# MFC/IMC Control Algorithm for Reduction of Load Torque Disturbance in PMSM Servo Drive Systems
## 4 Theoretical Analysis
### Models of Classical and MFC/IMC Cascade Control System  
The PMSM linear model \(\mathbf{W}_{M}(s)\) and current controllers in \(d\)-axis \(R_{id}(s)\) and \(q\)-axis \(R_{id}(s)\) were coupled and considered as one object. It was motivated by the fact that the current controllers' parameter values are strictly dependent on the motor type; the users usually are not able to manipulate those quantities. In Figure 1, element \(\hat{\mathbf{W}}_{M}(s)\) illustrates a simplified FOC current control loop. The system has two input signals: \(i_{q\_ref}(s)\)--reference current value in \(q\)-axis in \(d\)-\(q\) reference frame, and \(\tau_{load}(s)\)--load torque. The output signals of this system are: \(i_{d\_act}(s)\)--actual current in \(d\)-axis in \(d\)-\(q\) reference frame, \(i_{q\_act}(s)\)--actual current in \(q\)-axis in \(d\)-\(q\) reference frame, and \(\omega_{act}(s)\)--actual velocity of the PMSM motor shaft. Voltages generated by the current controllers in \(d\) and \(q\) axis are indicated by \(u_{d}(s)\) and \(u_{q}(s)\). It is also noteworthy that the third input signal, namely \(i_{d\_ref}(s)\), is equal to zero which is a consequence of the FOC control scheme. The above system can be presented in the following form Equation (7):  
\[\left[ \begin{array}{c}i_{d\_act}(s)\\ i_{q\_act}(s)\\ \omega_{act}(s)\end{array} \right]=\mathbf{W}_{M}(s)\left[ \begin{array}{c}u_{ d\_act}(s)\\ u_{q\_act}(s)\\ \tau_{load}(s)\end{array} \right] \tag{7}\]  
where \(\left[i_{d\_act}(s)\), \(i_{q\_act}(s)\), \(\omega_{act}(s)\right]^{T}\) is the PMSM model (\(\textbf{W}_{M}(s)\)) output signals vector, \(\left[u_{d\_act}(s)\), \(u_{q\_act}(s)\right)\), \(\tau_{load}(s)\big{]}^{T}\) is the PMSM model (\(\textbf{W}_{M}(s)\)) input signals vector, and \(\textbf{W}_{M}(s)\) is described by a set of transfer functions presented in Equation (8):  
\[\textbf{W}_{M}(s)=\left[ \begin{array}{cccc}W_{M\_id\_ud}(s)&W_{M\_id\_uq}(s)&W _{M\_id\_\tau\_load}(s)\\ W_{M\_iq\_ud}(s)&W_{M\_iq\_uq}(s)&W_{M\_iq\_\tau\_load}(s)\\ W_{M\_\_\omega\_ud}(s)&W_{M\_\omega\_\_\mu q}(s)&W_{M\_\omega\_\_\tau\_load}(s) \end{array} \right] \tag{8}\]  
Remark: the signal names in subscripts denote the signals upon which the transfer function is constructed, e.g., \(W_{M\_id\_ud}(s)\) denotes the transfer function as a ratio of input signal \(u_{d}(s)\) and output signal \(i_{d\_act}(s)\). The coupled system consisting of PMSM linear model \(\textbf{W}_{M}(s)\) and current controllers in \(d\)-axis \(R_{id}(s)\) and \(q\)-axis \(R_{iq}(s)\) can be described as Equation (9):  
\[\left[ \begin{array}{c}i_{d\_act}(s)\\ i_{q\_act}(s)\\ \omega_{act}(s)\end{array} \right]=\hat{\textbf{W}}_{M}(s)\left[ \begin{array}[ ]{c}i_{q\_ref}(s)\\ \tau_{load}(s)\end{array} \right], \tag{9}\]  
where \(\left[i_{d\_act}(s)\), \(i_{q\_act}(s)\), \(\omega_{act}(s)\right]^{T}\) is the combined system model (\(\hat{\textbf{W}}_{M}(s)\)) output signals vector, \(\left[i_{q\_ref}(s)\), \(\tau_{load}(s)\right]^{T}\) is the combined system model (\(\hat{\textbf{W}}_{M}(s)\)) input signals vector and \(\hat{\textbf{W}}_{M}(s)\) is described by a set of transfer functions shown in Equation (10):  
\[\hat{\textbf{W}}_{M}(s)=\left[ \begin{array}{cc}W_{M\_id\_iq\_ref}(s)&W_{M \_id\_\tau\_load}(s)\\ W_{M\_iq\_iq\_ref}(s)&W_{M\_iq\_\tau\_load}(s)\\ W_{M\_\_\omega\_\_\mu\_ref}(s)&W_{M\_\omega\_\_\tau\_load}(s)\end{array} \right] \tag{10}\]  
The influence on the velocity control is determined by a pair of transfer functions: \(\omega_{act}(s)\) and \(i_{q\_ref}(s)\), along with \(\omega_{act}(s)\) and \(\tau_{load}(s)\); both pairs are derived from \(\hat{\textbf{W}}_{M}(s)\). The coupling of PMSM and current regulators reduces the velocity control loop structure to a basic closed-loop scheme as shown in Figure 2.  
Figure 1: FOC current control loop in PMSM.