## 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 Variable Flux  
The software implementation carried out in Matlab/Simulink for the PMSM-SCS is based on the variable-flux DTC strategy. The Matlab/Simulink implementation presented is based on the SMC- and FOSMC-type controller description equations presented in Section 2.2. For the numerical simulation scenario performed in Matlab/Simulink using the SMC-type controller described in Section 2, the parameters in Equations (34)-(43) are as follows: \(\varepsilon\) = 300, \(q\) = 200, and \(c\) = 100. For the FOSMC-type controller described in Section 2, Equations (44)-(52), the parameters are as follows: \(k_{p}\) = 100, \(k_{d}\) = 1, and \(\mu\) = 0.55. Based the SMC-type controller for the PMSM-SCS with the DTC strategy at variable flux, the evolution of the parameters is shown in Figure 12. The response time is \(tr\) = 2.3 ms and the settling time is \(ts\) = 2.3 ms. There is an 80% reduction in response time and an approximate 96% reduction in settling time, compared to the original PI controller performance. Furthermore, using the FOSMC-type controller for the PMSM-SCS with the DTC strategy at variable flux, the evolution of the parameters is shown in Figure 13. The response time is \(tr\) = 2.2 ms and the settling time is \(ts\) = 2.2 ms. There is an 82% reduction in response time and an approximate 97% reduction in settling time compared to the initial performance of the PI controller. Figure 11: Evolution of speed comparison for PMSM-SCS based on DTC strategy using PI, PI-EOA, FOPI, TID, and FO-Lead-Lag speed controllers at constant flux. Figure 12: Parameter evolution of the PMSM-SCS based on DTC strategy using SMC-type controller at variable flux. In terms of parametric robustness, Figure 14 shows the parameter evolution of the PMSM-SCS using the DTC strategy with the FOSMC variable-flux controller, where the \(J\) parameter increases by 50% compared to the nominal value. Good dynamic performance is also observed in this case, in the sense that the overshoot is negligible and the steady-state error is less than 0.1%, just as in the case of the nominal value of the \(J\) parameter. Obviously, given the importance of the \(J\) parameter, which represents the combined inertia of the PMSM rotor and load, the response time increases to 4 ms and the stator currents increase by approximately 20%. The flux space-vector trajectory for the PMSM-SCS based on the DTC strategy using the FOSMC in the case of the variable-flux controller is shown in Figure 15a, and the evolution of the flux reference for the FOSMC-type controller is shown in Figure 15b. The comparison of the evolution of the PMSM rotor speed for the sensorless control system based on the DTC strategy using SMC and FOSMC variable-flux controllers is shown in Figure 16. Note that this comparison is made between the SMC- and FOSMC-type controllers and the FO-Lead-Lag controllers, which have the best performance of the  
Figure 14: Parameter evolution of the PMSM-SCS based on DTC strategy using FOSMC-type controller at variable flux—\(J\) parameter increased by 50%. Figure 13: Parameter evolution of the PMSM-SCS based on DTC strategy using FOSMC-type controller at variable flux. group of controllers presented in the previous section. Thus, the best performance of the PMSM-SCS, based on the DTC strategy, is achieved using the FOSMC-type controller. Description of the Fractal Dimension and Calculus for the Speed Signal Controlled by the Proposed Controllers Used in PMSM-SCS, Based on DTC Strategy  
For comparison of the performance of the PMSM-SCS using the DTC strategy, in the case of all the controllers presented in the previous sections, the FD of the signal controlled by the control system, in this case the PMSM speed, is proposed as a performance indicator. According to [28], an FD value of the controlled signal is higher if the control system provides better performance. One of the most widely used methods for calculating the FD of a one-dimensional signal is the box-counting method [28].