ENG540 Control System Analysis

ENG540 CONTROL SYSTEMS ANALYSIS
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Coursework: First and second order systems, identification and analysis
Aim: To determine the transfer function of a first and a second order system
obtained from experimental data and carry out analysis on the effect of feedback
on the systems.

Table of Content
First Order System........................................................................................................ 1
a) Determine the first order Transfer Function................................................................1
b) Closed loop transfer function (unity negative feed back, H(s) = 1)...................................2
c) Closed loop transfer function (negative feed back, H(s) = 0.5)........................................3
d) Performance of the open loop system compared with the closed loop systems.....................5
Second Order System.................................................................................................... 6
a) Determine the second order Transfer Function............................................................6
b) Closed loop transfer function (unity negative feed back, H(s) = 1)...................................7
c) Closed loop transfer function (negative feed back, H(s) = 0.5)........................................8
d) Performance of the open loop system compared with the closed loop systems.....................9
References............................................................................................................... 11
Appendices............................................................................................................... 12
Appendix 1: MATLAB Source code.........................................................................12
List of Figures
Figure 1 : Open loop transfer function response.......................................................................1
Figure 2 : Closed loop transfer function response (Unity feedback)...............................................3
Figure 3 : Closed loop transfer function response (H(s) = 0.5).....................................................4
Figure 4 : Open loop transfer function response.......................................................................7
Figure 5 : Closed loop transfer function response (Unity feedback)...............................................8
Figure 6 : Closed loop transfer function response (H(s) = 0.5).....................................................9
List of Tables
Table 1 : First order system response summary(from MATLAB response).....................................5
Table 2 : Second order system response summary (from MATLAB response)...............................10

1
First Order System
a) Determine the first order Transfer Function.
The general form a first order transfer function is:
G(s)= k
1+τs ...... (1)
where:
k = gain, and
τ = time constant.
Given a step response, k represents the steady state value of the response while τ
represents the value of time (in seconds) when the response gets to 63.2% (1 - e-1) of
its final (steady state) value (Nise, 2019) (Dorf & Bishop, 2017) . This approach is
used to obtain the numerical transfer function of the system whose response is given
in this exercise.
From the given step response, the final value is 1 and the time it takes the response to
get to 63.2% (0.632) is approximately 0.5 seconds. Hence, the open loop transfer
function of the system is given by:
G(s)= 1
1+0.5 s OR
G(s)= 1
0.5 s +1 ...... (2)
Figure 1: Open loop transfer function response

2
System information (as generated by MATLAB):
RiseTime: 1.0985
SettlingTime: 1.9560
SettlingMin: 0.9045
SettlingMax: 1.0000
Overshoot: 0
Undershoot: 0
Peak: 1.0000
PeakTime: 5.2729
b) Closed loop transfer function (unity negative feed back, H(s) = 1)
The transfer function of a closed loop control system with a general negative feedback
is given by (BELL, 1981) :
T (s )= G(s )
1+G( s) H (s) ...... (3)
Where G(s) is the open loop transfer function. From section (a), G( s)= 1
0.5 s +1 and
H(s) = 1 for unity feedback system.
Substituting G(s) in equation (3),
1+G( s)=1+ 1
0.5 s +1 =1+0.5 s+1
0.5 s +1 = 0.5 s +2
0.5 s +1 ...... (4a)
G(s)
1+ G(s)= 1
0.5 s +1 × 0.5 s+1
0.5 s+2 = 1
0.5 s +2 ...... (4b)
Hence, closed loop transfer function is:
T (s )= 1
0.5 s+ 2 ...... (5a) OR
T (s )= 0.5
0.25 s+1 ...... (5b)
k = 0.5