SPEED CONTROL OF SERVOMOTOR USING CONVENTIONAL CONTROLLER AND FUZZY LOGIC CONTROLLER

A Project report submitted in partial fulfillment of the requirement

For the Award of Degree of

BACHELOR OF ENGINEERING

IN

ELECTRICAL AND ELECTRONICS ENGINEERING

Submitted by

G.MULI NAIDU 311126514121

P.NARENDRA 311126514143

S V SUBRAHMANYESWARI 311126514089

T.SEKHAR 311126514109

Under the esteemed guidance of

Mrs.B.VANAJAKSHI, M.E

Assistant Professor

Department of Electrical and Electronics Engineering

ANIL NEERUKONDA INSTITUTE OF TECHNOLOGY AND SCIENCES

(Approved by AICTE, Affiliated to Andhra University, Accredited by NBA)

BHEEMUNIPATNAM, VISAKHAPATNAM

2011 – 2015

SPEED CONTROL OF D.C SERVOMOTOR USING CONVENTIONAL CONTROLLERS AND FUZZY LOGIC CONTROLLER

A Project report submitted in partial fulfillment of the requirement

For the Award of Degree of

BACHAELOR OF ENGINEERING

IN

ELECTRICAL AND ELECTRONICS ENGINEERING

Submitted by

G.MULI NAIDU 311126514121

P.NARENDRA 311126514143

S V SUBRAHMANYESWARI 311126514089

T.SEKHAR 311126514010

Under the esteemed guidance of

Mrs. B.VANAJAKSHI, M.E

Assistant Professor

Department of Electrical and Electronics Engineering

ANIL NEERUKONDA INSTITUTE OF TECHNOLOGY AND SCIENCES

(Approved by AICTE, Affiliated to Andhra University, Accredited by NBA)

BHEEMUNIPATNAM, VISAKHAPATNAM

2011 – 2015

DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING

ANIL NEERUKONDA INSTITUTE OF TECHNOLOGY & SCIENCES

(Affiliated to Andhra University, Visakhapatnam)

CERTIFICATE

This is to certify that the project work entitled “SPEED VONTROL OF D.C SERVOMOTOR BY

USING CONVENTIONAL CONTROLLERS AND FUZZY LOGIC CONTROLLER” is a bonafide

record of the work done by G.MULINAIDU,P.NARENDRA, S.V.SUBRAHMANYESWARI, SEKHAR

in partial fulfillment of the requirements for the award of degree of BACHAELOR OF ENGINEERING in

Electrical and Electronics Engineering. It is a record of bonafide work carried out under the guidance and

supervision of Assistant Prof. Mrs.VANAJAKSHIFaculty of the Department of EEE during the academic

year of 2010 – 2014

Signature of the project Guide Signature of the Head of the Department

Mrs. B.Vanajakshi, (M.E) Dr. G.RAJA RAO, M.E, PhD, MIEEE,

Department of EEE Department of EEE

ANITS ANITS

ACKNOWLEDGEMENT

We would like to articulate our deep gratitude to our project guide Assistant Prof.

Ms.B.VANAJAKSHI, M.E. who has been source of motivation and firm support to carry out the project. We

express our gratitude to Dr. G.RAJA RAO, M.E, PhD, MIEEE, MSITEProfessor and Head of the Department,

Electrical and Electronics Engineering for his valuable suggestion and constant encouragement all through

the thesis work.

We would also like to convey our sincere gratitude and indebtedness to all other faculty and staff of

the Department of Electrical and Electronics Engineering, ANITS, who bestowed their great effort and

guidance at appropriate times without which it would have been very difficult on our project work.

An assemblage of this nature could never have been attempted with our reference and inspiration

from the works of others whose details are mentioned in references section. We acknowledge our

indebtedness to all of them. Further, we would like to express our feelings towards our parents who directly

or indirectly encouraged and motivated us during this dissertation.

Chapter 1

Chapter 1

introduction :

1.1introduction

1.2 dc servo motor systems

1.3 Conventional controllers

Chapter 2

2.1 introduction

2.2 working principle

2.3 dc servo motor types

2.3.1 field controlled dc servo motor

2.3.2 armature controlled dc servo motor

2.4 advantages of dc servo motor

2.5 difference between ac and dc servo motor

2.6 applications of dc servo motor

2.7 mathematical modelling of dc servo motor

Chapter 3

Conventional controllers

3.1 proportional controllers

3.1.1 proportional action

3.2 integral controller

3.3 proportional plus integral control

3.3.1 time response

3.4 derivative action

3.5 proportional plus derivative (pd)control

3.5.1 time response

3.5.2 summary of effect of pd control

3.6 PID controller

3.6.1 proportional response

3.6.2 integral response

3.6.3 derivative response

3.6.4 time response

3.7 summary tables

Chapter 4

Fuzzy logic controller

4.1 introduction

4.2 unique features of fuzzy logic

4.3 applications for fuzzy logic

4.4 terminologies used in fuzzy logic controller

4.5 fuzzy controller model

4.5.1 fuzzification

4.5.2 knowledge base(KB)

