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MARWAN NAFEA MINJAL
MODELING AND CONTROL OF PIEZOELECTRIC STACK ACTUATORS WITH HYSTERESIS 2012/2013-3
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MODELING AND CONTROL OF PIEZOELECTRIC STACK ACTUATORS WITH HYSTERESIS
MARWAN NAFEA MINJAL
A project report submitted in partial fulfilment of the requirements for the award of the degree of Master of Engineering (Electrical - Mechatronics & Automatic Control)
Faculty of Electrical Engineering UniversitiTeknologi Malaysia
AUGUST 2013
ii
I declare that this project report entitled “Modeling and Control of Piezoelectric Stack Actuators with Hysteresis” is the result of my own research except as cited in the references. The project report has not been accepted for any degree and is not concurrently submitted in candidature of any other degree.
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iii
ACKNOWLEDGEMENT
First and foremost, I am thankful to Allah S.W.T for everything, especially for his guidance in accomplishing this level of study. I am grateful to my beloved parents and sister for their continuous support. Without their love and encouragement, I would never have succeeded in my study. I wish to thank my supervisor Assoc. Prof. Dr. Zaharuddin bin Mohamed for his continuous support and supervisions. A lot of creative thoughts and valuable discussion have had significantly influence this final year project report. This project would not have been possible without the support and supervision from him. Thanks to all my lecturers who taught me during my master study, the incorporation of the knowledge you gave is what brought all the pieces together. Last but not least, thanks to my friends and classmates for all of their support during my study and their assistance during the research, especially Amirah ‘Aisha Badrul Hisham.
iv
ABSTRACT
Piezoelectric actuators are popularly applied as actuators in high precision systems due to their small displacement resolution, fast response and simple construction. However, the hysteresis nonlinear behavior limits the dynamic modeling and tracking control of piezoelectric actuators. This thesis studies a dynamic model of a moving stage driven by piezoelectric stack actuator. The BoucWen model is introduced and analyzed to express the nonlinear hysteresis term of the piezoelectric stack actuator, where the values of the parameters of the model have been taken from a previous work. The simulated results using MATLAB/Simulink demonstrate the existence of the hysteresis phenomenon between the input voltage and the output displacement of the piezoelectric stack actuator, and validate the correctness of the model. Moreover, a Luenberger observer is designed to estimate the hysteresis nonlinearity of the system, and then combined with the voltage input signal to form a Luenberger-based feedforward controller to control the displacement of the system. Furthermore, a Proportional-Integral-Derivative (PID) feedback controller is integrated with the feedforward controller to achieve more accurate output displacement, where the gains of the PID controller are optimized using Particle Swarm Optimization. Several performance index formulas have been studied to get the best solution of the PID’s gains. An Integral Time Squared Error plus Absolute Error performance index formula has been proposed to achieve zero overshoot and steady-state error. The simulated results accomplished using MATLAB/Simulink show the ability of the designed controllers to vastly reduce the amount of error of the output displacement and the response time of the system.
v
ABSTRAK
Pemacu
piezoelektrik
popular
digunakan
sebagai
pemacu
system
berketepatan tinggi memandangkan ia memberikan resolusi sesaran yang kecil, tindak balas yang cepat dan konstruksi yang mudah. Namun, sifat histerisis yang tidak linear menghadkan pemodelan dinamik dan penjejakan bagi pemacu ini. Tesis ini mengkaji model dinamik bagi pemacu bergerak berperingkat dipacu oleh aktuator piezoelektrik bertingkat. Model Bouc-Wen diperkenalkan dan dianalisis untuk menyatakan terma histerisis tidak linear bagi aktuator piezoelektrik bertingkat, di mana nilai parameter yang digunakan bagi model ini diambil daripada projek yang terdahulu.
Keputusan
simulasi
dengan
menggunakan
MATLAB/Simulink
menunjukkan tentang kewujudan fenomena histerisis antara voltan input dan sesaran output bagi pemacu piezoelektrik berlapis, dan mengesahkan kesahihan model. Tambahan pula, pemerhati Luenberger telah direka untuk menganggarkan histerisis tidak
linear
bagi
sistem
isyarat input voltan membentuk
dan
kemudian
menggabungkan
satu pengawal suapbalik
hadapan
dengan berasaskan
Luenberger untuk mengawal sesaran sistem. Tambahan pula, satu pengawal suapbalik berasaskan Perkadaran-Pembezaan-Kamiran (PID) disepadukan dengan pengawal suapbalik hadapan untuk mencapai sesaran output yang lebih tepat, di mana peningkatan pengawal PID dioptimumkan menggunakan Particle Swarm Optimization. Beberapa indeks prestasi telah dikaji untuk mendapatkan penyelesaian yang terbaik untuk nilai gandaan PID. Formula gabungan indek kamiran ralat kuasa dua dan indeks ralat mutlak telah dicadangkan untuk mencapai lajakan sifar dan ralat keadaan mantap sifar. Keputusan-keputusan simulasi yang diperoleh dengan menggunakan MATLAB/Simulink menunjukkan keupayaan pengawal yang direka dengan mengurangkan jumlah ralat sesaran output yang besar dan mengurangkan masa tindak balas sistem.
vi
TABLE OF CONTENTS
CHAPTER
TITLE
DECLARATION
ii
ACKNOWLEDGEMENT
iii
ABSTRACT
iv
ABSTRAK
v
TABLE OF CONTENTS
vi
LIST OF TABLES
ix
LIST OF FIGURES
x
LIST OF ABBREVIATIONS 1
2
PAGE
xiii
INTRODUCTION 1.1. Piezoelectricity
1
1.2. Piezoelectric Actuators
4
1.3. Nonlinearities in Piezoelectric Actuators
5
1.4. Displacement Control of Piezoelectric Actuators
7
1.5. Thesis motivations
8
1.6. Problem Statement
8
1.7. Research Objectives
8
1.8. Scope of Work
9
1.9. Organization of the Thesis
9
LITERATURE REVIEW 2.1. Introduction
10
2.2. Modeling Overview
10
2.2.1. The Linear Model of Piezoelectric Actuators
11
2.2.2. The Nonlinear Models of Piezoelectric Actuators
11
vii
2.2.2.1. Lumped-Parameter Based on Maxwell-Slip Model 2.2.2.2. Duhem Hysteresis Model
13
2.2.2.3. Preisach Hysteresis Model
14
2.2.2.4. Bouc-Wen Hysteresis Model
15
2.3. Control Overview 2.3.1. Feedback control
4
5
18 19
2.3.1.1. PID Feedback Controller
20
2.3.1.2. PSO-Based PID Controllers
23
2.3.2. Feedforward control
24
2.3.3. Hybrid Feedback and Feedforward control
25
2.4. Summary 3
12
27
METHODOLOGY 3.1. Introduction
29
3.2. Project Methodology
29
3.3. Summary
33
MODELING 4.1. Introduction
34
4.2. Modeling the System
34
4.2.1. Bouc-Wen Hysteresis Model
35
4.2.2. Modeling the Piezo-Actuated Stage
38
4.3. Simulation Results of the Modeled System
42
4.4. Summary
50
CONTROL DESIGN AND RESULTS 5.1. Introduction
51
5.2. Feedforward Controller Design and Simulation
51
5.2.1. Luenberger Observer Design
52
5.2.2. Luenberger Observer-Based Feedforward Controller Design
55
5.2.3. Simulation Results of the Designed Feedforward Controller
57
viii
5.3. Hybrid Feedforward and Feedback Controllers Design and Simulation
60
5.3.1. Hybrid Feedforward and PSO-Based PID Feedback Controllers Design
60
5.3.2. Simulation Results of the Designed Hybrid Controllers67 5.4. Summary 6
72
CONCLUSION AND RECOMMENDATIONS 6.1. Conclusion
73
6.2. Recommendations
74
REFERENCES
76
APPENDICIES A-C
83
x
LIST OF TABLES
TABLE NO
TITLE
PAGE
2.1.
The effect of PID gains on the response of the system.
22
4.1.
Values of the system parameters
39
5.1.
The initial values of the PSO algorithm.
62
5.2.
Performance index formulas and their properties.
64
5.3.
Values of PID gains obtained by various fitness functions.
67
5.4.
The simulation results using different fitness functions.
69
x
LIST OF FIGURES
FIGURE NO. 1.1.
TITLE
PAGE
Simplified cell structure of quartz. - (a) arrangement of Si- and O-ions with the main crystal axes; (b) two- and three-fold axes.
1.2.
2
Direct piezoelectric effect in a cell structure of quartz. - (a) longitudinal piezoelectric effect; (b) transversal piezoelectric effect. 3
1.3.
Schematic of piezoelectric stack actuator.
4
1.4.
Schematic of piezoelectric bender actuator.
5
1.5.
Hysteresis curve of piezoelectric actuators.
6
1.6.
Creep curve of piezoelectric actuators.
6
1.7.
Schematic representation of the piezoelectric stack actuator model. 13
1.8.
Graph force versus displacement for a hysteresis functional [38].
16
1.9.
Feedback control loop.
19
2.1.
A block diagram of a PID controller in a feedback loop.
20
2.2.
Feedforward control loop.
24
2.3.
Hybrid feedback and feedforward control loop.
26
3.1.
Flow chart of the project methodology.
32
4.1.
Schematic diagram of a moving stage driven by a piezoelectric actuator.
4.2.
38
A MATLAB/Simulink block diagram of (a) the entire piezoactuated system. (b) the sub-model of Bouc-Wen hysteresis model. 40
4.3.
A MATLAB/Simulink block diagram of the system in state-space representation.
4.4.
The response of the system represented by differential equations to an 80 V step input.
4.5.
42 43
The response of the system represented by state space to an 80 V step input.
43
xi
4.6.
The error between the step reference input and the output displacement.
44
4.7.
A triangular input voltage with 80 V and 1 Hz.
45
4.8.
The output displacement response compared with the reference signal.
4.9.
45
The error between the triangular reference input and the output displacement.
4.10.
46
The hysteresis relationship between the input voltage and the output displacement of the system.
4.11.
47
The hysteresis loop using a triangular input voltage with a frequency of 1 Hz and a voltage of (a) 60 V. (b) 80 V. (c) 100 V. (d) 120 V.
4.12.
48
The hysteresis loop using a triangular input voltage with a voltage of 80 V and a frequency of (a) 1 Hz. (b) 2 Hz. (c) 10 Hz. (d) 100 Hz.
49
4.13.
The hysteresis loop from reference [6].
50
5.1.
The general structure of Luenberger observer.
53
5.2.
MATLAB/Simulink block diagram of Luenberger observer of the system.
5.3.
