# A Unified Approach for the Control of Power Electronics Converters. Part II: Tracking
## 6 Numerical Examples
### Full-Bridge Inverter for Stand-Alone Applications
#### 6.1.1 Sunsoidal Reference Tracking for a Resistive Load  
From a control point of view, the objective for such a converter operating in a stand-alone application is to enforce the output voltage \(v_{C}\) to track a sinusoidal reference of the form \(y_{ref}=V_{\max}\sin(\omega_{0}t)\). As in the case of PWM inverters, it is considered that the switches of the bridge can be commutated by using bipolar or unipolar strategies [43]. Here, only the case of a bipolar strategy is considered. In this case, the switches of each leg operate complementarily. The high side switch of a leg is off while the low side switch is on, and the high side switch is on when the low side switch is off, leading to a two-level commutation. If we define the control variable \(u\) such that \(u=1\) when \(S_{1}\) and \(S_{4}\) are on, and \(u=-1\) when \(S_{2}\) and \(S_{3}\) are on, the model can be written as follows:  
\[\left\{ \begin{array}{l}L\dfrac{di_{L}(t)}{dt}=-r_{L}\;i_{L}(t)-v_{C}(t)+u(t) \;V_{in}\\ C\dfrac{dv_{C}(t)}{dt}=i_{L}(t)-\dfrac{v_{C}}{R}(t)\end{array} \right.\]  
Remark that \(u(t)=1\) or \(-1\). To have a control \(u_{1}(t)=0\) or \(1\), we consider the change in variable \(u(t)=2u_{1}(t)-1\), and defining the state vector \(x=\left[ \begin{array}{cc}i_{L}&v_{C}\end{array} \right]^{T}\), the model of the full bridge can be translated into the canonical form (1). \[\left\{ \begin{array}{l}\dfrac{dx(t)}{dt}=\underbrace{\left[\begin{array}{ cc}-\dfrac{r_{L}}{L}&-\dfrac{1}{L}\\ \dfrac{1}{C}&-\dfrac{1}{RC}\end{array}\right]}_{A_{0}}x(t)+\left[\underbrace{ \left[\begin{array}{cc}-\dfrac{1}{L}\\ 0\end{array}\right]}_{B_{0}}+\underbrace{\left[\begin{array}{cc}\dfrac{2}{L} \\ 0\end{array}\right]}_{B_{n_{1}}}u_{1}(t)\right]V_{in}\\ y(t)=\underbrace{\left[\begin{array}{cc}0&1\\ \underbrace{C_{0}}_{C_{0}}\end{array}\right]}_{A}x(t)\end{array} \right.\]  
Matrices associated with the previous model are defined by  
\[A_{1}=A_{2}=0,\;\;\;B_{1}=\left[ \begin{array}{c}-\frac{1}{L}\\ 0\end{array} \right],\;\;B_{2}=\left[ \begin{array}{c}\frac{1}{L}\\ 0\end{array} \right],\;C_{1}=C_{2}=0\]  
Figure 1: H-bridge inverter circuit. We consider the full-bridge converter whose parameter values are defined in Table 1.  
The reference \(y_{ref}\) can be generated from a second-order exosystem whose state space model is  
\[\frac{dz(t)}{dt}=\left[ \begin{array}{cc}0&-\omega_{0}\\ \omega_{0}&0\end{array} \right]z(t)=\Theta(z)(t),\quad z(0)=\left[ \begin{array}[ ]{c}V_{\max}\\ 0\end{array} \right]\]  
with \(z\in\mathbb{R}^{2}\). It is possible to express the state reference signal as  
\[x_{ref}(t) = \left[ \begin{array}{c}i_{ref}(t)\\ v_{ref}(t)\end{array} \right]=\underbrace{\left[ \begin{array}{cc}\omega_{0} C&\frac{1}{R}\\ 0&1\end{array} \right]}_{\Pi_{1}}z(t)\]  
The control is expressed as  
\[\arg\left\{\min_{d\in A_{S}}e^{T}(t)PB_{0}(x(t))d\right\}\text{ with }B_{0}(x)=\left[ \begin{array}{cc}\frac{-V_{in}}{L}&\frac{V_{in}}{L}\\ 0&0\end{array} \right]\text{and }P=\left[ \begin{array}{cc}1.727&0.033\\ 0.033&0.033\end{array} \right]\]  
Then, an equivalent formulation is  
\[\left\{ \begin{array}{ll}\text{If }e^{T}P\left[\begin{array}{cc}2V_{in}/L&0 \end{array}\right]^{T}>0\text{ then }u_{1}=0\\ \text{Otherwise }u_{1}=1\end{array} \right.\]  
To test the potential and the robustness of the proposed approach, we consider the case in which the input \(V_{in}\) is equal to 240 V and changes at 0.02 s to 320 V, changing after to 230 V at 0.05 s before returning to 240 V at 0.08 s. Figure 2 shows the output voltage, inductor current, input voltage and the associated duty cycle. As can be observed, the output tracks the reference and the control is able to compensate the changes in the input voltage \(V_{in}\) by accommodating, in an appropriate way, the duty cycle.