Malab tutorial

An Introduction to
MATLAB and Simulink
Trondheim
February 26, 2014

Contents
1 Introduction 2
1.1 Download and installation of MATLAB . . . . . . . . . . . . . . . . . . . . . 2
1.2 References and Matlab tutorials . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 MATLAB basics 3
2.1 The MATLAB workstation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Getting help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Saving and loading data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Use MATLAB as a calculator 6
4 Matrix calculation 7
4.1 Creating Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.2 Indexing matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.3 Array operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.4 Elementwise operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.5 Array of function values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.6 Transposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5 Visualization 11
5.1 Plotting several graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.2 Graphical screen handeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.3 Other graph types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6 Programming in MATLAB 15
6.1 Script m-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6.2 User defined functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6.4 If statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6.5 For-loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.6 While-loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
7 Ordinary differential equations 22
7.1 Solving differential equations numerically . . . . . . . . . . . . . . . . . . . . 22
7.2 Sets of differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
7.3 Example of numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . 24
8 Eigenvalues and eigenvectors 25
9 Sumulink 28
9.1 Constructing a Simulink model . . . . . . . . . . . . . . . . . . . . . . . . . . 29
9.2 Creating a block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
9.3 Example of a block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
9.4 Running a simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1

1 Introduction
MATLAB is a computer program designed for technical calculations. Its name is an abbre-
viation of “MATrix LABoratory”. In this program, matrix computations are implemented in
a straightforward manner. However, many other pre-programmed routines are available. In
addition, it is possible to implement user-deﬁned calculation routines. MATLAB allows users
to implement calculations in relatively short programming time. When these calculations
have been performed, they can be visualized by means of several plot-routines.
1.1 Download and installation of MATLAB
You must be directly connected to the eduroam wireless network (at NTNU to be able to
follow this installation procedure).
1. Map network drive: //progdist.ntnu.no/progdist. If you are having trouble map-
ping the network drive, please follow the instructions on Innsida.
2. At the network drive, go to: /campus/Matlab/R2013b/Kilde/Windows_64-og-32-bit
to access the newest version of the MATLAB software.
3. Start setup.exe, and follow the instructions.
4. Read the lesmeg.txt in the “Matlab”-folder to get the right licenses and registration-
codes.
1.2 References and Matlab tutorials
This quick-start course is based on the Interactive Matlab Course (Martens et al., 2012) and
A Practical Introduction to Matlab (Gockenbach, 1999).
Here are some other good MATLAB tutorials:
URL Comments
Eindhoven University of
Technology
http://www.imc.tue.
nl/
The basis for this MATLAB course.
Michigan Tech http://www.math.
mtu.edu/~msgocken/
intro/intro.html
Parts of this tutorial is used here.
MathWorks http://www.
mathworks.se/
academia/student_
center/tutorials/
launchpad.html
Many tutorials available including
interactive and video tutorials.
Table 1: Useful Matlab links
2

2 MATLAB basics
MATLAB is a computer program designed for technical calculations. Its name is an abbre-
viation of “MATrix LABoratory”. In this program, matrix computations are implemented in
a straightforward manner. However, many other pre-programmed routines are available. In
addition, it is possible to implement user-deﬁned calculation routines. MATLAB allows users
to implement calculations in relatively short programming time. When these calculations
have been performed, they can be visualized by means of several plot-routines.
MATLAB can be extended with so called toolboxes. In general, these are collections of
MATLAB functions that are tailored for special classes of problems. MATLAB performs
calculations with the aid of matrices. These are rectangular schemes of numbers, which are
also called arrays.
2.1 The MATLAB workstation
A typical layout of the MATLAB workstation can be seen in Figure 1. The main window
in MATLAB is the command window, positioned at the center, in which commands can be
typed after the MATLAB prompt:
>>
Figure 1: The MATALB workstation.
3

