**3.2. Amplifier peripheral circuit design**

As can be seen in Fig. 7, the linear amplifier circuit is composed of resistance R� which determines the amplification ratio of the current amplifier and the amplifier output current, resistance R�, resistance R� which determines the dynamic characteristics of the linear amplifier circuit, condenser C�, R� which limits the linear amplifier circuit current, and the load(here, coil). When defining the form of output desired by the designer using time response characteristics, the amplifier peripheral device values can be determined in the following manner utilizing genetic algorithm.

204 Performance Evaluation of Bearings

**Figure 9.** Genetic algorithm

generation is defined as an algorithm.

process of generating the next generation population after going through the aforementioned series of processes is described as one generation and the method to finding an optimized solution to an objective function through operations in a specific

Genetic algorithms can be categorized into BCGA(Binary Coded Genetic Algorithm), SGA(Signal Genetic Algorithm), and RCGA(Real Coded Genetic Algorithm) depending on the expression of the chromosome. Generally, RCGA is used for optimization problems regarding continuous search domain variable with constraints. This is because if the chromosome is expressed by real code, genes that match perfectly with the variable in question could be used and the degree of precision of the calculation is only dependent on

As can be seen in Fig. 7, the linear amplifier circuit is composed of resistance R� which determines the amplification ratio of the current amplifier and the amplifier output current, resistance R�, resistance R� which determines the dynamic characteristics of the linear amplifier circuit, condenser C�, R� which limits the linear amplifier circuit current, and the

the calculation ability of the computer regardless of the length of the gene.

**3.2. Amplifier peripheral circuit design** 

First, the value of the part to solve is defined. Since the amplification ratio A of the current amplifier, current limiting resistance R�, load inductance L, and resistance R are unknown, the variables of the genetic algorithm to find are limited to the resistance R� that determines the amplification ratio of the amplifier output current, resistance R�, resistance R� which determines the dynamic characteristics of the linear amplifier current, and condenser C�. At this point, if the amplification ratio of the amplifier output current is given, one less genetic algorithm variable needs to be found as the resistances R� and R� have a proportional relationship.

Next, the searching range of the parameters to be identified is limited according to the characteristics of each device. As the resistance R� which determines the amplification ratio of the amplifier output current is a signal resistance, it is desirable to have a high resistance value. Therefore, in the case of resistance R� which determines the amplification ratio of the amplifier output current, it has to be sought in the kΩ range. In contrast, resistance R� which determines the dynamic characteristics of the linear amplifier circuit has to be sought in a wide range. For condenser C�, which determines the dynamic characteristics of the linear amplifier circuit, a value in the nF to μF range is ideal when considering the dynamic characteristics of the current amplifier.

After that, the objective function is determined to implement the genetic algorithm. The objective in this program is the design of a linear current amplifier that has a current output in the form that the designer seeks. Therefore, it has the form shown in Equation (23) and the response of the system that satisfies the time response characteristics defined by the designer is defined as shown in Equation (25).

$$\mathcal{G}\_{\rm r}(\mathbf{s}) = -\frac{\mathbf{e}\_1 \mathbf{s} + \mathbf{e}\_0}{\mathbf{s}^2 + \mathbf{d}\_1 \mathbf{s} + \mathbf{d}\_0} \tag{25}$$

At this point, the randomly given d�, d�, e� and e� are the coefficients of the system G� that satisfies the time response characteristics. The objective function to implement the genetic algorithm is defined as shown in Equation (26).

$$\mathbf{F}\_{\rm obj} = \int \mathbf{e}(\mathbf{t})^2 \mathbf{d}\mathbf{t} \tag{26}$$

Here, e�t� � g��t� � g����t�, g��t� is the step response of the system defined by the designer and g����t� is the step response of the current amplifier transfer function.

Finally, the parameters to operate the genetic algorithm, such as the size of the entity group, the maximum chromosome length, maximum number of generations, crossbreeding probability, and mutation probability, are defined. Here, in order to improve the performance of the implemented genetic algorithm, configuration for methods such as the penalty strategy, elite strategy, and scale fitting method is necessary.

Control of Magnetic Bearing System 207

**Figure 12.** Amplifier applied with the designed parameter value



Voltage[V]



0

manufactured amplifier

RCGA like that of Appendix 1.

**3.3. Magnetic bearing system identification** 

performing experimentation. Such a process is called identification.

