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EE 4314 Lab 1 Handout Control Systems Simulation with MATLAB and SIMULINK
Fall 2012
1. Lab Information
This is a take-home lab assignment. There is no experiment for this lab. You will study the tutorial
in the next section and do the examples to learn the basics of MATLAB and Simulink for control
systems simulation. You will need use to MATLAB, Simulink, and the control systems toolbox which
are available in some of the OIT managed computer labs of UTA. Also, you can use your own
installation of MATLAB.
After studying the examples, you will work on the problems in the Lab Report section on your own
and are required to submit a report by uploading it via EE4314 Blackboard. Please check the
information about the assignment policy in the Lab Report section.
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2. MATLAB and Simulink Tutorial
2.1. MATLAB Basics
The main elements of the interface are:
- Command Window
- Current Directory/Folder
- Workspace Window
- Command History Window
- Start Button
2.2. Basic Commands
- who - list current variables. who lists the variables in the current workspace.
- whos - is a long form of WHO. It lists all the variables in the current workspace, together
with information about their size, bytes, class, etc.
- help – display help text in command window.
- clc - clears the command window and homes the cursor.
- clf – clears current figure.
- clear - clears variables and functions from memory.
- clear all - removes all variables, globals, functions.
- save – saves workspace variables to disk. Use “help save” for details.
- load – loads workspace variables from disk.
- plot – PLOT(X,Y) plots vector Y versus vector X. If X or Y is a matrix, then the vector is
plotted versus the rows or columns of the matrix, whichever line up.
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2.3. MATLAB Variables
In general, the matrix is defined in the MATLAB command interface by input from the keyboard
and assigned a freely chosen variable name.
>> x = 1.00
A 1x1 matrix will be assigned to the variable x.
>> y = [1 5 3]
A row vector of length 3 will be defined.
>> z = [3 1 2; 4 0 5];
A 2x3 matrix will be defined.
An element of the matrix can be accessed by using the index (), for example:
>> z(2,3)
ans =
5
: can be used as an index to refer to the whole row or column, for example:
>> z(:,1)
ans =
3
4
Example 1: Simple 2D Plot
PLOT(X,Y) plots vector Y versus vector X. If X or Y is a matrix, then the vector is plotted versus
the rows or columns of the matrix, whichever line up.
If you do the following:
>> x = 0 : 1 : 10;
A vector called x of length 11 will be defined.
>> y = -1: 1 : 9;
Another vector called y of length 11 will be
defined.
>> plot(x,y)
As a result you will get the plot y versus x as
shown on the right.
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To put labels on the axes:
>> grid on;
>> xlabel('Time (sec)');
>> ylabel('Distance (meter)');
>> legend('Distance from point
A to B');
>> title('Distance Plot');
The result will be as shown on the right.
List of useful plot commands include:
- xlim - gets the x limits of the current axes.
- ylim - gets the y limits of the current axes.
- semilogx() - is the same as plot(), except a logarithmic (base 10) scale is used for the X-axis.
- semilogy() - is the same as plot(), except a logarithmic scale is used for the Y-axis.
- plot3(x,y,z) – similarly to plot(), plot3() plots a line in 3-space through the points whose
coordinates are the elements of x, y and z.
Example 2: 3D Plot
>> x = -1 : 0.1 : 1;
A vector of length 21.
>> y = -1 : 0.1 : 1;
>> Z = exp(-1*(x'*y));
Z is a 21x21 matrix.
>> mesh(x,y,Z)
As a result you will get the plot on the right.
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>> contour(x,y,Z)
As a result you will get the plot on the right.
Contour(Z) is a contour plot of matrix Z treating
the values in Z as heights above a plane.
2.4. Vector Operations
There are 2 types of vector operations in MATLAB
1) Element by element operations
.+ element by element addition operation
.- element by element subtraction operation
.* element by element multiplication operation
.^ element by element power operation
./ element by element division operation
2) Matrix operations
' matrix transpose
* matrix multiply
^ matrix power
eye(N) is the N-by-N identity matrix.
inv(X) is the inverse of the square matrix X.
det(X) is the determinant of the square matrix X.
trace(X) is the sum of the diagonal elements of X, which is also the sum of the eigenvalues
of X.
Example 3: Matrix Operations
>> X = eye(2)
X =
1 0
0 1
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>> Y = [1 3; 2 1]
Y =
1 3
2 1
>> Z = [3 2; 1 4]
Z =
3 2
1 4
Element by element multiplication VS matrix multiplication
>> Y.*Z
ans =
3 6
2 4
>> Y*Z
ans =
6 14
7 8
Element by element power VS matrix power
>> Z^2
ans =
11 14
7 18
>> Z.^2
ans =
9 4
1 16
Inverse, transpose and multiplication example
>> inv(Z)
ans =
0.4000 -0.2000
-0.1000 0.3000
>> inv(Z)*Y'
ans =
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-0.2000 0.6000
0.8000 0.1000
2.5. Complex Numbers
An imaginary can be defined using the i , j.