Commands are entered with the return key. The command is then executed by MATLAB. If
MATLAB is ready, after possible outputs a new prompt appears.
The window on the left is the “Current directory” window. The Current directory window
depicts the files in the current directory. This is the first user-governed directory where MAT-
LAB will look for files or functions. On the top right you will find the “Workspace” window,
in which all declared variables are shown. The bottom right window is the “Command his-
tory”, where you can find all commands you have (recently) entered. By double-clicking on
such a command, MATLAB reuses the command.
2.1.1 Interruption
A (long) MATLAB calculation can be interrupted with Ctrl-c. After this, a new prompt
appears and new commands can be entered. This interruption is especially useful when a
programmed loop is running that is corrupt.
2.1.2 Variables
The name of a variable in MATLAB has to start with a letter. After that, the name can
consist of an arbitrary number of letters, numbers or symbols like “ ” and “-”. MATLAB
does distinguish between upper and lower case letters.
Some other important commands for the management of variables are given in Table 2.
Command Explanation
who Gives a list of the variables in use.
whos Gives a list of the variables in use as well as some extra information.
clear Removes all variables.
clear x y Removes the variables x and y.
Table 2: MATALB commands
2.2 Getting help
In principle all information about MATLAB can be found in MATLAB’s help facilities. Since
the MATLAB commands and functions can often be used in more general situations, the in-
formation you will obtain through the help facilities will often be more general than necessary.
Below we list the most important ways to use the MATLAB help facilities.
When you know the name of a MATLAB command you want to use, e.g. the function plot,
the fastest way to obtain information is to type >> help ‘function name’, in this case >>
help plot. This will show you some help in the command window. An alternative is to
use the command doc plot, that will open the help browser containing a more extensive
explanation, sometimes containing examples.
If you do not know the name of a function or do not know whether such a function exists,
using the command >> lookfor ‘function’ or the Help Browser would be appropriate.
4

Help Functions By typing the command >> help ‘function name’ in the Command
Window, an M-help file appears in the Command Window. This file
contains a short description of the function, the syntax, and other closely
related help functions. This is MATLAB’s most direct help facility, and
you will often use this facility to quickly check how a certain function
works and which notation should be used. If more extensive results are
needed, try the command doc.
Look for By typing the command >> lookfor ‘topic’ in the Command Win-
dow, you can call up all help possibilities which contain the specific
search word. You can use this possibility to look for a function of which
you do not remember the function name exactly. In this manner, all
related topics appear in the Command Window.
2.3 Saving and loading data
Very often, results of MATLAB calculations need to be reused elsewhere. The values of
variables can be saved in a .mat file. This can be done with the command
>> save filename
A saved .mat file can be read into MATLAB with the command
>> load filename
where the extension .mat can be omitted. The saved variables and their values are stored in
the workspace and can now be used again.
Besides loading data stored in .mat files, MATLAB is useful for processing data which is
obtained from external sources, e.g., experimental measurements. Typically this data is
available as a plain text file organized into columns. MATLAB can easily handle tab or
space-delimited text.
There is more than one way to read data into MATLAB from a external data file. The
simplest, though least flexible, procedure is to use again the load command to read the entire
content of the file in a single step. The load command requires that the data in the file is
organized into a rectangular array. No column titles are permitted. One useful form of the
load command is:
>> load measurements.txt
where measurements.txt is the name of the file containing the data. The result of this op-
eration is that the data in measurements.txt is stored in a variable called ‘measurements’.
Files with various extension can be used. Note, that any extension except .mat indicates to
MATLAB that the data is stored as plain ASCII text. A .mat extension is reserved for a file
which has stored MATLAB variables.
Suppose you had a simple ASCII file named meas xy.txt that contained two columns of
numbers. The following MATLAB statements will load this data into the variable meas xy,
and then copy it into two vectors, x and y.
>> load meas xy.txt; % read data into the meas xy variable
5

>> x = meas xy(:,1); % copy first column of meas xy into x
>> y = meas xy(:,2); % and second column into y
After applying these commands, the data stored in column 1 of the text file is stored in the
MATLAB variable x, and the data stored in column 2 of the text file is stored in the MATLAB
variable y. You are now able to process this data with MATLAB.
Another option to import data in MATLAB is choose the Home-tab and press the Import
data-button. In the window opening, you can select your data file. Selecting next in this win-
dow, various import options are accessable. For example, you can choose whether MATLAB
should split columns between spaces, or between commas. Furthermore, you can choose the
number of header rows. On the righthand side of your window, you will see a preview how
MATLAB will import the data. If you are satisfied, select finish.
3 Use MATLAB as a calculator
The MATLAB “Command Window” can be used as an advanced calculator and to test parts
of your code.
By clicking on the Command Window, after the MATLAB prompt >>, a command can be
typed. After having pushed the return key, the command is executed, and possibly the result
appears on the screen. A command and its result looks like this:
>> a = 1 + 2 + 3
a =
6
The result of 1+2+3 is assigned to the variable a. You can now use this variable in successive
operations, such as:
>> b = 2*a + a/3
b =
14
In MATLAB, variables are introduced by assigning a value. Note, that the value can be
numerical values, matrices (called arrays), or other types. You can use the variable, and later
possibly assign a new value to the variable. It is not possible to perform calculations with
variables that have not been assigned a value. A command does not need to start with an
assignment of the form variable =. In such a case, the result is automatically assigned to the
variable ans. You can then use ans further.
>> 4*a + 1
ans =
25
>> ans*ans
ans =
625
If you now want to know what the value of a is, the following command suffices:
6