55
MATALB/Simulink block diagram of the feedforward controller and the system.
5.4.
56
The output displacement response of the feedforward-controlled system, compared with the reference signal
5.5.
58
The error between the reference and the output displacement of the feedforward-controlled system.
58
5.6.
The hysteresis loop of the feedforward-controlled system.
59
5.7.
The general structure of the PID controller.
60
5.8.
The system with the hybrid controllers.
65
5.9.
The flow chart of the process of tuning the PID’s gains.
66
5.10.
The response of the system to an 80 V step input when using and
performance index formulas.
5.11.
A Statistics chart of the results of the fitness functions used.
5.12.
The output displacement response compared with the reference signal of the hybrid-controlled system.
68 69 70
xii
5.13. 5.14.
The error between the reference and the output displacement of the hybrid-controlled system.
71
The hysteresis relationship of the hybrid-controlled system.
72
xiii
LIST OF ABBREVIATIONS
AE
-
Absolute Error
IAE
-
Integral Absolute Error
ISE
-
Integral Squared Error
ITAE
-
Integral Time Absolute Error
ITSE
-
Integral Time Squared Error
ITSE+AE
-
Integral Time Squared Error Plus Absolute Error
PI
-
Proportional-Integral
PID
-
Proportional-Integral-Derivative
PSO
-
Particle Swarm Optimization
CHAPTER 1
INTRODUCTION
Piezoelectric actuators are widely used for micro/nano manipulation systems [1], micro-robots [2], vibration active control [3], precision machining [4], and atomic force microscopy [5]. This is due to their special characteristics such as high resolution in nanometer range, fast response, and high stiffness. The major advantage of using piezoelectric actuators is that they do not have any frictional or static characteristics, which usually exist in other types of actuators. However, the main disadvantage of piezoelectric actuators is the nonlinearity that is mainly due to hysteresis behavior, creep phenomenon and high frequency vibration [6].
1.1.
Piezoelectricity
Piezoelectric effect was discovered for the first time in 1880 by the brothers Pierre and Jacques Curie. They noticed, that a mechanical deformation in certain directions causes opposite electrical surface charges at opposite crystal faces. This effect, which was also found afterwards in quartz and other crystals without symmetry center, has been called piezoelectric effect by Hankel. The prefix ‘piezo-’ is derived from the Greek word ‘piezein’, which means ‘press’. Thus, the word piezoelectricity means electricity resulting from pressure [7]. In 1881, Lippmann predicted the existence of the inverse piezoelectric effect from thermodynamic considerations, and then Pierre and Jacques Curie verified this in the same year [8].
2
Piezoelectric effect happens due to the existence of polar axes within the piezoelectric material structure. This means that there is an electrical dipole moment in axis directions caused by the distribution of the electrical charge in the chemical bond of the cell structure of the piezoelectric material. Figure 1.1 shows the cell structure of quartz.
Figure 1.1.
Simplified cell structure of quartz. - (a) arrangement of Si- and O-ions with the main crystal axes; (b) two- and three-fold axes.
The cell consists of negative charged O-ions and positive charged Si-ions and has three two-fold polar rotation axes fold rotation axis
,
and
in the drawing plane and a three-
vertical on the drawing plane. If there is a deformation of the
quartz structure along the polar
-axis, an additional electrical polarization
performs along this axis. The electrical polarization is caused by the displacement of the positive and negative ions of the crystal net against each other, resulting an electrical charge on the appropriate crystal surfaces that is vertical on the
-axis,
and thus an outside electrical polarization voltage. This effect is called direct longitudinal piezoelectric effect. Applying compression or tensile stresses vertically on the
-axis results an additional electrical polarization in an opposite sign on
-
axis direction. This behavior is called direct transversal piezoelectric effect. Figure 1.2 shows the direct piezoelectric effect in a cell structure of quartz [8].
3
Figure 1.2.
Direct piezoelectric effect in a cell structure of quartz. - (a)
longitudinal piezoelectric effect; (b) transversal piezoelectric effect.
Both longitudinal piezoelectric effects and transversal piezoelectric effect are reversible. This means that a contraction or an extension of the quartz structure can be achieved under the influence of electrical fields. This effect is called inverse piezoelectric effect [8, 9]. This effect is the working principle of all piezoelectric actuators. Piezoelectric materials can be divided into the following three types:
1. Single crystals, such as quartz. 2. Piezoelectric ceramics, such as lead zirconate titanate. 3. Polymers, such as polyvinyl fluoride.
Single crystals and polymers show a weak piezoelectric effect, which makes them limited to be used in sensor applications, while piezoelectric ceramics have large electromechanical coupling, which makes them suitable for actuator applications [10].
4
1.2.
Piezoelectric Actuators
Piezoelectric actuators are specific actuators using piezoelectric materials as active materials. They are several types of piezoelectric actuators, such as stacks, benders, flextensional, langevin transducers and various motors. The most popular ones are stacks and benders. A stack contains a pile of piezoceramic layers and electrodes mounted electrically in parallel and mechanically in series, which increases the maximum displacement. The focus of this study is piezoelectric stack actuators. Figure 1.3 shows a schematic of piezoelectric stack actuator [7, 11].
Figure 1.3.
Schematic of piezoelectric stack actuator.
Benders have mechanical motion amplification, where two piezoceramic layers are attached with opposing polarization, which makes the first layer expands while the other shrinks under voltage excitation. This causes the structure to bend, and the overall motion on the actuator tip is greater than the strain of the ceramics [7]. Figure 1.4 shows a schematic of piezoelectric bender actuator.
5
Figure 1.4.
1.3.
Schematic of piezoelectric bender actuator.
Nonlinearities in Piezoelectric Actuators
Piezoelectric actuators exhibit nonlinear behavior caused by hysteresis, creep and vibration. Hysteresis in piezoelectric actuators causes that the displacement depends on the current and the previous excitation voltage. Hysteresis phenomenon is based on the crystalline polarization effect and molecular friction [12]. The displacement generated by piezoelectric actuator depends on the applied electric field and the piezoelectric material constant which is related to the remnant polarization that is affected by the electric field applied to piezoelectric material. The deflection of the hysteresis curve depends on the previous value of the input voltage, which means that piezoelectric materials have memory because they remain magnetized after the external magnetic field is removed [13]. Figure 1.5 shows the hysteresis curve of piezoelectric actuators.
6
Figure 1.5.
Hysteresis curve of piezoelectric actuators.
Creep is a drift of the output displacement for a constant applied voltage, which increases over extended periods of time during low-speed operations. Creep is related to the effect of the applied voltage on the remnant polarization of the piezoelectric actuator. If the operating voltage of a piezoelectric actuator is increased, the remnant polarization continues to increase. This manifests itself in a slow creep after the voltage change is complete [14]. Figure 1.6 shows the creep curve of piezoelectric actuators.
Figure 1.6.
Creep curve of piezoelectric actuators.
7
Vibration effect is caused by exciting the resonant modes of the system. To avoid vibration effect, the frequency of the applied voltage should be smaller than the lowest resonant peak of the piezoelectric actuator [15].
The focus of this study is on hysteresis modeling of piezoelectric actuators, since creep and vibration can be negligible in high speed and low frequency applications [15].
1.4.
Displacement Control of Piezoelectric Actuators
Piezoelectric actuators are commonly used in applications requiring high resolution and precision. Their suitable dynamic properties extend the application areas into high speed areas. However, nonlinearities in piezoelectric actuators and external load effect decrease the open-loop positioning accuracy. If a high accuracy is required, nonlinearities and disturbances have to be compensated. The compensation is usually accomplished using six control types [7]:
1. Feedforward voltage control, where nonlinear models are normally used. 2. Feedback voltage control, where several displacement sensors are used. 3. Feedforward and feedback voltage control. 4. Feedforward charge control, where the operating current is controlled. 5. Feedback charge control, where charge is measured and controlled. 6. Feedforward and feedback charge control.
This study focuses on feedforward and feedback voltage control.
8
1.5.
Thesis motivations
Piezoelectric stack actuators are popularly applied as actuators in high precision systems due to their small displacement resolution, fast response and simple construction. However, the hysteresis nonlinear behavior limits the dynamic modeling and tracking control of piezoelectric actuators.
An accurate hysteresis model is needed to present the hysteresis nonlinear behavior of piezoelectric stack actuators, and effective controllers are required to achieve high precision and fast displacement of the systems that are driven by piezoelectric stack actuators.
1.6.
Problem Statement
Hysteresis has a high nonlinear effect on piezoelectric stack actuators. This effect causes difficulties in modeling and controlling piezoelectric stack actuators, and limits their applications in high precision positioning systems.
Development of an accurate model and efficient control of piezoelectric stack actuators is needed to achieve precise accuracy and better dynamic performance.
1.7.
Research Objectives
The objectives of this study are: 1. To derive a dynamic model of a moving stage driven by piezoelectric stack actuator with hysteresis. 2. To study the effect of hysteresis on the behavior of the systm
9
3. To design a feedforward controller with Luenberger observer, and a feedback PID controller to control the displacement of the moving stage.
1.8.
Scope of Work
This study focuses on the hysteresis modeling of a moving stage driven by piezoelectric stack actuator. Bouc-Wen hysteresis model is used to model the hysteresis in the system. The model is studied, derived and then implemented using MATLAB Simulink. A feedforward with Luenberger observer is designed and then combined with a PID controller that is tuned using PSO method. These two combined controllers are then used to control the displacement of the system.
1.9.
Organization of the Thesis
This thesis is organized as follows: Chapter 1 introduces the topic of this thesis. Chapter 2 discusses a literature review about the topic of this thesis. Chapter 3 presents the methodology that is used in this project. The results of modeling are presented and discussed in Chapter 4, while the results of control and optimization are presented and discussed in Chapter 5. Finally, a conclusion and future work are introduced in Chapter 6.
CHAPTER 2
LITERATURE REVIEW
2.1. Introduction
In this chapter, previous works regarding modeling of piezoelectric stack actuators are reviewed. Some of the well-recognized linear and nonlinear models are discussed. Moreover, several effective methods to control piezoelectric actuators are reviewed. Finally, a summary of modeling and control piezoelectric actuators is presented. Furthermore, a dynamic model to represent the piezoelectric stack actuator, and control methods to control the system are selected.
2.2. Modeling Overview
Several models have been developed to mathematically describe the behavior of piezoelectric actuators. These models are subdivided into two categories, linear models and nonlinear models. The following sections overview these models.
