Study of DC Motor Position Control using Root Locus and PID Controller in MATLAB

Study of DC Motor Position Control using Root Locus and PID

Controller in MATLAB

Khushboo Sao

1

Mr. Dharmendra Kumar Singh

2

Mr. Amit Agrawal

3

1

PG Scholar

2

HOD

3

Assistant Professor

1,2

Department of

Electrical & Electronics Engineering

3

Department of

Electrical Engineering

1,2,3

Dr.C.V. Raman Institute & Science & Technology Bilaspur, Chhattisgarh, India

Abstract—In this paper design an electric drive with a dc

motor, with the required angular position controlled in two regimes in the given desired reference command signal . In our practical world we have to use a lot of controlling process in industry or any engineering section. So, it does vary initiative attempt to design a control drive in corresponding to our practical field. 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 (position, speed, acceleration, currents). In this case the project is representation of a work on the control drive design. Here, focusing control based on the position and various method (root locus, state space, PID and frequency control method) was using to designing this position control drive. And all the designing was related with the DC motor .For using in the practical field, this project used the real world catalog value. DC motor system, the use of MATLAB for comprehensive study of analysis, performance and position control design methods has been demonstrated.

Key words:Position Control, Root locus, PID Controller

I. INTRODUCTION

DC motors are designed for use in industrial and commercial applications. Now a days many applications find the usage of DC motors as it provides excellent speed control for both acceleration and deceleration with effective and simple torque control. it is widely used in robot manipulators and home appliances where speed and position control are required. D.C motors have long been the primary means of electric traction and they are also used for mobile equipment such as golf carts, quarry and mining applications.

With the growing of technology, the new applications find the use of both speed control as well as position control for better and efficient performance. In addition to the existing control techniques, we introduce a control technique, which helps in the position control of DC motor by control system.

The purpose of developing a control system is to enable stable control since it has parameters tuning difficulties, non-linearity, poor stability and imprecise control. The mathematical model of the basic open loop system, to be derived and analyzed, will be followed by designing the closed loop system, controller models added, different control strategies will be applied with the closed loop, tested, analyzed and finally, based on system controller selection and design, overall controller effect, system performance and analysis, the control strategy will be selected, design and dynamic analysis will be verified by MATLAB/Simulink.

The purpose of developing a control system is to enable stable and reliable control. Once the control system has been specified and the type of control has been decided, then the design and analysis are done.

There are three major objectives of system analysis and design: producing the desired transient response, reducing steady-state error, and achieving stability. The objective is to design a DC Motor position controller using Root Locus Method and PID controller considering disturbance in the system and reduce the effect of the disturbance to zero.

The uncompensated motor may only rotate at 0.1 rad/sec with an input voltage of 1 V. Since the most basic requirement of a motor is that it should rotate at the desired speed, the steady-state error of 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, we want it to have a settling time of 2 seconds for example. Since a speed faster than the reference may damage the equipment, we want to have an overshoot of less than 16.In this project, we are to design, apply, verify and performance of different control strategies in order to improve the desired transient response, reducing steady state errors and achieving stability to control the output angular position, θ of a given DC motor, corresponding to applied input voltage Vin. If the simulation is done with the reference input r(t) by a unit step input, then the motor speed output should have

 Settling time less than 40 milliseconds

 Overshoot less than 16%

 No steady state error

 No steady state error due to a disturbance

Finally it has been observed that the values of peak overshoot, settling time and other parameters, attain values which improve the stability of the system.

II. PHYSICAL SETUP

[image:2.595.59.274.56.166.2]

Fig. 1 Schematic diagram of a DC Motor

A. System Equation

The motor torque T is related to the armature current, i , by a torque constant K;

ki

T



Equation 1

The generated voltage,

e

_a, is relative to angular velocity by;

dt d k K ea m

  

 Equation 2

From Figure 1 we can write the following equations based on the Newton‟s law combined with the Kirchhoff‟s law:

ki dt d b t d d

J 2    2

Equation 3

dt d K V Ri dt di

L     Equation 4

B. Transfer Function

Using the Laplace transform, equations 3 and 4 can be written as:

