active vibrration control of beam

ACTIVE VIBRATION CONTROL OF A SMART

BEAM USING LQR , PID AND FUZZY LOGIC

CONTROLLERS

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

April ,2014

ACKNOWLEDGMENT

I

ACKNOWLEDGMENT

I would like to give me 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 .Dr. Rabeea

H. Thjeel, the dean of the college, and for Prof. Dr. Amen A.

Nassar, the Head of Mechanical Engineering Department, for

their appreciated support and assistance to me in order to

complete my study. The author also wants to thanks all the

staff of the Mechanical Engineering Department for their

facilitating , support and encouragement in order to complete

this Thesis.

ABSTRACT

III

ABSTRACT

Theoretical model of a cantilever beam bonded with 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 verified

using ANSYS. The optimal 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 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. A comparison with the

results of other studies [18] , [19] and [24] showed that the controllers designed

in this study gave better performance for the same case studies.

CONTENTS

IV

CONTENTS

Title Page

Acknowledgment I

Abstract II

Contents III

Abbreviations VI

Symbols VII

List of Figures IX

List of Tables XII

Chapter one : Introduction 1

1.1Background 1

1.2 Smart Structures 2

1.3 Aim of this Thesis 3

1.4 Structure and layout of this Thesis 3

Chapter two : literature review 5

2.1 Introduction 5

2.2 Summary of literature review 14

Chapter three : Mathematical Modeling and Active

Control Techniques

16

3.1 Introduction 16

3.2 Mathematical model of the cantilever beam 17

CONTENTS

V

3.3 State space representation of the cantilever beam 23

3.4 Active vibration control 26

3.5 Linear Quadratic Regulator (LQR) State Feedback Design 27

3.6 Proportional – Integral –Derivative Controller (PID) 32

3.6.1 PID controller theory 33

3.6.2 Loop tuning 34

3.6.3 Stability 34

3.6.4 Manual tuning 35

3.6.5 PID Controller auto tuning 36

3.7 fuzzy logic controller(FLC) 37

3.7.1 Fuzzification 38

3.7.2 The rule of fuzzy control and the FLC 39

3.7.3 Defuzzification 41

chapter four : computer simulation using matlab, results and discussion 42

4.1 Introduction 42

4.2 mode shapes and natural frequencies of the cantilever beam 45

4.3 active vibration control using (LQR) controller 47

4.3.1 response of cantilever beam to initial condition 47

4.3.2 impulse response of cantilever beam 52

4.4 active vibration control using (PID) controller 56

4.4.1 Manually determination of (PID) controller constants 56

4.4.2 determination of (PID) controller constants using auto tuning 61

4.5 Active vibration control using (FL) controller 66

CONTENTS

VI

4.6 The optimal location of the piezoelectric patches 77

4.7 Results and discussion 79

4.8 comparison between the current research and other researches 81

chapter five : conclusion and recommendations for future works 82

5.1 Conclusion of this Thesis 82

5.2 Recommendations for future works 84

References

Appendix

ABBREVIATIONS

VII

ABBREVIATIONS

Term Description DISO Double-input, single-out

FE Finite element

FL CONTROLLER Fuzzy logic 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

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

3.1 Smart cantilever beam with piezoelectric patches 18

3.2 LQR block diagram controller 31

3.3 Block diagram of PID controller 33

3.4 The basic configuration of fuzzy logic system 39

4.1 The first four mode shapes of the cantilever beam 46

4.2 Response of the cantilever beam to initial condition when

piezoelectric patches near fixed end with LQR control

48

4.3 Bode plot of the cantilever beam subjected to initial

condition when piezoelectric patches near fixed end with

(LQR)controller

49

4.4 Rlocus plot of the controlled and uncontrolled cantilever

beam subjected to initial condition when piezoelectric

patches near fixed end with LQR controller

49

4.5 Response of the cantilever beam with initial condition when

piezoelectric patches at free end with LQR controller

50

4.6 Bode plot of the cantilever beam subjected to initial

condition when piezoelectric patches at free end with LQR

controller

51


beam subjected to initial condition when piezoelectric

patches at free end with LQR controller

51

4.8 Impulse response of the cantilever beam when piezoelectric

patches near fixed end with LQR controller

52

4.9 Bode plot of the cantilever beam subjected impulse force

when piezoelectric patches near fixed end with LQR

controller

53


beam subjected to impulse force when piezoelectric patches

near fixed end with LQR controller

54

4.11 Impulse response of the cantilever beam when piezoelectric

patches at free end with LQR controller

54

4.12 Bode plot of the cantilever beam subjected to impulse force

when piezoelectric patches at free end with LQR controller

55



at free end with LQR controller

56

LIST OF FIGURES

XI

4.14 Impulse response of cantilever beam when the patches near

fixed end with PID controller

57

4.15 Bode plot of the cantilever beam subjected to impulse force

when piezoelectric patches near fixed end using PID

controller

58



near fixed end using PID controller

59

4.17 Impulse response of cantilever beam when the patches at

free end with PID controller

59


when piezoelectric patches at free end using PID controller

60



at free end using PID controller

61

4.20 MATLAB (PID )tuner 62

4.21 Impulse response of cantilever beam when the patches near

fixed end with (PID parameter tuning with matlab )