11
2.2.1. The Linear Model of Piezoelectric Actuators
A well-known description of piezoelectric actuators behavior was published by a standards committee of the IEEE in 1966 and most recently revised in 1987 [16]. This description consists of two linear fundamental. The linearized fundamental relations are represented as follows:
)1.2( )1.1(
where
represents the mechanical strain,
is the elastic compliance matrix
when subjected to a constant electrical field,
represents the mechanical stress,
a matrix of piezoelectric material constant,
is the electric field vector,
electric displacement vector, and
is
is the
is the permittivity measured at a constant stress.
These equations state that the electrical displacement and material strain exhibited by a piezoelectric ceramic are both linearly affected by the electrical field and the mechanical stress to which the ceramic is subjected. These linearized fundamental relations fail to describe the hysteresis behavior that is presents in all piezoelectric actuators, which makes this description not accurate enough for recent modeling purposes [1].
This lack of accuracy led to a need to a nonlinear model to describe the nonlinearities in piezoelectric actuators, such as hysteresis, which is the main reason of nonlinear behavior of piezoelectric actuators. Some of the most recognized nonlinear models are reviewed in the following section.
2.2.2. The Nonlinear Models of Piezoelectric Actuators
12
Nonlinear models of piezoelectric actuators take hysteresis, creep and vibration in account to get accurate models of piezoelectric actuators. The majority of these nonlinear models focus on hysteresis, since it is the main reason of the nonlinear behavior of piezoelectric actuators. Moreover, it is possible to reduce the effect of creep and vibration when modeling piezoelectric actuators to drive systems at high speed and low frequency, respectively [15].
Several hysteresis model have been proposed to model piezoelectric actuators, such as Maxwell-slip model [1], Duhem model [17], Preisach model [18], Bouc-Wen [19] and Prandtl–Ishlinskii [20]. Furthermore, a combination of hysteresis, creep and vibration models have been proposed to achieve more accurate models of piezoelectric actuators [5, 21].
The following sections highlight some of the well-known nonlinear models of piezoelectric actuators.
2.2.2.1. Lumped-Parameter Based on Maxwell-Slip Model
Over the past few decades, several nonlinear models of piezoelectric actuators have been proposed. One of the most well-known models was proposed by Goldfarb and Celanovic in 1997 [1]. They proposed a lumped-parameter model to describe the nonlinear behavior of piezoelectric stack actuators, where a generalized Maxwell resistive capacitor was proposed as lumped-parameter casual representation of the hysteresis. This model is completely based on physical principles, and it consists of electric and a mechanical domain, as well as the connection between the two domains. Moreover, this model describes both the hysteresis nonlinearity and the linear dynamical aspects. The schematic resulting piezoelectric stack actuator is shown in Figure 2.1.
13
Figure 2.1.
Schematic representation of the piezoelectric stack actuator model.
The generalized Maxwell resistive capacitance, which is represented by the MRC element, locates in the electrical domain and thus relates the element's electrical voltage to charge. The piezoelectric actuator model has two ports of interaction, a force-velocity port on the mechanical side and a voltage-current port on the electrical side. Thus, the piezoelectric stack actuator is assumed to have a lumped mass and a linear material damping and stiffness.
This model expresses the nonlinear behavior of piezoelectric actuators as sets of equations that relate the electrical and the mechanical behaviors and combine them together in a mass-spring-damper second order differential equation.
2.2.2.2. Duhem Hysteresis Model
Duhem hysteresis model was proposed by Duhem and Stefanini in 1897 [22]. This model has been used to describe the hysteresis term of piezoelectric actuators, and then it received more attention after combining it with the lumped parameter model by Adriaens, Koning, and Banning in 2000 [17]. They presented an electromechanical piezoelectric actuator model, based on physical principles, which is a combination of a first-order differential equation to describe the hysteresis effect
14
by relating the current flowing through the circuit, the input voltage and other parameters that control the shape of the hysteresis loop, and a partial differential equation to describe the mechanical behavior. The description of the mechanical behavior of piezo-actuated positioning system is similar to the previous model [23].
This model has been studied by several recent studies on modeling piezoelectric actuators, such as designing a high-speed atomic force microscope [24], piezoelectric actuation systems for micro/nano manipulation systems [25], and fast atomic force microscopy using a piezoelectric actuator for positioning control [26].
2.2.2.3. Preisach Hysteresis Model
A hysteresis model was suggested by Preisach in 1935 [18]. Since then, it became a widely accepted model to describe the hysteresis behavior in piezoelectric actuators. Preisach model is based on some assumption regarding the physical mechanisms of magnetization. Although it was first regarded as a physical model of hysteresis, Preisach model is in fact a phenomenological model that has mathematical generality. Thus it is valid to many disciplines [27]. This model expresses the hysteresis nonlinear effect as a double integration equation of the input and output of the system, and the “up” and “down” switching values of the input.
Preisach hysteresis model has been applied and improved by several recent studies to model piezoelectric actuators using various aspects and applications, such as modeling a piezoelectric actuator to drive active antennas [27], and modeling a nano-positioner stage driven by piezoelectric actuator [28]. Other studies have also introduced modified models of Preisach hysteresis model, such as improving the prediction of Preisach model using bilinear interpolation [29], and a hybrid modeling using neural network hysteresis modeling with an improved Preisach model to predict the output of the piezoelectric actuator [30, 31].
15
2.2.2.4. Bouc-Wen Hysteresis Model
A well-known hysteresis semi-physical model was proposed initially early in 1971 [32], and then generalized by Wen in 1976 [19]. Since then, it was known as Bouc-Wen model and became widely applied in modeling piezoelectric actuators, and other different engineering fields applications, such as magneto-rheological dampers [33], base isolation devices for buildings [34], and wood joints [35]. This wide attention to this model is due to its simplicity and ability to describe a variety of different hysteretic curves [36, 37]. Moreover, Bouc-Wen model includes a set of differential equations, which give the advantage of using a first order differential equation to describe the hysteresis model that can reflect the hysteresis relationship between input and output. These differential equations can be easily converted into the form of state equation, which makes this model good from control point of view [37].
The function that describes the hysteresis behavior between the force and the displacement was proposed by Bouc [32]. Consider Figure 2.2, where and
is a displacement. By considering that
at instant time [38].
is a force
is a function of time, then the value of
will depend on the value of
at time
and the past value of
16
Figure 2.2.
Graph force versus displacement for a hysteresis functional [38].
The graph of Figure 1.1 remains the same for all increasing functions ( ) between 0 and
, and for all decreasing functions ( ) between the values
and
. This is called the rate-independent property [39]. Bouc proposed two equations to describe hysteresis oscillator, but it was found that it is hard to solve these equations due to their high nonlinearity. Thus, the use of the variant of the Stieltjes integral [40] was proposed to solve a function of bounded variations. This model is known as Bouc model [41]. In 1980, Bouc model has been extended by Wen to describe the relationship between the restoring force and the hysteresis. Then, it was called BoucWen hysteresis model [19, 42].
Some subsequent studies have proposed modifications to this model in order to take into account some physical properties in hysteretic systems. Baber and Wen modified Bouc-Wen model in order to model engineering structures [43], then Baber and Noori have considered pinching to modify Bouc-Wen model [44], while Foliente modified Bouc-Wen model to model wood joints systems [35]. In order to model piezoelectric actuators, Low and Guo have modified Bouc-Wen hysteresis model, based on the assumption of the elastic structure and material, and by using Newton
17
Laws and assuming that piezoelectric actuator can be assumed as a mass-springdamper system [45].
Several recent studies have applied different estimation methods to identify Bouc-Wen model parameters. Ha, Kung, Fung and Hsien applied genetic algorithm to identify the parameters of Bouc-Wen model [46]. Minase, Tien-Fu, Cazzolato, and Steven applied Unscented Kalman Filter to identify Bouc-Wen parameters for a precise operation of a micro-motion stage [47]. Wang, Zhu and Yang applied leastsquares method to identify the parameters of Bouc-Wen model for a linearized model of a piezoelectric stack actuator [48]. Rakotondrabe applied nonlinear filter system identification method to identify the parameters of Bouc-Wen model [36]. Qiang, Chao, Dong, Shunwei and Xueliang applied nueral networks to identify Bouc-Wen parameters for a high precision system driven by piezoelectric actuator [49]. Wang, Zhang, Mao and Zhou used particle swarm optimization to estimate the parameters of Bouc-Wen model for a piezoelectric stack actuator [50].
Bouc-Wen hysteresis model has been studied and applied by several recent works. Lin and Yang used Bouc-Wen hysteresis model to describe the hysteresis nonlinearity of a piezo-actuated stage that is driven by a piezoelectric stack actuator, and then verified the model by experiments [51]. Chang used Bouc-Wen hysteresis model to describe the hysteresis nonlinear dynamics of a precision positioning system, which is a moving stage driven by a piezoelectric stack actuator. Chang obtained a precise tracking performance as 10 nanometers [52].
Wang and Mao used Bouc-Wen hysteresis model based on intelligent optimization to describe the hysteresis nonlinearity of a piezoelectric actuator. Moreover they achieved high-quality fitting curve between experimental and modeled data [53]. Sofla, Rezaei, Zareinejad and Saadat used Bouc-Wen hysteresis model to represent the hysteresis nonlinearity, unknown system parameters and external load disturbance of a piezo-actuated stage driven by a piezoelectric stack actuator. Furthermore, they proved the correctness of the model by experiments [6].
18
Wei, C. Zhang, G. Zhang and Hu proposed a modified modeling method based on dynamic recurrent neural network combined with Bouc-Wen hysteresis model. Then, they used sets of data including driving voltages and corresponding displacements to train the network, and modified the parameters in the neural network by back propagation algorithm. Moreover, the results were verified with experiments [54]. Zhu and Wang introduced a non-symmetrical Bouc-Wen hysteresis operator for modeling the non-symmetrical hysteresis of a piezoelectric actuator. Then, a corresponding parameter identification method, which can identify the parameters by obtaining analytical solutions with a set of input-output experimental data [55]. Liu, Cai, Dong and Qu used Bouc-Wen hysteresis model to present the hysteresis nonlinear term of a piezo-actuated system. Then, a real-coded adaptive genetic algorithm is adopted to identify the model parameters simultaneously [37].
2.3. Control Overview
As discussed in Section 2.2, hysteresis phenomenon affects the response of piezoelectric actuators, causing the output displacement to have a nonlinear relationship with input voltage. This will lead to several challenges when designing a controller for any system that is driven by a piezoelectric actuator.
