**Author details**

**6. Conclusions and final remarks**

below:

80 Actuators

device.

The chapter presents the design and test methodology of wideband magnetostrictive resonators, that is, actuators and energy harvesting devices. Detailed conclusions are presented

• The magnetostriction phenomenon as magnetomechanical cross-effect enables new concepts of the research methods for designing of effective actuators utilizing mechanical

• Similarly to piezo transducers, magnetostrictive cores allow to create sensor-actuator reversible devices. The choice whether the transducer is a sensor or an actuator is dictated by the amount of active material. In the smallest scale, even a sensor can be used as an actuator.

• The concept of a mechanical resonator has been introduced, which can be understood as an actuator or an energy harvester. A wideband actuator and a harvester can be the same

• By evaluating the effects of polarization direction, it was found that composites with perpendicular polarization show the highest magnetostriction value in comparison to the others, such as parallel polarized and the ones without polarization in the entire frequency range. What is more, at higher values of frequencies the response of the composite material was comparable with monolithic Terfenol-D, and this is why it can be suggested that the

• The measurement of the deformation of the actuator core was carried out with implementation of the Fiber Bragg grating sensor. This solution allowed to increase the accuracy of

• The magnetostriction of the GMM composites achieved in this work can be further improved by more accurate fabrication parameters, especially by the investigation of polarization and differences in the volume fraction of particles. Moreover, research on different matrixes as binders could show if the behavior of a matrix-powder connection will have

• Magnetostrictive cores based on NdFeB magnets and pure Terfenol-D or its replacement composites were designed and verified on the real constructions of devices. Testing methods and dedicated systems have been developed for applications both as a wideband

• The described method of energy transformation allows the usage of mechanical shock to generate the electric current. The current signal frequencies spectrum, generated in a coil due to the wave movement, is determined by the magnetic resonance frequency. This fact allows the selection of a specific harvester depending on the working environment frequency.

• The authors design and prototype various groups of harvesters, mainly with the magnetic core ("Pulse Power Supply"), for use as power supplies which are capable of producing

measurements and neutralized the effect of the electromagnetic field on the results.

composite material provides the basis for applying it in actuating devices.

influence on the properties of these materials.

actuator or as an energy harvesting device.

tens of watts in a few milliseconds.

vibrations in a selected frequency band, even in an ultrasonic range.

Jerzy Kaleta\*, Rafał Mech and Przemysław Wiewiórski

\*Address all correspondence to: jerzy.kaleta@pwr.edu.pl

Department of Mechanics, Materials Science and Engineering, Wrocław University of Science and Technology, Wrocław, Poland

## **References**


[11] Dong X, Qi M, Guan X, Ou J. Fabrication of Tb0.3Dy0.7Fe2 /epoxy composites: Enhanced uniform magnetostrictive and mechanical properties using a dry process. Journal of Magnetism and Magnetic Materials. 2011;**323**:351-355

**Section 2**

**Control Systems**


**Section 2**

**Control Systems**

[11] Dong X, Qi M, Guan X, Ou J. Fabrication of Tb0.3Dy0.7Fe2

Magnetism and Magnetic Materials. 2011;**323**:351-355

Applied Physics A. 2005;**80**:1563-1566

82 Actuators

and Actuators: A Physical. 2009;**156**(2):350-358

Experimental Methods in Solid Mechanics; 2002. pp. 80-81

Theory and experiments. Acoustics. 2008;**08**:4629-4634

EMBS Annual International Conference; 2006

Geology of Industrial Minerals; 2004

Development Centre; 2000

Technology; 2015

2014;**598**:69-74

uniform magnetostrictive and mechanical properties using a dry process. Journal of

[12] Jianjun T, Zhijun Z, De'an P, Shengen Z. Bonded Terfenol-D composites with low eddy

[13] Kwon OY, Kim JC, Kwon YD, Yang DJ, Lee SH, Lee ZH, Hong SH. Magnetostriction

[14] Dai X, Wen Y, Li P, Yang J, Zhang G. Modeling, characterization and fabrication of vibration energy harvester using Terfenol-D/PZT/Terfenol-D composite transducer. Sensors

[15] Bomba J, Kaleta J. Giant magnetostrictive materials (GMM) as a functional material for construction of sensors and actuators. In: 19th Danubia-Adria Symposium on

[16] Kaleta J, Lewandowski D, Mech R, Wiewiórski P. Energy harvester based on Terfenol-D for low power devices. Interdisciplinary Journal of Engineering Sciences. 2014;**2**(1):8-12

[17] Kaleta J, Kot K, Rikitatt M, Wiewiórski P. Multidof wireless sensor system based on IMU-MEMS technology supported by energy harvesting methods. In: Recent Advances

[18] Monaco E, Lecce L, Natale C, Pirozzi S, May C. Active noise control in turbofan aircrafts:

[19] Pouponneau P, Yahia L, Merhi Y, Epure LM, Martel S. Biocompatibility of candidate materials for the realization of medical microdevices, In: Proceedings of the 28th IEEE

[20] Hedrick JB, Rare Earths in Selected U.S. Defense Applications, 40th Forum on the

[21] Quattrone RF, Berman JB, Trovillion JC, Feickert CA, Kamphaus JM, White SR, Giurgiutiu V, Cohen GL, Tech. Rep., US Army Corps of Engineers and Engineer Research and

[22] Mech R. Magnetomechanical properties of composites based on giant magnetostrictive material powders, (in polish) [PhD thesis]. Wroclaw University of Science and

[23] Kaleta J, Wiewiórski P. Magnetic field distribution detecting and computing methods

[24] Blazejewski W, Gasior P, Kaleta J. Advances in Composite Materials—Ecodesign and Analysis, Application of Optical Fibre Sensors to Measuring the Mechanical Properties

[25] Mech R, Kaleta J, Kot K, Wiewiórski P. High power actuator based on magnetostrictive composite core with temperature drift compensation. Key Engineering Materials.

for experimental mechanics. Engineering Transactions. 2010;**58**(3/4):97-118

of Composite Materials and Structures. Rijeka, Croatia: InTech; 2011

in Integrity-Reliability-Failure: IRF, Edições INEGI; 2013. pp. 359-360

current loss and high magnetostriction. Rare Metals. 2010;**29**(6):579-582

and magnetomechanical properties of grain-oriented Tb0.33Dy0.67Fey

/epoxy composites: Enhanced

/epoxy composite.

**Chapter 5**

Provisional chapter

**Modeling, System Identification, and Control of**

DOI: 10.5772/intechopen.75088

This chapter is dedicated to modeling, system identification, and control of electromagnetic actuators with the main focus on the actuators used in magnetic levitation, in fuel injection systems, and in variable valve timing (VVT). These actuators have a simple structure, good reliability, and low manufacturing costs. However, from control viewpoint, they are nonlinear systems and are open-loop unstable. Therefore, mathematical modeling, system identification-based parameter estimation, and control strategies are presented, when the moving armature is controlled around an equilibrium position or is

Keywords: electromagnetic actuator, modeling, identification, gain scheduled control

Electromagnetic actuators are widely used in the industry, and they transform the electric energy into linear motion. From the large variety of applications, in this chapter we are going

• Fuel injection systems and variable valve timing actuators used in internal combustion

These applications are relevant from control point of view: in the first case, the moving armature is controlled around an equilibrium position; in the second case, the armature might go under control between the two extreme positions—armature open and armature close.

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

controlled between the two extreme positions of the armature.

Modeling, System Identification, and Control of

**Electromagnetic Actuators**

Electromagnetic Actuators

http://dx.doi.org/10.5772/intechopen.75088

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Alexandru Forrai

Abstract

1. Introduction

• Magnetic levitation

to focus on:

engines

Alexandru Forrai

#### **Modeling, System Identification, and Control of Electromagnetic Actuators** Modeling, System Identification, and Control of Electromagnetic Actuators

DOI: 10.5772/intechopen.75088

#### Alexandru Forrai Alexandru Forrai

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.75088

#### Abstract

This chapter is dedicated to modeling, system identification, and control of electromagnetic actuators with the main focus on the actuators used in magnetic levitation, in fuel injection systems, and in variable valve timing (VVT). These actuators have a simple structure, good reliability, and low manufacturing costs. However, from control viewpoint, they are nonlinear systems and are open-loop unstable. Therefore, mathematical modeling, system identification-based parameter estimation, and control strategies are presented, when the moving armature is controlled around an equilibrium position or is controlled between the two extreme positions of the armature.

Keywords: electromagnetic actuator, modeling, identification, gain scheduled control

#### 1. Introduction

Electromagnetic actuators are widely used in the industry, and they transform the electric energy into linear motion. From the large variety of applications, in this chapter we are going to focus on:


These applications are relevant from control point of view: in the first case, the moving armature is controlled around an equilibrium position; in the second case, the armature might go under control between the two extreme positions—armature open and armature close.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Nevertheless, magnetic levitation—in particular a magnetically levitated train—is a good example, where closed-loop control plays a key role, since the open-loop system is unstable [1]. The system can be linearized around an operating point, and a linear controller can be designed.

Furthermore, magnetic bearings and their control are from a long time the focus of control system design community. Feedback linearization and asymptotically exact linearization of an active magnetic bearing are presented in [2, 3]. Advanced control strategies are discussed in detail in [4, 5].