4.5.3 inference mechanism

4.5.4 defuzzification and denormalization

4.6 defuzzification methods

4.6.1 centroid method

4.6.2 height method

4.6.3 mean of maxima method

4.6.4 sugeno method

LIST OF FIGURES

Fig.1.1 D.C servomotor circuit diagram

Fig 2.1 Servomechanism circuit

Fig.2.2 Field controlled of field controlled servomotor

Fig.2.3 Armature Controlled Servomotor

Fig. 3.1 Block Diagram of Integral controller

Fig.3.2 Diagram of Proportional-Integral control

Fig.3.3 Block diagram of servomotor drive using PI controller

Fig.3.4 Simulink model of PI controller

Fig. 3.5Block Diagram of Derivative Controller

Fig.3.6 Block Diagram of PD Controller

Fig.3.7 Simulink diagram Of servomotor drive using PD controller

Fig.3.8 Speed Response of PD Controller

Fig.3.9 Block Diagram of Controller and Plant

Fig.3.10 Block diagram of a basic PID controller

Fig.3.11 Simulink diagram of servomotor drive using PID controller

Fig.3.12 Speed Response of PID Controller

Fig.4.1 Membership functions (a) Triangular (b) Trapezoidal (c) Bell shaped

(d) Polynomial - Z (e) polynomial – S

Fig.4.2 Structure of Fuzzy Logic controller

Fig.4.3 Triangular membership functions

Fig. 4.4 Centre of – area, method of Defuzzification

Fig.4.5 Height method of Defuzzification

Fig.5.1 Block diagram of servomotor

Fig.5.1 Block diagram of servomotor

Fig.5.2 Simulink model comparing conventional controllers

Fig 5.3 Graphs comparingg performance of conventional controllers

Fig 5.4 Simulink model of fuzzy controller

Fig 5.5 Fuzzy Internal block controller

Fig5.6 FIS file of fuzzy block

Fig5.7 Membership functions of fuzzy controller

Fig.5.8 View ruler scale of fuzzy output

Fig 5.9 Performance output of servomotor using fuzzy controller

ABSTRACT

This paper is to design a controller for servo motor. The experiment is used to obtain the transfer function to design the PID controller. The effectiveness of the design is validated using MATLAB/Simulink. This new design method gives us a simple and powerful way to design a speed controller for a servo motor. In this study DC servo motor’s mathematical model and equation were extracted and there were three different motion controller designed for control the velocity It was created simulation model at the MATLAB program and proportional integral derivative

In order to obtain more compact speed control the design and application of a fuzzy logic controller to DC-servomotor is investigated. The proposed strategy is intended to improve the performance of the original control system by use of a fuzzy logic controller (FLC) as the motor load changes. Computer simulation demonstrates that FLC is effective in position control of a DC-servomotor comparing with conventional oneThe FLC structure consists of anintegrator and variable structure system. The integral control is introduced into it in order to eliminated steady state error due different command inputs and improves control precision, while thefuzzy control would maintain the insensitivity to parametervariation and disturbances. The FLC strategy is implementedand applied to a position control of a DC servomotor drives.Experimental results indicated that FLC system performancewith respect to the sensitivity to parameter variations is greatlyreduced. Also, excellent control effects and avoids the chattering phenomenon.

CHAPTER-1

INTRODUCTION

1.1 IntroductionAutomatic systems plays a common role in our daily life, they can be found in almost any electronic devices and appliances we use daily, starting from air conditioning systems, automatic doors, and automotive cruise control systems to more advanced technologies such as robotic arms, production lines and thousands of industrial and scientific applications.

DC servomotors are one of the main components of automatic systems; any automatic system should have an actuator module that makes the system to actually perform function. The most common actuator used to perform this task is the DC servomotor. Historically, DC servomotors also played a vital role in the development of the computer’s disk drive system; which make them one of the most important components in our life that we cannot live without it. Due to their importance, the design of controllers for these systems has been an interesting area for researchers from all over the world.

Servomotor is a motor used for position or speed control in closed loop control systems. The requirement from a servomotor is to turn over a wide range of speeds and also to perform position and speed instructions given. DC and AC servomotors are seen in applications by considering their machine structure in general. DC servo motors have been used generally at the computers, numeric control machines, industrial equipments, weapon industry, and speed control of alternators, control mechanism of full automatic regulators as the first starter, starting systems quickly and correctly. In the field of control of mechanical linkages and robots, research works are mostly found on DC motors. While some properties of DC servomotors are the same, like inertia, physical structure, shaft resonance and shaft characteristics, their electrical and physical Constants are variable and the more advantages of DC servo motor over the AC servo motors .The velocity and position tolerance of servo motors which are used at the control systems are nearly the same.

However, even with all of their useful applications and usage, servomotor systems still suffer from several non-linear behaviors and parameters affecting their performance, which may lead for the motor to require more complex controlling schemes, or having higher energy consumption and faulty functions in some cases. For these purposes the controlling speed of DC servomotor system is an interesting area that still offers multiple topics for research, especially after the discovery of conventional controllers and fuzzy logic at the variable working situations to the simulation model has prepared in Mat lab program for improvement the servo motor performance. From this point of view and the importance of having high efficient servomotor systems, played a vital role in designing smart controllers that can eliminate or cope with the non-linear effects found in servomotor systems and improve the functions they are used for.

1.2 DC Servomotor Systems:

A Servomotor system consists of different mechanical and electrical components, the different components are integrated together to perform the function of the servomotor, Figure 1.1 below shows a typical model of a servomotor system

Fig.1.1.d.c servomotor circuit diagram

DC Servomotors have good torque and speed characteristics; also they have the ability to be controlled by changing the voltage signal connected to the input. These characteristics made them powerful actuators used everywhere. The main concern about DC servomotors is how to eliminate the non-linear characteristics that affect both the output speed and position. Another important non-linear behavior in servomotors is the saturation effect, in which the output of the motor cannot reach the desired value.

For example, if we want to reach a 100 rpm angular speed when we supply a 12 volt input voltage, but the motor can only reach 90 rpm at this voltage. The saturation effect is very common in almost all servomotor systems. Other non-linear effect is the dead zone; in which the motor will not start to rotate until the input voltage reaches a specific minimum value, which makes the response of the system slower and requires more controllability. A mathematical type of non-linear effect found in the servomotors is the backlash in the motor gears. Some of the servomotors use internal gears connections in order to improve their torque and speed characteristics, but this improvement comes over the effect in the output speed and position characteristics.

The goal here is to find a smart controller that is capable of eliminating as much as possible from the disturbance.

1.3 Conventional and Fuzzy controllers

Conventional proportional–integral–derivative (PID) controllers have been well developed

and applied for about half a century, and are extensively used for industrial automation and

process control today. The main reason is due to their simplicity of operation, ease of design,

inexpensive maintenance, low cost, and effectiveness for most linear systems. Recently,

motivated by the rapidly developed advanced microelectronics and digital processors,

conventional PID controllers have gone through a technological evolution, from pneumatic

controllers via analog electronics to microprocessors via digital circuits. However, it has been

known that conventional PID controllers generally do not work well for nonlinear systems,

higher order and time-delayed linear systems, and particularly complex and vague systems

that have no precise mathematical models . To overcome these difficulties, various types of

modified conventional PID controllers such as auto tuning and adaptive PID controllers were

developed lately. Also, a class of nonconventional type of PID controller employing fuzzy

logic has been designed and simulated for this purpose. The fuzzy PID controller has the

following special features.

