*Nonlinear Average Model of the Converter*

The differential equations for each mode can be derived from Figures 5 and 6, respectively. The differential equations for mode I and mode II are :

$$\begin{aligned} \dot{i}\_{L\_1} &= \frac{1}{L\_1} \Big( E \Big) \\ \dot{i}\_{L\_2} &= \frac{1}{L\_2} \Big( -v\_{C\_p} + v\_0 \Big) \\ \dot{v}\_{C\_p} &= \frac{1}{v\_{C\_p}} \Big( i\_{L\_2} \Big) \\ \dot{v}\_{C\_0} &= \frac{1}{C\_0} \Big( -i\_{L\_2} - \frac{v\_0}{R} \Big) \\ \dot{i}\_{L\_1} &= \frac{1}{L\_1} \Big( v\_{C\_p} - v\_0 + E \Big) \\ \dot{i}\_{L\_2} &= \frac{1}{L\_2} \Big( -v\_{C\_p} + v\_0 \Big) \\ \dot{v}\_{C\_p} &= \frac{1}{v\_{C\_p}} \Big( i\_{L\_2} \Big) \\ \dot{v}\_{C\_0} &= \frac{1}{C\_0} \Big( -\dot{i}\_{L\_2} - \frac{v\_0}{R} \Big) \end{aligned} \tag{18}$$

By using the switching function as a weighting factor, the average non-linear model can be derived as:

$$
\begin{bmatrix}
\dot{i}\_{L\_1} \\
\dot{i}\_{L\_2} \\
\dot{v}\_{\mathcal{C}\_p} \\
\dot{v}\_{\mathcal{C}\_0} \\
\dot{v}\_{\mathcal{C}\_0}
\end{bmatrix} = \begin{bmatrix}
0 & 0 & \frac{(1-d)}{L\_1} & -\frac{(1-d)}{L\_1} \\
0 & 0 & -\frac{1}{L\_2} & \frac{d}{L\_2} \\
\frac{(1-d)}{\overline{\mathcal{C}\_0}} & -\frac{d}{\overline{\mathcal{C}\_0}} & 0 & -\frac{1}{\overline{\mathcal{K} \mathcal{C}\_0}}
\end{bmatrix} \begin{bmatrix}
i\_{L\_1} \\
i\_{L\_2} \\
v\_{\mathcal{C}\_p} \\
v\_{\mathcal{C}\_0} \\
\end{bmatrix} + \begin{bmatrix}
\frac{1}{L\_1} \\
0 \\
0 \\
0 \\
0
\end{bmatrix} e \tag{19}
$$

The above description can be generalized as:

$$\dot{\mathfrak{x}}(t) = A(d)\mathfrak{x} + B(d)e \tag{20}$$

where the state vector is *x*(*t*) = [*iL*<sup>1</sup> , *iL*<sup>2</sup> , *vC<sup>p</sup>* , *v*0] <sup>&</sup>gt; <sup>∈</sup> <sup>R</sup><sup>4</sup> <sup>+</sup>, the control signal *d* ∈ (0, 1), and the input voltage *e* ∈ R. The model described in (19) is bilinear, since the signal *d* is multiplying to all state variables.

Linealization of non-linear systems is a useful technique for analyzing and controlling complex high-order dynamical systems. This process describes the converter behaviour to small perturbations around an operation point, where the perturbations are applied to the input signals [25].

According to above, each state variable and the input signal are the sum of DC and AC components, which can be decomposed as:

$$\begin{aligned} \dot{i}\_{L\_1} &= I\_{L\_1} + \tilde{i}\_{L\_1} \\ \dot{i}\_{L\_2} &= I\_{L\_2} + \tilde{i}\_{L\_2} \\ \upsilon\_{\mathsf{C}\_p} &= V\_{\mathsf{C}\_p} + \mathfrak{v}\_{\mathsf{C}\_1} \\ \upsilon\_{\mathsf{C}\_0} &= V\_{\mathsf{C}\_0} + \mathfrak{v}\_{\mathsf{C}\_0} \\ d &= D + \tilde{d} \\ e &= E + \tilde{e} \end{aligned} \tag{21}$$

where *IL*<sup>1</sup> , *IL*<sup>2</sup> , *VC<sup>p</sup>* , *VC*<sup>0</sup> , *<sup>D</sup>*, *<sup>E</sup>* represent the DC components, and ˜*iL*<sup>1</sup> , ˜*iL*2 , *v*˜*C<sup>p</sup>* , *v*˜*C*<sup>0</sup> , ˜*d*, *e*˜ are the AC components. In steady state, the AC components are equal to zero.

Linearizing around an equilibrium point

$$\mathcal{E} := \left( I\_{\mathcal{L}\_1 \prime} I\_{\mathcal{L}\_2 \prime} V\_{\mathbb{C}\_{p'}} V\_{\mathbb{C}\_0} \right) \in \mathbb{R}\_+^4 \tag{22}$$

The linear representation of the systems shown in (19) can be rewritten as:

$$
\sharp = A\sharp + B\sharp \tag{23}
$$

where *<sup>x</sup>*˜ <sup>∈</sup> <sup>R</sup><sup>4</sup> <sup>+</sup> is the state vector, and *<sup>u</sup>*˜ = [ ˜*d*,*e*˜] <sup>&</sup>gt; <sup>∈</sup> <sup>R</sup><sup>2</sup> <sup>+</sup> is the vector with inputs. *A* is a constant matrix in *R* 4*X*4 , and *B* is a constant matrix in *R* 4*X*2 . The average linear timeinvariant model is:

$$
\begin{aligned}
\begin{bmatrix}
\dot{\tilde{t}}\_{L\_1} \\
\dot{\tilde{t}}\_{L\_2} \\
\dot{\tilde{t}}\_{C\_p} \\
\dot{\tilde{v}}\_{C\_0}
\end{bmatrix} &= \begin{bmatrix}
0 & 0 & \frac{(1-D)}{L\_1} & -\frac{(1-D)}{L\_1} \\
0 & 0 & -\frac{1}{L\_2} & \frac{D}{L\_2} \\
\frac{(1-D)}{C\_0} & -\frac{d}{C\_0} & 0 & -\frac{1}{RC\_0}
\end{bmatrix} \begin{bmatrix}
\dot{\tilde{t}}\_{L\_1} \\
\dot{\tilde{t}}\_{L\_2} \\
\dot{\tilde{v}}\_{C\_p} \\
\dot{\tilde{v}}\_{C\_0}
\end{bmatrix} \\ &+ \begin{bmatrix}
\frac{E}{L\_1(1-D)} & \frac{1}{L\_1} \\
\frac{E}{L\_2(1-D)^2} & 0 \\
\frac{E}{RC\_p(1-D)^4} & 0 \\
\end{bmatrix} \begin{bmatrix}
\dot{d} \\
\dot{\tilde{e}}
\end{bmatrix} \end{aligned} \tag{24}$$
