design of active vibration controller of a smart beam using lqr , pid and fuzzy logic techniques
DESCRIPTION
Theoretical model of An aluminum cantilever beam bonded with a pair ofpiezoelectric patches (sensor and actuator) used as a smart beam is derived. Thefirst four fundamental natural frequencies and mode shapes of this system aredetermined and verified using ANSYS. The optimal best location of thepiezoelectric patches (actuators) is determined to give best possible activevibration control performance. Then three different controllers, Linear QuadraticRegulator (LQR), Proportional-Integral- Derivative (PID) and Fuzzy Logic (FL)are implemented using MATLAB in order to achieve the required reduction inthe output velocity and displacement of the smart beam. A comparison betweensystem responses using these three controllers is carried out in order to realizewhich controller will give the desired performance, in terms of robust design,faster response, higher reliability and stability. Simulation results showed thatfuzzy logic controller gave the best results where the reduction of the outputamplitude of the cantilever beam displacement is found to be (90.5%) while inPID and LQR controllers where ( 90.4% ) and (49.65% ) respectively. Thesettling time of the system is found to be (0.3 sec) when using FL controllerwhile it were (0.9 sec ) and (0.209 sec) when using PID and LQR controllerrespectively . A comparison with the results of other studies showed that thecontrollers which designed in this study gave better performance for the samecase studies.TRANSCRIPT
DESIGN OF ACTIVE VIBRATION CONTROLLER
OF A SMART BEAM USING LQR , PID AND
FUZZY LOGIC TECHNIQUES
A THESIS
SUBMITED TO THE COLLEGE OF ENGINEERING
UNIVERSITY OF BASRAH
IN PARTIAL FULFILMENT OF
THE REQUIRMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
IN
MECHANICAL ENGINEERING
(Applied Mechanics)
By
Tahseen Hashim Saleh
B. SC .Mechanical Engineering
2014
ونريد أن نمن على الذين استضعفوا في الرض ونجعلهم أئمة ونجعلهم الوارثين ﴿5﴾
صدق اهلل العلي العظيم
سورة القصص
DEDICATION
DEDICATION
I would like to dedicate this thesis for all the
honest people whom work in shadow for my
beloved country Iraq
to my family
father and mother
to my wife
to my brothers and sisters
to my friends
ACKNOWLEDGMENT
I
ACKNOWLEDGMENT
I would like to give my deeply gratitude and appreciation to
my supervisors Dr. Qusai Talib Abdulwahab and Dr. Jaafar
Khalaf Ali for their assistance, support and guidance in order
to complete this thesis. thanks also owing to Prof . Rabee` H.
Thejel the dean of the college, and for Prof. Dr. Ameen A.
Nassar, the Head of Mechanical Engineering Department, for
their appreciated support and assistance in order to complete
my study. The author also wants to thank all the staff of the
Mechanical Engineering Department for their facilitating ,
supporting and encouragement in order to complete this thesis.
ABSTRACT
III
ABSTRACT
Theoretical model of An aluminum cantilever beam bonded with a pair of
piezoelectric patches (sensor and actuator) used as a smart beam is derived. The
first four fundamental natural frequencies and mode shapes of this system are
determined and verified using ANSYS. The optimal best location of the
piezoelectric patches (actuators) is determined to give best possible active
vibration control performance. Then three different controllers, Linear Quadratic
Regulator (LQR), Proportional-Integral- Derivative (PID) and Fuzzy Logic (FL)
are implemented using MATLAB in order to achieve the required reduction in
the output velocity and displacement of the smart beam. A comparison between
system responses using these three controllers is carried out in order to realize
which controller will give the desired performance, in terms of robust design,
faster response, higher reliability and stability. Simulation results showed that
fuzzy logic controller gave the best results where the reduction of the output
amplitude of the cantilever beam displacement is found to be (90.5%) while in
PID and LQR controllers where ( 90.4% ) and (49.65% ) respectively. The
settling time of the system is found to be (0.3 sec) when using FL controller
while it were (0.9 sec ) and (0.209 sec) when using PID and LQR controller
respectively . A comparison with the results of other studies showed that the
controllers which designed in this study gave better performance for the same
case studies.
CONTENTS
IV
CONTENTS
Title PAGE
Acknowledgment I
Certification II
Abstract III
Contents IV
Abbreviations VII
Symbols VIII
List Of Figures X
List Of Tables XIII
Chapter One : Introduction 1
1.1Background 1
1.2 Smart Structures 2
1.3 Applications Of Active Vibration Control 3
1.4 Aim Of This Thesis 7
1.5 Layout Of This Thesis 8
Chapter Two : Literature Review 9
2.1 Introduction 9
2.2 Summary Of Literature Review 17
Chapter Three : Mathematical Modeling And Active Control Techniques 18
3.1 Introduction 18
3.2 Mathematical Model Of The Cantilever Beam 19
3.3 State Space Representation Of The Cantilever Beam 25
3.4 Active Vibration Control 27
CONTENTS
V
3.5 Linear Quadratic Regulator (LQR) State Feedback Design 28
3.6 Proportional – Integral –Derivative Controller (PID) 33
3.6.1 PID Controller Theory 34
3.6.2 Loop Tuning 35
3.6.3 Stability 36
3.6.4 Manual Tuning 36
3.6.5 PID Controller Auto Tuning 37
3.7 Fuzzy Logic Controller(FLC) 38
3.7.1 Fuzzification 39
3.7.2 The Rule Of Fuzzy Control And The FLC 40
3.7.3 Defuzzification 42
Chapter Four : Computer Simulation Using MATLAB 43
4.1 Introduction 43
4.2 Mode Shapes And Natural Frequencies Of The Cantilever Beam 46
4.3 The Best Location Of The Piezoelectric Patches 48
4.4 Active Vibration Control Using (LQR) Controller 48
4.4.1 Response Of Cantilever Beam To Initial Condition 50
4.4.2 Impulse Response Of Cantilever Beam 55
4.5 Active Vibration Control Using (PID) Controller 59
4.5.1 Manually Determination Of (PID) Controller Constants 61
4.5.2 Determination Of (PID) Controller Constants Using Auto Tuning 66
4.6 Active Vibration Control Using (FL) Controller 71
Chapter Five :Results And Discussion 81
5.1 Introduction 81
5.2 Determination Of The Best Location Of The Piezoelectric Patches 81
5.3 Results And Discussion 83
CONTENTS
VI
5.4 Comparison Between The Current Research And Other Researches 86
Chapter Six : Conclusion And Recommendations For Future Works 92
6.1 Conclusion Of This Thesis 92
6.2 Recommendations For Future Works 94
References 95
Appendix 100
ABBREVIATIONS
VII
ABBREVIATIONS
Term Description DISO Double-input, single-out
FE Finite element
FL CONTROLLER Fuzzy logic controller
LQG CONTROLLER Linear-quadratic-Gaussian controller
LQR CONTROLLER Linear-quadratic-regulator controller
MIMO Multi input multi output
MOM Middle of Maximum
MV manipulated variable
NB Negative big
NM Negative medium
NS Negative small
PB Positive big
PDVF Polyvinylidene fluoride
PI Performance index
PID CONTROLLER Proportional -integral-derivative controller
PM Positive medium
POF Proportional output feedback
PS Positive small
PZT Lead zirconate titanate patches
RTOS real time operation system
rlocus root locus
SFD squeeze film damper
SISO Single input single output
SVFB State-variable feedback
ZO Zero
SYMBOLS
VIII
SYMBOLS
Term Description Si Units
A Cross sectional area of the beam m2
A State matrix -
Ac The closed-loop plant matrix -
B Input matrix -
baj Width of the j-th actuator m
bi Constant Vm3/N
bsi The width of the i- th sensor patch m
Cpi Piezoelectric capacitance of i-th sensor F
D
The matrix of the influence of actuators
on varied modal of structure
-
d31j Constant of the j-th actuator m/V
E Output matrix -
Eaj Young’s modulus of the j-th actuator N/m2
Eb Young’s modulus of smart beam N/m2
e 31i The piezoelectric stress constants of the i-
th sensor patch Vm/N
e(t) Error -
F feedforward matrix -
g(i) The electric charge measured at the i -th
sensor at instant (t) coulomb
g31i Piezoelectric constant of i-th sensor Vm/N
H(x) Heaviside function -
haj Thickness of the j-th actuator m
hb Thickness of smart beam m
hsi Thickness of sensor patch m
J Cross sectional area moments of inertia m4
K Gain of LQR controller -
Kd Derivative gain of PID controller -
Ki Integral gain of PID controller -
SYMBOLS
IX
Kj Constant N.m
Kp Proportional gain of PID controller -
Ksi Constant Vm3/N.F
L Length of smart beam m
M
The influence matrix of the sensor’s
output -
m The number of the actuator -
N The number of modes used -
Q Applied force from the actuator N
Q Matrix of LQR controller -
qk(t) The generalized modal coordinate -
R Matrix of LQR controller -
r The total number of sensors. -
rsi The average coordinate of the sensor
measured from the mid-plane of the beam m
t Time sec
u The electrical input vector V
V(t) The vector of the voltage subjected to
(m) actuator V
Vj The control voltage across piezoelectric
actuators V
W(x ,t) Transverse displacement of the beam m
Xj ,Xj+1 The locations of the ends of the j -th
piezoelectric patch along the x-axis. m
y The output vector m
ζk Damping ratio of the i -th mode -
ρ Density of the beam material kg/m3
ρA Mass per unit length of smart beam kg/m
ϕk(x) The mode shape function -
ωk The natural frequency of the i - th mode rad/sec
LIST OF FIGURES
X
LIST OF FIGURES
Figure No. Figure title Page
1.1 Active vibration control of vehicle suspension system 4
1.2 Squeeze film damper 5
1.3 Active bearing with piezo actuators 6
1.4 Active vibration control in milling machine 7
3.1 Smart cantilever beam with piezoelectric patches 20
3.2 LQR block diagram controller 33
3.3 Block diagram of PID controller 34
3.4 The basic configuration of fuzzy logic system 40
4.1 Beam dimension 43
4.2 The first four mode shapes of the cantilever beam 47
4.3 Flow chart of LQR controller 49
4.4 Response of the cantilever beam to initial condition when
piezoelectric patches near fixed end with LQR control
50
4.5 Bode plot of the cantilever beam subjected to initial
condition when piezoelectric patches near fixed end with
(LQR)controller
51
4.6 Rlocus plot of the controlled and uncontrolled cantilever
beam subjected to initial condition when piezoelectric
patches near fixed end with LQR controller
52
4.7 Response of the cantilever beam with initial condition when
piezoelectric patches at free end with LQR controller
53
4.8 Bode plot of the cantilever beam subjected to initial
condition when piezoelectric patches at free end with LQR
controller
54
4.9 Rlocus plot of the controlled and uncontrolled cantilever
beam subjected to initial condition when piezoelectric
patches at free end with LQR controller
54
4.10 Impulse response of the cantilever beam when piezoelectric
patches near fixed end with LQR controller
55
4.11 Bode plot of the cantilever beam subjected impulse force
when piezoelectric patches near fixed end with LQR
controller
56
4.12 Rlocus plot of the controlled and uncontrolled cantilever
beam subjected to impulse force when piezoelectric patches
57
LIST OF FIGURES
XI
near fixed end with LQR controller
4.13 Impulse response of the cantilever beam when piezoelectric
patches at free end with LQR controller
57
4.14 Bode plot of the cantilever beam subjected to impulse force
when piezoelectric patches at free end with LQR controller
58
4.15 Rlocus plot of the controlled and uncontrolled cantilever
beam subjected to impulse force when piezoelectric patches
at free end with LQR controller
59
4.16 Flow chart of PID controller 60
4.17 Impulse response of cantilever beam when the patches near
fixed end with PID controller
61
4.18 Bode plot of the cantilever beam subjected to impulse force
when piezoelectric patches near fixed end using PID
controller
63
4.19 Rlocus plot of the controlled and uncontrolled cantilever
beam subjected to impulse force when piezoelectric patches
near fixed end using PID controller
63
4.20 Impulse response of cantilever beam when the patches at
free end with PID controller
64
4.21 Bode plot of the cantilever beam subjected impulse force
when piezoelectric patches at free end using PID controller
65
4.22 Rlocus plot of the controlled and uncontrolled cantilever
beam subjected to impulse force when piezoelectric patches
at free end using PID controller
65
4.23 MATLAB (PID )tuner 66
4.24 Impulse response of cantilever beam when the patches near
fixed end with (PID parameter tuning with MATLAB )
controller
67
4.25 Bode plot of the cantilever beam subjected impulse force
when piezoelectric patches near fixed end using PID
controller MATLAB tuning
68
4.26 Rlocus plot of the controlled and uncontrolled cantilever
beam subjected to impulse force when piezoelectric patches
near fixed end using PID controller MATLAB tuning
68
4.27 Impulse response of cantilever beam when the patches at
free end with PID controller MATLAB tuning
69
4.28 Bode plot of the cantilever beam subjected impulse force
when piezoelectric patches at free end using PID controller
MATLAB tuning
70
LIST OF FIGURES
XII
4.29 Rlocus plot of the controlled and uncontrolled cantilever
beam subjected to impulse force when piezoelectric patches
at free end using PID controller MATLAB tuning
71
4.30 Fuzzy logic controller page in MATLAB 75
4.31 The input displacement member function of cantilever beam 75
4.32 The input velocity member function of cantilever beam 76
4.33 The output controlled displacement member ship of
cantilever beam
76
4.34 Rule viewer of the designed FLC 77
4.35 Surface viewer of the designed FLC 77
4.36 Inputs displacement and velocity of sine wave 78
4.37 Output controlled displacement when the input is sine wave 78
4.38 Inputs displacement and velocity of unit step 79
4.39 Output controlled displacement when the inputs are unit
steps
79
4.40 The Simulink page of FLC in MATLAB 80
5.1 The best location of piezoelectric patches on the cantilever
beam
83
5.2 Response of cantilever beam –different type of controller at
best Location
84
5.3 Response of cantilever beam –different type of controller at
best Location(zoomed)
85
5.4 Response of the cantilever beam to initial condition with
LQR controller at best location of the patches.