controller

62


when piezoelectric patches near fixed end using PID

controller MATLAB tuning

63



near fixed end using PID controller MATLAB tuning

64

4.24 Impulse response of cantilever beam when the patches at

free end with PID controller MATLAB tuning

64


when piezoelectric patches at free end using PID controller

MATLAB tuning

65



at free end using PID controller MATLAB tuning

66

4.27 Fuzzy logic controller page in MATLAB 70

4.28 The input displacement member function of cantilever beam 70

4.29 The input velocity member function of cantilever beam 71

4.30 The output controlled displacement member ship of

cantilever beam

71

4.31 Rule viewer of the designed FLC 72

LIST OF FIGURES

XII

4.32 Surface viewer of the designed FLC 73

4.33 Inputs displacement and velocity of uniform random number 74

4.34 Output controlled displacement when the inputs are

uniform random number

74

4.35 Inputs displacement and velocity of sine wave 75

4.36 Output controlled displacement when the input is sine wave 75

4.37 Inputs displacement and velocity of unit step 76

4.38 Output controlled displacement when the inputs are unit

steps

76

4.39 The Simulink page of FLC in MATLAB 77

4.40 The optimal location of piezoelectric patches on the

cantilever beam

79

4.41 Response of cantilever beam- different type of controller at

optimal Location

80

4.42 Comparison between Ref.[18] and current study for LQR

controller

82

4.43 Comparison between Ref. [19] and current study for PID

controller

83

4.44 Comparison between Ref [24] and current study for FL

controller

84

LIST OF TABLES

XIII

LIST OF TABLES

Table No. Table Title Page

3.1 Effects of increasing PID parameters independently 36

3.2 Fuzzy logic controller rule base 40

4.1 Properties of cantilever beam 43

4.2 Properties of piezoelectric patches (sensor and actuator) 44

4.3 Natural frequencies of the cantilever beam 45

4.4 Properties of controlled and uncontrolled cantilever beam

subjected to initial conditions when the patches near fixed

end with LQR controller

48


subjected to initial conditions when the patches at free


50


subjected to impulse force when the patches near fixed


53


subjected to impulse force when the patches at free end

with LQR controller

55

4.8 Properties of impulse response of cantilever beam with

PID controller when the patches near fixed end

58


(PID)controller when the patches at free end

60


(PID)controller when the patches near fixed end matlab

tuning

63


(PID)controller when the patches at free end matlab

tuning

65

4.12 FLC expanded rules 67

4.13 Effect of piezoelectric patches location on the beam

response

78

Chapter

one

CHAPTER ONE INTRODUCTION

1

CHAPTER ONE

INTRODUCTION

1.1Background

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.


2

1.2 Smart Structures

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 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.


3

1.3 Aim of this 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 optimal 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.

1.4 Structure and layout of this Thesis

This thesis consists of five chapters , chapter one give an introduce to the

active vibration control of smart beam . Chapter two discuss the literature of the

past researcher whose researches were dedicated to the control of the smart

beam with many different types of controller .

Chapter three introduce the methodology of getting the mathematical model of

the smart beam and discuss the active vibration control techniques .


4

Chapter four discuss the computerized modeling and simulation of the smart

beam using MATLAB software and introduce a discussion of the results

obtaining in this chapter.

Finally , chapter five is dedicated to discuss the conclusions from this thesis and

give the recommendation for futures works.

Chapter

two

CHAPTER TWO LITERATURE REVIEW

5

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


6

affecting significantly its passive behavior. Some recent works are reported here.

