Ninevah University
College of Electronics Engineering
Systems & Control Engineering Department
Digital Control -I Lab.
'Third stage'
Group : B
Exp. No .5
Step Response of a Discrete Time
System and Effect of Sampling Time on
System Response
REPRESENTED BY:
‫الوليد خالد موسى‬
‫محمد عزيز‬
‫ايه محمد حازم‬
Date 2021\11\29
1
OBJECTIVE :
To study
I. Closed loop response of a discrete time system
II. Comparison of time responses of continuous time and discrete time systems
III. Effect of sampling time on system response and system parameters
Software Used: MATLAB/ SIMULINK
Procedure
Example : Consider a unity feedback control system having forward path transfer
function
1
𝑠(𝑠+1)
Determine
(i) step response in continuous and discrete domain.
(ii) effect of sampling time on system response
Clc;clear all;close all;
n=[1];
d=[1 1 0];
sys=tf(n,d)
sysd=c2d(sys,1,'zoh')
syscl=feedback(sys,1)
syscld=feedback(sysd,1)
step(syscl,'y')
stepinfo(syscl)
hold on
step(syscld,'b')
stepinfo(syscld)
hold on
sysd1=c2d(sys,.6,'zoh')
syscld2=feedback(sysd1,1)
step(syscld2,'r')
stepinfo(syscld2)
hold on
sysd2=c2d(sys,.1,'zoh')
syscld3=feedback(sysd2,1)
step(syscld3,'m')
stepinfo(syscld3)
hold on
grid
legend('sysc','sysd_T=1','sysd_T=0.6','sysd_T=0.1')
2
= sys
1
------s^2 + s
.Continuous-time transfer function
Sample time: 0.6 seconds
Discrete-time transfer function
= syscld2
= sysd
0.1488 z + 0.1219
-------------------z^2 - 1.4 z + 0.6707
z + 0.2642 0.3679
---------------------z^2 - 1.368 z + 0.3679
Sample time: 0.6 seconds
.Discrete-time transfer function
Sample time: 1 seconds
.Discrete-time transfer function
= syscl
1
----------s^2 + s + 1
.Continuous-time transfer function
= ans
RiseTime: 1.8000
SettlingTime: 11.4000
SettlingMin: 0.9037
SettlingMax: 1.3133
Overshoot: 31.3289
Undershoot: 0
Peak: 1.3133
PeakTime: 3.6000
= syscld
0.3679 z + 0.2642
----------------z^2 - z + 0.6321
= sysd2
Sample time: 1 seconds
.Discrete-time transfer function
0.004837 z + 0.004679
---------------------z^2 - 1.905 z + 0.9048
= ans
RiseTime: 1.6390
SettlingTime: 8.0759
SettlingMin: 0.9315
SettlingMax: 1.1629
Overshoot: 16.2929
Undershoot: 0
Peak: 1.1629
PeakTime: 3.5920
Sample time: 0.1 seconds
.Discrete-time transfer function
= syscld3
0.004837 z + 0.004679.
--------------------z^2 - 1.9 z + 0.9095
= ans
Sample time: 0.1 seconds
.Discrete-time transfer function
RiseTime: 1
SettlingTime: 16
SettlingMin: 0.8015
SettlingMax: 1.3996
Overshoot: 39.9576
Undershoot: 0
Peak: 1.3996
PeakTime: 3
= ans
RiseTime: 1.6000
SettlingTime: 8.3000
SettlingMin: 0.9090
SettlingMax: 1.1837
Overshoot: 18.3712
Undershoot: 0
Peak: 1.1837
PeakTime: 3.6000
= sysd1
0.1488 z + 0.1219
---------------------z^2 - 1.549 z + 0.5488
3
Step Response
4
DISCUSSION :
1. How sampling time affects rise time, peak time, % overshoot, settling time of a
system?
Sampling time
1
0.6
0.1
Rise time
1
1.8
1.6
Peak time
3
3.6
3.6
Overshoot %
39.97
31.32
18.37
Settling time
16
11.4
8.3
We see an increase of peak time with a decrease of sampling time, that is, there is a direct
relationship between them.
But there are decrease in (overshoot, Rise time and Settling time). That is, there is an inverse
relationship between them.