It has the same linear structure as the conventional PID controller, but has constant

coefficient, self - tuned control gains: the proportional, integral, and derivative gains are

nonlinear functions of the input signals. The controller is designed based on the classical

discrete PID controller, from which the fuzzy control law is derived. Membership functions

are simple triangular ones with only twenty five rules. The fuzzification, control-rule

execution, and defuzzification steps are all embedded in the final formulation of the fuzzy

control law. The resulting control law is an explicit conventional formula, so the controller

works just like a conventional PID controller, while the fuzzification–rules–Defuzzification

routine is not needed throughout the entire control process.

Fuzzy PID controller has superior performance than PID controller. However, despite the

significant improvements made in the fuzzy PID controllers over its classical counterparts.

The fuzzy PID controller is not able to meet certain optimality criteria . The constant control

gains of the controller are termed manually, hence effective performance due to the lack of

optimization. A lot of research and the huge number of different solutions proposed, most

industrial control systems are still based on conventional PID regulators. Different sources

estimate the share taken by PID controllers at between 90% and 99%. Some of the reasons

for this situation are PID controllers are robust and simple to design. There exists a clear

relationship between PID and system response parameters. As a PID controller has only three

parameters, plant operators have a deep knowledge about the influence of these parameters

and the specified response characteristics on each other. Many PID tuning techniques have

been elaborated during recent decades, which facilitate the operator’s task. Because of its

flexibility, PID control could benefit from the advances in technology. Most of the classical

industrial controllers have been provided with special procedures to automate the adjustment

of their parameters (tuning and self-tuning). However, PID controllers cannot provide a

general solution to all control problems. The processes involved are in general, complex and

time-variant, with delays and non-linearity, and often with poorly defined dynamics. When

the process becomes too complex to be described by analytical models, it is unlikely to be

efficiently controlled by conventional approaches. Servomotor speed Control is done with

Proportional Derivative (PD), Proportional Integral (PI), Proportional Integral derivative

(PID), Fuzzy Logic PID (FLPID) controllers are used and performances of controllers are

improved.

CHAPTER-2

2. SERVOMOTOR

2.1 Introduction

Servomotor:

 An electrical motor, controlled with the help of servomechanism. If the motor as controlled device, associated with servomechanism is DC motor, then it is commonly known DC Servo Motor. If the controlled motor is operated by AC, it is called AC Servo Motor. 

Servomechanism:

A servo system mainly consists of three basic components – a controlled device, a output sensor, a feedback system.

This is an automatic closed loop control system. Here instead of controlling a device by applying variable input signal, the device is controlled by a feedback signal generated by comparing output signal and reference input signal.

When reference input signal or command signal is applied to the system, it is compared with output reference signal of the system produced by output sensor, and a third signal produced by feedback system. This third signal acts as input signal of controlled device. This input signal to the device presents as long as there is a logical difference between reference input signal and output signal of the system. After the device achieves its desired output, there will be no longer logical difference between reference input signal and reference output signal of the system. Then, third signal produced by comparing theses above said signals will not remain enough to operate the device further and to produce further output of the system until the next reference input signal or command signal is applied to the system. Hence the primary task of a servomechanism is to maintain the output of a system at the desired value in the presence of disturbances.

Fig 2.1 Servomechanism circuit

2.2 Working principle:

A servo motor is basically a d.c motor (in some special cases it is AC motor) along with some other special purpose components that make a DC motor a servo. In a servo unit, you will find a small d.c motor, potentiometer, gear arrangement and an intelligent circuitry. The intelligent circuitry along with the potentiometer makes the servo to rotate according to our wishes.

As we know, a small d.c .motor will rotate with high speed but the torque generated by its rotation will not be enough to move even a light load. This is where the gear system inside a servomechanism comes into picture. The gear mechanism will take high input speed of the motor (fast) and at the output; we will get a output speed which is slower than original input speed but more practical and widely applicable.

Say at initial position of servo motor shaft, the position of the potentiometer knob is such that there is no electrical signal generated at the output port of the potentiometer . This output port of the potentiometer is connected with one of the input terminals of the error detector amplifier. Now an electrical signal is given to another input terminal of the error detector amplifier. Now difference between these two signals, one comes from potentiometer and another comes from external source, will be amplified in the error detector amplifier and feeds the motor. This amplified error signal acts as the input power of the d.c motor and the motor starts rotating in desired direction. As the motor shaft progresses the potentiometer knob also rotates as it is coupled with motor shaft with help of gear arrangement. As the position of the potentiometer knob changes there will be an electrical signal produced at the potentiometer port. As the angular position of the potentiometer knob progresses the output or feedback signal increases. After desired angular position of motor shaft the potentiometer knob is reaches at such position the electrical signal generated in the potentiometer becomes same as of external electrical signal given to amplifier. At this condition, there will be no output signal from the amplifier to the motor input as there is no difference between external applied signal and the signal generated at potentiometer. As the input signal to the motor is nil at that position, the motor stops rotating. This is how a simple conceptual servo motor works.

2.3 DC Servo Motor types:

The motors which are utilized as DC servo motors, generally have separate DC source for field winding and armature winding. The control can be archieved either by controlling the field current or armature current. Field control has some specific advantages over armature control and on the other hand armature control has also some specific advantages over field control. Depending on specific applications they are used.

2.3.1. Field Controlled DC Servo Motor :

The figure below illustrates the schematic diagram for a field controlled DC servo motor. In this arrangement the field of d.c motor is excited be the amplified error signal and armature winding is energized by a constant current.

Fig.2.2 Field controlled of field controlled servomotor

The field is controlled below the knee point of magnetizing saturation curve. At that portion of the curve the mmf linearly varies with excitation current. That means torque developed in the motor is directly proportional to the field current below the knee point of magnetizing saturation curve. From general torque equation it is found that, torque T ∝φIa. Where, φ is field flux and Ia is armature current. But in field controlled DC servo motor, the armature is excited by constant current, hence Ia is constant here. Hence, T ∝ φ

As field of this DC servo motor is excited by amplified error signal, the torque of the motor i.e. rotation of the motor can be controlled by amplified error signal. If the constant armature current is large enough then, every little change in field current causes corresponding change in torque on the motor shaft.

