*3.2.3 Disturbance compensator*

third pole location, α = 10. After calculated the PID controller parameters, the derivative gain, *Kd* and some portion of the proportional gain, *Kp* are moved to the feedback path for acquiring the PI-PD control in enhancing the transient response. Even though both PI-PD and conventional PID controllers show an approximately similar closed loop characteristic equation, both of them comprised of different control law

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

*Ki s*

Based on Eqs (17) and (18), the conventional PID controller is functioned based on the error signal, *E*(*s*) only, whereas the PI-PD controller is operated based on the error signal, *E*(*s*) and the output signal *X*(*s*). Hence, the PI-PD controller tends to

The model-based feedforward control is employed to improve the overshoot reduction characteristic of the PI-PD control. The control law of the model-based

where *Kff* and *Xr*ð Þ*s* represent the linearized feedforward gain and reference input. From Eq. (19), the model-based feedforward control is acted based on the desired output or reference input. Hence, by using the feedforward control, the desired output is known in advance and it can synthesize an adequate control signal to the closed loop system for moving the mechanism to the targeted output. Thus, the model-based feedforward control is used to enhance the system following characteristic and provide a better overshoot reduction characteristic. It also leads

To design the model-based feedforward control, the relationship between the controlled voltage and the levitation displacement is obtained via experiments. First, a ramp input voltage with gradient, m = 0.1 V/t is applied to the system at different levitation displacement from 0 mm to 15 mm with every 1 mm incremental displacement. Then, the minimum voltage to levitate the steel ball at various displacements is determined. The quantitative comparisons of ten (10) repeatability

þ *Kds*

*E s*ð Þ (17)

*E s*ðÞ� *Kpb* <sup>þ</sup> *Kds X s*ð Þ (18)

*Uff*ðÞ¼ *s KffXr*ð Þ*s* (19)

*UPID*ðÞ¼ *s Kp* þ

act faster than the conventional PID controller to compensate the error.

*Ki s* 

*UPI*�*PD*ðÞ¼ *s Kp* þ

*3.2.2 Model-based feedforward control*

feedforward control is expressed as

to a faster positioning time.

**Figure 4.**

**94**

*The maglev system driving characteristic in open loop.*

In order to enhance the disturbance rejection characteristic of the proposed controller, a disturbance compensator is designed and incorporated with the FF PI-PD control, via lowering the magnitude of sensitivity function. In practical, the external disturbance and parameter uncertainties are lumped as an equivalent disturbance. A simple way to attenuate the equivalent disturbance is through introducing a cancelation term to it. The proposed disturbance compensator considers the difference between the actual output and the reference input as an equivalent disturbance. Then, an adequate voltage is applied to suppress the equivalent disturbance. The control law of the disturbance compensator is expressed as

$$U\_x(\mathbf{s}) = [X(\mathbf{s}) - X\_r(\mathbf{s})]K\_{\mathcal{f}} \tag{20}$$

where *X*(*s*) = *K*z*I*(*s*), *Kff*, *Kz, X(s)* and *Xr*ð Þ*s* represent the linearized feedforward gain, linearized disturbance compensator gain, levitation height and reference input.

#### **Figure 5.**

*Experimental step responses of the FF PI-PD and PI-PD control system. (a) Responses to a 0.5 mm step input. (b) Responses to a 1.0 mm step input.*

From Eq. (20), the difference between the actual output and the reference input is considered as an estimated disturbance. Then, a sufficient voltage is applied to the control signal to attenuate the estimated disturbance.

To design the disturbance compensator, the relationship between the controlled current and the levitation displacement is attained experimentally. First, the mechanism is stabilized by a control system. Next, the required current to levitate the steel ball at different levitation displacement from �2.5 mm to 2.5 mm with every 0.5 mm interval displacement is measured. The quantitative comparisons of ten (10) repeatability tests are conducted at various levitation displacements. **Figure 6** shows the system current dynamic where 0 mm denotes the system datum or initial position which is at 10.5 mm. The system current dynamic is employed as a disturbance compensator in the proposed controller.

**Figure 7** depicts the experimental impulse disturbance rejection performance of the FF PI-PD and FF PI-PD + *K*<sup>z</sup> controllers. It can be seen clearly that the FF PI-PD + *K*<sup>z</sup> controller is less sensitive to the external disturbance. The disturbance rejection characteristic of the FF PI-PD + *K*<sup>z</sup> controller is proven theoretically by using the closed loop sensitivity function. The sensitivity functions of the FF PI-PD and FF PI-PD + *K*<sup>z</sup> controllers are

$$S\_{\rm FFPI-PD}(\mathfrak{s}) = \frac{1}{\mathfrak{1} + \left(K\_p + \frac{K\_i}{s} + K\_{pb} + K\_d \frac{sa\_k}{s + a\_k}\right) G\_p(\mathfrak{s})} \tag{21}$$

$$\mathbf{S}\_{\text{FFPI}-\text{PD}+K\_{\varepsilon}}(\mathbf{s}) = \frac{1}{\left(\mathbf{K}\_a - \mathbf{K}\_{\overline{f}\overline{f}}\mathbf{K}\_x\right) + \left(\mathbf{K}\_p + \frac{\mathbf{K}\_i}{s} + \mathbf{K}\_{pb} + \mathbf{K}\_d \frac{\mathbf{s} \mathbf{a}\_c}{s + w\_c}\right)\mathbf{G}\_p(\mathbf{s})}} \tag{22}$$

