**4. Experimental performance**

Experiments are conducted to evaluate the effectiveness of the FF PI-PD + *K*<sup>z</sup> control system. Two types of motion control that are positioning and tracking controls are experimentally examined. The full-state feedback (FSF) controller is designed and compared with the proposed control. FSF control is chosen for the comparison purpose is because it is an advanced controller that has been regularly applied to the nonlinear applications such as maglev system [18, 19], inverted pendulum system [20] and others. Besides, the FF PI-PD control is compared with the proposed one in order to prove the usefulness of the disturbance compensator. The robust performance of the proposed control is examined by injecting an impulse disturbance to the system, and followed by increasing the mass of the ball by 25%.

**Figure 12** illustrates the block diagram of the FSF controller. An integral action is added into the FSF controller to eliminate the steady state error by increasing the transfer function to type one system. From **Figure 12**, the system state-space model is written as

$$
\begin{bmatrix}
\dot{\mathbf{x}}(t) \\
\dot{e}(t)
\end{bmatrix} = \begin{bmatrix}
A & \mathbf{0} \\
\end{bmatrix} \begin{bmatrix}
\mathbf{x}(t) \\
e(t)
\end{bmatrix} + \begin{bmatrix}
B \\
\mathbf{0}
\end{bmatrix} u(t) + \begin{bmatrix}
\mathbf{0} \\
\mathbf{1}
\end{bmatrix} r(t) \tag{24}
$$

where.

$$\mathbf{A} = \begin{bmatrix} \mathbf{0} & \mathbf{1} & \mathbf{0} \\ \mathbf{K}\_{\mathbf{x}}/M & \mathbf{0} & -\mathbf{K}\_{\mathbf{c}}/M \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \end{bmatrix}, \ \mathbf{B} = \begin{bmatrix} \mathbf{0} \\ \mathbf{0} \\ \mathbf{1}/K\_{a} \end{bmatrix} \text{ and } \ \mathbf{C} = \begin{bmatrix} \mathbf{K}\_{\mathbf{x}} & \mathbf{0} & \mathbf{0} \end{bmatrix}$$

*T z*ð Þ¼ *X z*ð Þ

*Gp*ð Þ¼ *z*

*Gc*ð Þ¼ *z*

*<sup>μ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>β</sup>=γ*2, *<sup>μ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>μ</sup>*<sup>1</sup> *<sup>e</sup>γ<sup>T</sup>* <sup>þ</sup> *<sup>e</sup>*�*γ<sup>T</sup>* � *<sup>μ</sup>*<sup>1</sup> � *<sup>μ</sup>*<sup>1</sup>

*KiTTd* � *Kd*, *μ*<sup>6</sup> ¼ *Kpb* � *KffKz* � *Kp*

*Discrete-time of FF PI-PD +* K*z control system.*

2*Kd*, *μ*<sup>7</sup> ¼ *Kp* þ *KffKz* � *Kpb*

b*-coefficient of mass parameter variation.*

*Td* = 1.60 � <sup>10</sup>�<sup>3</sup> s.

**Figure 10.**

**98**

*<sup>μ</sup>*<sup>4</sup> <sup>¼</sup> *<sup>e</sup>γ<sup>T</sup>* <sup>þ</sup> *<sup>e</sup>*�*γ<sup>T</sup>*, *<sup>μ</sup>*<sup>5</sup> <sup>¼</sup> *KiT*<sup>2</sup> <sup>þ</sup> *Kp* � *Kpb* <sup>þ</sup> *KffKz*

where

**Figure 9.**

*R z*ð Þ <sup>¼</sup> *<sup>z</sup>*�<sup>1</sup>*Gp*ð Þ*<sup>z</sup> Gc*ð Þ*<sup>z</sup>*

*Control Based on PID Framework - The Mutual Promotion of Control and Identification…*

*μ*2*z* þ *μ*<sup>3</sup> *z*<sup>2</sup> � *μ*4*z* þ 1

*<sup>μ</sup>*5*z*<sup>2</sup> <sup>þ</sup> *<sup>μ</sup>*6*<sup>z</sup>* <sup>þ</sup> *<sup>μ</sup>*<sup>7</sup> *μ*8*z*<sup>2</sup> � *μ*9*z* þ *μ*<sup>10</sup>