>> a = 2 + i
a =
2.0000 + 1.0000i
Useful commands for complex number operations include:
- real(X) is the real part of X.
- imag(X) is the imaginary part of X.
- conj(X) is the complex conjugate of X.
- abs(X) is the absolute value of the elements of X. When X is complex, abs(X) is the
complex modulus (magnitude) of the elements of X.
- angle(X) returns the phase angles, in radians, of a matrix with complex elements.
- cart2pol(X,Y) transforms corresponding elements of data stored in Cartesian coordinates
X,Y to polar coordinates (angle TH and radius R).
- pol2cart(TH,R) transforms corresponding elements of data stored in polar coordinates
(angle TH, radius R) to Cartesian coordinates X,Y.
Example 4: Imaginary Number Operations
>> real(a)
ans =
2
>> imag(a)
ans =
1
>> conj(a)
ans =
2.0000 - 1.0000i
>> abs(a)
ans =
2.2361
>> angle(a)
ans =
0.4636
>> [th,r] = cart2pol(real(a), imag(a))
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th =
0.4636
r =
2.2361
>> [x, y] = pol2cart(th, r)
x =
2
y =
1
2.6. Linear Systems / Control Systems Toolbox
At the MATLAB command prompt, type “help control” to learn about the control systems toolbox.
Some useful commands include:
- tf(num, den) - creates a continuous-time transfer function with numerator(s) num and
denominator(s) den.
- ss(a,b,c,d) - creates the continuous-time state-space model
- impulse() - calculates the unit impulse response of a linear system.
- step() - calculates the unit step response of a linear system.
- lsim() - simulates the time response of continuous or discrete linear systems to arbitrary
inputs.
- residue(b,a) - finds the residues, poles, and direct term of a partial fraction expansion of the
ratio of two polynomials, b(s) and a(s).
- ode23() - solves initial value problems for ordinary differential equations.
- Dsolve () – solves differential equations symbolically.
Example 5: Time responses of a linear system
Create a system with transfer function
G s s
s2 2s10
>> Num = [1 0];
>> Den = [1 2 10];
>> sys = tf(Num,Den)
Transfer function:
s
--------------
s^2 + 2 s + 10
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Simulate the impulse response of
G s
>> impulse(sys)
The result is shown on the right.
Simulate a step response of the same system
>> step(sys)
The result is shown on the right.
Now, lets simulate the time response of the above
system using different input function.
>> t = 0 : 0.1 : 10;
>> u = sin(2.*t);
>> lsim(sys,u,t)
The result is shown on the right.
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Example 6: Partial Fraction Expansion
Suppose that you want to write
G s s
s2 2s10 in pole-residue form of
skps
r
ps
r
ps
rsG
n
n
2
2
1
1 . You can use residue() command to solve the problem.
>> Num = [1 0];
>> Den = [1 2 10];
>> [r,p,k] = residue(Num, Den)
r =
0.5000 + 0.1667i
0.5000 - 0.1667i
p =
-1.0000 + 3.0000i
-1.0000 - 3.0000i
k =
[]
It means that you can now rewrite G(s) as
G s 0.5 0.1667i
s1 3i0.50.1667i
s1 3i.
Example 7: First Order ODE Solver
Given 02 xx , plot the time response of x(t) from t = 0 – 10 for x(0) = 5. You will first open up
m-file editor to create the function. First, click on the “New” button and type the following function
code. Then, save the function as “myfunction.m” in your current working directory.
function dx = myfunction(t,x)
dx = -2*x;
Note that the name of the m-file has to be the same as the name that you use inside the code after
“function” statement. Also, when you call that function, it should be in the MATLAB current directory
(unless you add its directory path to MATLAB path from MATLAB>>File>>Set Path).
>> [t,x] = ode23(@myfunction, [0 10], 5);
>> plot(t,x)
>> grid on
>> xlabel('time (seconde)');
>> ylabel('Amplitude');
>> title('Simple ODE Solver');
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The result is:
2.7. Simulink
Simulink® is an environment for multidomain simulation and Model-Based Design for dynamic
and embedded systems. Simulink programming consists of manipulating or connecting block diagrams
in a graphical user interface environment, and therefore it is more intuitive and descriptive than the
conventional script-based environment of MATLAB. Simulink can be used to simulate very complex
dynamical systems and evaluate the performance of controllers for such systems.