>> a
a =
6
Normally, MATLAB displays the output below the command. The output is suppressed by
ending the command with a semicolon. After having entered the command s=1+2, the screen
looks as follows
>> s = 1 + 2;
The value 3 has been assigned to the variable s. If you want to know the value of this variable,
you can either look in the “Workspace” window on the top left of your MATLAB window, or
request the value of s as follows
>> s
s =
3
If a command is longer than one line, you can end the line with three dots, and then press
the return key. You can then continue on the following line. After completing the command,
the screen looks as follows:
>> s = 1 + 2 + ...
3 + 4
s =
10
4 Matrix calculation
MATLAB uses matrices (or arrays) as the basic calculation unit. An array is a rectangular
scheme of numbers, called elements, arranged in m rows and n columns. The simplest array
contains only one column or one row.
4.1 Creating Matrices
An array with one row is also called a row or a row vector, and an array with one column is
also called a column or a column vector. If the difference between consecutive elements in a
row is always the same, the row can be formed with the command:
‘variable’ = ‘initial value’:‘step size’:‘final value’
Here ‘initial value’ is the first element of the row, ‘final value’ is the last element of the row,
and ‘step size’ is the difference between the consecutive elements. For example:
>> x = 1:-0.2:0
gives the row
x =
1.0000 0.8000 0.6000 0.4000 0.2000 0
7

If the difference between consecutive elements of the row is +1 , ‘step size’ can be omitted.
For example,
>> x = 1:5
gives the row
x =
1 2 3 4 5
For arrays with m rows and n columns, the numbers or elements aij are sorted in different
rows and columns:
A =





a11 a12 · · · a1n
a21 a22 · · · a2n
...
...
...
...
am1 am2 · · · amn





. (1)
Such an array can be entered in different ways:
Between ‘[’ and ‘]’ the elements are separated by spaces or commas, and the rows are separated
by a semicolon. The commands are:
>> a = [1 2 3;4 5 6;7 8 9]
or
>> a = [1,2,3;4,5,6;7,8,9]
or
>> a = [1 2 3
4 5 6
7 8 9]
In each of the cases above, the result is
a =
1 2 3
4 5 6
7 8 9
4.2 Indexing matrices
Elements of arrays can be indicated by means of their index. In the case of row and column
vectors, one index suffices. This can be used according to Table 3.
Arrays (both matrices and vectors) can be concatenated into new arrays. If the arrays are
arranged in a row, the number of rows in the arrays have to be equal. If the arrays are arranged
in column, the number of columns in the arrays have to be equal. The concatenation of the
arrays a=[2 3 4] and b=[6 5] proceeds as follows:
8

Command Result
a(1,2) Gives the element on the first row and sec-
ond column of a.
a(1,:) Gives the first row of a.
a(:,3) Gives the third column of a.
a(:,2)=[10;10;10] Changes the second column of a into a col-
umn with 10’s only. Only works if a is a
n × 3 array, n ≥ 2.
b(1) The first element of b in the cases when b
contains either one row or one column.
a(1,[3,1]) An array consisting of the 3rd and 1st el-
ements of the first row of a.
Table 3: MATALB matrix indexes.
>> [a,b]
ans =
2 3 4 6 5
4.3 Array operations
An ‘array operation’ is an operation (like addition or subtraction) that is applied to cor-
responding elements of two arrays of the same form. When an operation is applied to all
elements of one array, one also speaks of an ‘array operation’.
The operations addition and subtraction are automatically interpreted as array operations.
After the assignments
>> a = [1,2,3]; b = [6,5,4];
addition of the arrays results in
>> a+b
ans =
7 7 7
and subtraction of the arrays results in
>> a-b
ans =
-5 -3 -1
Note that the arrays a and b should be of the same size.
4.4 Elementwise operations
Array operations, such as multiplication ‘*’, division ‘/’ and raising to a power ‘ˆ’, are in-
dicated by putting a dot ‘.’ in front of the operation sign. In these array operations, the
9

operation is performed elementwise, e.g., the (1,1)-element of a.*b, yields a(1,1)*b(1,1)
when a and b are arrays of appropriate sizes. Hence
>> a.*b
ans =
6 10 12
>> a./b
ans =
0.1667 0.4000 0.7500
>> a.^b
ans =
1 32 81
For all these operations, the following conventions hold:
• If in an array operation one of the arrays is a number, this number is interpreted as an
array of which each element equals this number, and of which the size is equal to the
size of the other array.
• If each element of an array is to be multiplied by a scalar, one does not need to put a
dot in front of the multiplication sign.
Examples:
>> 2.^b
ans =
64 32 16
>> [2 2 2].^b
ans =
64 32 16
>> a./2
ans =
0.5000 1.0000 1.5000
>> 3*a
ans =
3 6 9
Detailed information about these operations can be obtained with the command help arith.
4.5 Array of function values
Consider the function f(x) = x3
1+x2 . We want to assign the row vector [f(0), f(0.2), · · · ,
f(1.8), f(2)] to the variable y. First we make an array with the arguments [0, 0.2, · · · ,
1.8, 2].
>> x = 0:0.2:2
x =
Columns 1 through 7
0 0.2000 0.4000 0.6000 0.8000 1.0000 1.2000
10