The problem of controlling the hysteresis effect can be highly minimized by using charge or current to drive the piezoelectric actuator. It has been proven that the piezoelectric effect in a charge-driven piezoelectric actuator is minimal [56, 57]. However, this type of controlling piezoelectric actuators has not been widely used due to the difficulties associated with driving highly capacitive loads with available charge/current amplifiers. The main problem is the presence of offset voltages in the charge or current source circuit, and the uncontrolled behavior of the output voltage, which charges up the capacitive load. When the output voltage reaches the power supply rails, the signal applied to the actuator saturates and distortions occur [14].
19
Several studies propose new structures for charge and current sources capable of regulating the DC profile of the piezoelectric actuator [58]. However, the need for additional electric circuits increases the complexity of the control hardware, the feedback loops and the building cost [58, 59]. Thus, this study focuses on voltage control of piezoelectric actuators.
The following sections review the literature on voltage control development for piezoelectric actuators. The different approaches are generally classified into feedback, feedforward, iterative, and sensor-less control.
2.3.1. Feedback control
In feedback control, the controlled variable is measured and compared with a desired value. This difference between the actual and desired value is called the error. Feedback control manipulates an input to the system to minimize this error [14]. Figure 2.3 shows a block diagram of a feedback control loop.
Figure 2.3.
Feedback control loop.
Precision positioning can be achieved at high frequencies if feedback controller gains can be chosen to be suitably high at those frequencies to overcome nonlinearities effect. However, there are limits to the improvements that can be achieved in positioning performance with high-gain controllers due to the low gain margins that occur in piezoelectric actuators. This behavior is a result of a rapid loss in phase at the sharp resonant peak in the frequency response as well as a loss in
20
phase due to higher frequency dynamics and filters used with sensors and actuators [14].
One method to increase the gain margin is by modifying the first sharp resonant peak of the system with a notch filter, and then using it to design feedback controllers that improve the closed-loop performance in piezoelectric actuators even at high frequencies [21, 60]. Another method is by using integral controllers, which gives the advantage of providing high gain feedback at low frequencies. Thus, integral controllers can overcome creep and hysteresis effects and lead to precision positioning. This makes traditional Proportional-Integral-Derivative (PID) feedback controllers suitable to control piezoelectric actuators [14, 61].
2.3.1.1. PID Feedback Controller
The first published theoretical analysis of a PID controller was in 1922 by Minorsky [62]. This controller is still used in many control applications, even in piezo-actuated systems that need high level of control precision. This is due to its simplicity, and its ability to improve transient and steady-state response [63], and it has been used by several recent studies to control piezo-actuated systems [61]. Figure 2.4 shows a block diagram of a PID controller in a feedback loop.
Figure 2.4.
A block diagram of a PID controller in a feedback loop.
21
Proportional control is the control action that occurs in direct proportion with the system error. Proportional control responds to only the present error, and it can be adjusted by multiplying the error by the proportional gain constant. If the proportional gain is too high, the system can become unstable. In contrast, if the proportional gain is too low, the control action may be too small to respond to the system disturbances [64].
Integral control is used in systems where using proportional control only is not enough for reducing the steady-state error within acceptable bounds. The integral controller is based on the principle that the control action should exist as long as the error is different from zero, which means that the integral term is proportional to both the magnitude and the duration of the error. The summation of the error over time is accumulated and multiplied by the integral gain, and then added to the controller output. In some cases, it is responsible for introducing undesirable effects into the control loop in the form of increase settling time, reduced stability and integral windup [64].
Derivative control is only active when the error is changing, which means that its contribution will be zero for static errors. Derivative action on its own will therefore allow uncontrolled steady-state errors. Thus, the derivative control is usually combined with either Proportional control or Proportional-Integral control. Derivative controller can be considered as a high-pass filter that is sensitive to setpoint changes and process noise when operating in the forward path [64].
Table 2.1 summarizes the effect of each gain of the PID controller on the rise time, overshoot, settling time and the steady-state error. “NT” means no definite trend, or minor change.
22
Table 2.1. Gain
The effect of PID gains on the response of the system. Response Rise Time
Overshoot
Settling Time
Steady-State Error
Proportional
Decrease
Increase
NT
Decrease
Integral
Decrease
Increase
Increase
Eliminate
Derivative
NT
Decrease
Decrease
NT
The parameters (gains) of the PID controller need to be tuned correctly in order to get the desired response of the system.
Several methods have been proposed to tune the parameters PID controllers, such as Ziegler-Nichols method [65] and Cohen-Coon method [66], but due to the high nonlinearity of piezoelectric actuators, these methods are neither applicable nor efficient to tune the parameters of PID controllers in high precision systems. Thus, recent research effort has gone into the automated tuning of the parameters of PID controllers, such as Genetic Algorithms (GA) [67] and Particle Swarm Optimization (PSO) [68].
Both techniques are inspired by nature, and have proved themselves to be effective solutions to optimization problems. From literature, it is found that, GA is faster in terms of computational time, since the computational time increases linearly with the number of generations for GA, while for PSO the computational time increases almost exponentially with the number of generations, due to the communication between the particles after each generation. However, the PSO seems to arrive at its final parameter values in fewer generations than the GA. Furthermore, PSO can be implemented simply without too many parameters, and it has good global searching ability, because the information of particle is single-directional, each particle remembers the past position, and the convergence is very quick [69, 70]. Thus, this study will focus on PSO method to tune the parameters of PID controllers.
23
2.3.1.2. PSO-Based PID Controllers
In 1995 Kennedy and Eberhart
proposed PSO, which is a heuristic
optimization method that is inspired by the social behavior of bird flocking or fish schooling [68]. While searching for food, the birds are either scattered or go together before they detect the place where they can find the food. While birds are looking for food from one place to another, usually there is a bird that can find food faster. Since these birds are passing the information between each other at any time while searching the food from one place to another, the birds will finally gather at the place where food can be found. In PSO algorithm, the solution swam is compared to the bird swarm, the moving birds from one place to another are equal to the development of the solution swarm, good information is equal to the most optimist solution, while the food resource is equal to the most optimist solution during the whole progress [71].
Each particle in the PSO moves through the search space with an adaptable velocity that is modified dynamically according to its own moving experience and other particles moving experience as well. In PSO, each particle tries to improve itself by simulating the characteristics of its successful peers. Furthermore, each particle has a memory, which means it would be capable of remembering the best position in the search space that it has tried. The position corresponding to the best fitness is known as “best position”, and the best one among all particles in the population is called “global best position” [71, 72].
Different modified algorithms of PSO have been proposed, such as PSO with inertia weight [73], PSO with construction factor [74], fully informed PSO [75], and hybrid PSO with mutation [76]. This study focuses on PSO with inertia weight due to its simplicity and high ability to explore wider range of solutions.
Several recent studies have applied PID controllers to control the displacement of piezoelectric actuators [61]. From literature, it can be noticed that the results of controlling piezo-actuated systems using traditional PID controllers
24
only were not accurate enough for high precision positioning systems, which led to the need of using improved PID techniques, such as PID sliding mode controllers [77, 78]. Nevertheless, the results were not accurate enough for high precision systems due to the existence of the nonlinear effects in piezoelectric actuators, such hysteresis. Those nonlinear effects could be minimized by combining feedforward controllers with feedback controllers [79].
2.3.2. Feedforward control
In a feedforward control, the control variable signal is not based on the error. Instead, it is based on the knowledge about the mathematical model of the process, or measurements of the process disturbances [14]. Figure 2.5 shows a feedforward control loop.
Figure 2.5.
Feedforward control loop.
Using feedforward control can lead to improving the output tracking performance in piezo-actuated systems [80]. Furthermore, major improvements in precision positioning at high frequencies can be achieved by using feedforward techniques [81]. This ability to increase the bandwidth with model-based feedforward while achieving sub-nanometer scale positioning precision was introduced experimentally in [80]. The feedforward method uses a mathematical model of the system’s dynamics to determine vibration-compensating inputs for piezoelectric positioners. This is compound with the hysteresis inverse to invert the system model.
25
Several methods have been proposed to deal with hysteresis in piezoelectric actuators, most of which are based on feedforward inverse compensation, which invert mathematical models of the hysteresis to determine the hysteresis compensating inputs. Hysteresis inversion is suitable in low frequency operation since creep can be corrected using feedback and vibrations are negligible at low frequencies. The inversion can be done by fitting the hysteresis with polynomials [82] or any hysteresis nonlinear model, such Bouc-Wen hysteresis model. Then, the model is inverted to achieve the inverted hysteresis model. An alternative approach is to directly take the inverse model and use it to find the input [83-85].
Another proposed approach to overcome the complexity and inaccuracy that occur in some cases when applying the previous methods, such as inaccurate results due to asymmetric hysteresis loops [51]. This approach uses a hysteresis observer to estimate the hysteresis effect. Besides that, the problem of velocity measurement can also be solved by using the hysteresis observer to estimate the velocity of the positioning stage from input signal and output position measurements. This will offer the opportunity to emit velocity sensors, and thus reduce the cost and eliminate measurement noise [86].
Several recent studies have applied the previous approaches, but it has been noted that modeling uncertainties and external disturbance usually exist, so a feedback controller is needed to enhance the robustness of the systems and improve the tracking performance [86]. Thus, an integration of feedforward and feedback control is required to achieve the desired accuracy and precision.
2.3.3. Hybrid Feedback and Feedforward control
Feedforward control improves performance without affecting the stability problems associated with feedback design. However, it is not able to account modeling errors. Therefore, it is necessary to use an integration of feedback and
26
feedforward control to reduce the errors that are caused by uncertainty in the inverse input. The use of feedforward inputs can improve the tracking performance compared with the use of feedback alone, even in the presence of plant uncertainties [14]. Figure 2.6 shows a hybrid feedback and feedforward control loop.
Figure 2.6.
Hybrid feedback and feedforward control loop.
This hybrid control method was used by several recent studies due to the high accurate results that were accomplished when applying it piezo-actuated high precision positioning systems. Different combinations of controllers were used in this type of hybrid control of piezo-actuated systems. Song, Zhao, Zhou and AbreuGarcía applied a combination of a PD/lead-lag feedback controller with an inverse hysteresis feedforward controller to control a piezo-actuated system. Furthermore, they accomplished accurate experimental results by applying this type of hybrid control [27].
Gu and Zhu applied a combination of a model-based feedforward controller and a PID feedback controller for high-accuracy and high-speed tracking control of piezoelectric actuators. The authors proved the accuracy of this method by experimental results [85].
Sofla, Rezaei, Zareinejad and Saadat proposed a hysteresis observer to estimate the hysteresis effect. Then for real-time compensation of the observer error, parametric uncertainties and external disturbances, the sliding mode control strategy
27
with a perturbation estimation function is utilized as a feedback controller. The control method was proved by experimental results [6].
