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Question 1
The transfer function that is to be studied in this question is
( )
The discreet time transfer functions were calculated for Ts = 0
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1
The output is as follows,
Transfer function:
9
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98e-006
----------------------z^2 - 1
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997
Sampling time (seconds): 0
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09056 z + 0
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724 z + 0
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1

Clearly there is a difference between the two discreet transfer functions, specifically in the
numerator and denominator coefficients
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The difference because of the time
Table 1 - Tabulated data

Ts
a1
a2
b0
b1

0
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1
-1
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72357
0
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740818
9
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090559
9
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081941

While the table shows the change in the values, it does not show the trend of the values either
increasing or decreasing
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The coefficient of z
square has an increasing trend with respect to the sampling time while all the other coefficients
show a decreasing trend
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This is shown in figure 2 as the poles move towards the centre with
increasing sample time
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Therefore, discretization would affect stability and performance which heeds to be
considered and is discussed in later parts of this report
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001); % small sampling time
Gd2 = c2d(Gs,0
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den{1}(2),Gd2
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den{1}(3),Gd2
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num{1}(2),Gd2
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num{1}(3),Gd2
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001,0
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The system with and without a controller will be simulated using MATLAB
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There is no controller but the system
digital response is plotted in figure 4 to figure 7
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2s, 0
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6s, and 0
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Figures 3 to 7 show that in the three methods of discretization, the difference increases as the time
sample increases from 0
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8s
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Figure 3 - closed loop unity feedback

Figure 4 - 0
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04

Figure 6 - 0
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08

clc;
clear all;
% transfer function
s = tf('s');
Gs = 20/((s+1)*(s+2));
for v = 1:5;
Ts = 0
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The PID is coded
with the following code,
% PID constants
Kp = 10;
Ki = 20;
Kd = 10;
% PID equation
C = Kp + Ki/s + Kd*s;
dC = c2d(C,Ts,'tustin');
Gd_cl_control = feedback(dC*Gd_Tus,1);
step(Gd_cl_control,20);

Figures 8 shows this response which makes it clear that the PID response is better
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08) and is stable
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1;
figure;
hold on;
% Tustin
Gd_Tus = c2d(Gs,Ts,'tustin');
Gd_Tus_cl = feedback(Gd_Tus,1);
step(Gd_Tus_cl,12);
% PID constants
Kp = 10;
Ki = 20;
Kd = 10;
% PID equation
C = Kp + Ki/s + Kd*s;
dC = c2d(C,Ts,'tustin');
Gd_cl_control = feedback(dC*Gd_Tus,1);
step(Gd_cl_control,20);

% complete labelling of graph
legend('No PID', 'PID');
xlabel('Time')
ylabel('Output')
title('response behaviour')



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