Columns 8 through 11
1.4000 1.6000 1.8000 2.0000
For making the row of function values, we use array operations. Here, using dots is necessary,
since the function f(x) should be evaluated per element of x.
>> y = x.^3./(1+x.^2)
y =
Columns 1 through 7
0 0.0077 0.0552 0.1588 0.3122 0.5000 0.7082
Columns 8 through 11
0.9270 1.1506 1.3755 1.6000
4.6 Transposition
In an array (matrix), the rows and columns can be interchanged. This is called transposition.
The result is called the transposed array. In MATLAB, transposition can be performed with
the function transpose, or simply the accent (’). For example
>> A = [1 2;3 4;5 6]
A =
1 2
3 4
5 6
>> transpose(a)
or
>> A’
ans =
1 3 5
2 4 6
The commands transpose(A) and A’ give the same result for arrays containing real valued
elements, but not for matrices having complex elements.
5 Visualization
To draw a graph, you ﬁrst need to open a graphic screen. This is done with the command:
>> figure
Graphs are drawn with the command plot. The command
>> plot(v)
where v is a row vector containing the numbers v1 until vn, generates a plot where the points
(1, v1), (2, v2) until (n, vn) are connected by straight lines. The command
>> plot(v,w)
11

where v and w are two row vectors with the same number of elements n, plots the points
(vi, wi), i ∈ {1, · · · , n}, and connects them by straight lines.
5.1 Plotting several graphs
The command
>> plot(v,w,x,y)
plots the points (vi, wi) and (xi, yi) in the same figure.
To determine the appearance of your graph, after plotting a vector you can add the color and
linestyle of the graph. For example, the command
>> plot(x,y,’r--’)
will plot y versus x, with a red dashed line. Type help plot to see options for other lines,
markers and colors.
Suppose that we want to draw the graph of the function f(x) = sin(x)+2x
1+x2 on the interval
[0, 2π] consisting of 100 linearly spaced points. Note that the smoothness of the graph is of
course determined by the number of point. The graph can be plotted by using the following
commands:
>> x = 0:2*pi/99:2*pi;
>> y = (sin(x)+2*x)./(1+x.^2);
>> figure
>> plot(x,y)
When drawing a graph, the axes are chosen automatically. You can choose the lengths of the
axes yourself with
>> axis([XMIN XMAX YMIN YMAX])
which takes care that the graph is drawn in the rectangle XMIN ≤ x ≤ XMAX and
Y MIN ≤ y ≤ Y MAX.
After each new plot command, the old plot will in general disappear. If you want to draw
different plots with different plot commands in the same figure, you have to give the following
command after the first plot command
>> hold on
MATLAB then holds the graphs. The command
>> hold off
restores the state in which every new plot command makes the previous plot disappear.
5.2 Graphical screen handeling
The commands in Table 4 are also important for handling the graphical screen.
12

Command Result
shg Show current graph window.
clf Clear the current figure.
grid Add a grid to the figure.
title(’title text’) Gives the figure a title called ’title text’.
text(x,y,’string’) puts the text ‘string’ at the point (x,y) of
the last plot.
xlabel(’text’) writes information along the x-axis.
ylabel(’text’) writes information along the y-axis.
legend(’graph1’,’graph2’,...) show a legend-window in your figure.
Table 4: Graph commands.
5.2.1 An illustrative example
To illustrate the above methods, we plot the functions f1(t) = 2 sin(2πt) and f2(t) = cos(πt)
on the interval [0, 1] in one figure. The sine-function is marked with dashed yellow line, while
the cosine-function is dot-dashed in black. Moreover, we give the figure a suitable title, and
x- and y-labels. Finally, we add legends to the figure and specify the y-axis to be between
−2.2 and 2.2. The result is seen in Figure 2 and the code is seen below.
>> t = 0:0.05:1;
>> f 1 = 2*sin(2*pi*t);
>> f 2 = cos(pi*t);
>> figure
>> plot(t,f 1,’y--’)
>> hold on
>> plot(t,f 2,’k.-’)
>> hold off
>> xlabel(’t’)
>> ylabel(’f(t)’)
>> title(’Sine and cosine functions’)
>> legend(’f 1(t)=2sin(2pit)’,’f 2(t)=cos(pit)’)
>> axis([0 1 -2.2 2.2])
5.3 Other graph types
You can use other plot functions in MATLAB as well. The most important are shown in
Figure 3 and listed in Table 5.
You can save a figure by selecting File → Save as and select the required file directory, filename
and file type. An other option is to use the print command. Type help print for additional
information. When you want to be able to open and modify your figures again in MATLAB,
you should save your figure as a .fig file.
13

