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


incorporated with the PI-PD control to enhance the system following characteristic. This improvement leads to a better overshoot reduction characteristic and shorten the positioning time. Lastly, a disturbance compensator is introduced for robustness enhancement of the FF PI-PD control. The proposed disturbance compensator estimates the disturbances and imparts an adequate voltage to compensate them. Overall, the proposed FF PI-PD + *K*<sup>z</sup> control system provides the advantages which are: (i) a better overshoot reduction characteristic; and (ii) low sensitivity to exter-

> *sω<sup>c</sup> s* þ *ω<sup>c</sup>*

where *Kp*, *Ki*, *Kpb*, *Kd*, *Kff*, *Kz*, *ωc, and Xr* denote the proportional gain, integral gain, feedback proportional gain, derivative gain, linearized feedforward gain, linearized disturbance compensator gain, system cut-off frequency and reference

There are three (3) major parts in the design procedure of the FF PI-PD + *K*z

The PI-PD controller is a modified PID controller, where it consists of derivative gain, *Kd* and some portion of the proportional gain, *Kpb* at the feedback path. Both are evidenced to have an approximately similar closed loop characteristic equation,

The desired characteristic equation of a general second-order system is denoted as

where *β*, *γ*, *α*, *ζ* and *ω<sup>n</sup>* represent the open loop gain, open loop pole, third pole

<sup>2</sup>*ζ*<sup>2</sup> <sup>þ</sup> *<sup>ω</sup><sup>n</sup>*

3*ζ*

*Ki* <sup>¼</sup> *αω<sup>n</sup>*

*Kd* <sup>¼</sup> <sup>2</sup>*ζω<sup>n</sup>* <sup>þ</sup> *αζω<sup>n</sup>*

To achieve a fast positioning with low overshoot performance, the design specifications are set as: settling time, *ts* = 0.5 s, percentage of overshoot, *%OS* < 10% and

location, desired damping ratio and desired natural frequency, respectively. By comparing Eqs (11) and (13), the PID parameters (*Kp*, *Ki* and *Kd*) are

*X s*ðÞþ *KffXr*ðÞþ*<sup>s</sup> Kff* ½ � *KzI s*ðÞ� *Xr*ð Þ*<sup>s</sup>*

<sup>2</sup> <sup>þ</sup> *Kp<sup>β</sup>* � *<sup>γ</sup>*<sup>2</sup> *<sup>s</sup>* <sup>þ</sup> *Ki<sup>β</sup>* (11)

<sup>2</sup> (13)

*<sup>β</sup>* (14)

*<sup>β</sup>* (15)

*<sup>β</sup>* (16)

<sup>2</sup> <sup>þ</sup> *Kp* <sup>þ</sup> *Kpb <sup>β</sup>* � *<sup>γ</sup>*<sup>2</sup> *<sup>s</sup>* <sup>þ</sup> *Ki<sup>β</sup>* (12)

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*ζωns* <sup>þ</sup> *<sup>ω</sup><sup>n</sup>*

<sup>2</sup> <sup>þ</sup> *<sup>γ</sup>*<sup>2</sup>

(10)

nal disturbance and parameter variation.

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

*E s*ðÞ� *Kpb* <sup>þ</sup> *Kd*

*Ki s*

*U s*ðÞ¼ *Kp* þ

input, respectively.

control system.

**93**

**3.2 Design procedure**

*3.2.1 PI-PD controller*

as proved in Eqs (11) and (12).

*δPID*ð Þ¼ s *s*

*δPI*�*PD*ðÞ¼ *s s*

<sup>3</sup> <sup>þ</sup> *<sup>β</sup>Kds*

<sup>3</sup> <sup>þ</sup> *<sup>β</sup>Kds*

*δdesired*ðÞ¼ *s* ð Þ *s* þ *αζω<sup>n</sup> s*

*Kp* <sup>¼</sup> <sup>2</sup>*αω<sup>n</sup>*

The control law of the FF PI-PD + *K*z control system is

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

**Table 1.** *Model parameters.*

**Figure 3.** *Block diagram of the FF PI-PD +* K*z control system.*

is designed under the following considerations: (i) the PI-PD control is designed to improve the transient response of the conventional PID controller, (ii) the modelbased feedforward control is integrated to obtain a better overshoot reduction characteristic and (iii) the disturbance compensation control is employed for robustness enhancement.