**Figure 13.** Step response comparison between the system that obtained from the RCGA results and the

<sup>0</sup> 0.01 0.02 0.03 0.04 0.05 0.06 -2.5

Time[s]

yr y experiment

Appendix 1 is a program that estimates the amplifier peripheral circuit part values in accordance with the response of the system defined arbitrarily, and the result is shown in Fig. 10, Fig. 9, Fig. 11 and Fig. 12 shows the step response measurement graphs of the system that was produced by designing and manufacturing the amplifier circuit using

In order to design the controller for the designed magnetic bearing system, there needs to be a process to estimate the parameter values that exist in the given system and is difficult to measure. In case a magnetic bearing system model with the same response as that of experimentation results can be found, the designer can design the desired controller without

**Figure 10.** Trends of the optimum parameters of each generation

**Figure 11.** Step response comparison between the system obtained from the RCGA results and the system defined by the designer

**Figure 12.** Amplifier applied with the designed parameter value

3 3.5 4 4.5 5 x 10-9

Cf

Rd

Ri

**Figure 10.** Trends of the optimum parameters of each generation

system defined by the designer




Current[A]



0

. **Figure 11.** Step response comparison between the system obtained from the RCGA results and the

Time[s]

0 0.005 0.01 0.015

<sup>0</sup> <sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> <sup>100</sup> <sup>120</sup> <sup>140</sup> <sup>160</sup> <sup>180</sup> <sup>200</sup> <sup>8000</sup>

Generation

<sup>0</sup> <sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> <sup>100</sup> <sup>120</sup> <sup>140</sup> <sup>160</sup> <sup>180</sup> <sup>200</sup> <sup>2</sup>

Generation

<sup>0</sup> <sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> <sup>100</sup> <sup>120</sup> <sup>140</sup> <sup>160</sup> <sup>180</sup> <sup>200</sup> 2.5

Generation

Ri

Rd

Cf

yout yr

**Figure 13.** Step response comparison between the system that obtained from the RCGA results and the manufactured amplifier

Appendix 1 is a program that estimates the amplifier peripheral circuit part values in accordance with the response of the system defined arbitrarily, and the result is shown in Fig. 10, Fig. 9, Fig. 11 and Fig. 12 shows the step response measurement graphs of the system that was produced by designing and manufacturing the amplifier circuit using RCGA like that of Appendix 1.

## **3.3. Magnetic bearing system identification**

In order to design the controller for the designed magnetic bearing system, there needs to be a process to estimate the parameter values that exist in the given system and is difficult to measure. In case a magnetic bearing system model with the same response as that of experimentation results can be found, the designer can design the desired controller without performing experimentation. Such a process is called identification.

$$\frac{\mathbf{u}(\mathbf{s})}{\mathbf{e}(\mathbf{s})} = \mathbf{K\_p} + \frac{\mathbf{K\_l}}{\mathbf{s}} + \mathbf{K\_D}\mathbf{s} \tag{27}$$

$$\frac{\mathbf{u}(\mathbf{z})}{\mathbf{e}(\mathbf{z})} = \mathbf{K}\_{\mathbf{p}} + \frac{\mathbf{K}\_{\mathbf{l}}\mathbf{T}(\mathbf{z} + \mathbf{1})}{\mathbf{z}(\mathbf{z} - \mathbf{1})} + \frac{\mathbf{K}\_{\mathbf{D}}(\mathbf{z} - \mathbf{1})}{\mathbf{z}\mathbf{z}} \tag{28}$$

$$\mathbf{u}(\mathbf{z}) = \frac{2\mathbf{K}\_{\mathrm{p}}\mathrm{Tr}(\mathbf{z}-\mathbf{1}) + \mathbf{K}\_{\mathrm{I}}\mathrm{T}^{2}\mathrm{z}(\mathbf{z}-\mathbf{1}) + 2\mathbf{K}\_{\mathrm{D}}\mathrm{z} - \mathbf{1})^{2}}{2\mathrm{T}\mathrm{z}(\mathbf{z}-\mathbf{1})}$$

$$\mathbf{f}(\mathbf{z}^{2}-\mathbf{z})\mathbf{u}(\mathbf{z}) = \left(\left(\mathrm{K}\_{\mathrm{P}} + \frac{\mathrm{T}}{2}\mathrm{K}\_{\mathrm{I}} + \frac{\mathrm{K}\_{\mathrm{D}}}{\mathrm{T}}\right)\mathrm{z}^{2} + \left(\frac{\mathrm{T}}{2}\mathrm{K}\_{\mathrm{I}} - \mathrm{K}\_{\mathrm{P}} - \frac{2\mathrm{K}\_{\mathrm{D}}}{\mathrm{T}}\right)\mathrm{z} + \frac{\mathrm{K}\_{\mathrm{D}}}{\mathrm{T}}\right)\mathrm{e}(\mathbf{z})\tag{29}$$