From Eqs (21) and (22), the proposed controller consists of the additional elements to reduce sensitivity of the system and hence to accomplish better robustness to disturbance. **Figure 8** presents the frequency responses of the FF PI-PD and FF PI-PD + *K*<sup>z</sup> from the disturbance to the displacement. To decrease the effect of disturbance, the sensitivity functions of the closed-loop system must have sufficiently low magnitude. As can be seen clearly in **Figure 8**, the proposed controller consists of lower sensitivity magnitude than the FF PI-PD controller up to the range of 70 Hz. Thus, it can be expressed that the disturbance compensation control scheme of the proposed controller tends to improve the system disturbance rejection characteristic. In short, the FF PI-PD + *K*<sup>z</sup> control system is less sensitive to the external disturbance and parameter variation in comparison to the FF PI-PD controller.

**3.3 Stability analysis**

**Figure 8.**

**97**

**Figure 7.**

*Experimental impulse disturbance rejection performance.*

control system is expressed as

Basically, digital control systems are used for motion control. Thus, the stability of the FF PI-PD + *K*<sup>z</sup> control system is discussed in discrete-time. The nonlinear elements of the controller are undergone linearization and the linearized gains are used for the stability analysis. After linearized, the feedforward and disturbance compensator gains are 0.26 V/mm and 30.67 mm/A, respectively. The stability analysis using the linearized model is adequate to provide the important knowledge of stability. The

Using backward difference rule, the pulse transfer function of the FF PI-PD + *K*<sup>z</sup>

discrete-time FF PI-PD + *K*<sup>z</sup> control system is illustrated in **Figure 9**.

*Simulated frequency response for sensitivity of FF PI-PD and FF PI-PD +* K*z controllers.*

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

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

**Figure 6.** *The maglev system current dynamic in closed loop.*

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

**Figure 7.** *Experimental impulse disturbance rejection performance.*

**Figure 8.** *Simulated frequency response for sensitivity of FF PI-PD and FF PI-PD +* K*z controllers.*

### **3.3 Stability analysis**

Basically, digital control systems are used for motion control. Thus, the stability of the FF PI-PD + *K*<sup>z</sup> control system is discussed in discrete-time. The nonlinear elements of the controller are undergone linearization and the linearized gains are used for the stability analysis. After linearized, the feedforward and disturbance compensator gains are 0.26 V/mm and 30.67 mm/A, respectively. The stability analysis using the linearized model is adequate to provide the important knowledge of stability. The discrete-time FF PI-PD + *K*<sup>z</sup> control system is illustrated in **Figure 9**.

Using backward difference rule, the pulse transfer function of the FF PI-PD + *K*<sup>z</sup> control system is expressed as

From Eq. (20), the difference between the actual output and the reference input is considered as an estimated disturbance. Then, a sufficient voltage is applied to the

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

To design the disturbance compensator, the relationship between the controlled current and the levitation displacement is attained experimentally. First, the mechanism is stabilized by a control system. Next, the required current to levitate the steel ball at different levitation displacement from �2.5 mm to 2.5 mm with every 0.5 mm interval displacement is measured. The quantitative comparisons of ten (10) repeatability tests are conducted at various levitation displacements. **Figure 6** shows the system current dynamic where 0 mm denotes the system datum or initial position which is at 10.5 mm. The system current dynamic is employed as a distur-

**Figure 7** depicts the experimental impulse disturbance rejection performance of the FF PI-PD and FF PI-PD + *K*<sup>z</sup> controllers. It can be seen clearly that the FF PI-PD + *K*<sup>z</sup> controller is less sensitive to the external disturbance. The disturbance rejection characteristic of the FF PI-PD + *K*<sup>z</sup> controller is proven theoretically by using the closed loop sensitivity function. The sensitivity functions of the FF PI-PD

<sup>1</sup> <sup>þ</sup> *Kp* <sup>þ</sup> *Ki*

<sup>þ</sup> *Kp* <sup>þ</sup> *Ki*

1

1

From Eqs (21) and (22), the proposed controller consists of the additional elements to reduce sensitivity of the system and hence to accomplish better robustness to disturbance. **Figure 8** presents the frequency responses of the FF PI-PD and FF PI-PD + *K*<sup>z</sup> from the disturbance to the displacement. To decrease the effect of disturbance, the sensitivity functions of the closed-loop system must have sufficiently low magnitude. As can be seen clearly in **Figure 8**, the proposed controller consists of lower sensitivity magnitude than the FF PI-PD controller up to the range of 70 Hz. Thus, it can be expressed that the disturbance compensation control scheme of the proposed controller tends to improve the system disturbance rejection characteristic. In short, the FF PI-PD + *K*<sup>z</sup> control system is less sensitive to the external disturbance and parameter variation in comparison to the FF PI-PD

