## 2 Modeling
### Discrete Small-Signal Model  
The discrete small-signal model of the converter is obtained by assuming a perturbation with respect to an operating point in the state variables, output variables, input variables and duty cycle. This allows for the variables to be written as (62), i.e., as the sum of their value at the operating point \((X,\ Y,\ U,\ D)\), in addition to a perturbation that generates \((\hat{x},\hat{y},\hat{u},\hat{d})\).  
\[x=X+\hat{x};\ y=Y+\hat{y};\ u=U+\hat{u};\ d=D+\hat{d} \tag{62}\]  
where \(x\) is given by (50) and the rest by (63).  
\[\begin{array}{c}X=[I_{L_{p_{1}}},I_{L_{p_{2}}},I_{L_{M}},I_{L_{S_{1}}},I_{L_{ S_{2}}},I_{L_{F}},V_{C_{p_{1}}},V_{C_{p_{2}}},V_{C_{OS_{1}}},V_{C_{OS_{2}}},V_{C_{S_{1}}},V_{C_{S_{2}}},V_{C_{F}}]^{T}\\ \hat{x}=[i_{L_{p_{1}}},i_{L_{p_{2}}},i_{L_{M}},i_{L_{S_{1}}},i_{L_{S_{2}}},i_{L_{ F}},i_{L_{F}},\hat{v}_{C_{D_{2}}},\hat{v}_{C_{OS_{1}}},\hat{v}_{C_{OS_{2}}},\hat{v}_{C_{S_{1}}},\hat{v}_{C_{S_{2}}},\hat{v}_{C_{S_{1}}},\hat{v}_{C_{S_{2}}},\hat{v}_{C_{F}}]^{T }\\ y=[v_{R}\ i_{R}]^{T};\ \ \ \ Y=[V_{R}\ I_{R}]^{T};\ \ \ \ \hat{y}=[\hat{v}_{R}\ \hat{v}_{R}]\\ u=v_{in};\ \ \ \ U=V_{in};\ \ \ \hat{u}=\hat{v}_{in}\end{array} \tag{63}\]Now, from (59) and (60), the following equation obtained (64).  
\[\begin{array}{c}x(k+1)=X+\hat{x}(k+1)=\Big{[}A_{1}\cdot\Big{(}D+\hat{d}\Big{)}+A _{2}\cdot\Big{(}D+\hat{d}\Big{)}+A_{3}\cdot\Big{(}1-2\Big{(}D+\hat{d}\Big{)} \Big{)}\Big{]}\cdot[X+\hat{x}(k)]+[B\cdot\Big{(}D+\hat{d}\Big{)}+\\ B\cdot\Big{(}D+\hat{d}\Big{)}+B\cdot\Big{(}1-2\Big{(}D+\hat{d}\Big{)}\Big{)} \Big{]}\cdot[U+\hat{u}(k)]+\Big{[}E\cdot\Big{(}D+\hat{d}\Big{)}+E\cdot\Big{(}1- 2\Big{(}D+\hat{d}\Big{)}\Big{)}\Big{]}\\ Y+\hat{y}(k)=C[X+\hat{x}(k)]+D[U+\hat{u}(k)]\end{array} \tag{64}\]  
From term to term, (64) can be written as (65), i.e., a constant term that depends on the system at its operating point (\(OP_{x}/OP_{y}\)), plus a linear term that depends on the perturbed system (\(L_{x}/L_{y}\)), plus a nonlinear term that depends on the perturbed system (\(NL_{x}\)).  
\[\begin{array}{c}X+\hat{x}(k+1)=OP_{x}(X,U)+L_{x}\Big{(}\hat{x}(k),\hat{u}(k ),\hat{d}\Big{)}+NL_{x}(\Big{(}\hat{x}(k),\hat{d}\Big{)}\\ Y+\hat{y}(k)=OP_{y}(X,U)+L_{y}(\hat{x}(k),\hat{u}(k))\end{array} \tag{65}\]  
where OPx/y, Lx/y and NL are given, respectively, by (66).  
\[\begin{array}{c}OP_{x}=[A_{1}\cdot D\cdot X+A_{2}\cdot D\cdot X+A_{3}\cdot( 1-2D)X+[B\cdot U]+E]\\ L_{x}=[A_{1}\cdot D+A_{2}\cdot D+A_{3}\cdot(1-2D)]\hat{x}(k)+B\cdot\hat{u}(k )+[A_{1}\cdot X+A_{2}\cdot X-2A_{3}\cdot X]\cdot\hat{d}\\ NL_{x}=[A_{1}+A_{2}-2A_{3}]\hat{d}\cdot\hat{x}(k)\\ \text{And for the output,}\\ OP_{y}=C\cdot X+D\cdot U\\ L_{y}=C\cdot\hat{x}(k)+D\cdot\hat{u}(k)\end{array} \tag{66}\]  
Since the model sought is dynamic, the constant terms (equilibrium state) are of no interest, because, without any loss of generality, they can be taken to point 0 of the state-space. On the other hand, the nonlinear term consists of the product of two very small magnitudes, so it can be ignored without losing practical accuracy. As a result, (65) can be written as (67), which represents the small-signal linear model of the converter in Figure 2.  
\[\begin{array}{c}\hat{x}(k+1)=\underbrace{[A_{1}\cdot D+A_{2}\cdot D+A_{3} \cdot(1-2D)]}_{\hat{x}}\hat{x}(k)+B\cdot\hat{u}(k)[\underbrace{(A_{1}+A_{2}-2 A_{3})X]}_{\hat{x}}\hat{d}\\ \hat{y}(k)=C\cdot\hat{x}(k)+D\cdot\hat{u}(k)\end{array} \tag{67}\]  
Regarding the perturbations, both \(\hat{u}\) and \(\hat{d}\) can be considered inputs, so an extended input vector (\(\hat{u}_{e}\)) including both can be considered. Thus, (67) can be written as in (68).  
\[\begin{array}{c}\hat{x}(k+1)=\hat{A}\cdot\hat{x}(k)+\hat{B}\cdot\hat{u}_{e }(k)\\ \hat{y}(k)=C\cdot\hat{x}(k)+D\cdot\hat{u}(k)\end{array} \tag{68}\]  
where  
\[\hat{B}=\big{[}B,\hat{E}\big{]};\hat{u}_{e}(k)=\Big{[}\hat{v}_{in};\hat{d} \Big{]}\]