 

s KI s bs s

Js2( ) ( )

Equation 5

)

(

)

(

)

(

)

(

I

RI

s

V

s

Ks

s

Ls









_{Equation 6}

Where s denotes the Laplace operator. From 6 we can express I(s):

Ls R

s Ks S V s I

 

 ( ) ( )

)

( 

Equation 4.29 and substitute it in (5) to obtain:

Ls R

s Ks s V K s bs s Js

  

 ( ) ( ( ) ( )) )

(

2  

EQUATION 7

This equation for the DC motor is shown in the block diagram in Fig.4.43, From equation (4.30), the transfer function from the input voltage, V (s), to the output angle, θ, directly follows:



{( )( ) }



) (

) ( )

( ₂

K b Js sL R s

K s

V s s G

  





[image:2.595.61.269.653.748.2]

Before any consideration of the above equations, we must know the constant values of data, K, J, b, V, L and R. This is very important to the application of DC motor which we will be used.

Fig. 2: A Closed- Loop System that Representing the DC Motor

Assume the following values for the physical parameters. (J) Moment of inertia of the rotor 3.2284E-6 kg.m^2 (b) Motor viscous friction constant 3.5077E-6 N.m.s (Kb) Electromotive force constant 0.0274 V/rad/sec (Kt) Motor torque constant 0.0274 N.m/Amp (R) Electric resistance 4 Ohm

(L) Electric inductance 2.75E-6H Assume that

1) The input of the system is the voltage source (V) applied to the motor's armature.

2) The output is the position of the shaft (theta). 3) The rotor and shaft are assumed to be rigid. 4) Viscous friction model, that is, the friction torque is

proportional to shaft angular velocity.

III. MATLAB CODING

A. Transfer Function

Running this code in the command window produces the output shown below.

1) For Open Loop Transfer Fuction J = 3.2284E-6;

b = 3.5077E-6; K = 0.0274; R = 4; L = 2.75E-6; s = tf('s');

P_motor = K/(s*((J*s+b)*(L*s+R)+K^2)) 2) Transfer function:

0.0274

--- 8.878e-12 s^3 + 1.291e-05 s^2 + 0.0007648 s 3) For Closed Loop Transfer Function

J = 3.2284E-6; b = 3.5077E-6; K = 0.0274; R = 4; L = 2.75E-6; s = tf('s');

P_motor = K/(s*((J*s+b)*(L*s+R)+K^2)) sys_cl = feedback(P_motor,1)

4) Transfer function:

0.0274

--- 8.878e-012 s^3 + 1.291e-005 s^2 + 0.0007648 s + 0.027

B. Step Response for Open Loop System J = 3.2284E-6;

b = 3.5077E-6; K = 0.0274; R = 4; L = 2.75E-6; s = tf('s');

[image:3.595.53.280.57.260.2]

Fig. 3 Step Response Curve for Open Loop System The open-loop response of the system is not even stable. Stability of a system can be verified with the MATLAB command isstable where a returned value of TRUE (1) indicates that the system is stable and a returned value of FALSE (0) indicates that the system is not stable. isstable(P_motor)

ans = 0

C. Step Response for Closed Loop J = 3.2284E-6;

b = 3.5077E-6; K = 0.0274; R = 4; L = 2.75E-6; s = tf('s');

P_motor = K/(s*((J*s+b)*(L*s+R)+K^2)); sys_cl = feedback(P_motor,1);

[image:3.595.308.546.93.362.2]

step(sys_cl)

Fig. 4 Step Response Curve for Closed Loop System The steady-state error appears to be driven to zero and the overshoot is less than 16%, though the settle time requirement is not met.

Final Value: 1

Rise Time: 0.0426 Settling Time: 0.13 Over shoot: 7.16

D. Drawing the open-loop root locus

rlocus(P_motor)

title('Root Locus – Open Loop Control') sgrid(.5, 0)

sigrid(100)

Fig. 5: Root Locus – Open Loop Control

From the above figure, the two open-loop poles near the origin cannot be distinguished because the scale of the axes is set to show the third pole which is much farther to the left than the other two poles.