The active vibration control of simple cantilever beams is studied by many

researchers. Piezoelectric patches as actuators are mounted on the beams. The

system identification and pole placement control method is used by[1] . The

beam with piezo-patches Finite Element (FE) model of the structure is

constructed and the closed loop control is applied by [2] and [3]. Ref.[ 4] also

used the beam with piezo-patches FE model, but applied modal control

strategies. Ref.[5] reported results on active vibration control of cantilever beam

type of structures by using the commercial FE package ANSYS. The influence

of sensor/actuator location is studied for a cantilever type beam. Ref.[6] extended

the work of Ref.[5] and proposed the procedure for the simulation of active

vibration control in ANSYS, for cantilever and plate type of structures. Ref.[7]

used a reduced model of a cantilever beam to design the optimal controller using

Linear Quadratic Regulator (LQR) algorithm with state feedback control law.

Ref.[ 8] studied the vibration control of several modes of a clamped square plate

by locating discrete sensor/actuator devices at points of maximum strain. Ref.[ 9]

presented an optimal placement strategy of piezoelectric sensor/actuator pairs for

the vibration control of laminated composite plates. Ref.[10] studied active

vibration control in a four-bar linkage. Numerical simulations are reported in all

the references given. Experimental results are also reported in some studies

refs.[1], [3] and [5]. In the present study piezoelectric patches are used as both

sensor and actuator to control the vibration of a cantilever beam. The controller

used here is a proportional- integral-derivative (PID) based output feedback

controller. The state space mathematical model required for designing the

controller in MATLAB is extracted from the results of modal analysis of a

cantilever beam done in ANSYS. Both full and reduced models of the cantilever


7

beam are used for the analysis and their responses are obtained in SIMULINK

environment.

Piezoelectric sensing and control with distributed piezoelectric transducers

have been intensively studied, e.g. [11], [12], [13]. Application of piezoelectric

materials in active structural control requires appropriate simulation and design

tools ( such Matlab , Labview , Ansys ,etc.) , Vibration control of cantilever

beam based on the piezoelectric smart component has become the hot issue in

the research of vibration engineering A numerical study concerning the active

vibration control of smart piezoelectric beams is presented by many researchers

as some of their researches are mentioned below:

In (2001) ,Yavuz Yaman , et al ,[13], investigates 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 .

In( 2002), Ulrich Gabbert, et al, [14], introduce 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


8

vibration suppression discrete-time control design tools were used, such as

optimal LQR controller incorporated in a tracking system .

In (2007), Gou Xinke and Tian Haimin ,[11], 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 this 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..

In (2008) , Zhang Jing-jun , et al , [15] , 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


9

control method, robust H∞ control has strong robustness to modal parameters

variation and has a good closed-loop dynamic performance.

In (2010), Dong Jingshi ,et al , [16] , 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.

Also in (2010), K. B. Waghulde ,et al ,[17], 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.

In (2011), S.M. Khot ,et al ,[18], used reduced model for cantilever beam and

the design of optimal controller is achieved using Linear Quadratic Regulator


10

(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 .

Also in (2011) , Deepak Chhabra , et al ,[19], 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.

Also in(2011) , S.M. Khot , et al ,[20], dealt with the extraction of the full and

reduced mathematical models of a cantilever beam into MATLAB from its FE

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


11

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.

In(2012), Deepak Chhabra ,et al ,[21], 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.

Also in (2012), Tamara Nestorović , et al ,[22] , introduced the concept of an

active vibration control for piezoelectric light weight structures and presented

through several subsequent steps: model identification, controller design,


12

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.

In (2013) , A.P. Parameswaran , et al ,[23], 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

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


13

real time operating system (RTOS) platform wherein deterministic and reliable

control could be achieved.

Also in (2013 ) , Preeti Verma , et al ,[24] , 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.


14

2.2 Summary of literature review

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 , 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.


15

2. There was no sufficient work done on the active vibration control using

fuzzy logic controller method.

3. 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

16

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 modelling 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.


17

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 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


18

considered as a Bernoulli-Euler beam , the governing equations of motion and

associated boundary conditions are derived as follows [11]:

( ) ( ) ( )

( ) ( )

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


19

The total electric charge of the i-th piezoelectric sensor can be obtained as

follows [11]:

( ) ∫ ( )

( )

( )

( )

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 [11]

( )

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 [11]

( ) ∑ [ ( ) ( ) ] ( )

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 :


20

( )

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 [11]:

( ) ∑ ( ) ( )

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 [25]:

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


21

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[11]:

∑ ( j ) ( j)

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 [11] :


22

( ) ( ) ( ) ( ) ( )

( ) ( )

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 matri of the sensor’s output .

And they are given as[11] :

( i , j) j ( j ) ( j) i , , j , , ,m ( )

And

(i , ) si ( i ) ( i) i , , , r , , , ( )

Where

N :total number of mode used.

Ksi : constant given as(V.m3/N.F) [15]:


23

( )

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 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

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

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


24

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[26].