*Td* � *Kd*, *<sup>μ</sup>*<sup>8</sup> <sup>¼</sup> ð Þ *<sup>T</sup>* <sup>þ</sup> *Td* , *<sup>μ</sup>*<sup>9</sup> <sup>¼</sup> *<sup>T</sup>* <sup>þ</sup> <sup>2</sup>*Td*, *<sup>μ</sup>*<sup>10</sup> <sup>¼</sup> *Td* and

*<sup>T</sup>* <sup>þ</sup> <sup>2</sup>*Kpb* � <sup>2</sup>*KffKz* � <sup>2</sup>*Kp*

1 þ *z*�<sup>1</sup>*Gp*ð Þ*z Gc*ð Þ*z*

<sup>2</sup> *<sup>e</sup>γ<sup>T</sup>* <sup>þ</sup> *<sup>e</sup>*�*γ<sup>T</sup>* , *<sup>μ</sup>*<sup>3</sup> ¼ �*μ*<sup>1</sup> <sup>þ</sup> *<sup>μ</sup>*<sup>1</sup>

*<sup>T</sup>* <sup>þ</sup> *Kp* � *Kpb* <sup>þ</sup> *KffKz*

(23)

<sup>2</sup> *<sup>e</sup>*�*γ<sup>T</sup>*,

<sup>2</sup> *<sup>e</sup>γ<sup>T</sup>* <sup>þ</sup> *<sup>μ</sup>*<sup>1</sup>

*Td* <sup>þ</sup>

*Td* � *KiTTd* <sup>þ</sup>

**Figure 12.** *Block diagram of the full-state feedback (FSF) controller.*

Based on Eq. (24), the state feedback control law, *ufsf* (*t*) is defined as

$$u\_{f\!\!\!f}(t) = -\mathbf{K}\mathbf{x}(t) + K\_{\!\!\!e}e(t) \tag{25}$$

takes longer settling time than the FF PI-PD and FF PI-PD + *K*<sup>z</sup> controllers to reach steady-state that less than 100 μ m. **Figure 16** presents the simulated closed-loop frequency response for the three control systems. As can be seen in **Figure 16**, the FF PI-PD and FF PI-PD + *K*<sup>z</sup> controls demonstrate wider bandwidth as compared to the FSF control. Therefore, it can be explained that the both FF PI-PD and FF PI-PD + *Kz*

*Experimental step responses of the three control systems at positive side direction. (a) Responses to a 0.5 mm step*

**Controller** *K1 K2 K3 Kp Ki Kpb Kd* FSF 0.80 0.02 0.01 — 2.36 — — FF PI-PD ——— 0.45 1.20 0.15 0.03 FF PI-PD + Kz ——— 0.45 1.20 0.15 0.03

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

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

**Table 3** shows the quantitative comparison of twenty (20) repeatability experimental results for the point-to-point motion of the three controllers. The settling time, *ts* is determined as the time where the system is stabilized within 100 μm. All three controllers demonstrate zero overshoot at every step input. Although FSF performs zero overshoot in all the step input, it takes long settling time to reach steady state that of less than 100 μm. The settling time of the FF PI-PD + *K*<sup>z</sup> is

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

controllers could perform shorter settling time than the FSF controller.

*input (default mass). (b) Responses to a 1.0 mm step input (default mass).*

65.6% shorter than the FSF controller.

**4.2 Tracking performance**

**Table 2.**

**Figure 14.**

**101**

*Controller parameters.*

where **K**, *Ki*, *x*(t) and *e*(t) represent the state feedback gain matrix, integral gain, levitation displacement and error, accordingly.