Therefore, it makes sense to develop high-accuracy mathematical models, to investigate methods for parameter identification, and finally to apply control strategies to improve performance and reliability of the system.

We will discuss these topics in the next sections, but before that let us focus on applications, where electromagnetic actuators are widely used.

#### 1.1. Magnetic levitation

Magnetic bearings in combination with high-speed electric motors are used across many industries, from oil and gas industry to electric power generation industry (i.e., high-speed electric generators) and from the semiconductor industry to nuclear industry, etc.

The main structure of the magnetic bearing is shown in Figure 1 (reproduced from [6]).

Another well-known application of the magnetic levitation is the magnetically levitated highspeed train (Maglev; see Figure 1), having speeds over 500 [km/h] [7, 8].

#### 1.2. Fuel injection and variable valve timing (VVT)

The main purpose of the fuel injection system is to deliver fuel to the cylinders. However, how that fuel is delivered is that it makes the difference in engine performance, emissions, and noise characteristics.

Most notable advances achieved in diesel engines resulted directly from superior fuel injection

Modeling, System Identification, and Control of Electromagnetic Actuators

http://dx.doi.org/10.5772/intechopen.75088

87

Unlike its spark-ignited engine counterpart, the diesel fuel injection system delivers fuel under extremely high injection pressures (e.g., around 2000 [bar]). This means that the system com-

The actuators used in diesel fuel injection systems can be either electromagnetic (our focus) or piezoelectric [11]. A diesel fuel injection system using an electromagnetic actuator, from Bosch

Nowadays, most of the fuel injection systems are electronically controlled. However, it is still not enough to deliver an accurate amount of fuel at the proper time to achieve good combustion. Additional aspects are critical to ensure proper fuel injection system performance, such as [9]:

• Fuel atomization—ensuring that fuel atomizes into very small fuel particles is a primary

• Bulk mixing—while fuel atomization and complete evaporation of fuel are critical, ensuring that the evaporated fuel has sufficient oxygen during combustion is equally important

• Air utilization—effective utilization of the air in the combustion chamber is closely tied to bulk mixing and can be accomplished by dividing the total injected fuel into a number of jets.

While conventional fuel injection systems employ a single injection event for every engine

Using multiple injections—during every engine cycle—higher engine performance and lower engine noise can be achieved. However, the injector lifetime might be reduced, and therefore advanced control algorithms as well as malfunction detection and fault isolation algorithms

ponent designs and materials should be selected to withstand higher stresses [10].

design objective for diesel fuel injection systems.

cycle, newer systems can use multiple injection events [12].

to ensure optimum engine performance.

can be applied (see next sections).

Figure 2. Diesel fuel injection system from Bosch.

system designs [9].

[12], is shown in Figure 2.

Figure 1. Applications of magnetic levitation.

Most notable advances achieved in diesel engines resulted directly from superior fuel injection system designs [9].

Unlike its spark-ignited engine counterpart, the diesel fuel injection system delivers fuel under extremely high injection pressures (e.g., around 2000 [bar]). This means that the system component designs and materials should be selected to withstand higher stresses [10].

The actuators used in diesel fuel injection systems can be either electromagnetic (our focus) or piezoelectric [11]. A diesel fuel injection system using an electromagnetic actuator, from Bosch [12], is shown in Figure 2.

Nowadays, most of the fuel injection systems are electronically controlled. However, it is still not enough to deliver an accurate amount of fuel at the proper time to achieve good combustion. Additional aspects are critical to ensure proper fuel injection system performance, such as [9]:


While conventional fuel injection systems employ a single injection event for every engine cycle, newer systems can use multiple injection events [12].

Using multiple injections—during every engine cycle—higher engine performance and lower engine noise can be achieved. However, the injector lifetime might be reduced, and therefore advanced control algorithms as well as malfunction detection and fault isolation algorithms can be applied (see next sections).

Figure 2. Diesel fuel injection system from Bosch.

Nevertheless, magnetic levitation—in particular a magnetically levitated train—is a good example, where closed-loop control plays a key role, since the open-loop system is unstable [1]. The system can be linearized around an operating point, and a linear controller can be designed.

Furthermore, magnetic bearings and their control are from a long time the focus of control system design community. Feedback linearization and asymptotically exact linearization of an active magnetic bearing are presented in [2, 3]. Advanced control strategies are discussed in

Therefore, it makes sense to develop high-accuracy mathematical models, to investigate methods for parameter identification, and finally to apply control strategies to improve per-

We will discuss these topics in the next sections, but before that let us focus on applications,

Magnetic bearings in combination with high-speed electric motors are used across many industries, from oil and gas industry to electric power generation industry (i.e., high-speed

Another well-known application of the magnetic levitation is the magnetically levitated high-

The main purpose of the fuel injection system is to deliver fuel to the cylinders. However, how that fuel is delivered is that it makes the difference in engine performance, emissions, and noise

electric generators) and from the semiconductor industry to nuclear industry, etc.

speed train (Maglev; see Figure 1), having speeds over 500 [km/h] [7, 8].

The main structure of the magnetic bearing is shown in Figure 1 (reproduced from [6]).

detail in [4, 5].

86 Actuators

formance and reliability of the system.

1.1. Magnetic levitation

characteristics.

Figure 1. Applications of magnetic levitation.

where electromagnetic actuators are widely used.

1.2. Fuel injection and variable valve timing (VVT)

Another relevant application is the electromechanical valve actuators used in automotive engines, to achieve variable valve timing (VVT). With VVT, larger valve overlap, valve lift, duration, and timing adjustments can be achieved depending on engine speed, load, and temperature.

Variable valve timing leads to improved fuel economy and lower emissions by decoupling the valve timing from the piston motion [13]. This is especially valid in case of advanced combustion technologies, as described in [14, 15].

However, the moving components of the valve actuators create unnecessary wear and excessive noise. The armature landing speed shall be kept, e.g., under 0.1 [m/s]; otherwise, they are excessively loud and are damaging to the actuator and engine valve.

Whenever high-performance and high-accuracy control is required, the electromagnetic actuator is driven by a half H-bridge (see Figure 3), which might be equipped optionally with a current sensing resistor RSENSE.

Typical voltage and current waveforms as well as the switching order of the commutation elements T<sup>1</sup> and T<sup>2</sup> are shown in Figure 3.

After the electromagnetic armature is pulled up, the actuator current is reduced, and the commutation elements are controlled via pulse-width modulation (PWM).

During armature movement, due to the induced electromotive force (e.m.f.), a small current dip as well as a small current peak might be observed (see Figure 3).

The duty factor of the actuator—specified on the data sheet—is defined as

$$[Duty] \text{[\%]} = \frac{T\_{ON}}{T\_{ON} + T\_{OFF}} 100 \text{[\%]} \tag{1}$$

2. Mathematical modeling

and by the motion equation.

The voltage equation is

Then, we can also write

flux linkage for one fixed position z is calculated by

written as

The mathematical model of the electromagnetic actuator is described by the voltage equation

∂Ψ ∂i di dt þ

where vin is the applied voltage, i is the armature current, Ψ is the armature flux, z is the

If we note with v ¼ z\_, the armature speed and then the equation of the motion can be

where m is the moving mass, Fm is the electromagnetic force, and FS is the spring force. Since the armature displacement often is very short, the spring force can be considered constant. In case of magnetic levitation, the spring force is replaced by the weight of the moving mass.

> ði 0

The electromagnetic force can be expressed based on the electromagnetic co-energy Wco:

Wco ¼

Fm ¼ � <sup>∂</sup>Wco ∂z � � � � i¼ct

Fm ¼ � <sup>ð</sup><sup>i</sup>

ð∞ 0

Ψð Þ¼ i; z

where at t ¼ 0 and the following conditions hold: vin ¼ 0, i 6¼ 0, and di=dt ¼ 0.

model can be very accurate and might be written formally into a nonlinear form:

0 ∂Ψ ∂z

The Ψ ¼ Ψð Þ i; z and Fm ¼ Fmð Þ i; z static characteristics can be measured. The flux-linkage characteristic is derived by integration (very often the current decay test is used). Thus, the

Although the model does not take into account the effect of eddy currents, the numerical

∂Ψ ∂z dz

dt (2)

http://dx.doi.org/10.5772/intechopen.75088

mz€ ¼ FS � Fm (3)

Modeling, System Identification, and Control of Electromagnetic Actuators

Ψdi (4)

di (6)

½ � vinð Þ� t R � i tð Þ dt (7)

(5)

89

vin ¼ Ri þ

armature position, and R is the electrical resistance of the coil.

In practice, exceeding this value might shorten significantly the lifetime of the actuator.

Figure 3. Electromagnetic actuator driven by a half H-bridge.

#### 2. Mathematical modeling

The mathematical model of the electromagnetic actuator is described by the voltage equation and by the motion equation.

The voltage equation is

Another relevant application is the electromechanical valve actuators used in automotive engines, to achieve variable valve timing (VVT). With VVT, larger valve overlap, valve lift, duration, and

Variable valve timing leads to improved fuel economy and lower emissions by decoupling the valve timing from the piston motion [13]. This is especially valid in case of advanced combus-

However, the moving components of the valve actuators create unnecessary wear and excessive noise. The armature landing speed shall be kept, e.g., under 0.1 [m/s]; otherwise, they are

Whenever high-performance and high-accuracy control is required, the electromagnetic actuator is driven by a half H-bridge (see Figure 3), which might be equipped optionally with a

Typical voltage and current waveforms as well as the switching order of the commutation

After the electromagnetic armature is pulled up, the actuator current is reduced, and the

During armature movement, due to the induced electromotive force (e.m.f.), a small current

TON þ TOFF

100½ � % (1)

timing adjustments can be achieved depending on engine speed, load, and temperature.

excessively loud and are damaging to the actuator and engine valve.

commutation elements are controlled via pulse-width modulation (PWM).