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 [26]:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

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 :feedforward 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


25

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

by applying the state space representation procedure on cantilever beam ,

equations (3.12)and (3.13) can be rewritten in state form as :

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

Where

A :the system matrix.

B :the electrical input matrix.

E :the output matrix.

F :feedforward matrix.

u :the electrical input vector.

y :the output vector.

and matrices are given by [11] :

[

] ( )

[

] ( )

( )

F = [0] (3.28)


26

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 ,

passive techniques were used. These include traditional vibration dampers, shock

absorbers, and base isolation.

The typical active vibration control system uses several components:

A massive platform suspended by several active drivers (that may use voice

coils, hydraulics, pneumatics, piezoelectric or other techniques)

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.

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

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

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

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

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


27

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 − (3.29)

that gives desirable closed-loop properties. The closed-loop system using this

control becomes :

( ) ( )

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

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


28

∫ ( )

( )

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

(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


29

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 :

( ) ( ) ( )

Then, substituting into equation (3.31) 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.


30

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 :

( ) ( ) ( )

( )

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 ( ) ( )

( )

( ) ( )

( )


31

This result is of extreme importance in modern control theory. Equation (3.41) is

nown as the “ algebraic iccati equation “(A E) It is a matri 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.

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.

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


32

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.

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[27].

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

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

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

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

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

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

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

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


33

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:

( ) ( ) ( ) ∫ ( )

( ) ( )

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).


34

In this study a design of PID controller with the help of auto tuning in

MATLAB software.

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, 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


35

unstable [27], 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 .

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 [28].


36

Table (3.1) Effects of increasing PID parameters independently.

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 [29] ,

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.


37

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 [30], 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 logical controller is

made of four main components [31]:

1. Fuzzification.

2. Rule base.

3. Decision making .

4. Defuzzification.


38

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 when received step response.

( ) lot the scopes of displacement’s and the force of control’s out (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 (Negative Big) , NS(Negative Small), ZO (ZerO) ,

PS(Positive Small) , PB(Positive Big) .

(3) According to the fuzzy rule of [32], the process of fuzzy illation can be

determined.

(4) At last, using the way of MOM method to calculate in order to obtain the

result.


39

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.


40

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.

Ref.[35] 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


41

Where

NB :Negative Big.

NM :Negative Medium.

NS :Negative Small.

ZO :ZerO.

PS :Positive Small.

PM :Positive Medium.

PB :Positive Big.

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, RESULTS AND DISCUSSION

42

CHAPTER FOUR

COMPUTER SIMULATION USING MATLAB, RESULTS AND

DISCUSSION

4.1 Introduction

This chapter covers the simulation of vibration control of fixed – free aluminum

cantilever beam using MATLAB . firstly , the formula described in [25] 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 used in this

study is shown in table (4.1) and the properties of piezoelectric patches (sensor

and actuator ) which are used in this chapter is shown in table (4.2) below , 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 , 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).


43

Table (4.1) properties of cantilever beam.

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 6x10

-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

-14 m

4

mass of the beam m 4.21 g


44

Table (4.2) properties of piezoelectric patches (sensor and actuator).

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


45

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 [25]

( ) √

( )

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 frequency(rad/s)

1st mode

32.1

2nd

mode

289.7

3rd

mode

804.7

4th

mode

1577.2


46

Figure (4.1) below shows the first four mode shapes of the cantilever beam using

ANSYS software.

Figure (4.1) The first four mode shapes of the cantilever beam.


47

4.3 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 :

[ ]

4.3.1 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 (4.2 ) below :


48

Figure (4.2 ) 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:

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

controlled peak amplitude

-5.48x10-5

m

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 .


49

Figure (4.3 ) bode plot of the cantilever beam subjected to initial condition when piezoelectric

patches near fixed end with LQR controller.

Figure (4.4 ) rlocus plot of the controlled and uncontrolled cantilever beam subjected to initial

condition when piezoelectric patches near fixed end with LQR controller.


50

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 ( 4.5) below :

Figure ( 4.5) response of the cantilever beam with initial condition when piezoelectric

patches at free end with LQR controller.


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

Table (4.5) properties of controlled and uncontrolled cantilever beam subjected to initial

conditions when the patches at free end with LQR controller.



10.6x10-5

m


8.44x10-5

m


51


uncontrolled cantilever beam when the beam subjected to initial condition and

the piezoelectric patches at free end with LQR controller .

Figure (4.6 ) bode plot of the cantilever beam subjected to initial condition when piezoelectric


Figure (4.7 ) rlocus plot of the controlled and uncontrolled cantilever beam subjected to initial

condition when piezoelectric patches at free end with LQR controller.


