## 3 Control of the Inverter
### Inner Current Control Loops
By applying the invariance conditions \(S(x)=0\) and replacing in Equations (9) and (10), the equivalent controls are given by:  
\[1-u_{1_{eq}}=\frac{v_{in}(t)-L\frac{di_{1e}(t)}{dt}}{v_{1}(t)}>0\Rightarrow \frac{di_{1e}(t)}{dt}<\frac{v_{in}(t)}{L} \tag{16}\]  
\[1-u_{2_{eq}}=\frac{v_{in}(t)-L\frac{di_{2e}(t)}{dt}}{v_{2}(t)}>0\Rightarrow \frac{di_{2e}(t)}{dt}<\frac{v_{in}(t)}{L} \tag{17}\]  
Replacing Equations (16) and (17) in Equations (12) and (13) and because on the surface \(i_{1}(t)=i_{1e}(t)\) and \(i_{2}(t)-i_{2e}(t)\), the following equations are obtained defining the ideal sliding dynamic of the inverter:  
\[\frac{dv_{1}}{dt}=\frac{1}{C}\frac{v_{in}i_{1e}}{v_{1}}+\frac{1}{r_{0}C}(v_{2 }-v_{1})-\frac{L}{C}\frac{i_{1e}}{v_{1}}\frac{di_{1e}}{dt} \tag{18}\]  
\[\frac{dv_{2}}{dt}=\frac{1}{C}\frac{v_{in}i_{2e}}{v_{2}}-\frac{1}{r_{0}C}(v_{2 }-v_{1})-\frac{L}{C}\frac{i_{2e}}{v_{2}}\frac{di_{2e}}{dt} \tag{19}\]  
Now, consider that converter variables has incremental variations around one instantaneous operation point \(v_{1}=V_{1}+\overline{v}_{1}\), \(v_{2}=V_{2}+\overline{v}_{2}\), \(i_{1e}=I_{1e}+\overline{i}_{1e}\), \(i_{2e}=I_{2e}+\overline{i}_{2e}\), \(\frac{di_{1e}}{dt}=I^{\prime}_{1e}+\overline{i}^{\prime}_{1e^{\prime}}\), \(\frac{di_{2e}}{dt}=I^{\prime}_{2e}+\overline{i}^{\prime}_{2e^{\prime}}\)\(v_{in}=V_{in}+\overline{v}_{in}\) and \(r_{0}=R_{0}+\overline{r}_{o}\). By following the conventional linearization procedure preserving only first-order terms, it is obtained that:  
\[\frac{d\overline{v}_{1}}{dt}=W_{11}\overline{i}_{1e}-LW_{12}\overline{i}^{ \prime}_{1e}-(CW_{11}W_{12}+b)\overline{v}_{1}+b\overline{v}_{2}+W_{12} \overline{v}_{in}-W_{3}\overline{r}_{o} \tag{20}\]  
\[\frac{d\overline{v}_{2}}{dt}=W_{21}\overline{i}_{2e}-LW_{22}\overline{i}^{ \prime}_{2e}-(CW_{21}W_{22}+b)\overline{v}_{1}+b\overline{v}_{1}+W_{22} \overline{v}_{in}+W_{3}\overline{r}_{o} \tag{21}\]  
\[ \begin{array}{c}W_{11}=\frac{1}{CV_{1}}\big{(}V_{in}-L^{\prime}_{1e}\big{)} \hskip 28.452756 ptW_{12}=\frac{I_{1e}}{CV_{1}}\hskip 28.452756 ptb=\frac{1}{R_{0}C} \\ W_{21}=\frac{1}{CV_{2}}\big{(}V_{in}-L^{\prime}_{2e}\big{)}\hskip 28.452756 ptW_{22}= \frac{I_{2e}}{CV_{2}}\hskip 28.452756 ptW_{3}=\frac{V_{2}-V_{1}}{R_{0}^{2}C} \end{array}  \tag{22}\]  
By applying the Laplace transform to Equations (20) and (21), Equations (23) and (24) are obtained with which the linear model of whole system can be represented using the block diagram in Figure 5.  
Transfer functions are defined using the polynomial Equations (25)-(30) considering \(G_{1}(s)=B_{1}(s)/A(s)\), \(G_{2}(s)=B_{2}(s)/A(s)\), \(H_{12}(s)=C_{1}(s)/A(s)\), \(H_{21}(s)=C_{2}(s)/A(s)\), \(H_{v1}(s)=D_{1}(s)/A(s)\), \(H_{v2}(s)=D_{2}(s)/A(s)\), \(H_{r1}(s)=E_{1}  
\[C_{1}(s)=b(W_{21}-LW_{22}s) C_{2}(s)=b(W_{11}-LW_{12}s) \tag{28}\] \[D_{1}(s)=W_{12}s+W_{12}(CW_{21}W_{12}+b)+bW_{22} D_{2}(s)=W_{22}s+W_{22}(CW_{11}W_{12}+b)+bW_{12}\] (29) \[E_{1}(s)=W_{3}(s+CW_{21}W_{22}) E_{2}(s)=W_{3}(s+CW_{11}W_{12}) \tag{30}\]  
As it can be noted, the parameters of the plant transfer functions \(G_{1}(s)\) and \(G_{2}(s)\). have a high dependence not only on the operation point defined by the input voltage and the load, but also on the shape of the converter waveforms. In order to accurately model the behavior of these parameters, the inductor currents and their derivatives are analyzed in the next section through application of harmonic balance technique facilitating the synthesis of the simplest controller.