Control Systems | Assignment | Answers

1
Control systems
Student’s Name
Institutional Affiliation
Date

2
Q1
a)
The general transfer function is given as,
G ( s )= k
(1+ τs)
To obtain the required constants, the given graph is used, using this graph, the time constant τ
can be obtained from the point where the magnitude of the response curve is about 63 % of
the final value which is 0.63 ×1=0.63. The time that corresponds to a response time of about
0.63 is about 0.5 seconds. Therefore,
τ =0.5
The DC gain k is the maximum value of the response which is just¿ 1. The transfer function is
therefore,
G ( s )= k
(1+ τs)= 1
1+0.5 s

3
G ( s )= 2
s +2
Figure 1: Response of the open-loop system
b)
The closed loop transfer function with unity feedback is given by,
Gc ( s )= G ( s )
1+ H ( s ) Gs ¿ ¿
Where H ( s )=1
Gcl ( s ) =
2
s+2
1+ 2
s +2
=
2
s+2
s+ 2+ 2
s+2
Gcl ( s )= 2
s+ 4

4
Gcl ( s )= 0.5
1+0.25 s
From which,
τ =0.25
k =0.5
Figure 2: Response of the unity feedback system
c)
The transfer function with H ( s )=0.5 is given by,
Gl ( s)= G ( s )
1+G ( s ) H ( s )−G ( s )