The direction of rotation can be changed by changing polarity of the field. The direction of rotation can also be altered by using split field motor where the field winding is divided into two parts, one half of the winding is wound in clockwise direction and other half in wound in anticlockwise direction. The amplified error signal is fed to the junction point of these two halves of the field as shown below. The magnetic field of both halves of the field winding opposes each other. During operation of the motor, magnetic field strength of one half dominates other depending upon the value of amplified error signal fed between these halves. Due to this, the DC servo motor rotates in a particular direction according to the amplified error signal voltage.

The main disadvantage of field control DC servo motors, is that the dynamic response to the error is slower because of longer time constant of inductive field circuit. The field is an electromagnet so it is basically a highly inductive circuit hence due to sudden change in error signal voltage, the current through the field will reach to its steady state value after certain period depending upon the time constant of the field circuit. That is why field control DC servo motor arrangement is mainly used in small solar tracking system

The main advantage of using field control scheme is that, as the motor is controlled by field – the controlling power requirement is much lower than rated power of the motor.

2.3.2 Armature Controlled DC Servo Motor :

The figure below shows the schematic diagram for an armature controlled DC servo motor. Here the armature is energized by amplified error signal and field is excited by a constant current source

fig.2.3 Armature Controlled Servomotor

The field is operated at well beyond the knee point of magnetizing saturation curve. In this portion of the curve, for huge change in magnetizing current, there is very small change in mmf in the motor field. This makes the servo motor is less sensitive to change in field current. Actually for armature controlled DC servo motor, we do not want that, the motor should response to any change of field current.

Again, at saturation the field flux is maximum. Now if φ is large enough, for every little change in armature current Ia there will be a prominent changer in motor torque. That means mechanism becomes much sensitive to the armature current.

As the armature of d.c motor is less inductive and more resistive, time constant of armature winding is small enough. This causes quick change of armature current due to sudden change in armature voltage. That is why dynamic response of armature controlled DC servo motor is much faster than that of field controlled DC servo motor.

The direction of rotation of the motor can easily be changed by reversing the polarity of the error signal.

2.4 Mathematical modelling of dc servo motor

ea (t )=¿Input armature voltage

ia (t )=¿Armature current

Ra=¿Armature resistance

La=¿Armature inductance

eb (t )=¿Back EMF

T m (t )=¿Developed motor torque

ωm (t )=¿Rotor angular velocity

J=¿Motor moment of inertia

B=¿Friction coefficient

Kb=¿BackEMF constant

K t=¿Torque constant

T l=¿Load torque

Figure 2.3 depicts a generic model of a DC motor that includes two windings; a stationary field winding on the stator and a second winding for the rotating armature. This type of motor can be controlled by varying either the field current or the armature current. Most modernServo motor are somewhat different in construction. The field winding is replaced with two or more powerful rare‐earth magnets on the stator. Since the field strength of these motors is constant, they can only be controlled by varying the armature current ia.

The armature current,ia , can be expressed in terms of the applied motor voltage ,ea, byApplying Kirchhoff's Voltage Law (KVL) to the armature circuit in Figure 2.3. As with mostwindings, the impedance of the armature can be represented by a resistive component,Ra , in series with an inductive component,la.Here the differential equation for armature control (for armature control the field is constant)

Apply KVL, from first loop equation:

ea (t )=Raia (t )+ la

di (a )dt

+eb (t ) (2.1)

The torque generated by the motor accelerates the armature of the motor as well asany additional load inertia on the motor shaft. Some of the torque also goes towardsovercoming friction. In order to maintain a linear system model, only the viscous motor friction will be considered for the time being. For a mechanical system undergoing pure rotational motion, Newton’s second law states that the sum of the applied torques is equal to

the product of the mass moment of inertia, J, and the angular acceleration of the body, d ωm ( t )

dx

The torque developed in motor

T m (t )=Jd ωm ( t )

dx+B ωm (t ) (2.2)

And the torque developed by the motor is proportional to the product of armature current and the field flux for armature control field is constant

T m (t )∝∅ ia ( t )

For armature control field is constant

T m (t )∝ia (t )

T m (t )=K t ia (t ) (2.3)

Where K t is torque constant

The back EMF is proportional to the speed

eb ( t )∝ωm ( t )

eb ( t )=Kb ωm (t ) (2. 4 )

WhereKb is back EMF constant.

Takinkg laplace transform on both sides of all equations

Ea ( s)=Ra I a (s )+s La I a (s )+Eb ( s) (2.5)

T m (s )=Js ωm ( s)+B ωm ( s) (2.6)

T m (s )=K t I a (s ) (2.7)

Eb ( s)=Kb ωm (s ) (2.8)

From the equation (2.5)

I a (s )=Ea ( s)−Eb (s )

Ra+¿s La¿

(2.9)

From the equation(2.6)

ωm (s )=T m (s )Js+B

(2.10)

Model for dc servo motor

For the particular load, the speed change with respect to the input signal

Final transfer function:

Ea (s )ωm ( s )

=K t

¿¿

CHAPTER-3

3.CONVENTIONAL CONTROLLERS

3.1 Proportional Controller

The proportional term makes a change to the output that is proportional to the current

error value. The proportional response can be adjusted by multiplying the error by a constant

K p, called the proportional gain. The transfer function of a proportional controller is simply a

gain sayK p. If the input of the controller is e (t ) then the output is u (t )=K p e (t ) or in a Laplace

transform domainU (s )=K P E (s ). As K P increases the unit-step response may becomes

faster and eventually the feedback system may becomes unstable. For the same unit-step

reference input the steady-state plant outputs are different for differentK P.

