(t_{1})+\mathbf{\Psi}_{ 100}(t-t_{1})&\text{for}&t\in(t_{1},t_{2})\\ \mathbf{x}(t)&=&\mathbf{\[PERSON]}_{110}(t-t_{2})\mathbf{x}(t_{2})+\mathbf{\[PERSON]}_{ 110}(t-t_{2})&\text{for}&t\in(t_{2},t_{3})\\ \mathbf{x}(t)&=&\mathbf{\[PERSON]}_{010}(t-t_{3})\mathbf{x}(t_{3})+\mathbf{\[PERSON]}_{ 010}(t-t_{3})&\text{for}&t\in(t_{3},t_{4})\\ \mathbf{x}(t)&=&\mathbf{\[PERSON]}_{011}(t-t_{4})\mathbf{x}(t_{4})+\mathbf{\[PERSON]}_{ 011}(t-t_{4})&\text{for}&t\in(t_{4},t_{5})\\ \mathbf{x}(t)&=&\mathbf{\[PERSON]}_{001}(t-t_{5})\mathbf{x}(t_{5})+\mathbf{\[PERSON]}_{ 001}(t-t_{5})&\text{for}&t\in(t_{5},T)\end{array}  \tag{8}\]  
where \(\mathbf{\[PERSON]}_{u_{1}u_{2}u_{3}}\) and \(\mathbf{\[PERSON]}_{u_{1}u_{2}u_{3}}\) are given by  
\[\mathbf{\[PERSON]}_{u_{1}u_{2}u_{3}}(t) = e^{\mathbf{A}u_{1}u_{2}u_{3}t}\] \[\mathbf{\[PERSON]}_{101}(t) = \int_{0}^{t_{1}}\mathbf{\[PERSON]}_{101}(\alpha)d\alpha\] \[\mathbf{\[PERSON]}_{100}(t-t_{1}) = \int_{t_{1}}^{t}\mathbf{\[PERSON]}_{101}(\alpha)d\alpha\] \[\mathbf{\[PERSON]}_{110}(t-t_{2}) = \int_{t_{2}}^{t}\mathbf{\[PERSON]}_{110}(\alpha)d\alpha\] \[\mathbf{\[PERSON]}_{010}(t-t_{3}) = \int_{t_{3}}^{t}\mathbf{\Phi}_{010}(\alpha)d\alpha\] \[\mathbf{\Psi}_{011}(t-t_{4}) = \int_{t_{4}}^{t}\mathbf{\Phi}_{011}(\alpha)d\alpha\] \[\mathbf{\Psi}_{001}(t-t_{5}) = \int_{t_{5}}^{t}\mathbf{\Phi}_{001}(\alpha)d\alpha\]  
It is worth to note that the limit cycle can be determined once its initial state \(\mathbf{x}(0)\) is obtained. This initial state \(\mathbf{x}(0)\) can be computed by imposing \(T\)-periodicity hence getting  
\[\mathbf{x}(0)=(1-\mathbf{\Phi})^{-1}\mathbf{\Psi} \tag{9}\]  
where \(\mathbf{\Phi}\) and \(\mathbf{\Psi}\) are given by the following expressions  
\[\mathbf{\Phi} = \mathbf{\Phi}_{001}(T-t_{5})\mathbf{\Phi}_{011}(t_{5}-t_{4}) \mathbf{\Phi}_{010}(t_{4}-t_{3})\mathbf{\Phi}_{110}(t_{3}-t_{2})\mathbf{\Phi} _{100}(t_{2}-t_{1})\mathbf{\Phi}_{101}(t_{1})\] \[\mathbf{\Psi} = \mathbf{\Phi}_{001}(T-t_{5})\mathbf{\[PERSON]}_{011}(t_{5}-t_{4}) \mathbf{\[PERSON]}_{010}(t_{4}-t_{3})\mathbf{\[PERSON]}_{110}(t_{3}-t_{2})\mathbf{\[PERSON]} _{100}(t_{2}-t_{1})\mathbf{\[PERSON]}_{101}(t_{1})+\] \[\mathbf{\[PERSON]}_{101}(t) = \int_{0}^{t_{1}}\mathbf{\[PERSON]}_{101}(\alpha)d\alpha\]0 \[\mathbf{\[PERSON]}_{101}(t) = \int_{0}^{t_{1}}\mathbf{\[PERSON]}_{101}(\alpha)d\alpha\]1 \[\mathbf{\[PERSON]}_{101}(t) = \int_{0}^{t_{1}}\mathbf{\[PERSON]}_{101}(\alpha)d\alpha\]2  
Figure 7 shows an illustration of how to construct the monodromy matrix using the state transition and the saltation matrices.