The MATLAB command pole can be employed to determine the exact values of the open-loop poles.

poles = pole(P_motor) poles =

1.0e+06 *

0

-1.4545

-0.0001

The open-loop pole located very far to the left does not affect the closed-loop dynamics unless very large gains are used.

These large gains place two of the closed-loop poles in the right-half complex s-plane where the system becomes unstable.

The transient closed-loop response for small gains will not be affected much by the open-loop pole at -1.45e6.

The correct way to neglect this pole in order to maintain its steady-state contribution is to keep the DC gain of the transfer function the same, as follows:

p s for ), ( 1 /

) ( )

(  



 G s

p s

s G s H

poles(1), poles(3) ans =

0 ans = -59.2260

Reduced transfer function to neglect the pole at 1.45e-6 without affecting the steady-state behavior of the system.

0 500 1000 1500

0 1 2 3 4 5 6x 10

4

Step Response

Time (sec)

A

m

p

lit

u

d

e

Step Response

Time (sec)

A

m

p

lit

u

d

e

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0

0.2 0.4 0.6 0.8 1 1.2 1.4

System: sys_cl Final Value: 1

System: sys_cl Settling Time (sec): 0.13

System: sys_cl Peak amplitude: 1.07 Overshoot (%): 7.16 At time (sec): 0.089

[image:3.595.46.292.307.714.2]

poles = pole(P_motor);

rP_motor = minreal(P_motor*(s/max(abs(poles)) + 1)) rP_motor =

2122 --- s^2 + 59.23 s Continuous-time transfer function pole(rP_motor)

E. Draw the root locus of the reduced system

rlocus(rP_motor)

title('Root Locus - P Control') axis([ -300 100 -200 200]) sgrid(.5, 0)

sigrid(100)

Fig. 6: Root Locus - P Control

F. Integral control

Use integral control to remove the steady-state error due to a constant disturbance.

Add a 1 / s term to the forward path of the system. C = 1/s;

rlocus(C*rP_motor)

title('Root Locus - I Control') axis([ -300 100 -200 200]) sgrid(.5, 0)

sigrid(100)

Fig. 7 Root Locus - I Control

G. PI CONTROL

Using PI instead of I control adds a zero to the open-loop system. We'll place this zero at s = -20. The zero must lie between the open-loop poles of the system in this case so that the closed-loop system will be stable.

C = (s + 20) / s; rlocus(C*rP_motor)

title('Root Locus - PI Control') axis([ -300 100 -200 200]) sgrid(.5, 0)

sigrid(100)

Fig. 8: Root Locus - PI Control

Managed to stabilize the system and achieve zero steady-state error to a constant disturbance, but the system is still not fast enough.

H. PID control Determining Gain

In order to pull the root locus further to the left, to make it faster, we need to place a second open-loop zero, resulting in a PID controller. place the two PID zeros at s = -60 and s = -70.

C = (s + 60)*(s + 70) / s; rlocus(C*rP_motor)

title('Root Locus - PID Control') axis([ -300 100 -200 200]) sgrid(.5, 0)

[image:4.595.42.556.80.764.2]

sigrid(100)

Fig. 9: Root Locus - PID Control

-300 -250 -200 -150 -100 -50 0 50 100

-200 -150 -100 -50 0 50 100 150 200

System: untitled1 Gain: 0.103 Pole: 0.0403 - 1.92i Damping: -0.021 Overshoot (%): 107 Frequency (rad/sec): 1.92 System: untitled1

Gain: 0 Pole: -59.2 Damping: 1 Overshoot (%): 0 Frequency (rad/sec): 59.2

0.5

0.5

Root Locus - I Control

Real Axis

Im

a

g

in

a

ry

A

x

is

-300 -250 -200 -150 -100 -50 0 50 100

-200 -150 -100 -50 0 50 100 150 200

System: untitled1 Gain: 0 Pole: -59.2 Damping: 1 Overshoot (%): 0 Frequency (rad/sec): 59.2 System: untitled1

