**4.2 Tracking performance**

For tracking motion, periodic trapezoidal reference input is utilized to command the maglev system. The maximal tracking error is stated as *Emax* = max |*xr* - *x*| where

#### **Figure 15.**

*Experimental step responses of the three control systems at negative side direction. (a) Responses to a* � *0.5 mm step input (default mass). (b) Responses to a* �*1.0 mm step input (default mass).*

error of the FSF controller at 0.5 mm amplitude is around 1.5 times larger than the FF PI-PD + *K*<sup>z</sup> controller. Meanwhile, as the amplitude increased to 1.0 mm, the FSF controller maximum tracking error is about 2 times larger than the FF PI-PD + *K*<sup>z</sup> controller (see error signal in **Figure 17(b)**). The maximum tracking error occurred at the slope of the trapezoidal signal where the velocity changes. Thus, the experimental results proved that the FSF controller has low adaptability to the velocity change. The FSF controller comprised of narrow bandwidth (see **Figure 16**). Hence, it can explain that the FSF controller does not have sufficient speed to cope with the variation of velocity effectively. The average of *Emax* and *Erms* values of twenty (20) experiments for the tracking motion is summarized in **Table 4**. At amplitude 0.5 mm, the *Emax* and *Erms* values of the FSF controller are 48.2% and 46.7% larger than the FF PI-PD + *K*<sup>z</sup> controller. Besides, the *Emax* and *Erms* values of the FF PI-PD + *K*<sup>z</sup> controller are 47.1% and 58.9% smaller than the FSF controller at 1.0 mm amplitude. On the other hand, the difference of *Emax* and *Erms* values between the FF PI-PD and the FF PI-PD + *K*<sup>z</sup> controllers are insignificant. Overview, the FF PI-PD and FF PI-PD + *K*<sup>z</sup> control systems track the trapezoidal signal more accurately and precisely with the smaller *Emax* and *Erms* values as compared to the FSF

*Experimental positioning performances of twenty (20) experiments for three controllers (default mass).*

**Step height Performance index FF PI-PD + Kz FF PI-PD FSF** 0.5 mm *OS*, % Average 0.00 0.00 0.00

*Enhanced Nonlinear PID Controller for Positioning Control of Maglev System*

*DOI: http://dx.doi.org/10.5772/intechopen.96769*

1.0 mm *OS*, % Average 0.00 0.00 0.00

0.5 mm *OS*, % Average 0.00 0.00 0.00

1.0 mm *OS*, % Average 0.00 0.00 0.00

Standard deviation 0.00 0.00 0.00 *<sup>t</sup>*s, s Average 1.45 <sup>10</sup><sup>1</sup> 1.17 <sup>10</sup><sup>1</sup> 4.35 <sup>10</sup><sup>1</sup>

Standard deviation 3.01 <sup>10</sup><sup>2</sup> 1.91 <sup>10</sup><sup>2</sup> 6.37 <sup>10</sup><sup>2</sup>

standard deviation 0.00 0.00 0.00 *<sup>t</sup>*s, s Average 2.02 <sup>10</sup><sup>1</sup> 1.60 <sup>10</sup><sup>1</sup> 5.48 <sup>10</sup><sup>1</sup>

Standard deviation 3.82 <sup>10</sup><sup>2</sup> 1.19 <sup>10</sup><sup>2</sup> 5.45 <sup>10</sup><sup>2</sup>

Standard deviation 0.00 0.00 0.00 *<sup>t</sup>*s, s Average 1.25 <sup>10</sup><sup>1</sup> 1.90 <sup>10</sup><sup>1</sup> 3.83 <sup>10</sup><sup>1</sup>

Standard deviation 3.22 <sup>10</sup><sup>2</sup> 1.19 <sup>10</sup><sup>1</sup> 8.64 <sup>10</sup><sup>2</sup>

Standard deviation 0.00 0.00 0.00 *<sup>t</sup>*s, s Average 1.77 <sup>10</sup><sup>1</sup> 2.03 <sup>10</sup><sup>1</sup> 5.06 <sup>10</sup><sup>1</sup>

Standard deviation 2.35 <sup>10</sup><sup>2</sup> 6.07 <sup>10</sup><sup>2</sup> 5.36 <sup>10</sup><sup>2</sup>

The robust performance of the proposed controller is evaluated in the presence of mass variation. The 25% extra load is added to the default load of the mechanism. In this experiment, the control performance is examined in two type of motions: point-to-point and tracking motions. The robust performance of the FF PI-PD + *K*<sup>z</sup>

controller is then compared with the FF PI-PD and FSF controllers.

controller.

**103**

**4.3 Robustness performance**

OS*: overshoot, ts: settling time.*

**Table 3.**

**Figure 16.** *Simulated closed-loop frequency response.*

*xr* is the reference input and *x* is the levitation height. In addition, the root mean square error, *Erms* is calculated as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1*=N*P*<sup>N</sup> <sup>k</sup>*¼<sup>1</sup>*e*<sup>2</sup> q where *N* represents the number of data samples and *e* is the tracking error.

**Figure 17** illustrates the trapezoidal tracking performance of the FSF, FF PI-PD and FF PI-PD + *K*<sup>z</sup> control systems to 0.5 mm and 1.0 mm amplitudes. Both FF PI-PD and FF PI-PD + *K*<sup>z</sup> controllers demonstrate almost identical tracking performance. The tracking error difference between them is insignificant. On the other hand, the FSF controller demonstrates the worst tracking performance with the largest tracking error among the compared controllers. The maximum tracking


*Enhanced Nonlinear PID Controller for Positioning Control of Maglev System DOI: http://dx.doi.org/10.5772/intechopen.96769*

#### **Table 3.**

*Experimental positioning performances of twenty (20) experiments for three controllers (default mass).*

error of the FSF controller at 0.5 mm amplitude is around 1.5 times larger than the FF PI-PD + *K*<sup>z</sup> controller. Meanwhile, as the amplitude increased to 1.0 mm, the FSF controller maximum tracking error is about 2 times larger than the FF PI-PD + *K*<sup>z</sup> controller (see error signal in **Figure 17(b)**). The maximum tracking error occurred at the slope of the trapezoidal signal where the velocity changes. Thus, the experimental results proved that the FSF controller has low adaptability to the velocity change. The FSF controller comprised of narrow bandwidth (see **Figure 16**). Hence, it can explain that the FSF controller does not have sufficient speed to cope with the variation of velocity effectively. The average of *Emax* and *Erms* values of twenty (20) experiments for the tracking motion is summarized in **Table 4**. At amplitude 0.5 mm, the *Emax* and *Erms* values of the FSF controller are 48.2% and 46.7% larger than the FF PI-PD + *K*<sup>z</sup> controller. Besides, the *Emax* and *Erms* values of the FF PI-PD + *K*<sup>z</sup> controller are 47.1% and 58.9% smaller than the FSF controller at 1.0 mm amplitude. On the other hand, the difference of *Emax* and *Erms* values between the FF PI-PD and the FF PI-PD + *K*<sup>z</sup> controllers are insignificant. Overview, the FF PI-PD and FF PI-PD + *K*<sup>z</sup> control systems track the trapezoidal signal more accurately and precisely with the smaller *Emax* and *Erms* values as compared to the FSF controller.
