Compensator Design for Speed Control of DC Motor by Root Locus Approach using MATLAB

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 03 Issue: 01 | Jan-2016 www.irjet.net p-ISSN: 2395-0072
© 2016, IRJET | Impact Factor value: 4.45 | ISO 9001:2008 Certified Journal | Page 1245
Compensator Design for Speed Control of DC Motor by Root Locus
Approach using MATLAB
Akshay C. Mahakalkar1, Gaurav R. Powale2, Yogita R. Ashtekar3, Dinesh L. Mute4
1, 2 B.E. 4th Year Student of Electrical Engineering Department, DMIETR, Sawangi (Meghe),Wardha(MH), India
3, 4 Asst. Prof, Electrical Engineering Department, DMIETR, Sawangi (Meghe),Wardha(MH), India
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Abstract - In this paper we focus on stability and
performance of the System. Motors do not give full
stability in its running operations. So as to maintain
maximum accuracy and to get desired response the
compensator circuit is use with system i.e. DC motor.
Thus to get desired output from system it is necessary to
introduce a compensator in system. This can be done by
reducing overshoots and time domain parameters. It is
not possible to study a practical system and evaluate all
minor and major parameters, hence all the study is
done on the basis of Laplace domain. Another problem
encountered is that it becomes difficult to evaluate all
the responses on the basis of first order system.
Therefore to control speed of DC motor we have
assumed the system to be performing on position
control. In this paper compensator is designed by using
MATLAB programming.
Key Words: Compensators, Computer Aided Design,
Desired Response Specifications, MATLAB, Root Locus.
1. INTRODUCTION
Today control systems are playing an important role in
development of modern civilization and technology. Every
system we come across today has some type of control
engineering involved in it. In any sector of industry, you
will find control systems everywhere. Say, inventory
controlled of manufactured products, automatic assembly
line, machine tool control, the space technology and
weapon systems, the robotics, the power plants all are the
industrial sectors. The system to be controlled is given
different names. The most common being a process or a
plant or the controlled system itself.
No system can be ideal and hence there are some power
consuming elements. These causes the fall in efficiency
and performance of the system. The system do not gives
its actual performance. In a DC motor as considered has
various power consuming elements such as inductance,
resistance, friction, inertia, etc. It becomes possible to use
a controller with the system which helps in providing
desired response at the output. This can help to maintain
speed, accuracy and performance of device (motor, etc.).
In this project we are concentrating on compensator as a
controller to a DC motor. A compensator is a component in
a control system that improves an undesirable frequency
response in a feedback and control system. It is a
fundamental building block in classical control theory.
There are two general types of compensators: Lead
Compensators, and Lag Compensators. If we combine the
two types, we can get a special Lead-Lag Compensator
system.
2. PROBLEM STATEMENT
To control the speed of DC motor, primary task is calculate
the transfer function of motor. The transfer function
obtained will be in the form of first order as given below,
Where,
Speed of rotation (Output) Voltage
(Input)
Calculating the response and time domain parameters is
difficult. Thus to make it easier converting the above
equation in second order,
After calculating responses by considering position
control the model will again be converted to velocity control
by using derivative block as stated below,
Fig-1: System block diagram of compensated system
In root locus technique, graphical calculations are more
and becomes a long process to obtain a compensator

model. Thus here we have formed a complete program for
which makes it easy to calculate.
3. DESIGNING PROCEDURE:
3.1 Lead Compensator:
1. Compute the transfer function of the system. Here
computed is,
2. Get required or available value of ζ and ωn through
responses of system or from given parameters and
decide required Ts and Mp and then calculate and
ωn which is required for designing as per
specifications.
3. Plot root locus of open loop T.F. of the system
computed and Plot ‘Sd’= -ζ.ωn .
4. Calculate φ= .
5. Plot compensating zero by assuming it (Assume it just
after system pole).OR Find the by triangle
inequality theorem, from root locus graph.
Fig-2: Root Locus graph for calculation of pole and zero
6. Taking angle φ plot compensating pole to the left of
compensating zero. Find value of gain k by using
formula,
and Form the final transfer function for the lead
compensated system.
3.2 Lag Compensator:
The procedure of lag compensator design is same as
lead only the change in procedure is of calculations for
poles and zeros. So we will use steps of lead upto step 5 as
shown in 3.1. Following are the remaining steps.
1. Calculation for velocity error constant of system by
using,
2. Value of = Assumed greater than factor.
3. Compensating zero is assumed at an angle of less than
10 i.e. 6 taken with ζ line. Calculate compensating
pole by triangle inequality, then find value of gain k
and form the transfer function.
3.3 Lag-Lead Compensator:
The procedure of lag-lead compensator design is same
as lead and lag compensators designed individually.
Following are the steps which give a sequential idea for
designing Lag-Lead compensator.
1. Compute the transfer function of the system. Here
computed is,
2. Get required or available value of ζ and ωn through
responses of system or from given parameters and
decide required Ts and Mp and then calculate and
ωn which is required for designing as per
specifications.
3. Plot root locus of open loop T.F. of the system
computed and Plot ‘Sd’= -ζ.ωn .
4. Calculate φ= .
5. Now, calculate lead branch as per part (3.1) and lag
branch as per part (3.2), then calculate combined gain
to compensate and form the final compensated
system.
4. PROGRAM FLOW CHART:
Input Transfer Function
=17.146/s(0.043s+1)
Evaluation of Response Parameters
Mp=10% Ts=0.7sec
Input Desired Parameters
ζ=0.8261 ωn=48.4212
Calculation of Complex Dominant Pole
Sd=-40.00+27.2875j
Calculation of Pole & Zero
Cz = 41.3220 Cp = infinity
Calculation of Gain
K= 0.1423
Form the Transfer Function
=(56.74s+2345)/(s^2 + 23.25s)
Evaluation of Compensated
Response
Fig-3 Algorithm for Lead Compensator