A high proportional gain results in a large change in the output for a given change in the

error. If the proportional gain is too high, the system can become unstable. In contrast, a

small gain results in a small output response to a large input error, and a less responsive or

less sensitive controller. If the proportional gain is too low, the control action may be too

small when responding to system disturbances. Tuning theory and industrial practice indicate

that the proportional term should contribute the bulk of the output change

3.1.1 Proportional Action

Proportional action provides an instantaneous response to the control error. This is

useful for improving the response of a stable system but cannot control an unstable system

with a nonzero steady-state error. By using this controller rise time increases and also steady

state error decreases. And peak overshoots increases. This will be done only by proper

selection of K value.

3.2 Integral Controller

In this controller the output u(t) is altered at a rate proportional to the error signal e(t). The

output u(t) depends upon the integral of the error signal e(t).

Mathematically

du(t )dt

=K . e (t)u ( t )=K∫0

1

e ( t )dt

Basic block diagram for integral controller is shown in Fig.4.1

Fig. 3.1 Block Diagram of Integral controller

3.3Proportional plus Integral Control:

Integral control action itself is not sufficient, as it introduces hunting in the system. Therefore

a combination of Proportional and integral control action is introduced to improve the system

performance. In this type of system, the actuating signal consists of proportional error signal

added with the integral of the error signal.

Fig.3.2 Block Diagram of Proportional-Integral control

Mathematically,

u (t ) = e (t) =K∫0

t

e ( t )dt (4.4 )

Where e (t) = error signal;

And ∫0

t

e ( t )dt = integral of error signal (4.5 )

Or U(S) = E(s) [1+KS ](4.6 )

Proportional plus Integral control increases the order and type of the system by one,

respectively. Therefore, it improves steady state performance. The effect of proportional and

integral control improves system steady state response with in less time and rise time also

increases.

3.3.1 Time response

Fig.3.3 Block diagram Of servomotor drive using PI controller

Transfer function of servomotor drive when conventional PI speed controller technique is used

as shown in Fig.4.3 is given by

꞊ 0.052 s+0.0416

0.0029 s3+0.010119 s2+0.0052 s+2.8692

The response for above transfer function is shown in Fig. 4.4

Fig.3.4 Simulink model of PI controller

3.4.1 Derivative action

Derivative action acts on the derivative or rate of change of control error. This

provides a fast response, as opposed to the integral action, but cannot accommodate constant

errors (i.e., the derivative of a constant, nonzero error is zero).

Fig. 3.5 Block Diagram of Derivative Controller

Derivatives have a phase of +90 degrees leading to an anticipatory or predictive

response. However, derivative control will produce large control signals in response to high

frequency control errors such as set point changes (set command) and measurement noise. In

order to use derivative control the transfer function must be proper. This often requires a pole

to be added to the controller (this pole is not present in the equations below). Block diagram

of derivative controller is shown in Fig.4.5

3.5 Proportional Plus derivative (PD) control

Proportional – derivative or PD control combines proportional control and derivative control in parallel.

C pd ( s )=K p (1+T d s ) (4.7 )

The proportional plus derivative controller produces an output signal consisting of two terms:

One proportional to error signal and the other proportional to the derivative signal.

In PD – controller,

u (t )αe ( t )+ ddt

e ( t ); (4.8 )

U ( t )=K p e ( t )+K pT dddt

e ( t ) (4.9 )

WhereK p → proportional gain;

T d →derivative time.

On taking Laplace transform of equations with zero initial conditions.

We get,

U (s )=K P E (s )+KP T D sE ( s ) (4.10 )

U (s )=E ( s ) [ (1+T D s ) ] (4.11 )

U (s ) / E ( s)=K P (1+T D s) (4.12 )

Fig.3.6 Block Diagram of PD Controller

Where T D=K D/ KP is called the derivative time, during which interval the

proportional control action takes effect. The anticipatory characteristic of derivative control

action is found in PD control action. This means, in transient mode, PD can anticipate the

direction of the error in making adjustments before excessive overshoot occurs. During the

stead-state mode, PD has an effect on the stead-state error only when the error changes with

respect to time. In the design of a PD controller, we want to place the controller’s corner

frequency ω=1/T P so that the phase margin improved with the new gain crossover

frequency. Block diagram of PD controller is shown in Fig.4.6

3.5.1 Time Response

Fig.3.7Simulink diagram of servomotor drive using PD controller

Transfer function of servomotor drive when conventional PD speed controller technique is

used as shown in Fig.4.3 is given by

0.0052 s+2.86

0.0029 s2+0.01531916 s+2.8692

The response of above transfer function is shown

Fig.3.8 Speed Response of PD Controller

3.5.2 Summary of effects of PD control

(1) Improves damping and reduce maximum overshoot.

(2) Reduces rise time and settling time.

(3) Improving the transient response.

(4) Increases Band width and an accentuation of high-frequency noiseas PD

controlleracts like a high-pass filter.

(5) Improves Gain Margin (GM), Phase Margin (PM), Resonant Peak ( M r ).

(6) May accentuate noise at higher frequencies.

(7) Not effective for lightly damped or initially unstable systems.

(8) May require a relatively large capacitor in circuit implementation.

3.6 PID controller

Fig.3.9 Block Diagram of Controller and Plant

The output of a PID controller, equal to the control input to the plant, in the time-

domain is as follows:

u (t )=K p e ( t )+K i∫ e ( t )dt +K pdedt

( 4.13 )

The working of PID controller in a closed-loop system, schematic arrangement is

shown in Fig.4.9. The variable (e ) represents the tracking error, the difference between the

desired input value (r ) and the actual output( y ) .

This error signal (e ) will be sent to the PID controller, and the controller computes

both the derivative and the integral of this error signal. The control signal (u ) to the plant is

equal to the proportional gain ( K p ) times the magnitude of the error plus the integral gain ( K i )

times the integral of the error plus the derivative gain ( Kd )times the derivative of the error.

This control signal(u )is sent to the plant, and the new output ( y ) is obtained. The new output

( y ) is then fed back and compared to the reference to find the new error signal (e ). The

controller takes this new error signal and computes its derivative and it’s integral again, ad

infinitum.

The transfer function of a PID controller is found by taking the Laplace transform of Eq.