Go to MATLAB Help and find simulink examples in the Demos tab. Take some time go over
general demos.
Example 8: Transfer Function Simulation using Simulink
Create a simulink model of
G s s
s2 2s10 and then simulate and plot a step response of this
transfer function.
Step1: Launch Simulink by typing simulink in
the command window. Then, open a new blank
simulink model.
Step2: Go to >>Simulink>>Continuous, then drag
the transfer function block from the panel and drop it
onto the blank model window.
Step3: Double-click the transfer function block to
edit the parameters. Enter numerator coefficient and
denominator coefficient as show in the picture on the
right.
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Step4: Go to >>Simulink>>Sources, then drag the
“Step” block onto the model window.
Step5: Go to >>Simulink>>Sinks, then drag the
“Scope” block onto the model window.
Step6: Connect the terminals as show in the
picture on the right.
Step7: Run the simulation, by pressing the play
button or go to >>Simulation>>Start.
Step8: Double-click the “Scope” block to view the
simulation result.
Step9: Press the auto-scale button (binocular) on
the scope window and get a nicely scaled plot as
shown on the right.
Step10: If you want to save the data shown on the
scope, you can click on the parameters button of the
scope window and setup the scope such that it saves
the plot data in the workspace. In the History tab, you
should check “Save data to workspace” option and
give it a variable name and select the “Array” format
as shown in the picture on the right. After that, all the
plots on the scope get saved when the simulation
stops. The first column of the output matrix provides
the time data and the other columns provide the
corresponding signal data. Then, you can plot the
signals with respect to time using the plot command.
For instance, the following command plots the second
column of “output” (signal data) versus the first
column (time data).
>> plot(output(:,1),output(:,2))
Step11: You can save the data in the workspace to a file by selecting all the workspace variables of
interest, right clicking on them, and choosing “Save As…” This saves them together as a .mat file
which is the file format that MATLAB uses for matrices. You can later import this data into the
workspace using the “Import data” button on the Workspace window or the “load” command. Note that
when you close MATLAB, you loose the data in the workspace.
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Example 9: Simulating an ODE Model using Simulink
You will create a Simulink model for
tfxxx 4.1
using continuous blocks. And you will use a user defined function block to handle arbitrary f(t). In
this example, you will also demonstrate how to send Simulink variables to workspace.
Step1: Open a new blank simulink model.
Step2: Go to >>Simulink>>Continuous, then drag
the 2 integrator blocks from the panel and drop it onto
the blank model window.
Step3: Go to >>Simulink>>Sources, then a clock
block onto the model window.
Step4: Go to >>Simulink>>User-Defined
Functions, then a “Embedded MATLAB Function”
block onto the model window.
Step5: Go to >>Simulink>>Math Operations,
then the “Sum” block onto the model window.
Step6: Go to >>Simulink>>Sinks, then drag the “To Workspace” block onto the model window.
Double click the block and select “Array” under Save Format.
Step7: Edit the “Sum” block so that it takes 3 inputs with signs +, −, −. Go to Math Operations,
then drag the “Gain” block to model window. Edit its gain to 1.4.
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Step8: Connect the terminals as show in the picture below.
Step9: You can now set the Embedded MATLAB Function block to reflect any function. In this
example, you can use f(t) = sin(t).
Step10: You can set the initial value of the integrators by double clicking the integrator block. In
this example, you can leave them at 0.
Step11: Run the simulation, by pressing the play button or go to >>Simulation>>Start.
Step12: Now go to MATLAB command
window, notice 2 new variables in the
workspace “simout” and “tout”. You can use
these variables as normal MATLAB variables.
Plot of “simout” versus “tout” is shown in the
picture on the right.
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3. Lab Report
Submit your individual report for the following four problems via EE4314 Blackboard
(https://elearn.uta.edu) in one of the following file formats: .doc, .docx, or .pdf. Your report should
include –if any– your formulations, listings of MATLAB code, pictures of Simulink block diagrams,
and resulting graphs for each problem. Make sure that you show all your work and provide all the
information needed to make it a standalone and self-sufficient submission. Have an appropriate report
format for your submission. This lab handout is a good example as to how you should format your
report. Make sure that you include the following information in your report:
- Report title, your name, ID number, lab section, report due date for you.
- Answers to the problems with your
o Mathematical derivations and formulations. It is highly recommended that you use the
Equation Editor of Microsoft Word, MathType, or a similar editor to write your
equations. You can also scan handwritten equations and merge them into your report.
o Pictures of Simulink block diagrams and property windows of important blocks in the
simulation.
o Listings of MATLAB codes with inline comments and explanation in text.
o Pictures of data plots with appropriate axis labeling, titles, and clearly visible axis
values. Screen printing is not well accepted.
