## 5 Hierarchical Control Development
### Voltage Regulation Controller  
A control linear technique called state feedback is selected for local controllers design. Evaluating model boost converter obtained in (1) as a static model, i.e., with all the variables in steady state, the conclusion is the boost converter is a nonlinear system. It is necessary to obtain a linearized model in a fixed operation point. First step to linearization is to obtain matrix **A** and **B** through Jacobian for the average model evaluated in a fixed operation point. For matrix **A**: \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Version** & \(V_{ac}\) & \(V_{DC}\) & \(R_{o}\) & \(L\) & \(r\) & \(C\) \\
**(kW)** & **(\(V_{RMS}\))** & **(V)** & **(f)** & **(mH)** & **(f)** & **(mF)** \\ \hline
3.3 & 220 & 380 & 43.76 & 7 & 0.27 & 7200 \\
1.4 & 127 & 190 & 43.76 & 4.1 & 0.2 & 4576 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Parameter values design for single phase DC–AC full bridge converter. \[\mathbf{A}= \begin{vmatrix}\frac{\partial f_{1}}{\partial v_{\text{C1}}}&\frac{ \partial f_{1}}{\partial i_{L}}&\frac{\partial f_{1}}{\partial v_{\text{C2}}} \\ \end{vmatrix}  \tag{5}\]  
For matrix \(\mathbf{B}\):  
\[\mathbf{B}= \begin{vmatrix}\frac{\partial f_{1}}{\partial u}\\ \frac{\partial f_{2}}{\partial u}&u=D\\ \frac{\partial f_{3}}{\partial u}&i_{L}=I_{L}\\ \frac{\partial f_{3}}{\partial u}&v_{\text{C2}}=V_{\text{C2}}\end{vmatrix}  \tag{6}\]  
Particular structure of matrix \(\mathbf{A}\) in linearized model obtained is:  
\[\mathbf{A}= \begin{vmatrix}\frac{-1}{R_{t}\text{C}_{1}}&\frac{-1}{\text{C}_{1} }&0\\ \frac{1}{L}&\frac{[(R_{D1}-R_{D2})D+R_{D2}]}{L}&\frac{-(1-D)}{L}\\ 0&\frac{(1-D)}{\text{C}_{2}}&\frac{-1}{\text{C}_{2}R_{\text{O}}}\end{vmatrix}  \tag{7}\]  
and for matrix \(\mathbf{B}\)  
\[\mathbf{B}= \begin{vmatrix}0\\ -\frac{[(R_{D1}-R_{D2})D+R_{D2}]i_{L}}{L}+\frac{V_{\text{C2}}}{L}\\ \frac{-I_{L}}{\text{C}_{2}}\end{vmatrix}  \tag{8}\]  
Now, matrix \(\mathbf{A}\) and \(\mathbf{B}\) are incorporated to linealized state equation:  
\[\widehat{x}=\mathbf{A}\widehat{x}+\mathbf{B}\widehat{u}\]  
with:  
\[\widehat{x}= \begin{vmatrix}\frac{\partial\widehat{v}_{\text{C1}}}{\partial t} \\ \frac{\partial\widehat{u}_{L}}{\partial t}\\ \frac{\partial\widehat{v}_{\text{C2}}}{\partial t}\end{vmatrix} \]  
and:  
\[\widehat{x}= \begin{vmatrix}\widehat{v}_{\text{C1}}\\ \widehat{i}_{L}\\ \widehat{v}_{\text{C2}}\end{vmatrix} \]  
Input vector \(\widehat{u}\), in this case, has a single element \(u\), it corresponds to modulating signal. The output equation is given by:\[\widehat{y}=\mathbf{C}\widehat{x}+\mathbf{D}\widehat{u}\]  
For this application, direct transfer matrix has not elements \(\mathbf{D}=0\), and matrix \(\mathbf{C}\) is defined as:  
\[\mathbf{C}= \begin{bmatrix}0&0&1\end{bmatrix} \]  
State feedback controller structure for boost converter is shown in Figure 6.  
Control scheme includes a gain-integrator trajectory to lead steady state error to zero and gain trajectories to feedback system state. The first step is to build up the augmented system, and the controllability matrix is required. For the second step, it is necessary to calculate gains values; the Ackermann method is applied to define gain values \(K_{I}\), \(K_{1}\), \(K_{2}\) and \(K_{3}\) as elements of the vector:  
\[\mathbf{K_{a}}= \begin{bmatrix}K_{1}&K_{2}&K_{3}&K_{I}\end{bmatrix} \]  
Parameters such as overshoot percentage, %\(OS\), and settle time, \(T_{s}\), are considered to calculate the desired polinomy. Two auxiliar poles are necessary to balance system order and the gain vector \(\mathbf{K_{a}}\). The product of desired polynomial evaluated in terms of the augmented system and controllability matrix brings out specific values for gains in matrix \(\mathbf{K_{a}}\).