The duty factor of the actuator—specified on the data sheet—is defined as

Duty½ �¼ % TON

In practice, exceeding this value might shorten significantly the lifetime of the actuator.

dip as well as a small current peak might be observed (see Figure 3).

tion technologies, as described in [14, 15].

elements T<sup>1</sup> and T<sup>2</sup> are shown in Figure 3.

Figure 3. Electromagnetic actuator driven by a half H-bridge.

current sensing resistor RSENSE.

88 Actuators

$$w\_{in} = Ri + \frac{\partial \Psi}{\partial \mathbf{i}} \frac{d\mathbf{i}}{dt} + \frac{\partial \Psi}{\partial z} \frac{dz}{dt} \tag{2}$$

where vin is the applied voltage, i is the armature current, Ψ is the armature flux, z is the armature position, and R is the electrical resistance of the coil.

If we note with v ¼ z\_, the armature speed and then the equation of the motion can be written as

$$
\hbar \mathbf{m} \ddot{\mathbf{z}} = F \mathbf{s} - F\_m \tag{3}
$$

where m is the moving mass, Fm is the electromagnetic force, and FS is the spring force. Since the armature displacement often is very short, the spring force can be considered constant. In case of magnetic levitation, the spring force is replaced by the weight of the moving mass.

The electromagnetic force can be expressed based on the electromagnetic co-energy Wco:

$$\mathcal{W}\_{\text{co}} = \int\_0^i \Psi di\tag{4}$$

$$F\_m = -\frac{\partial W\_{co}}{\partial z}\Big|\_{i=ct} \tag{5}$$

Then, we can also write

$$F\_m = -\int\_0^i \frac{\partial \Psi}{\partial z} di\tag{6}$$

The Ψ ¼ Ψð Þ i; z and Fm ¼ Fmð Þ i; z static characteristics can be measured. The flux-linkage characteristic is derived by integration (very often the current decay test is used). Thus, the flux linkage for one fixed position z is calculated by

$$\Psi(i,z) = \int\_0^\infty [v\_{in}(t) - R \cdot i(t)]dt\tag{7}$$

where at t ¼ 0 and the following conditions hold: vin ¼ 0, i 6¼ 0, and di=dt ¼ 0.

Although the model does not take into account the effect of eddy currents, the numerical model can be very accurate and might be written formally into a nonlinear form:

$$\dot{\mathbf{x}} = f(\mathbf{x}) + \sum\_{i=1}^{m} \mathbf{g}(\mathbf{x})u \tag{8}$$

If we approximate the exponential term by Taylor series, we have

c<sup>1</sup> þ c2z � � <sup>≈</sup> <sup>1</sup> � <sup>i</sup>

Thus, the magnetic force—based on the analytical model—can be expressed as

2

2 1 <sup>2</sup> � <sup>i</sup>

c<sup>1</sup> þ c2z

3ð Þ c<sup>1</sup> þ c2z

The model set above is validated against the measured static (flux and force) characteristics. The "dots" in Figure 4 represent the measured data, and the solid lines represent the calculated

The above parameters are derived using nonlinear least squares, fitting the measured data (obtained using the current decay test and force measurements) with the analytical model.

Next, let us introduce the following notations, which help us to rewrite the model in a

� <sup>i</sup> � <sup>χ</sup><sup>z</sup> χi � v þ 1 χi

dz

<sup>m</sup> � <sup>i</sup> � <sup>k</sup>

<sup>m</sup> � <sup>z</sup> <sup>þ</sup>

1 m

model using the above model set, with <sup>Ψ</sup>max <sup>¼</sup> <sup>0</sup>:45½ � Wb , <sup>c</sup><sup>1</sup> <sup>¼</sup> <sup>0</sup>:4½ � <sup>A</sup> , and <sup>c</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>:<sup>375</sup> � 103

þ

þ

" #

i 2 2ð Þ c<sup>1</sup> þ c2z

Modeling, System Identification, and Control of Electromagnetic Actuators

i 2 8ð Þ c<sup>1</sup> þ c2z

2

2

dt <sup>¼</sup> <sup>v</sup> (16)

<sup>2</sup> (13)

http://dx.doi.org/10.5772/intechopen.75088

, when the current is zero

vin (15)

FSð Þ0 (17)

(14)

91

½ � A=m .

exp � <sup>i</sup>

Fm <sup>≈</sup> <sup>Ψ</sup>max � <sup>c</sup><sup>2</sup> � <sup>i</sup>

convenient form: χ<sup>i</sup> ¼ ∂Ψ=∂i, χ<sup>z</sup> ¼ ∂Ψ=∂z, and χ<sup>f</sup> ¼ Fm=i.

Thus, the voltage and motion equations are written as

Figure 4. Flux and force characteristics.

Since the magnetic force Fm depends on the square of the current i

di dt ¼ � <sup>R</sup> χi

dv dt ¼ � <sup>χ</sup><sup>f</sup>

i ¼ 0 and then χ<sup>f</sup> ¼ 0, there is no division by zero in the model.

ð Þ c<sup>1</sup> þ c2z

where <sup>x</sup> <sup>¼</sup> ½ � ivz <sup>T</sup> represents the state of the nonlinear system, <sup>u</sup> <sup>¼</sup> ½ � vin FS <sup>T</sup> is the input vector, and f xð Þ and g xð Þ are nonlinear functions of the state x. The output vector y is

$$y = h(\mathbf{x})\tag{9}$$

where h xð Þ in the most general case is a nonlinear function.

Finally, in the aim to illustrate our investigations, let us consider an electromagnetic actuator with parameters (catalog data) mentioned in Table 1 [16].

#### 2.1. Nonlinear model and piecewise linearization

The mathematical model described above is too general and is difficult to handle in analytical form. Therefore, we define an analytical model set, which describes the flux-linkage characteristic as

$$\Psi(i,z) = \Psi\_{\text{max}} \left[ 1 - \exp\left(-\frac{i}{c\_1 + c\_2 z}\right) \right] \tag{10}$$

where the parameters of the model set are Ψmax, c1, and c2.

Furthermore, the partial derivatives of the flux-linkage are

$$\frac{\partial \Psi}{\partial i} = \frac{\Psi\_{\text{max}}}{c\_1 + c\_2 z} \exp\left(-\frac{i}{c\_1 + c\_2 z}\right) \tag{11}$$

$$\frac{\partial \Psi}{\partial z} = -\frac{\Psi\_{\text{max}} \cdot c\_2 \cdot i}{\left(c\_1 + c\_2 z\right)^2} \exp\left(-\frac{i}{c\_1 + c\_2 z}\right) \tag{12}$$

The approach presented in this section is reproduced from [17].


Table 1. The solenoid parameters.

If we approximate the exponential term by Taylor series, we have

<sup>x</sup>\_ <sup>¼</sup> f xð ÞþX<sup>m</sup>

where <sup>x</sup> <sup>¼</sup> ½ � ivz <sup>T</sup> represents the state of the nonlinear system, <sup>u</sup> <sup>¼</sup> ½ � vin FS

where h xð Þ in the most general case is a nonlinear function.

with parameters (catalog data) mentioned in Table 1 [16].

where the parameters of the model set are Ψmax, c1, and c2. Furthermore, the partial derivatives of the flux-linkage are

∂Ψ

∂Ψ

The approach presented in this section is reproduced from [17].

Table 1. The solenoid parameters.

<sup>∂</sup><sup>i</sup> <sup>¼</sup> <sup>Ψ</sup>max c<sup>1</sup> þ c2z

<sup>∂</sup><sup>z</sup> ¼ � <sup>Ψ</sup>max � <sup>c</sup><sup>2</sup> � <sup>i</sup> ð Þ c<sup>1</sup> þ c2z

2.1. Nonlinear model and piecewise linearization

istic as

90 Actuators

vector, and f xð Þ and g xð Þ are nonlinear functions of the state x. The output vector y is

i¼1

Finally, in the aim to illustrate our investigations, let us consider an electromagnetic actuator

The mathematical model described above is too general and is difficult to handle in analytical form. Therefore, we define an analytical model set, which describes the flux-linkage character-

c<sup>1</sup> þ c2z

c<sup>1</sup> þ c2z � �

> c<sup>1</sup> þ c2z � �

� � � �

exp � <sup>i</sup>

Type Solenoid valve Stroke length 10 [mm] Operating voltage 24 [V] d.c. Maximum current 0.6 [A] Resistance 40 [Ω] Inductance 0.35–1.1 [H] Number of turns 2240

<sup>2</sup> exp � <sup>i</sup>

<sup>Ψ</sup>ð Þ¼ <sup>i</sup>; <sup>z</sup> <sup>Ψ</sup>max <sup>1</sup> � exp � <sup>i</sup>

g xð Þu (8)

y ¼ h xð Þ (9)

<sup>T</sup> is the input

(10)

(11)

(12)

$$\exp\left(-\frac{i}{c\_1 + c\_2 z}\right) \approx 1 - \frac{i}{c\_1 + c\_2 z} + \frac{i^2}{2(c\_1 + c\_2 z)^2} \tag{13}$$

Thus, the magnetic force—based on the analytical model—can be expressed as

$$F\_m \approx \frac{\Psi\_{\max} \cdot c\_2 \cdot i^2}{\left(c\_1 + c\_2 z\right)^2} \left[\frac{1}{2} - \frac{i}{3\left(c\_1 + c\_2 z\right)} + \frac{i^2}{8\left(c\_1 + c\_2 z\right)^2}\right] \tag{14}$$

The model set above is validated against the measured static (flux and force) characteristics. The "dots" in Figure 4 represent the measured data, and the solid lines represent the calculated model using the above model set, with <sup>Ψ</sup>max <sup>¼</sup> <sup>0</sup>:45½ � Wb , <sup>c</sup><sup>1</sup> <sup>¼</sup> <sup>0</sup>:4½ � <sup>A</sup> , and <sup>c</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>:<sup>375</sup> � 103 ½ � A=m . The above parameters are derived using nonlinear least squares, fitting the measured data (obtained using the current decay test and force measurements) with the analytical model.