87
5.5 Response of the cantilever beam to initial condition with
LQR controller at best location of the patches [8]
87
5.6 Response of the cantilever beam to impulse force with PID
controller at best location of the patches
88
5.7 Response of the cantilever beam to impulse force with PID
controller at best location of the patches [9]
89
5.8 Response of the cantilever beam to unit step force with FL
controller at best location of the patches
90
5.9 Response of the cantilever beam to unit step force with FL
controller at best location of the patches [14]
90
LIST OF TABLES
XIII
LIST OF TABLES
Table No. Table Title Page
3.1 Ziegler–Nichols Tuning Method 37
3.2 Fuzzy logic controller rule base 41
4.1 Properties of cantilever beam 44
4.2 Properties of piezoelectric patches (sensor and actuator) 45
4.3 Natural frequencies of the cantilever beam 46
4.4 Properties of controlled and uncontrolled cantilever beam
subjected to initial conditions when the patches near fixed
end with LQR controller
51
4.5 Properties of controlled and uncontrolled cantilever beam
subjected to initial conditions when the patches at free
end with LQR controller
53
4.6 Properties of controlled and uncontrolled cantilever beam
subjected to impulse force when the patches near fixed
end with LQR controller
56
4.7 Properties of controlled and uncontrolled cantilever beam
subjected to impulse force when the patches at free end
with LQR controller
58
4.8 Properties of impulse response of cantilever beam with
PID controller when the patches near fixed end
62
4.9 Properties of impulse response of cantilever beam with
(PID)controller when the patches at free end
64
4.10 Properties of impulse response of cantilever beam with
(PID)controller when the patches near fixed end matlab
tuning
67
4.11 Properties of impulse response of cantilever beam with
(PID)controller when the patches at free end matlab
tuning
70
4.12 FLC expanded rules 72
5.1 Effect of piezoelectric patches location on the beam
response
82
Chapter
one
CHAPTER ONE INTRODUCTION
1
CHAPTER ONE
INTRODUCTION
1.1 Background
In diverse areas like space and aircraft structures, satellites, cars, bridges etc,
undesired vibrations are a major cause of problems. The effects of such
vibrations are varied. Minor effects may include annoyance due to noise in
automobiles, machines etc. Major effects are felt in areas like space structures
where precise behavior of the structure is desired and any deviation from the
required behavior may result in major expense. Under such conditions ,vibration
control becomes very important. Vibration control is an upcoming and
challenging branch of mechanical engineering. It has wide ranging applications
in diverse fields. A lot of research is being conducted in this area and new
methods of control are being proposed and applied to the practical systems. In
particular, active vibration control using smart materials is attracting much
interest around the world. Active vibration control is the process of using smart
materials for controlling vibrations in real time. Smart materials are materials
that respond with significant change in a property upon application of an external
driving force. Such materials can act as sensors, which sense the disturbances in
the structures, and actuators, which are capable of applying the controlling force.
Examples of smart materials include piezoelectric materials, shape memory
alloys, etc. Among various smart materials, piezoceramics materials have the
advantages of being lightweight, low-cost, and easy-to-implement and offer the
CHAPTER ONE INTRODUCTION
2
sensing and actuation capabilities that can be utilized for passive and active
vibration control [1].
1.2 Smart Structures
Smart structures consist of highly distributed active device which comprises
sensors and actuators either embedded or attached with an existing passive
structure coupled by controller. The piezoelectric sensor senses the disturbance
and generates an electric charge due to the direct piezoelectric effects. The
piezoelectric actuator in turn produces a control force/moment due to the
converse piezoelectric effects. If the control force is appropriate, the structural
vibration may be suppressed. This technology has several applications such as
active vibration and buckling control, shape control and active noise control. The
finite element method is powerful tool for designing and analyzing smart
structures. Both structural dynamics and control engineering need to be dealt to
demonstrate smart structures, In most of present researches, FEM formulation of
smart cantilever beam usually done in ANSYS and design of control laws are
carried out in MATLAB control system toolbox. Hence, for designing
piezoelectric smart structures with control laws, it is necessary to develop a
general design scheme of actively controlled piezoelectric smart structures. In
order to achieve good, fast and robust controller , the researcher used many
different control method such as proportional-integral-derivative (PID), H∞
norm and Linear Quadratic Regulator (LQR) algorithm with state feedback
control law. Piezoelectric materials could be divided, from structural viewpoint,
into ceramic and polymeric forms. The most popular piezoelectric ceramics (or
in short, piezoceramics) are compounds of lead zirconate titanate (PZT), the
properties of which can be optimized to suit specific applications by appropriate
CHAPTER ONE INTRODUCTION
3
adjustment of the zirconate–titanate ratio. The polymeric form of the
piezoelectric materials as polyvinylidene fluoride (PDVF) having low stiffness
and electromechanical coupling coefficients (when compared to ceramics like
PZT, for instance). Structures with added functionality over and above the
conventional purpose of providing strength by reinforcement or stiffness may be
regarded as smart. Smart or adaptive structures, based on using a small change in
the structure geometry at critical locations induced by internally generated
control signals, can result in a non-linear amplification of the shape, stiffness or
strength, and so the structure will adapt to a functional need. In practice, smart
structures may be classified depending on their functionality and adaptation to
the changing situation :
1. Passive smart
2. Active smart
3. Intelligent.
Where the " passive smart" materials can act only as sensors , the " active smart"
materials can act as sensors and actuators and finally , the " intelligent "
materials have the ability to change their dimensions in more than one
coordinates [1].
1.3 Applications of Active Vibration Control
The development of high strength to weight ratio of mechanical structures
are attracting engineers to build light weight aerospace structures as well as to
build tall buildings and long bridges. The development of piezoelectric materials
has been used as sensors and actuators because if the forces are applied on that
CHAPTER ONE INTRODUCTION
4
material it produces voltage and this voltage goes to active devices and controls
the vibration. Their reliability, nearly linear response with applied voltage and
their low cost make piezoelectric materials the most widely preferred one as
collocated sensor and actuator pair. Active vibration control is the active
application of force in an equal and opposite fashion to the forces imposed by
external vibration . as an example of the application of active vibration control in
Practical life is in car suspension system, the main objective on the active
vibration control problem of vehicles suspension systems is to get security and
comfort for the passengers by reducing to zero the vertical acceleration of the
body of the vehicle. An actuator incorporated to the suspension system applies
the control forces to the vehicle body of the automobile for reducing its vertical
acceleration in active or semi-active way.
Figure (1.1) active vibration control of vehicle suspension system[2].
Another example of the active vibration control is in aircraft engine rotors , in
such engine, aircraft rotors vibrate due to unbalance and flow processes in
turbines and compressors. Without vibration reducing measures, vibrations of
CHAPTER ONE INTRODUCTION
5
the rotating components reach unacceptably high amplitudes. For this reason, so-
called squeeze film dampers (SFDs) are used for vibration reduction in today’s
engines to ensure safe operation and long life. however, SFDs have some
negative characteristics. The frequency range, in which passive elements can
significantly increase system damping, is smaller than the operating frequency
range of the engines. SFDs do not have a static load capacity and non-linear
characteristics. The units for the oil supply also need a lot of space and have high
masses. Research is being done for alternatives to SFDs to achieve the long-term
objective of a “More-Electric Engine” with a view to substitute conventional
with electrical components.
Figure (1.2) Squeeze film damper[2].
Piezoelectric actuators are appropriate for vibration reduction in aircraft engines
because they can provide high forces and have high stiffness while being
relatively compact. In addition, the operating frequency range of these elements
is very large; with appropriate control, it is possible to reduce the vibration
broadband. Another advantage is the ability to isolate the vibrations with the
actuator, for example to increase comfort inside the aircraft. Because of these
advantageous properties, the researchers in Mechanical Engineering are
researching for solutions to reduce rotor vibrations in aircraft engines with
CHAPTER ONE INTRODUCTION
6
piezoelectric actuators. So far, a concept has been designed to influence the
vibrations of the low-pressure shaft of an engine with four piezoelectric
actuators. The shaft of the low-pressure turbine is (due to its length) the most
critical rotating component in an engine with respect to vibration. The subject of
consideration are the unbalance-induced vibrations of the shaft, because they
provide the most significant share of the total vibration of the shaft during
operation. Since the system is operated beyond the first critical speed, the
oscillation amplitudes of the forward whirl are of particular interest.
Figure (1.3) active bearing with piezo actuators [2].
The other application of active vibration control is in milling machine , In which
the main problem is vibration in machine tool which affects on quality of
machined part. Hence these vibrations needed to suppressed during machining
[2].
CHAPTER ONE INTRODUCTION
7
Figure (1.4) active vibration control in milling machine [2].
1.4 Aim of The Thesis
The aim of this study can be summarized into the following points :
1. Drive a mathematical model of a cantilever beam with a pair of
piezoelectric patches (sensor and actuator) considering the whole
structure as a smart beam .
2. Apply different types of vibration control techniques such as (LQR , PID
and FL) on the smart beam in order to achieve the best design to control
its vibration.
3. Comparing the results obtaining from the three control methods for the
same structure to find the best method to be used in this application.
CHAPTER ONE INTRODUCTION
8
1.5 Layout of The Thesis
THESIS
CHAPTER ONE
INTRODUCTION
CHAPTER FOUR
COMPUTER
SIMULATION
CHAPTER TWO
LITERATURE
REVIEW
CHAPTER THREE
MATHEMATICAL
MODELING
CHAPTER FIVE
RESULTS AND
DISCUSSIONS
CHAPTER SIX
RECOMMENDATION
FOR FUTURE
WORKS
Chapter
two
CHAPTER TWO LITERATURE REVIEW
9
CHAPTER TWO
LITERATURE REVIEW
2.1 Introduction
An increasing interest in the possibilities of active control of structures has
given rise to new achievements in this field of research in many branches of
engineering over the past few years. In comparison with passive structures, smart
structures (or active structures, or structronic systems as they are referred to in
different literature) offer a great variety of possibilities for the structural behavior
control under changing environment conditions in the sense of adjusting or
adapting the structure parameters and behavior to new conditions. From this
point of view the term adaptive structure is also used to denote the possibility of
altering the structural response in the presence of disturbances or changed
working conditions. The ability of the structure to change its response in
accordance with the changed environment conditions comes from the presence
of active materials integrated with the structure. Such active materials (acting as
sensor and/or actuators) in connection with the control system enable automatic
adaptation of the structure to changing environment conditions. An important
role among active materials belongs to piezoelectric materials (such as thin
wafers, fibers or piezoelectric rods) used as actuators and sensors integrated in a
structure providing thus the adaptability of the smart structure, while not
affecting significantly its passive behavior.
CHAPTER TWO LITERATURE REVIEW
10
Vibration control of cantilever beam based on the piezoelectric smart component
has become the hot issue in the research of vibration engineering , a different
studies which concerning the active vibration control of smart piezoelectric
beams is presented by many researchers as some of their researches are
mentioned below:
Yavuz Yaman , et. al. , 2001 , [3] , investigated the effects of element
selection in finite element modeling and The effects of the piezoelectric
patches on the resonance frequencies of the smart structure the results of their
work were the design of ( H∞) controller was which effectively suppresses the
vibrations of the smart beam due to its first two modes. The suitability of the
(H∞) design technique in the modeling of uncertainties and in evaluating the
robust performance of the system was demonstrated . the results of their research
shows the reduction in vibration amplitude of controlled beam up to 52% than
the amplitude of the uncontrolled beam .
Ulrich Gabbert , et. al. , 2002 , [4] , introduced a development in modelling
and numerical analysis of piezoelectric material systems and controlled smart
structures based on a general purpose finite element software with the
possibilities of static and dynamic analyses and simulation. Design and
simulation of controlled smart structure is also presented, using a state-space
model of a structure obtained through the finite element analysis as a starting
point for the controller design. For the purpose of the control design for the
vibration suppression discrete-time control design tools were used, such as
optimal LQR controller incorporated in a tracking system .the result was the
CHAPTER TWO LITERATURE REVIEW
11
reduction of vibration amplitude by 66.6 % for all the controlled beam in all the
study cases under concerned.
Gou Xinke and Tian Haimin , 2007 , [1] , used a general method of active
vibration control and suppression for intelligent structures is put forth on the
basis of a negative state feedback control law. Actuator on different positions is
performed in order to investigate their effectiveness to suppress vibration in
intelligent structures, and the optimal position of piezoelectric cantilever beam in
vibration control is pointed out and the Analytical results are verified with
numerical simulations the conclusion of research was that the vibration of a
cantilever beam has been actively suppressed by applying control voltage to the
piezoceramic actuator, and the optimal control theory lead to a useful controller
design methodology for the design of robust controllers for the vibration control
of cantilever beam-like structures. the vibration amplitude was suppressed by
50% when the patches where located near fixed end.
Zhang Jing-jun , et. al. , 2008 , [5] , used the fuzzy logic controller to control
the smart structures vibration. The fuzzy IF-THEN rules are established on
analysis of the motion traits of cantilever beam. The fuzzy logic controller (FLC)
designs on using the displacement and the velocity of the cantilever beams tip as
the inputs, the control force on cantilever beams as the output. This new method
improves calculation efficiency and reduces calculation complexity and have the
better effects than which uses the acceleration and its rate of the cantilever beams
tip as the inputs. The simulation results with MATLAB illustrate that the
proposed method has a better control performance than existing methods , their
simulation results prove the effectiveness of theoretical analysis and achieve
good effects. It also demonstrates that compared with the LQG control method,
CHAPTER TWO LITERATURE REVIEW
12
robust H∞ control has strong robustness to modal parameters variation and has a
good closed-loop dynamic performance .the (FLC) reduced the vibration
amplitude to approximately 80.3% .
Dong Jingshi , et. al. , 2010 , [6] , introduced active vibration control system
in which The dynamic model of the cantilever beam is established by finite
element method (FEM). The piezoelectric actuator excited by control signal is
bonded near the fixed end of the cantilever beam to suppress the beam's first
vibration mode. The control signal is collected by a sensor and processed by the
quadratic optimal control theory. Experiment results show that the amplitude of
the active controlled cantilever beam is reduced to 48.2% than the uncontrolled
one under constant external excitation.
