Research Directions
Volume 1 , Issue 6 / Dec 2013
ISSN:-2321-5488
Research Article
COMPARATIVE STUDY ON DC MOTOR SPEED
CONTROL USING VARIOUS CONTROLLERS
K.VENKATESWARLU AND CH.CHENGAIAH
PG Student
Associate Professor
Abstract:
Modern manufacturing systems are automated machines that perform the
required tasks. The electric motors are perhaps the most widely used energy converters
in the modern machine tools and robots. These motors require automatic control of their
main parameters such as speed, position, acceleration etc..Electrical machines like DC
motors, brushless DC motors, permanent magnet DC motors are being controlled with
power electronics converters. The control has become precise with invention of Micro
Controllers and power devices like IGBT , Power MOSFET .The introduction of
MATLAB made the designers to simulate complex circuits with the modern electric
drives or power converter circuits. In this project the attempt is made to simulate a speed
control of separately excited DC motor with PID and FUZZY Controllers.The aim of
development of this project is towards providing efficient method for control speed of DC
motor using analog Controller.Withthe availability of MATLAB/SIMULINK,
FuzzyController for comprehensive study of modeling analysis and speed control
design methods has been demonstrated.
KEY WORDS:
DCmotor, open loop, closed loop, system, speed control,PID controller and FUZZY controller..
INTRODUCTION:
The field of electrical energy may be divided into three areas of specialization: Electronics, Power
and Control. Electronics basically deals with the study of semiconductor devices and circuits at lower
power .Power involves generation transmission and distribution of electrical energy .the stability and
response characteristics of closed loop transfer function is the concerning of control area of specialization.
In recent times many new modernelectromechanical energy conversion machines have been invented.
Electrical machines like dc motors, brushless dc motors, permanent magnet dc motors are being controlled
with power electronics converters. Power electronics deals with use of electronics for the control of large
power . The control has become precise with invention of microcontrollers and power devices like IGBTs ,
Power MOSFET .
In this project dc motor isused ,since it has various applications such as defense, industries,
Robotics etc. In this project separately excited DC drive system is used, because of their simplicity, ease of
application, reliability and favorable cost have long been a backbone of industrial applications and it will
have a long tradition of use as adjustable speed machines and a wide range of options have evolved for this
purpose. In these applications, the motor should be precisely controlled to give the desired performance.
Many varieties of control schemes such asproportional,integral,derivative, proportional integral
(PI), PID, adaptive, and FLCs, have been developed for speed control of dc motors[8]. The speed of dc
motor is controlled by varying the voltage to its armature or field.by varying the voltage to the field , one can
only run the motors at speed higher than base speed at the expense of the falling torque. The important
aspect of the speed control dc motor is the armature voltage control method .By varying voltage to the
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COMPARATIVE STUDY ON DC MOTOR SPEED CONTROL USING VARIOUS CONTROLLERS
armature of the dc motor the speed of the motor should be varied. Speed of DC motor can be controlled by
PID controller.
In this project mainly concentrated on speed control of separately excited DC motor with
designed a mathematical model of the separately excited DC motor using MATLAB code and SIMULINK
model is used for studying the performance characteristics of dc motor and concentrated on the design of
PID controller using MATLAB SIMULINK model for improving the drawbacks in normal system(dead
time ,rise time etc. remains large).
In order to eliminate the dead time present in the model and to improve the system performance
,FUZZY controller (PID) using MATLAB was proposed.
MATHEMATICAL MODELING OF SEPARATELY EXCITED DC MOTOR AND STABILITY
ANALYSIS
In order to build the DC motor's transfer function, its simplified mathematical model has been
used.This model consists of differential equations for the electrical part,mechanical part and the
interconnection between them. Theelectric circuit of the armature and the free body diagram of therotor in
the Fig 3.1 below.[1]
Fig.1: The electric circuit of armature and the free body diagram
motor system of the rotor for a DC motor
Table1:Physical parameters of the DC motor
Here consider 600 rpm as reference speed and it is considered as 1 pu.The motor torque, Tm, is
related to the armature current, i, by a constant factor Kt. The back emfem is related to the rotational speed
by the following equations
= I
=
………….(1)
………….(2)
Assuming that, Kt(torque constant) =Ke(electromotive force constant) =Km (motor constant).