$$\mathbf{u}\{\mathbf{n}+\mathbf{2}\} = \left(\mathbf{K\_{P}} + \frac{\mathbf{T}}{2}\mathbf{K\_{I}} + \frac{\mathbf{K\_{D}}}{\mathbf{T}}\right)\mathbf{e(n+2)} + \left(\frac{\mathbf{T}}{2}\mathbf{K\_{I}} - \mathbf{K\_{P}} - \frac{2\mathbf{K\_{D}}}{\mathbf{T}}\right)\mathbf{e(n+1)} + \frac{\mathbf{K\_{D}}}{\mathbf{T}}\mathbf{e(n)} + \mathbf{u(n+1)}\tag{30}$$

$$\mathbf{u(n+2)} = 6.001\mathbf{e(n+2)} - 10.999\mathbf{e(n+1)} + 5\mathbf{e(n)} + \mathbf{u(n+1)}\tag{31}$$

$$\mathbf{G}\_{\rm T} = \frac{\mathbf{b}\_{\rm m3} \mathbf{s}^3 + \mathbf{b}\_{\rm m2} \mathbf{s}^2 + \mathbf{b}\_{\rm m1} \mathbf{s} + \mathbf{b}\_{\rm m0}}{\mathbf{a}\_{\rm m5} \mathbf{s}^5 + \mathbf{a}\_{\rm m4} \mathbf{s}^4 + \mathbf{a}\_{\rm m3} \mathbf{s}^3 + \mathbf{a}\_{\rm m2} \mathbf{s}^2 + \mathbf{a}\_{\rm m1} \mathbf{s} + \mathbf{a}\_{\rm m0}} \tag{32}$$

Appendix 2 is the genetic algorithm program that allows the calculation of the unknown parameters from the above processes. At this point, the program part that is the same with the program to solve the amplifier peripheral circuit was excluded. Fig. 15 and Fig. 16 shows the graphs of the output results of the magnetic bearing system identification using RCGA similarly with Appendix 2.

Control of Magnetic Bearing System 211

Here, i��� is the control current necessary to support the levitating object with only the upper electromagnet and α is the current corresponding to the attractive force caused by the

At this point, the magnetic bearing system has a form symmetric about each axis. Therefore, when assuming the axis parallel to the direction vertical from the Earth as the x-axis, the attractive force control of the y- and z-axes is the same as the x-axis control case including

Fig. 17 is step response test result that is an example. When the program is work, levitated object is attracted by upper electric magnet. After that, the control signal is separated between upper and lower electric magnet. In the step response test, the disturbance mass is 150g.

> Control signal separate

current flowing in the lower electromagnet coil.

the neutral state and excluding the control current I��.

**Figure 17.** Step response test result for levitating system

0

0.2

0.4

0.6

Displacement[mm]

0.8

1

Program start

1.2

motion.

(36) can be obtained.

**3.5. Levitating object and equation of rotational motion** 

Jθ�

� � J�ωθ�

motion about the x-z plane is as shown in Equation (34).

Fig. 18 shows a schematic of a magnetic bearing system taking into consideration rotational

0 10 20 30 40

Reference input change

Time[s]

When the levitating object undergoes rotational motion, the torque caused by the rotational

Here, J is the moment of inertia of the levitating object about the y-axis and J� is the moment of inertia of the rotating levitating object about the x-axis. From Fig. 16, Equations (35) and

� � ����� � ���� (34)

Mass disturbance

**Figure 15.** Trends of optimum parameters for each generation

**Figure 16.** Experiment data comparison with the system obtained through RCGA results

### **3.4. Control signal division**

The system of Fig. 14 is a modeling of the case where the magnetic bearing system is assumed to a horizontally symmetric based on the center point so that the levitating object is supported using an upper based electromagnet about the left or right parts. In order to properly levitate the levitating object of this system, a control signal equal to that of Fig. 14 has to be implemented and consistently supplied to the left and right magnetic bearing.

Also, to divide the control signal that supports the levitating object using only the upper electromagnet like that of Fig. 14 into the upper and lower electromagnet, the current i�� flowing in the upper electromagnet coil has to include the attractive force caused by the current i���� flowing in the lower electromagnet coil, where Equation (33) has to be followed for the design.

$$\mathbf{i}\_{\rm up} = \mathbf{I}\_{\rm ss} + \mathbf{i}\_{\rm up0} + \mathbf{a} \tag{33}$$

Here, i��� is the control current necessary to support the levitating object with only the upper electromagnet and α is the current corresponding to the attractive force caused by the current flowing in the lower electromagnet coil.