K. B. Waghulde , et. al. , 2010 , [7], optimized the performance metric
corresponding to the mode of interest. This methodology is ideal for the design
of low-order controllers. A smart structure involves distributed actuators and
sensors along the structure and some type of processor that can analyze the
response from the sensor and use control theory to output commands to the
actuator. The actuator applies local stresses/strains to alter the behavior of the
system and they took into consideration that piezoelectric materials must be
bonded to the beam in a uniform fashion along with the fact that both materials
must have electrical contact on each side of the material, the results of their work
was a 30 % reduction in 1st - mode vibration response and there conclusion was
that the focus on the first mode will allow the creation of better controllers
through more accurate models of best fit.
CHAPTER TWO LITERATURE REVIEW
13
S.M. Khot , et. al. , 2011 , [8] , used reduced model for cantilever beam and
the design of optimal controller is achieved using Linear Quadratic Regulator
(LQR) algorithm with state feedback control law. The responses are obtained in
both MATLAB and ANSYS based on the obtained optimal control gains and
compared. Effect of selection of weighting matrices of performance index of
LQR on the performance of optimal controller is also reported. Validity of using
reduced model for designing optimal controller is checked by comparing its
response with that of full model. and they realize that reduced models are used
for designing controllers for active vibration control of real life complicated
systems, a lot of computational time can be saved .
Deepak Chhabra , et. al. , 2011 , [9] , addressed a general design and analysis
scheme of piezoelectric smart structures with control laws. The classical control
law, pole placement technique and LQR optimal control approach using state
feedback and arbitrary value of gain by output feedback has analyzed to achieve
the desired control. Numerical examples are presented to demonstrate the
validity of the proposed design scheme, Their study revealed that the LQR
control scheme is very effective in controlling the vibration as the optimal gain
is obtained by minimizing the cost function. Numerical simulation showed that
modeling a smart structure by including the sensor / actuator mass and stiffness
and by varying its location on the beam from the free end to the fixed end
introduced a considerable change in the system’s structural vibration
characteristics.
S.M. Khot , et. al. , 2011 , [10] , dealt with the extraction of the full and
reduced mathematical models of a cantilever beam into MATLAB from its FE
CHAPTER TWO LITERATURE REVIEW
14
model. The full model of the beam is reduced by discarding those modes which
do not contribute to the overall response. It is found that the frequency and
transient responses of the full and reduced models match closely. Hence the
reduced model may be used to represent the system which in turn reduces the
computational time. The controller is designed using proportional-integral-
derivative(PID) theory with output feedback. SIMULINK is then used to create
a working block diagram of the control system and perform the control action.
The result of their work that The transient responses of the controlled full and
reduced models are then plotted which are found to be in close agreement.
Deepak Chhabra , et. al. , 2012 , [11], Developed a smart structure with
patches located at the different positions to determine the better control effect.
The piezoelectric patches are placed on the free end, middle end and fixed end.
The study is demonstrated through simulation in MATLAB for various
controllers like Proportional Controller by Output Feedback (POF) , Proportional
Integral Derivative controller (PID) and Pole Placement technique. A smart
cantilever beam is modeled with SISO system. The entire structure is modeled
using the concept of piezoelectric theory, Euler-Bernoulli beam theory, Finite
Element Method (FEM) and the State Space techniques. The numerical
simulation shows that the sufficient vibration control can be achieved by the
proposed method, their conclusion was that From the responses of the various
locations of sensor/actuator on beam, it has been observed that best performance
of control is obtained, when the piezoelectric element is placed at fixed end
position.
CHAPTER TWO LITERATURE REVIEW
15
Tamara Nestorović , et. al. , 2012 , [12] , introduced the concept of an active
vibration control for piezoelectric light weight structures and presented through
several subsequent steps: model identification, controller design, simulation,
experimental verification and implementation on a particular object
piezoelectric smart cantilever beam. Special attention is paid to experimental
testing and verification of the results obtained through simulations. The
efficiency of the modeling procedure through the subspace based system
identification along with the efficiency of the designed optimal controller are
proven based on the experimental verification, which results in vibration
suppression to a very high extent not only in comparison with the uncontrolled
case, but also in comparison with previously achieved results. The experimental
work demonstrates a very good agreement between simulations and experimental
results and their result was the design of an optimal LQ feedback strategy is used
for the controller design, which provides the designer with lots of flexibility to
perform trade-offs among various performance criteria. The optimal LQ
controller requires a full knowledge of the state variables ,in order to generate
the control input. Therefore, a Kalman filter is used as an observer, in order to
estimate the unmeasurable state variables.
A.P. Parameswaran , et. al. , 2013 , [13] , introduced the principle of direct
output feedback based active vibration control which has been implemented on
a cantilever beam using Lead Zirconate-Titanate (PZT) sensors and actuators.
Three PZT patches were used, one as the sensor, one as the exciter providing the
forced vibrations and the third acting as the actuator that provides an equal but
opposite phase vibration/force signal to that of sensed so as to damp out the
vibrations. The designed algorithm is implemented on LabVIEW 2010 on
CHAPTER TWO LITERATURE REVIEW
16
Windows 7 Platform , the results showed inconsistent transient as well as steady
state characteristics in the dynamics of the beam. Hence, it was concluded that
experimental control of the vibrating smart beam needed to be performed on a
real time operating system (RTOS) platform wherein deterministic and reliable
control could be achieved.
Preeti Verma , et. al. , 2013 , [14] , presented a design of fuzzy logic
controller for identification of cracks and vibration control of cantilever beam
and an identification of the location and depth of creaks in beam using
measured the vibration data is introduced . Fuzzy controller is applied to
attenuate vibrations in a cantilever beam structure with large varying parameters.
The fuzzy logic controller used here comprises of two input parameters and one
output parameters. Gaussian and triangular, trapezoidal membership functions
are used for the fuzzy controller. The input parameters to the fuzzy- Gaussian
controller and fuzzy- triangular controller are relative deviation of first three
natural frequencies. The output parameters of the fuzzy inference system are
relative crack depth and relative crack location. At the beginning theoretical
analyses have been outlined for cracked cantilever beam to calculate the
vibration parameters such as natural frequencies. A set of boundary conditions
are considered involving the effect of crack location. A series of fuzzy rules are
derived from vibration parameters which are finally used for prediction of crack
location and its intensity. The comparison is made between Gaussian and
triangular membership functions by calculating deviation from expected values
of crack depth and crack location , By the Membership Function, they have been
detecting the crack depth and crack location, here, the fuzzy logic controller is
used for vibration control of cracks through the fuzzy parameters. So the damage
cracks in cantilever beam would be identified.
CHAPTER TWO LITERATURE REVIEW
17
2.2 Summary of Literature Review
The researcher used many different control method such as proportional-integral-
derivative (PID), H∞ norm and Linear Quadratic Regulator (LQR) algorithm
with state feedback control law , from the literature review it can be observe and
conclude the following points of interesting :
1. There was many work done in order to achieve a robust and fast
controllers using different methods.
2. There was no comparison made between the controllers (i.e. PID , LQR
and FL controllers ).
In the current thesis , a design of a controller based on LQR controller , PID
auto-tuning controller and FL controller with (35 rules) will be achieved.
Chapter
three
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
18
CHAPTER THREE
MATHEMATICAL MODELING AND ACTIVE
CONTROL TECHNIQUES
3.1 Introduction
A mathematical model is a description of a system using mathematical
concepts and language, The process of developing a mathematical model is
termed mathematical modeling which considered as the first and the important
step in any analysis , in this study the derivation of the mathematical model of a
cantilever beam based on newton’s second law of motion . Vibration control
techniques can be classified into three main categories:
1. Passive vibration control .
2. Semi active vibration control.
3. Active vibration control.
The Passive vibration control refers to vibration control or mitigation of
vibrations by passive techniques such as rubber pads or mechanical springs, as
opposed to "active vibration control" or "electronic force cancellation"
employing electric power, sensors, actuators, and control systems. Passive
vibration isolation is a vast subject, since there are many types of passive
vibration isolators used for many different applications. A semi active vibration
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
19
controller can be defined as a passive device in which the properties (stiffness,
damping, etc.) can be varied in real time with a low power input. As they are
inherently passive, they cannot destabilize the system. Active vibration controller
can be defined as an active device which reacts on the vibrations. In this case, it
can destabilize the system if the smart structure is not correctly tuned but as the
system is active the response versus large bandwidth disturbances is better. In
this study, the widely used techniques in active vibration control which are
Linear Quadratic Regulator (LQR) controller, Proportional-Derivative - Integral
(PID) controller and Fuzzy logic controller (FLC) would be described in this
chapter.
3.2 Mathematical Model of The Cantilever Beam
The smart structure is modeled based on the concept of piezoelectric theory
and Bernoulli-Euler beam theory, Figure (3.1) shows a schematic diagram of a
cantilever beam laminated with piezoelectric layers, such as PZT. The beam is
assumed to be initially straight, of length, of length (L) , width (b) , thickness
(hb), and of constant mass (ρA) per unit length and constant stiffness. W(x, t)
denotes the transverse displacement of the beam. The quantity ( EbJ) is the
bending stiffness of the beam , where (Eb) is Young’s modulus of the material
and (J ) is the principal cross sectional area moments of inertia, (Q) the applied
load, and the time (t). It is also assumed that the thickness of piezoelectric layers
is much thinner than that of the elastic plate, the smart structure in Fig.(3.1). is
considered as a Bernoulli-Euler beam , the governing equations of motion and
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
20
associated boundary conditions are derived as follows [1]:
( ) ( ) ( )
( ) ( )
Where:
ρ :density of the beam material (Kg/m3) .
A :cross sectional area of the beam (m2).
W(x ,t) :transverse displacement of the beam(m).
Cn :damping coefficient of the beam(N.s/m).
Eb :Young’s modulus of the beam(N/m2).
J :moments of inertia of the beam (m4).
Q(x ,t) :applied force from the actuator (N).
Figure (3.1) smart cantilever beam with piezoelectric patches [1].
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
21
The total electric charge of the i-th piezoelectric sensor can be obtained as
follows [1]:
( ) ∫ ( )
( )
( )
( )
where :
g(i) :the electric charge measured at the i -th sensor at instant (t)(coulomb).
rsi : The average coordinate of the sensor measured from the mid-plane of
plate (m) and given by [1]
( )
hbi : thickness of the beam(m) .
hsi :thickness of sensor patch(m).
bsi :the width of the i- th sensor patch(m) .
e 31i :the piezoelectric stress constants of the i- th sensor patch (Vm/N).
r :the total number of sensors.
The moment resultant can be obtained from [1]
( ) ∑ [ ( ) ( ) ] ( )
Where :
m :is the number of the actuator.
Vj=V(x, t) :the control voltage across piezoelectric actuators(V).
H(x) :Heaviside function.
Kj :is a constant (N.m) and can be calculated from the equation below :
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
22
( )
And
( )
( )
Where:
haj :thickness of the j-th actuator(m).
baj :width of the j-th actuator(m).
Eaj :young’s modulus of the j-th actuator(N/m2).
d31j :constant of the j-th actuator (m/V).
The transverse displacement w(x , t) of the composite beam can be expressed as
a linear superposition of the modes of the beam as [1]:
( ) ∑ ( ) ( )
Where:
qk(t) :the generalized modal coordinate.
ϕk(x) :the mode shape function.
N :the number of modes used.
The mode shape function of a cantilever beam is given by [15]:
( ( ) ( )) ( ( ) ( )) ( )
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
23
Where
( ) ( )
( ) ( ) ( )
L :length of cantilever beam(m).
and ( ) is given by :
( )
Substituting equation (3.7) into the sensor equation (3.2), lead to :
( ) ∫
( )
( )
( )
∑ ( ) ( )
( ) ( )
Where :
bi :bsie31irsi (Vm3/N)
After substituting equation (3.7) and equation(3.4) in equation (3.1) with the
help of equation (3.11)and after arrangement the equation become[1]:
∑ k(xj ) k(xj)
j
( )
Where
ωk :the natural frequency of the i - th mode (rad/s).
ζk :damping ratio of the i -th mode.
Xj ,Xj+1 :the locations of the ends of the j -th piezoelectric patch along the X-
axis(m).
Systematic equation of the cantilever beam can be rewritten as [1] :
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
24
( ) ( ) ( ) ( ) ( )
( ) ( )
where
( )) ( )
(
) ( )
Where
V(t) ϵ Rm : the vector of the voltage subjected to (m) actuator .
D ϵ RNxm
: the matrix of the influence of actuators on varied modal of structure.
M ϵ RrxN
:the influence matrix of the sensor’s output .
And they are given as[1] :
( i , j) j k(xj ) k(xj) i , , j , , ,m ( )
And
(i , k) si k(xi ) k(xi) i , , , r k , , , ( )
Where
R : real number sets.
Ksi : constant given as(V.m3/N.F) [15]:
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
25
( )
Where
bi :width of i-th sensor(m).
hsi :thickness of i-th sensor(m).
g31i :piezoelectric constant of i-th sensor(Vm/N).
Cpi :piezoelectric capacitance of i-th sensor(F).
3.3 State Space Representation of The Cantilever Beam
A state space representation is a mathematical model of a physical system as
a set of input, output and state variables related by first-order differential
equations. To abstract from the number of inputs, outputs and states, the
variables are expressed as vectors. Additionally, if the dynamical system is linear
and time invariant, the differential and algebraic equations may be written in
matrix form. The state space representation (also known as the "time-domain
approach") provides a convenient and compact way to model and analyze
systems with multiple inputs and outputs. With (p) inputs and (q) outputs, it
would otherwise have to write down (q x p) Laplace transforms to encode all the
information about a system. Unlike the frequency domain approach, the use of
the state space representation is not limited to systems with linear components
and zero initial conditions. "State space" refers to the space whose axes are the
state variables. The state of the system can be represented as a vector within that
space. The internal state variables are the smallest possible subset of system
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
26
variables that can represent the entire state of the system at any given time .The
minimum number of state variables required to represent a given system, (n) , is
usually equal to the order of the system's defining differential equation. If the
system is represented in transfer function form, the minimum number of state
variables is equal to the order of the transfer function's denominator after it has
been reduced to a proper fraction [16].
The most general state-space representation of a linear system
with (p) inputs, (q) outputs and (n) state variables is written in the following
form for continuous time – invariant [16]:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
Where
x(t) :state vector.
y(t) :output vector.
u(t) :input vector.
A :state matrix with dimension (n x n).
B :input matrix with dimension( n x p).