Figure 2: An illustrative example of a MATLAB plot.
Figure 3: Examples of other graphs in MATLAB.
14

Command Result
bar(x) Produces a bar-plot of vector x.
plot3(x,y,z) Produces a three-dimensional curve between the
points, whose coordinates are given in the vectors x,
y, and z.
mesh(X,Y,Z) Produces a three-dimensional surface between the grid
points, whose coordinates are given in the matrices X,
Y , and Z.
contour(X,Y,Z) Produces level curves in (x,y)-plane, such that all
points on a line have the same value for z. Since the
inputs in this functions are gridpoints, the inputs X,
Y , and Z should be matrices.
Table 5: Other MATLAB graphs.
6 Programming in MATLAB
6.1 Script m-Files
If you have to execute a series of commands repeatedly, it might be better to write the
commands in a so called script file. The file should have a name with the extension ’.m’. As
an example, let us consider the drawing of the function f(x) = |x sin(2πx)| on the interval [0, 2]
by using a script file: Under the ’Home’-tab, press the New script-button and the MATLAB
Editor/Debugger is opened. Type the following lines:
x = 0:0.1:2;
y = abs(x.*sin(2*pi*x));
plot(x,y)
title(’f(x)= |x sin(2 pi x)| on the interval [0,2]’)
xlabel(’x’)
ylabel(’f(x)’)
axis([0 2 0 2])
Save the file by choosing the ’Editor’-tab and press the Save-button. Save it with the name
picture.m. If you now give the command
>> picture
in the MATLAB prompt, then the graph of f(x) on the interval [0, 2] will appear. An other
option to save and run a file written in your editor window is to press the ’play’-button when
in the ’Editor’-tab, or simply pressing the short-key (F5). Be careful, that this will overwrite
a possible previous file.
Note that it is better in general to save the commands that generate a figure than to save the
figure itself.
15

6.2 User defined functions
6.2.1 Function m-files
In MATLAB you can also define your own functions. MATLAB assumes by default that all
functions act on arrays. Therefore, you must keep in mind the rules for array operations
when writing your own functions. You can then combine your own functions with MATLAB
functions. A function definition has to be saved in a function file, which is a file with the
extension .m. The name of the file has to be the same as the name of the function, and should
have the extension .m.
Consider the function f(x) = x2 + ex. In MATLAB we will write a function with the same
name. The definition of the function has to be saved in the file f.m. Choose the Home-tab
and press the New → Function and the ’MATLAB Editor/Debugger’ is opened. If you now
type the lines:
function y = f(x)
y = x.^2 + exp(x);
and you save the file under the name f.m, then within MATLAB the function f(x) is available.
• The first line of the file has to contain the word ’function’.
• The variables used are local; they will not be available in your ’Workspace’.
• If x is an array, then y becomes an array of function values.
• The semicolon at the end prevents that at every function evaluation unnecessary output
appears on your screen.
• The name of the function, i.e., f, must be the same as the file name, i.e., f.m.
You can change the definition of f(x) by typing the command edit f in the command prompt.
Warning: m-files should not be given the name of existing variables or MATLAB functions.
Conversely, after you have defined a function yourself, you should not give variables the same
name as your function, otherwise the function will not work any more.
6.2.2 Anonymous functions
Sometimes it is more convenient to define your function at the command line. MATLAB func-
tions produced at the command line are called anonymous functions. For example, consider
again the function f(x) = x2 + ex. We can create the anonymous function as follows:
>> f = @(x)(x.^2+exp(x))
Here, f is the name of the function, @ is the function handle, x is the input argument and
x.^2+exp(x) is the function.
Anonymous functions can have multiple input arguments. The general structure of anony-
mous functions is
name = @(input arguments) (functions).
16