Figure 3 shows the algorithm flow chart for the lead
compensator. In that explained step by step procedure for
evaluation of parameter for lead compensator. Similarly
figure 4 and figure 5 shows the algorithm flow charts for
lag and lag-lead compensator respectively.
=17.146/s(0.043s+1)
Mp=10% Ts=0.7sec
ζ=0.69 ωn=16.10
Sd=-11.11 +11.65j
Calculation of Pole & Zero
Cz = 2.1253 Cp = 1.2150
Calculation of Gain
K= 0.705
=(281.5 s + 598.2)/(s^3 + 24.46 s^2 + 28.25 s)
Response
Input Velocity Error Constant
Kv= 30
Fig-4 Algorithm for Lag Compensator
=17.146/s(0.043s+1)
Mp=10% Ts=0.7sec
ζ=0.8261 ωn=48.4212
Sd=-40.00+27.2875j
Calculation of Lag Branch
Cz2 = 7.8252 Cp2 = 7.4557
Calculation of Gain
K= 0.143
=(57.13 s^2 + 2807 s + 1.847e004)/(s^3 + 30.71 s^2 +
173.3 s)
Response
Input Velocity Error Constant
Kv= 18
Calculation of Lead Branch
Cz1 = 41.322 Cp1 = Infinity
Fig-5 Algorithm for Lag-Lead Compensator

5. RESULTS AND DISCUSSION
The responses obtained for Lead Compensator are plotted
as shown in figure 6 and the performance parameters
calculated and compared in table 1 which is directly
calculated from MATLAB program designed as per the
flow chart shown in figure 3. In lead compensator the pole
obtained by the formula for solving triangle is found to be
infinity, as this value is not possible we considered its
impedance to be infinity and behaves as open circuit.
The response obtained for Lag compensated system with
comparison to original one is shown in figure 7. The
evaluated values of different performance parameters of
responses is tabulated in table 1. Also figure 4 shows the
flow chart for the MATLAB program through which it is
evaluated. Lag compensator reduces the distortion but
takes a large settling time in compare to lead, thus this
proves that lag improves steady state time of response.
The steps as evaluated are as similar to lead. As per the
study it is found that the response of lag is not very
flexible which proves that lag is invalid for higher order
systems.
In Lag-Lead compensator the respective response and the
evaluated results of different performance parameters is
as shown in the figure 8 and table 1 with respect to results
evaluated as per MATLAB program as per flow chart figure
5. From the response it is clear that lag-lead compensator
is having both properties of lag of improving steady state
and lead for improving transient response.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Step Response for Compensated and Uncompensated System
Time (sec)
Amplitude
Compensated Lead Response
Uncompensated Response
Fig-6 Step response for compensated and uncompensated
system. (Lead Compensator)
0 0.5 1 1.5 2 2.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (sec)
Amplitude
Lag Compensated System
Uncompensated System
system. (Lag Compensator)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (sec)
Amplitude
Compensated System
Uncompensated System
system. (Lag-Lead Compensator)

Param
eters
Uncomp
ensated
System
Lead
Compen
sated
System
Lag
Compensa
ted System
Lag-Lead
Compensat
ed System
Peak
oversh
oot
10% 6% 12% 7%
Settlin
g time
(sec)
0.7 0.129 1.5 0.131
Rise
time
(sec)
0.135 0.038 0.2 0.037
Peak
time
(sec)
0.194 0.06 0.25 0.057
Table: 1: Comparison between parameters of Lead, Lag
and Lag-Lead compensator.
6. CONCLUSIONS
From the above results, the program is giving optimum
value of compensation. After obtaining the compensated
values it can be added with derivative block to get speed
control response. From the evaluation it is found that the
value of lead compensator can be varied as per conditions, but
choosing value of lag compensation is limited to certain
proportion only. From the simulation result of lead and lag
compensator it’s seen that lead compensator improves the
transient response and lag compensator improves the steady
state response.
As the lag-lead compensator response shown in figure 8 is
giving improved as compared to lag and lead compensators
acting individually and improves both transient and steady
state response.
REFERENCES
[1] Godhwani Arjun ”feedback control system”, IEEE
transaction, pp 1758-1764, 2003
[2] Wang Fei-Yue, Huang Yue,”A non-trail-and-error
method for phase-lead and phase lag
compensator Design”, IEEE transaction, pp 1654-
1660.2001
[3] Huey Yang Horng, “Lead-Lag Compensator Design
Based on Genetic Algorithms”, Conference on
Technologies and Applications of Artificial
Intelligence, IEEE, 2012
[4] Ishaq Sheikh, ’’Computer Aided Design of Lead
compensator using Root Locus Method”, National
conference on “Recent Research & Development in
Core Discipline of Engineering, Vadodara, April 2015
[5] C. S. Linda, K. K, ’’ Analysis and Design of Conventional
Controller for Speed Control of DC Motor -A MATLAB
Approach”, Int. Journal of Engineering Research and
Applications Vol. 5, Issue 2, (Part -5) February 2015,
pp.56-61
[6] Mute Dinesh L., et al. "System Identification Using
Neural Network Model for Speed Control of DC
Motor." International Research Journal of Engineering
and Technology, Vol 2 Issue 1(2015).
[7] K. Ogata, “Modern Control Engineering”, PHI learning
publications pvt. Ltd.
[8] William J. Paim-3, “A concise Introduction to
MATLAB”, Tata McGraw-Hill Edition.
[9] Prof. M. Gopal, ”Control Systems”, IIT Delhi, video
lectures NPTEL.

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