## 3 Numerical Simulations Based on Matlab/Simulink for the Proposed PMSM-SCS Using DTC Strategy
### Numerical Simulation for PMSM-SCS Based on DTC Strategy, Using Constant Flux
Based on the second FO-type controller of the PMSM-SCS based on the constant-flux DTC strategy, i.e., the TID controller described in Equation (33), whose tuning parameters are \(K_{t}\) = 0.035, \(K_{i}\) = 0.31, \(n\) = 10 and \(K_{d}\) = 0, the following transfer function is obtained:  
\[H_{TID}(s)=\frac{0.035^{0.9}+0.31}{s} \tag{55}\]  
For the FO-Lead-Lag speed controller shown in Equations (19) and (20), the following transfer function is obtained for the tuning parameters \(k^{\prime}\) = 300, \(x\) = 50, \(\lambda\) = 1.4, and \(\alpha\) = 0.11:  
\[H_{\text{FO-Lead-Lag}}(s)=\frac{3.606\cdot 10^{8}}{s^{2.1}+73730\ \text{s}^{1.1}+4.7 \cdot 10^{6}} \tag{56}\]  
For real-time implementation in an embedded system, it is necessary to achieve the equivalent transfer function of integer order of the transfer function of the FO-type speed controllers given in Equations (53), (55), and (56) in fractional form. For this, Appendix B presents the procedure to obtain these equivalent integer-order transfer functions using Oustaloup recursive filters [15]. Based on the nominal PMSM parameters in the case of the sensorless control system based on the DTC strategy using PI, PI-EOA, FOPI, TID, and FO-Lead-Lag speed controllers at constant flux, the numerical simulations are presented below. Plots of the following parameters of interest are presented for each sensorless control system based, on the DTC strategy using speed controllers: reference speed and estimated PMSM rotor speed, load and electromagnetic torque, stator currents and fluxes in the \(\alpha\)-\(\beta\) stationary reference frame. The load torque reference \(T_{L}\) is of 1 Nm and the speed reference signal \(\omega_{ref}\) used is expressed as a sequence of steps: [400 600 900] rpm\(\rightarrow\) [0 0.1 0.2] s. For the speed, electromagnetic torque and flux estimation observed are presented in Section 2.1.2; the tuning parameters are \(k_{1}\) = 150, \(k_{2}\) = 50, and \(K_{ISMO}\) = \(K_{ZSMO}\) = 0.1. Figure 6 shows the parameter evolution of the PMSM-SCS using the DTC strategy with PI speed controller at constant flux. Note that the response time is \(tr\) = 12 ms and the settling time is \(ts\) = 72 ms. The parameter evolution of the PMSM-SCS using the DTC strategy when using the PI-EOA speed controller is shown in Figure 7. In this case, the response time is \(tr\) = 9.2 ms and the settling time is \(ts\) = 30 ms. There is a 25% reduction in response time and a reduction in settling time of around 60%. Figure 6: Parameter evolution of the PMSM-SCS based on DTC strategy using PI controller at constant flux. Using the FOPI speed controller for the PMSM-SCS with DTC strategy at constant flux, the evolution of the parameters is shown in Figure 8. This gives a response time of \(tr=7.9\) ms and a settling time of \(ts=15\) ms. There is a 34% reduction in response time and a 79% reduction in settling time. Using the TID speed controller for the PMSM-SCS with DTC strategy at constant flux, the evolution of the parameters is shown in \[H_{FOPI}(s)=\frac{6.2s^{1.1}+24}{s^{1.1}} \tag{53}\]0 A response time of \(tr=5.9\) ms and a settling time of \(ts=14.5\) ms are obtained. There is a 50% reduction in response time and an 80% reduction in settling time. Using the FO-Lead-Lag speed controller for the PMSM-SCS with DTC strategy at constant flux, the evolution of the parameters is shown in \[H_{FOPI}(s)=\frac{6.2s^{1.1}+24}{s^{1.1}} \tag{53}\]1 This gives a response time of \(tr=3.2\) ms and a settling time of \(ts=9.1\) ms. There is a 73% reduction in response time and an approximate 87% reduction in settling time. \[H_{FOPI}(s)=\frac{6.2s^{1.1}+24}{s^{1.1}} \tag{53}\]2 Parameter evolution of the PMSM-SCS based on DTC strategy using FOPI controller at constant flux. \[H_{FOPI}(s)=\frac{6.2s^{1.1}+24}{s^{1.1}} \tag{53}\]3 Parameter evolution of the PMSM-SCS based on DTC strategy using PI-EOA controller at constant flux.