# A Unified Approach for the Control of Power Electronics Converters. Part II: Tracking
## 7 Approximated State Reference Signal from a Periodic Output Reference Signal
### Numerical Example  
Consider the converter depicted in Figure 5, whose parameters are given in Table 4. We consider the case in which the control \(u_{2}\) is equal to \(1-u_{1}\). As explained, it is not easy to determine the state reference signal. The idea is to use the method of the previous paragraph. Recall that the model is given by (22) and that the objective is to impose a reference signal at the output equal to \(y_{ref}(t)=v_{\mathcal{C}_{1_{ref}}}(t)-v_{\mathcal{C}_{2_{ref}}}(t)=V_{\max} \sin(\omega_{0}t)\). From \(y_{ref}(t)\), we must deduce the associated references \(v_{\mathcal{C}_{1_{ref}}}(t)\), \(v_{\mathcal{C}_{2_{ref}}}(t)\), \(i_{1_{ref}}(t)\) and \(i_{2_{ref}}(t)\). Using model (22), after extensive calculation, the expression of the output of the converter (27) is given by  
\[y=G(\lambda)V_{in}=\frac{-\lambda R(2\lambda-1)(\lambda-1)}{\lambda^{2}R( \lambda-1)^{2}+r_{L}(2\lambda^{2}-2\lambda+1)}V_{in}\ \ 0<\lambda<1\]  
Following the approach proposed in the previous paragraph, \(\lambda_{app}(t)\) solution of (25) is obtained solving  
\[-\lambda(t)RV_{in}\frac{(2\lambda(t)-1)(\lambda(t)-1)}{\lambda(t)^{2}R( \lambda(t)-1)^{2}+r_{L}(2\lambda(t)^{2}-2\lambda(t)+1)}=y_{ref}(t)\]  
Recall that only the admissible values of \(\lambda(t)\) are real, such that \(0<\lambda(t)<1\). The previous equation can also be written as  
\[1+\frac{y_{ref}(t)}{RV_{in}}\ \frac{\lambda(t)^{2}R\ (\lambda(t)-1)^{2}+r_{L} \big{(}2\lambda(t)^{2}-2\lambda(t)+1\big{)}}{\lambda(t)(2\lambda(t)-1)( \lambda(t)-1)}=0\]\(\lambda_{app}(t)\) is then given by  
\[\lambda_{app}(t)=\left[ \begin{array}{c}1-\lambda(t)\\ \lambda(t)\end{array} \right]\]  
Invoking the simple root locus building rules, we can easily see that, depending on the sign of \(y_{ref}(t)\), only one root is of interest and varies with \(y_{ref}(t)\) around \(1/2\). Then, it is possible to compute the value of \(\lambda(t)\) for a given value of \(y_{ref}(t)\). All the values of \(\lambda(t)\) for a given function \(y_{ref}(t)\) can be determined off-line. Integrating the differential system (28), we deduce the approximated reference \(x_{a}(t)\) for an output reference given by \(y_{ref}(t)=220\sin(2\pi 50t)\). The resulting waveforms were approximated as Fourier series to implement the simulation in PSIM software as follows:  
\[i_{1_{ref}} \simeq 0.99-4.99\sin\omega t-0.116\cos\omega t+0.068\sin 2 \omega t-1.0145\cos 2\omega t+0.199\sin 3\omega t+0.0056\cos 3\omega t\] \[i_{2_{ref}} \simeq 0.99+4.99\sin\omega t+0.116\cos\omega t+0.068\sin 2 \omega t-1.0145\cos 2\omega t-0.199\sin 3\omega t-0.0056\cos 3\omega t\] \[v_{C_{1_{ref}}} \simeq 318-110\sin\omega t-0.903\cos\omega t-0.622\sin 2\omega t-17. 387\cos 2\omega t+0.0015\sin 3\omega t-0.189\cos 3\omega t\] \[v_{C_{2_{ref}}} \simeq 318+110\sin\omega t-0.903\cos\omega t-0.622\sin 2\omega t-17. 387\cos 2\omega t-0.0015\sin 3\omega t+0.189\cos 3\omega t\]  
To determine the control, we solve the optimization problem (30). The solution is given by  
\[\bar{S}=\left[ \begin{array}{cccc}0.0054&0.0000&0.0000&0.0000\\ 0.0000&0.0001&0.0000&-0.0000\\ 0.0000&0.0000&0.0054&0.0000\\ 0.0000&-0.0000&0.0000&0.0001\end{array} \right],\ \ \bar{\mu}=1.39\times 10^{-7}\]  
and the control is given by  
\[\text{If }e^{T}\bar{S}A_{2}x>0,\ u_{1}=0,\text{ otherwise }u_{1}=1\text{ with }A_{2}=\left[ \begin{array}{cccc}0.&1/L&0&0\\ -1/C&0&0&0\\ 0&0&0&-1/L\\ 0&0&1/C&0\end{array} \right]\]  
Figure 7 shows the currents and the voltages \(i_{L_{i}}\), \(v_{C_{i}}\), \(i=1,2\), the output voltage and the duty cycle obtained by filtering the control \(u_{1}\). We can see that the proposed approximated reference trajectories and the resulting control allow us to obtain the expected result.