At this point, the magnetic bearing system has a form symmetric about each axis. Therefore, when assuming the axis parallel to the direction vertical from the Earth as the x-axis, the attractive force control of the y- and z-axes is the same as the x-axis control case including the neutral state and excluding the control current I��.

Fig. 17 is step response test result that is an example. When the program is work, levitated object is attracted by upper electric magnet. After that, the control signal is separated between upper and lower electric magnet. In the step response test, the disturbance mass is 150g.

**Figure 17.** Step response test result for levitating system

210 Performance Evaluation of Bearings

similarly with Appendix 2.

**3.4. Control signal division** 

followed for the design.

**Figure 15.** Trends of optimum parameters for each generation

3 x 10-4

1

2

Distance[m]

**Figure 16.** Experiment data comparison with the system obtained through RCGA results

The system of Fig. 14 is a modeling of the case where the magnetic bearing system is assumed to a horizontally symmetric based on the center point so that the levitating object is supported using an upper based electromagnet about the left or right parts. In order to properly levitate the levitating object of this system, a control signal equal to that of Fig. 14 has to be implemented and consistently supplied to the left and right magnetic bearing.


Step Response Estimated Value

Time[s]

Also, to divide the control signal that supports the levitating object using only the upper electromagnet like that of Fig. 14 into the upper and lower electromagnet, the current i�� flowing in the upper electromagnet coil has to include the attractive force caused by the current i���� flowing in the lower electromagnet coil, where Equation (33) has to be

i�� � ��� � i��� � � (33)

Appendix 2 is the genetic algorithm program that allows the calculation of the unknown parameters from the above processes. At this point, the program part that is the same with the program to solve the amplifier peripheral circuit was excluded. Fig. 15 and Fig. 16 shows the graphs of the output results of the magnetic bearing system identification using RCGA

### **3.5. Levitating object and equation of rotational motion**

Fig. 18 shows a schematic of a magnetic bearing system taking into consideration rotational motion.

When the levitating object undergoes rotational motion, the torque caused by the rotational motion about the x-z plane is as shown in Equation (34).

$$\text{lbf}\_{\text{y}} - \text{l\_p}\omega\dot{\theta}\_{\text{x}} = -\text{lf}\_{\text{xl}} + \text{lf}\_{\text{xr}} \tag{34}$$

Here, J is the moment of inertia of the levitating object about the y-axis and J� is the moment of inertia of the rotating levitating object about the x-axis. From Fig. 16, Equations (35) and (36) can be obtained.

$$
\Delta \text{sin}\Theta\_\mathbf{x} = \frac{\Delta \mathbf{x}}{\mathbf{l}} \approx \Theta\_\mathbf{x} \tag{35}
$$

Control of Magnetic Bearing System 213

**1. Attractive force calculation of the electromagnet using probable flux paths method** 

the electromagnet is 400turns, and the current flowing in the coil is 1A.

**Figure 19.** Electromagnet core drawing

**Figure 20.** Magnetic circuit of the electromagnet

electromagnet is as shown in Equation (38).

It is assumed that the levitating object is supported by the electromagnet. Here, the gap between the electromagnet and the levitating object is 0.6mm, the number of coil winding to

Fig. 19 shows the drawing of the electromagnet core. Fig. 20 shows the magnetic circuit that satisfies the Probable Flux Paths Method in the core of Fig. 19. The attractive force of the

F� � ������

� �� �� �����

Here, the length of the magnetic path is as shown in Equation (39) according to Fig. 2.

� �

� (38)

$$
\Delta \text{sin}\Theta\_\mathbf{y} = \frac{\Delta \mathbf{y}}{\mathbf{l}} \approx \Theta\_\mathbf{y} \tag{36}
$$

**Figure 18.** Magnetic bearing system taking into consideration rotational motion

When the levitating object undergoes rotational motion, the torque caused by the rotational motion about the y-z plane is as shown in Equation (37).

$$\text{lbf}\_{\text{x}} - \text{l\_{p}}\omega\text{o}\dot{\theta}\_{\text{y}} = -\text{lf}\_{\text{yl}} + \text{lf}\_{\text{yr}} \tag{37}$$

In order to control the rotating levitating object, application of a multi-variable controller using state-space expression is necessary.