Next, let us introduce the following notations, which help us to rewrite the model in a convenient form: χ<sup>i</sup> ¼ ∂Ψ=∂i, χ<sup>z</sup> ¼ ∂Ψ=∂z, and χ<sup>f</sup> ¼ Fm=i.

Since the magnetic force Fm depends on the square of the current i 2 , when the current is zero i ¼ 0 and then χ<sup>f</sup> ¼ 0, there is no division by zero in the model.

Thus, the voltage and motion equations are written as

$$\frac{di}{dt} = -\frac{R}{\chi\_i} \cdot i - \frac{\chi\_z}{\chi\_i} \cdot \upsilon + \frac{1}{\chi\_i} \upsilon\_{in} \tag{15}$$

$$\frac{dz}{dt} = v\tag{16}$$

$$\frac{d\upsilon}{dt} = -\frac{\chi\_f}{m} \cdot \mathbf{i} - \frac{\mathbf{k}}{m} \cdot \mathbf{z} + \frac{1}{m} F\_S(\mathbf{0}) \tag{17}$$

Figure 4. Flux and force characteristics.

Finally, using a piecewise approximation, the system can be written in state-space form as

$$\begin{aligned} \dot{\mathbf{x}} &= A(\mathbf{i}, \mathbf{z}) \cdot \mathbf{x} + B(\mathbf{i}, \mathbf{z}) \cdot \mathbf{u} \\ \mathbf{y} &= \mathbf{C} \cdot \mathbf{x} \end{aligned} \tag{18}$$

2.2. Linearized mathematical model

Let us approximate the magnetic force as

The equation of motion can be written as

M zð Þ¼ ; z€; i M zð Þþ <sup>0</sup>; z€0; i<sup>0</sup>

M zð Þ¼� ; <sup>z</sup>€; <sup>i</sup> <sup>2</sup>γ<sup>i</sup>

which can be further written as

where γ is a constant.

From control engineering viewpoint—in case of some applications (e.g., magnetic levitation) the piecewise linearized model might be too sophisticated. Therefore, in this section a linear-

> i 2 ð Þ c<sup>1</sup> þ c2z

> > i 2 ð Þ c<sup>1</sup> þ c2z

> > > ∂M ∂z€

 p0

2γi<sup>0</sup> ð Þ c<sup>1</sup> þ c2z<sup>0</sup>

ð Þþ z€ � z€<sup>0</sup>

Modeling, System Identification, and Control of Electromagnetic Actuators

2γi<sup>0</sup> ð Þ c<sup>1</sup> þ c2z<sup>0</sup>

<sup>2</sup> � <sup>a</sup><sup>2</sup> <sup>Δ</sup>z sð Þþ <sup>k</sup>Δi sð Þ¼ <sup>0</sup> (30)

∂M ∂i p0

<sup>2</sup> (25)

http://dx.doi.org/10.5772/intechopen.75088

93

<sup>2</sup> ¼ 0 (26)

ð Þ i � i<sup>0</sup> (27)

<sup>2</sup> ð Þ¼ i � i<sup>0</sup> 0 (28)

<sup>2</sup> Δi ¼ 0 (29)

ð Þ <sup>s</sup> � <sup>a</sup> ð Þ <sup>s</sup> <sup>þ</sup> <sup>a</sup> (31)

<sup>2</sup> (32)

Fm ≈ γ

M zð Þ¼ ; z€; i mz€ � Fs þ γ

∂M ∂z <sup>p</sup><sup>0</sup>

2 0 ð Þ c<sup>1</sup> þ c2z<sup>0</sup>

> 2 0 ð Þ c<sup>1</sup> þ c2z<sup>0</sup>

> > s

<sup>Δ</sup>i sð Þ ¼ � <sup>k</sup>

Δz sð Þ

where k and a are varying with the equilibrium point ð Þ i0; z<sup>0</sup> :

If we denote with Δz ¼ z � z<sup>0</sup> and Δi ¼ i � i0, we obtain

� <sup>2</sup>γ<sup>i</sup>

 

The equation above can be linearized around an operating point p<sup>0</sup> ¼ ð Þ z0; z€0; i<sup>0</sup> as follows:

ð Þþ z � z<sup>0</sup>

<sup>3</sup> ð Þþ z � z<sup>0</sup> mð Þþ z€ � z€<sup>0</sup>

<sup>3</sup> Δz þ mΔz€ þ

If we divide the equation with the moving mass m and apply the Laplace transform, we obtain

<sup>s</sup><sup>2</sup> � <sup>a</sup><sup>2</sup> ¼ � <sup>k</sup>

<sup>k</sup> <sup>¼</sup> <sup>2</sup>γi<sup>0</sup> m cð Þ <sup>1</sup> þ c2z<sup>0</sup>

ized mathematical model around an operating point is derived [1, 18].

where <sup>x</sup> <sup>¼</sup> ½ � izv <sup>T</sup> is the state-space vector, <sup>u</sup> <sup>¼</sup> ½ � vin FSð Þ<sup>0</sup> <sup>T</sup> is the input vector, <sup>y</sup> is the output vector, the A ið Þ ; z and B ið Þ ; z are current and position dependent matrices and C ¼ ½ � 110 if the armature current and position are sensed. We remark, that in practice sensing the armature speed and/or positions with sensor(s) might be expensive solution. Therefore, often only the armature current can be sensed in a cost-effective manner.

The terms A ið Þ ; z can be written as

$$A(i,z) = \begin{bmatrix} -\mathcal{R}/\chi\_i & 0 & -\chi\_z/\chi\_i \\ 0 & 0 & 1 \\ -\chi\_f/m & -k/m & 0 \end{bmatrix} \tag{19}$$

where

$$-\frac{\chi\_z}{\chi\_i} = \frac{c\_2 \cdot i}{c\_1 + c\_2 z} \tag{20}$$

The term B ið Þ ; z can be written as

$$B(i,z) = \begin{bmatrix} 1/\chi\_i & 0\\ 0 & 0\\ 0 & 1/m \end{bmatrix} \tag{21}$$

where

$$\frac{1}{\chi\_i} \approx \frac{2(c\_1 + c\_2 z)^2 + 2i(c\_1 + c\_2 z) + i^2}{2 \cdot \Psi\_{\text{max}} \cdot (c\_1 + c\_2 z)}\tag{22}$$

or a coarser approximation will be

$$\frac{1}{\chi\_i} \approx \frac{c\_1 + c\_2 z + i}{\Psi\_{\text{max}}} \tag{23}$$

Last but not least, the armature movement is subject to the following constraints:

$$w(t) = \begin{cases} 0 & \text{if} \quad z \ge z\_{\text{max}} \quad and \quad F\_s - F\_m \ge 0\\ 0 & \text{if} \quad z \le 0 \quad \text{and} \quad F\_s - F\_m \le 0 \end{cases} \tag{24}$$

as well as zmin ≤ z tð Þ ≤ zmax, where zmin and zmax are the minimum and maximum displacements of the armature.

#### 2.2. Linearized mathematical model

From control engineering viewpoint—in case of some applications (e.g., magnetic levitation) the piecewise linearized model might be too sophisticated. Therefore, in this section a linearized mathematical model around an operating point is derived [1, 18].

Let us approximate the magnetic force as

$$F\_m \approx \gamma \frac{\dot{\mathbf{r}}^2}{\left(c\_1 + c\_2 \mathbf{z}\right)^2} \tag{25}$$

where γ is a constant.

Finally, using a piecewise approximation, the system can be written in state-space form as

y ¼ C � x

2 6 4

> � χz χi

B ið Þ¼ ; z

<sup>≈</sup> <sup>2</sup>ð Þ <sup>c</sup><sup>1</sup> <sup>þ</sup> <sup>c</sup>2<sup>z</sup>

1 χi

Last but not least, the armature movement is subject to the following constraints:

�

<sup>≈</sup> <sup>c</sup><sup>1</sup> <sup>þ</sup> <sup>c</sup>2<sup>z</sup> <sup>þ</sup> <sup>i</sup> Ψmax

v tðÞ¼ <sup>0</sup> if z <sup>≥</sup> zmax and Fs � Fm <sup>≥</sup> <sup>0</sup>

as well as zmin ≤ z tð Þ ≤ zmax, where zmin and zmax are the minimum and maximum displacements

0 if z ≤ 0 and Fs � Fm ≤ 0

armature current can be sensed in a cost-effective manner.