From Fig.1 and physical parameters of system mentioned above, the following equations can be
writtenbased on Newton's law combined with Kirchoff's laws.
J +b = i- …… (3)
+ i=V..(4)
Research Directions • Volume 1 Issue 6 • Dec 2013
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COMPARATIVE STUDY ON DC MOTOR SPEED CONTROL USING VARIOUS CONTROLLERS
a)Transfer FunctionModel of DC Motor:
Using Laplace Transforms, the above equations can be expressed in terms of s-domain
s(Js+b) (s)= I(s)- (s)……….(5)
( (s)+ )I(s)=V(s)- s (s)…(6)
By eliminating I(s), the following open-loop transfer function can be obtained, where the
rotational speed is the output and the voltage V is the input.
When the motor is used as a component in a system, it is desired to describe it by the appropriate
transfer function between the motor voltage and its speed. For this purpose assuming (load torque) TL=0 and
(friction torque) Tf=0, since
neither affects the transfer function.
(7)
b)State-SpaceModel of DC Motor:
In the state-space form, Eqs.3 and 4 can be expressed by choosing speedandi as the state variable
and V as an input.
Basic form of state equation is
=Ax(t)+Bu(t). ..............(8)
Output equation
y(t)=Cx(t)+Du(t).................(9)
wherex(t)is the state variable matrix; r(t)is the input; y(t)is the output; and a, b, c, and d are
constant coefficient matrices.
For the system considered in this project state equation and output equation are
=
=
+
V.....(10)
..............................(11)
c)DESIGN REQUIREMENTS:
The design requirements for the separately excited DC motor are as follows.
Since the most basic requirements of a motor are that it should rotate at the desired speed, the
steady-state error essof the motor speed should be less than 1%. The other performance requirement is that
the motor must accelerate to its steady-state speed as soon as it turns on. In this case, it is desirable to have a
settling time Ts in less than 2sec. Since speed faster than the reference may damage the equipment, a peak
overshoot Mp ofless than 5% is wanted.
c.1)Open-loop Response:
The transfer function in Eq. 7 can be represented into MATLAB by defining the numerator (num)
and denominator (den) matrices as follows: num=Km and den=(Js+b)(Lm s + Rm) + Km2 The motor
parameters were determined on the basis of its step response about a steady-state operating point, namely
input voltage V= 6 volts and speed è = 600 r/s. Also, it makes the motor takes 3sec to reach its steady-state
speed; this does not satisfy the 2sec settling time criterion.
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COMPARATIVE STUDY ON DC MOTOR SPEED CONTROL USING VARIOUS CONTROLLERS
c.2)Closed Loop Control System:
In a closed-loop system the control input is affected by the system output.By usingoutput
information to affect in some way the control input ofthe system, feedback is being applied to that system. It
is often the case that the signal fed back from the system output is compared with a reference input signal,
the result of this comparison (the difference) then being used to obtain the control or actuating system input.
Fig 2:Closed loop control system
Such a closed-loop system is shown in Fig.2,where the error = reference input - system output.
Very often the reference input is directly related to the desired value ofsystem output, and where this is a
steady value with respect to time it is called a set point input. By means ofthe negative feedback loop output
is subtracted from the reference input the accuracy of the system output in relation to a desired value can be
much improved when compared to the response of an open-loop system. This is simply because the purpose
of the controller will most likely be to minimize the error between the actual system output and the desired
(reference input) value.
c.3)Open Loop and Closed Loop Response Results Using MATLAB Coding:
With the help of MATLAB programming , the performance of separately excited DC motor in time
domain was obtained and the results are shown in table 2 below.