(4.13)

K p+K i

s+K d s=

Kd s2+K p s+ K i

s(4.14 )

K p=¿Proportional gain K i=¿ Integral gain = Derivative gain

3.6.1 Proportional Response

In general, increasing the proportional gain will increase the speed of the control

system response. However, if the proportional gain is too large, the process variable will

begin to oscillate. If K p is increased further, the oscillations will become larger and the

system will become unstable and may even oscillate out of control. Block diagram of basic

PID controller is shown in Fig.4.10

Fig.3.10 Block diagram of a basic PID controller

3.6.2 Integral Response

The integral component sums the error term over time. The result is that even a small

error term will cause the integral component to increase slowly. The integral response will

continually increase over time unless the error is zero, so the effect is to drive the Steady-

State error to zero. A phenomenon called integral windup results when integral action

saturates a controller without the controller driving the error signal toward zero.

3.6.3 Derivative Response

The derivative component causes the output to decrease if the process variable is

increasing rapidly. The derivative response is proportional to the rate of change of the process

variable.

Increasing the derivative time (Td) parameter will cause the control system to react

more strongly to changes in the error term and will increase the speed of the overall control

system response. Most practical control systems use very small derivative time (Td), because

the Derivative Response is highly sensitive to noise in the process variable signal. If the

sensor feedback signal is noisy or if the control loop rate is too slow, the derivative response

can make the control system unstable.

3.6.4 Time Response

Fig.3.11Simulink diagram Of servomotor drive using PID controller

Transfer function of servomotor drive when conventional PI speed controller technique is used

as shown in Fig.4.3 is given by

0.156 s2+2.6 s+0.16640.0029 s3+0.166119 s2+2.6092 s+0.1664

The response of above transfer function

Fig.3.12 Speed Response of PID Controller

3.7 Summary Tables

A summary of the advantages and disadvantages of the three controllers is shown in Table 1.

Proportional (P) Integral (I) Derivative (D)

Advantages

Fast response time.

Minimizes fluctuation.

Contains small offset.

Returns system to steady state.

Keeps system at consistent setting.

Controls processes with rapidly changing outputs.

Disadvantages

Contains large offset.

Does not bring system to desired set point.

Slow response time.

Slow response time.

Requires combined use with another controller.

Table.1 Advantages and disadvantages of conventional controllers

4. FUZZY LOGIC CONTROLLER

4.1 Introduction

The most popular controller in industry is the PID controller because of its simple

structure and effective control. But due to its simple structure and effective control. But due

to its linear structure, conventional PID controllers are usually not effective if the processes

involved are higher order and time delay systems, Non-linear systems, complex systems

without precise mathematical models, and systems with uncertainties.

Fuzzy control is based on fuzzy logic, a logical system which is much closer in spirit

to human thinking and natural language than traditional logical systems. The fuzzy logic

controller (FLC) based on fuzzy logic provides a means of converting a linguistic control

strategy based on expert knowledge into an automatic control strategy. During the past

several years fuzzy controller has emerged as one of the most active and fruitful area for

research in the application of fuzzy set theory. The pioneering research of Mamdani and his

colleagues on fuzzy control was motivated by Zadeh’s seminal papers on the linguistic

approach and system analysis based on the theory of fuzzy sets. The idea of fuzzy logic was

invented by Professor L. A. Zadeh of the University of California at Berkeley in 1965. He

was a mathematician, electrical engineer, computer scientist, artificial intelligence researcher

and professor emeritus. FL is a problem-solving control system methodology that lends itself

to implementation in systems ranging from simple, small, embedded micro-controllers to

large, networked, multi-channel PC or workstation-based data acquisition and control

systems. It can be implemented in hardware, software, or a combination of both. FL provides

a simple way to arrive at a definite conclusion based upon vague, ambiguous, imprecise,

noisy, or missing input information.

The logic of an approximate reasoning continues to grow in importance, as it provides

an in expensive solution for controlling know complex systems. Fuzzy logic controllers are

already used in appliances washing machine, refrigerator, vacuum cleaner etc. Computer

subsystems (disk drive controller, Power management) consumer electronics (video, camera,

battery charger) C.D. Player etc.And so on in last decade, fuzzy controllers have converted

adequate attention in motion control systems. As the later possess non-linear characteristics

and a precise model is most often unknown.

Remote controllers are increasingly being used to control a system from a distant place due to

inaccessibility of the system or for comfort reasons.

4.2 Unique features of fuzzy logic

The unique features of fuzzy logic that made it a particularly good choice for many

control problems are as follows, It is inherently robust since it does not require precise, noise –

free inputs and can be programmed to fail safely is a feedback sensor quits or is destroyed. The

output control is a smooth control function despite a wide range of input variations. Since the

fuzzy logic controller processes user-defined rules governing the target control system. It can be

modified and tweaked easily to improve or drastically alter system performance. New sensors

can easily be incorporated into the system simply by generating appropriate governing rules.

4.3 Applications for Fuzzy Logic

Automatic control of dam gates for hydro electric-power plants.

Simplified control of robots.

Camera aiming for the telecast of sporting events.

Substitution of an expert for the assessment of stock exchange actives.

Preventing unwanted temperature fluctuations in air-conditioning systems. Efficient

and stable control of car-engines.

Cruise control for automobiles.

Improved efficiency and optimized function of industrial control applications.

Positioning of water-steppers in production of semiconductors.

Optimized planning of bus timetables.

Archiving system for documents.

Prediction system for early recognition of earth quakes Medicine technology cancer

diagnosis.

Combination of fuzzy logic and neural nets.

Recognition of handwritten symbols with pocket computers.

Recognition of motives in pictures with video cameras.

Automatic motor-control for vacuum cleaners with recognition of surface condition and

degree of soiling back light control for camcorders.

Compensation against vibrations in camcorders.

Single button control for washing machines.

Recognition of handwriting, objects, voice.

Flight aid for helicopters.

Simulation for legal proceedings.

Software-design for industrial processes.

Controlling of machinery speed and temperature for steel works.

Controlling of subway systems in order to improve driving comfort, precision of halting

and power economy.

Improved fuel-consumption for automobiles.

Improved sensitiveness and efficiency foe elevator control.

Improved safety for nuclear reactors.

