## 2 A Generalized Converter Model  
The main assumptions concerning the converters under study can be summarized as follow  
1. The circuits are composed of inductors, capacitors, resistors, diodes, and semiconductor switches. 2. The circuits are fed by voltage sources and fed loads of constant resistance, current, or power nature. 3. The converters is assumed to be operating in continuous conduction mode (CCM). 4. The different controlled semiconductor switches can be replaced by an association of appropriate variables \(u_{i}\), which take values \(0\) and \(1\) (open and closed, respectively). - \(x(t)\in\mathbb{R}^{n}\) is the state-vector essentially composed of inductor currents and capacitor voltages. - \(y(t)\in\mathbb{R}^{p}\) is the output vector. - \(u_{i}(t)\) is a variable associated to the \(i\)th switch, \(u_{i}=0\) (open or off) or \(1\) (closed or on), \(i=1,...,m\), which correspond to control inputs. - \(A_{0}\), \(B_{0}\), \(C_{0}\), \(A_{u_{i}}\), \(B_{u_{i}}\), and \(C_{u_{i}}\), \(i=1,...,m\) are constant matrices of appropriate dimensions. - \(v_{in}(t)\) is a voltage source (when \(v_{in}(t)\) is a constant voltage, we replace \(v_{in}(t)\) by \(V_{in}\)). \(A_{0}\) is supposed to be Hurwitz. As is shown below, this assumption is not restrictive for almost all the converters of interest. We also introduce the following matrices associated with a configuration of variables \(u_{1}u_{2}...u_{m}\). \[A_{u_{1}(t)u_{2}(t)...u_{m}(t)}=\sum\limits_{i=1}^{m}u_{i}(t)A_{u_{i}},\ \ B_{u_{1}(t)u_{2}(t)...u_{m}(t)}=\sum\limits_{i=1}^{m}u_{i}(t)B_{u_{i}},\ \ C_{u_{1}(t)u_{2}(t)...u_{m}(t)}=\sum\limits_{i=1}^{m}u_{i}(t)C_{u_{i}} \tag{2}\]  
Model (1) introduced in [1] is nonlinear, more precisely bilinear, the control is discrete and belongs to the set \(\{0,1\}^{m}\). To solve many control problems based on model (1), it is possible to write it in a more appropriate way. To obtain such a model, remark that matrix \(A_{u_{1}(t)u_{2}(t)...u_{m}(t)}\) can also be written as  
\[\begin{array}{ll}A_{u_{1}(t)u_{2}(t)...u_{m}(t)}=&(1-u_{1}(t))(1-u_{2}(t))...(1-u_{m}  
We can remark that \(A_{i}=A_{\text{bin}(i-1)}\) where \(\text{bin}(i-1)\) is the \(2^{m}\)-digits binary expression of \(i-1\). Functions \(\lambda_{i}(t)\) have the following important property:  
\[\sum\limits_{i=1}^{2^{m}}\lambda_{i}(t)=[1-u_{1}(t)+u_{1}(t)][1-u_{2}(t)+u_{2}( t)]\cdots[1-u_{m}(t)+u_{m}(t)]=1 \tag{6}\]  
Doing the same for matrices \(B_{u_{1}(t)u_{2}(t)\ldots u_{m}(t)}\) and \(C_{u_{1}(t)u_{2}(t)\ldots u_{m}(t)}\), a new expression for model (1) is obtained as  
\[\left\{ \begin{array}{l}\dfrac{dx(t)}{dt}=(A_{0}+\sum\limits_{i=1}^{M} \lambda_{i}(t)A_{i})x(t)+\sum\limits_{i=1}^{M}\lambda_{i}(t)B_{i}v_{in}(t)\\ y(t)=(C_{0}+\sum\limits_{i=1}^{M}\lambda_{i}(t)C_{i})x(t)\end{array} \right. \tag{7}\]  
where \(M=2^{m}\), \(\lambda(t)=\left[ \begin{array}{cc}\lambda_{1}(t)&\cdots&\lambda_{M}(t)\end{array}  \right]^{T}\in\Lambda_{S}\), \(\Lambda_{S}\) being defined as  
\[\Lambda_{S}=\left\{\lambda\in\{0,1\}^{M}:\sum\limits_{i=1}^{M}\lambda_{i}=1\right\} \tag{8}\]  
Noting that, for \(\lambda\in\Lambda_{S}\), \(B_{0}=\sum\limits_{i=1}^{M}\lambda_{i}B_{0}\), matrix \(B_{0}\) has been included in matrices \(B_{i}\). **Remark 1**. : _As previously done for \(B_{0}\), matrices \(A_{0}\) and \(C_{0}\) can be included in matrices \(A_{i}\) and \(C_{i}\), respectively. In that case, model (7) becomes_  
\[\left\{ \begin{array}{l}\dfrac{dx(t)}{dt}=\sum\limits_{i=1}^{M}\lambda_{i}(t )A_{i}x(t)+\sum\limits_{i=1}^{M}\lambda_{i}(t)B_{i}v_{in}(t)\\ y(t)=\sum\limits_{i=1}^{M}\lambda_{i}(t)C_{i}x(t)\end{array} \right. \tag{9}\]  
_where \(\lambda:\mathbb{R}\rightarrow\Lambda_{S}\). This model has previously been considered in works related to power converters (see, for example, [15; 17]). Finally, it is possible to write the previous model in another classical form corresponding to what is called a \"switched model\" [15; 18], given by_  
\[\left\{