Table 2:Comparison of results with and without feedback
Results
without
feedback
RiseTim e: 1.1362 sec
SettlingTime:
2.0653
sec
Settling Min: 0.0901
Settling Max: 0.0999
Overshoot: 0
Undershoot: 0
Peak: 0.0999
PeakT im e:5.2388 sec
Research Directions • Volume 1 Issue 6 • Dec 2013
Results with feedback
RiseTime: 1.017 sec
SettlingTim e:
1.847
sec
SettlingMin: 0.0825
SettlingMax: 0.090
Overshoot: 0
Undershoot: 0
Peak: 0.090
PeakTim e:4.6398 sec
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COMPARATIVE STUDY ON DC MOTOR SPEED CONTROL USING VARIOUS CONTROLLERS
Fig 3.open loop and closed loop
Here we have observed that gain in closed loop response decreases as we have employed -ve
feedback for the system ( from fig 3) . But system performance has been improved i.e system reached its
steady state output much before the open loop system. Here in the above case settling time is about 4
seconds which is very high .this much amount of settling time is not desirable in the system. There is a dead
time in the system also about 1 sec which is also very high. So by employing this methods we have not
reached our objectives ,
d)Root Locus Control Design Method:
For designing a controller using the root locus method, it should follow the following steps:
i. Open-loop Root Locus:
The main idea of root locus design is to find the closed-loop response from the open-loop root
locus plot. Then by adding zeros and/or poles to the original plant, the closed-loop response can be
modified. So, first, it is necessary viewing in the root locus for the plant. Three arguments must be founded
are the damping ratio (zeta) term (î> 0.8 corresponds to an overshoot the natural frequency (ùn) term (= 0
corresponds to no rise time criterion), and the sigma term (4.6/2 = 2.3 sec). Using these criteria results a
root locus plot shown in Fig. 8. The following three equations are used in continuous system designs: îùn≥
４．６　／
Ts (Sigma term) (11)
ùn≥
１．８　／　Tr ………..….(12)
Results
without
feedback
RiseTime: 1.1362 sec
SettlingTime: 2.0653
sec
Settling Min: 0.0901
Settling Max: 0.0999
Overshoot: 0
Undershoot: 0
Peak: 0.0999
PeakTime:5.2388 sec
î?
Results with feedback
RiseTime: 1.017 sec
SettlingTime:
1.847
sec
SettlingMin: 0.0825
SettlingMax: 0.090
Overshoot: 0
Undershoot: 0
Peak: 0.090
PeakTime:4.6398 sec
..(13)
where î: Damping ratio, ùn: Naturalfrequency, Ts: Settling time, Tr: Risetime and Mp: Peak
overshoot [11].
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COMPARATIVE STUDY ON DC MOTOR SPEED CONTROL USING VARIOUS CONTROLLERS
Original Bode Plot:
The main idea of frequency-baseddesign is to use the Bode plot of theopen-loop transfer function
toestimate the closed-loop response.
Adding a controller to the systemchanges the open-loop Bode plot,therefore changing the closedloopresponse. First, the Bode plot for theoriginal open-loop transfer function From Bode Plot we can obtain
gain margin and phase margin and from these two values we can analyse the stability. If both GM and PM
are greater than zero then the system is stable.If both GM and PM are equal to zero then the system is
marginally stable. If both GM and PM are less than zero then the system is unstable.
With the help of MATLAB programming , the performance of separately excited DC motor in time
domain and frequency domain was obtained and the results are shown in table 3 below and various plots of
the system are shown in fig 4.
Table 3 :Time and Frequency responses:
In the following figures various plots were discussed:
Fig 4: various plots of system funcions.
PID CONTROLLER DESIGN:
Types ofControllers :
The basic controllers depending upon the manner in which the control signal is produced are
classified as
1) Proportional Controller
2) Derivative Controller
3) Integral Controller
4) Proportional - Plus - Integral Controller (PI)
5)ProportionalPlus-Derivative Controller (PD)
6) Proportional Plus - Integral Plus - Derivative Controller (PID)
1 Proportional Controller(P):
The controller for which the control signal at the output of the controller is simply related to the
input of the controller by a proportional constant is known as proportional controller. If u(t) is the control
Research Directions • Volume 1 Issue 6 • Dec 2013
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COMPARATIVE STUDY ON DC MOTOR SPEED CONTROL USING VARIOUS CONTROLLERS
Time
response
characteristics
RiseTime: 1.1362
sec
SettlingTime:
2.0653 sec
SettlingMin:
0.0901
SettlingMax :
0.0999
Overshoot: 0
Undershoot: 0
Peak: 0.0999
PeakTime: 5.2388
sec
Frequency
response
characteristics
GainMargin: Inf
GMFrequency: Inf
PhaseMargin: Inf
PMFrequency: Inf
DelayMargin: Inf
DMFrequency :Inf
Stable: 1
signal at the output of the controller and e(t) is theinput of the controller, then u(t) á e(t) or u(t) =Kp
e(t) where Kp is termed as proportional gain. The block diagram shown in figure4.1 represents the
proportional controller
4.1 Block diagram representation ofProportional Controller
Fig 5:Characteristics of Proportional Controller
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COMPARATIVE STUDY ON DC MOTOR SPEED CONTROL USING VARIOUS CONTROLLERS
In the above Fig.5, PB, Proportional band is a band of error where every controller output has
proportional error signal or vice-versa. The proportional band should not be narrow or wide because if it is
narrow, the controller mode exhibits the characteristics of the discontinuous mode of controller. If it is
mode, it has permanent residual error known as offset in its operating point. Proportional controller with
very high gain acts as two -position controller. The forward gain Kp has effect on both transient response
and the steady state response .