4.4 Terminologies used in Fuzzy logic controller

Some of the basic concepts and basic definitions of fuzzy set theory and fuzzy logic

are summarized as below;

Universe of Discourse:

Universe of discourse U is the space where the fuzzy variables are defined, which

could be discrete are continuous. It is a collection of objects all having the same

characteristics. The individual elements in the universe ‘U’ will be defined as “u”.

Fuzzy set:

A fuzzy set ‘F’ in a universe of discourse ‘U’ is characterized by a membership

function μF :U → [ 0 ,1 ] . Thus a fuzzy set F in U may be represented as a set of ordered pairs

of a element u and its grade of membership function:

F={(u , μF (u ) )∙ uϵU } . (4.1)

Continuous Fuzzy sets:

A fuzzy set ‘F’ is said to be continuous if its membership function is continuous.

Height of Fuzzy set:

The largest membership value of fuzzy set ‘F’ is called the Height of Fuzzy set.

Domain:

The domain of a fuzzy set is the range of allowable values of the variable.

Support:

The support of a fuzzy set F is the crisp set of all points in the Universe of Discourse

U such that μF (u )≥ 0.

Cross Over Point:

The element u in the universe of Discourse U at which μF=0 ∙5, is called the cross

over point.

Cross over ( F )={ μFu⃒� (u )=0.5 } (4.2)

Core:

The core of a fuzzy set F is the set of all points u in such that μF (u )=1

Core ( F )={ μFu⃒� (u )=1} (4.3)

α cut:

The α cut or α level set of a fuzzy set F is a crisp set defined by

Fa={ μFu⃒� (u )≥ α } (4.4)

Strong α cut:

Strong α cut is defined by

Fa={ μFu⃒� (u )>α } (4.5)

Fuzzy Singleton:

A fuzzy set whose support a single point in universe of discourse U is with

μF (u )=1.0is referred to as fuzzy singleton.

Union:

Let A and B be two fuzzy sets in U with membership functions μ Aand B ,

respectively.

The membership function of μ A∪B of A∪B is a point wise defined for all uϵU by

μ A∪B (u )=max {μ A (u ) , μ B (u ) } (4.6)

Intersection:

The membership function of μ A ∩B of A ∩ B is a point wise defined for all u uϵU by

μ A ∩B (u )=min {μ A (u ) , μ B (u ) } (4.7)

Complement:

The membership function μ A of the complement of a fuzzy set A is point wise

defined for all u ϵU by

μ A (u )=1−μA (u ) (4.8)

Connectives:

The connectives are used in rule formations when number of inputs is greater than

one. The connectives ‘and’ and ‘or’ are always defined in pairs, for example,

a∧b=¿min(a ,b) – minimum

a∨b=¿max(a ,b)−¿ maximum

a∧b=a . b – algebraic product

a∨b=a+b−a .b−¿algebraic or probabilistic sum

Membership Functions:

Every element in the universe of discourse is a member of fuzzy set to some grade,

may be even zero. The grade of membership for all its members describes a fuzzy set. In

fuzzy sets elements are assigned a grade of membership, such that the transition from

membership to non membership is rather than abrupt. It forms a crucial part in a fuzzy rule

base model because they only actually define the fuzziness of a control variable or process

variable. The most popular choices for the shape of membership functions are triangular,

trapezoidal and bell shaped functions [15, 35].

(i)Triangular Membership function: it is one of the most popular among the scientists in this

field. It can be generally defined using a left point, center point and right point. Overlap and

sensitivity are the two parameters that can be used to adjust the shape of the triangles for

better performance. The triangular membership function is shown in Fig.4.1 (a).

The triangular curve is a function of a vector, x, and depends on three scalar parameters a, b,

c and is given by

f (x : a , b , c)={x−ab−a

,a≤ x ≤b }{c−x

c−b, b≤ x ≤ c}

{0 , x ≤ a∧c ≤ x } (4.9)

(ii)Trapezoidal membership function: as the name suggests of this class of membership

function is that of a trapezoidal as shown in Fig.5.1 (b). The maximum membership value

1.0 occurs over a small range about the central point of the function. The trapezoidal curve is

a function of a vector, x, and depends on four scalar parameters a, b, c and d is given by

f (x : a , b , c , d )={0 , x≤ a }

{(x−a)/ (b−a) , a≤ x≤ b }

{1 , b ≤ x ≤ c }

{d−xd−c

, c≤ x≤ d }{0 , d ≤ x } (4.10)

(iii) Bell shaped membership function: as the name suggests of this class of membership

function is that of bell shaped one as shown in Fig.5.1(c). The maximum membership value

for bell shaped is 1 at x=c, and the membership ‘ x ’ decreases as its derivative from this

central value of ‘0’. In the case of bell shape functions the oscillations are the minimum and

rise time is also reduced.

f (x : a ,b , c , d )={ 1

1+(x−c )2, a≤ x ≤b }

{1 x=c } (4.11)

(iv) Polynomial–Z: Polynomial-based curves can have several functions, including

asymmetrical polynomial curve open to the left (polynomial-Z) and its mirror image, as

shown in Fig.5.1 (d), (e) open to the right (polynomial-S) [35].

(a)

(b)

(c)

(d)

(e)

Fig.5.1 Membership functions. (a)Triangular (b) Trapezoidal (c) Bell shaped

(d) polynomial - Z (e) polynomial – S

4.5 Fuzzy Controller Model

Fuzzy modeling is the method of describing the characteristics of a system using

fuzzy inference rules. The method has a distinguishing feature in that it can express

linguistically complex non-linear system. It is however, very hand to identify the rules and

tune the membership functions of the reasoning. Fuzzy Controllers are normally built with

fuzzy rules. These fuzzy rules are obtained either from domain experts or by observing the

people who are currently doing the control. The membership functions for the fuzzy sets will

be derive from the information available from the domain experts and/or observed control

actions. The building of such rules and membership functions require tuning. That is,

performance of the controller must be measured and the membership functions and rules

adjusted based upon the performance. This process will be time consuming [38].

The basic configuration of Fuzzy logic control based as shown in Fig.5.2consists of

four main parts (i) Fuzzification

(ii) Knowledge Base

(iii) Inference Engine and

(iv)Defuzzification.