*<sup>s</sup>* <sup>þ</sup> *Kpb* <sup>þ</sup> *Kd <sup>s</sup>ω<sup>c</sup>*

*s*þ*ω<sup>c</sup>*

*<sup>s</sup>* <sup>þ</sup> *Kpb* <sup>þ</sup> *Kd <sup>s</sup>ω<sup>c</sup>*

*Gp*ð Þ*s*

*s*þ*ω<sup>c</sup>*

*Gp*ð Þ*s*

(21)

(22)

control signal to attenuate the estimated disturbance.

bance compensator in the proposed controller.

*SFFPI*�*PD*ðÞ¼ *s*

*Ka* � *KffKz*

and FF PI-PD + *K*<sup>z</sup> controllers are

*SFFPI*�*PD*þ*Kz* ðÞ¼ *s*

controller.

**Figure 6.**

**96**

*The maglev system current dynamic in closed loop.*

#### **Figure 9.**

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

$$T(z) = \frac{X(z)}{R(z)} = \frac{z^{-1} \mathbf{G}\_p(z) \mathbf{G}\_c(z)}{\mathbf{1} + z^{-1} \mathbf{G}\_p(z) \mathbf{G}\_c(z)}\tag{23}$$

The Jury stability test is used to examine the stability limit of the FF PI-PD + *K*<sup>z</sup> control system. The characteristic in Eq. (23) is used to identify the stability limit of the FF PI-PD + *K*<sup>z</sup> control system. **Figures 10** and **11** show the minimum and

maximum values of the mass parameter variation, respectively. The results show that the parameter that influences the stability of the control system is the object mass, *M*. To maintain the system stable, the object mass, *M* must be kept between 3.9 g < *M* < 190 g. In short, the Jury test proves that the FF PI-PD + *K*<sup>z</sup> control system remains

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

**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

*e t*ð Þ � � <sup>þ</sup>

3 7 <sup>5</sup>, B <sup>¼</sup> *B* 0

0 0 1*=Ka*

2 6 4 � �*u t*ðÞþ <sup>0</sup>

3 7 5 1

and C ¼ ½ � *Ks* 0 0

� �*r t*ð Þ (24)

stable with the increment of mass weight to two times of its default one.

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

**4. Experimental performance**

d*-coefficient of mass parameter variation.*

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

*x t* \_ð Þ

*e t* \_ð Þ � � <sup>¼</sup> *<sup>A</sup>* <sup>0</sup>

01 0 *Kx=M* 0 �*Kc=M* 00 0

�*C* 0 � � *x t*ð Þ

by 25%.

**Figure 11.**

is written as

where.

**99**

A ¼

2 6 4

where

$$G\_p(z) = \frac{\mu\_2 z + \mu\_3}{z^2 - \mu\_4 z + 1}$$

$$G\_\varepsilon(z) = \frac{\mu\_\\$z^2 + \mu\_6 z + \mu\_7}{\mu\_\\$z^2 - \mu\_\\$z + \mu\_{10}}$$

*<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> <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>2</sup> *<sup>e</sup>γ<sup>T</sup>* <sup>þ</sup> *<sup>μ</sup>*<sup>1</sup> <sup>2</sup> *<sup>e</sup>*�*γ<sup>T</sup>*, *<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 <sup>T</sup>* <sup>þ</sup> *Kp* � *Kpb* <sup>þ</sup> *KffKz Td* <sup>þ</sup> *KiTTd* � *Kd*, *μ*<sup>6</sup> ¼ *Kpb* � *KffKz* � *Kp <sup>T</sup>* <sup>þ</sup> <sup>2</sup>*Kpb* � <sup>2</sup>*KffKz* � <sup>2</sup>*Kp Td* � *KiTTd* <sup>þ</sup> <sup>2</sup>*Kd*, *<sup>μ</sup>*<sup>7</sup> <sup>¼</sup> *Kp* <sup>þ</sup> *KffKz* � *Kpb 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 *Td* = 1.60 � <sup>10</sup>�<sup>3</sup> s.

**Figure 10.** b*-coefficient of mass parameter variation.*

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

**Figure 11.** d*-coefficient of mass parameter variation.*

The Jury stability test is used to examine the stability limit of the FF PI-PD + *K*<sup>z</sup> control system. The characteristic in Eq. (23) is used to identify the stability limit of the FF PI-PD + *K*<sup>z</sup> control system. **Figures 10** and **11** show the minimum and maximum values of the mass parameter variation, respectively. The results show that the parameter that influences the stability of the control system is the object mass, *M*. To maintain the system stable, the object mass, *M* must be kept between 3.9 g < *M* < 190 g. In short, the Jury test proves that the FF PI-PD + *K*<sup>z</sup> control system remains stable with the increment of mass weight to two times of its default one.