Lin and Yang proposed a hybrid feedback and feedforward control structure to compensate the hysteresis and friction nonlinearities of a piezo-actuated stage. A Proportional-Integral feedback control associated with feedforward compensating based on the hysteresis observer were used in this method [51]. The hysteresis observer used is based on Luenberger observer [87], since it has an easy structure, easy to implement and it gives the benefit of eliminating the phase lag from the control loop. This elimination increases the margin of the stability of the system [88]. Several experiments were carried out to prove the accuracy of this method.
In this study, the last proposed control method by Lin and Yang will be applied and improved by studying its efficiency when using different parameters of the system. Furthermore, a PSO-based PID controller will be used to achieve more accurate results, as demonstrated in the following chapters.
2.4.
Summary
In this chapter, linear and nonlinear methods of modeling piezoelectric actuators were reviewed. From literature, it is found that the linear model of piezoelectric actuators is not accurate enough for high precision positioning systems. Thus, nonlinear model should be chosen for this purpose, since they are capable of describing the nonlinear behavior of piezoelectric actuators. Bouc-Wen hysteresis model were chosen to model the piezoelectric actuator in this study due to its high ability to represent the nonlinear behavior of piezoelectric actuators, and to model several hysteresis loops. Bouc-Wen hysteresis model and its modified models were reviewed in this chapter.
28
Furthermore, this chapter highlighted different methods to control piezoactuated systems. It is found that PID controllers are frequently used as feedback controllers with piezo-actuated systems due to their control the nonlinearity of these systems, and their high feedback gain. Moreover, this chapter reviewed one of the efficient methods to tune the parameters of PID controllers, which is PSO. This method proved itself as a very accurate method to tune PID controllers, but so far, it is noted that there is a lack of applying this method to tune PID controllers in piezoactuated systems. Thus, this study will focus on this method to explore this part of knowledge.
Besides that, this chapter also reviewed feedforward control methods in piezo-actuated systems, and it is found that feedforward controllers can improve the stability of these systems, but they are not able to account modeling uncertainties. Hybrid control methods between feedback and feedforward control methods were reviewed. It is found that the hybrid methods give more accurate results than feedback or feedforward control only. This study will apply the hybrid control method to control piezoelectric stack actuators. Furthermore, the method will be improved by using a PSO-based PID controller to achieve more accurate results.
CHAPTER 3
METHODOLOGY
3.1. Introduction
This chapter discusses the methodology used to derive the dynamic hysteresis model of the piezo-actuated system, and the designed controllers for the system.
3.2. Project Methodology
As discussed in the Scope of Work and the Literature Review, this study focuses on the hysteresis modeling and displacement control of piezoelectric stack actuators. To get an accurate dynamic model of systems that are driven by piezoelectric actuators, those systems are modeled together with the piezoelectric actuator as one piezo-mechanism system. These systems are usually called piezoactuated systems.
In this thesis, the piezo-actuated system under study is a moving stage driven by a piezoelectric stack actuator. The dynamic model of the system is based on Bouc-Wen hysteresis model to express the nonlinear behavior of the piezoelectric stack actuator, while creep and vibration nonlinearities will not be modeled in this study, since they have a minor effect on the response of the piezoelectric stack
30
actuator compared to the hysteresis effect, and they can be greatly minimized by fast positioning conditions and low frequency voltage input, respectively [15]. To give better understanding about Bouc-Wen hysteresis model, the model and its theoretical and physical origins and developments are studied and discussed in this thesis.
Bouc-Wen hysteresis model is utilized to express the dynamic model of piezo-actuated system under study, and the values of the parameters of the system are taken from a previous work [6]. Furthermore, the modeled system is simulated and tested using MATLAB/Simulink. Then, the simulated modeling results are compared with the results achieved by the same reference.
In order to control the displacement of the piezo-actuated system and improve its stability, the hysteresis nonlinear effect is estimated by a Luenberger-like observer to get the hysteresis state [87]. Then, this state is combined with the original reference signal to form a feedforward controller to compensate the hysteresis nonlinearity. The resulting controller is then used to control the simulated piezoactuated system using MATLAB/Simulink.
The simulated results of the feedforward controller show that it is able to improve the tracking of the reference signal and the stability of the piezo-actuated system, but still not accurate enough for high precision positioning systems and modern control purposes. In addition, the feedforward controller is not able to control modeling uncertainties and external disturbances. Thus, a PID feedback controller is combined with the feedforward controller to form hybrid feedforward and feedback controllers.
Due to the high nonlinearity of the controlled system, the parameters of the PID controller are optimized using PSO method while it is combined with the feedforward controller to get the best design for the hybrid controllers.
31
The hybrid controllers are applied on the simulated piezo-actuated system using MATLAB/Simulink. Then, the results of the simulation are compared with previous works that use similar and different types of controllers to control similar systems. The achieved results from this study show higher tracking accuracy, less overshoot and faster response and settling times, as demonstrated later in Chapter 5.
The methodology that is applied in the project methodology is illustrated briefly in a flow chart, as shown in Figure 3.1.
32
Figure 3.1.
Flow chart of the project methodology.
33
3.3. Summary
In this chapter, the methodology used in this study is explained and discussed. The methodology of proposing Bouc-Wen hysteresis model and its evolution is explained theoretically and mathematically. Moreover, the steps required to design controllers for the system are discussed, where a Luenberger observer for the system is designed first, and the combined with the voltage input to form a feedforward controller, which is then integrated with PSO-based PID feedback controller to form hybrid controllers to control the displacement of the system.
CHAPTER 4
MODELING
4.1. Introduction
This chapter discusses the origin of Bouc-Wen hysteresis model and its improved versions over the years, and then way it is applied on the system under study to obtain its dynamic hysteresis model. Moreover, this chapter discusses the results accomplished by modeling the piezo-actuated system and simulating it using MATLAB/Simulink. Section 4.2 discusses the derivation of Bouc-Wen hysteresis model and using on the system under study. Section 4.3 discusses the simulated results accomplished using MATLAB/Simulink, where several tests are done using different voltage sources, amplitudes and frequencies. The results are compared with a previous work, and then further analyses are done to test the response of the system in different cases. Finally, the summary of this chapter is highlighted in Section 4.4.
4.2. Modeling the System
In order to model the system understudy, the dynamic model of the piezoelectric stack actuator is studied based on Bouc-Wen hysteresis model. Then piezo-actuated system is introduced and combined with the piezoelectric stack actuator’s model to obtain the final hysteresis model of the whole system. The
35
following two sections detail the methodology adopted in achieving the final recent Bouc-Wen hysteresis model for piezoelectric actuators after being modified in past few decades, and the structure of the piezo-actuated system, respectively.
4.2.1. Bouc-Wen Hysteresis Model
To give better understanding of how Bouc-Wen hysteresis model was proposed and developed until it reached to its final well-known form, this section details the derivation based on the theoretical and physical principles of the original proposed model by Bouc, and its modified models as well.
In 1971, Bouc proposed the following form to define the form of the functional :
(
(
))
)1.2(
Consider the equation
( )
where ( ) is a function of
( )
)1.1(
and , ( ) is the input. Equations 4.1 and 4.2
describe completely the hysteresis oscillator. Bouc found that it is difficult to give an explicit solution of Equation 4.1 duo to the nonlinearity of the function
( ).
Therefore, the use of variant of the Stieltjes integral to define the functional
was
proposed, and to solve the function ( ), which is a function of bounded variations [41]:
36
( )
where phenomena, are defined,
( )
∫ (
)
( )
)1.4(
is the hereditary kernel function that takes into account hysteretic [
] is the time instant after which displacement and force
is the total variation of
in the time interval [
] [89]. The function
satisfies the mathematical properties and the hysteresis properties, thus:
( )
∑
)1.1(
Equations 4.2 to 4.3 can then be rewritten as:
|
)1.4(
( )
∑
)1.4(
|
Equations 4.5 and 4.6 are known as Bouc model. Then, Equation 4.6 has been extended by Wen to describe the relationship between the restoring force and the hysteresis as follows [19]:
̇ ̇
| ̇| | ̇|
̇| | | ̇
|
)1.4( ̇
)1.4( ̇
Equations 4.7 and 4.8 are the earliest version of Bouc-Wen model, where ̇ represents the time derivative of the hysteresis nonlinear term, 0
is the post
to pre-yielding stiffness ratio that controls the shape of the hysteresis loop,
and
37
are parameters that control the shape and the magnitude of the hysteresis loop, while > 1 is a scalar that controls the smoothness of the transition from elastic to plastic response [42].
In 1980, Wen introduced the following modified hysteresis equation of BoucWen model [90]:
̇
where
| ̇| | | ̇
̇| |
)1.4(
is a parameter that controls the shape of the hysteresis loop. In order
to model piezoelectric actuators, Low and Guo assumed that
, since the
assumption of the elastic structure and material [45]. Thus, Equation 4.9 can be rewritten as follows:
̇
| ̇| ̇
̇| |
)1.24(
By modifying Equation 4.10 slightly to express the piezoelectric actuator dynamic equation and by using Newton Laws and assuming that piezoelectric actuator can be represented as a mass-spring-damper system:
̈ ̇
where
( ̇ ̇
| ̇|
)
)1.22(
̇| |
)1.21(
is the displacement of the piezoelectric actuator,
nonlinear term, and ̇ ̈ and ̇ are the derivatives of
and
is the hysteresis
respectively.
and
are the mass, damper coefficient and stiffness factor of the piezoelectric actuator. the is the applied voltage, piezoelectric ceramic, while
is
is the excitation force that generated by the is the piezoelectric material constant. Equations 4.11
and 4.12 represent the dynamic equations of piezoelectric stack actuators according
38
to Bouc-Wen hysteresis model. These equations have been applied widely to model different piezo-actuated systems with slight modifications in certain cases when using different types of piezoelectric actuators or when the systems require different assumptions.
4.2.2. Modeling the Piezo-Actuated Stage
The model is a moving stage driven by a piezoelectric actuator. One end is fixed and the other is sliding horizontally. By assuming a high generated force by a piezoelectric actuator comparing to the frictional force, the schematic diagram of the moving stage system is as shown in Figure 4.1.
Figure 4.1.
Schematic diagram of a moving stage driven by a piezoelectric actuator.
In this system, the piezoelectric stack actuator is regarded as a force generator that generates force due to the applied voltage. The whole system can be modeled using Equations 4.11 and 4.12, where
and
in this case are the mass, damper
coefficient and stiffness factor of the whole positioning mechanism combined together.
39
As mentioned in Chapter 3, the parameters of the system are taken from previous work [6]. The values of these parameters are given in Table 4.1.
Table 4.1.