- Comments –if any– to let us know how we can make your learning experience better in this
lab.
Below is the assignment policy:
This assignment is due 9/19/2012 by 11:59 pm.
You must upload a single file in .doc, .docx, or .pdf format via the “Lab Assignments” link at
EE4314 Blackboard (https://elearn.uta.edu).
Late reports will get 20% deduced score from the normal score for each late day (0-24 hr)
starting right after the due date and time. For example, a paper that is worth 80 points and is 2
days late (24hr – 48hr) will get 80 – 80 × 2 × (20/100) = 48 points. A paper that is late for 5 or
more days will get 0 score.
You will have two chances of attempt to submit your report via Blackboard and only the last
submission will be considered.
Grading is out of 100 points and that includes 20% (20 points total) for the format. For
example, part (b) of Problem 1 is 10 points and 2 points of that is for the format of your answer.
A nice format refers to a clear, concise, and well organized presentation of your work.
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Problem 1 (25 pts). Let
(1)
be a second order dynamical system, in which f = f(t) is the input with unit (m/s2), x = x(t) is the
output in (m), ζ = 2 is the damping ratio (dimensionless) and ωn = 8 (rad/s) is the natural frequency.
a) (15 pts) Use ode23() to simulate the time response of the system from t = 0 to t = 5 s with zero
input (i.e. f(t) = 0), and initial conditions of (0.4 m, 0.2 m/s) for x and the first time derivative of x,
respectively. Write the program using m-file(s) that output a plot of the change of x and with time.
Plot both variables on the same graph with proper labeling and annotation such that x and are clearly
distinguishable.
(Hint for ode23: You need to use the state space representation of the system to create a function
that returns the derivatives of x. For instance, let . Hence, your function implementation
will return dx such that dx=A*x+B*f. Note that x=[x1; x2] is a vector of two elements so A is 2×2
and B is 2×1. Note also that ode23 requires a 2×1 vector of initial values.)
b) (10 pts) Use ode23() to simulate the time response of the system from t = 0 to t = 5 s where
and
. Write the program using m-file(s) that output a
plot of the change of x and with time. Plot both variables on the same graph with proper labeling and
annotation such that x and are clearly distinguishable.
Problem 2 (25 pts). For the system in equation (1) with ζ = 2, ωn = 8 rad/s, and zero initial
conditions,
a) (8 pts) Transform the differential equation to frequency domain representation using Laplace
transform. Show the steps of your formulation and indicate what Laplace transform properties that you
use. (You do not need to use MATLAB for this part.)
b) (5 pts) Use the tf command to create the system transfer function. Show your commands/codes
and the transfer function you get.
c) (6 pts) Plot the step response of the system obtained in part (b) for . Indicate which
variable the plot shows: .
d) (6 pts) Using lsim command, plot the time response of the system where f is a sinusoidal wave of
amplitude 1.2 m/s2 and 0.75 Hz frequency for .
Problem 3 (25 pts). For the system in equation (1) with ζ = 2 and ωn = 8 rad/s,
a) (8 pts) Create a Simulink model of the 2nd
order differential equation where f(t) is a function
generator. Show the block diagram and indicate which signal line is .
b) (7 pts) Simulate the time response of the system from t = 0 to t = 5 s where with f(t) as a sinusoidal function of 1.2 m/s
2 amplitude and 0.75 Hz frequency. Is the result the
same as the result of Problem 2-d, why?
c) (10 pts) Pick two different amplitudes and frequencies for f(t) such as f1(t) and f2(t). Simulate the
time response of the system for both f1 and f2 from t = 0 to t = 5 s with zero initial conditions. Show by
using the simulation results that the response of the system to f1(t)+f2(t) is the same as the sum of the
individual responses to f1(t) and f2(t). What is this property of the system called?
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Problem 4 (25 pts). In Simulink,
a) (7 pts) Using only two Step sources and a summer, create a single pulse signal that rises to an
amplitude of 10 at t = 1 and falls back to zero at t = 2.5 as shown in part (a) of the following picture.
Show the block diagram of your design and the plot of the pulse signal with clearly visible axes values.
b) (10 pts) Using only Ramp sources and a summer, create the signal shown in part (b) of the
following picture. Show the block diagram of your design and the plot of the signal with clearly visible
axes values.
c) (8 pts) Input the signal in part (b) to the system in Problem 3-a by replacing the function
generator with the summed ramp sources. Apply zero initial conditions. Show the response of the
system for and compare it with the step response obtained in Problem 2-c.
t=1 t=2.5
10
0
(a)
t=1
1.2
1
0 t=1.1 t=1.3 t=1.4
1.0
(b)