To design the FF PI-PD + *K*<sup>z</sup> controller, the PI-PD control is designed at the first place. The PI-PD controller is proposed to improve the positioning performance of the conventional PID controller [17]. By moving the derivative action and some portion of the proportional gain to the feedback path, the resonance peak of conventional PID controller in the closed-loop frequency response can be reduced. Thus, it explains that the PI-PD controller demonstrates a better transient response than the conventional PID controller. Then, a low pass filter is adopted to improve the positioning accuracy by attenuating the amplification of the measurement noise. However, the PI-PD control transient response is unsatisfied because the overshoot remains high. To solve this problem, a model-based feedforward control is

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

incorporated with the PI-PD control to enhance the system following characteristic. This improvement leads to a better overshoot reduction characteristic and shorten the positioning time. Lastly, a disturbance compensator is introduced for robustness enhancement of the FF PI-PD control. The proposed disturbance compensator estimates the disturbances and imparts an adequate voltage to compensate them. Overall, the proposed FF PI-PD + *K*<sup>z</sup> control system provides the advantages which are: (i) a better overshoot reduction characteristic; and (ii) low sensitivity to external disturbance and parameter variation.

The control law of the FF PI-PD + *K*z control system is

$$U(\mathbf{s}) = \left(K\_p + \frac{K\_i}{\mathfrak{s}}\right) E(\mathbf{s}) - \left(K\_{pb} + K\_d \frac{\mathfrak{so}\_c}{\mathfrak{s} + o\_\ell}\right) \mathbf{X}(\mathfrak{s}) + K\_{\tilde{\mathcal{H}}} \mathbf{X}\_r(\mathfrak{s}) + K\_{\tilde{\mathcal{H}}} [\mathbf{K}\_z \mathbf{I}(\mathfrak{s}) - \mathbf{X}\_r(\mathfrak{s})],\tag{10}$$

where *Kp*, *Ki*, *Kpb*, *Kd*, *Kff*, *Kz*, *ωc, and Xr* denote the proportional gain, integral gain, feedback proportional gain, derivative gain, linearized feedforward gain, linearized disturbance compensator gain, system cut-off frequency and reference input, respectively.

#### **3.2 Design procedure**

There are three (3) major parts in the design procedure of the FF PI-PD + *K*z control system.

## *3.2.1 PI-PD controller*

The PI-PD controller is a modified PID controller, where it consists of derivative gain, *Kd* and some portion of the proportional gain, *Kpb* at the feedback path. Both are evidenced to have an approximately similar closed loop characteristic equation, as proved in Eqs (11) and (12).

$$\delta\_{\rm PID}(\mathbf{s}) = \mathbf{s}^3 + \beta \mathbf{K}\_d \mathbf{s}^2 + (\mathbf{K}\_p \beta - \gamma^2)\mathbf{s} + \mathbf{K}\_i \beta \tag{11}$$

$$\delta \delta\_{\rm Pl-PD}(\mathbf{s}) = \mathbf{s}^3 + \beta \mathbf{K}\_d \mathbf{s}^2 + \left[ \left( \mathbf{K}\_p + \mathbf{K}\_{pb} \right) \beta - \mathbf{y}^2 \right] \mathbf{s} + \mathbf{K}\_i \beta \tag{12}$$

The desired characteristic equation of a general second-order system is denoted as

$$\delta\_{desired}(\mathfrak{s}) = (\mathfrak{s} + a\mathfrak{s}'o\mathfrak{n})\left(\mathfrak{s}^2 + 2\mathfrak{s}'o\mathfrak{s}\mathfrak{s} + o\mathfrak{n}^2\right) \tag{13}$$

where *β*, *γ*, *α*, *ζ* and *ω<sup>n</sup>* represent the open loop gain, open loop pole, third pole location, desired damping ratio and desired natural frequency, respectively.

