DESIGN OF ACTIVE VIBRATION CONTROLLER OF A SMART BEAM USING LQR , PID AND FUZZY LOGIC TECHNIQUES

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


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


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



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



near fixed end using PID controller

63

4.20 Impulse response of cantilever beam when the patches at

free end with PID controller

64


when piezoelectric patches at free end using PID controller

65



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


when piezoelectric patches near fixed end using PID

controller MATLAB tuning

68



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


when piezoelectric patches at free end using PID controller

MATLAB tuning

70

LIST OF FIGURES

XII



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


subjected to initial conditions when the patches at free


53


subjected to impulse force when the patches near fixed


56


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


(PID)controller when the patches at free end

64


(PID)controller when the patches near fixed end matlab

tuning

67


(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


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


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


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


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


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].


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.


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.


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


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,


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.


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


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.


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


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.


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


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


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].


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 :


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]:

( ( ) ( )) ( ( ) ( )) ( )


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] :


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]:


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

http://en.wikipedia.org/wiki/Dynamical_system

http://en.wikipedia.org/wiki/State_variable


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


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 ,

http://en.wikipedia.org/wiki/Vibration

http://en.wikipedia.org/wiki/Machinery

http://en.wikipedia.org/wiki/Vibration_control


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 :

http://en.wikipedia.org/wiki/Shock_absorber

http://en.wikipedia.org/wiki/Shock_absorber

http://en.wikipedia.org/wiki/Mass

http://en.wikipedia.org/wiki/Voice_coil

http://en.wikipedia.org/wiki/Voice_coil

http://en.wikipedia.org/wiki/Piezo-electric

http://en.wikipedia.org/wiki/Accelerometer

http://en.wikipedia.org/wiki/Amplifier

http://en.wikipedia.org/wiki/Amplifier

http://en.wiktionary.org/wiki/invert

http://en.wikipedia.org/wiki/Signal_(information_theory)

http://en.wikipedia.org/wiki/PID_controller

http://en.wikipedia.org/wiki/Frequency


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


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 :


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 :

( ) ( ) ( )

( )


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.


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.

http://en.wikipedia.org/wiki/Control_loop

http://en.wikipedia.org/wiki/Industrial_control_system

http://en.wikipedia.org/wiki/Control_valve

http://en.wikipedia.org/wiki/Damper_(flow)


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]:

http://en.wikipedia.org/wiki/Optimal_control


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,


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].


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.


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]:


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) .


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.


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


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].


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


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


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


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.


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 .


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


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:


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


- 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.


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 .


piezoelectric patches at free end are given in table (4.5) below.


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


controlled peak amplitude 8.44x10

-5 m



amplitude 20.3% -



54


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


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.


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.




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


-6 m



amplitude 49.65% -



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.


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.


58



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


-6 m



-6 m



amplitude 32.09% -




piezoelectric patches at free end using LQR controller.

Figure ( 4.14) bode plot of the cantilever beam subjected to impulse force when piezoelectric



59


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.


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


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.


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.


uncontrolled


m


controlled


m



amplitude 80.93% -




piezoelectric patches near fixed end using PID controller.


63

Figure (4.18 ) bode plot of the cantilever beam subjected to impulse force when piezoelectric

patches near fixed end using PID controller.


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) :


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.


uncontrolled


m



-6 m



amplitude 60.15% -



65



piezoelectric patches at free end using PID controller.


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.


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.


67

Figure (4.24) impulse response of cantilever beam when the patches near fixed end with (PID

parameter tuning with matlab ) controller.


beam when the piezoelectric patches near fixed end are given in table (4.10)

below:


patches near fixed end MATLAB auto tuning.

cantilever

beam property value unit


-6 m



-6 m



amplitude 90.4% -



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.


patches near fixed end using PID controller MATLAB auto tuning.


impulse force when piezoelectric patches near fixed end using PID controller MATLAB auto

tuning.


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.


beam when the piezoelectric patches at free end are given in table (4.11) below:


70


patches at free end MATLAB auto tuning.

cantilever

beam

property value unit


-6 m



-6 m



amplitude 88.4% -


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.


patches at free end using PID controller MATLAB auto tuning.


71


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) :


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)


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)


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)


75

Figure (4.30) fuzzy logic controller page in MATLAB

Figure (4.31) the input displacement member function of cantilever beam


76

Figure (4.32) the input velocity member function of cantilever beam

Figure (4.33) the output controlled displacement member ship of cantilever beam


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


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


79

Figure (4.38) inputs displacement and velocity of unit step

Figure (4.39) output controlled displacement when the inputs are unit steps


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.


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


2.48 80.93%


1.02 60.15%

PID-

MATLAB

auto

tuning


0.9 90.4%


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%


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


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


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).


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):


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] .


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.


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 :


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].


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.


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 .


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))));



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)))-


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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% % Active Vibration Control of a Cantilever Beam Using Bonded%%%

%% Piezoelectric Sensors and Actuators with PID controller %%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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)));





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))))-


ph2(m)=((am(m)*(sinh(am(m)*x2)))+(sin(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

حفاضه -حكايه -حاظش انخط انظى انخشبؼ (

ضباب(ان انطق

يقذيت ان سعانت

خايؼت انبصشة - كهت انذعت

اناخغخش دسخت كدضء ي يخطهباث م

انذعت انكاكت ف

)انكاك انخطبق(

باحقذو

ححغ اشى صانح

تكاكانذعت ان ػهو ف بكانسط

4102

انغخخهص

حى اعخؼال ػخبت ركت يصػت ي االنو يثبج انشعانتف ز

ػها صج ي انهاصق انكشضغطت انخ اعخؼهج كخحغظ

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

خضاص الانزكت ي اخم انحصل ػه افضم اداء نهغطشة انفؼانت ػه ا

ز انؼخبت، حى بؼذا اعخخذاو ثالثت ااع ي انخحكاث انفؼانت ف

انخحكى انخفاضه -انخكايه -انخاظشانخحكى انظى انخشبؼ انخط

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

ف انذساعاث انصتاالعخقشاس ي حهك االعخدابتحث االداء عشػت

. انزكسة اػال

Documents

DESIGN OF ACTIVE VIBRATION CONTROLLER OF A SMART BEAM USING LQR , PID AND FUZZY LOGIC TECHNIQUES