Values of the system parameters
Parameter
Value
Parameter
Value
m
2.17 kg
d
9.013×10-7 m/V
b
4378.67 Ns/m
α
0.38
k
3×105 N/m
β
0.0335
Fext
0N
γ
0.0295
Substituting the values of the parameters into Equations 4.11 and 4.12 gives the system dynamic equation, which can be simulated in MATLAB/Simulink. Figure 4.2 shows a MATLAB/Simulink block diagram used to simulate the system. The parameters of the system are utilized in MATLAB as shown in Appendix A.
40
Figure 4.2.
A MATLAB/Simulink block diagram of (a) the entire piezo-actuated system. (b) the sub-model of Bouc-Wen hysteresis model.
The voltage input is multiplied by the gain displacement. Since the unit of
so it can be converted to
is m/V, then doing this multiplication will give the
corresponding displacement of each input voltage at any time of the simulation. This is needed to compare the response with reference input. The transfer function refers to the left side of Equation 4.11 (
̈
̇
) in Laplace domain, and thus make
the block diagram simpler. The system is tested using MATLAB/Simulink to prove the correctness of the modeling and simulation process.
The differential equations of the system can be represented in state-space form as
41
̇ ̇
[ [
[ ̇]
|
][ ]
[
]
]
| |
| ][ ]
[
[
| |
)1.24(
] [ ]
[
] [ ]
̇
where the states represent represent
̇ and
; the input
̇ . Equation 4.13 can be represented in matrix form as
and
̇
|
|
| | )1.21(
Thus, the system in Figure 4.2 can be represented using Equation 4.14 as shown in Figure 4.3. Then, it is compared with the system in Figure 4.2 using MATLAB/Simulink to prove that they represent the same system and they have the same response.
42
Figure 4.3.
A MATLAB/Simulink block diagram of the system in state-space representation.
Several tests are made on the system using different voltage sources, amplitudes and frequencies. The results of these tests are shown and discussed in the following section.
4.3. Simulation Results of the Modeled System
This section presents and analyzes the results of simulating the piezo-actuated system under study using MATLAB/Simulink, and tests its response under several cases. To simulate the model of the piezo-actuated stage system, MATLAB/Simulink block diagram that is shown in Figure 4.2 is simulated. A step voltage input with amplitude of 80 V is applied to the input of the system, and the result of the simulation is shown in Figure 4.4.
43
Figure 4.4.
The response of the system represented by differential equations to an 80 V step input.
To examine the correctness of the state-space representation of the piezoactuated system, the MATLAB/Simulink block diagram in Figure 4.3 is then simulated. The same step voltage input that was applied to the previous model is used. The result of the simulation is shown in Figure 4.5, and it shows that both results in Figures 4.4 and 4.5 are identical. This proves that both representations of the system can be used in modeling piezo-actuated systems.
Figure 4.5.
The response of the system represented by state space to an 80 V step input.
44
The error between the reference input and the output displacement of both representations is shown in Figure 4.6.
Figure 4.6.
The error between the step reference input and the output displacement.
It can be noted from Figures 4.4 to 4.6 that the response of the system looks stable, but with a high error (about 72.1
) at the beginning of the simulation.
Analyzing the system using a step input does not give full detailed results about its performance. This is because of the hysteresis behavior that cannot be seen clearly when the input voltage is constant. Thus, a triangular input voltage with amplitude of 80 V and frequency of 1 Hz is used to test the performance of the system. The input voltage signal is shown in Figure 4.7.
45
Figure 4.7.
A triangular input voltage with 80 V and 1 Hz.
The input voltage is multiplied by the piezoelectric material constant ( ) to give the value of the corresponding reference displacement signal. Then, the output displacement response is compared with the reference signal as shown in Figure 4.8.
Figure 4.8.
The output displacement response compared with the reference signal.
From Figure 4.8, it is clear that the output displacement evinces a distortion on both rising and falling slopes, which indicates a nonlinear relationship between
46
the input voltage and the output displacement of the piezoelectric actuator. Based on Bouc-Wen model, the nonlinear relationship between the input and the output is caused by the hysteresis phenomenon that exists in piezoelectric ceramics [13]. The resultant error between the reference signal and the output displacement response, which is caused by the hysteresis effect, is demonstrated clearly in Figure 4.9.
.
Figure 4.9.
The error between the triangular reference input and the output displacement.
It is noted from Figure 4.9 that error of the displacement keeps changing between the range of -7.43
to 7.5
, where the negative sign refers to the
direction of movement. This permanent error is due to the hysteresis relationship between the input voltage and the output displacement. The hysteresis phenomenon is demonstrated in Figure 4.10, which shows the nonlinear relationship between the input voltage and the output displacement of the system.
47
Figure 4.10. The hysteresis relationship between the input voltage and the output displacement of the system.
The hysteresis phenomenon is based on the crystalline polarization effect and molecular friction. The displacement generated by piezoelectric actuator depends on the applied electric field and the piezoelectric material constant which is related to the remnant polarization that is affected by the electric field applied to piezoelectric material. The deflection of the hysteresis curve depends on the previous value of the input voltage, which means that piezoelectric materials have memory because they remain magnetized after the external magnetic field is removed. This magnetization makes the output displacement response to the increased input voltage from 0 V to 80 V differs from that one to the decreased input voltage from 80 V to 0 V. Furthermore, it is also indicated that the initial ascending curve starts from the origin, but the loops do not go back to the origin even if the applied voltage is back to zero. This is caused by the polarization and elongation that occurs in the piezoelectric material under positive voltages cannot be completely retrieved even if the input voltage returns to zero [13].
To understand the hysteresis behavior between the input voltage and the output displacement, triangular input voltages with different amplitudes of [60 V, 80 V, 100 V and 120 V], and similar frequency of 1 Hz, were applied on the system, as shown in Figure 4.11.
48
Figure 4.11. The hysteresis loop using a triangular input voltage with a frequency of 1 Hz and a voltage of (a) 60 V. (b) 80 V. (c) 100 V. (d) 120 V.
Figure 4.11demonstrate that increasing the input voltage leads to increasing the maximum output displacement, and decreases the distance between the initial ascending curve and the other loops. On the other hand, changing the frequency of the input voltage leads to make the hysteresis loop narrower and sharper [91], as shown in Figure 4.12, where triangular input voltages with 80 V and different frequencies of [1 Hz, 2 Hz, 10 Hz and 100 Hz] are applied on the system.
49
Figure 4.12. The hysteresis loop using a triangular input voltage with a voltage of 80 V and a frequency of (a) 1 Hz. (b) 2 Hz. (c) 10 Hz. (d) 100 Hz.
From the accomplished results, it is noted that there is a phase lag between the input voltage and the output displacement of the waveforms peaks. Changing the input voltage leads to changing the phase lag between the input and the output. This phase lag indicates the existence of the hysteresis behavior in piezoelectric actuators. In other words, hysteresis can be defined as a phase lag between a periodic input and its corresponding output [92].
According to Sofla et al [6], which is the reference that the parameters of this project are taken from, the hysteresis relationship between the voltage input and the output displacement are shown in Figure 4.13.
50
Figure 4.13. The hysteresis loop from reference [6].
By comparing the Figures 4.10 and 4.13, it is clearly that the results achieved in this chapter are similar to the results achieved by Sofla et al [6], which proves the correctness of the modeling process.
4.4. Summary
This chapter discussed the derivation of Bouc-Wen hysteresis model and its improved versions, and their application on the piezo-actuated system under study to obtain its dynamic hysteresis model. Several simulations using MATLAB/Simulink have been carried out to test the response of the system to different voltage input signals. The results prove the existence of the hysteresis behavior in the piezoelectric stack actuator used to drive the system, and prove that Bouc-Wen hysteresis model is applicable for this system. Furthermore, the results obtained are similar to the previously published results, which prove the correctness of the modeling steps.
CHAPTER 5
CONTROL DESIGN AND RESULTS
5.1. Introduction
This chapter discusses the steps used to design a feedforward controller and a hybrid feedforward and feedback controllers for the piezo-actuated system. The results achieved by using the designed controllers are presented, discussed and compared. Section 5.2 presents and the design method of the Luenberger observerbased feedforward controller, and discusses the results accomplished by using this controller. Section 5.3 presents the designing steps of the hybrid Luenberger observer-based feedforward and the PSO-based PID feedback controllers, and discusses and compares the results of the designed controllers and the results of the uncontrolled system as well. Finally, the summary of this chapter is highlighted in Section 5.4.
5.2. Feedforward Controller Design and Simulation
This section presents the steps taken to design the Luenberger observer-based feedforward controller for the piezo-actuated system, and then discusses the simulated results of the feedforward-controlled system.
52
5.2.1. Luenberger Observer Design
Designing the feedforward controller for the piezo-actuated system requires designing an observer to estimate the hysteresis state. Thus, the system is tested to make sure it is observable, which means it is possible to design an observer of the states. The observability test is made by checking the rank of observability matrix defined by Equation 5.1.
)4.2(
[
where
is the system matrix,
]
is the output matrix of the system, and
is
the dimention of martix . For a system to be observable, the observability matrix of this system should have full column rank. Alternatively, if the dimension of matrix is
then the column rank of the observability matrix should also be m to be
observable [93].
The observability matrix of the system is then tested by finding the rank of the matrix in Equation 5.1.
)4.1( [
]
The results of testing the system show that it is observable, since it has a full column rank, where the test is done using a MATLAB code shown in Appendix B.
53
Since the system is observable, then it is possible to design an observer for it. The observer chosen is a Luenberger-like observer, due to its simplicity of implementation, easiness of construction and its ability to eliminate the phase lag that in the control loop [87, 88]. The structure of Luenberger observer is illustrated in Figure 5.1.
Figure 5.1.
The general structure of Luenberger observer.
where the system plant is represented by Equation 5.3,
̇ }
)4.4(
and the general form of Luenberger observer is,
̂̇
̂
( ̂
̂
̂)
}
)4.1(
54
where ̂ represents the estimated state vector, ̂ represents the estimated output (the hysteresis), and
is the observer gain.
Based on Equation 5.3, and by modifying Equation 5.4 to represent the piezoactuated system under study, the Luenberger observer representation for this system is as
̂̇
|
̂
| ̂
| ̂|
̂
Substituting ̂
(
̂) )4.4(
̂
̂ yields
̂̇
|
̂
| ̂
| ̂|
̂
(
̂) )4.4(
̂
The estimation error vector is defined in Equation 5.7, while the dynamics of the observer error is defined in Equation 5.8.
