**4.3. Simulation results (regulation)**

In this section the simulated behavior of IM control under the FOC-FOPI scheme, when Test 1 is applied, is shown in Figs. 18 to 21, for different values of the integration order of the integral part of PI controller ( = 0.7, = 1.0, = 1.7 and = 2.0). Note that the classical FOC-PI scheme is obtained from the FOC-FOPI strategy by setting = 1. In all the figures, only the controlled variable (motor speed) is shown. The results obtained are quite satisfactory, as can be seen from Figures 18 to 21. The evolution of the controlled variable tries to follow the speed reference at all times, in spite of the perturbation being applied.

From Figure 18 it can be observed that for integration orders less than 1.0 the response presents no overshoot although the response is slower. It can be concluded, from information contained in Figure 20, that the response is faster when the integration order is greater than 1.0 but an overshoot is observed. When is chosen as 2.0 critically stable behavior is attained, as shown in Figure 21

**Figure 18.** Simulation results for ߥ ൌ ͲǤͲ

**Figure 17.** Description of Test 1 (Regulation)

50 100 150

1

2

Resistive torque [Nm]

3

Speed reference [rad/s]

**Table 1.** Variation of load torque

**4.3. Simulation results (regulation)** 

behavior is attained, as shown in Figure 21

simulation level (Mira & Duarte-Mermoud, 2009). See also Figure 21. Many other integration orders have been analyzed at the simulation level but they are not shown here

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>0</sup>

Time [s]

Interval [s] Level of load 0 – 45 50% 45 – 90 0% 90 – 135 100% 135 – 180 0% 180 – 225 50% 225 – 270 100% 270 – 315 50%

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>0</sup>

Time[s]

In this section the simulated behavior of IM control under the FOC-FOPI scheme, when Test 1 is applied, is shown in Figs. 18 to 21, for different values of the integration order of the integral part of PI controller ( = 0.7, = 1.0, = 1.7 and = 2.0). Note that the classical FOC-PI scheme is obtained from the FOC-FOPI strategy by setting = 1. In all the figures, only the controlled variable (motor speed) is shown. The results obtained are quite satisfactory, as can be seen from Figures 18 to 21. The evolution of the controlled variable tries to follow

From Figure 18 it can be observed that for integration orders less than 1.0 the response presents no overshoot although the response is slower. It can be concluded, from information contained in Figure 20, that the response is faster when the integration order is greater than 1.0 but an overshoot is observed. When is chosen as 2.0 critically stable

the speed reference at all times, in spite of the perturbation being applied.

for the sake of space. The reader is referred to Mira (2008) for more details.

**Figure 19.** Simulation results for ߥ ൌ ͳǤͲͲ

**Figure 20.** Simulation results for ߥ ൌ ʹǤͲͲ

Advanced Control Techniques for Induction Motors 321

Since the BCS has fixed controller parameters its behavior is not as good as the adaptive

From simulation results obtained in this study it is possible to state that the integration order of FOPI controller plays a central role in speed control of an IM, when compared with the BCS. Choosing a suitable value of the integration order allows obtaining fast/slow responses and over/under damped responses. For this particular case of IM speed control , it was observed that for integration orders lesser than 1 the stabilization time is rather large and the controlled variable may not present overshoot. On the contrary, for values over 1 the stabilization time is small (and diminishes as integration order increases); the overshoot increases as the integration order does, reaching instability for integration order equal to 2. It was observed that the best results obtained from this study correspond to integration orders near to 1.40, presenting small rise and stabilization times, though with certain degree

In conclusion, the adaptive strategies studied present clear advantages with respect to the BCS used as basis of comparison. Amongst the adaptive schemes the APBC with time-

The results reported here have been supported by CONICYT-Chile under grants Fondecyt

Al-Nimma, D.A. and Williams, S. (1980). Study of rapid speed-changing methods on A.C. motor drives. *IEE Proceedings, Part B, Electric Power Applications*, Vol.127, No.6, pp. 382 –

Araujo, R. E. & Freitas, D. (2000), "Non-linear control of an induction motor : sliding mode theory leads to robust and simple solution ". *International Journal of Adaptive Control and* 

Bose, B.K. (1997). *Power Electronics and Variable Frequency Drives; Technology and Applications*.

Bose, B.K. (2002). *Modern Power Electronics and AC Drives*, Prentice Hall PTR, ISBN-13: 978-

Byrnes, C.I., Isidori, A. & Willems, J.C. (1991), "Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems". *IEEE Transactions on* 

*Automatic Control*, Vol. 36, No. 11, pp. 1228-1240, November, ISSN: 0018-9286.

*Signal Processing*, Vol.14, No. 2, pp. 331-353, MES, ISSN: 0890-6327.

IEEE Press Marketing, ISBN-13: 978-0780310841, New York, USA

strategies studied and presented here.

varying adaptive gains is the one that behaves better.

Manuel A. Duarte-Mermoud & Juan C. Travieso-Torres

1061170, Fondecyt 1090208 and FONDEF D05I-10098.

0130167439, Upper Saddle River, USA

*Department of Electrical Engineering, University of Chile, Santiago, Chile* 

of overshoot.

**Author details** 

**Acknowledgement** 

385, ISSN: 0143-7038

**6. References** 

**Figure 21.** Simulation results for ߥ ൌ ͳǤ͵Ͳ

Experimental analysis of the FOC-FOPI scheme is currently underway. These results will be reported, compared and discussed in the near future.
