EE4107 - Cybernetics Advanced
Exercise 6: State-space models (Solutions) A state-space model is a structured form or representation of a set of differential equations. Statespace models are very useful in Control theory and design. The differential equations are converted in matrices and vectors, which is the basic elements in MathScript. Assume we have the following differential equations:
̇ ̇ ̇̇ ̇ ̇
This gives on vector form:
A general linear State-space model may be written on the following general form:
̇
MathScript has several functions for creating state-space models: Function ss Sys_order1
Sys_order2
Description Constructs a model in state-space form. You also can use this function to convert transfer function models to state-space form. Constructs the components of a first-order system model based on a gain, time constant, and delay that you specify. You can use this function to create either a state-space model or a transfer function model, depending on the o utput parameters you specify. Constructs the components of a second-order system model
Example >A = [1 2; 3 4] >B = [0; 1] >C = B' >ssmodel = ss(A, B, C) >K = 1; >T = 1; >H = sys_order1 (K, T)
>dr = 0.5 >wn = 20
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2
step
lsim
based on a damping ratio and natural frequency you specify. You can use this function to create either a state-space model or a transfer function model, depending on the output parameters you specify. Creates a step response plot of the system model. You also can use this function to return the step response of the model outputs. If the model is in state-space form, you also can use this function to return the step response of the model states. This function assumes the initial model states are zero. If you do not specify an output, this function creates a plot. Creates the linear simulation plot of a system model. This function calculates the output of a system model when a set of inputs excite the model, using discrete simulation. If you do not specify an output, this function creates a plot.
Example:
Given the following state-space model:
[̇̇]
The MathScript code for implementing the model is:
% Creates a state-space model A = [1 2; 3 4]; B = [0; 1]; C = [1, 0]; D = [0]; model = ss(A, B, C, D)
Task 1: State-space models Task 1.1 Given the following system:
̇ ̇ ̇
Set the system on the following state-space form:
Solutions:
State-space model: EE4107 - Cybernetics Advanced
>[A, B, C, D] = sys_order2 (wn, dr) >ssmodel = ss(A, B, C, D)
>num=[1,1]; >den=[1,-1,3]; >H=tf(num,den); >t=[0:0.01:10]; >step(H,t);
>t = [0:0.1:10] >u = sin(0.1*pi*t)' >lsim (SysIn, u, t)
[̇̇] ⏟ ⏟ Task 1.2 Given the following system:
̇ ̇ ̇ ̇
Set the system on the following state-space form:
Solution:
The state-space model becomes:
̇̇ ̇ ⏟
Task 1.3 Given the following system:
̇ ̇ ̇
Set the system on the following state-space form:
EE4107 - Cybernetics Advanced
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Solutions:
State-space model:
[̇̇]
Task 2: Mass-spring-damper system Given a mass-spring-damper system:
Using Newtons 2. law:
̈ ( ̇ )
The model of the system can be described as:
Where
̇
̈
– position, – speed/velocity, – acceleration
- damping constant,
Task 2.1
- mass, - spring constant,
– force
Set the system on the following state-space form:
̇
EE4107 - Cybernetics Advanced
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Assuming the control signal is equal to the force Solution:
We set:
This gives:
Finally:
5 and that we only measure the position.
̇ ̇ ̇̇ ̈ ̇ ̇ [̇̇]
Where the control signal is equal to the force We only measure the position
, i.e.:
The state-space model for the system then becomes:
i.e.:
Task 2.2
Define the state-space model above using the ss function in MathScript. Apply a step in u and use the step function in MathScript to simulate the result. EE4107 - Cybernetics Advanced
Start with Solution:
6
,
,
, then explore with other values.
MathScript Code:
clear clc % c k m
Define Parameters in State-space model = 1; = 1; = 1;
% A B C D
Define State-space model = [0, 1; -(k/m), -(c/m)]; = [0; 1/m]; = [1, 0]; = [0];
ss_model = ss(A,B,C,D) step(ss_model) This gives:
Task 2.3 Convert the state-space model defined in the previous task t o a transfer function MathScript code.
EE4107 - Cybernetics Advanced
() ()()
using
Use
7
,
,
.
Do the same using “pen and paper” and Laplace. Do you get the same answer?
Solution: MathScript:
In MathScript we can simple use the tf function in order to convert the state-space model to a transfer function.
… H = tf(ss_model) This gives:
() (())
“Pen and paper”:
We have the equations:
We set
̇ ̇ ̇ ̇ ,
,
:
Laplace gives:
Further (setting eq. 1 into eq. 2):
This gives:
or:
EE4107 - Cybernetics Advanced
( ) () (()) ̇ ̇ () ()() () () () ( )
Which gives:
and
gives:
Alternative metod:
The state-space model with
,
,
:
Given the state-space model on the form:
We can use the following formula to find the transfer functions:
where:
This gives:
Where:
EE4107 - Cybernetics Advanced
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() [ ] ] ()[ () (()) ()
This means:
, which is the same answer.
Task 3: Block Diagram Given the following system:
Task 3.1 Find the state-space model from the block diagram. Note!
̇ ̇ and
.
Solution:
We get the following state-space model from the block diagram:
̇ ̇
EE4107 - Cybernetics Advanced
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10 This gives:
[̇̇] ⏟
Task 3.2 Implement the state-space model in MathScript and simulate the system using the step function in MathScript. Set
And
,
Solution:
Task 4: State-space model to Transfer functions Given the following system:
̇
EE4107 - Cybernetics Advanced
Task 4.1
̇
Find the state-space model on the form (pen and paper):
Solution:
First we do:
This gives:
̇
̇ ̇ [̇̇]
Task 4.2 Define the state-space model in MathScript and find the step response for the system. Discuss the results. Solution:
MathScript code:
clear,clc A = [0 1; -1 -3]; B = [0 0; 2 4]; C = [5 6]; D = [7 0]; ssmodel = ss(A,B,C,D) step(ssmodel) Step response:
EE4107 - Cybernetics Advanced
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12
As you see we get 2 transfer functions because this is a so-called MISO system (Multiple Input, Single Output).
Task 4.3 Find the following transfer functions:
Solution:
() (()) () (())
In MathScript we can simple do as follows:
H = tf(ssmodel) This gives the following): Transfer Function Input:1 Output:1 7,000s^2+33,000s+17,000 -----------------------
EE4107 - Cybernetics Advanced
13 1,000s^2+3,000s+1,000 Input:2 Output:1 24,000s+20,000 --------------------1,000s^2+3,000s+1,000
As you see we get 2 transfer functions because this is a so-called MISO system (Multiple Input, Single Output):
) ( () () () (())
Additional Resources
http://home.hit.no/~hansha/?lab=mathscript
Here you will find tutorials, additional exercises, etc.
EE4107 - Cybernetics Advanced