52

4.3.2 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 ( 4.8) below :

Figure (4.8 ) impulse response of the cantilever beam when piezoelectric patches near fixed

end with LQR controller.




53

Table (4.6) properties of controlled and uncontrolled cantilever beam subjected to impulse

force when the patches near fixed end with LQR controller



2.9x10-6

m

settling time 5.4 sec


1.46x10-6

m



uncontrolled cantilever beam when the beam subjected to impulse force and the

piezoelectric patches near fixed end with LQR controller.

Figure (4.9 ) bode plot of the cantilever beam subjected impulse force when piezoelectric

patches near fixed end with LQR controller.


54

Figure (4.10 ) 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 (

4.11) below :

Figure (4.11 ) impulse response of the cantilever beam when piezoelectric patches at free end

with LQR controller.


55



Table (4.7) properties of controlled and uncontrolled cantilever beam subjected to impulse

force when the patches at free end with LQR controller.



4.05x10-6

m



2.75x10-6

m




piezoelectric patches at free end using LQR controller.

Figure ( 4.12) bode plot of the cantilever beam subjected to impulse force when piezoelectric



56


impulse force when piezoelectric patches at free end with LQR controller.

4.4 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 , there are two method used in order to

determine the value of the three controller (Kp , Ki and Kd) which are described

below.

4.4.1 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 :


57

The figure( 4.14) 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 (4.14) impulse response of cantilever beam when the patches near fixed end with PID

controller.

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:


58

Table (4.8) properties of impulse response of cantilever beam with PID controller when the

patches near fixed end.



2.45x10-6

m



0.467x10-6

m




piezoelectric patches near fixed end using PID controller.

Figure (4.15 ) bode plot of the cantilever beam subjected to impulse force when piezoelectric

patches near fixed end using PID controller.


59


impulse force when piezoelectric patches near fixed end using PID controller.

The figure(4.17) 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) :

Figure (4.17) impulse response of cantilever beam when the patches at free end with PID

controller.


60

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.



2.61x10-6

m



1.04x10-6

m




piezoelectric patches at free end using PID controller.


patches at free end using PID controller.


61

Figure ( 4.19) rlocus plot of the controlled and uncontrolled cantilever beam subjected to

impulse force when piezoelectric patches at free end using PID controller.

4.4.2 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 (4.20) 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 :


62

Figure( 4.21) 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 (4.20) MATLAB PID tuner.

Figure (4.21) impulse response of cantilever beam when the patches near fixed end with (PID

parameter tuning with matlab ) controller.


63


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

below:


patches near fixed end MATLAB auto tuning.



2.45x10-6

m



0.234x10-6

m


Figures (4.22,4.23) 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.


64


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

tuning.

Figure(4.24) 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 (4.24) impulse response of cantilever beam when the patches at free end with PID

controller MATLAB auto tuning.


65


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


patches at free end MATLAB auto tuning.



2.6x10-6

m



0.3x10-6

m


Figures (4.25,4.26) 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 free end using PID controller MATLAB

auto tuning.


66


patches at free end using PID controller MATLAB auto tuning.


impulse force when piezoelectric patches at free end using PID controller MATLAB auto

tuning.

4.5 Active vibration control using (FL) controller

In this section ,the 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. (4.26) 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. (4.27) and (4.28) shows the displacement and

velocity ranges respectively ,while Fig.(4.29 ) 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) :


67

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)


68

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)


69

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)


70

Figure (4.27) fuzzy logic controller page in MATLAB

Figure (4.28) the input displacement member function of cantilever beam


71

Figure (4.29) the input velocity member function of cantilever beam

Figure (4.30) the output controlled displacement member ship of cantilever beam


72

The rule viewer and the rule surface of the designed (FLC) are shown below in

figure (4.30) and (4.31) respectively .

Figure (4.31) rule viewer of the designed FLC


73

Figure (4.32) surface viewer of the designed FLC

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 (uniform

random number, sine wave and unit step) , (4.33,4.34,4.35,4.36,4.37 and 4.38)

below shows each type of inputs and the response (output) of the controller to

these different types of inputs while Fig. (3.37) shows the Simulink page for the

FLC in MATLAB .