E :output matrix with dimension (q x n).
F :feed forward matrix ( q x p ).
The state space approach is considered as the basic of the modern control
theories and is strongly recommended in the design and analysis of control
systems with a great amount of inputs and outputs. the system equations are
expressed by state-space equations, and then decoupled using the procedure
outlined in the following , let
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
27
( ) ( ) ( ( ) ( ) ( ) ( ) ( ) ( )) ( )
by applying the state space representation procedure on cantilever beam , the
matrices are given by [1] :
[
] ( )
[
] ( )
( )
F = [0] (3.28)
3.4 Active Vibration Control
Active vibration control is the active application of force in an equal and
opposite fashion to the forces imposed by external vibration. With this
application, a precision industrial process can be maintained on a platform
essentially vibration-free, Many precision industrial processes cannot take place
if the machinery is being affected by vibration. For example, the production
of semiconductor wafers requires that the machines used for the
photolithography steps be used in an essentially vibration-free environment or
the sub-micrometer features will be blurred. Active vibration control is now also
commercially available for reducing vibration in helicopters, offering better
comfort with less weight than traditional passive technologies , In the past ,
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
28
passive techniques were used. These include traditional vibration dampers, shock
absorbers, and base isolation[2].
The typical active vibration control system uses several components:
A massive platform suspended by several active drivers (that may use voice
coils, hydraulics, pneumatics and piezoelectric )
Three accelerometers that measure acceleration in the three degrees of
freedom
An electronic amplifier system that amplifies and inverts the signals from the
accelerometers. A PID controller or any other controller can be used to get
better performance than a simple inverting amplifier.
For very large systems, pneumatic or hydraulic components that provide the
high drive power required.
If the vibration is periodic, then the control system may adapt to the ongoing
vibration, thereby providing better cancellation than would have been provided
simply by reacting to each new acceleration without referring to past
accelerations.
3.5 Linear Quadratic Regulator (LQR) State Feedback Design
A system can be expressed in state variable form as shown in equation
( 3.22) with x(t)∈Rn , u(t)∈R
m . The initial condition is x(0). assuming here that
all the states are measurable and seek to find a state-variable feedback (SVFB)
control :
u − x (3.29)
that gives desirable closed-loop properties. The closed-loop system using this
control and after substation it in equ.(3.20) the equ. becomes :
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
29
( ) ( )
Where
Ac :the closed-loop plant matrix
Note that the output matrices C and D are not used in SVFB design. If there is
only one input so that m=1, then Ackermann's formula gives a SVFB K that
places the poles of the closed-loop system as desired. However, it is very
inconvenient to specify all the closed-loop poles, and it is also liked a technique
that works for any number of inputs. Since many naturally occurring systems are
optimal, it makes sense to design man-made controllers to be optimal as well. To
design a SVFB that is optimal, performance index (PI) may be defined the
∫ ( )
( )
Substituting the SVFB control into this yields
∫
( ) ( )
The objective in optimal design is to select the (SVFB) gain ( K) that minimizes
the performance index J.
The performance index J can be interpreted as an energy function, so that
making it small keeps small the total energy of the closed-loop system. Note that
both the state x(t) and the control input u(t) are weighted in J, so that if J is small,
then neither x(t) nor u(t) can be too large. Note that if J is minimized, then it is
certainly finite, and since limits of integral of x(t) goes to infinity, which implies
that x(t) goes to zero as (t) goes to infinity, This in turn guarantees that the
closed loop system will be stable., The two matrices Q (an n× n matrix) and R
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
30
(an m×m matrix) are selected by the design engineer. Depending on how these
design parameters are selected, the closed-loop system will exhibit a different
response.
Generally speaking, selecting Q large means that, to keep J small and the state
x(t) must be smaller. On the other hand selecting R large means that the control
input u(t) must be smaller to keep J small. it means that larger values of Q
generally result in the poles of the closed-loop system matrix Ac= (A - BK)
being further left in the s-plane so that the state decays faster to zero. On the
other hand, larger R means that less control effort is used, so that the poles are
generally slower, resulting in larger values of the state x(t). One should select Q
to be positive semi-definite and R to be positive definite. This means that the
scalar quantity (xT
Q x) is always positive or zero at each time t for all functions
x(t), and the scalar quantity (uT R u) is always positive at each time t for all
values of u(t) , This guarantees that J is well-defined. In terms of eigenvalues, the
eigenvalues of Q should be non-negative ,while those of R should be positive. If
both matrices are selected diagonal, this means that all the entries of R must be
positive while those of Q should be positive, with possibly some zeroes on its
diagonal, then R is invertible.
Since the plant is linear and the PI is quadratic, the problem of determining the
(SVFB) gain (K) to minimize J is called the Linear Quadratic Regulator (LQR).
The word 'regulator' refers to the fact that the function of this feedback is to
regulate the states to zero. This is in contrast to tracker problems, where the
objective is to make the output follow a prescribed (usually nonzero) reference
command. to find the optimal feedback (K) it should be proceed as follows.
Suppose there existing of a constant matrix (P) such that :
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
31
( ) ( ) ( )
Then, substituting into equation (3.32) yields :
∫
( )
( ) ( ) ( )
where assuming that the closed-loop system is stable so that x(t) goes to zero as
time (t ) goes to infinity. Equation (3.34) means that (J ) is now independent of
K. It is a constant that depends only on the auxiliary matrix P and the initial
conditions.
Now, SVFB gain (K ) can be found so that assumption (3.32) does indeed
hold. To accomplish this, differentiate (3.32) and then substitute from the closed-
loop state equation (3.29) to see that (3.32) is equivalent to :
( )
( )
( ) ( )
Now note that the last equation has to hold for every x(t). Therefore, the term in
brackets must be identically equal to zero. Thus, proceeding one sees that :
( ) ( ) ( )
( )
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
32
This is a matrix quadratic equation. Exactly as for the scalar case, one may
complete the squares. Though this procedure is a bit complicated for matrices,
suppose that:
( )
Then, there results :-
A A ( ) ( )
( )
( ) ( )
( )
This result is of extreme importance in modern control theory. Equation (3.41) is
known as the “ algebraic iccati equation “(A E) It is a matrix quadratic
equation that can be solved for the auxiliary matrix P given (A,B ,Q ,and R).
Then, the optimal SVFB gain is given by (3.40). The minimal value of the PI
using this gain is given by (3.34), which only depends on the initial
condition[18].
This mean that the cost of using the SVFB (3.40) can be computed from the
initial conditions before the control is ever applied to the system. The design
procedure for finding the LQR feedback K is:
• Select design parameter matrices Q and R.
• Solve the algebraic Riccati equation for P.
• Find the SVFB using K = R−1
BT P.
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
33
There are very good numerical procedures for solving the ARE. The MATLAB
routine that performs this is named LQR(A ,B, Q ,and R).
Figure (3.2) LQR block diagram controller
3.6 Proportional – Integral –Derivative Controller (PID)
A proportional-integral-derivative controller is a control loop feedback
mechanism (controller) widely used in industrial control systems. A PID
controller calculates an "error" value as the difference between a
measured process variable and a desired set point. The controller attempts to
minimize the error in outputs by adjusting the process control inputs, The PID
controller algorithm involves three separate constant parameters , and is
accordingly sometimes called three-term controller, the proportional , the
integral , the derivative values, denoted (P, I, and D), ( P ) depends on
the present error,( I ) on the accumulation of past errors , and (D ) is a prediction
of future errors, based on current rate of change, The weighted sum of these
three actions is used to adjust the process via a control element such as the
position of a control valve, a damper, or the power supplied to a heating element.
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
34
In the absence of knowledge of the underlying process, a PID controller has
historically been considered to be the best controller . By tuning the three
parameters in the PID controller algorithm , the controller can provide control
action designed for specific process requirements. The response of the controller
can be described in terms of the responsiveness of the controller to an error, the
degree to which the controller overshoots the set point, and the degree of system
oscillation. Note that the use of the PID algorithm for control does not
guarantee optimal control of the system or system stability[17].
Figure (3.3) block diagram of PID controller
3.6.1 PID Controller Theory
The PID control scheme is named after its three correcting terms, whose sum
constitutes the manipulated variable (MV). The proportional, integral, and
derivative terms are summed to calculate the output of the PID controller.
Defining L(t) as the controller output, the final form of the PID algorithm is[17]:
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
35
( ) ( ) ( ) ∫ ( )
( ) ( )
Where
Kp :Proportional gain, a tuning parameter.
Ki :Integral gain, a tuning parameter.
Kd :Derivative gain, a tuning parameter.
e(t) :Error.
t :Time or instantaneous time (the present).
In this study a design of PID controller with the help of auto tuning in
MATLAB software would be achieved.
3.6.2 Loop Tuning
Tuning a control loop is the adjustment of its control parameters
(proportional band/gain, integral gain/reset, derivative gain/rate) to the optimum
values for the desired control response. Stability (bounded oscillation) is a basic
requirement, but beyond that, different systems have different behavior, different
applications have different requirements, and requirements may conflict with one
another. Designing and tuning a PID controller appears to be conceptually
intuitive, but can be hard in practice, if multiple (and often conflicting)
objectives such as short transient and high stability are to be achieved. Usually,
initial designs need to be adjusted repeatedly through computer simulations until
the closed-loop system performs or compromises as desired. Some processes
have a degree of nonlinearity and so parameters that work well at full-load
conditions don't work when the process is starting up from no-load; this can be
corrected by gain scheduling (using different parameters in different operating
regions). PID controllers often provide acceptable control using default tunings,
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
36
but performance can generally be improved by careful tuning, and performance
may be unacceptable with poor tuning.
3.6.3 Stability
If the PID controller parameters (the gains of the proportional, integral and
derivative terms) are chosen incorrectly, the controlled process input can be
unstable [17], i.e., its output diverges, with or without oscillation, and is limited
only by saturation or mechanical breakage. Instability is caused by excess gain,
particularly in the presence of significant lag . Generally, stabilization of
response is required and the process must not oscillate for any combination of
process conditions and set points, though sometimes marginal stability (bounded
oscillation) is acceptable or desired [17] .
3.6.4 Manual Tuning
If the system must remain online, one tuning method is to first
set (Ki) and (Kd ) values to zero. Increase the (Kp) until the output of the loop
oscillates, then the (Kp ) should be set to approximately half of that value for a
"quarter amplitude decay" type response. Then increase (Ki) until any offset is
corrected in sufficient time for the process. However, too much (Ki) will cause
instability. Finally, increase (Kd ) , if required , until the loop is acceptably quick
to reach its reference after a load disturbance. However, too much (Kd ) will
cause excessive response and oscillation. A fast PID loop tuning usually
overshoots slightly to reach the set point more quickly; however, some systems
cannot accept overshoot, in which case an over-damped closed-loop system is
required , which will require a (Kp) setting significantly less than half that of the
(Kp) setting that was causing oscillation . in table (3.1) , the effect of each
increment or reducing of parameters to the final response is shown [18].
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
37
Table (3.1) Ziegler–Nichols Tuning Method .
parameter rise time Overshoot Settling
time
Steady-
state error
Stability
Kp decrease increase small
change
Decrease Degrade
Ki decrease increase increase Eliminate Degrade
Kd minor
change
decrease degrease no effect Improve if
(Kd)small
3.6.5 PID Controller Auto Tuning
MATLAB software enable us to get the optimal gains of PID controller
[19] , Simulink Control Design provides automatic gain-tuning
capabilities for Simulink PID Controller blocks. It can be accomplished that the
initial tuning of a PID controller with a single click. The product linearizes a
Simulink model to obtain a linear plant model, MATLAB software then uses the
linear plant model and a proprietary tuning method to compute the PID gains
based on the closed-loop performance that desired. An initial controller is
suggested based on an analysis of our system dynamics. it can then interactively
adjust the response time and transient behavior in the PID Tuner. The PID Tuner
also provides several plots that can use to analyze the controller behavior. For
example, a step reference tracking plot and an open-loop Bode plot can be used
to compare the performance of the current design with the design corresponding
to initial gain values.
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
38
3.7 Fuzzy Logic Controller(FLC)
The field of fuzzy system and control has been making a big progress
motivated by the practical success in industrial process control. Fuzzy systems
can be used in as closed-loop controllers. In this case the fuzzy system measures
the outputs of the process and takes control actions on the process continuously.
The fuzzy logic controller uses a form of quantification of imprecise information
(input fuzzy sets) to generate by an inference scheme, which is based on a
knowledge base of control force to be applied on the system [20], The advantage
of this quantification is that the fuzzy sets can be represented by a unique
linguistic expression such as small, medium, and large etc. The linguistic
representation of a fuzzy set is known as a term, and a collection of such terms
defines a term-set, or library of fuzzy sets. Fuzzy control converts a linguistic
control strategy usually based on expert knowledge into an automation control
strategy. There are three functions required to be performed by fuzzy logic
controller before the controller can generate the desired output signals. The first
step is to fuzzify each input. This can be realized by associating each input with
a set of fuzzy variables. In order to give semantics of a fuzzy variable a
numerical sense, a membership function is assigned with each variable. The
second step is the decision making which is depending on the rule base .the last
step is the Defuzzification process which is totally the invers of the
Fuzzification process The logical controller is made of four main components
[21]:
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
39
1. Fuzzification.
2. Rule base.
3. Decision making .
4. Defuzzification.
In this study , fuzzy logic controller is designed as the double-input, single-out
(DISO) system: The inputs are the displacement and the velocity of the tip of
cantilever beams, and the output is the control force on cantilever beams.
3.7.1 Fuzzification
In Fuzzification , the displacement is defined as (S) the velocity is defined
as (V) and the control force is defined as (U). Two types of membership
functions commonly adopted in fuzzy logic control are triangle and trapezoidal
shape. these two type membership functions can be used. In this study , as
compared with other methods, the method of the Middle of Maximum ( MOM )
was more effective. Accordingly, a way of establish fuzzy system is proposed as
following:
(1) At first, the scope of the displacement and the velocity are the maximal
response of beam when received step response.
(2) Plot the scopes of displacement’s and the force of control’s out as
NB(Negative Big) , NM (Negative Medium) , NS (Negative Small ) , ZO
(ZerO) , PS (Positive Small ) , PM (Positive Medium ) , PB (Positive Big) ; Then
plot the scopes of velocity ’s out as NB(Negative Big) , NS(Negative Small), ZO
(ZerO) , PS(Positive Small) , PB(Positive Big) .
