# Adaptive Feedback Control for Four-Phase Interleaved Boost Converter Used with PEM Fuel Cell +
## 3 Controller Design
### Proposed Control Design
#### 3.2.1 Current Loop Gains Synthesis  
The state representation of the current loop (inner loop) can be expressed as follows:  
\[\left\{\begin{array}{c}\dot{x}_{i}(t)=A_{i}x_{i}+B_{i}u(t)\\ y_{i}=C_{i}x_{i}(t)\end{array}\right. \tag{15}\]  
Based on Equation (13), the transfer function \(F_{iu}\) can be expressed in the following form:  
\[F_{iu}=\frac{m_{1}s^{-1}+m_{0}s^{-2}}{1+n_{1}s^{-1}+n_{0}s^{-2}} \tag{16}\]  
where \(A_{i}\), \(B_{i}\), and \(C_{i}\) are the state matrices. The state vector can be written as  
\[x_{i}(t)=\begin{bmatrix}x_{1i}&x_{2i}\end{bmatrix}^{T} \tag{17}\]  
The matrices \(A_{i}\), \(B_{i}\), and \(C_{i}\) can be given as  
\[A_{i}=\begin{bmatrix}0&1\\ -n_{0}&-n_{1}\end{bmatrix} \tag{18}\]  
\[B_{i}=\begin{bmatrix}0\\ 1\end{bmatrix} \tag{19}\]\[C_{i}=\begin{bmatrix}m_{0}&m_{1}\end{bmatrix} \tag{20}\]  
The objective of current feedback control is to maintain the force-measuring inductor current \(i_{k}\) (\(k=1,\dots,4\)) in accordance with the reference current \(I_{ref}\) generated by the voltage controller. The state-feedback control law for the current loop can be formulated as  
\[u(t)=-K_{i}x_{i}(t)-M_{i}e_{i}(t)+N_{i}I_{ref} \tag{21}\]  
where \(I_{ref}\) represents the reference current, \(e_{i}=\int(y_{i}(t)-I_{ref})dt\) denotes the current error, and \(K_{i}\), \(M_{i}\), and \(N_{i}\) are the respective controller parameters designed for the system.  
To synthesize the controller gains, the superposition theorem is applied to derive the gains individually. The resulting control law \(u(t)\) can be expressed in the following manner:  
\[u(t)=u_{1}(t)+u_{2}(t) \tag{22}\]  
where  
\[u_{1}(t)=-K_{i}x_{i}(t)-M_{i}e_{i}(t) \tag{23}\]  
and  
\[u_{2}(t)=N_{i}I_{ref} \tag{24}\]  
Step 1: \(u_{1}\) controller synthesis  
According to Equation (15), the closed-loop system can be written as follows:  
\[\begin{bmatrix}\dot{x}_{i}\\ \dot{e}_{i}\end{bmatrix}=\begin{bmatrix}A_{i}-B_{i}K_{i}&B_{i}M_{i}\\ C_{i}&0\end{bmatrix}\begin{bmatrix}\dot{x}_{i}\\ \dot{e}_{i}\end{bmatrix}-\begin{bmatrix}0\\ 1\end{bmatrix}I_{ref} \tag{25}\]  
The next goal is to determine the gains \(K_{i}\) and \(M_{i}\) to enhance system stability, ensuring that, for any initial state \(\tilde{\xi}_{i}(0)\), the control law \(u_{1}\) minimizes the performance index  
\[J_{i}=\int_{0}^{\infty}\tilde{\xi}_{i}^{T}Q\tilde{\xi}_{i}+u_{1}^{T}Ru_{1} \tag{26}\]  
where \(\tilde{\xi}_{i}=\begin{bmatrix}x_{i}\\ \dot{e}_{i}\end{bmatrix}\). In this scenario, both \(Q\) and \(R\) are positive values. These matrices are represented as diagonal matrices, with their elements carefully selected to reduce tracking errors to a minimum.  
Step 2: \(u_{2}\) controller synthesis  
Given that the steady-state static error is zero, expressed as \(e_{i}(t\longrightarrow\infty)=y_{i}(t)-I_{ref}=0\), the closed-loop system can be formulated as follows:  
\[\dot{x}_{i}(t)=(A_{i}-B_{i}K_{i})x_{i}+B_{i}N_{i}I_{ref} \tag{27}\]  
The closed-loop transfer function can be given by  
\[F_{iuc_{CL}}=C_{i}(s-A_{i}+B_{i}K_{i})^{-1}+B_{i}N_{i} \tag{28}\]  
In steady state, it is necessary that \(F_{iuc_{CL}}(0)=1\); thus,  
\[N_{i}=\frac{1}{C_{i}(-A_{i}+B_{i}K_{i})^{-1}B_{i}} \tag{29}\]