A ið Þ¼ ; z

1 χi

The terms A ið Þ ; z can be written as

The term B ið Þ ; z can be written as

or a coarser approximation will be

where

92 Actuators

where

of the armature.

x\_ ¼ A ið Þ� ; z x þ B ið Þ� ; z u

where <sup>x</sup> <sup>¼</sup> ½ � izv <sup>T</sup> is the state-space vector, <sup>u</sup> <sup>¼</sup> ½ � vin FSð Þ<sup>0</sup> <sup>T</sup> is the input vector, <sup>y</sup> is the output vector, the A ið Þ ; z and B ið Þ ; z are current and position dependent matrices and C ¼ ½ � 110 if the armature current and position are sensed. We remark, that in practice sensing the armature speed and/or positions with sensor(s) might be expensive solution. Therefore, often only the

> �R=χ<sup>i</sup> 0 �χz=χ<sup>i</sup> 001 �χ<sup>f</sup> =m �k=m 0

> > <sup>¼</sup> <sup>c</sup><sup>2</sup> � <sup>i</sup>

1=χ<sup>i</sup> 0 0 0 0 1=m

<sup>2</sup> <sup>þ</sup> <sup>2</sup>i cð Þþ <sup>1</sup> <sup>þ</sup> <sup>c</sup>2<sup>z</sup> <sup>i</sup>

3 7

2

<sup>2</sup> � <sup>Ψ</sup>max � ð Þ <sup>c</sup><sup>1</sup> <sup>þ</sup> <sup>c</sup>2<sup>z</sup> (22)

2 6 4 3 7

<sup>c</sup><sup>1</sup> <sup>þ</sup> <sup>c</sup>2<sup>z</sup> (20)

<sup>5</sup> (21)

<sup>5</sup> (19)

(18)

(23)

(24)

The equation of motion can be written as

$$M(z, \ddot{z}, \dot{\mathbf{i}}) = m\ddot{z} - F\_s + \gamma \frac{\dot{\mathbf{i}}^2}{\left(c\_1 + c\_2 z\right)^2} = \mathbf{0} \tag{26}$$

The equation above can be linearized around an operating point p<sup>0</sup> ¼ ð Þ z0; z€0; i<sup>0</sup> as follows:

$$M(z,\ddot{z},\mathrm{i}) = M(z\_0,\ddot{z}\_0,\mathrm{i}\_0) + \frac{\partial M}{\partial \boldsymbol{z}}\Big|\_{p\_0} (z - z\_0) + \frac{\partial M}{\partial \ddot{z}}\Big|\_{p\_0} (\ddot{z} - \ddot{z}\_0) + \frac{\partial M}{\partial \boldsymbol{i}}\Big|\_{p\_0} (\mathbf{i} - \mathbf{i}\_0) \tag{27}$$

which can be further written as

$$M(\mathbf{z}, \ddot{\mathbf{z}}, \dot{\mathbf{i}}) = -\frac{2\gamma \dot{\mathbf{r}}\_0^2}{\left(\mathbf{c}\_1 + \mathbf{c}\_2 \mathbf{z}\_0\right)^3} (\mathbf{z} - \mathbf{z}\_0) + m(\ddot{\mathbf{z}} - \ddot{\mathbf{z}}\_0) + \frac{2\gamma \dot{\mathbf{r}}\_0}{\left(\mathbf{c}\_1 + \mathbf{c}\_2 \mathbf{z}\_0\right)^2} (\mathbf{i} - \mathbf{i}\_0) = \mathbf{0} \tag{28}$$

If we denote with Δz ¼ z � z<sup>0</sup> and Δi ¼ i � i0, we obtain

$$-\frac{2\gamma i\_0^2}{\left(c\_1 + c\_2 z\_0\right)^3} \Delta z + m \Delta \ddot{z} + \frac{2\gamma i\_0}{\left(c\_1 + c\_2 z\_0\right)^2} \Delta \dot{i} = 0 \tag{29}$$

If we divide the equation with the moving mass m and apply the Laplace transform, we obtain

$$(s^2 - a^2)\Delta z(s) + k\Delta \dot{z}(s) = 0\tag{30}$$

$$\frac{\Delta z(s)}{\Delta \dot{i}(s)} = -\frac{k}{s^2 - a^2} = -\frac{k}{(s-a)(s+a)}\tag{31}$$

where k and a are varying with the equilibrium point ð Þ i0; z<sup>0</sup> :

$$k = \frac{2\gamma i\_0}{m(c\_1 + c\_2 z\_0)^2} \tag{32}$$

Figure 5. Plant gain and pole variation with the equilibrium position.

$$a^2 = \frac{2\gamma r\_0^2}{m(c\_1 + c\_2 z\_0)^3} \tag{33}$$

θ ¼ ½a1…an b0…bm�

Now, suppose for a given system that we do not know the values of the parameters in θ, but we have recorded inputs and outputs over the time interval. If the input signal is persistently exciting—condition described in details in [19, 20]—then the solution can easily be computed

In this section a clustering-based identification method, proposed by Ferrari-Trecate et al. (2003) is used (see [21, 22]), where the plant is assumed to be described by piecewise linear

where w tð Þ is white noise, θ<sup>i</sup> i ¼ 1, …, s are the parameter vectors, φð Þt is a regression vector,

It is assumed that the order of each sub-model is the same, and u(t) and y(t) are the input and

An important phase of the system identification experiment is input signal design. In case of nonlinear systems, a multilevel random signal is often used [23, 24], and a bi-level pseudoran-

The generation of the multilevel random signal—using shift registers—is done according to [25]. Figure 6 shows a five-level random signal with maximal length, using four-shift registers

This input signal is applied—when the armature is fixed—in order to identify the dynamic inductance denoted by χ<sup>i</sup> and the electrical resistance R. The output signal—armature current

vinð Þþ <sup>k</sup> <sup>χ</sup><sup>i</sup>

χ<sup>i</sup> þ RTS

⋮

<sup>φ</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>θ</sup><sup>1</sup> <sup>þ</sup> w tð Þ, if <sup>φ</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>∈</sup>C<sup>1</sup>

<sup>φ</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>θ</sup><sup>s</sup> <sup>þ</sup> w tð Þ, if <sup>φ</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>∈</sup>C<sup>1</sup>

To emphasize that the calculation of the y tð Þ is from the past data, we will write

by modern software tools.

output, respectively.

∪s

models having s sub-models, such as

Furthermore, it is assumed that Cs

<sup>i</sup>¼<sup>1</sup>Ci <sup>¼</sup> <sup>C</sup>, Ci <sup>∩</sup> Cj <sup>¼</sup> <sup>∅</sup>, and <sup>∀</sup><sup>i</sup> 6¼ <sup>j</sup>.

where TS is the sampling time.

y tðÞ¼

and n is the order of the piecewise ARX (PWARX) model.

dom binary signal (PRBS) is not suitable for nonlinear systems.

—when the multilevel random signal is applied is shown in Figure 6.

i kð Þ¼ TS

Next, the voltage equation is written in discrete form at the time moment t ¼ t kð Þ:

χ<sup>i</sup> þ RTS

with coefficients a<sup>1</sup> ¼ 1, a<sup>2</sup> ¼ �1, a<sup>3</sup> ¼ 1, a<sup>4</sup> ¼ �2 [25].

8 ><

>:

T

http://dx.doi.org/10.5772/intechopen.75088

Modeling, System Identification, and Control of Electromagnetic Actuators

<sup>b</sup>y tðÞ¼ <sup>φ</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>θ</sup> (36)

<sup>i</sup>¼<sup>1</sup> are polytopic and they satisfy the well-posed condition:

i kð Þ � 1 (38)

(37)

95

<sup>φ</sup>ð Þ ¼ ½� <sup>t</sup> y tð Þ � <sup>1</sup> … � y tð Þ � <sup>n</sup> u tð Þ…u tð Þ� � <sup>m</sup> <sup>T</sup> (35)

It means that a family of transfer functions are obtained and the system can be viewed as a linear parameter-varying (LPV) system.

The variation of k and a values with the equilibrium position z<sup>0</sup> is shown in Figure 5 and can be well approximated by quadratic functions.

#### 3. System identification

#### 3.1. Clustering-based system identification

In the previous section, we have seen that using the current decay test and the nonlinear least squares method, the parameters of the mathematical model can be identified.

However, the current decay test is time-consuming, since measurements shall be performed for each grid point defined by armature current and position ð Þ i; z . Thus, the obvious question might arise: is there a faster solution to identify the parameters?

During the system identification process, we will note the system's input and output at time t by u tð Þ and y tð Þ, respectively [19].