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
40
(3) According to the fuzzy rule of [22], the process of fuzzy illation can be
determined.
(4) At last, using the method of MOM method in calculation to obtain the result.
3.7.2 The rule of fuzzy control and the FLC
The fuzzy rule shows the fuzzy relation between the input and output. The
inputs and output are connected with this relationship. The basic configuration of
the fuzzy system with Fuzzifier and Defuzzifier is shown in Figure (3.4) . In this
study , the displacement of the tip of cantilever beam is chosen for the one input,
the velocity is the other. In tradition method, the inputs usually are the velocity
and the rate of the velocity. In this way, the time of calculation has been
improved.
Figure (3.4) shows the basic configuration of fuzzy logic system.
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
41
Basing on the control rules, the signal is translated to the driver. The function of
fuzzy logic controller is making the inputs fuzz up. In other words, it is the fuzzy
control that executes the process of Fuzzification he fuzzy control’s basis is
the rule database which was composed of several rules. The final purpose of the
fuzzy logic controller is to make the fuzzy rule come true. The aim of vibration
control is to minimize the response of the cantilever beam. The function of the
fuzzy logic controller is to provide a force to reduce the vibration of the beam.
Fuzzy IF-THEN rule base is obtained by the analysis with many trial-and-
errors. Table(3.2) shows the rule base used to control the cantilever beam. [23]
used only (8 rules) and a modification and addition of a new rule in order to
obtain the desired response would be used and the result is (35 rules ) instead of
(8 rules). Fuzzy IF-THEN rule is the center of control system. The fuzzy rule
base is not invariable, it could be modify in practice.
Table (3.2) fuzzy logic controller rule base.
Control
force (U)
Displacement (S)
VELOCITY
(V)
NB NM NS ZO PS PM PB
NB PS PM PB ZO NB NM NS
NS PS PM PB ZO NB NM NS
ZO ZO ZO ZO ZO ZO ZO ZO
PS PS PM PB ZO NB NM NS
PB PS PM PB ZO NB NM NS
CHAPTER THREE MATHEMATICAL MODELING AND ACTIVE CONTROL TECHNIQUES
42
3.7.3 Defuzzification
Defuzzification is the process of producing a quantifiable result in fuzzy
logic, given fuzzy sets and corresponding membership degrees. It is typically
needed in fuzzy control systems. These will have a number of rules that
transform a number of variables into a fuzzy result, that is, the result is described
in terms of membership in fuzzy sets. A common and useful Defuzzification
technique is middle of maximum (MOM) which is used in this study, First, the
results of the rules must be added together in some way. The most typical fuzzy
set membership function has the graph of a triangle. Now, if this triangle were to
be cut in a straight horizontal line somewhere between the top and the bottom,
and the top portion were to be removed, the remaining portion forms a trapezoid.
The first step of Defuzzification typically "chops off" parts of the graphs to form
trapezoids (or other shapes if the initial shapes were not triangles).
Chapter
four
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
43
CHAPTER FOUR
COMPUTER SIMULATION USING MATLAB
4.1 Introduction
This chapter covers the simulation of vibration control of fixed – free
aluminum cantilever beam using MATLAB . firstly , the formula described in
[15] would be used to obtain the first four mode shapes and natural frequencies ,
then verification of this result with the result obtain for the same cantilever
beam from ANSYS software , The properties of the aluminum cantilever beam
shown in table (4.1) and the properties of piezoelectric patches (sensor and
actuator ) is shown in table (4.2) which are used in this chapter , In this study, the
masses of the piezoelectric patches and the Epoxy layer between the patches and
the cantilever beam were neglected because they were very small compared
with the mass of the cantilever beam [1] , also assuming that the mounting of
the patches on the cantilever beam is in perfect way which wouldn’t allow to any
relative motion between the cantilever beam and the piezoelectric patches
(sensor and actuator) [1].
Figure (4.1) beam dimension [1].
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
44
Table (4.1) properties of cantilever beam [1].
parameter symbol value unit
Thickness of the
beam hb 0.006 m
Length of the beam lb 0.26 m
Width of the beam b 0.01 m
Cross sectional area
of the beam A 60x10
-6 m
2
Young’s modulus of
the beam Eb 70x10
9 N/m
2
damping ratio ζ 0.002 -
density ρ 2700 Kg/m3
second moment of
area I 18x10
-11 m
4
mass of the beam m 42.1 g
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
45
Table (4.2) properties of piezoelectric patches (sensor and actuator) [1].
parameter symbol value unit
Thickness of the
piezoelectric patch
hp
0.0005
m
Length of the
piezoelectric patch lp 0.045 m
Width of the
piezoelectric patch b 0.01 m
Capacitance
CP 65x10-9
F
Young’s modulus
of the piezoelectric
patch
Ep 5.3x1010
N/m2
Piezoelectric
constant d31 -270x10
-12 m/V
Piezoelectric
constant g31 -9.2x10
-3 Vm/N
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
46
4.2 Mode Shapes and Natural Frequencies of The Cantilever Beam
The natural frequencies of the cantilever beam can be obtaining from the
equation (4.1) shown below [15]
( ) √
( )
Where
Ib :second moment of area of the cantilever beam(m4)
m :mass of the cantilever beam(Kg)
Eb :young’s modulus of the beam(N/m2)
L :length of cantilever beam(m)
And (ak ) is giving in equation (3.10).
The natural frequencies for the first four modes of the cantilever beam shown in
table (4.3) below .
Table (4.3) natural frequencies of the cantilever beam .
mode natural frequencies (rad/s)
(Ansys)
natural frequencies(rad/s)
(equ. (4.1))
1st mode
32.1 33.4
2nd
mode
289.7 290.2
3rd
mode
804.7 806.1
4th
mode
1577.2 1579.1
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
47
Figure (4.2) below shows the first four mode shapes of the cantilever beam using
ANSYS software.
Figure (4.2) The first four mode shapes of the cantilever beam.
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
48
4.3 The best location of the piezoelectric patches
The best location of the piezoelectric patches refer to the position in the
cantilever beam where the actuation and sensing process is optimal , i.e. the
location where the better response of the cantilever beam is achieved , this is
reflected as small amplitudes of displacement and velocity and small settling
time with relatively reasonably output phase change.
4.4 Active Vibration Control Using (LQR) Controller
In this section , a design of a controller to the cantilever beam based on “
linear quadratic regulator” (LQR) theory would be achieved , according to
equation ( 3.31), the design of the controller depends on the weight of the matrix
(Q)and (R) , after many trial and error , the desired value of (Q) and (R) which
gives the optimal response of the cantilever beam finally determined , the value
of (Q and R) are given below :
[ ]
The flow chart for the MATLAB program used is shown in figure (4.3) and the
MATLAB codes used to calculate the response of the beam is mentioned in the
appendix .
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
49
Figure (4.3) flow chart of LQR controller
START
Read the beam constants
(Eb, hb ,b , hp , Ib , ρ , Ab , Lb , ζ , d31 , Lp , Ep , cp , g31) and the
location of the patches( x1 ,x2)
Calculate the natural frequencies (wn , k ,c , d , M) and
matrices (A , B , C, D )
Solve Riccati equation to determine (P)
Read the matrices (R and Q)
Determine SVFB
K=R-1 B-T P
Determine the control matrix
Ac =( A-B*K)
Plot the response of controlled and uncontrolled system
END
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
50
4.4.1 Response of Cantilever Beam to Initial Condition
The response of the cantilever beam when the piezoelectric patches (sensor
and actuator) at position near fixed end ( x1=0.01 m , x2=0.055 m) to initial
condition of x0=[ 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0.01] {i.e. displacement of (1cm) at
free end} is shown in figure (4.4 ) below :
Figure (4.4 ) response of the cantilever beam to initial condition when piezoelectric patches
near fixed end with LQR controller.
The properties of the controlled and uncontrolled cantilever beam when the
piezoelectric patches near fixed end are given in table (4.4) below:
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
51
Table (4.4) properties of controlled and uncontrolled cantilever beam subjected to initial
conditions when the patches near fixed end with LQR controller.
cantilever beam property value unit
uncontrolled peak amplitude -6.43x10
-5 m
Settling time 0.1 sec
controlled peak amplitude -5.48x10
-5 m
Settling time 0.04 sec
- Reduction in vibration
amplitude 14.77% -
- Max. Phase change -45 degree
The figures below shows the bode plot and the rlocus plot of the controlled and
uncontrolled cantilever beam when the beam subjected to initial condition and
the piezoelectric patches near fixed end with LQR controller .
Figure (4.5) bode plot of the cantilever beam subjected to initial condition when piezoelectric
patches near fixed end with LQR controller.
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
52
Figure (4.6 ) rlocus plot of the controlled and uncontrolled cantilever beam subjected to initial
condition when piezoelectric patches near fixed end with LQR controller.
The response of the cantilever beam when the piezoelectric patches (sensor and
actuator) at free end ( x1=0.215 m , x2=0.26 m) to initial condition of x0=[ 0 ; 0
; 0 ; 0 ; 0 ; 0 ; 0 ; 0.01] {i.e. displacement of (1cm) at free end} is shown in
figure ( 4.7) below .
The properties of the controlled and uncontrolled cantilever beam when the
piezoelectric patches at free end are given in table (4.5) below.
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
53
Figure ( 4.7) response of the cantilever beam with initial condition when piezoelectric
patches at free end with LQR controller.
Table (4.5) properties of controlled and uncontrolled cantilever beam subjected to initial
conditions when the patches at free end with LQR controller.
cantilever
beam
property value unit
uncontrolled
peak amplitude 10.6x10-5
m
Settling time 0.06 sec
controlled peak amplitude 8.44x10
-5 m
Settling time 0.02 sec
- Reduction in vibration
amplitude 20.3% -
- Max. Phase change -45 degree
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
54
The figures below shows the bode plot and the rlocus plot of the controlled and
uncontrolled cantilever beam when the beam subjected to initial condition and
the piezoelectric patches at free end with LQR controller .
Figure (4.8 ) bode plot of the cantilever beam subjected to initial condition when piezoelectric
patches at free end with LQR controller.
Figure (4.9 ) rlocus plot of the controlled and uncontrolled cantilever beam subjected to initial
condition when piezoelectric patches at free end with LQR controller.
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
55
4.4.2 Impulse Response of Cantilever Beam
The impulse response of the cantilever beam when the piezoelectric patches
(sensor and actuator) at position near fixed end ( x1=0.01 m , x2=0.055 m) to is
shown in figure ( 4.10) below :
Figure (4.10 ) impulse response of the cantilever beam when piezoelectric patches near fixed
end with LQR controller.
The properties of the controlled and uncontrolled cantilever beam when the
piezoelectric patches near fixed end are given in table (4.6) below:
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
56
Table (4.6) properties of controlled and uncontrolled cantilever beam subjected to impulse
force when the patches near fixed end with LQR controller
cantilever
beam
property value unit
uncontrolled peak amplitude 2.9x10
-6 m
settling time 5.4 sec
controlled peak amplitude 1.46x10
-6 m
settling time 0.209 sec
- Reduction in vibration
amplitude 49.65% -
- Max. Phase change -45 degree
The figures below shows the bode plot and the rlocus plot of the controlled and
uncontrolled cantilever beam when the beam subjected to impulse force and the
piezoelectric patches near fixed end with LQR controller.
Figure (4.11 ) bode plot of the cantilever beam subjected impulse force when piezoelectric
patches near fixed end with LQR controller.
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
57
Figure (4.12 ) rlocus plot of the controlled and uncontrolled cantilever beam subjected to
impulse force when piezoelectric patches near fixed end with LQR controller.
The impulse response of the cantilever beam when the piezoelectric patches
(sensor and actuator) at free end ( x1=0.215 m , x2=0.26 m) is shown in figure
(4.13) below :
Figure (4.13 ) impulse response of the cantilever beam when piezoelectric patches at free end
with LQR controller.
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
58
The properties of the controlled and uncontrolled cantilever beam when the
piezoelectric patches near fixed end are given in table (4.7) below:
Table (4.7) properties of controlled and uncontrolled cantilever beam subjected to impulse
force when the patches at free end with LQR controller.
cantilever
beam
property value unit
uncontrolled peak amplitude 4.05x10
-6 m
settling time 4.25 sec
controlled peak amplitude 2.75x10
-6 m
settling time 0.19 sec
- Reduction in vibration
amplitude 32.09% -
- Max. Phase change -45 degree
The figures below shows the bode plot and the rlocus plot of the controlled and
uncontrolled cantilever beam when the beam subjected to impulse force and the
piezoelectric patches at free end using LQR controller.
Figure ( 4.14) bode plot of the cantilever beam subjected to impulse force when piezoelectric
patches at free end with LQR controller.
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
59
Figure (4.15 ) rlocus plot of the controlled and uncontrolled cantilever beam subjected to
impulse force when piezoelectric patches at free end with LQR controller.
4.5 Active vibration control using (PID) controller
In this section , a design of an active vibration controller for the cantilever
beam base on (PID) controller would be attained, according to equation(3.43) the
proportional- integral-derivative (PID) controller depend fundamentally on the
constant (Kp , Ki and Kd) , therefore in order to reach the desired response and
properties for the control system , the flow chart shown in figure (4.16) illustrate
the step used for writing the MATLAB program codes which used to determine
the response of the cantilever beam with PID controller ,the codes of the
program is shown in the appendix. there are two method used in order to
determine the value of the three controller (Kp , Ki and Kd) which are described
below.
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
60
Large
Very small
Figure (4.16) flow chart of PID controller
START
Read the beam constants
(Eb, hb ,b , hp , Ib , ρ , Ab , Lb , ζ , d31 , Lp , Ep , cp , g31) and the
location of the patches( x1 ,x2)
Calculate the natural frequencies (wn , k ,c , d , M) and
matrices (A , B , C, D )
Read the PID controller constant (Kp , ki , kd )
or adjust their value
calculate the error
Insert the set point (reference )
Error=?