Gain: 2.34 Pole: -71.1 Damping: 1 Overshoot (%): 0 Frequency (rad/sec): 71.1

System: untitled1 Gain: 2.21e-005 Pole: -0.0287 - 1.82i Damping: 0.0158 Overshoot (%): 95.2 Frequency (rad/sec): 1.82

0.5

0.5

Root Locus - PID Control

Real Axis

Im

a

g

in

a

ry

A

x

Two of the closed-loop poles can be placed well within both the settling time and percent overshoot requirements. The third closed-loop pole moves from the open-loop pole at s = -59.2 to the open-loop zero at s = -60. This closed-loop pole nearly cancels with the zero (which remains in the closed-loop transfer function) because they are so close together. Therefore, we can neglect its effect. Let's reduce our new model again by performing the zero-pole cancelation using the minreal command. We pass 0.1 as a tolerance parameter as follows. The root locus for this further reduced system with controller is shown below. rsys_ol = minreal(C*rP_motor, 0.1);

rlocus(rsys_ol)

title('Root Locus - PID Control') axis([ -300 100 -200 200]) sgrid(.5, 0)

[image:5.595.304.538.53.674.2]

sigrid(100)

Fig. 10 Root Locus - PID Control

I. Determining Gain

Need the settling time and the overshoot to be as small as possible, particularly because of the effect of the extra zero. Large damping corresponds to points on the root locus near the real axis. A fast response corresponds to points on the root locus far to the left of the imaginary axis. To find the gain corresponding to a point on the root locus, we can use the rlocfind command. Specifically, enter the command [k,poles] = rlocfind(rsys_ol) in the MATLAB command window.

Then go to the plot and select a point on the root locus on left side of the loop, close to the real axis as shown below with the small + marks. This will ensure that the response will be nearly as fast as possible with minimal overshoot. These pole locations indicate that the response would have almost no overshoot if it were a canonical second-order system.

Fig. 11: rlocfind command selected_point = -1.4028e+002 +1.4286e+001i k = 0.1307

poles = 1.0e+002 * -1.3862 + 0.1384i -1.3862 - 0.1384i

sys_cl = feedback(k*rsys_ol,1); t = 0:0.0001:0.1;

step(sys_cl, t) grid

ylabel('Position, \theta (radians)')

[image:5.595.309.539.56.298.2]

title('Response to a Step Reference with PID Control')

Fig. 12: Response to a Step Reference with PID Control dist_cl = feedback(P_motor,k*C);

figure;step(dist_cl, t) grid

ylabel('Position, \theta (radians)')

title('Response to a Step Disturbance with PID Control')

Root Locus - PID Control

Real Axis

Im

a

g

in

a

r

y

A

x

is

-300 -250 -200 -150 -100 -50 0 50 100 -200

-150 -100 -50 0 50 100 150 200

System: rsys_ol Gain: 0.841 Pole: -73 Damping: 1 Overshoot (%): 0 Frequency (rad/sec): 73

System: rsys_ol Gain: 6.94e-005 Pole: -0.0962 - 3.21i Damping: 0.03 Overshoot (%): 91 Frequency (rad/sec): 3.21

0.5

0.5

Response to a Step Reference w ith PID Control

Time (sec)

P

o

s

it

io

n

,

(

r

a

d

ia

n

s

)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0

0.2 0.4 0.6 0.8 1 1.2 1.4

System: sys_cl Final Value: 1 System: sys_cl

Settling Time (sec): 0.0387 System: sys_cl

Peak amplitude: 1.14 Overshoot (%): 13.6 At time (sec): 0.0144 System: sys_cl

[image:5.595.52.292.250.508.2]

[image:6.595.52.284.55.276.2]

Fig. 13: Response to a Step Disturbance with PID Control In response to a step reference the system has an overshoot of approximately 14%, a settling time just under 0.04 seconds, and no steady-state error. Also, the response to a step disturbance reaches a steady-state value of zero. Therefore, all of the design requirements have been met.

IV. RESULT & DISCUSSION

A. Result

Figure 13 shows the time response specifications which indicates clearly that using the Root Locus Techniques as a position controller for the DC motor is a very efficient. In response to a step reference the system has an overshoot of approximately 14%, a settling time just under 0.04 seconds, and no steady-state error.