Fig.5.2 Structure of Fuzzy Logic controller

4.4.1 Fuzzification

Fuzzificationmaps from the crisp input space to fuzzy sets in certain, input universe of

discourse. So for a specific input value x, it is mapped to the degree of membership A(x).

The fuzzification involves the following functions. Measures the value of input variables.

1. Performs a scale mapping that transfers the range of values of input variables into

corresponding universe of discourse.

2. Performs the function of fuzzification that converts input data into suitable linguistic

variables, which may be viewed as labels of fuzzy sets.

The input variables to fuzzifier are the crisp controlled variables. Selection of the

control variables relies on the nature of the system and its desired output. It is more common

in the literature to use the output error and the derivative of output. Each of the fuzzy logic

control (FLC) input and output signal is interpreted into a number of linguistic variables. The

number of linguistic variables specifies the quality of control which can be achieved using the

fuzzy controller. As the number of linguistic variables increases, the computational time and

required memory increases. Therefore a compromise between the quality of control and

computational time is needed to choose the number is seven.

Each linguistic variables NB, NM, NS, ZE, PS, PM, PB which stands for negative

big, negative medium, negative small, zero positive small, positive medium, positive big

respectively. For simplicity it is assumed that the membership functions are symmetrical and

each one overlaps the adjacent functions by 50% i.e., triangle shaped function, the other type

of functions used are trapezoidal-shaped and Bell-shaped. Fig.5.3 shows the seven linguistic

variable and the triangular membership function with 50% overlap and the universe of

discourse from – a to a [19].

Fig.4.3 Triangular membership functions

4.4.2Knowledge Base (KB)

Knowledge base comprises of the definitions of fuzzy MFs for the input and output

variables and the necessary control rules, which specify the control action by using linguistic

terms.

It consists of a database and linguistic control rule base.

1. The database provides necessary definitions, which are used to define linguistic

control rules and fuzzy data, manipulation in a FLC.

2. The rule base characterizes the control goals and control policy of the domain

experts by means of a set of a set of linguistic control rules.

4.4.3 Inference Mechanism

A Fuzzy interface process consists of following steps:

Step 1:Fuzzification of input variables.

Step 2: Application of Fuzzy operator.

(AND, OR, NOT) In t he If (antecedent) part of the rule.

Step 3: Implication from the antecedent to the consequent (Then part of the rule).

Step 4: Aggregation of the consequents across the rules.

4.4.4Defuzzification and Denormalization

The function of a Defuzzification module (DM) is as follows:

Performs the so-called Defuzzification. which converts the set of modified control

Output values into single point – wise values.

Performs an output denormalization. This maps the point-wise value of the control

output onto its physical domain. This step in not needed if non normalized fuzzy sets is

used.

A Defuzzification strategy is aimed at producing a non-fuzzy control action that best

represents the possibility of an inferred fuzzy control action. Seven strategies in the

literature. among the many that have been proposed by investigators, are popular for

defuzziffying fuzzy output functions:

4.6 Defuzzification methods

Centroid method

Height method

Mean-max method

Sugeno method

The best well-known Defuzzification method is Centroid method.

4.6.1Centroid Method

The Centroid method also referred to as centre-of-area or centre – of – gravity is the

most popular Defuzzification method.

In the discrete case Z ={u1, ….u}) this results in

Zo

=

∑i=1

l

zi( μu( zi )

∑i=1

l

μ( zi)

(4.12)

In the continuous case we obtain

u* =∫u

u−μo( μ )du ¿∫u

μo (μ )du ¿

¿¿

(4.13)

Where ∫ is the classical integral – so this method determines the centre off area below the

combined membership function.

Fig. 5.4 Centre of – area, method of Defuzzification

Fig.5.4 shows the above operation in a graphical way, it can be seen that this

Defuzzification method takes into account the area of U as whole. Thus if the area of two

clipped fuzzy sets constituting ‘U’ overlap, then the over lapping is not reflected in the above

formula. This operation is computationally rather complex and therefore results in quite slow

inference cycles.

5.6.2 Height Method

The COA method is simplified to consider only the height of each contributing

membership function at the midpoint of the base. Graph for height method is shown in Fig.5.5

Fig.5.5 Height method ofDefuzzification.

5.6.3. Mean of Maxima Method

The height method is further simplified in the MOM method, where only the highest

membership function component in the output is considered. If M such maxima are present,

then the formula is

Z0=¿∑m=1

MZmM ❑❑

(4.14)

Zm=mthelement in the universe of discourse, where the output MF is at the maximum value.

And M= number of such elements.

5.6.4. Sugeno method

In this method Defuzzification is very simple. The Defuzzification formula

Z0=K 1DOF 1+¿ K 2 DOF2

DOF1+¿ DOF2¿

¿ (5.15)

CHAPTER-5

CONTROLLER DESIGN:

5.1 D.C. Servo Motor Parameter:

The motor used in this experiment is a 25V D.C. motor with no load speed of 4050 rpm.

Parameter Value

R-resistance 1 Ω

L-inductance 29.79 mH

J-moment of inertia 0.01 kg.m2

Kt -torque constant 0.052 Nm/A

Kb-electromotive force constant 0.1 V/rad/s

B-viscous friction coefficient 0.004 N.m/rad/s

5.2 Block diagram:

Fig.5.1 Block diagram of servomotor

Explanation:

Fig.5.2 Simulink model comparing conventional controllers

Fig 5.3 Graphs comparing performance of conventional controllers

Fig 5.4 Simulink model of fuzzy controller

Fig5.6 FIS file of fuzzy block

Fig5.7 Membership functions of fuzzy controller

Fig.5.8 View ruler scale of fuzzy output

Fig 5.9 Performance output of servomotor using fuzzy controller

Controller Peak time

(t p¿

Rise time

(t r ¿

Delay time

(t d)

Settling time

(t s)

Steady state error

(ess)

Peak over shoot

(%M p)

Without controller

0.74 0.315 0.6456 2.35 0.85 28.1

PD 0.101 0.0355 0.4573 1.44 0.997 77.1

PI 0.802 0.33 0.8869 3.19 1 30.1

PID 0.0776 0.0302 0.612 0.22 1 22

Fuzzy

controller

0.0723 0.028 0.036 0.169 1 11.4