If Kp increases, steady state error decreases. Rise time and setting time decreases and overall
steady state response improves.The main drawback of proportional control system is it has maximum
overshoots and is more oscillatory. This isobjectionable for many of the control system
applications.Therefore, while going in for proportional control action forimproving system performance, a
control system engineer will have to make a compromise in choosing a proper gain K, so that the steady
state error and the maximum overshoot of the output response are within the acceptable limits. This will
become a difficult proposition.
2. Integral Controller(I):
In the proportional control of a plant whose transfer function does not possess an integrator 1/s,
there is a steady state error or offset, the response to a step input such an included in the controller. In the
integral control of a plant, the control signal, the output signal from the controller, at any instant is the area
under the actuating error signal curve up to that instant. The control signal u(t) can have a non zero value
when the actuating error signal e(t) is zero. This is impossible in the case of proportional controller since a
non zero control signal requires a non zeroactuating error signal.
Therefore u(t) á ∫e(t)d(t) or u(t) = Ki e(t)dt
Where Ki — Integral gain constant
Genetic algorithm that yields good results in many practical problems is composed of three
operators:Selection:Theindividualsare selected randomly at initial stage & organized pair-wise using
Roulatte wheel technique, for the implementation of crossover. Crossover: The individuals, randomly
organized pair-wise, have their space locations combined, in such a way that each former pair of individuals
gives rise to a new pair.Mutation: Some individuals are randomly modified, in order to reach other points of
the search space.
Block diagram representation of integral controller.
3.Derivative Controller (D):
The derivative controllers are implemented to account for future values, by taking the derivative,
and controlling based on where the signal is going to be in the future. Derivative controllers should be used
with care, because even small amount of high-frequency noise can cause very large derivatives, which
appear like amplified noise. Also, Derivative controllers are difficult to implement perfectly in hardware or
software, so frequently solutions involving only integral controllers or proportional controllers are
preferred over using derivative controllers
Block diagram representation of derivative controller.
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COMPARATIVE STUDY ON DC MOTOR SPEED CONTROL USING VARIOUS CONTROLLERS
4.PID Controller :
The PD controller could add damping to a system, but the steady-state response is not affected.
The PI controller could improve the relative stability and improve the steady-state error at the same time,
but the rise time is increased. This leads to the motivation of using a PID controller so that the best features
of each of the PI and PD controllers are utilized
Fig 6. Block diagram representation of PID Controller.
(s)=
+
s+ ................(14)
The PID controllers are the standard tools for theindustrialautomation. The flexibility of the
controller makes it possible to use PID control in many situations. The controllers can also be used in
cascade control and other controller configurations. Many simple control problems can be handled very
well by PID control, provided that the performance requirements are not too high. The PID algorithm is
packaged in the form of standard regulators for process control and is also the basis of many tailormade
control systems. Transfer function of PID controller is As an alternative, the PI portion of the controller can
be designed first for a portion ofthe requirement on relative stability, and, finally, the PD portion is
designed.
With the help of MATLAB programming , the performance of separately excited DC motor with
and without PID controller was obtained and results are tabulated in table 3 and corresponding MATLAB
coding was presented.
