Block diagrams are much in use in control theory, and we can have block diagrams in the time plane
(differential equations) and in the s plane (transfer functions).
Often we need to find the total transfer function (from input to output) from a block diagram that contains of several blocks. The most used rules are for serial, parallel and feedback blocks:
Serial:
Parallel:
Feedback:
For simple systems we can do this using pen and paper, but for more complex systems we need to use a computer tool like e.g. MathScript.
MathScript have built-in functions for manipulating block diagrams and transfer functions, e.g.:
Serial:
Faculty of Technology , Postboks 203, Kjølnes ring 56, N-3901 Porsgrunn, Norway. Tel: +47 35 57 50 00 Fax: +47 35 57 54 01

…
H = series (h1,h2)
Parallel:
…
H = parallel (h1,h2)
Feedback:
…
H = feedback (h1,h2)
Task 1.1
Find the transfer function from the following block diagram (pen and paper):
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Define the transfer function in MathScript and find the step response for the total system.
Solution:
The total transfer function becomes (pen and paper):
MathScript: clear clc
% H1 num=[1]; den=[1, 1];
H1= tf(num, den);
% H2 num=[1]; den=[1, 1, 1];
H2 = tf(num, den);
H_series = series(H1,H2) figure(1) step(H_series)
EE4107 - Cybernetics Advanced

We get the same transfer function in MathScript as we get with pen and paper.
Step Response:
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Task 1.2
Find the transfer function from the following block diagram (pen and paper):
Define the transfer function in MathScript and find the step response for the total system.
Solution:
The total transfer function becomes (pen and paper):
MathScript:
…
H_parallel = parallel(H1,H2) figure(2)
EE4107 - Cybernetics Advanced

step(H_parallel)
We get the same transfer function in MathScript as we get with pen and paper.
Step Response:
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Task 1.3
Find the transfer function from the following block diagram (pen and paper):
Define the transfer function in MathScript and find the step response for the total system.
Solution:
The total transfer function becomes (pen and paper):
MathScript:
EE4107 - Cybernetics Advanced

…
H_feedback = feedback(H2,H1) figure(3) step(H_feedback)
We get the same transfer function in MathScript as we get with pen and paper.
Step Response:
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Given the following system:
̈ ̇
is the position
̇
is the speed/velocity
̈
is the acceleration
F is the Force (control signal, u) d and k are constants
Task 2.1
Draw a block diagram for the system using pen and paper.
EE4107 - Cybernetics Advanced

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Solution:
The block diagram becomes:
You may also use this notation:
Task 2.2
Based on the block diagram, find the transfer function for the system .
Where the force may be denoted as the control signal .
Solution:
In order to find the transfer function for the system, we need to use the serial and feedback rules.
We start by using the serial rule:
Next, we use the feedback rule:
EE4107 - Cybernetics Advanced

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Next, we use the serial rule:
Finally, we use the feedback rule:
Given the following system:
̇
̇
Task 3.1
Draw a block diagram for the system using pen and paper
Solution:
The block diagram becomes: b u
-
1 s x2 a2
-
1 s x1 a1 c y
EE4107 - Cybernetics Advanced

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Task 4.1
Given the following block diagram:
Find the transfer function (“pen and paper”):
See if you get the same answer using MathScript. Plot the step response as well.
You may also use MathScript to find poles and zeroes.
Discuss the results.
Solutions:
We use the parallel rule:
This gives:
Then we get:
EE4107 - Cybernetics Advanced

Numerator:
Denominator:
[ ]
Finally we get:
MathScript: clear clc num = 2; den = [3, 1];
H1 = tf(num, den) num = [1, 2]; den1 = [1, 0]; den2 = [-3, 1]; den = conv(den1, den2);
H2 = tf(num, den)
H = parallel(H1, H2) poles(H) zero(H) figure(1) step(H) figure(2) pzmap(H)
We get the following results:
-3,000s^2+9,000s+2,000
----------------------
-9,000s^3+1,000s
This should be the same as we found using “pen and paper”.
Step Response:
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EE4107 - Cybernetics Advanced

We see both from the transfer function, poles and the step response that the system is unstable.
Task 4.2
Do the same for the following block diagrams as well: a)
Solutions:
MathScript: clear clc num = [1];
EE4107 - Cybernetics Advanced
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den = [10, 1];
H1 = tf(num, den) num = [1]; den = [1, 1];
H2 = tf(num, den)
H = feedback(H1, H2) poles(H) zero(H) figure(1) step(H) figure(2) pzmap(H) b)
Solutions:
MathScript: clear clc num = [1]; den = [10, 1];
H1 = tf(num, den)
H = feedback(H1, 1) poles(H) zero(H) figure(1) step(H) figure(2) pzmap(H)
EE4107 - Cybernetics Advanced
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c)
Solutions:
MathScript:
Similar as previous tasks
 http://home.hit.no/~hansha/?lab=mathscript
Here you will find tutorials, additional exercises, etc.
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EE4107 - Cybernetics Advanced