# Analysis of Subharmonic Oscillation and Slope Compensation for a Differential Boost Inverter
## 5 Accurate Stability Analysis Using Floquet Theory  
The differential equations describing the dynamics of switching converters are time periodic with the switching period \(T\) determining the periodicity of solutions at the fast switching scale. DC-AC inverters are also time periodic with the switching period \(T\) and the voltage reference period \(T_{g}=1/f_{g}\). For such time periodic systems Floquet theory can be used to study the stability of periodic orbits [53]. Here, this theory will be applied using a quasi-static approximation treating the DC-AC inverter as a DC-DC converter with a slowly varying reference voltage and duty cycle. With this approximation, the reference voltage \(v_{\text{ref}}\) is considered constant within a switching cycle. Floquet theory has been widely used in the analysis of stability of dynamical systems [53] in general and switching converters in particular [38; 39; 40]. For DC-DC converters, the stability dynamics at the fast switching cycle can be accurately predicted by analyzing the stability of the fixed points of the Poincare map of the system using its Jacobian matrix or using Floquet theory combined with Filippov method which leads to the same results as the Poincare map [38]. The main tool for studying the stability of periodic orbits using Floquet theory is the principal fundamental matrix or the monodromy matrix \(\mathbf{M}\). This matrix plays a key role in the accurate stability analysis of switching systems [38; 39; 40; 53]. The monodromy matrix is such that the dynamics in the vicinity of a quasi-static periodic orbit can be expressed as follows  
\[\mathbf{\hat{x}}(t+T)=\mathbf{M}\mathbf{\hat{x}}(t)\qquad\forall t \tag{18}\]  
where the overhat stands for small signal variations. Its eigenvalues are called the _characteristic multipliers_ or _Floquet multipliers_ and it can be seen that they determine the amount of contraction or expansion near a periodic orbit and hence they determine the stability of these periodic orbits. Let us start by finding the monodromy matrix \(\mathbf{M}\). Let \(\mathbf{x}(t)\approx\mathbf{x}(t+T)\) the quasi-steady-state value of the state vector. Let \(\mathbf{x}(DT)=(\mathbf{I}-\mathbf{\Phi})^{-1}\mathbf{\Psi}\approx\mathbf{x}( t)\approx\mathbf{x}(DT)\) be the value of \(\mathbf{x}(t)\) at time instant \(DT\), where \(\mathbf{\Phi}=\mathbf{\Phi}_{1}\mathbf{\Phi}_{0}\), \(\mathbf{\Phi}_{1}=e^{\mathbf{A}_{1}DT}\), \(\mathbf{\Phi}_{0}=e^{\mathbf{A}_{0}(1-D)T}\), \(\mathbf{\Psi}_{1}=(e^{\mathbf{A}_{1}DT}-\mathbf{I})\mathbf{A}_{1}^{-1} \mathbf{B}V_{g}\), \(\mathbf{\Psi}_{0}=(e^{\mathbf{A}_{0}(1-D)T}-\mathbf{I})\mathbf{A}_{0}^{-1} \mathbf{B}V_{g}\), \(\mathbf{\Psi}=\mathbf{\Phi}_{1}\mathbf{\Psi}_{0}+\mathbf{\Psi}_{1}\). Let \(\mathbf{m}_{1}(\mathbf{x}(t))=\mathbf{A}_{1}\mathbf{x}(t)+\mathbf{B}_{1}V_{g}\) and \(\mathbf{m}_{0}(\mathbf{x}(t))=\mathbf{A}_{0}\mathbf{x}(t)+\mathbf{B}_{0}V_{g}\) be the vector fields for \(u=1\) and \(u=0\) respectively. Let us define the augmented state vector \(\mathbf{x}_{a}=(i_{1},\,i_{2},\,v_{01},\,v_{02},\,v_{i})^{\intercal}\). Let \(\mathbf{A}_{a1}\), \(\mathbf{A}_{a0}\), \(\mathbf{B}_{a1}\), \(\mathbf{B}_{a0}\), \(\mathbf{w}_{a}\) and \(\mathbf{K}_{a}\) be, respectively, the associated augmented state matrices, input vectors, vector of external parameters and vector of feedback coefficients that are expressed as follows  
\[\mathbf{A}_{a1} = \begin{pmatrix}\mathbf{A}_{1}&\mathbf{0}\\ -1&\mathbf{0}\end{pmatrix} ,\,\,\mathbf{B}_{a1}= \begin{pmatrix}\mathbf{B}_{1} &0\\ \mathbf{0}&1\end{pmatrix}