# A Unified Approach for the Control of Power Electronics Converters. Part II: Tracking
## 6 Numerical Examples
### Boost Inverter for Stand-Alone Applications
Retaining the first and third harmonics, and because, usually, the circuit is constituted by two boost converters with the same components, the approximation of currents \(i_{1_{ref}}(t)\) and \(i_{1_{ref}}(t)\) is given by  
\[i_{1_{ref}}(t)\simeq I_{L}+I_{s_{1}}\sin(\omega_{0}t)+I_{c_{1}} \cos(\omega_{0}t)+I_{s_{2}}\sin(2\omega_{0}t)+I_{c_{2}}\cos(2\omega_{0}t)+I_{s _{3}}\sin(3\omega_{0}t)+I_{c_{3}}\cos(3\omega_{0}t)\] \[i_{2_{ref}}(t)\simeq I_{L}-I_{s_{1}}\sin(\omega_{0}t)+I_{c_{1}} \cos(\omega_{0}t)-I_{s_{2}}\sin(2\omega_{0}t)+I_{c_{2}}\cos(2\omega_{0}t)-I_{s _{3}}\sin(3\omega_{0}t)+I_{c_{3}}\cos(3\omega_{0}t)\]  
and then the exosystem is defined as  
\[\frac{dz(t)}{dt}=\Theta z(t)=diag\bigg{(}\bigg{[} \begin{array}{cc}0&-\omega _{0}\\ \omega_{0}&0\end{array} \bigg{]},\bigg{[} \begin{array}{cc}0&-2\omega_{0}\\ 2\omega_{0}&0\end{array} \bigg{]},\bigg{[} \begin{array}{cc}0&-3\omega_{0}\\ 3\omega_{0}&0\end{array} \bigg{]},0\bigg{)}z(t)\]  
with the initial condition  
\[z(0)=\big{[} \begin{array}{cccccccc}1&0&1&0&1&0&1\end{array} \big{]}^{T}\]  
The state reference writes  
\[x_{ref}(t)=\Pi_{1}(z)(t)=\left[ \begin{array}{cccccccc}I_{c_{1}}&I_{s_{1}}&I _{c_{2}}&I_{s_{2}}&I_{c_{3}}&I_{s_{3}}&I_{L}\\ 0&V_{\max}/2&0&0&0&0&V_{dc}\\ I_{c_{1}}&-I_{s_{1}}&I_{c_{2}}&-I_{s_{2}}&I_{c_{3}}&-I_{s_{3}}&I_{L}\\ 0&-V_{\max}/2&0&0&0&0&V_{dc}\end{array} \right]z(t)\]  
The amplitudes are obtained by solving the following algebraic system of nonlinear equations:  
\[F(I_{L},I_{s_{1}},I_{c_{1}},I_{s_{2}},I_{c_{2}},I_{s_{3}},I_{c_{3}})=0\]  
where the components of \(F\) are given in Appendix A. For the desired output reference, Table 5 gives the values obtained using the MATLAB function _fsolve_. The control is defined by  
\[(u_{1},u_{2})=bin_{2}(j-1)\text{ with }j=\arg\biggl{\{}\min_{i=1,2,3,A}e^{T}PA_{ i}x\biggr{\}}\]  
where \(A_{1}=0,\ A_{2}=A_{u_{2}},\ A_{3}=A_{u_{1}},\ A_{4}=A_{u_{1}}+A_{u_{2}}\) and \(P\) solution of (8) given by  
\[P=\left[ \begin{array}{cccc}2.0933&0.0817&0.4567&0.0183\\ 0.0817&0.0458&0.0183&0.0092\\ 0.4567&0.0183&2.0933&0.0817\\ 0.0183&0.0092&0.0817&0.0458\end{array} \right]\]  
Figure 6 depicts the output voltage \(y\) and the duty cycles obtained by filtering control \(u_{1}\) and \(u_{2}\) besides the inductor currents \(i_{1},\ i_{2}\) and the capacitor voltages \(v_{C_{1}},\ v_{C_{2}}\). We can see that the proposed method leads to satisfactory results for this non-trivial tracking problem. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\mathbf{I_{L}}\) & \(\mathbf{I_{s_{1}}}\) & \(\mathbf{I_{c_{1}}}\) & \(\mathbf{I_{s_{2}}}\) & \(\mathbf{I_{c_{2}}}\) & \(\mathbf{I_{s_{3}}}\) & \(\mathbf{I_{c_{3}}}\) \\ \hline
0.9653 & 4.5779 & \(-0.1390\) & 0.0197 & \(-0.9718\) & \(-0.0609\) & \(-0.0020\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Amplitudes of reference signals. Figure 6: Simulated waveform of the boost inverter feeding a resistive load and using two control actions. Another means of controlling the inverter is to commute among the configurations associated with \(u_{1}(u_{2})=0(1)\) and \(u_{1}(u_{2})=1(0)\). In this case, it is possible to describe this operation mode using only the variable \(u_{1}\), remarking that \(u_{1}(u_{2})=0,(1=1-u_{1})\) and \(u_{1}(u_{2})=1(0=1-u_{1})\)--in other words, replacing \(u_{2}\) with \(1-u_{1}\) in the model above. The model becomes  
\[\frac{dx(t)}{dt} = \left[\left[ \begin{array}{cccc}-\frac{p_{1}}{L_{1}}&-\frac{1}{L _{1}}&0&0\\ \frac{1}{C_{1}}&-\frac{1}{RC_{1}}&0&\frac{1}{RC_{1}}\\ 0&0&-\frac{r_{2}}{L_{2}}&0\\ 0&\frac{1}{RC_{2}}&0&-\frac{1}{RC_{2}}\end{array} \right]+u_{1}(t)\underbrace{ \left[ \begin{array}{cccc}0&\frac{1}{L_{1}}&0&0\\ -\frac{1}{C_{1}}&0&0&0\\ 0&0&0&-\frac{1}{L_{2}}\\ 0&0&\frac{1}{C_{2}}&0\end{array} \right]}_{A_{u_{1}}}x(t)+\underbrace{\left[ \begin{array}{c}\frac{1}{L_{1}}\\ 0\\ \frac{1}{L_{2}}\\ 0\end{array} \right]}_{B_{0}}V_{in} \tag{22}\] \[y(t) = \underbrace{\left[ \begin{array}{cccc}0&1&0&-1\\ \hline\end{array} \right]}_{C_{0}}x(t)\]  
This operation mode is more constrained than the previous one.