## 3 Stability Analysis
### Lyapunov Exponents  
Lyapunov exponents provide an additional analysis tool to study the orbits in power converters [40]. Recall that Floquet multipliers are restricted to the study of periodic orbits. However, it can be the case that a power converter shows more complex phenomena that go beyond a Period 1 solution, such as quasi-periodicity and chaos. To quantify this, one has to make use of Lyapunov exponents. For a system whose evolution is governed by \(\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})\), given an infinitesimally small perturbation \(\delta\mathbf{x}(0)\) around a reference orbit (notice that here we do not restrict the perturbation to a given periodic solution as in Section 3.1), the maximal Lyapunov exponent \(\lambda_{max}\) is defined as:  
\[\lambda_{max}=\lim_{t\to\infty}\frac{1}{t}\log\frac{||\delta\mathbf{x}(t)||}{ ||\delta\mathbf{x}(0)||}. \tag{19}\]  
The maximal Lyapunov exponent quantifies the rate of convergence (\(\lambda_{max}<0\)) or divergence (\(\lambda_{max}>0\)) of the reference orbit and the infinitesimally perturbed one. Hence, \(\lambda_{max}\) describes the nature of a given orbit as follows:  
* If \(\lambda_{max}<0\), the attractor is a stable fixed point. * If \(\lambda_{max}=0\), the attractor is a stable periodic (or quasi-periodic) orbit. * If \(\lambda_{max}>0\), the attractor is chaotic. In Equation (19), \(\delta\mathbf{x}(t)\) is the evolution in time of the infinitesimal perturbation (also called tangent space) and its dynamics is governed by the variational equation  
\[\delta\dot{\mathbf{x}}=\frac{\partial\mathbf{f}(\mathbf{x})}{\partial\mathbf{ x}}\delta\mathbf{x} \tag{20}\]  
As in Section 3.1, the tangent evolution needs to be specially treated at the times where the system changes from a vector field to another. To solve this, we make use of the Mueller algorithm to solve the tangent evolution for systems with discontinuities [50]. We first describe the general procedure. Let \(\mathbf{x}^{-}\) be the state vector right before a discontinuous event, which occurs when an indicator function \(h(\mathbf{x})=0\). In addition, let \(\mathbf{x}^{+}\) be the state right after the discontinuous event which may undergo a discontinuous change given by a so-called impact function \(\mathbf{g}(\mathbf{x}^{-})\) that maps the state of the system right before/right after the discontinuous event, namely \(\mathbf{x}^{+}=\mathbf{g}(\mathbf{x}^{-})\). Similarly, \(\mathbf{f}^{-}\) and \(\mathbf{f}^{+}\) are the vector fields that govern the dynamics before and after the discontinuity, respectively. With these considerations, the correction of the tangent space at the discontinuity is:  
\[\delta\mathbf{x}^{+}=G(\mathbf{x}^{-})\delta\mathbf{x}^{-}+[G(\mathbf{x}^{-}) \mathbf{f}^{-}(\mathbf{x}^{-})-\mathbf{f}^{+}(\mathbf{x}^{+})]\delta t. \tag{21}\]  
In Equation (21), \(\delta t\) is defined as:  
\[\delta t=\frac{-\mathbf{H}(\mathbf{x}^{-})\delta\mathbf{x}^{-}}{\mathbf{H}( \mathbf{x}^{-})\mathbf{f}^{-}(\mathbf{x}^{-})}. \tag{22}\]  
and  
\[G(\mathbf{x})=\frac{\partial\mathbf{g}(\mathbf{x})}{\partial\mathbf{x}}; \qquad\mathbf{H}(\mathbf{x})=\frac{\partial h(\mathbf{x})}{\partial\mathbf{x}}. \tag{23}\]  
In order to apply the Mueller algorithm to the sZETA converter controlled via ramp compensation, it is convenient to express the dynamics of the system in an autonomous wayincluding the evolution of the ramp. To do this, we extend the original system (3) to account for the ramp dynamics, making \(\mathbf{\tilde{x}}:=[i_{1},\ i_{2},\ v_{1},\ v_{2},\ \int edt,\ x_{6}]^{T}\) where \(x_{6}=-\frac{\dot{A}}{T}t\), hence: \[\mathbf{\tilde{f}}_{ON}= \begin{bmatrix}V_{in}/L_{1}\\ v_{1}/L_{2}-v_{2}/L_{2}+V_{in}/L_{2}\\ -i_{2}/C_{1}\\ i_{2}/C_{2}-v_{2}/(RC_{2})\\ V_{ref}-v_{2}\\ -A/T\end{bmatrix} \mathbf{\tilde{f}}_{OFF}= \begin{bmatrix}-v_{1}/L_{1}\\ -v_{2}/L_{2}\\ i_{1}/C_{1}\\ i_{2}/C_{2}-v_{2}/(RC_{2})\\ V_{ref}-v_{2}\\ -A/T\end{bmatrix}