## 4 Simulation Results
### Rejection of Measured Perturbation
To impose a constant output \(y_{e}(t)\) equal to 350 V, from (92), a simple calculation leads to  
\[1-\lambda_{e_{2}}(t)=\frac{1}{2}\frac{V_{in}}{y_{e}+r_{C}w(t)}\left[1+\sqrt{1 -\frac{4r_{L}(y_{e}+r_{C}w(t))}{V_{in}^{2}}}\right] \tag{93}\]  
and the state of the converter has to track  
\[x_{e}(t)=\Bigg{[} \begin{array}{cc}\frac{w(t)}{1-\lambda_{e_{2}}(t)}&y_{e} \end{array} \Bigg{]}^{T}\]  
Note that neglecting the parasitic circuit elements (i.e., \(r_{L}=r_{C}=0\)) simplifies drastically expression (93), which can be written as  
\[1-\lambda_{e_{2}}(t)=\frac{V_{in}}{y_{e}}\]  
Figure 6 depicts the considered perturbation \(w(t)=i_{0}(t)=3.5+sin(200\pi t)\) A, the output voltage \(y\), and the inductor current \(i_{L}\). The proposed control is efficient and the objectives are attained. Case 2: Constant output voltage while the converter fed a constant power load \(P_{ref}\). In this paragraph, the objective is to maintain the output voltage \(y_{e}\) at 350 V, the power load being constant and equal to \(P_{ref}\). In that case, the load current is \(i_{0}=w=P_{ref}/y_{e}\) and then  
\[1-\lambda_{e_{2}}=\frac{1}{2}\frac{V_{in}}{y_{e}+P_{ref}/y_{e}}\Bigg{[}1+ \sqrt{1-\frac{4r_{L}(y_{e}+P_{ref}/y_{e})}{V_{in}^{2}}}\Bigg{]}\]  
which leads to  
\[x_{e}(t)=\Bigg{[} \begin{array}{cc}\frac{P_{ref}/y_{e}}{1-\lambda_{e_{2}}}&y _{e}\end{array} \Bigg{]}^{T} \tag{94}\]  
Figure 7 depicts the output voltage and inductor current for a power load which is piecewise constant 1, 0.5, and 1.2 kW. The presence of parasitic resistances \(r_{L}\) and \(r_{C}\) justifies that \(V_{in}w=V_{in}i_{L}\
eq y_{e}w=y_{e}i_{0}=P_{ref}\). Indeed, from (92), we have  
\[y_{e}w=\frac{V_{in}w}{1-\lambda_{e_{2}}}-\frac{w^{2}r_{L}}{\left(1-\lambda_{e _{2}}\right)^{2}}-r_{C}w^{2} \tag{95}\]  
However, because the inductor current in steady-state is equal to \(i_{L_{e}}=w/(1-\lambda_{e_{2}})\), we deduce that  
\[P_{ref}=V_{in}i_{L_{e}}-r_{L}i_{L_{e}}^{2}-r_{C}i_{0}^{2}\]The proposed control is efficient and allows to maintain a constant output voltage and power load \(P_{ref}\). Case 3: Constant power at the output and variable input voltage. In this paragraph, a variant of the previous scenario is considered. The problem consists in having a constant power at the output and possible variations of the input voltage. Of course, this is possible if an adequate power balance is maintained by the control. Figure 6: Response to a periodic disturbance in the output current \(i_{0}\). Figure 7: Response to disturbances in the output power \(P_{ref}\). We suppose that the input voltage is measurable and is interpreted as a perturbation. We also suppose that a given output power reference denoted \(P_{ref}\) is given. From the value of \(P_{ref}\) and the measurement of the output voltage, it is possible to deduce the value of the load current \(i_{0}\) which can be considered as a given perturbation. In that case, the vector of perturbations is defined as \(w(t)=\left[ \begin{array}{cc}i_{0}&v_{in}(t)\end{array} \right]^{T}=\left[ \begin{array}{cc}w_{1}(t)&w_{2}(t)\end{array} \right]^{T}\) and the model given by  
\[\left\{ \begin{array}{l}\frac{dx(t)}{dt}=\left[\underbrace{\left[ \begin{array}{cc}-\frac{\mu_{1}+\tau_{C}}{L}&-\frac{1}{L}\\ \frac{1}{L}&0\end{array}\right]}_{A_{0}}+\underbrace{\left[\begin{array}{cc} \frac{\tau_{C}}{L}&\frac{1}{L}\\ -\frac{1}{L}&0\end{array}\right]}_{A_{0}}\right]_{A_{1}}\right]x(t)+\underbrace{ \left[\begin{array}{cc}\left[\begin{array}{cc}\frac{\tau_{C}}{L}&\frac{1}{L} \\ -\frac{1}{L}&0\end{array}\right]+\underbrace{\left[\begin{array}{cc}-\frac{ \tau_{C}}{L}&0\\ 0&0\end{array}\right]}_{B_{\text{eqn}_{1}}}\right]}_{B_{\text{eqn}_{1}}}w(t)\\ y(t)=\left[\underbrace{\left[\begin{array}{cc}\tau_{C}&1\end{array}\right]+ \underbrace{\left[\begin{array}{cc}-\tau_{C}&0\\ \end{array}\right]}_{C_{\text{eqn}_{1}}}u_{1}(t)\right]x(t)+\underbrace{\left[ \begin{array}{cc}-\tau_{C}&0\\ \end{array}\right]}_{D_{\text{eqn}}}w(t)\\ \end{array}\right]\end{array} \right.