Ackermann's formula is used to determine the state feedback gain matrix, **K** and the integral gain, *Ki*. For comparative purpose, the design specifications of FSF controller are set as: settling time, *ts* = 0.5 s, percentage of overshoot, *%OS* < 10% as well as third and fourth poles location, α = 10. The frequency of first-order low-pass filter, *ω<sup>c</sup>* is selected based on the system cut-off frequency at around 600 rad/s. The FSF controller parameters are tuned to have the best positioning performance at 1.0 mm step response as similar to the FF PI-PD + *K*<sup>z</sup> controller (see **Figure 13**). **Table 2** shows the controller parameters for FSF, FF PI-PD and FF PI-PD + *K*<sup>z</sup> control systems.

### **4.1 Positioning performance**

In this experiment, the initial position is set at 10.5 mm and the working range is within �2.5 mm. **Figures 14** and **15** show the experimental positioning performance of the FSF, FF PI-PD and FF PI-PD + *K*<sup>z</sup> control systems to 0.5 mm, 1.0 mm, �0.5 mm and � 1.0 mm step inputs, respectively. As observed clearly, the FF PI-PD + *K*<sup>z</sup> controller shows almost identical positioning performance, with no overshoot as the FF PI-PD and FSF control systems. However, the FSF control system

**Figure 13.** *Experimental step responses of controllers at 1.0 mm reference input.*


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

**Table 2.**

Based on Eq. (24), the state feedback control law, *ufsf* (*t*) is defined as

*Control Based on PID Framework - The Mutual Promotion of Control and Identification…*

levitation displacement and error, accordingly.

*Experimental step responses of controllers at 1.0 mm reference input.*

*Block diagram of the full-state feedback (FSF) controller.*

**4.1 Positioning performance**

**Figure 12.**

**Figure 13.**

**100**

where **K**, *Ki*, *x*(t) and *e*(t) represent the state feedback gain matrix, integral gain,

Ackermann's formula is used to determine the state feedback gain matrix, **K** and the integral gain, *Ki*. For comparative purpose, the design specifications of FSF controller are set as: settling time, *ts* = 0.5 s, percentage of overshoot, *%OS* < 10% as well as third and fourth poles location, α = 10. The frequency of first-order low-pass filter, *ω<sup>c</sup>* is selected based on the system cut-off frequency at around 600 rad/s. The FSF controller parameters are tuned to have the best positioning performance at 1.0 mm step response as similar to the FF PI-PD + *K*<sup>z</sup> controller (see **Figure 13**). **Table 2** shows the controller parameters for FSF, FF PI-PD and FF PI-PD + *K*<sup>z</sup> control systems.

In this experiment, the initial position is set at 10.5 mm and the working range is within �2.5 mm. **Figures 14** and **15** show the experimental positioning performance of the FSF, FF PI-PD and FF PI-PD + *K*<sup>z</sup> control systems to 0.5 mm, 1.0 mm, �0.5 mm and � 1.0 mm step inputs, respectively. As observed clearly, the FF PI-PD + *K*<sup>z</sup> controller shows almost identical positioning performance, with no overshoot as the FF PI-PD and FSF control systems. However, the FSF control system

*ufsf*ðÞ¼� *t* **K**xðÞþ*t Kie t*ð Þ (25)

*Controller parameters.*

**Figure 14.**

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

takes longer settling time than the FF PI-PD and FF PI-PD + *K*<sup>z</sup> controllers to reach steady-state that less than 100 μ m. **Figure 16** presents the simulated closed-loop frequency response for the three control systems. As can be seen in **Figure 16**, the FF PI-PD and FF PI-PD + *K*<sup>z</sup> controls demonstrate wider bandwidth as compared to the FSF control. Therefore, it can be explained that the both FF PI-PD and FF PI-PD + *Kz* controllers could perform shorter settling time than the FSF controller.

**Table 3** shows the quantitative comparison of twenty (20) repeatability experimental results for the point-to-point motion of the three controllers. The settling time, *ts* is determined as the time where the system is stabilized within 100 μm. All three controllers demonstrate zero overshoot at every step input. Although FSF performs zero overshoot in all the step input, it takes long settling time to reach steady state that of less than 100 μm. The settling time of the FF PI-PD + *K*<sup>z</sup> is 65.6% shorter than the FSF controller.
