# Adaptive Feedback Control for Four-Phase Interleaved Boost Converter Used with PEM Fuel Cell +
## 3 Controller Design
### Proposed Control Design
#### 3.2.2 Voltage Loop Gains Synthesis  
Similar to the process for the current loop, the initial stage in determining the gains of the voltage controller involves converting the open-loop transfer function of the voltage loop into a state representation. This transfer function is the product of the closed-loop  
The state matrix in Equation (31) can be expressed as follows:  
\[A_{v}=\begin{bmatrix}0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&1&0\\ -n_{0}&-n_{1}&-n_{2}&-n_{3}&-n_{4}&-n_{5}\end{bmatrix} \tag{41}\]  
\[B_{v}=\begin{bmatrix}0\\ 0\\ 0\\ 0\\ 0\\ 1\end{bmatrix} \tag{42}\]  
\[C_{v}=\begin{bmatrix}m_{0}&m_{1}&m_{2}&m_{3}&m_{4}&m_{5}\end{bmatrix} \tag{43}\]  
The approach employed to derive gains for the current loop is also applied to determine gains for the voltage control. The state-feedback control law for the current loop can be formulated as  
\[I_{ref}=-K_{v}x_{v}(t)-M_{v}e_{v}(t)+N_{v}V_{ref} \tag{44}\]  
where \(V_{ref}\) represents the reference voltage, \(e_{v}=\int(y_{v}(t)-V_{ref})dt\) denotes the voltage error, and \(K_{v}\), \(M_{v}\), and \(N_{v}\) represent the designed controller parameters, respectively.  
The control law for \(I_{ref}\) can be expressed as  
\[I_{ref}=I_{ref_{1}}+I_{ref_{2}} \tag{45}\]  
where  
\[I_{ref_{1}}=-K_{v}x_{v}(t)-M_{v}e_{v}(t) \tag{46}\]  
and  
\[I_{ref_{2}}=N_{v}V_{ref} \tag{47}\]  
_Step 1: \(I_{ref_{1}}\) controller synthesis_  
According to Equation (31), the augmented system can be written as  
\[\begin{bmatrix}\dot{x}_{v}\\ \dot{e}_{v}\end{bmatrix}=\begin{bmatrix}A_{v}-B_{v}K_{v}&-B_{v}M_{v}\\ C_{v}&0\end{bmatrix}\begin{bmatrix}x_{v}\\ e_{v}\end{bmatrix}-\begin{bmatrix}0\\ 1\end{bmatrix}V_{ref} \tag{48}\]  
The gains \(K_{v}\) and \(M_{v}\) are determined based on the control law for \(I_{ref}\), aiming to minimize the performance index  
\[J_{v}=\int_{0}^{\infty}\xi_{v}^{T}Q\xi_{v}+I_{ref_{1}}^{T}RI_{ref_{1}} \tag{49}\]  
where \(Q>0\), \(R>0\), and \(\xi_{v}=\begin{bmatrix}x_{v}\\ e_{v}\end{bmatrix}\).  
_Step 2: \(I_{ref_{2}}\) controller synthesis_  
Assuming steady-state static error is zero \(e_{v}(t\longrightarrow\infty)=y_{v}(t)-V_{ref}=0\). The closed-loop system can therefore be written as follows:  
\[\dot{x}_{v}(t)=(A_{v}-B_{v}K_{v})x_{v}+B_{v}N_{v}V_{ref} \tag{50}\]  
The closed-loop transfer function between the reference voltage and the output voltage can be expressed as follows:  
\[F_{vV_{ref_{1}}}=C_{v}(s-A_{v}+B_{v}K_{v})^{-1}+B_{v}N_{v} \tag{51}\]In the steady state, it is essential that \(F_{vV_{ref}/CL}(0)=1\); thus,  
\[N_{v}=\frac{1}{C_{v}(-A_{v}+B_{v}K_{v})^{-1}B_{v}} \tag{52}\]  
With the system parameters outlined in Table 1, we can derive the parameters for both controllers. The current and voltage gains, denoted as \(N_{i}\), \(M_{i}\), \(K_{i}\), \(N_{v}\), \(M_{v}\), and \(K_{v}\), respectively, are as follows:  
\[\left\{\begin{array}{c}N_{i}=0.0398\\ M_{i}=31.6228\\ K_{i}=\begin{bmatrix}-109060&1700\end{bmatrix}\end{array}\right. \tag{53}\]  
\[\left\{\begin{array}{c}N_{v}=0.0746\\ M_{v}=31.6228\\ K_{v}=\begin{bmatrix}1.9245&0.142&10.0027&0&0&0\end{bmatrix}\end{array}\right. \tag{54}\]