For single-input single-output linear systems, we can write

$$y(t) = \phi^T(t)\theta \tag{34}$$

where θ is the parameter vector (unknown) and the φ is the recorded (known) input-output data vector:

$$\begin{aligned} \boldsymbol{\theta} &= [\boldsymbol{a}\_1 \dots \boldsymbol{a}\_n \quad & \boldsymbol{b}\_0 \dots \boldsymbol{b}\_m]^T\\ \boldsymbol{\varphi}(t) &= [-\boldsymbol{y}(t-1) \dots -\boldsymbol{y}(t-n) \quad \boldsymbol{u}(t) \dots \boldsymbol{u}(t-m)]^T \end{aligned} \tag{35}$$

To emphasize that the calculation of the y tð Þ is from the past data, we will write

$$
\widehat{y}(t) = \boldsymbol{\varphi}^T(t)\boldsymbol{\theta} \tag{36}
$$

Now, suppose for a given system that we do not know the values of the parameters in θ, but we have recorded inputs and outputs over the time interval. If the input signal is persistently exciting—condition described in details in [19, 20]—then the solution can easily be computed by modern software tools.

In this section a clustering-based identification method, proposed by Ferrari-Trecate et al. (2003) is used (see [21, 22]), where the plant is assumed to be described by piecewise linear models having s sub-models, such as

$$y(t) = \begin{cases} \begin{aligned} \boldsymbol{\varrho}^{\top}(t)\boldsymbol{\theta}\_{1} + \boldsymbol{w}(t), & \quad \boldsymbol{\text{if}} & \quad \boldsymbol{\varrho}^{\top}(t) \in \mathsf{C}\_{1} \\ & \vdots & \\ \boldsymbol{\varrho}^{\top}(t)\boldsymbol{\theta}\_{s} + \boldsymbol{w}(t), & \quad \boldsymbol{\text{if}} & \quad \boldsymbol{\varrho}^{\top}(t) \in \mathsf{C}\_{1} \end{aligned} \tag{37}$$

where w tð Þ is white noise, θ<sup>i</sup> i ¼ 1, …, s are the parameter vectors, φð Þt is a regression vector, and n is the order of the piecewise ARX (PWARX) model.

It is assumed that the order of each sub-model is the same, and u(t) and y(t) are the input and output, respectively.

Furthermore, it is assumed that Cs <sup>i</sup>¼<sup>1</sup> are polytopic and they satisfy the well-posed condition: ∪s <sup>i</sup>¼<sup>1</sup>Ci <sup>¼</sup> <sup>C</sup>, Ci <sup>∩</sup> Cj <sup>¼</sup> <sup>∅</sup>, and <sup>∀</sup><sup>i</sup> 6¼ <sup>j</sup>.

An important phase of the system identification experiment is input signal design. In case of nonlinear systems, a multilevel random signal is often used [23, 24], and a bi-level pseudorandom binary signal (PRBS) is not suitable for nonlinear systems.

The generation of the multilevel random signal—using shift registers—is done according to [25]. Figure 6 shows a five-level random signal with maximal length, using four-shift registers with coefficients a<sup>1</sup> ¼ 1, a<sup>2</sup> ¼ �1, a<sup>3</sup> ¼ 1, a<sup>4</sup> ¼ �2 [25].

This input signal is applied—when the armature is fixed—in order to identify the dynamic inductance denoted by χ<sup>i</sup> and the electrical resistance R. The output signal—armature current —when the multilevel random signal is applied is shown in Figure 6.

Next, the voltage equation is written in discrete form at the time moment t ¼ t kð Þ:

$$\dot{\mathbf{u}}(k) = \frac{T\_S}{\chi\_i + RT\_S} v\_{in}(k) + \frac{\chi\_i}{\chi\_i + RT\_S} \dot{\mathbf{u}}(k-1) \tag{38}$$

where TS is the sampling time.

<sup>a</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup>γ<sup>i</sup>

linear parameter-varying (LPV) system.

Figure 5. Plant gain and pole variation with the equilibrium position.

well approximated by quadratic functions.

3.1. Clustering-based system identification

3. System identification

94 Actuators

by u tð Þ and y tð Þ, respectively [19].

data vector:

It means that a family of transfer functions are obtained and the system can be viewed as a

The variation of k and a values with the equilibrium position z<sup>0</sup> is shown in Figure 5 and can be

In the previous section, we have seen that using the current decay test and the nonlinear least

However, the current decay test is time-consuming, since measurements shall be performed for each grid point defined by armature current and position ð Þ i; z . Thus, the obvious question

During the system identification process, we will note the system's input and output at time t

where θ is the parameter vector (unknown) and the φ is the recorded (known) input-output

squares method, the parameters of the mathematical model can be identified.

might arise: is there a faster solution to identify the parameters?

For single-input single-output linear systems, we can write

2 0 m cð Þ <sup>1</sup> þ c2z<sup>0</sup>

<sup>3</sup> (33)

y tðÞ¼ <sup>φ</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>θ</sup> (34)

<sup>J</sup> <sup>¼</sup> min <sup>χ</sup>ið Þ� <sup>i</sup>; <sup>z</sup> <sup>χ</sup>b<sup>i</sup> ð Þ ð Þ <sup>i</sup>; <sup>z</sup>

and we can find out the estimated parameters of the model <sup>Ψ</sup><sup>b</sup> max <sup>¼</sup> <sup>0</sup>:437½ � Wb , <sup>c</sup>b<sup>1</sup> <sup>¼</sup> <sup>0</sup>:37½ � <sup>A</sup> ,

Having the χ<sup>i</sup> ¼ χið Þ i; z function identified, the parameters of the model set, namely, Ψmax, c1,

This identification is repeated only around different positions z, when the armature is fixed and thus is much faster than identifying Ψ ¼ Ψð Þ i; z around different current and position

In practice, it might be the case that the system is open-loop unstable; thus, system identification experiments have to be performed under closed-loop (for more details see [26, 27]).

Closed-loop identification is a very challenging task. Due to the presence of feedback loop, the input signal might not be persistently exciting. In the aim to achieve a persistent excitation of the system, it is recommended in [19] to switch between different simple controller structures. First, the system shall be stabilized under feedback around an equilibrium position, as shown

Under closed loop, the reference input (armature position) is disturbed by a persistently exciting input signal (i.e., pseudorandom binary signal)—as shown in Figure 9, and three

in Figure 8; details about the controller design K sð Þ are described in the next section.

linear transfer functions are identified, which are defined as

½ � A=m . Values, which are in good accordance with the values, are found via

Modeling, System Identification, and Control of Electromagnetic Actuators

and <sup>c</sup>b<sup>2</sup> <sup>¼</sup> <sup>0</sup>:<sup>36</sup> � <sup>103</sup>

and c2, are found.

the current decay test.

values using the current decay test.

3.2. Identification under closed-loop

Figure 8. Closed-loop system identification.

<sup>2</sup> (39)

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97

Figure 6. Multilevel random input and corresponding output signal.

The equation above defines the regression space (see also references [21, 22]) having in this case two axis, defined by i kð Þ � 1 and vinð Þk . The data collected during the system identification experiment is shown in the regression space in Figure 7.

The regression space is clustered in five different regions for i ¼ 0:1, …; 0:5½ � A , and the parameters are identified for each case using the least squares method. In each defined cluster, we assume that χ<sup>i</sup> is constant, and basically we use a piecewise linear approximation of χi.

The system identification experiments are repeated around different positions, when the armature is fixed; thus, the function <sup>χ</sup>b<sup>i</sup> <sup>¼</sup> <sup>χ</sup>b<sup>i</sup>ð Þ <sup>i</sup>; <sup>z</sup> can be estimated.

Now, using the nonlinear least squares, we can minimize the J objective function:

Figure 7. Regression space—Clustering-based identification.

$$J = \min(\chi\_i(\mathbf{i}, z) - \widehat{\chi}\_i(\mathbf{i}, z))^2\tag{39}$$

and we can find out the estimated parameters of the model <sup>Ψ</sup><sup>b</sup> max <sup>¼</sup> <sup>0</sup>:437½ � Wb , <sup>c</sup>b<sup>1</sup> <sup>¼</sup> <sup>0</sup>:37½ � <sup>A</sup> , and <sup>c</sup>b<sup>2</sup> <sup>¼</sup> <sup>0</sup>:<sup>36</sup> � <sup>103</sup> ½ � A=m . Values, which are in good accordance with the values, are found via the current decay test.

Having the χ<sup>i</sup> ¼ χið Þ i; z function identified, the parameters of the model set, namely, Ψmax, c1, and c2, are found.

This identification is repeated only around different positions z, when the armature is fixed and thus is much faster than identifying Ψ ¼ Ψð Þ i; z around different current and position values using the current decay test.

#### 3.2. Identification under closed-loop

The equation above defines the regression space (see also references [21, 22]) having in this case two axis, defined by i kð Þ � 1 and vinð Þk . The data collected during the system identification

The regression space is clustered in five different regions for i ¼ 0:1, …; 0:5½ � A , and the parameters are identified for each case using the least squares method. In each defined cluster, we assume that χ<sup>i</sup> is constant, and basically we use a piecewise linear approximation of χi.

The system identification experiments are repeated around different positions, when the

Now, using the nonlinear least squares, we can minimize the J objective function:

experiment is shown in the regression space in Figure 7.

Figure 7. Regression space—Clustering-based identification.

Figure 6. Multilevel random input and corresponding output signal.

96 Actuators

armature is fixed; thus, the function <sup>χ</sup>b<sup>i</sup> <sup>¼</sup> <sup>χ</sup>b<sup>i</sup>ð Þ <sup>i</sup>; <sup>z</sup> can be estimated.

In practice, it might be the case that the system is open-loop unstable; thus, system identification experiments have to be performed under closed-loop (for more details see [26, 27]).