The anonymous function will be evaluated in the same way as the function m-ﬁles. Thus the
value of f(1.2) will be obtained by the command
>> f(1.2)
ans =
4.7601
6.3 Comments
The most important way to make a program understandable to others is the use of comments.
In MATLAB, this can be done in three ways, all including the %-sign:
• Single line comment: Place the %-sign on a line in your program, and everything after
the %-sign will not be executed. The shortcut for commenting out a line in MATLAB
is Ctrl+r while Ctrl+t can be used to remove the comment.
• Cell mode: You can subdivide a MATLAB program into cells. These are similar to
sections in a normal texts. To start a cell, one should type two %-signs followed by
the ’cellheader’, i.e., %% ’cellheader’, where ’cellheader’ can be replaced with a caption.
For example, one can divide a program in a cell ’Initialization’, a cell ’Calculations’ and
a cell ’Output Processing’. When you are editing an m-ﬁle, you can run the active cell
(i.e. the cell containing your cursor) by using Ctrl+Enter.
• Multiple line comment: To comment several lines, use the syntax:
%{
First commented line
Second commented line
...
Last commented line
%}
6.4 If statements
One of the most commonly used programming language constructs is the if-construct. With
this construct it is possible to decide whether or not to execute certain program lines, based
on a relational test of logical variables. The general form of this construct is:
if logical expression
program lines
elseif logical expression
program lines
else
program lines
end
The elseif and else statements are optional, so they can also be omitted. A logical expres-
sion is either true or false. In MATLAB, the value 1 is given to a true expression, and the
17

value 0 is given to a false expression. When evaluating logical expressions, we use relational
and logical operators given in Table 6.
Relation Explanation
< Less than
<= Less or equal than
> Greater than
>= Greater or equal than
== Equal to
~= Not equal to
Logical
& and
| or
~ not
Table 6: MATLAB logicals.
Examples:
• 1 < 2 is true,
• 1 == 1 is true,
• 1 == 2 is false,
• 1 ~= 2 is true.
Example 1. Write a MATLAB program that simulates the sign function. Use the MATLAB
help to see what this function means.
% make a row with time steps
t = -10:0.1:10
% determine the number of time steps in array t with the command size
n = max(size(t))
% initialize the values of f at zero
f = zeros(1,n)
% determine the function values in a FOR loop
for k = 1:n
if t(k) < 0
f(k) = -1;
elseif t(k) == 0
f(k) = 0;
elseif t(k) > 0
f(k) = 1;
end
end
18

% plot the function
plot(t,f)
Check that, due to the fact that we have initialized f at 0, this if-loop can be simplified.
Example 2. Using a for-loop and conditional tests to calculate the inner product of two
vectors.
To this end, we write a new function m-file, i.e., we create our own MATLAB command to
calculate the inner product. We call our new command inprod. To realize this, we write a
new function m-file inprod.m. We will call the two vectors we want to multiply in this file a
and b. We will call the output argument result.
function result = inprod(a,b)
% test whether the vectors a and b have the same length
%(otherwise it is not possible to determine the inner product.)
[ra ca] = size(a); % the command size gives the size of the matrix a by
assigning the number of rows of a to ra and the number of columns of a to ca.
[rb cb] = size(b);
if ca~=1 % Note, that ’~’ represents ’not’
error(’first argument is not a vector’)
end
if cb~=1
error(’second argument is not a vector’)
end
if ra~=rb
error(’vectors cannot be multiplied: they do not have the same length’)
end
% initialization of result (a number)
result = 0;
for p = 1:ra
result = result+a(p)*b(p);
end
6.5 For-loop
MATLAB knows two repeating constructs. The first is the for-loop, and the second one is
the while-loop. The for-loop construct offers you the possibility to execute certain pieces of
a program several times, while the value of certain variables is changing. A so-called loop
variable keeps track of how often the program lines have to be repeated.
The general form of a for-loop is:
for loopvariable = start:step:end
command lines
end
When step is omitted, a step size of 1 is used by default.
19

Example 3. A simple for-loop
% definition of constant n
n = 10;
% producing an array with n rows
% and 1 column, filled with zeroes
A = zeros(n,1);
% fill this column with values from 1 up to
% n in a FOR loop
for k = 1:n
A(k,1) = k;
end
A % print the result on the screen.
This loop works as follows: after the for command, the loop variable k is given the value 1,
and the program lines until the end are executed. Thus, A(1, 1) is given the value 1. After
the end command, MATLAB jumps back to the for line, and the loop variable k is given the
value 2. After this, the program lines are executed again: A(2, 1) is given the value 2. This
is repeated until the last step, where the loop variable k is given the value 10. After this, the
program continues with the lines after end, where A is printed on the screen.
for k = 1:n means ‘for k is 1 up to n with steps of 1’. You can also use larger steps to go
from 1 to n, or start with another value: for example, for k = 2:2:10 means ‘for k is 2 up
to 10 with steps of 2’ (i.e., 2,4,6,8,10). Of course you are free to choose other loop variables
like, for instance, a and/or b.
Example 4.
% initialization of the variable p
p = 0;
% run loop n times
for k = 1:n
p = p+1; % increase counter variable p
end
p
In this loop the value of p is raised by one every time the program goes through the loop.
When the program goes through the loop for the ﬁrst time, the new value of p becomes equal
to the old value of p (which was 0) plus 1, so p becomes 1. So this is an assignment statement
and not an equation. This kind of assignment construction is convenient to use to keep track
of a counter or a summation within a loop construction.
It is also possible to nest one or more for loops. By using this, you can easily ﬁll an n × m
matrix by going through the matrix element by element:
Example 5. Give each element of a matrix a random value between 0 and 1
% definition of the size of the matrix
nr = 4; % number of rows
nc = 3; % number of columns
% producing an n x m matrix with nr rows and nc columns, which is filled
20

