Exercise 4: 2.order Systems (Solutions) - Telemark …home.hit.no/~hansha/documents/subjects/EE4107/exercises...2 EE4107 - Cybernetics Advanced The overshoot factor (“oversvingsfaktoren”)

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Exercise 4: 2.order Systems (Solutions)

A second order transfer function is given on the form:

( )

( )

Where

is the gain

zeta is the relative damping factor

[rad/s] is the undamped resonance frequency.

The value of is critical for stability of the system:

2


The overshoot factor (“oversvingsfaktoren”) of the step response is defined as:

√

MathScript: We can easily implement and analyze 2.order systems in MathScript using built-in

functions.

Example:

We have the following 2.order system:

( )

i.e.,

We can use the tf function or the sys_order2 function in MathScript:

num=[1];

den=[1, 1, 1];

H = tf(num, den)

step(H)

or:

dr = 1

wn = 1

[num, den] = sys_order2(wn, dr)

H = tf(num, den)

step(H)

This should give the same results.

[End of Example]

Task 1: Basic 2.order properties

Given the following transfer function:

( ) ( )

( )

Task 1.1

Find the following parameters (pen and paper):

The gain

The relative damping factor

The undamped resonance frequency [rad/s]

3


Solution:

Based on the general case:

( )

We get:

The undamped resonance frequency [rad/s]:

√

The relative damping factor :

√

The gain

Task 2: Response Time


( ) ( )

( )

( )( )

Task 2.1

Find the total response time for the given system.

Note! The response time for a 2.order system is approximately:

Solution:

We do the following:

( ) ( )

( )

( )( )

The total response time for the given system is:

Where

4


We need to find :

We have:

( )

This means:

Then we get:

Task 3: Transfer function to Differential equation


( ) ( )

( )

Task 3.1

Find the differential equation for the system.

Solution:

We do as follows:

( )[ ] ( )[ ]

This gives:

( ) ( ) ( ) ( ) ( )

This gives the following differential equation:

Task 4: 2.order transfer functions

Task 4.1

Define the transfer function below using the tf and the sys_order2 functions (2 different methods

that should give the same results).

5


( )

( )

Set

Do you get the same results using tf() and sys_order2()?

Solution:

clear

clc

K = 1;

w = 1;

z = 1;

num = [K];

den = [(1/w)^2, 2*z*(1/w), 1];

H = tf(num, den)

step(H)

or:

clear

clc

dr = 1

wn = 1

H = sys_order2(wn, dr)

step(H)

Task 4.2

Plot the step response (use the step function in MathScript) for different values of . Select as

follows:

Explain the results.

Solution:

6


We see the results are as expected.

gives a “underdamped” system

gives a “critically damped” system

gives a “overdamped” system

Task 5: More 2.order transfer functions

For the transfer functions given below, find the following parameters:

The gain

The relative damping factor

The undamped resonance frequency [rad/s]

You may also try to implement the systems in MathScript and perform a step response.

Task 5.1

( )

Solution:


( )

We get:

7


1. The undamped resonance frequency [rad/s]:

( has no relevance)

2. The relative damping factor :

3. The gain

4. The overshoot factor :

√

Task 5.2

( )

Solution:


( )

We transform our transfer function as follows:

( )

Then we get:

1. The undamped resonance frequency [rad/s]:

√

√

( √

has no relevance)

2. The relative damping factor :

8


√

√

√

3. The gain

4. The overshoot factor :

√

MathScript Code:

clear

clc

% System 1

num1 = [5];

den1 = [1, 4, 1]

H1 = tf(num1, den1)

figure(1)

step(H1)

% System 1

num2 = [9];

den2 = [3, 4, 2]

H2 = tf(num2, den2)

figure(2)

step(H2)

Task 6: Differential equation to Transfer function

Given the following differential equation:

Task 6.1

Find the transfer function:

( ) ( )

( )

Solution:

We get:

9


( ) ( ) ( ) ( )

Further:

( )[ ] ( )

This gives the following transfer function:

( )

( )

Task 7: Stability

Given the following system:

( ) ( )

( )( )

Task 7.1

Find poles and zeroes for the system (check your answer using MathScript) and draw them in the

complex plane.

Tip! In MathScript you can use the built-in functions poles(), zero() and pzgraph().

Solutions:

Zeros:

Poles:

10


MathScript:

MathScript code:

clear

clc

% Transfer function

num = 5*[10, -1];

den1 = [2, 1];

den2 = [5, 1];

den = conv(den1,den2);

H = tf(num, den)

p = poles(H)

z = zero(H)

pzmap(H)

We get the same answer in MathScript.

Task 7.2

Is the system stable or not? Why/Why not?

Solution:

The system is stable because both the poles are in the left half plane.

Task 8: 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 8.1

Draw a block diagram for the system using pen and paper.

11


Solution:

The block diagram becomes:

You may also use this notation:

Task 8.2

Based on the block diagram, find the transfer function for the system ( ) ( )

( ).

Where the force may be denoted as the control signal .

Set the transfer function on the on the following standard form:

( )

Find , and as functions of , and .

Solution:

In order to find the transfer function for the system, we need to use the serial and feedback rules.

12


We start by using the serial rule:

( )

Next, we use the feedback rule:

( )

Next, we use the serial rule:

( )

( )

Finally, we use the feedback rule:

( ) ( )

( )

( )

( )

( )

Or if we want it on the standard 2.order form:

( )

( )

We get:

( ) ( )

( )

This means:

√

√

Task 8.3

Simulate the system in MathScript (step response).

Try with different values for , and .

13


Solution:

MathScript code:

% Mass-spring-damper system

clear

clc

% Define variables

m = 1;

d = 1;

k = 1;

% Define Transfer function

num = 1/m ;

den = [1, (d/m), (k/m)];

H = tf(num, den);

% Step Response

step(H)

This gives the following results:

Additional Resources

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

Here you will find tutorials, additional exercises, etc.

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

Documents

Exercise 4: 2.order Systems (Solutions) - Telemark …home.hit.no/~hansha/documents/subjects/EE4107/exercises...2 EE4107 - Cybernetics Advanced The overshoot factor (“oversvingsfaktoren”)