By comparing Eqs (11) and (13), the PID parameters (*Kp*, *Ki* and *Kd*) are

$$K\_p = \frac{2a\alpha\_n^{-2}\zeta^2 + a\alpha\_n^{-2} + \gamma^2}{\beta} \tag{14}$$

$$K\_i = \frac{a o o\_n{}^3 \zeta}{\beta} \tag{15}$$

$$K\_d = \frac{2\zeta a\_n + a\zeta a\_n}{\beta} \tag{16}$$

To achieve a fast positioning with low overshoot performance, the design specifications are set as: settling time, *ts* = 0.5 s, percentage of overshoot, *%OS* < 10% and

is designed under the following considerations: (i) the PI-PD control is designed to improve the transient response of the conventional PID controller, (ii) the modelbased feedforward control is integrated to obtain a better overshoot reduction characteristic and (iii) the disturbance compensation control is employed for

**Symbol Description, unit Value** *<sup>M</sup>* Steel ball mass, Kg 9.40 <sup>10</sup><sup>2</sup> *xo* Nominal displacement, m 1.00 <sup>10</sup><sup>2</sup> *io* Nnominal current, A 3.94 <sup>10</sup><sup>1</sup>

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

*Ka* Power amplifier gain, V/A 6.51 *Ks* Sensor sensitivity, V/m 1.67 <sup>10</sup><sup>2</sup> *g* Gravitational acceleration, m/s<sup>2</sup> 9.81

/A2 2.31 <sup>10</sup><sup>4</sup>

*K* Electromagnetic constant, Nm<sup>2</sup>

To design the FF PI-PD + *K*<sup>z</sup> controller, the PI-PD control is designed at the first place. The PI-PD controller is proposed to improve the positioning performance of the conventional PID controller [17]. By moving the derivative action and some portion of the proportional gain to the feedback path, the resonance peak of conventional PID controller in the closed-loop frequency response can be reduced. Thus, it explains that the PI-PD controller demonstrates a better transient response than the conventional PID controller. Then, a low pass filter is adopted to improve the positioning accuracy by attenuating the amplification of the measurement noise. However, the PI-PD control transient response is unsatisfied because the overshoot

remains high. To solve this problem, a model-based feedforward control is

robustness enhancement.

*Block diagram of the FF PI-PD +* K*z control system.*

**Figure 3.**

**92**

**Table 1.** *Model parameters.* 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

$$U\_{PID}(\mathbf{s}) = \left(K\_p + \frac{K\_i}{\mathfrak{s}} + K\_d \mathfrak{s}\right) E(\mathbf{s}) \tag{17}$$

tests are carried out at various levitation displacements. **Figure 4** depicts the system driving characteristic that is measured and adopted as a model-based feedforward

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

**Figure 5** shows the step responses of the PI-PD and FF PI-PD controllers at 0.5 mm and 1.0 mm step inputs. In contrast to the PI-PD controller, the FF PI-PD controller demonstrates a better overshoot reduction characteristic. Besides, the FF PI-PD controller positioning time is shorter than the PI-PD controller. The comparative experimental performances show that the model-based feedforward control

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 distur-

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.

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

*Uz*ðÞ¼ *s* ½ � *X s*ðÞ� *Xr*ð Þ*s Kff* (20)

bance. The control law of the disturbance compensator is expressed as

control in the proposed controller.

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

*3.2.3 Disturbance compensator*

**Figure 5.**

**95**

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

improves the overshoot reduction characteristic.

$$(U\_{PI-PD}(\mathbf{s}) = \left(K\_p + \frac{K\_i}{\mathfrak{s}}\right) \mathbf{E}(\mathbf{s}) - \left(K\_{pb} + K\_d \mathbf{s}\right) \mathbf{X}(\mathbf{s}) \tag{18}$$

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

## *3.2.2 Model-based feedforward control*

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 feedforward control is expressed as

$$U\_{\tilde{\mathcal{f}}}(\mathbf{s}) = K\_{\tilde{\mathcal{f}}} X\_r(\mathbf{s}) \tag{19}$$

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 a faster positioning time.

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

**Figure 4.** *The maglev system driving characteristic in open loop.*
