practical case

7/29/2019 Practical Case

1/13

SYSTEM THEORY

Submitted by

SREERAG.K.S

S1 IDC, Mtech

RIT

1


2/13

Problem

Position control of a Rotary Arm

Drive: Separately Excited DC motor

Control Mechanism: Armature Control

Assumption: 1) Field Flux is maintained constant.

2) Shaft and arm move in unison without any jerk.

Specifications:

1. Supply voltage, Vs= 1V

2.Viscous friction Coeff., B=0.005

3.Motor Torque constant, km=0.1Nm/A.

4. Speed constant, kb=0.1V/rad/s.

5. Armature Resistance, Ra= 1.35 .

6. Armature Inductance, La= 0.56mH.

7. Moment of Inertia of motor, Jm=0.0019kg/m2.

8. Gain sensor, ks=1.

Value Courtesy: Automatic Control Systems by Benjamin C Kuo

Plan Elevatiton

2


3/13

Solution:

Mathematical model of DC motor (Armature Control)

)()()()(

tVtERtIdt

tdIL

sbaa

a

a=++

)()(

tdt

tdm

=

J

T

dt

d

J

B

dt

tdm

=+

2

2 )(

)(tIkTaim

=

)(tkEmbb

=

Therefore the transfer function will be:

( )iaibaaa

i

sksBRkksBLJRsJL

k

sV

s

+++++

=

)()(

)(23

3


4/13

Matlab Coding

%Modeling of a Rotary arm using separately excited DC motor.%Assuming Field flux to be constant and neglecting armature reaction.clear allclckm=0.1;kb=0.1;

Ra=1.35;La=0.56*10^-3;J=0.0019;Tl=1;tm=20*10^-6;Bm=0.0001;disp('State Space model of Open loop system in continuous domain')A=[-1*Ra/La,-1*kb/La,0;km/J,-1*Bm/J,0;0,1,0];B=[1/La;0;0];C=[0,0,1];D=0;sys=ss(A,B,C,D)[n,d]=ss2tf(A,B,C,D);disp('Open loop Transfer Function of the system')TFo=tf(n,d)subplot(4,2,1)margin(TFo)disp('Transfer function of the system')[num,den]=cloop(n,d,-1);TF_cl=tf(num,den)%Closed loop transfer function with unity feedback ofpotentiometric encoderPoles=pole(TF_cl)TF_cld=c2d(TF_cl,1)[Ac,Bc,Cc,Dc]=tf2ss(num,den);disp('State space representation of the system')sys_cl=ss(Ac,Bc,Cc,Dc)sys_cld=c2d(sys_cl,1)subplot(4,2,2)rlocus(TF_cl)%Root locus of the systemsubplot(4,2,3)nyquist(TF_cl)%Nyquist plot of the systemsubplot(4,2,4)pzmap(TF_cl)%Pole Zero plot of the systemsubplot(4,2,5)step(TF_cl)disp('Specifications of the system in continuous domain')specs_cl=stepinfo(TF_cl)disp('Specifications of the system in Discrete domain')specs_cld=stepinfo(TF_cld)M=ctrb(sys);

disp('The rank of controllability matrix')RCo=rank(M)%Rank of Controllability matrix

4


5/13

if RCo==3disp('The System is controllable')end

Output

State Space model of Open loop system in

continuous domain

sys =a =

x1 x2 x3

x1 -2411 -178.6 0

x2 52.63 -0.05263 0

x3 0 1 0

b =

u1

x1 1786

x2 0

x3 0

c =

x1 x2 x3

y1 0 0 1

5


6/13

d =

u1

y1 0

Continuous-time state-space model

Open loop Transfer Function of the system

TFo =

9.398e04

-----------------------

s^3 + 2411 s^2 + 9525 s

Continuous-time transfer function

Transfer function of the system

TF_cl =

9.398e04

----------------------------------

s^3 + 2411 s^2 + 9525 s + 9.398e04

6


7/13

Continuous-time transfer function.

Poles=

1.0e+03 *

-2.4068-0.0020 + 0.0059i

-0.0020 - 0.0059i

TF_cld =

0.8854 z^2 - 0.1275 z + 1.311e-07

----------------------------------------

z^3 - 0.2615 z^2 + 0.01942 z + 3.533e-21

Sample time: 1 seconds

Discrete-time transfer function.

7


8/13

State space representation of the system

sys_cl =

a =

x1 x2 x3x1 -2411 -9525 -9.398e+04

x2 1 0 0

x3 0 1 0

b =

u1

x1 1

x2 0

x3 0

c =

x1 x2 x3

y1 0 0 9.398e+04

d =

8


9/13

u1

y1 0

Continuous-time state-space model.

sys_cld =

a =x1 x2 x3

x1 -0.0001077 -0.2616 -5.736

x2 6.103e-05 0.147 0.3197

x3 -3.402e-06 -0.00814 0.1146

b =

u1

x1 6.103e-05

x2 -3.402e-06

x3 9.421e-06

c =

x1 x2 x3

9


10/13

y1 0 0 9.398e+04

d =

u1

y1 0

Sample time: 1 seconds

Discrete-time state-space model.

Specifications of the system in continuous

domain

specs_cl =

RiseTime: 0.2152

SettlingTime: 1.7903

SettlingMin: 0.8761

SettlingMax: 1.3520

Overshoot: 35.1998

Undershoot: 0

10


11/13

Peak: 1.3520

PeakTime: 0.5324

Specifications of the system in Discrete domain

specs_cld =

RiseTime: 1

SettlingTime: 2

SettlingMin: 0.9894

SettlingMax: 1.0001

Overshoot: 0.0065

Undershoot: 0

Peak: 1.0001

PeakTime: 4

The rank of controllability matrix

RCo =

11


12/13

3

The System is controllable

Simulink Model

12


13/13

13

practical case

Documents