Also, the response to a step disturbance reaches a steady-state value of zero. Therefore, all of the design requirements have been met.

B. Discussion

Position response characteristics of separately excited dc motor were obtained by mathematical model using MATLAB coding 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 Kd we

have improved the characteristics like steady state error. In this project study the performance of PID and Root Locus Techniques has been studied. In these techniques has been included into the system in to improve the response of the dc motor without affecting the other parameters. There are some design constraints like settling time, rise time, maximum overshoot, damping ratio; steady state error under which the PID controller are designed. Steady state parameter is the primary key which has to be taken into under consideration. We have to use conventional controller to control the position of motor.

By appropriate parameter of the dc motor transfer function closed loop system are constructed and response of the closed loop response has been observed. The mathematical formulation of the transfer function has been done and location of the poles and zero has been plotted. The stability of the system has been analyzed by plotting poles and zeros on the root loci plot. The closed loop response of the dc motor with PID controller has also been studied.

In this work Root Locus technique is used in Matlab in order to achieve the desired results. It has been concluded that Root locus Technique improves the steady state behavior of the DC Motor system. In addition to steady state behavior other parameters like rise time, settling time , overshoot also get improved by assuming suitable values of pole and zero the corresponding transfer function are constructed. The open loop and closed loop response are also plotted and the behavior can be easily analyzed in terms of rise time, settling time overshoot as described in the corresponding curve.

To achieve the better results root locus is employed and it can further be concluded that the time domain parameter get improved to a large extent.

Infect the overshoot can be seen in the graph but improvement in the rise time and settling time compensated for the effect the overshoot.

The stability of a DC motor system can further be analyzed in terms of root loci plot. In order for a liner system to be stable all of its pole must have a negative real part that is they must all lie within left half S plane.

V. CONCLUSION & FUTURE SCOPE

A. Conclusion

Speed response characteristics of separately excited dc motor were obtained by mathematical model using MATLAB coding 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 Kd we

have improved the characteristics like steady state error. In this project study the performance of PID and Root Locus Techniques has been studied. In these techniques has been included into the system in to improve the response of the dc motor without affecting the other parameters. There are some design constraints like settling time, rise time, maximum overshoot, damping ratio; steady state error under which the PID controller are designed .Steady state parameter is the primary key which has to be taken into under consideration.

We have to use conventional controller to control the position of motor. By appropriate parameter of the dc motor transfer function closed loop system are constructed and response of the closed loop response has been observed. The mathematical formulation of the transfer function has been done and location of the poles and zero has been plotted. The stability of the system has been analyzed by plotting poles and zeros on the root loci plot. The closed Response to a Step Disturbance w ith PID Control

Time (sec)

P

o

s

it

io

n

, 

(

ra

d

ia

n

s

)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

System: dist_cl Settling Time (sec): 0.0975

System: dist_cl Final Value: 0 System: dist_cl

[image:6.595.306.534.404.441.2]

loop response of the dc motor with PID controller has also been studied.

In this work Root Locus technique is used in Matlab in order to achieve the desired results. It has been concluded that Root locus Techinque improves the steady state behavior of the DC Motor system. In addition to steady state behavior other parameters like rise time, settling time , overshoot also get improved by assuming suitable values of pole and zero the corresponding transfer function are constructed. The open loop and closed loop response are also plotted and the behavior can be easily analyzed in terms of rise time, settling time overshoot as described in the corresponding curve.

To achieve the better results root locus is employed and it can further be concluded that the time domain parameter get improved to a large extent.

Infect the overshoot can be seen in the graph but improvement in the rise time and settling time compensated for the effect the overshoot.

The stability of a DC motor system can further be analyzed in terms of root loci plot. In order for a liner system to be stable all of its pole must have a negative real part that is they must all lie within left half S plane.

B. Future Scope

 Tuning of PID controller for position control using Artificial Intelligence techniques.

 Tuning of PID controller for position control using Fuzzy Logic techniques.

 Implementation of real time hardware system for DC Motor position control.

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