Calculate the response of the beam
END
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
61
4.5.1 Manually determination of (PID) controller constants
In this method , the effect of each constant mentioned in table (3.1) will be used
to determine the optimal value of each constant. After many trial and error ,
finally the three constant values were determined and whom shown below :
The figure( 4.17) below show the response of cantilever beam with (PID)
controller to impulse force when the piezoelectric patches near fixed end
(x1=0.01 m ,x2=0.055 m) :
Figure (4.17) impulse response of cantilever beam when the patches near fixed end with PID
controller.
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
62
The properties of the controlled with (PID parameter tuning with MATLAB)
controller and uncontrolled cantilever beam when the piezoelectric patches near
fixed end are given in table (4.8) below:
Table (4.8) properties of impulse response of cantilever beam with PID controller when the
patches near fixed end.
cantilever beam property value unit
uncontrolled
peak amplitude 2.45x10-6
m
settling time 5.33 sec
controlled
peak amplitude 0.467x10-6
m
settling time 2.48 sec
- Reduction in vibration
amplitude 80.93% -
- Max. Phase change -90 degree
The figures below shows the bode plot and the rlocus plot of the controlled and
uncontrolled cantilever beam when the beam subjected to impulse force and the
piezoelectric patches near fixed end using PID controller.
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
63
Figure (4.18 ) bode plot of the cantilever beam subjected to impulse force when piezoelectric
patches near fixed end using PID controller.
Figure (4.19 ) rlocus plot of the controlled and uncontrolled cantilever beam subjected to
impulse force when piezoelectric patches near fixed end using PID controller.
The figure(4.20) below show the response of cantilever beam with (PID)
controller to impulse force when the piezoelectric patches at free end (x1=0.215
m , x2=0.26 m) :
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
64
Figure (4.20) impulse response of cantilever beam when the patches at free end with PID
controller.
The properties of the controlled with (PID) controller and uncontrolled cantilever
beam when the piezoelectric patches at free end are given in table (4.9) below:
Table (4.9) properties of impulse response of cantilever beam with (PID)controller when the
patches at free end.
cantilever beam property value unit
uncontrolled
peak amplitude 2.61x10-6
m
settling time 4.49 sec
controlled peak amplitude 1.04x10
-6 m
settling time 1.02 sec
- Reduction in vibration
amplitude 60.15% -
- Max. Phase change -90 degree
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
65
The figures below shows the bode plot and the rlocus plot of the controlled and
uncontrolled cantilever beam when the beam subjected to impulse force and the
piezoelectric patches at free end using PID controller.
Figure (4.21 ) bode plot of the cantilever beam subjected impulse force when piezoelectric
patches at free end using PID controller.
Figure ( 4.22) rlocus plot of the controlled and uncontrolled cantilever beam subjected to
impulse force when piezoelectric patches at free end using PID controller.
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
66
4.5.2 Determination of (PID) controller constants using auto tuning
MATLAB software enable us to determine the (PID) controller constants
with a very accurate values and gives the optimal values of the constants in
order to reach the desired response , Figure (4.23) shows the main window in
MATLAB used to “ parameter tuning” , when using the MATLAB window of
(parameter tuning ) and applying our system , the following values of the (PID)
controller were determined :
Figure( 4.24) below show the response of cantilever beam with (PID)with
MATLAB auto tuning controller to impulse force when the piezoelectric
patches near fixed end (x1=0.01 m ,x2=0.055 m) :
Figure (4.23) MATLAB PID tuner.
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
67
Figure (4.24) impulse response of cantilever beam when the patches near fixed end with (PID
parameter tuning with matlab ) controller.
The properties of the controlled with (PID) controller and uncontrolled cantilever
beam when the piezoelectric patches near fixed end are given in table (4.10)
below:
Table (4.10) properties of impulse response of cantilever beam with (PID)controller when the
patches near fixed end MATLAB auto tuning.
cantilever
beam property value unit
uncontrolled peak amplitude 2.45x10
-6 m
settling time 5.33 sec
controlled peak amplitude 0.234x10
-6 m
settling time 0.9 sec
- Reduction in vibration
amplitude 90.4% -
- Max. Phase change -90 degree
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
68
Figures (4.25 and 4.26) below shows the bode plot and the rlocus plot of the
controlled and uncontrolled cantilever beam when the beam subjected to impulse
force and the piezoelectric patches near fixed end using PID controller
MATLAB auto tuning.
Figure (4.25 ) bode plot of the cantilever beam subjected impulse force when piezoelectric
patches near fixed end using PID controller MATLAB auto tuning.
Figure (4.26 ) rlocus plot of the controlled and uncontrolled cantilever beam subjected to
impulse force when piezoelectric patches near fixed end using PID controller MATLAB auto
tuning.
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
69
Figure(4.27) below show the response of cantilever beam with (PID) controller
to impulse force when the piezoelectric patches at free end (x1=0.215 m
,x2=0.26 m) with MATLAB auto tuning .
Figure (4.27) impulse response of cantilever beam when the patches at free end with PID
controller MATLAB auto tuning.
The properties of the controlled with (PID) controller and uncontrolled cantilever
beam when the piezoelectric patches at free end are given in table (4.11) below:
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
70
Table (4.11) properties of impulse response of cantilever beam with (PID)controller when the
patches at free end MATLAB auto tuning.
cantilever
beam
property value unit
uncontrolled peak amplitude 2.6x10
-6 m
settling time 2.3 sec
controlled peak amplitude 0.3x10
-6 m
settling time 0.637 sec
- Reduction in vibration
amplitude 88.4% -
- Max. Phase change -90 degree
Figures (4.28 and 4.29) below shows the bode plot and the rlocus plot of the
controlled and uncontrolled cantilever beam when the beam subjected to impulse
force and the piezoelectric patches free end using PID controller MATLAB
auto tuning.
Figure (4.28 ) bode plot of the cantilever beam subjected impulse force when piezoelectric
patches at free end using PID controller MATLAB auto tuning.
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
71
Figure (4.29 ) rlocus plot of the controlled and uncontrolled cantilever beam subjected to
impulse force when piezoelectric patches at free end using PID controller MATLAB auto
tuning.
4.6 Active vibration control using (FL) controller
In this section , a design of a vibration controller for the cantilever beam
based on fuzzy logic controller theory will be achieved , the design of the
controller would be created using MATLAB software . the main page in
MATLAB for fuzzy logic controller is shown in Fig. (4.30) below . the inputs
for the controller will be the displacement and the velocity while the output will
be the controlled displacement of the beam , Fig. (4.31) and (4.32) shows the
displacement and velocity ranges respectively ,while Fig.(4.33 ) shows the
output controlled displacement of the cantilever beam , the rules base used are
contain (35 rules) based on table (3.2) which are obtained as follows in table
(4.12) :
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
72
Table (4.12) FLC expanded rules
NO. RULE
1 If (velocity is NB) and (displacement is NB) then (controlled cantilever
displacement is PS)
2 If (velocity is NB) and (displacement is NM) then (controlled cantilever
displacement is PM)
3 If (velocity is NB) and (displacement is NS) then (controlled cantilever
displacement is PB)
4 If (velocity is NB) and (displacement is ZO) then (controlled cantilever
displacement is ZO)
5 If (velocity is NB) and (displacement is PS) then (controlled cantilever
displacement is NB)
6 If (velocity is NB) and (displacement is PM) then (controlled cantilever
displacement is NM)
7 If (velocity is NB) and (displacement is PB) then (controlled cantilever
displacement is NS)
8 If (velocity is NS) and (displacement is NB) then (controlled cantilever
displacement is PS)
9 If (velocity is NS) and (displacement is NM) then (controlled cantilever
displacement is PM)
10 If (velocity is NS) and (displacement is NS) then (controlled cantilever
displacement is PB)
11 If (velocity is NS) and (displacement is ZO) then (controlled cantilever
displacement is ZO)
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
73
12 If (velocity is NS) and (displacement is PS) then (controlled cantilever
displacement is NB)
13 If (velocity is NS) and (displacement is PM) then (controlled cantilever
displacement is NM)
14 If (velocity is NS) and (displacement is PB) then (controlled cantilever
displacement is NS)
15 If (velocity is ZO) and (displacement is NB) then (controlled cantilever
displacement is ZO)
16 If (velocity is ZO) and (displacement is NM) then (controlled cantilever
displacement is ZO)
17 If (velocity is ZO) and (displacement is NS) then (controlled cantilever
displacement is ZO)
18 If (velocity is ZO) and (displacement is ZO) then (controlled cantilever
displacement is ZO)
19 If (velocity is ZO) and (displacement is PS) then (controlled cantilever
displacement is ZO)
20 If (velocity is ZO) and (displacement is PM) then (controlled cantilever
displacement is ZO)
21 If (velocity is ZO) and (displacement is PB) then (controlled cantilever
displacement is ZO)
22 If (velocity is PS) and (displacement is NB) then (controlled cantilever
displacement is PS)
23 If (velocity is PS) and (displacement is NM) then (controlled cantilever
displacement is PM)
24 If (velocity is PS) and (displacement is NS) then (controlled cantilever
displacement is PB)
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
74
25 If (velocity is PS) and (displacement is ZO) then (controlled cantilever
displacement is ZO)
26 If (velocity is PS) and (displacement is PS) then (controlled cantilever
displacement is NB)
27 If (velocity is PS) and (displacement is PM) then (controlled cantilever
displacement is NM)
28 If (velocity is PS) and (displacement is PB) then (controlled cantilever
displacement is NS)
29 If (velocity is PB) and (displacement is NB) then (controlled cantilever
displacement is PS)
30 If (velocity is PB) and (displacement is NM) then (controlled cantilever
displacement is PM)
31 If (velocity is PB) and (displacement is NS) then (controlled cantilever
displacement is PB)
32 If (velocity is PB) and (displacement is ZO) then (controlled cantilever
displacement is ZO)
33 If (velocity is PB) and (displacement is PS) then (controlled cantilever
displacement is NB)
34 If (velocity is PB) and (displacement is PM) then (controlled cantilever
displacement is NM)
35 If (velocity is PB) and (displacement is PB) then (controlled cantilever
displacement is NS)
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
75
Figure (4.30) fuzzy logic controller page in MATLAB
Figure (4.31) the input displacement member function of cantilever beam
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
76
Figure (4.32) the input velocity member function of cantilever beam
Figure (4.33) the output controlled displacement member ship of cantilever beam
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
77
The rule viewer and the rule surface of the designed (FLC) are shown below in
figure (4.34) and (4.35) respectively .
Figure (4.34) rule viewer of the designed FLC
Figure (4.35) surface viewer of the designed FLC
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
78
The designed (FL) controller is tested using different type of inputs to ensure its
validity to control many type of inputs . The types of inputs used are ( sine wave
and unit step) , (4.36,4.37,4.38 and 4.39) below shows each type of inputs and
the response (output) of the controller to these different types of inputs while
Fig. (4.40) shows the Simulink page for the FLC in MATLAB .
Figure (4.36) inputs displacement and velocity of sine wave
Figure (4.37)output controlled displacement when the inputs are sine wave
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
79
Figure (4.38) inputs displacement and velocity of unit step
Figure (4.39) output controlled displacement when the inputs are unit steps
CHAPTER FOUR COMPUTER SIMULATION USING MATLAB
80
Figure (4.40) the Simulink page of FLC in MATLAB
Chapter
five
CHAPTER FIVE RESULTS AND DISCUSSION
18
CHAPTER FIVE
RESULTS AND DISCUSSION
5.1 Introduction
In this chapter , the results obtain from the computer simulation will be
reviewed and discussed , the best location of the piezoelectric patches( sensor
and actuator) will be determined . in order to investigate the proper and faster
designed controller , a comparison between the three designed controller
(LQR,PID and FL controller) will be carried out. finally , a comparison
between the controllers designed in this study and another controller designed in
another past studies will be achieved .
5.2 Determination of The Best Location of The Piezoelectric Patches
In order to obtain best location and get a robust and fast controller , an
examination of the effect of the location on each controller ( i.e. LQR , PID and
FL controllers ) on the response of the cantilever beam will be carried out . Table
(5.1) below illustrated the properties of each used controller and the effect of the
patches location on the response of the cantilever beam .it can be clearly
recognizing that the best location for the patches locate near fixed end (given
better reduction in displacement amplitude with relatively decent settle time ).
Fig. (5.1) shows the best location of the patches on the cantilever beam.
CHAPTER FIVE RESULTS AND DISCUSSION
18
Table (5.1) effect of piezoelectric patches location on the beam response.
type of
controller response to
location of the
piezoelectric
patches
peak amplitude of
the output
displacement(m)
settle
time(sec)
Reduction in
output
amplitude
LQR
impulse force near fixed end 1.46x10-6
0.209 49.65%
impulse force at free end 2.75x10-6
0.19 32.09%
initial condition near fixed end -5.48x10-5
0.04 14.77%
initial condition at free end 8.44x10-5
0.06 20.3%
PID
Manually
tuning
impulse force near fixed end 0.467x10-6
2.48 80.93%
impulse force at free end 1.04x10-6
1.02 60.15%
PID-
MATLAB
auto
tuning
impulse force near fixed end 0.234x10-6
0.9 90.4%
impulse force at free end 0.3x10-6
0.637 88.4%
FLC
Unit step near fixed end 0.1 x10-6
0.3 90.5%
Unit step at free end 0.23x10-6
0.51 88.5%
Sine wave near fixed end 0.8x10-6
0.2 84%
Sine wave at free end 0.9x10-6
0.4 82%
CHAPTER FIVE RESULTS AND DISCUSSION
18
Figure (5.1) the best location of piezoelectric patches on the cantilever beam
5.3 Results and Discussion
In order to conclude which vibration control technique is better to achieve a
robust , faster and reliable controller , a comparison between (LQR , PID ,PID –
MATLAB auto tuning and FL ) controller has been made in figure (5.2) below .
It is obvious from figure (5.3) and table (5.1) that fuzzy logic controller (FLC) is
the best controller compared with the other controller according to the suitable
reduction in the output displacement of the cantilever beam and the relatively
CHAPTER FIVE RESULTS AND DISCUSSION
18
smaller settle time which give the desired properties to control the vibration of
the cantilever beam , PID-MATLAB auto tuning also provide acceptable
performance.
Figure (5.2) response of cantilever beam –different type of controller at best Location
CHAPTER FIVE RESULTS AND DISCUSSION
18
Figure (5.3) response of cantilever beam –different type of controller at best
Location(zoomed).