74

Figure (4.33) inputs displacement and velocity of uniform random number

Figure (4.34) output controlled displacement when the inputs are uniform random number


75

Figure (4.35) inputs displacement and velocity of sine wave

Figure (4.36)output controlled displacement when the inputs are sine wave


76

Figure (4.37) inputs displacement and velocity of unit step

Figure (4.38) output controlled displacement when the inputs are unit steps


77

Figure (4.39) the Simulink page of FLC in MATLAB

4.6 The optimal location of the piezoelectric patches

The optimal 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. In order to obtain this location and get a robust and fast controller , by

examine the effect of the location on each controller ( i.e. LQR controller , PID

controller ,and FL controller ) . Table (4.13) 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 optimal location for the

patches locate near fixed end (given better reduction in displacement amplitude

with relatively decent settle time ). Fig. (4.40) shows the optimal location of the

patches on the cantilever beam.


78

Table (4.13) 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)

LQR

impulse force near fixed end 1.46x10-6

0.209

impulse force at free end 2.75x10-6

0.19

initial

condition near fixed end -5.48x10

-5 0.35

initial

condition at free end 8.44x10

-5 0.23

PID


2.48


1.02

PID-

MATLAB

auto

TUNING


0.9


0.637

FLC

uniform

random

number

near fixed end 0.5x10-6

1.3

uniform

random

number

at free end 0.7x10-6

0.9

sine wave near fixed end 0.71x10-6

2.01

sine wave at free end 0.82x10-6

1.7


79

Figure (4.40) the optimal location of piezoelectric patches on the cantilever beam

4.7 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 (4.41) below ,


80

Figure (4.41) response of cantilever beam –different type of controller at optimal

Location.

It is obvious from figure (4.41) and table (4.13) 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

smaller settle time which give the desired properties to control the vibration of

the cantilever beam , PID-MATLAB auto tuning also provide acceptable

performance.


81

In figures(4.22 and 4.23) which showing the bode plot of the controlled with

PID-MATLAB auto tuning and uncontrolled system , it can be noticed that the

original system(uncontrolled ) and the controlled system are at the same phase

( there were no change in phase between input and output ) with less

amplitude(dB) for the controlled system which giving as the optimal vibration

control of the system and the same results also can be realized from

figures(4.3, 4.4, 4.6, 4.7, 4.9,4.10, 4.12, 4.13, 4.15, 4.16, 4.18, 4.19, 4.25 and

4.26).

Also in Figures (4.32 , 4.34 and 4.36) 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 .

4.8 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 Ref. [18] are shown below in figure (4.42):


82

From Ref.[18] current study

Figure (4.42) comparison between Ref. [18] and current study for LQR

controller.

It is obvious that our designed LQR controller is much better performing than

the controller in Ref. [18] since the amplitude of the output in our controller is

(-5.48*10-6

m) while in Ref. [18] is (-6*10-6

m) and the settle time in the current

study is (0.18 sec ) while in Ref. [18] is (20 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 Ref. [19] are shown in figure (4.43) below:


83

Current study Ref. [19]

Figure (4.43)comparison between Ref [19] and current study for PID

controller.

The response of the current PID controller give us an output peak amplitude of

(0.234*10-6

m) with settle time of (0.9 sec ) while in Ref. [19] the output peak

amplitude is (4*10-6

m ) therefore the current study gives much better controller

design than that of Ref. [19].

The last comparison will be made to compare the performance of the fuzzy logic

controller FLC of the current study with the controller from Ref.[24], figure

(4.44) shows the response of the cantilever beam of both FLC of Ref.[24] and

FLC of the current study to the impulse force :


84

Current study Ref. [24]

Figure (4.44)comparison between Ref. [24] and current study for FL

controller.

Again , the designed FLC of the current give us a peak amplitude of the output

of (0.6*10-6

m) while the FLC controller from Ref.[24] give a peak amplitude of

the output of (4*10-6

m) , therefore , our designed controller is better than the

controller of Ref.[24] .

Chapter

five

CHAPTER FIVE CONCLUSION AND RECOMMENDATIONS FOR FUTURE WORKS

85

CHAPTER FIVE

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.

The conclusions of this study can be summarized into the following points:

1. Wide range of Active controllers can be applied in order to achieve the

desired response of the cantilever beam .


86

2. Active vibration control using (LQR , PID and FL) controllers give strong

robustness to modal parameters variation and has a good closed-loop

dynamic performance compared to (H∞) controller .

3. 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.

4. 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.

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.


87

3. An experimental study of active vibration control of a cantilever beam is

extremely recommended to verifying the theoretical results.

references

REFERENCES

REFERENCES

[1] Manning Wj, Plummerar and Levesleymc, (2000),”Vibration Control of A

Flexible Beam with Integrated Actuators and Sensors”. Smart Materials and

Structures 9: Pp 932–939.