̃ ̃̇
(
)̃
̂ |
| ̃
)4.4( | ̃|
)4.4(
Then, the problem of the hysteresis observer is to design an observer gain to force the observer error to approach zero as quickly as possible, and the observer gain
should be chosen such that
is stable (has negative real eigenvalues)
[86, 94].
Designing the observer is done using a MATLAB code shown in Appendix B, where the gain of the Luenberger observer is designed to be as in Equation 5.9.
55
[
which makes
]
)4.4(
stable, since its eigenvalues are [-10;-11;-12].
Thus, the general block diagram of a Luenberger observer illustrated in Figure 5.1 is modified to be suitable for the piezo-actuated system under study, and then implemented in a MATALB/Simulink block diagram as in Figure 5.2.
Figure 5.2.
MATLAB/Simulink block diagram of Luenberger observer of the system.
5.2.2. Luenberger Observer-Based Feedforward Controller Design
To design the feedforward controller, the controllability of the system is tested to make sure it is possible to design a controller for this system. The controllability test is made by checking the rank of controllability matrix defined by Equation 5.10.
56
[
where
is the input matrix, and
]
)4.24(
is the dimension of matrix . For a system
to be observable, the controllability matrix of this system should have full row rank. Alternatively, if the dimension of matrix
is
then the row rank of the
controllability matrix should also be n to be controllable [93].
The controllability matrix of the system is then tested by finding the rank of the matrix in Equation 5.11.
)4.22( [
]
The results of testing the system show that it is controllable, since it has full row rank, where the test is done using a MATLAB code shown in Appendix B. Since the system is controllable, then it is possible to design a feedforward controller for it. Then the feedforward controller is implemented on the piezo-actuated system using MATALB/Simulink as shown in Figure 5.3.
Figure 5.3.
MATALB/Simulink block diagram of the feedforward controller and the system.
57
The gain of the feedforward controller is chosen to be
to convert the
output of Luenberger observer from a hysteresis unit (meters) to a voltage unit (Volts). Thus, the feedforward control signal can be written as
( )
where
( )
̂( )
)4.21(
( ) is the control signal of the feedforward controller, ( ) is the
input voltage of the system as a function of time, and ̂ is the estimated hysteresis, which
is
the
output
of
the
Luenberger
observer.
Several
tests
using
MATLAB/Simulink are done on the system with the feedforward controller to test the accuracy of the results and the efficiency of the controller. These results are shown and discussed in Section 5.2.3.
5.2.3. Simulation Results of the Designed Feedforward Controller
To test the effect of the Luenberger observer-based feedforward controller on the performance of the system, the same triangular input voltage with amplitude of 80 V and frequency of 1 Hz that was used to test the open-loop model in Chapter 4 is used in this case. Then, the output displacement response is compared with the reference signal as shown in Figure 5.4.
58
Figure 5.4.
The output displacement response of the feedforward-controlled system, compared with the reference signal
Figure 5.4 demonstrate that there is a distortion on both rising and falling slopes of the output displacement. However, it is less than that of the open-loop system which was illustrated in Figure 4.8. The resultant error between the reference signal and the output displacement response, which is caused by the hysteresis effect, is demonstrated in Figure 5.5.
Figure 5.5.
The error between the reference and the output displacement of the feedforward-controlled system.
59
By comparing the error of the feedforward-controlled system in Figure 5.5 with the error of the open-loop system that is illustrated in Figure 4.9, it is can be seen that the amount of error of the open loop has reduced from -7.43 to -3.25
and 3.25
and 7.5
with the feedforward controller. The hysteresis
relationship between the input voltage and the output displacement is demonstrated in Figure 5.6.
Figure 5.6.
The hysteresis loop of the feedforward-controlled system.
By comparing the hysteresis relationship of the open loop system in Figure 4.10 with the one of the feedforward-controlled system in Figure 5.6, it is shown that the hysteresis loop of the feedforward-controlled system is less nonlinear than that of the open-loop system. The hysteresis loop in Figure 5.6 shows almost straight ascending and descending lines in the middle ranges of voltages, but it still has curved slopes on the peak values of the voltages. Furthermore, there are more than one value of output displacements that correspond to the same input voltage, which proves the existence of the hysteresis nonlinear relationship between the input voltage and the output displacement. Thus, a hybrid feedforward and feedback controllers are needed to control the output displacement, and achieve more accurate results.
60
5.3. Hybrid Feedforward and Feedback Controllers Design and Simulation
As discussed in Chapter 3, the system is controlled using an integration of a Luenberger observer-based feedforward controller and a PSO-based PID feedback controller. The following two sections highlight the design steps and the simulation results achieved by using the designed controllers. Furthermore, the results of the open-loop system are compared with the results of the feedforward-controlled system and the hybrid feedforward and feedback controlled system.
5.3.1. Hybrid Feedforward and PSO-Based PID Feedback Controllers Design
In order to design a PID controller for this system, the proportional gain ( the integral gain ( ) and the derivative gain (
),
) are designed according to Table
2.1 so they eliminate the overshoot and minimize the rise time, settling time and steady-state error. By analysing the response of the system and studying Table 2.1, it is found that all the gains of the PID controller are needed. Thus, the structure of the PID controller is as illustrated in Figure 5.7.
Figure 5.7.
The general structure of the PID controller.
The formula that describes the PID controller can be written as
61
( )
( )
∫ ( )
( )
( ) is the controller output signal,
Where
)4.24(
is the error,
is the time,
while is a variable of integration that takes value from zero to the present .
Due to the high nonlinearity of piezoelectric actuators, using ordinary methods such as Ziegler-Nichols method and Cohen-Coon is neither applicable nor efficient to tune the parameters of PID controllers in high precision. Thus, designing the PSO-based PID controller is done simultaneously while it is connected to piezoactuated system and the feedforward controller. This method improves the accuracy of the results, since it takes the performance of the feedforward controller in account while finding the best gains values of the PID controller.
PSO tries to find the optimal solution for PID gains using a population of particles. Each position in the search space is a possible solution of the problem. Particles collaborate to find the best position (best solution) in the search space (solution space). Each particle moves according to its velocity. At each iteration, the particle movement is computed as in Equations 5.14 to 5.16 [95, 96].
( (
)
) ( ) (
( ) ( ( )
( ) ( )
)4.21( ( ))
)4.24(
( ))
)4.24(
where
,
is the number of particles in a group,
,
is the number of members in a particle (dimension of the problem), is the pointer of iteration,
is the current position of a particle at iteration ,
is the velocity of
a particle
at iteration
is the cognitive
,
is the inertia weight factor,
62
acceleration factor,
is the social acceleration factor,
between zero and one,
is the best position of the
best particle among all particles in the population, inertia weight,
and
are random numbers particle,
is the maximum value of the
is minimum value of the inertia weight,
maximum number of iterations, and
is the
is the
is the current number of iteration.
By applying the PSO equations on the PID tuning problem, the position of the particles represents the gains of the PID controller, which means that [
and
]. Several steps should be done in order to compute the solution of
each problem. The first step is assigning the initial values of some variables in Equations 5.14 to 5.16. These variables and their chosen initial values are stated in Table 5.1.
Table 5.1. Variable
The initial values of the PSO algorithm. Value
Variable
Value
( )
Random
3
( )
Random
4
,
Random
20
1.4
0.4
1.4
0.9
Selecting the initial values of
( ) depends on the previous experience
about the best approximate values of the gains of the PID controller that give the optimal response of the system. On the other hand, choosing the initial values of ( ) depends on expecting how close are the optimal values from the initial assumed values of the PID gains. In this case, choosing high random initial values of the velocity leads to explore further spaces of the solutions, and vice versa. Choosing the initial values of
and
depends on the complexity of the problem, where
choosing high numbers of these two variables leads to a more accurate solution in
63
most cases, unless the solution is trapped in a local optima, or leading to a heavy computational time. Choosing the values of
and
depends on the range of
solution, desired convergence velocity, and on previous experience about the solution [97, 98].
The solution of the PSO is based on the performance index, which is a quantitative measure to measure the system performance of the designed PID controller. The PSO tries to get the best minimized fitness of the solution of performance criterion. The most well-known performance index formulas are Integral Squared Error ( Error (
), Integral Time Squared Error (
), Integral Time Absolute Error (
), Integral Absolute
), and Absolute Error (
). The
formulas of these performance indexes are described in Equations 5.17 to 5.21 [99].
∫ ∫
( ) ( )
∫ | ( )| ∫
| ( )| | ( )|
)4.24( )4.24( )4.24( )4.14( )4.12(
The properties and advantages of each performance index are shown in Table 5.2 [99].
Table 5.2.
Performance index formulas and their properties.
64
Performance Index
Properties
Penalizes large control errors. Settling time longer than ITSE. Suitable for highly damped systems.
Penalizes long settling time and large control errors. Suitable for highly damped systems.
Penalizes control errors.
Penalizes long settling time and control errors.
Penalizes all errors equally regardless of direction.
To reduce the overshoot of the system, and to achieve better steady-state error, a performance index formula of the summation of
and
is proposed as
in Equation 5.22.
| ( )|
In this research,
∫
and
( )
)4.11(
performance index
formulas are used to tune the gains of the PID feedback controller, and then the results are compared in Section 5.3.2. Thus, the feedforward controller and the PSObased PID feedback controller are combined together to form the hybrid controllers loop with the piezo-actuated system. Then, the whole system is developed in MATLAB/Simulink, as shown in Figure 5.8.
65
Figure 5.8.
The system with the hybrid controllers.
After building the system, a suitable PSO algorithm is written as MATLAB code, and then simulated synchronously with system built MATLAB/Simulink file to get the best gain values of the PID controller’s gains. The MATLAB code used in the optimization is shown in Appendix C.
The steps that are used to tune the PID controller are illustrated in a flow chart as in Figure 5.9 [100].
66
Figure 5.9.
The flow chart of the process of tuning the PID’s gains.
The results of applying the MATLAB code and simulating the hybrid controllers system are shown and discussed in Section 5.3.2.
67
5.3.2. Simulation Results of the Designed Hybrid Controllers
This section presents and discusses the results accomplished by using the hybrid controllers to control the piezo-actuated system, and then compare all the results of the open-loop system, feedforward-controlled system, and the hybrid controllers system.
By using
and
performance index formulas
that were selected in Section 5.3.1, the PSO-based feedback controller is tuned while it is integrated with the Luenberger observer-based feedforward controller, and using a step input of an 80 V. Table 5.3 presents the PID gains obtained using PSO for various fitness functions.
Table 5.3.
Values of PID gains obtained by various fitness functions.
IAE
80.0408
79.1211
0.0251
ITAE
80.0642
72.0295
0.0245
ISE
92.3313
82.2100
0.0148
ITSE
113.3382
80.2728
0.0151
ITSE+AE
62.6100
80.0102
0.0301
The output displacements achieved using these performance index formulas are illustrated in Figure 5.10.