with zeroes. It is not really necessary to make the matrix beforehand and
fill it with zeroes, but it is neater, it prevents errors, and sometimes it
may be useful. Especially for large matrices, as it is much faster.
A = zeros(nr,nc);
% fill the matrix with random values between 0 and 1 in a nested FOR loop
for r = 1:nr
for c = 1:nc
A(r,c) = rand(1);
end
end
A
6.6 While-loop
The second repeating construct in MATLAB is the while-loop. With the for-loop you have
to specify beforehand how many times you want to execute the loop. With the while-loop
this number of executions is not speciﬁed beforehand, but rather depends on whether or not
a certain expression is true or false. The general form of a while-loop is:
while logical expressions
command lines
end
Example 6. A ‘fair’ lottery. Here we use the while-loop. Write a script m-ﬁle with the name
lottery.m and the following contents:
% let MATLAB determine a random value between 1 and 10
n = round(rand(1)*10+0.5);
% initialization of stopping variable
stop = 0;
while stop == 0
choice = input(’Enter an integer between 1 and 10’)
if choice == n
stop = 1;
disp(’*** ***’)
disp(’You have won!!!!’)
disp(’*** ***’)
else
disp(’Sorry, incorrect: next player please’)
end
end
21

7 Ordinary differential equations
In this section we will solve ordinary differential equation (ODE) such as
˙x = f(x). (2)
When the analytic solution is not available one has to use numerical solution techniques to
approximate a solution.
The following section explains the use of numerical solvers, and how to interpret the results.
7.1 Solving differential equations numerically
For the numerical calculation (or rather, approximation) of the solution of a differential
equation, one uses algorithms that are related to the Euler algorithm. Briefly, the Euler
algorithm boils down to the following. Even though one does not know the solution of the
differential equation explicitly, one does know the derivative of the solution in every point
(this is given by the right hand side of the differential equation), or, in other words, one
knows what the direction field of the differential equation is. This direction field is now used
to approximate the solution. From a given initial value, the next point of the solution is
calculated by moving from the initial value in the direction of the direction field. From the
point obtained in this way, one does the same again, etc. So the quality of the approximation
of the solution depends on the initial value, the direction field, and the step size.
In MATLAB, differential equations can be solved numerically with the commands such as
ode45, ode23 or ode15s. The underlying algorithms of ode45 and ode23 make use of Runge-
Kutta-Fehlberg integration with variable step size, i.e., the algorithm increases the step size
when the solution varies less. ode23 uses the second and third order formulas, while ode45
uses the fourth and fifth order formulas. The routine ode15s uses another integration routine.
This routine is specialized in problems, that the previous routines have problems to deal with:
so-called stiff differential equations.
The mentioned routines can be applied to sets of first order differential equations of the form:
dy
dt
= f(t, y). (3)
Here y is the state vector and f the vector valued function that gives the time-derivative of
the state vector as a function of t and y.
Throughout this section, extensive use is made of function-files to facilitate numerical solv-
ing of differential equation. For an explanation of these, see “User defined functions” in
Section 6.2.
The form in which ode23 is used, is:
>> [t,y] = ode23(’filename’,[t0,t1],y0)
Here ’filename’ is the name of the .m file in which the differential equation is defined, [t0, t1]
the time-interval over which the differential equation is solved, and y0 the vector with initial
values. The algorithm in the program ode23 first determines in a point the derivative dy
dt of the
22