[2] Bruant I, Coffignal G, Lene F And Verge M, (2000),”Active Control Of

Beam Structures With Piezoelectric Actuators And Sensors: Modeling

And Simulation”. Smart Materials and Structures 10: pp 404–408.

[3] Gaudenzi P, Carbonaro R and Benzi E, (2000),”Control Of Beam Vibrations

By Means Of Piezoelectric Devices: Theory and Experiments”. Composite

Structures 50(4): pp 373–379.

[4] Singh Sp, Pruthi Hs and Agarwal Vp, (2003),” Efficient Modal Control

Strategies For Active Control Of Vibrations”. Journal Of Sound And

Vibration 262: pp 563–575.

[5] Xu Sx And Koko Ts, (2004),” Finite Element Analysis And Design Of

Actively Controlled Piezoelectric Smart Structures”. Finite Elements In

Analysis And Design 40: pp 241–262.

[6] Karagulle H, Malgaca L And Oktem Hf, (2004), “Analysis Of Active

Vibration Control In Smart Structures By Ansys”. Smart Materials And

Structures 13: pp 661–667.

REFERENCES

[7] Khot Sm, Yelve Np And Iyer R, (2008),” Active Vibration Control Of

Cantilever Beam By Using Optimal (LQR) Controller”. Proceedings Of

International Conference On ‘Total Engineering, Analysis And

Manufacturing Technologies’ (Team Tech 2008), September 2008, Jn Tata

Auditorium, Iisc Bangalore, India: pp 22–24.

[8] Lim Yh, (2003),”Finite Element Simulation Of Closed Loop Vibration

Control Of A Smart Plate Under Transient Loading”. Smart Materials And

Structures 12: pp 272–286.

[9] Quek St, Wang Sy And Ang Kk, (2003),” Vibration Control Of Composite

Plates Via Optimal Placement Of Piezoelectric Patches”. Journal Of

Intelligent Material, Systems And Structures 14: pp 229–245.

[10] Xianmin Z, Changjian S And Erdman Ag, (2002),”Active Vibration

Controller Design And Comparison Study Of Flexible Linkage Mechanism

Systems”. Mechanical Machine Theory 37: pp 985–997.

[11] Gou Xinke Tian Haimin, (2007), “Active Vibration Control Of A

Cantilever Beam Using Bonded Piezoelectric Sensors And Actuators”

College Of Electrical And Information Engineering, Lanzhou University Of

Technology, Lanzhou, 730050, China : pp 85–88.

[12] Juntao Fei, Yunmei Fang And Chunyan Yan, (2010), “The Comparative

Study Of Vibration Control Of Flexible Structure Using Smart Materials”

Volume 2010 : pp 2–12.

[13] Yavuz Yaman, Tarkan Çalişkan, Volkan Nalbantoğlu, Eswar Prasad, And

David Waechter ,(2001) ,” Active Vibration Control Of A Smart Beam”

REFERENCES

Paper Published In The Proceedings Of Cansmart Symposium Held At

Montreal, October 2001: pp 125–133.

[14] Ulrich Gabbert, Tamara Nestorović Trajkov And Heinz Köppe , (2002) ,”

Modelling, Control And Simulation Of Piezoelectric Smart Structures

Using Finite Element Method And Optimal LQR Control” Facta

Universities Series : pp 418–430.

[15] Zhang Jing-Jun , Cao Li-Ya And Yuan Wei-Ze ,(2008),” Active Vibration

Control For Smart Structure Base On The Fuzzy Logic” International

Conference On Advanced Computer Control : pp 231–234

[16] Dong Jingshi , Guo Kang , Zheng Wei , Shenchuanliang, Cheng Guangming

, Ding Ji ,(2010),” Study On Active Vibration Control Technique Based On

Cantilever Beam” 2010 International Conference On Computer,

Mechatronics, Control And Electronic Engineering (Cmce) : pp 468–471

[17] K. B. Waghulde , Dr. Bimleshkumar Sinha , M. M. Patil And Dr. S. Mishra

,(2010),” Vibration Control Of Cantilever Smart Beam By Using

Piezoelectric Actuators And Sensors” K. B. Waghulde Et Al. /International

Journal Of Engineering And Technology Vol.2(4), 2010: pp 259-262.

[18] S.M. Khot, Nitesh P. Yelve And Ramya Iyer ,(2011),” Active Vibration

Control Of Cantilever Beam By Using Optimal (LQR) Controller ”: pp 2–

10.

[19] Deepak Chhabra1, Pankaj Chandna And Gian Bhushan,(2011),” Design

And Analysis Of Smart Structures For Active Vibration Control Using

Piezo-Crystals” International Journal Of Engineering And Technology

Volume 1 No. 3, December, 2011 : pp 153–163.

