**2.2 Dynamic modeling**

**Figure 2** illustrates the principle diagram of maglev system. The dynamic equation of the maglev system is

$$M\frac{d^2\mathcal{X}}{dt^2} = -F\_m(i,\mathcal{x}) + F\_\mathcal{g} \tag{1}$$

*Fm*ð Þ *i*, *x* ≈*Fm*ð Þþ *io*, *xo*

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

governed by

respectively.

*Kx* = 2*Kio*

*2 /xo 3* .

gain, *Ks*, it becomes

The Eq. (8) is amplified to

where *β* = -*KsKc*/*KaM* and *γ*

ter values are shown in **Table 1**.

**system framework**

**3.1 Control structure**

**91**

Eq. (1), the linear transfer function is

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

*<sup>∂</sup>Fm*ð Þ *<sup>i</sup>*, *<sup>x</sup>*

During levitating, the relationship between the *Fm* (*i*, *x*) and the *Fg* is

where *io* and *x*<sup>o</sup> denote the nominal current and nominal displacement,

*X s*ð Þ

*Gp*ðÞ¼ *s*

Substitute Eqs (5) and (6) into Eq. (1), and undergoes Laplace transform on

*I s*ð Þ <sup>¼</sup> � *Kc*

*I*(s) denote the levitation height and current, respectively, where *Kc* = 2*Kio/ xo*

where *Kc* and *Kx* represent the current and position coefficient, while *X*(s) and

Rewrite Eq. (7) by involving power amplifier gain, *Ka* and sensor sensitivity

*Vs*ð Þ*s Vi*ð Þ*<sup>s</sup>* <sup>¼</sup> � *Ks*

*Gp*ðÞ¼ *s*

As shown in Eq. (9), it proves that the uncompensated system is unstable in open loop because it comprises of one pole located at the right half *s*-plane. Thus, a feedback control system is a vital need to stabilize the system. The system parame-

**feedforward and disturbance compensations (FF PI-PD +** *K***z) control**

This section is devoted to explaining the formulation of FF PI-PD + *K*<sup>z</sup> control approach for the maglev system. Next, the control strategy of FF PI-PD + *K*<sup>z</sup> controller is discussed and followed by the design procedure of the proposed control.

The block diagram of FF PI-PD + *K*<sup>z</sup> control system for positioning and robust control of 1-DOF maglev system is depicted in **Figure 3**. The feedback loop consists of PI-PD control and disturbance compensation scheme, whereas the feedforward loop contains a model-based feedforward control. The FF PI-PD + *K*<sup>z</sup> control system

*<sup>2</sup>* = *Kx*/*M.*

**3. Proportional integral-proportional derivative control with**

Lastly, the stability of the proposed control is examined.

*M <sup>s</sup>*<sup>2</sup> � *Kx M*

> *Ka Kc M <sup>s</sup>*<sup>2</sup> � *Kx M*

*β*

*<sup>∂</sup><sup>i</sup> i t*ðÞþ *<sup>∂</sup>Fm*ð Þ *<sup>i</sup>*, *<sup>x</sup>*

*∂x*

*Mg* ¼ *Fm*ð Þ *io*, *xo* (6)

(7)

(8)

*<sup>s</sup>*<sup>2</sup> � *<sup>γ</sup>*<sup>2</sup> (9)

*x t*ð Þ (5)

*<sup>2</sup>* and

where *M*, *F*m, *Fg, i* and *x* denote the steel ball mass, electromagnetic force, gravitational force, current and levitation height respectively.

The *Fm* (*i*, *x*) is in negative sign indicates that it always functioning in opposite direction against the gravitational force, *Fg.*

The electromagnetic force, *Fm*(*i*, *x*) is described as

$$F\_m(i, \mathbf{x}) = K \frac{i^2}{\mathbf{x}^2} \tag{2}$$

where *K* represents the electromagnetic constant. The gravitational force, *Fg* is denoted as

$$F\_{\mathbf{g}} = \mathbf{M} \mathbf{g} \tag{3}$$

where *g* represents the gravitational acceleration.