68
Figure 5.10. The response of the system to an 80 V step input when using and
performance index formulas.
The results shown in Figure 5.10 prove that PSO can be utilized to obtain optima PID gains to control the displacement of the piezo-actuated system. It is shown that using
performance index formula to tune the gains of the PSO-
based PID controller gives the fastest response time, but with the highest overshoot, while using
gives slower response time and lower overshoot. Using the proposed performance index formula gives the slowest response time, but without
overshoot, while using
and
performance index formulas give medium
response time and medium overshoot.
To understand the results achieved using the previous performance index formulas, the results are summarized as in Table 5.4 based on rise time ( ), overshoot ( (
).
), settling time ( ), steady-state error (
), and simulation time
69
Table 5.4.
The simulation results using different fitness functions. (ms)
(%)
(ms)
(nm)
(s)
IAE
0.92
2.5
2
1.2
154.7
ITAE
0.9
2.6
2
1.8
155.9
ISE
0.6
9.6
1.8
1.1
153.2
ITSE
0.5
13.7
2
1.9
155.3
ITSE+AE
1
0
1.6
0
157.5
To further examine the performance of each performance index formula, the results in Table 5.4 are presented in a statistic chart to show the difference of the results achieved in each case, as shown in Figure 5.11.
Figure 5.11. A Statistics chart of the results of the fitness functions used.
Figure 5.11 shows the advantage of the fast rise time when using on the other hand it has a disadvantage of a high overshoot. Using overshoot and a bit higher rise time than
.
and
, but gives less
give almost similar
results, and they are considered to give medium quality results in terms of control
70
aspects. A good performance can be achieved when using the proposed fitness function of
, where it is clearly has zero overshoot, zero steady-state error
and the lowest settling time of 1.6 ms. However, the response shows the highest rise time of 1 ms, and the highest simulation time of 2.625 minutes (157.5 seconds). The disadvantage of the simulation time may be ignored since the difference between all the results is considerably small, and it can be minimized by using faster computers.
Thus,
performance index formula is used to tune the gains of the
PSO-based PID feedback controller while it is integrated with the Luenbergerobserver-based feedforward controller. The values of the gains of the PSO-based PID feedback controller are deduced as [
] = [62.6, 80, 0.03]. By applying the
hybrid controllers on the piezo-actuated system, the output displacement response of the system compared with the reference signal is as shown in Figure 5.12.
Figure 5.12. The output displacement response compared with the reference signal of the hybrid-controlled system.
Figure 5.12 demonstrates that the output displacement is almost exactly similar to the reference signal, where there is a considerably small value of error
71
between them. The error between the reference signal and the output displacement is illustrated in Figure 5.13.
Figure 5.13. The error between the reference and the output displacement of the hybrid-controlled system.
By comparing the error of the hybrid controllers system in Figure 5.13 with the error of the feedforward-controlled system in Figure 5.5, it is can be seen that the amount of error of the feedforward-controlled system has reduced from [-3.25 and 3.25
] to [-0.079
and 0.071
] using the hybrid controllers. This
indicates a reduction of the maximum error from 10.402% in open-loop system, to 4.507% using Luenberger observer-based feedforward controller, to 0.109% using the designed hybrid Luenberger observer-based feedforward controller and PSObased PID feedback controller.
The hysteresis relationship between the input voltage and the output displacement is demonstrated in Figure 5.14. It is shown that as the error reduces, the relationship between the input voltage and the output displacement has become linear. This proves the effectiveness of the designed controllers, which makes the system suitable to be used in high-precision positioning systems.
72
Figure 5.14. The hysteresis relationship of the hybrid-controlled system.
5.4. Summary
This chapter discussed the steps of designing the Luenberger observer-based feedforward controller and the results achieved when applying it on the piezoactuated system. Then, the steps of designing the hybrid Luenberger observer-based feedforward controller with the PSO-based PID feedback controller are discussed, then the results accomplished by using the hybrid controllers are discussed and compared with the results accomplished by using the Luenberger observer-based feedforward controller. The results achieved when using the hybrid controllers show high displacement accuracy and low value of error with a linear relationship between the input voltage and the output displacement, which proves the effectiveness of the designed controllers.
CHAPTER 6
CONCLUSION AND RECOMMENDATIONS
6.1. Conclusion
Bouc-Wen hysteresis model and its evolution were studied, analyzed and then applied to represent the hysteresis nonlinear behavior of piezoelectric actuators. The equivalent dynamic model of the moving stage driven by a piezoelectric stack actuator was derived based on Bouc-Wen hysteresis model, and then simulated using MATLAB/Simulink. Furthermore, the results were verified by comparing with previous work.
A Luenberger observer was designed to estimate the hysteresis of the system, and subsequently combined with the voltage input to form a Luenberger observerbased feedforward controller. The results of using the feedforward controller showed improvement compared to the open-loop system, but they were not accurate enough for high precision positioning systems. Thus, a PSO-based PID feedback controller was integrated with the Luenberger observer-based feedforward controller to control the displacement of the piezo-actuated system, where several performance index formulas were utilized to test the response of the system. Finally, a proposed performance index formula of
was used to tune the PSO-based PID
feedback controller, since it has zero overshoot, zero steady-state error and the lowest settling time of 1.6 ms. The accomplished simulated results show high displacement accuracy and low value of error with a linear relationship between the
74
input voltage and the output displacement. This demonstrates the effectiveness of the designed controllers and the ability of using this piezo-actuated system in highprecision positioning systems.
6.2. Recommendations
Several research works can be conducted in this area. These works can be listed as follows:
1. Further verifications and improvements of the modeled system can be achieved in the future by building and testing a physical system.
2. It is recommended to model the system using other hysteresis models, such as Preisach, Duhem and IEEE models, and then compare all the results to know which model is closer to the physical system. Furthermore, studying the creep and vibration models, and then applying them on the system can give more accurate results.
3. Better response of the controlled system might be achieved by tuning the PID controller using other metaheuristic methods, such as Genetic Algorithms. After that, the results can be compared to decide which method is the most suitable to one for this system.
4. The control part might be extended by designing robust or adaptive controllers for the system while using external load. This will give the ability of designing systems for more complicated applications.
75
5. It is recommended to study smart structures of piezoelectric stack actuators, such as bi-directional stack actuators, or using other types of piezoelectric actuators, such as benders, and then use those actuators to drive the system.
76
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83
APPENDIX A
INITIALIZING THE PARAMETERS OF THE DIFFERENTIAL EQUATIONS AND STATE-SPACE REPRESENTATIONS SYSTEMS – MATLAB CODE
clear;clc; m=2.17 b=4378.67 k=3*10^5 d=9.013*10^-7 a=0.38 beta=0.0335 gamma=0.0295 A=[0 B=[0 C=[1 D=0; N=[0 I=[0
1 0;-(k/m) -(b/m) -(k/m);0 0 0] 0;k*d/m 0;0 a*d] 0 0] 0 0;0 0 0;0 0 -beta] 0 0;0 0 0;0 0 -gamma]
84
APPENDIX B
OBSERVABILITY AND CONTROLLABILITY TEST AND LUENBERGER OBSERVER DESIGN
clear;clc; m=2.17 b=4378.67 k=3*10^5 d=9.013*10^-7 a=0.38 beta=0.0335 gamma=0.0295 A=[0 1 0;-(k/m) -(b/m) -(k/m);0 0 0] B=[0 0;k*d/m 0;0 a*d] C=[1 0 0] D=0; N=[0 0 0;0 0 0;0 0 -beta] I=[0 0 0;0 0 0;0 0 -gamma] R_Obs=rank(obsv(A,C)) R_Con=rank(ctrb(A,B)) R_obs2=rank([C;C*A;C*A^2]) R_Con2=rank([B A*B A^2*B]) poles = [-10 -11 -12] L=place(A',C',poles)' eig(A-L*C)
85
APPENDIX C
PSO PROGRAM – MATLAB CODE
clear;clc tic % -----------------------------------------------------------------% System parameters: m=2.17; b=4378.67; k=3*10^5; d=9.013*10^-7; a=0.38; beta=0.0335; y=0.0295; A=[0 B=[0 C=[1 D=0; N=[0 I=[0
1 0;-(k/m) -(b/m) -(k/m);0 0 0]; 0;k*d/m 0;0 a*d]; 0 0]; 0 0;0 0 0;0 0 -beta]; 0 0;0 0 0;0 0 -y];
poles = [-10 -11 -12]; L=place(A',C',poles)'; % -----------------------------------------------------------------% PSO: n=4; max=20; dim = 3; c1=1.4; c2=1.4; wmin=0.4; wmax =0.9; R1 = rand(dim,n); R2 = rand(dim,n); c_fit =zeros(n,1); x(1:2,:) = abs(rand(2,n)); x(3,:) = abs(.01*rand(1,n));
86 v(1:2,:) = abs(rand(2,n)); v(3,:) = abs(.01*rand(1,n)); l_best_x = x; for i=1:max w(i)=wmax-((wmax-wmin)/max)*i; end for i = 1:n Kp = x(1,i); Ki = x(2,i); Kd = x(3,i); simopt = simset('solver','ode3','SrcWorkspace','Current','DstWorkspace','Curr ent'); [tout,xout,yout] = sim('PSO_System',[0 1],simopt); c_fit(i) = err; end l_best_fit = c_fit; [g_best_fit,g] = min(l_best_fit); for i=1:n g_best_x(:,i)=l_best_x(:,g); end v = w(1) *v + c1*(R1.*(l_best_x-x)) + c2*(R2.*(g_best_x-x)); x = x + v; for iter=2:max for i=1:n Kp = x(1,i); Ki = x(2,i); Kd = x(3,i); simopt = simset('solver','ode3','SrcWorkspace','Current','DstWorkspace','Curr ent'); [tout,xout,yout] = sim('PSO_System',[0 1],simopt); c_fit(i) = err; end for i=1:n if c_fit(i) < l_best_fit(i) l_best_fit(i)=c_fit(i); l_best_x(:,i)=x(:,i); end end [c_g_best_fit,g]=min(l_best_fit); if c_g_best_fit < g_best_fit g_best_fit=c_g_best_fit; for i=1:n g_best_x(:,i)=l_best_x(:,g); end end
87
v = w(iter) *v + c1*(R1.*(l_best_x-x)) + c2*(R2.*(g_best_x-x)); x = x + v; end Kp=g_best_x(1,1) Ki=g_best_x(2,1) Kd=g_best_x(3,1) toc time=toc;
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