Exercise 3b: Block Diagrams...

EE4107 - Cybernetics Advanced

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Exercise 3b: Block Diagrams (Solutions)

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:

2


…

H = series(h1,h2)

Parallel:

…

H = parallel(h1,h2)

Feedback:

…

H = feedback(h1,h2)

Task 1: Transfer functions

Task 1.1

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:

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)

3


We get the same transfer function in MathScript as we get with pen and paper.

Step Response:

Task 1.2




Solution:


MathScript:

…

H_parallel = parallel(H1,H2)

figure(2)

4


step(H_parallel)


Step Response:

Task 1.3




Solution:


MathScript:

5


…

H_feedback = feedback(H2,H1)

figure(3)

step(H_feedback)


Step Response:

Task 2: Mass-spring-damper system

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.

6


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:

7


Next, we use the serial rule:

Finally, we use the feedback rule:

Task 3: Differential Equations

Given the following system:

Task 3.1

Draw a block diagram for the system using pen and paper

Solution:

The block diagram becomes:

1

sa2

1

s

b

a1

c

y

u - --

x1x2

8


Task 4: More Block Diagrams

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:

9


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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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];

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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)

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c)

Solutions:

MathScript:

Similar as previous tasks

Additional Resources

http://home.hit.no/~hansha/?lab=mathscript

Here you will find tutorials, additional exercises, etc.

http://home.hit.no/~hansha/?lab=mathscript

Exercise 3b: Block Diagrams...

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