# A Unified Approach for the Control of Power Electronics Converters. Part II: Tracking
## 6 Numerical Examples
### Full-Bridge Inverter for Stand-Alone Applications
#### 6.1.2. Sinusoidal Reference Tracking for Nonlinear Loads  
Now, the problem is to track a sinusoidal reference for the case in which the loads are nonlinear, illustrating the method proposed in Section 5.2. The output current is supposed to be measured and is considered a measurable perturbation. Now, the basic equations describing the full bridge are given by  
\[\left\{ \begin{array}{l}L\frac{di_{L}(t)}{dt}=-r_{L}\;i_{L}(t)-v_{C}(t)+u(t)\;V_ {in}\\ C\frac{dv_{C}(t)}{dt}=i_{L}(t)-i_{0}(t)\end{array} \right.\]  
Figure 2: Simulated waveforms of the full-bridge inverter feeding a resistive load. where \(w(t)=i_{0}(t)\) is the output current and the state space model writes  
\[\left\{ \begin{array}{l}\frac{dx(t)}{dt}=\underbrace{\left[\begin{array}{cc}- \frac{r_{L}}{L}&-\frac{1}{L}\\ \frac{1}{C}&0\end{array}\right]}_{A_{0}}x(t)+\left[\underbrace{\left[\begin{array} []{cc}-\frac{1}{L}\\ 0\end{array}\right]}_{B_{0}}+\underbrace{\left[\begin{array}{cc}\frac{2}{L} \\ 0\end{array}\right]}_{B_{n_{1}}}u_{1}(t)\right]V_{in}+\underbrace{\left[ \begin{array}{cc}0\\ -\frac{1}{C}\end{array}\right]}_{E_{0}}w(t)\\ y(t)=\underbrace{\left[\begin{array}{cc}0&1\end{array}\right]x(t)}_{C_{0}}x(t) \end{array} \right.\]  
The reference is the same \(y_{ref}(t)=V_{\max}\sin(\omega_{0}t)\). From the second state equation, the current reference can be deduced. It is expressed as  
\[i_{t_{ref}}(t)=\omega_{0}CV_{\max}\cos(\omega_{0}t)+i_{0}(t)\]  
and then the state reference is  
\[x_{ref}(t) = \left[ \begin{array}{c}i_{ref}(t)\\ v_{ref}(t)\end{array} \right]=\underbrace{\left[ \begin{array}{cc}\omega_{0} C&0\\ 0&1\end{array} \right]}_{\Pi_{1}}z(t)+\underbrace{\left[ \begin{array}{cc}1\\ 0\end{array} \right]}_{\Pi_{p}}w(t)\]  
The control is expressed as \(\arg\biggl{\{}\min_{d\in\Lambda_{S}}e^{T}(t)P[B_{0}(x(t))+E_{0}(w(t))]d\biggr{\}}\) with  
\[B_{0}(x)=\left[ \begin{array}{cc}-\frac{V_{in}}{L}&\frac{V_{in}}{L}\\ 0&0\end{array} \right]x,E_{0}(w)=\left[ \begin{array}{cc}0&0\\ -\frac{1}{C}&-\frac{1}{C}\end{array} \right]w\text{ and }P=\left[ \begin{array}{cc}1.828&0.041\\ 0.041&0.029\end{array} \right]\]  
An equivalent formulation is  
\[\left\{ \begin{array}{l}\text{If }e^{T}P\left[\begin{array}{cc}2V_{in}/L&0 \end{array}\right]^{T}>0\text{ then }u_{1}=0\\ \text{Otherwise }u_{1}=1\end{array} \right.\]  
A first nonlinear load is considered, which corresponds to a Thyristor-Based Rectifier (TBR). The output current in this case has a total harmonic distortion (THD) of approximately 66.34% (see the sinusoidal components used in PSIM simulation in Table 2). Figure 3 represents the output voltage and the inductor current for a numerical experiment in which, at the beginning, \(V_{in}\) is equal to 240 V; it then increases to 320 V at 0.02 s, decreases to 230 V at 0.05 s and returns to 240 V at 0.08 s. The control is robust, the output is not affected by the changes in the input voltage and its THD is lower than 0.04%. The associated duty cycle, also shown in Figure 3, shows how the control accommodates the duty cycle to tackle the changes in the input voltage and the nonlinearities of the load. A second nonlinear load corresponding to a compact fluorescent lamp (CFL) is also considered. The output current in this case has a THD of approximately 99.84% (see the sinusoidal components used in PSIM simulation in Table 3). The output voltage, inductor current, input voltage and the associated duty cycle are shown in Figure 4. The input voltage \(V_{in}\) changes as above. For this load, too, the control is efficient and robust, guaranteeing a THD for the output that is lower than 0.04%. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Frequency (Hz)** & **Amplitude (A)** & **Phase (Degrees)** \\ \hline
60 & 1.2864 & 0 \\
180 & 0.8911 & 180 \\
300 & 0.4690 & 26 \\
420 & 0.4623 & \(-\)105 \\
540 & 0.4623 & 90 \\
660 & 0.3484 & \(-\)69 \\
780 & 0.2613 & 143 \\
900 & 0.2010 & \(-\)8 \\
1020 & 0.1273 & \(-\)147 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the TBR-type nonlinear load.