Table 4:Comparision of results for open ,closed loop and PID controller responses
Open
loop Closed loop Response
response
response
with
PID
controller
Rise Time:
1.1362 sec
Settling
Time:
2.0653 sec
Settling
Min: 0.0901
Settling
Max: 0.0999
Overshoot:
0
Undershoot:
0
Peak:
0.0999
Peak Tim e:
5.2388 sec
Rise Time:
1.0173 sec
Settling
Time:
1.8476 sec
Settling
Min: 0.0825
Settling
Max: 0.0908
Overshoot:
0
Undershoot:
0
Peak:
0.0908
Peak Time:
4.6398 sec
Research Directions • Volume 1 Issue 6 • Dec 2013
Rise Tim e:
0.1195 sec
Settling
Time:
16.0857 sec
Settling
Min: 0.8841
Settling
Max: 1.0878
Overshoot:
8.7813
Undershoot:
0
Peak:
1.0878
Peak Tim e:
0.2337 sec
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COMPARATIVE STUDY ON DC MOTOR SPEED CONTROL USING VARIOUS CONTROLLERS
Here a clear cut comparison is given for various values like rise time,dead time etc.. in table 3 . By
employing this PID controller we can see(Fig 4.9) that it has drastically reduced from 1.1362 in openloop to
0.1195 in closed loop system. Coming to peak time there is significant change from open loop to PID
controlled system i.e from 5.2388 seconds to 0.2337. but coming to settling time there is a drastic change
from 2.0653 to 16.0857 , this is not desirable . There is change in dead time i.e improvement in dead time
from 4 sec to 2 seconds in this system .
SIMULINK Model for Implementation of PIDController
Fig 7:Simulink model of PID controller
Fig 8:Response with PID controller
FUZZY CONTROLLER DESIGN:
Fuzzy logic is a problem-solving control system methodology that leads itself to implementation
in systems ranging from simple, small, embedded micro controllers to large, networked, multi-channel
PCor workstation-based data acquisition and control systems. Fuzzylogic was conceived as a better method
for sorting and handling data but has proven to be an excellent choice for many control system applications
since it mimics human control logic.
a)Fuzzy Logic Controllers :
Once the system behavior is thoroughly studied, comprehensiveperformance objectives of the
controller can be speculated. Then the help of intelligent trial and error science as well as self-institution
learned from experience can formulate the heuristic rules that describe the system behavior equation. This
forms the basis for the design of a Fuzzy logic controller as shown in Fig 9 gives an idea of the construction
of a Fuzzy logic based controller. The principal design parameters for a Fuzzy logic controller arethe
following:
Data Base :
1. Discretization / normalization of universe of discourse,
2. Fuzzy partition of the input and output spaces,
3. Completeness,
4. Choice of the membership function of a primary Fuzzy set;
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COMPARATIVE STUDY ON DC MOTOR SPEED CONTROL USING VARIOUS CONTROLLERS
Rule base:
1. Choice of process state (input) variables and control (output)variables of Fuzzy control rules,
2. Source and derivation of Fuzzy control rules,
3. Types of Fuzzy control rules,
4. Consistency interactivity, completeness of Fuzzy control rules Fuzzy inference mechanism,
Defuzzification strategies and the interpretation of a defuzzificationoperator.
Fig 9:Control System with Fuzzy Logic Controller
b)Membership functions :
Membership functions form a crucial part in a Fuzzy rule base model because they only actually
define the fuzziness of a control variable or a process variable. The most popular choices for the shape of the
membership functions are triangular and trapezoidal functions. It is easy to see that parametric choice of the
triangular function is the most economical one. This explains the predominant use of this type of
membership functions. Once the shape of the function is selected, one has to map each element of the term
set on the domain of the correspondinglinguistic variable. This mapping can affect the nature of the Fuzzy
controller in a number of ways Triangular Membership Function The triangular membership function can
be generally defined using aleft point, center point, and right point. Using these we can calculate the slopes
of the sides of the triangle. Once the triangle is thus constructed, membership values can be calculated.
Trapezoidal Membership Function
The maximum membership value of 1.0 occurs over small rangeabout the central point of the
function. Studies show in general that these functions give a better performance than triangular membership
functions. Oscillations encountered when triangular functions were used greatly reduces. This can be
attributed to the flat top of the membership function. The flat top actually stabilizes the decision over a range
and there forethereis not great change in the decision as would happened in a triangular membership
function.
c)DefuzzificationModule :
The functions of a defuzzification module (DM) are as follows :
Performs the defuzzification, which converts the set of modified control output values into single pointwise values.