Substitute Eqs (2) and (3) into Eq. (1), the dynamic equation of the maglev system can be accordingly rewritten as

$$\frac{d^2x}{dt^2} = -\frac{K}{M}\frac{i^2}{x^2} + \text{g} \tag{4}$$

The Eq. (2) shows the inherent nonlinearities characteristic of the *Fm* (*i*, *x*) which can be linearized by using the Taylor Series approximation at the equilibrium position where

**Figure 2.** *The principal diagrams of the maglev system.*

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

$$F\_m(i, \boldsymbol{\omega}) \approx F\_m(i\_o, \boldsymbol{\omega}\_o) + \frac{\partial F\_m(i, \boldsymbol{\omega})}{\partial \boldsymbol{i}} \dot{\boldsymbol{\epsilon}}(t) + \frac{\partial F\_m(i, \boldsymbol{\omega})}{\partial \boldsymbol{\omega}} \boldsymbol{\omega}(t) \tag{5}$$

During levitating, the relationship between the *Fm* (*i*, *x*) and the *Fg* is governed by

$$\mathcal{M}\_{\mathfrak{g}} = F\_m(i\_o, \mathfrak{x}\_o) \tag{6}$$

where *io* and *x*<sup>o</sup> denote the nominal current and nominal displacement, respectively.

Substitute Eqs (5) and (6) into Eq. (1), and undergoes Laplace transform on Eq. (1), the linear transfer function is

$$\frac{X(s)}{I(s)} = \frac{-\frac{K\_c}{M}}{\left[s^2 - \frac{K\_x}{M}\right]}\tag{7}$$

where *Kc* and *Kx* represent the current and position coefficient, while *X*(s) and *I*(s) denote the levitation height and current, respectively, where *Kc* = 2*Kio/ xo <sup>2</sup>* and *Kx* = 2*Kio 2 /xo 3* .

Rewrite Eq. (7) by involving power amplifier gain, *Ka* and sensor sensitivity gain, *Ks*, it becomes

$$\mathbf{G}\_p(s) = \frac{V\_s(s)}{V\_i(s)} = \frac{-\frac{K\_s}{K\_s}\frac{K\_c}{M}}{\left[s^2 - \frac{K\_x}{M}\right]} \tag{8}$$

The Eq. (8) is amplified to

(Magnelab hall effect current sensor HCT-0010-005) with resolution of 0.38 mA is

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

**Figure 2** illustrates the principle diagram of maglev system. The dynamic equa-

where *M*, *F*m, *Fg, i* and *x* denote the steel ball mass, electromagnetic force,

The *Fm* (*i*, *x*) is in negative sign indicates that it always functioning in opposite

*Fm*ð Þ¼ *<sup>i</sup>*, *<sup>x</sup> <sup>K</sup> <sup>i</sup>*

Substitute Eqs (2) and (3) into Eq. (1), the dynamic equation of the maglev

The Eq. (2) shows the inherent nonlinearities characteristic of the *Fm* (*i*, *x*) which can be linearized by using the Taylor Series approximation at the equilibrium

*d*2 *x dt*<sup>2</sup> ¼ � *<sup>K</sup> M i* 2 2

*dt*<sup>2</sup> ¼ �*Fm*ð Þþ *<sup>i</sup>*, *<sup>x</sup> Fg* (1)

*<sup>x</sup>*<sup>2</sup> (2)

*Fg* ¼ *Mg* (3)

*<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>g</sup>* (4)

used. The controller is implemented at a sampling rate of 1 kHz.

*<sup>M</sup> <sup>d</sup>*<sup>2</sup> *x*

gravitational force, current and levitation height respectively.

The electromagnetic force, *Fm*(*i*, *x*) is described as

where *K* represents the electromagnetic constant.

where *g* represents the gravitational acceleration.

direction against the gravitational force, *Fg.*

The gravitational force, *Fg* is denoted as

system can be accordingly rewritten as

position where

**Figure 2.**

**90**

*The principal diagrams of the maglev system.*

**2.2 Dynamic modeling**

tion of the maglev system is

$$G\_p(\mathfrak{s}) = \frac{\beta}{\mathfrak{s}^2 - \mathfrak{y}^2} \tag{9}$$

where *β* = -*KsKc*/*KaM* and *γ <sup>2</sup>* = *Kx*/*M.*

As shown in Eq. (9), it proves that the uncompensated system is unstable in open loop because it comprises of one pole located at the right half *s*-plane. Thus, a feedback control system is a vital need to stabilize the system. The system parameter values are shown in **Table 1**.
