## 3 Stability Analysis
### Lyapunov Exponents
\] (24) In this framework, two discontinuous events occur and therefore two different corrections to the tangent evolution must be done: * The first one corresponds to the transition of the switch from ON to OFF position, which is given by the indicator function \[\tilde{h}_{r}(\mathbf{\tilde{x}})=k_{V}(V_{ref}-v_{2})+k_{int}x_{5}+x_{6}-i_{1}\] (25) Here, the fields \(\mathbf{f}^{-}=\mathbf{\tilde{f}}_{ON}\), \(\mathbf{f}^{+}=\mathbf{\tilde{f}}_{OFF}\), and the states do not change their values abruptly, hence the impact function \(\mathbf{\tilde{g}}_{r}(\mathbf{\tilde{x}})\) is trivially \[\mathbf{\tilde{g}}_{r}(\mathbf{\tilde{x}})=[i_{1},\ i_{2},\ v_{1},\ v_{2},\ x_{5},\ x_{6}].\] (26) With these ingredients, the different terms to be applied in Equations (21) and (22) are: \[\mathbf{H}_{r}=[-k_{1},\ 0,\ 0,\ -k_{V},\ k_{int},1]\] (27) and \[G_{r}=I\] (28)
* The second discontinuous event occurs when the ramp resets to 0 at the end of every cycle, or equivalently when \(x_{6}=-A\) which corresponds to the indicator function \[\tilde{h}_{c}(\mathbf{\tilde{x}})=x_{6}+A\] (29) Special care must be taken at this point. If the duty cycle is not saturated, then at each \(T\)-cycle \(\mathbf{f}^{-}=\mathbf{\tilde{f}}_{OFF}\) and \(\mathbf{f}^{+}=\mathbf{\tilde{f}}_{ON}\). However, it might happen that the cycle is saturated and the switch remained in the ON position during the whole period, in which case \(\mathbf{f}^{-}=\mathbf{f}^{+}=\mathbf{\tilde{f}}_{ON}\). In both cases, \(x_{6}\) changes its value abruptly to a constant value \(x_{6}=0\), hence the impact function is: \[\mathbf{\tilde{g}}_{c}(\mathbf{\tilde{x}})=[i_{1},\ i_{2},\ v_{1},\ v_{2},\ x_{5},\ 0],\] (30) which leads to \[\mathbf{H}_{c}=[0,\ 0,\ 0,\ 0,\ 0,\ 0,1]\] (31) and \[G_{c}= \begin{bmatrix}1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&0\end{bmatrix} \] (32) With these corrections, the \(\lambda_{max}\) calculation can be summarized in the following steps:1. \(\quad\) Evolve the extended system: \[ \begin{bmatrix}\mathbf{\hat{x}}\\ \dot{\mathbf{\hat{x}}}\end{bmatrix} =\left\{ \begin{array}{ll}\left[\begin{array}[ ]{c}\mathbf{\tilde{t}}_{ON}(\mathbf{\tilde{x}})\\ \frac{\partial\Omega_{ON}(\mathbf{\tilde{x}})}{\partial\mathbf{\hat{x}}}\delta \mathbf{\hat{x}}\end{array}\right]\quad\text{if}\quad\quad u=1\\ \left[\begin{array}{c}\mathbf{\tilde{t}}_{OFF}(\mathbf{\tilde{x}})\\ \frac{\partial\Omega_{OFF}(\mathbf{\tilde{x}})}{\partial\mathbf{\hat{x}}} \delta\mathbf{\hat{x}}\end{array}\right]\quad\text{if}\quad\quad u=0\end{array} \right.\] (33) in the interval \([t_{0},\ t_{0}+\tau]\) with randomly chosen initial conditions for \(\delta\mathbf{\tilde{x}}(t_{0})\), such that \(||\delta\mathbf{\tilde{x}}(t_{0})||=1\). 2. During the evolution, identify the times of each of the possible discontinuous events given by (25) or (29) and apply the correction (21) in accordance with the nature of the discontinuous event. 3. When \(t=t_{0}+\tau\), calculate and store \[\lambda_{i}=\frac{1}{\tau}\log\frac{||\delta\mathbf{\tilde{x}}(t)||}{|| \delta\mathbf{\tilde{x}}(t_{0})||}\]
4. Set \(t_{0}=t_{0}+\tau\) and renormalize \(||\delta\mathbf{\tilde{x}}(t_{0})||=1\). 5. Repeat Steps 1-4 for a sufficiently large number of steps \(N\). 6. Calculate the maximal Lyapunov exponent as: \[\lambda_{max}=\frac{1}{N}\sum_{i=1}^{N}\lambda_{i},\]