In figures (4.5 and 4.6) which showing the bode plot of the controlled with LQR
and uncontrolled system , it can be noticed that the original system(uncontrolled
) and the controlled system are approximately at the same phase ( the
maximum phase change is -45 degrees ) with less amplitude (dB) for the
controlled system (-150 dB) which giving as the best vibration control of the
system and the same results also can be realized from figures(4.8, 4.9, 4.11,
4.12, 4.14,4. 15, 4.18, 4.19, 4.21, 4.22, 4.25, 4.26, 4.28 and 4.29).
CHAPTER FIVE RESULTS AND DISCUSSION
18
Also in Figures (4.37 and 4.39) which illustrate the response of the fuzzy logic
control to different type of inputs , it is obvious that the controller is very robust
and reliable . the use of (35 rules) instead of (8 rules) gave more reliability and
stability for the system and allow the system to be more robust with faster
response time and gave the system a less settling time . Simulation results
indicate that the vibration of a cantilever beam has been actively suppressed by
applying control voltage to the piezoceramic actuator and the optimal control
theory lead to a useful controller design methodology for the design of robust
controllers for the vibration control of cantilever beam-like structures. It also
demonstrated that when application of vibration control theory, the arrangement
of sensors and actuators should considering the result of other assemblies, but
the best position is near the fixed point, which is also called maximal
deformation of the beam.
5.4 Comparison Between the Current Research and Other Researches
In this section , a comparison between the different type of controller used
in this study with the controllers from other studies will be illustrated in the
followed paragraphs . The response of the system with LQR controller to initial
condition of (1cm at free end ) when the patches near fixed end and the response
of the same system taken from [8] are shown below in figures (5.4 and 5.5):
CHAPTER FIVE RESULTS AND DISCUSSION
18
Figure (5.4) response of the cantilever beam to initial condition with LQR controller at best
location of the patches.
Figure (5.5) response of the cantilever beam to initial condition with LQR controller at best
location of the patches [8] .
CHAPTER FIVE RESULTS AND DISCUSSION
11
It is obvious that the current designed LQR controller is much better
performing than the controller in [8] since the reduction in the output amplitude
of the current study is (14.77 %) while in [8] is ( 7.3 %) and the settle time in
the current study is (0.04 sec ) while in [8] is (0.08 sec) .
In the other hand , the response of the system to response force with PID-
MATLAB auto tuned controller in the current study and the response of the same
system to the same force given in [9] are shown in figures (5.6 and 5.7) below:
Figure (5.6) response of the cantilever beam to impulse force with PID controller at best
location of the patches.
CHAPTER FIVE RESULTS AND DISCUSSION
18
Figure (5.7) response of the cantilever beam to impulse force with PID controller at best
location of the patches [9].
The response of the current PID controller gave an output amplitude
reduction of (90.4 %) while in [9] the percentage was (80.4 %) and the settle
time in current study was found to be (0.9 sec) while in [9] was (3.56 sec)
therefore the current study gives much better controller design than that of [9].
The last comparison will be made to compare the performance of the fuzzy
logic controller FLC of the current study with the controller from [14], figures
(5.8 and 5.9 ) shows the response of the cantilever beam of both FLC of [14] and
FLC of the current study to the unit step force :
CHAPTER FIVE RESULTS AND DISCUSSION
89
Figure (5.8) response of the cantilever beam to unit step force with FL controller at best
location of the patches .
Figure (5.9) response of the cantilever beam to unit step force with FL controller at best
location of the patches [14].
CHAPTER FIVE RESULTS AND DISCUSSION
88
.
Again , the designed FLC of the current study gave a reduction in output
amplitude of (90.5 % ) while in [14] gave (40 % ) and the settled time for current
study FL controller was (0.3 sec) and for [14] was (0.8 sec ) so the current
designed FL controller is better than the controller of [14] .
Chapter
six
CHAPTER SIX CONCLUSION AND RECOMMENDATIONS FOR FUTURE WORKS
92
CHAPTER SIX
CONCLUSION AND RECOMMENDATIONS FOR FUTURE
WORKS
5.1 Conclusion of This Thesis
This study presents a theoretical analysis of the flexural response of a beam
with bonded piezoelectric sensor/actuator patches based on the classical beam
theory. A controller is designed by the optimal control theory. Piezoelectric
patches have been bonded at the root and the tip of the cantilevered beam
investigated respectively. Simulation results indicate that the vibration of a
cantilever beam has been actively suppressed by applying control voltage to the
piezoceramics actuators, and the optimal control theory lead to a useful
controller design methodology for the design of robust controllers for the
vibration control of cantilever beam-like structures. It also demonstrated that
when application of vibration control theory, the arrangement of sensors and
actuators should considering the result of other assemblies, but the optimal
position is near the fixed point, which is also called maximal deformation of the
beam.
CHAPTER SIX CONCLUSION AND RECOMMENDATIONS FOR FUTURE WORKS
93
The conclusions of this study can be summarized into the following points:
1. Active vibration control using (LQR , PID and FL) controllers give strong
robustness to modal parameters variation and has a good closed-loop
dynamic performance .
2. It has been demonstrated that the optimal location for the piezoelectric
patches(sensor/actuator) and investigation of the response in different
location on the cantilever beam and finally the optimal location obtaining
to be near the fixed end.
3. Controlling of the cantilever beam using fuzzy logic controller (FLC)
gives a better results than proportional-integral-derivative(PID) and linear
quadratic regulator(LQR) in terms of response and settling time.
4. Active vibration control gives a reduction in vibration amplitude of
(49.65 %) when using LQR controller , (80.93% ) when using PID
manually tuning ,( 90.4%) when using PID auto tuning and (90.5% ) when
using FL controller.
5. The settling time was (0.209 , 2.48 , 0.9 and 0.3 sec ) for impulse force
response when using (LQR , PID manually tuning , PID auto tuning ) and
unit step force for FL controllers respectively .
CHAPTER SIX CONCLUSION AND RECOMMENDATIONS FOR FUTURE WORKS
94
5.2 Recommendations for Future Works
The recommendations for the other future works can be summarized in to the
following :
1. The future researches can cover another types of active controllers to
control the cantilever beam such as (H∞ controller and Fuzzy-PID
controller ).
2. A comparison between the results from MATLAB software and ANSYS
software can be done in order to verifying the theoretical results.
3. An experimental study of active vibration control of a cantilever beam is
extremely recommended to verifying the theoretical results.
references
REFERENCES
95
REFERENCES
[1] Gou Xinke and Tian Haimin , “Active Vibration Control of a Cantilever
Beam Using Bonded Piezoelectric Sensors and Actuators” , College Of
Electrical and Information Engineering , Lanzhou University of
Technology, Lanzhou , China , pp. 85-88, 2007.
[2] S. Bennett , ” A History Of Control Engineering “ , Peter Peregrinus
Hitchin Corp. , Herts. , UK , pp. 48-110 , 1993.
[3] Yavuz Yaman , Tarkan Çalişkan , Volkan Nalbantoğlu , Eswar Prasad and
David Waechter , ” Active Vibration Control of A Smart Beam ” , Paper
Published In The Proceedings Of Cansmart Symposium Held at Montreal,
pp.125-133, 2001.
[4] Ulrich Gabbert , Tamara Nes.Trajkov and Heinz Köppe, ” Modelling
,Control and Simulation of Piezoelectric Smart Structures Using Finite
Element Method and Optimal LQR Control ” , Facta Universities Series ,
pp. 418-430, 2002 .
[5] Zhang Jing-Jun , Cao Li-Ya and Yuan Wei-Ze , ” Active Vibration Control
for Smart Structure Base on The Fuzzy Logic ” International Conference on
Advanced Computer Control , pp. 231-234, 2008.
[6] Dong Jingshi , Guo Kang , Zheng Wei , Shenchuanliang , Ch. Guangming
and Ding Ji , ” Study on Active Vibration Control Technique Based On
Cantilever Beam ” International Conference On Computer, Mechatronics,
Control and Electronic Engineering (Cmce) , pp. 468-471, 2010.
REFERENCES
96
[7] K. B. Waghulde , Bimleshkumar Sinha , M. M. Patil and S. Mishra “
Vibration Control of Cantilever Smart Beam by Using Piezoelectric
Actuators And Sensors ” , International Journal of Engineering and
Technology, Vol.2(4) ,pp. 259-262, 2010.
[8] S.M. Khot, Nitesh P. Yelve and Ramya Iyer “ Active Vibration Control of
Cantilever Beam by Using Optimal (LQR) Controller ” , pp. 2-10, 2011.
[9] Deepak Chhabra1, Pankaj Chandna and Gian Bhushan , ” Design and
Analysis of Smart Structures for Active Vibration Control Using Piezo-
Crystals ” , International Journal of Engineering and Technology ,Vol.1,
No.3, pp. 153-163, 2011.
[10] S.M. Khot ,Nitesh P Yelve, Rajat Tomar, Sameer Desai and S. Vittal,”
Active Vibration Control of Cantilever Beam by Using PID Based Output
Feedback Controller ” , Journal of Vibration and Control , pp. 366-372,
2011.
[11] Deepak Chhabra, Kapil Narwal and Pardeep Singh, “ Design and Analysis
of Piezoelectric Smart Beam for Active Vibration Control” , International
Journal of Advancements in Research and Technology, Vol.1, pp. 1-5,
2012.
[12] Tamara Nestorović, Navid Durrani and M. Trajkov, “ Experimental Model
Identification and Vibration Control of a Smart Cantilever Beam Using
Piezoelectric Actuators and Sensors ” J .Electroceram ,pp. 42-55, 2012.
[13] A.P. Parameswaran, A.B. Pai , P.K.Tripathi and K.V.Gangadharan ,”Active
Vibration Control of a Smart Cantilever Beam on General Purpose
Operating System ” Defence Science Journal, Vol. 63, No. 4, pp. 413-417,
2013.
REFERENCES
97
[14] Preeti Verma , Manish Rathore and Rajeev Gupta , “ Vibration Control of
Cantilever Beam Using Fuzzy Logic Controller ” , International Journal of
Science, Engineering and Technology Research (IJSETR) ,Vol. 2, pp. 906-
909, 2013.
[15] Singiresu S. Rao , “ Vibration of Continuous Systems ” , John Wiley and
Sons. Inc. , pp. 563-570, 2007.
[16] Friedland and Bernard , “ Control System Design: an Introduction to State
Space Methods ” , McGraw-Hill , pp. 103-130 , 1986.
[17] Ang , K.H., Chong , G.C.Y. and Li. Y. , “ PID Control System Analysis,
Design and Technology ” , IEEE Trans Control Systems Tech, pp. 559-576,
2005.
[18] Jinghua Zhong , “ PID Controller Tuning: a Short Tutorial ”, Springer , pp.
408-412, 2006.
[19] Brian R. Hunt, Ronald L. Lipsman, Jonathan M. Rosenberg, Kevin R.
Coombes, John E. Osborn and Garrett J. Stuck , ” A Guide to
MATLAB: for Beginners and Experienced Users “ , Cambridge University
Press , pp.102-300, 2006.
[20] R.S.Subramaniam , A.M.Reinhorn, M.A.Riley and S. Nagarajaiah, “
Hybrid Control of Structures Using Fuzzy Logic ”, Microcomput, Civil
Eng. , pp. 1-17, 1996.
[21] J. Yen and R. Langari , “ Fuzzy Logic-Intelligence Control and Information
” , Prentice Hall, Englewood Cliffs , pp. 101-125, 1998.
[22] L.A. Zadeh and R. R. Yager , “ Fuzzy sets and applications: selected “ ,
Wiley , pp. 338- 353, 1987.
REFERENCES
98
[23] Wedad Ibraheem Majeed , Shibly Ahmed Al-Samarraie and Mohanad
Mufaq Al-Saior , “ Vibration Control Analysis of A Smart Flexible
Cantilever Beam Using Smart Material ” Journal of Engineering ,Vol.19 ,
pp. 82-95, 2013.
[24] Jingjun Zhang, Liya Cao, Weize Yuan and Ruizhen Gao , ” Active
Vibration Control for Smart Structure Base on The Fuzzy Logic ” , Second
International Symposium On Intelligent Information Technology
Application , pp. 901-905, 2008.
[25] Jingjun Zhang, Lili He, Ercheng Wang and Ruizhen Gao , “ Active
Vibration Control of Flexible Structures Using Piezoelectric Materials ” ,
Hebei University Of Engineering , Handan, China , pp.540-544, 2008.
[26] Varun Kumar and Deepak Chhabra , ” Design of Fuzzy Logic Controller
for Active Vibration Control of Cantilever Plate With Piezo -Patches as
Sensor/Actuator ” , International Journal of Emerging Research In
Management and Technology , pp. 34-44, 2013.
[27] Sellers and David , " An Overview of Proportional Plus Integral Plus
Derivative Control and Suggestions for Its Successful Application and
Implementation ” , McGraw-Hill , pp. 212–226, 2001.
[28] Wj .Manning , Plummerar and Levesleymc , “ Vibration Control of a
Flexible Beam with Integrated Actuators and Sensors ” , Journal of
Intelligent Material ,Smart Materials and Structures , pp. 932–939, 2000.
[29] I. Bruant , G. Coffignal , F. Lene and M. Verge, “ Active Control of Beam
Structures With Piezoelectric Actuators And Sensors:Modeling and
Simulation ” , Journal of Intelligent Material ,Smart Materials and
Structures , pp. 404-408, 2000.
REFERENCES
99
[30] P.Gaudenzi , R.Carbonaro and E.Benzi , “ Control of Beam Vibrations by
Means of Piezoelectric Devices:Theory and Experiments” , Journal of
Intelligent Material ,Composite Structures , pp. 373-379, 2000.
[31] Sp. Singh, Hs. Pruthi and Vp. Agarwal , “ Efficient Modal Control
Strategies for Active Control of Vibrations ” , Journal of Sound and
Vibration , pp. 563-575 , 2003.
[32] Xu Sx and Ts. Koko, ” Finite Element Analysis and Design of Actively
Controlled Piezoelectric Smart Structures ” , Journal of Intelligent Material
, Finite Elements In Analysis And Design , pp. 241-262, 2004.
[33] H. Karagulle, L. Malgaca and Hf. Oktem , “ Analysis of Active Vibration
Control in Smart Structures by Ansys ”, Journal of Intelligent Material
,Smart Materials And Structures , pp. 661-667, 2004.
