## 2 2 DOF IMC for SRM
### Design of the 2 DOF IMC Current Regulator  
In this subsection, according to the characteristics of SRMs, a 2 DOF IMC controller is designed. Then, the stability analysis and parameter tuning of the controller are given. Figure 1 depicts the block diagram of the 2 DOF IMC based on vector control. The 2 DOF internal model control is used for the current loop and the traditional PI controller is adopted for the speed loop. In Figure 2, the structure of the 2 DOF IMC is displayed. The 2 DOF IMC can be regarded as a special feedback structure consisting of a set-point controller \(Q_{1}(s)\), disturbance rejection controller \(Q_{2}(s)\), plant \(G(s)\) and process model \(G_{n}(s)\). In the figure, the \(R(s)\), \(U(s)\), \(Y(s)\), \(d(s)\) are the reference input, process output, system output and external interference signal, respectively. The input and output relationship of a 2 DOF IMC can be described as:  
\[Y(s) =\frac{R(s)G(s)Q_{1}(s)+d(s)(1-Q_{2}(s)G_{n}(s))}{1+Q_{2}(s)(G(s)- G_{n}(s))} \tag{7}\]  
If the corresponding process model \(G_{n}(s)\) is perfect, i.e., \(G(s)=G_{n}(s)\), then  
\[Y(s)=R(s)Q_{1}(s)G_{n}(s)+d(s)(1-Q_{2}(s)G_{n}(s)) \tag{8}\]  
Figure 1: Structure diagram of SRM control system based on the 2 DOF IMC. Figure 2: Principle block diagram of the 2 DOF IMC. From (8), the set-point tracking and disturbance rejection are decoupled and independently adjusted by the \(Q_{1}(s)\), \(Q_{2}(s)\), respectively. Based on the above assumptions, if \(Q_{1}(s)=Q_{2}(s)=G_{n}^{-1}(s)\) can be realized, then \(Y(s)=R(s)\) can be obtained. It means that the system is characterized by strong robustness. However, in practical application, this is difficult to achieve. Hence, in this paper, the controllers \(Q_{1}(s)\) and \(Q_{2}(s)\) are supplemented with low-pass filters \(f_{1}(s)\), \(f_{2}(s)\), which can be further expressed as:  
\[\left\{ \begin{array}{l}Q_{1}(s)=G_{n}^{-1}(s)f_{1}(s)\\ Q_{2}(s)=G_{n}^{-1}(s)f_{2}(s)\end{array} \right. \tag{9}\]  
where the \(f_{1}(s)\) and \(f_{2}(s)\) are designed as:  
\[\left\{ \begin{array}{l}f_{1}(s)=\frac{1}{1+\lambda_{1}s}\\ f_{2}(s)=\frac{1}{1+\lambda_{2}s}\end{array} \right. \tag{10}\]  
where the filter parameters \(\lambda_{1}\) and \(\lambda_{2}\) should satisfy \(\lambda_{1}>0\) and \(\lambda_{2}>0\), respectively. By simplifying the structure in Figure 2, the 2 DOF IMC can be equivalent to the traditional feedback structure, shown in Figure 3, and the corresponding IMC controllers \(T(s)\) and \(F(s)\) can be developed as:  
\[\left\{ \begin{array}{l}T(s)=\frac{Q_{1}(s)}{Q_{2}(s)}=\frac{1+ \lambda_{2}s}{1+\lambda_{1}s}\\ F(s)=\frac{Q_{2}}{1-Q_{2}G_{n}(s)}\end{array} \right. \tag{11}\]  
Based on the above design process, a 2 DOF IMC controller for the SRM is constructed. To simplify the design of the controller, the average model is chosen as process model \(G_{n}\), as illustrated in the following equation. \[ \begin{bmatrix}u_{d}\\ u_{q}\\ u_{0}\end{bmatrix} = R \begin{bmatrix}i_{d}\\ i_{q}\\ i_{0}\end{bmatrix} +\omega_{e} \begin{bmatrix}0&-L_{dc}&0\\ L_{dc}&0&\frac{\sqrt{2}}{2}L_{ac}\\ 0&0&0\end{bmatrix}\begin{bmatrix}i_{d}\\ i_{q}\\ i_{0}\end{bmatrix} + \begin{bmatrix}L_{dc}&0&\frac{\sqrt{2}}{2}L_{ac}\\ 0&L_{dc}&0\\ \frac{\sqrt{2}}{2}L_{ac}&0&L_{dc}\end{bmatrix} p \begin{bmatrix}i_{d}\\ i_{q}\\ i_{0}\end{bmatrix} \end{bmatrix} \tag{12}\]  
For the current loop system, the SRM can be described as a three-input and three-output system. According to (12) and using the Laplace transform, the system can be equivalent to:  
\[Y(s)=G(s)U(s) \tag{13}\]  
where:  
\[ \begin{array}{l}U(s)\!=\!\begin{bmatrix}u_{d}(s)&u_{q}(s)&u_{0}(s)\end{bmatrix} ^{T}\\ Y(s)=[i_{d}(s)&i_{q}(s)&i_{0}(s)]^{T}\\ G_{n}(s)=\begin{bmatrix}R+L_{dc}s&-L_{dc}\omega e&\frac{\sqrt{2}}{2}L_{ac}s\\ L_{dc}\omega e&R+L_{dc}&\frac{\sqrt{2}}{2}L_{ac}\omega e\\ \frac{\sqrt{2}}{2}L_{ac}s&0&R+L_{dc}s\end{bmatrix}^{-1}.\end{array}  \tag{14}\]  
Figure 3: Simplified block diagram of the 2 DOF-IMC. Substituting (13) into (11), the controllers \(T(s)\) and \(F(s)\) for the SRM system can be deduced.