Performs an output denormalization, which maps the point-wise value of the control output onto its
physical domain. This step is not needed if non normalized Fuzzy sets are 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 defuzzifyingFuzzy output functions:
1. Max-membership principle
2. Centroid method
3. Mean-max membership
4. Center of sums
5. First (or last) of maxima
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COMPARATIVE STUDY ON DC MOTOR SPEED CONTROL USING VARIOUS CONTROLLERS
The best well-known defuzzification method is centroid method.
c.1)Centroid Method :
The centroid method also referred to as center - of - area or center - of - gravity is the most popular
defuzzification method.
In the discrete case (U = {u1,…………… un} this results in
=
...........
(15)
In the continuous case we obtain classical integral
Fig. 10.shows the above operation in a graphical way. It can be seen that this defuzzification
method takes into account the area of U as a whole. Thus the areas of two clipped Fuzzy sets continuing U
overlap, then the overlapping area is not reflected in the above formula. This operation is computationally
rather complex and therefore results in quiteslowinferencecycles.
Fig 10:Centre-of-area method of Defuzzification
d)Design of PID FuzzyController :
As stated earlier, for a PID controller, the inputs to Fuzzy controller are Kp,Kd and Ki. and the
output of Fuzzy controller in u. To start with, we have to activate Basic Fuzzy Inference System (FIS) editor
either by typing FIS editor viewer in the Fuzzy logic tool box of the launchpad
Fig 11:Simulink model for FUZZY controller
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COMPARATIVE STUDY ON DC MOTOR SPEED CONTROL USING VARIOUS CONTROLLERS
O
u
t
p
u
t
Time in sec
Fig 12:Response with FUZZY controller
Any Fuzzy controller involves the following:
Basic FIS editor
Membership function
Rule editor
Rule viewer
Surface viewer
Table 5: Comparision of results.
Open Loop
Closed
P ID
FUZZY
R isetime
Rise time
1.5sec
0.2sec
S ettling
Settling
time
time 0.6
Loop
R iseTime:
RiseTime:
1.1362 sec
1.017 sec
S ettling
SettlingTi
T ime:
me:1.847s
2.0653sec
ec
P eak:
Peak:
0.0999
0.090
P eakTime:
Peak
5.2388 sec
Time:4.63
1.2sec
Deadtime
Dead
0 sec
time 1sec
98sec
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COMPARATIVE STUDY ON DC MOTOR SPEED CONTROL USING VARIOUS CONTROLLERS
all these shows a great change in the performance of the system. However there is peak overshoot
and steady state error .This stedy state error can be removed by increasing the gain and peak overshoot
automatically will reduce as load is employed on the system. Hence these will not pose any problem on
system performance .
The above Simulink results is tabulated in Table 5.
From fig12 there is no dead time in the system i.e dead time is 0. Hence we have reached one of our
aim .there is a considerable decrese in risee time which is in the order of 0.2sec . This shows how fastly the
system is responding .system has reached its steady state before 0.6 seconds , all these shows a great change
in the performance of the system. However there is peak overshoot and steady state error . This stedy state
error can be removed by increasing the gain and peak overshoot automatically will reduce as load is
employed on the system. Hence these will not pose any problem on system performance .
CONCLUSIONS
Speed response characteristics of separately excited dc motor were obtained by mathematical
model using MATLABcoding and SIMULINK model .The response is found to be not satisfactory i.e
response doesn't satisfy the desired design requirements like rise time ,settling time ,peak value ,steady state
error and dead time etc. There exists a dead time of 1 sec which is a major drawback to the system by
conventional method.
To overcome the above drawback we employed PID controller design, by proper tuning of KP, KI
and Kdwe have improved the characteristics like steady state error .But the above designed system failed to
reduce the dead time of the system. Hence in order to reduce the dead time modern technique like FUZZY
controller was employed.
FUZZY controller is proposed to replace conventional PID controller to improve the system
characteristics.. The corresponding step response is very smooth and ripple free The rule base adopted is of
MAMDANItype and its rule viewer is presented .A thorough tuning of rule base may be required for
improvement of the response further.
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[10] A. Klee, Development of a Motor Speed Control System Using MATLAB and Simulink, Implemented
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