REFERENCES

[20] Sm Khot, Nitesh P Yelve, Rajat Tomar, Sameer Desai And S Vittal,(2011),”

Active Vibration Control Of Cantilever Beam By Using PID Based Output

Feedback Controller” Journal Of Vibration And Control 18(3): pp 366–372.

[21] Deepak Chhabra, Kapil Narwal And Pardeep Singh,(2012),”Design And

Analysis Of Piezoelectric Smart Beam For Active Vibration Control”

International Journal Of Advancements In Research & Technology,

Volume 1, Issue1, June-2012 1 Issn 2278-7763 : pp 1–5.

[22]Tamara Nestorović, Navid Durrani And Miroslav

Trajkov,(2012),”Experimental Model Identification And Vibration Control

Of A Smart Cantilever Beam Using Piezoelectric Actuators And Sensors” J

Electroceram (2012) 29: pp 42–55.

[23] A.P. Parameswaran, A.B. Pai, P.K. Tripathi And K.V. Gangadharan ,(2013)

,”Active Vibration Control Of A Smart Cantilever Beam On General

Purpose Operating System” Defence Science Journal, Vol. 63, No. 4, July

2013: pp 413-417.

[24] Preeti Verma , Manish Rathore And Dr Rajeev Gupta,(2013),”Vibration

Control Of Cantilever Beam Using Fuzzy Logic Controller” International

Journal of Science, Engineering and Technology Research (IJSETR)

Volume 2, Issue 4, April 2013: pp 906-909.

[25] Vibration Of Continuous Systems By Singiresu S. Rao,(2007), John

Wiley & Sons. Inc. pp 563-570.

[26] Friedland And Bernard,(2005), “Control System Design: An Introduction

To State Space Methods”,Dover. Isbn 0-486-44278-0 : pp 103–130.

REFERENCES

[27] Ang, K.H., Chong, G.C.Y., And Li, Y., (2005),.”PID Control System

Analysis, Design, And Technology” , IEEE Trans Control Systems Tech,

13(4): pp 559-576.

[28] Jinghua Zhong ,(Spring 2006),” PID Controller Tuning: A Short Tutorial”,

Retrieved 2011-04-04 : pp 408–412.

[29] [Http://Www.Mathworks.Com]

[30] Subramaniam R. S., Reinhorn A. M., Riley M. A., Nagarajaiah S. ,(1996),

“Hybrid Control Of Structures Using Fuzzy Logic”, Microcomput, Civil

Eng: pp 1-17.

[31]Yen J., Langari R. ,(1998), “Fuzzy Logic-Intelligence Control And

Information” , Prentice Hall, Englewood Cliffs, Nj : pp 101-125.

[32] Zadeh L.A., (1965), “Fuzzy Set”. In Format. Control, Vol. 8: pp 338- 353.

[33] Dr. Wedad Ibraheem Majeed , Dr. Shibly Ahmed Al-Samarraie And

Mohanad Mufaq Al-Saior ,(2013) , “Vibration Control Analysis Of A

Smart Flexible Cantilever Beam Using Smart Material” Journal Of

Engineering, Volume 19 January 2013, Number 1 : pp 82-95.

[34] Jingjun Zhang, Lili He, Ercheng Wang And Ruizhen Gao ,(2008),” Active

Vibration Control Of Flexible Structures Using Piezoelectric Materials”

Hebei University Of Engineering, Handan, China :pp540–544.

[35] Jingjun Zhang, Liya Cao, Weize Yuan And Ruizhen Gao,(2008),” Active

Vibration Control For Smart Structure Base On The Fuzzy Logic” Second

REFERENCES

International Symposium On Intelligent Information Technology

Application : pp 901–905.

[36] Varun Kumar And Deepak Chhabra ,(2013), ” 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 &Technology : pp 34–44.

[37] Bennett, Stuart, (1993),.”A History Of Control Engineering”, 1930-1955.

Iet. Isbn 978-0-86341-299-8: pp 48.

[38] Sellers, David. "An Overview Of Proportional Plus Integral Plus Derivative

Control And Suggestions For Its Successful Application And

Implementation”. Archived From The Original On March 7, 2007.

Retrieved 2007-05-05 : pp 212–226.

appendix

APPENDIX

APPENDIX

1-MATLAB program 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

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

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

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

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 program used for PID controller

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

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

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

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

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

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

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

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

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

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

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

APPENDIX

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

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

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

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)')

active vibrration control of beam

Documents