## 5 Results
### Model Precision  
This section shows how the different models proposed produce, in fact, very good approximations of the machine flux both at slow and high speeds. The CNT model (Continuous) is used again as the reference output and the others are compared to it. Figure 11 presents, as an example, the outputs of all the models for the machine with rotor turning at the same angular velocity of 6 \(\frac{\text{rad}}{\text{s}}\) as the voltage input: the continuous time (CNT), the discrete time with Simulink integrator (SIMULINK-DSCR), the discrete time in natural reference frame (NAT-DSCR), with fast inverse transformation (INV-DSCR), in rotor reference frame (ROT-DSCR) and with fast transformation (FAST-DSCR). The last 4 are all conducted with just 1 sub-interval integration. Overall, it is easy to see how all the models correctly solve the flux equation and give macroscopically equal results. Figure 11b shows a magnified view of the difference between the CNT solution and the proposed methods. These errors may impact the whole control system since they are predictions used by almost all other functions of the motor control algorithm afterwards and their relative error increases with the flux frequency, thus making it much more problematic at high speeds. As stated before, in the real control system a current error  
Figure 10: Execution time comparison increasing the discrete sub-interval integration number. CNT and SIMULINK-DSCR do not depend on the number of sub-intervals. Time is normalized to the CNT execution time. feedback is necessary to compensate for the small integration errors, but it is not described in this paper. Detailed description and error evaluation is going to be analyzed only for the FAST-DSCR algorithm, given that it has been verified in Section 5.1 that it is the simplest and fastest solver among those proposed. Moreover, since comparing a continuous signal with a discrete one would produce a discontinuous error value, the difference is shown with a very short mean filter (on 2 data points) to better visualize the results. Since the proposed solution is also a prediction of the flux at the next iteration step \(\varphi_{k+1}\), the error is calculated between the current algorithm iteration value and the value at the next step of the CNT model, considered to be the true approximation. This is carried out also for the SIMULINK-DSCR which is not predictive, thus its error is intrinsically larger. However, this decision reflects the ultimate use of the algorithms proposed, which is the best flux estimator for high performance motor control that must implement predictive calculations. Figure 12 shows the difference between the CNT and FAST-DSCR algorithms as percentage of the maximum value of flux. The error of the SIMULINK-DSCR algorithm is added as a reference. The four plots are for the four fluxes estimated by the model which are the stator and rotor flux in the \(d\) and \(q\) axis. In this example the rotational speed of the stator field and the electrical angular speed of the rotor are the same and equal to 6 \(\frac{\text{rad}}{\text{s}}\), therefore slow compared to common values. Figure 11: Simulink flux machine model comparison example between the model presented: CNT, SIMULINK-DSCR, NAT-DSCR, INV-DSCR, ROT-DSCR and FAST-DSCR. (**a**) overview, (**b**) zoom to highlight differences. The proposed solver is expected to have better accuracy than SIMULINK-DSCR the faster the motor rotation. Therefore Figure 13 shows the error percentage when the stator field rotates at 6200 \(\frac{\text{rad}}{\text{s}}\) and the rotor has a mechanical angular velocity of 5700 \(\frac{\text{rad}}{\text{s}}\), modelling a condition of 8.1% slip close to the nominal maximum mechanical speed of 15,000 rpm. At this high speed, a clear improvement of the model precision can be seen over the standard integration of SIMULINK-DSCR. To quantify the exact level of improvement, Table 2 shows the mean error squared of the percentage error for the different fluxes and integration methods at both low and high speed.