Closed-loop identification is a very challenging task. Due to the presence of feedback loop, the input signal might not be persistently exciting. In the aim to achieve a persistent excitation of the system, it is recommended in [19] to switch between different simple controller structures.

First, the system shall be stabilized under feedback around an equilibrium position, as shown in Figure 8; details about the controller design K sð Þ are described in the next section.

Under closed loop, the reference input (armature position) is disturbed by a persistently exciting input signal (i.e., pseudorandom binary signal)—as shown in Figure 9, and three linear transfer functions are identified, which are defined as

Figure 8. Closed-loop system identification.

Figure 9. Input and output signal used to identify T sð Þ.

$$T(s) = \frac{Y(s)}{R(s)} = \frac{P(s)K(s)}{1 + P(s)K(s)}\tag{40}$$

In this section, we are looking for a linear controller, which can stabilize the plant and can

T sð Þ¼ kkD

K sð Þ¼�kPI

T sð Þ¼ kkPI

Since a is a positive value, we observe that the PI controller cannot stabilize the plant P sð Þ.

Let us consider a PID controller—having a single tuning parameter KPID—in the form

K sð Þ¼�kPID

T sð Þ¼ kkPIDð Þ <sup>s</sup> <sup>þ</sup> <sup>a</sup>

(large overshoot), which can be canceled with a prefilter KPREð Þ¼ s a=ð Þ s þ a .

The closed-loop transfer function has a zero at s ¼ �a, which might affect the system response

s þ a

s<sup>2</sup> � as þ kkPI

ð Þ s þ a 2

The system is stable if kkD > a; however, the steady-state error might be significant, since the

K sð Þ¼�kDð Þ s þ a (45)

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<sup>s</sup> <sup>þ</sup> kkD � <sup>a</sup> (46)

<sup>s</sup> (47)

<sup>s</sup> (49)

<sup>s</sup><sup>2</sup> <sup>þ</sup> ð Þ kkPID � <sup>a</sup> <sup>s</sup> <sup>þ</sup> kkPIDa (50)

(48)

fulfill performance and robustness requirements [18].

The closed-loop transfer function shows that

controller gain kD cannot be made arbitrarily large.

The next option is to consider a PI controller such as

In this case, the closed-loop transfer function becomes

The block diagram of the control system is shown in Figure 10.

A very simple PD controller, which can stabilize the plant, is

4.1. PD controller

4.2. PI controller

4.3. PID controller

The closed-loop transfer function is

$$H(\mathbf{s}) = \frac{\mathcal{U}(\mathbf{s})}{E(\mathbf{s})} = \frac{\mathcal{K}(\mathbf{s})}{1 + P(\mathbf{s})\mathcal{K}(\mathbf{s})} \tag{41}$$

$$P(\mathbf{s}) = \frac{Y(\mathbf{s})}{U(\mathbf{s})} = \frac{T(\mathbf{s})}{H(\mathbf{s})(1 - T(\mathbf{s}))} \tag{42}$$

where we used the well-known identity S sð Þþ T sð Þ¼ 1:

$$S(\mathbf{s}) = \frac{E(\mathbf{s})}{R(\mathbf{s})} = 1 - T(\mathbf{s}) \tag{43}$$

The procedure can be repeated around different equilibrium positions; thus, a family of transfer functions can be obtained.

#### 4. Control of electromagnetic actuators

Let us start with the easier case: the moving armature is controlled around an equilibrium position—magnetic bearings and magnetically levitated high-speed trains are typical applications.

The linearized mathematical model, around an equilibrium position, can be written as

$$P(s) = \frac{\Delta z(s)}{\Delta \dot{i}(s)} = -\frac{k}{(s-a)(s+a)}\tag{44}$$

where k and a are strictly positive values, varying with the equilibrium point ð Þ i0; z<sup>0</sup> .

In this section, we are looking for a linear controller, which can stabilize the plant and can fulfill performance and robustness requirements [18].

#### 4.1. PD controller

A very simple PD controller, which can stabilize the plant, is

$$K(\mathbf{s}) = -k\mathbf{\bar{o}}\,(\mathbf{s} + \mathbf{a})\tag{45}$$

The closed-loop transfer function shows that

$$T(s) = \frac{kk\_D}{s + kk\_D - a} \tag{46}$$

The system is stable if kkD > a; however, the steady-state error might be significant, since the controller gain kD cannot be made arbitrarily large.

#### 4.2. PI controller

T sð Þ¼ Y sð Þ

H sð Þ¼ U sð Þ

P sð Þ¼ Y sð Þ

S sð Þ¼ E sð Þ

where we used the well-known identity S sð Þþ T sð Þ¼ 1:

Figure 9. Input and output signal used to identify T sð Þ.

transfer functions can be obtained.

tions.

98 Actuators

4. Control of electromagnetic actuators

R sð Þ <sup>¼</sup> P sð ÞK sð Þ

E sð Þ <sup>¼</sup> K sð Þ

U sð Þ <sup>¼</sup> T sð Þ

The procedure can be repeated around different equilibrium positions; thus, a family of

Let us start with the easier case: the moving armature is controlled around an equilibrium position—magnetic bearings and magnetically levitated high-speed trains are typical applica-

<sup>Δ</sup>i sð Þ ¼ � <sup>k</sup>

The linearized mathematical model, around an equilibrium position, can be written as

where k and a are strictly positive values, varying with the equilibrium point ð Þ i0; z<sup>0</sup> .

P sð Þ¼ <sup>Δ</sup>z sð Þ

<sup>1</sup> <sup>þ</sup> P sð ÞK sð Þ (40)

<sup>1</sup> <sup>þ</sup> P sð ÞK sð Þ (41)

H sð Þð Þ <sup>1</sup> � T sð Þ (42)

R sð Þ <sup>¼</sup> <sup>1</sup> � T sð Þ (43)

ð Þ <sup>s</sup> � <sup>a</sup> ð Þ <sup>s</sup> <sup>þ</sup> <sup>a</sup> (44)

The next option is to consider a PI controller such as

$$K(\mathbf{s}) = -k\_{Pl} \frac{\mathbf{s} + \mathbf{a}}{\mathbf{s}} \tag{47}$$

In this case, the closed-loop transfer function becomes

$$T(s) = \frac{k k\_{Pl}}{s^2 - as + k k\_{Pl}}\tag{48}$$

Since a is a positive value, we observe that the PI controller cannot stabilize the plant P sð Þ.

#### 4.3. PID controller

Let us consider a PID controller—having a single tuning parameter KPID—in the form

$$K(\mathbf{s}) = -k\_{\rm PID} \frac{\left(\mathbf{s} + \mathbf{a}\right)^2}{\mathbf{s}} \tag{49}$$

The block diagram of the control system is shown in Figure 10.

The closed-loop transfer function is

$$T(s) = \frac{k k\_{\rm PID} (s+a)}{s^2 + (k k\_{\rm PID} - a)s + k k\_{\rm PID} a} \tag{50}$$

The closed-loop transfer function has a zero at s ¼ �a, which might affect the system response (large overshoot), which can be canceled with a prefilter KPREð Þ¼ s a=ð Þ s þ a .

Figure 10. The block diagram of the control system.

Then, the closed-loop transfer function becomes

$$T(s) = \frac{k k\_{\rm PID} a}{s^2 + (k k\_{\rm PID} - a)s + k k\_{\rm PID} a} \tag{51}$$

However, we observed that the plant parameters k and a are varying with the operating point ð Þ i0; z<sup>0</sup> ; thus, for good performance and robustness, the controller should take into account that

A survey of linear parameter-varying control applications can be found in [28], and control applications validated by experiments are presented in [27, 29] for actuators and for medical Xray systems in [30]. High-accuracy mathematical modeling and a linear parameter-varying

where both the controller gain and controller zero are dependent on the equilibrium position. Finally, stability and robustness (quadratic stability) of such a control system can be analyzed

If a simpler approach is preferred, a gain-scheduled controller might be a good choice, which is

Since perfect cancelation of the varying plant pole at s ¼ �a with a fixed controller zero is not possible, we choose to place the controller fixed zero left to the varying poles, such as

easier to implement in real time, and its stability and robustness are easier to analyze.

K sð Þ¼ ; z<sup>0</sup> kPIDð Þ z<sup>0</sup>

kPIDð Þ¼ z<sup>0</sup>

where the variation of the values k ¼ k zð Þ<sup>0</sup> and a ¼ a zð Þ<sup>0</sup> were shown already in Figure 5.

Next, the gain and phase margins are calculated for different equilibrium positions z<sup>0</sup> and shown in Figure 11. We can observe that—due to the gain-scheduled controller—the gain and phase margins do not change significantly with the equilibrium position, and such a robust-

The system response using the gain-scheduled controller is investigated, considering the

• The moving armature is controlled around an equilibrium position, and the set point (reference position) is changed Δz<sup>0</sup> ¼ 1½ � mm (see Figure 12, left plot). The control system —including the prefilter—exhibits approximately P:O: ≈ 25% overshoot and settling time

where the only one tuning parameter is controller gain KPIDð Þ z<sup>0</sup> , defined as

ness is difficult to achieve with a single controller, having fixed parameters.

following two cases:

Tset ¼ 1:5½ �s .