[34] S.M. Khot, Np. Yelve and R. Iyer , “ Active Vibration Control of Cantilever
Beam by Using Optimal (LQR) Controller ” , Proceedings Of International
Conference On “Total Engineering” Analysis and Manufacturing
Technologies , Bangalore, India , pp. 22-24, 2008.
[35] Yh. Lim , “ Finite Element Simulation of Closed Loop Vibration Control of
a Smart Plate Under Transient Loading ” , Journal of Intelligent Material ,
Smart Materials And Structures , pp. 272-286, , 2003.
[36] St .Quek , Sy.Wang and Kk. Ang , “ Vibration Control of Composite
Plates Via Optimal Placement of Piezoelectric Patches ” , Journal of
Intelligent Material , Journal f Intelligent Material , Systems And Structures
, pp. 229-245, 2003.
appendix
APPENDIX
100
APPENDIX
1-MATLAB Codes used for LQR controller
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % Active Vibration Control of a Cantilever Beam Using Bonded %%
%% Piezoelectric Sensors and Actuators with LQR controller %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Eb=70e9; % young's modulus of the beam
hb=0.006; % thickness of the beam
b=0.01; % width of the beam = width of piezoelectric patches
hp=0.0005; % thickness of piezoelectric patches
Ib=(b*hb^3)/12; % second moment of area of the beam
ro=2700; % density of the beam
Ab=b*hb; % cross sectional area of the beam
lb=0.26; % length of the beam
z=0.02; % damping ratio of the beam
d31=-270e-12; % piezoelectric constant
lp=45e-3; % length of piezoelectric patches
APPENDIX
101
Ep=5.3e10; % young's modulus of piezoelectric patches
cp=65e-9; % capacitance of piezoelectric patches
g31=-9.2e-3; % piezoelectric constant
% location of piezoelectric patches is x1=0.01 , x2=0.05 if the patches
locate near fixed end and x1=0.215 , x2=0.26 if the patches locate at
free end
x1=0.215 ; % location of the piezoelectric patches
x2=x1+(lp); % location of the piezoelectric patches
w=zeros(1,4);
bl=zeros(1,4);
for i=1:4
bl(i)=((2*i)-1)*pi/2;
w(i)=(bl(i))^2*(Eb*Ib/(ro* Ab *lb^4 ))^(1/2);
end
k=[w(1,1)^2 w(1,2)^2 w(1,3)^2 w(1,4)^2];
k=-diag(k);
c=[2*z*w(1,1) 2*z*w(1,2) 2*z*w(1,3) 2*z*w(1,4)];
c=-diag(c);
p=-((Ep/Eb)*(6*hp*hb*(hp+hb))/(hb^3+8*hp^3+(6*hb*hp^2)));
kj=-(b*d31*Eb*hb^2*p)/(12*hp*(1-p));
an=zeros(1,1);
phi1=zeros(1,1);
phi2=zeros(1,1);
d=zeros(4,1);
s1=zeros(1,1);
for j=1:4
for g=1:1
APPENDIX
102
an(j)=(j*pi/lb);
s1(j)=((cos(an(j)*lb))+(cosh(an(j)*lb)))/((sin(an(j)*lb))+(sinh(an(j)*lb)));
phi1(j)=((an(j)*(sinh(an(j)*x1))+(sin(an(j)*x1))))-
((an(j)*s1(j)*(cosh(an(j)*x1))-(cos(an(j)*x1))));
phi2(j)=((an(j)*(sinh(an(j)*x2))+(sin(an(j)*x2))))-
((an(j)*s1(j)*(cosh(an(j)*x2))-(cos(an(j)*x2))));
d(j,g)=kj*(phi2(j)-phi1(j));
end
end
ks=(b*hb*g31)/(2*cp);
am=zeros(1,1);
ph1=zeros(1,1);
ph2=zeros(1,1);
M=zeros(1,4);
s2=zeros(1,1);
for n=1:1 %here we are considering only one sensor output y(1). This
%sensor extends from x1 to x2
% therefore, we will get response of this sensor only
% y is the response of the sensor (output voltage of it)
for m=1:4
am(m)=(m*pi/lb);
s2(m)=((cos(am(m)*lb))+(cosh(am(m)*lb)))/((sin(am(m)*lb))+(sinh(am(m)*
lb)));
ph1(m)=((am(m)*(sinh(am(m)*x1))+(sin(am(m)*x1))))-
((am(m)*s2(m)*(cosh(am(m)*x1))-(cos(am(m)*x1))));
ph2(m)=((am(m)*(sinh(am(m)*x2)))+(sin(am(m)*x2)))-
((am(m)*s2(m)*(cosh(am(m)*x2))-(cos(am(m)*x2))));
APPENDIX
103
M(n,m)=ks*(ph2(m)-ph1(m));
end
end
A=[0 0 0 0 1 0 0 0;0 0 0 0 0 1 0 0;0 0 0 0 0 0 1 0;0 0 0 0 0 0 0 1;k(1,1)
k(1,2) k(1,3) k(1,4) c(1,1) c(1,2) c(1,3) c(1,4);k(2,1) k(2,2) k(2,3) k(2,4)
c(2,1) c(2,2) c(2,3) c(2,4);k(3,1) k(3,2) k(3,3) k(3,4) c(3,1) c(3,2) c(3,3)
c(3,4);k(4,1) k(4,2) k(4,3) k(4,4) c(4,1) c(4,2) c(4,3) c(4,4)];
B=[0;0;0;0;d(1,1);d(2,1);d(3,1);d(4,1)];
C=[M(1,1) M(1,2) M(1,3) M(1,4) 0 0 0 0 ];
D=0;
sys_without_controller=ss(A,B,C,D);
x0=[ 0 ;0 ; 0 ; 0; 0; 0 ; 0 ;0.01];
t=0:0.1:10;
impulse(sys_without_controller,'b')%plot impulse response without
controller
% if we want the initial condition response ,we should use the command
below
% initial(sys_without_controller,'b',x0) %plot response before control in
blue color
R=1e-10; %here we want to minimize the effect of u, because u is very
large %(in quantity) as compared to x
% we can play with weights to achieve better controller design
and
% smaller response
APPENDIX
104
Q=30e1*[1 0 0 0 0 0 0 0;0 1 0 0 0 0 0 0;0 0 1 0 0 0 0 0;0 0 0 1 0 0 0 0;0 0
0 0 1 0 0 0;0 0 0 0 0 1 0 0;0 0 0 0 0 0 1 0;0 0 0 0 0 0 0 1];
KK=lqr(A,B,Q,R);
ACONT=(A-(B*KK));
sys_with_controller_LQR=ss(ACONT,B,C,D);
hold on
impulse(sys_with_controller_LQR,'r')
% if we want the initial condition response ,we should use the command
below
%initial(sys_with_controller_LQR,'r',x0) % plot response after control
title('response of cantilever beam ')
ylabel('displacement (m)')
2-MATLAB Codes used for PID controller
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% % Active Vibration Control of a Cantilever Beam Using Bonded%%%
%% Piezoelectric Sensors and Actuators with PID controller %%%
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APPENDIX
105
Eb=70e9; % young's modulus of the beam
hb=0.006; % thickness of the beam
b=0.01; % width of the beam = width of piezoelectric patches
hp=0.0005; % thickness of piezoelectric patches
Ib=(b*hb^3)/12; % second moment of area of the beam
ro=2700; % density of the beam
Ab=b*hb; % cross sectional area of the beam
lb=0.26; % length of the beam
z=0.02; % damping ratio of the beam
d31=-270e-12; % piezoelectric constant
lp=45e-3; % length of piezoelectric patches
Ep=5.3e10; % young's modulus of piezoelectric patches
cp=65e-9; % capacitance of piezoelectric patches
g31=-9.2e-3; % piezoelectric constant
% location of piezoelectric patches is x1=0.01 , x2=0.05 if the patches
locate near fixed end and x1=0.215 , x2=0.26 if the patches locate at
free end
x1=0.215 ; % location of the piezoelectric patches
x2=x1+(lp); % location of the piezoelectric patches
w=zeros(1,4);
bl=zeros(1,4);
for i=1:4
bl(i)=((2*i)-1)*pi/2;
w(i)=(bl(i))^2*(Eb*Ib/(ro* Ab *lb^4 ))^(1/2);
end
k=[w(1,1)^2 w(1,2)^2 w(1,3)^2 w(1,4)^2];
k=-diag(k);
APPENDIX
106
c=[2*z*w(1,1) 2*z*w(1,2) 2*z*w(1,3) 2*z*w(1,4)];
c=-diag(c);
p=-((Ep/Eb)*(6*hp*hb*(hp+hb))/(hb^3+8*hp^3+(6*hb*hp^2)));
kj=-(b*d31*Eb*hb^2*p)/(12*hp*(1-p));
an=zeros(1,1);
phi1=zeros(1,1);
phi2=zeros(1,1);
d=zeros(4,1);
s1=zeros(1,1);
for j=1:4
for g=1:1
an(j)=(j*pi/lb);
s1(j)=((cos(an(j)*lb))+(cosh(an(j)*lb)))/((sin(an(j)*lb))+(sinh(an(j)*lb)));
phi1(j)=((an(j)*(sinh(an(j)*x1))+(sin(an(j)*x1))))-
((an(j)*s1(j)*(cosh(an(j)*x1))-(cos(an(j)*x1))));
phi2(j)=((an(j)*(sinh(an(j)*x2))+(sin(an(j)*x2))))-
((an(j)*s1(j)*(cosh(an(j)*x2))-(cos(an(j)*x2))));
d(j,g)=kj*(phi2(j)-phi1(j));
end
end
ks=(b*hb*g31)/(2*cp);
am=zeros(1,1);
ph1=zeros(1,1);
ph2=zeros(1,1);
M=zeros(1,4);
s2=zeros(1,1);
APPENDIX
107
for n=1:1 %here we are considering only one sensor output y(1). This
%sensor extends from x1 to x2
% therefore, we will get response of this sensor only
% y is the response of the sensor (output voltage of it)
for m=1:4
am(m)=(m*pi/lb);
s2(m)=((cos(am(m)*lb))+(cosh(am(m)*lb)))/((sin(am(m)*lb))+(sinh(am(m)*
lb)));
ph1(m)=((am(m)*(sinh(am(m)*x1))+(sin(am(m)*x1))))-
((am(m)*s2(m)*(cosh(am(m)*x1))-(cos(am(m)*x1))));
ph2(m)=((am(m)*(sinh(am(m)*x2)))+(sin(am(m)*x2)))-
((am(m)*s2(m)*(cosh(am(m)*x2))-(cos(am(m)*x2))));
M(n,m)=ks*(ph2(m)-ph1(m));
end
end
A=[0 0 0 0 1 0 0 0;0 0 0 0 0 1 0 0;0 0 0 0 0 0 1 0;0 0 0 0 0 0 0 1;k(1,1)
k(1,2) k(1,3) k(1,4) c(1,1) c(1,2) c(1,3) c(1,4);k(2,1) k(2,2) k(2,3) k(2,4)
c(2,1) c(2,2) c(2,3) c(2,4);k(3,1) k(3,2) k(3,3) k(3,4) c(3,1) c(3,2) c(3,3)
c(3,4);k(4,1) k(4,2) k(4,3) k(4,4) c(4,1) c(4,2) c(4,3) c(4,4)];
B=[0;0;0;0;d(1,1);d(2,1);d(3,1);d(4,1)];
C=[M(1,1) M(1,2) M(1,3) M(1,4) 0 0 0 0 ];
D=0;
sys_without_controller=ss(A,B,C,D);
x0=[ 0 ;0 ; 0 ; 0; 0; 0 ; 0 ;0.01];
t=0:0.1:10;
impulse(sys_without_controller,'b') % plot impulse response without
controller
APPENDIX
108
sys_without_controller_ss=ss(A,B,C,D);
[num,den]=ss2tf(A,B,C,D);
sys_without_controller=tf(num,den);
% the value of PID controller constants which determined using MATLAB
% are:
kp=0.0002 ;
ki=0.03;
kd=0.004;
% if we used the values of the constant determined manually then
% kp=1e-4 ;
% ki=1e-1;
% kd=1e-1;
Controller = tf([kp,ki,kd],[1,0]);
sys_with_controller=feedback(Controller*sys_without_controller,1);
impulse(sys_without_controller,'r',t)
hold on
impulse(sys_with_controller,'b',t)
title('response of cantilever beam ')
ylabel('displacement (m)')
حقاثنؼخبت ركت باعخخذاو فؼال اخضاصحصى يغطش
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اناخغخش دسخت كدضء ي يخطهباث م
انذعت انكاكت ف
)انكاك انخطبق(
باحقذو
ححغ اشى صانح
تكاكانذعت ان ػهو ف بكانسط
4102
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حى اعخؼال ػخبت ركت يصػت ي االنو يثبج انشعانتف ز
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ف انبذات، حى اداد انخشدداث انطبؼت االعاعت االسبؼت يغ ااط .يشغم
. ) Ansys حى انخحقق يا باعخخذاو بشايح) ثىاالخضاص نهؼخبت انزكت
بؼذا حى اداد افضم يقغ نخثبج انهاصق انكشضغطت ػه انؼخبت
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باعخخذاو ي اخم انغطشة ػه اخضاص انؼخبت انزكت انضبابانطق
.حى ػم يقاست ب االاع انثالثت ي (MATLAB)بشايح
انضباب ع انطقانحث خذ ا انخحكى ي انغخخذيت انخحكاث
افضم االاع اناعبت نالعخخذاو ف زا انخطبق حث ا ؼط
حث ا غبت حخذ قت االخضاص اناحح اعخدابت عشؼت يغخقشة فاػهت
( كا صي 51.9انخحكاث )%زا انع ي باعخخذاو
( ثات 1.5 51.2 ثات( با كاج انخائح )% 1.5االعخقشاس)
-انخاظشثات ( باعخخذاو انخحكاث ي ع 1.415 25.99)%
حى بؼذ رنك اخشاء .ػه انخان انطق انضباب انخفاضه -انخكايه
ف يصت انذساعت اخش ف ز انصتيقاست ب انخحكاث
ف ز انذساعت افضم ي انصتدساعاث اخش خذ ا انخحكاث
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