## 2 A Generalized Converter Model
\begin{array}{l}\dfrac{dx(t)}{dt}=A_{\sigma(t)}x(t)+B_{\sigma(t)}v_{ in}(t)\\ y(t)=C_{\sigma(t)}x(t)\end{array} \right. \tag{10}\]  
_where \(\sigma(t)\) is the switching control strategy defined by_  
\[ \begin{array}{ccc}\sigma:&\mathbb{R}&\rightarrow&\{1,2,\cdots,M\}\\ t&\rightarrow&\sigma(t)=i\end{array}  \tag{11}\]  
**Remark 2**. : _Model (7) can represent numerous converters of interest. Indeed, when \(v_{in}(t)\) and \(y(t)\) are DC voltages, the converter is a DC-DC converter. If \(y(t)\) is an AC voltage and \(v_{in}(t)\) is a DC one, the converter is a DC-AC converter. If \(v_{in}(t)\) is an AC voltage, depending on the nature of \(y(t)\), the converter will be an AC-DC or an AC-AC converter. It is possible to extend all the considered models to the case of multi-sources. In that case, \(v_{in}(t)\) is a vector whose dimension is equal to the number of sources. However, for simplicity, only mono-source converters are considered, the extension to multi-sources converters being possible without any difficulty._  
As is shown below, the advantage of model (7) is to propose a decomposition of the dynamical equation as a sum of a linear term and a bilinear matrix function of state and control variables \(x\) and \(\lambda\). This specificity suggests writing model (7) in the following form, more adapted to deal with the control design problems investigated in the sequel:  
\[\left\{ \begin{array}{l}\frac{dx(t)}{dt}=A_{0}x(t)+\sum\limits_{i=1}^{M}B_{i}(x (t),v_{in}(t))\lambda_{i}(t)\\ y(t)=C_{0}x(t)+\sum\limits_{i=1}^{M}D_{i}(x(t))\lambda_{i}(t),\quad\lambda: \mathbb{R}\rightarrow\Lambda_{S}\end{array} \right. \tag{12}\]  
where \(B_{i}(x(t),v_{in}(t))=A_{i}x(t)+B_{i}v_{in}(t)\) and \(D_{i}(x(t))=C_{i}x(t)\). For simplicity, when \(v_{in}(t)=V_{in}\), we denote \(B_{i}(x(t),v_{in}(t))\) by \(B_{i}(x(t))\). As pointed out above, the model is bilinear with a discrete control belonging to \(\Lambda_{S}\). An associated _relaxed_ or _embedded_ model can be defined as follows:  
\[\left\{ \begin{array}{l}\frac{dx(t)}{dt}=A_{0}x(t)+\sum\limits_{i=1}^{M}B_{i }(x(t),v_{in}(t))\lambda_{i}(t)\\ y(t)=C_{0}x(t)+\sum\limits_{i=1}^{M}D_{i}(x(t))\lambda_{i}(t),\quad\lambda: \mathbb{R}\rightarrow\Lambda_{R}\end{array} \right. \tag{13}\]  
where now  
\[\Lambda_{R}=\left\{\lambda\in[0,1]^{M}:\sum\limits_{i=1}^{M}\lambda_{i}=1\right\} \tag{14}\]  
There is a close connection between models (12) and (13). As developed in [25], for arbitrary close initial conditions, the solution set of initial value problems of (12) is dense in the solution set of initial value problems of (13) for the \(C^{0}\) Whitney topology on the infinite interval \([0,\infty)\). More precisely, there exist approximating trajectories of (12), of a given trajectory of (13), approximations in the weighted norm \([0,\infty)\), \(|f(t)|=sup_{t\geq 0}\{|f(t)|r(t)\}\) where \(r:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{>0}\). This result is of importance for practical applications because it is possible to connect controls designed from models (12) and (13). A trajectory of (13) can be arbitrarily approached by a trajectory of (12). A practical way can be to average a control developed from (12) leading to a control applicable for (13). Inversely, a control law developed from model (13) can be translated into a control compatible with model (12) using, for example, a PWM or a hysteresis comparator. Some care has to be taken to prevent some undesirable effects [28]. These connections and other ones are discussed in details, for example, for linear switched systems, in [29]. However, here, the important difference is that the modes are affine.