ð Þ s þ a zð Þ<sup>0</sup>

ð Þ <sup>s</sup> <sup>þ</sup> amax <sup>2</sup>

8 þ a zð Þ<sup>0</sup> Tset k zð Þ<sup>0</sup> Tset

2

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<sup>s</sup> (56)

(57)

observer for fault detection and fault isolation are presented in [17]. We can design a linear parameter-varying controller having the form

using modern software tools, and details are described in [27].

z ¼ �amax. Then, the gain-scheduled controller can be written as

K sð Þ¼ ; z<sup>0</sup> kPIDð Þ z<sup>0</sup>

the plant parameters are varying.

Next, based on performance and robustness specifications, we would like to find a suitable value for the controller gain KPID. Usually, performance specifications are given in terms of settling time Tset and percent of overshoot P:O:

For a second-order system

$$T(s) = \frac{\omega\_n^2}{s^2 + 2\pi\omega\_n s + \omega\_n^2} \tag{52}$$

where ω<sup>n</sup> is the natural frequency and the τ is the damping factor; we have Tset ≈ 4=ð Þ τω<sup>n</sup> and <sup>P</sup>:O: <sup>¼</sup> <sup>100</sup> � <sup>e</sup>�τπ<sup>=</sup> ffiffiffiffiffiffi <sup>1</sup>�<sup>τ</sup> <sup>p</sup> .

Since we have only one tuning parameter KPID, the performance specifications are given only in terms of settling time Tset ¼ 1:25½ �s . Therefore, the PID controller gain can be calculated as

$$1\,\text{k}\text{k}\_{\text{PID}} - a = 2\pi\omega\omega\_n \approx \frac{8}{T\_{\text{set}}}\tag{53}$$

Thus, we obtain

$$k\_{PID} = \frac{8 + aT\_{set}}{kT\_{set}}\tag{54}$$

Next, the stability and robustness in a classical framework can be assessed. We calculate the gain and phase margins, obtaining Gm ¼ 0:47½ � m=A and Pm ¼ 37 deg ½ � for the equilibrium position <sup>z</sup><sup>0</sup> <sup>¼</sup> <sup>3</sup> � <sup>10</sup>�<sup>3</sup> ½ � m .

#### 4.4. Gain-scheduled controller

We have seen that the actuator can be stabilized around, and equilibrium point and performance and robustness can be guaranteed.

However, we observed that the plant parameters k and a are varying with the operating point ð Þ i0; z<sup>0</sup> ; thus, for good performance and robustness, the controller should take into account that the plant parameters are varying.

A survey of linear parameter-varying control applications can be found in [28], and control applications validated by experiments are presented in [27, 29] for actuators and for medical Xray systems in [30]. High-accuracy mathematical modeling and a linear parameter-varying observer for fault detection and fault isolation are presented in [17].

We can design a linear parameter-varying controller having the form

Then, the closed-loop transfer function becomes

Figure 10. The block diagram of the control system.

settling time Tset and percent of overshoot P:O:

<sup>1</sup>�<sup>τ</sup> <sup>p</sup> .

½ � m .

mance and robustness can be guaranteed.

For a second-order system

100 Actuators

<sup>P</sup>:O: <sup>¼</sup> <sup>100</sup> � <sup>e</sup>�τπ<sup>=</sup> ffiffiffiffiffiffi

Thus, we obtain

position <sup>z</sup><sup>0</sup> <sup>¼</sup> <sup>3</sup> � <sup>10</sup>�<sup>3</sup>

4.4. Gain-scheduled controller

T sð Þ¼ kkPIDa

T sð Þ¼ <sup>ω</sup><sup>2</sup>

Next, based on performance and robustness specifications, we would like to find a suitable value for the controller gain KPID. Usually, performance specifications are given in terms of

where ω<sup>n</sup> is the natural frequency and the τ is the damping factor; we have Tset ≈ 4=ð Þ τω<sup>n</sup> and

Since we have only one tuning parameter KPID, the performance specifications are given only in terms of settling time Tset ¼ 1:25½ �s . Therefore, the PID controller gain can be calculated as

kkPID � <sup>a</sup> <sup>¼</sup> <sup>2</sup>τω<sup>n</sup> <sup>≈</sup> <sup>8</sup>

kPID <sup>¼</sup> <sup>8</sup> <sup>þ</sup> aTset kTset

Next, the stability and robustness in a classical framework can be assessed. We calculate the gain and phase margins, obtaining Gm ¼ 0:47½ � m=A and Pm ¼ 37 deg ½ � for the equilibrium

We have seen that the actuator can be stabilized around, and equilibrium point and perfor-

n s<sup>2</sup> þ 2τωns þ ω<sup>2</sup>

n

Tset

<sup>s</sup><sup>2</sup> <sup>þ</sup> ð Þ kkPID � <sup>a</sup> <sup>s</sup> <sup>þ</sup> kkPIDa (51)

(52)

(53)

(54)

$$K(\mathbf{s}, z\_0) = k\_{\rm PID}(z\_0) \frac{\left(\mathbf{s} + a(z\_0)\right)^2}{\mathbf{s}} \tag{55}$$

where both the controller gain and controller zero are dependent on the equilibrium position.

Finally, stability and robustness (quadratic stability) of such a control system can be analyzed using modern software tools, and details are described in [27].

If a simpler approach is preferred, a gain-scheduled controller might be a good choice, which is easier to implement in real time, and its stability and robustness are easier to analyze.

Since perfect cancelation of the varying plant pole at s ¼ �a with a fixed controller zero is not possible, we choose to place the controller fixed zero left to the varying poles, such as z ¼ �amax. Then, the gain-scheduled controller can be written as

$$K(s, z\_0) = k\_{\rm PID}(z\_0) \frac{\left(s + a\_{\rm max}\right)^2}{s} \tag{56}$$

where the only one tuning parameter is controller gain KPIDð Þ z<sup>0</sup> , defined as

$$k\_{\rm PID}(z\_0) = \frac{8 + a(z\_0)T\_{\rm set}}{k(z\_0)T\_{\rm set}} \tag{57}$$

where the variation of the values k ¼ k zð Þ<sup>0</sup> and a ¼ a zð Þ<sup>0</sup> were shown already in Figure 5.

Next, the gain and phase margins are calculated for different equilibrium positions z<sup>0</sup> and shown in Figure 11. We can observe that—due to the gain-scheduled controller—the gain and phase margins do not change significantly with the equilibrium position, and such a robustness is difficult to achieve with a single controller, having fixed parameters.

The system response using the gain-scheduled controller is investigated, considering the following two cases:

• The moving armature is controlled around an equilibrium position, and the set point (reference position) is changed Δz<sup>0</sup> ¼ 1½ � mm (see Figure 12, left plot). The control system —including the prefilter—exhibits approximately P:O: ≈ 25% overshoot and settling time Tset ¼ 1:5½ �s .

5. Conclusions

control were presented.

scheduled PID controller).

TNO, Helmond, The Netherlands

Technology. 2003;11(2):185-195

[6] http://mechanicstips.blogspot.nl/2015/06/

Address all correspondence to: alexandru.forrai@tno.nl

Author details

Alexandru Forrai

References

18(4):18-25

204-215

This chapter dealt with mathematical modeling, system identification, and control of electromagnetic actuators. Actuators are often used in industrial applications such as magnetic levitation, electromagnetic bearings, as well as in fuel injectors in the automotive industry.

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After a detailed mathematical model was presented, two different parameter identification techniques were described. The first one is based on the classical current decay test, and the second one is a clustering-based system identification approach. Since the actuator is openloop unstable, the main steps of system identification of the actuators under closed-loop

Finally, very simple and easy-to-apply control strategies were discussed, when the armature is controlled around a fixed equilibrium position (PID controller) as well as when the armature is controlled between two extreme positions, armature open and armature closed (gain-

[1] Bittar A, Sales RM. H<sup>2</sup> and H<sup>∞</sup> control for MagLev vehicles. IEEE Control Systems. 1998;

[2] Lindlau JD, Knospe CR. Feedback linearization of an active magnetic bearing with voltage

[3] Li L, Shinshi T, Shimokohbe A. Asymptotically exact linearizations for active magnetic bearing actuators in voltage control configuration. IEEE Transaction on Control Systems

[4] Fittro RL, Knospe CR. Rotor compliance minimization via μ-control of active magnetic bearings. IEEE Transaction on Control Systems Technology. 2002;10(2):238-250

[5] Duan G-R, Howe D. Robust magnetic bearing control via eigenstructure assignment dynamical compensation. IEEE Transaction on Control Systems Technology. 2003;11(2):

control. IEEE Transaction on Control Systems Technology. 2002;10(1):21-31

Figure 11. Gain and phase margin variation with z0.

Figure 12. System response with the gain-scheduled controller.

• The moving armature is controlled between the two extreme positions, armature open and armature close, Δz<sup>0</sup> ¼ 10½ � mm (see Figure 12, right plot). In this case the main goal is to achieve so-called soft landing of the moving armature to reduce wear and noise.

It is important to highlight that controller design is made based on the linearized plant, but validation of the controller in simulations or during hardware-in-the-loop (HIL) experiments shall be done using the nonlinear plant model.

During our control design investigations, we considered that the armature position can be measured. In practice, there are applications, where the armature position cannot be measured in a cost-effective way.

Therefore, we remark that controlling the moving armature without measuring the armature position (e.g., measuring only the current) remains a challenging research topic, which exceeds the goals and the limits of this chapter.