The sampling time has affecting on stability
5
2. How practical sample and hold circuit works?
Practical Sample and Hold Circuit Control input open and closes solid-state switch at
sampling rate fs . Modes of operation - tracking ( switch closed) hold (switch open) Sample
and Hold Parameters acquisition time - time for instant switch closes until Vi within defined
% of input. Determined by input time constant τ = RinC 5τ value = 99.3% of final value
aperture time - time it takes switch to open decay rate - rate of discharge of C when circuit is
in hold mode
The zero-order hold (ZOH) is a mathematical model of the practical signal reconstruction done by a
conventional digital-to-analog converter (DAC). That is, it describes the effect of converting a discrete-time
signal to a continuous-time signal by holding each sample value for one sample interval. It has several
applications in electrical communication.
6
7
3.Write down matlab code to obtain step respons of a unity feedback system
having forward path transfer function of.Also show effect of sampling time
on time response specification parameters.
G(s )=
1
𝑠2 +4𝑠+3
clc
clear all
close all
n=[1];
d=[1 4 3];
sys=tf(n,d)
sysd=c2d(sys,1,'zoh')
syscl=feedback(sys,1)
syscld=feedback(sysd,1)
step(syscl,'y')
stepinfo(syscl)
hold on
step(syscld,'b')
stepinfo(syscld)
hold on
sysd1=c2d(sys,.6,'zoh')
syscld2=feedback(sysd1,1)
step(syscld2,'r')
stepinfo(syscld2)
hold on
sysd2=c2d(sys,.1,'zoh')
syscld3=feedback(sysd2,1)
step(syscld3,'m')
stepinfo(syscld3)
hold on
grid on
legend('sysc','sysd_T=1','sysd_T=0.6','sysd_T=0.1')
8
= sys
1
-------------
= sysd1
s^2 + 4 s + 3
.Continuous-time transfer function
= sysd
0.1577 z + 0.04252
z^2 - 0.7141 z + 0.09072
0.08648 z + 0.03906
-----------------------Sample time: 0.6 seconds
.Discrete-time transfer function
= syscld2
0.08648 z + 0.03906
-----------------------z^2 - 0.4177 z + 0.01832
Sample time: 1 seconds
.Discrete-time transfer function
= syscl
----------------------z^2 - 0.6276 z + 0.1298
Sample time: 0.6 seconds
.Discrete-time transfer function
= ans
RiseTime: 1.2000
SettlingTime: 3
SettlingMin: 0.2272
SettlingMax: 0.2505
Overshoot: 0.1922
Undershoot: 0
Peak: 0.2505
PeakTime: 3.6000
= sysd2
0.004384 z + 0.003837
1
------------s^2 + 4 s + 4
.Continuous-time transfer function
= syscld
0.1577z + 0.04252
---------------------z^2 - 0.26 z + 0.06084
Sample time: 1 seconds
.Discrete-time transfer function
= ans
RiseTime: 1.6790
SettlingTime: 2.9170
SettlingMin: 0.2252
SettlingMax: 0.2498
Overshoot: 0
Undershoot: 0
Peak: 0.2498
PeakTime: 4.6900
= ans
RiseTime: 1
SettlingTime: 3
SettlingMin: 0.2412
SettlingMax: 0.2533
Overshoot: 1.3326
Undershoot: 0
Peak: 0.2533
PeakTime: 3
---------------------z^2 - 1.646 z + 0.6703
Sample time: 0.1 seconds
.Discrete-time transfer function
= syscld3
0.004384 z + 0.003837
---------------------z^2 - 1.641 z + 0.6742
Sample time: 0.1 seconds
.Discrete-time transfer function
= ans
RiseTime: 1.7000
SettlingTime: 2.9000
SettlingMin: 0.2284
SettlingMax: 0.2500
Overshoot: 0
Undershoot: 0
Peak: 0.2500
PeakTime: 8.9000
9
4. Build Simulink model to obtain step response of a unity
feedback system whose closed loop transfer function
1
is given by: G(s)=𝑠2+1𝑠+1
. Also show effect of
sampling time on time response specification parameters.
10
Sampling time
1
0.6
0.1
Rise time
1
1.2
1
Peak time
2
2.4
2.4
Overshoot %
46.7243
44.9308
32.5997
Settling time
16
10.8
7.9
We see an increase of peak time with a decrease of sampling time, that is, there is a direct
relationship between them.
But there are decrease in (overshoot, Rise time and Settling time). That is, there is an inverse
relationship between them.
The sampling time has affecting on stability
11