function defined in the function file ’filename’. With this, the next point is determined, etc.
The algorithm itself determines with which time steps it goes through the interval [t0, t1].
Since ode45 works with higher order formulas, less integration steps are needed with this
command. As a consequence of this, ode23 in general gives less smooth figures than ode45.
The output of ode gives a vector t with the time instances at which the solution x has been
calculated, and a row vector y with associated solution values.
7.1.1 Choice of solver
For most differential equations, both ode23 and ode45 are suited. The difference between
ode23 and ode45 is rather subtle: with the same accuracy, ode23 will need more time steps,
though each time step is calculated faster.
However, for a certain class of differential equations, the stiff differential equations, the pre-
viously mentioned solvers will not be able to find an accurate solution, or may need excessive
computation times for taking very small time steps. In that case, the problem might be of
a class called “Stiff differential equations”. For these differential equations, the ode15s is a
better choice.
7.2 Sets of differential equations
The commands ode23 and ode45 can also be applied for sets of differential equations. For
example, consider the set of differential equations
˙x1 = 5x1 − 2x2 (4)
˙x2 = 7x1 − 4x2 (5)
The initial values x1(0) = 2 and x2(0) = 8 define exactly one solution.
With the commands ode23 and ode45 this type of differential equations can be tackled. To
start, we write the function file deq.m in the “MATLAB Editor/Debugger”, with the contents:
function xdot = deq(t,x)
xdot = [5*x(1)-2*x(2);7*x(1)-4*x(2)];
Note, that xdot has to be a column vector. The command:
>> [t,x] = ode23(’deq’,[0,1],[2,8])
now calculates on the time interval [0, 1] the values of x1(t) and x2(t) with initial conditions
x1(0) = 2 and x2(0) = 8. The variable x becomes an array with two columns, where the first
column contains the values of x1(t) and the second column contains the values of x2(t). The
variable t is a vector consisting of all time-instances where the solution is calculated. You
can plot the solutions with the command:
>> plot(t,x)
Since x consists of two columns, two graphs are drawn in the same figure. If you only enter
the command:
23

>> ode23(’deq’,[0,1],[2,8])
the graphs of x1(t) and x2(t) are drawn, since by default the “OutputFcn” odeplot is used.
Note that you could also have written the file deq.m in the following way:
function xdot = deq(t,x)
A = [5,-2;7,-4];
xdot = A*x;
7.3 Example of numerical solution
The Duffing equation is a model for mechanical oscillations with a nonlinear spring. The
dynamics of this system, at a certain parameter is described by:
¨x + 0.1 ˙x − x + x3
= 0. (6)
To use MATLAB to compute the direction field, trajectories and solutions of this system, we
rewrite this system in the state space form ˙q = f(t, q), where q = [q1, q2] = [x, ˙x] . The
nonlinear function f(t, q) becomes
˙q = f(t, q) =
q2
−0.1q2 + q1 − q3
1
. (7)
This equation is implemented in the MATLAB routine duffing.m, containing
function qdot=duffing(t,q);
qdot=[q(2); -0.1*q(2)+q(1)-q(1)^3];
In the following script file, the above mentioned function file is used, to plot first the direction
field, afterwards the trajectory over time, and finally the solution curves in phase plane,
together with the direction field. Since f(t, q) is not explicitly dependent on t, the script uses
duffing(0,q) to compute the direction field, this is valid at all times.
%% script file to analyse duffing equation.
clear all;
close all;
%% create grid for direction plot
[Q1,Q2]=meshgrid(-1.5:.3:1.5,-0.7:.14:0.7);
%% create matrices with elements of vector Qd
Qd1=zeros(size(Q1));
Qd2=zeros(size(Q2));
for i=1:11;
for j=1:11;
Qdot=duffing(0,[Q1(i,j);Q2(i,j)]);
Qd1(i,j)=Qdot(1);
Qd2(i,j)=Qdot(2);
24

end
end
%% plot direction plot
figure(1);
quiver(Q1,Q2,Qd1,Qd2);
xlabel(’q 1’);
ylabel(’q 2’)
title(’Direction field of Duffing equation’);
%% create time vector and numerical solution with ode45
[t, Q]=ode45(’duffing’,[0,20],[1.5;0]);
%% plot trajectory in time
figure(2)
plot(t,Q(:,1))
xlabel(’t’);
ylabel(’q 1’);
figure(3)
plot(t,Q(:,2))
xlabel(’t’);
ylabel(’q 2’);
%% plot trajectory in phase space
figure(4);
quiver(Q1,Q2,Qd1,Qd2);
xlabel(’q 1=x’);
ylabel(’q 2=dotx’)
title(’Solution curve in phase plane’);
hold on;
plot(Q(:,1),Q(:,2),’k’);
hold off;
Running these ﬁles yields the plots in Figure 4–6.
8 Eigenvalues and eigenvectors
In addition to solving linear systems (with the backslash operator), Matlab performs many
other matrix computations. Among the most useful is the computation of eigenvalues and
eigenvectors with the eig command. If A is a square matrix, then
ev = eig(A)
returns the eigenvalues of A in a vector, while
[V,D] = eig(A)
25

Figure 4: Direction field of Duffing equation.
Figure 5: Time trajectory of Duffing equation.
26

Figure 6: Trajectory of Duﬃng equation in Phase plane.
returns the spectral decomposition of A: V is a matrix whose columns are eigenvectors of A,
while D is a diagonal matrix whose diagonal entries are eigenvalues. The equation AV = V D
holds. If A is diagonalizable, then V is invertible, while if A is symmetric, then V is orthogonal.
Example 7.
>> A = [1 3 2;4 5 6;7 8 9]
A =
1 3 2
4 5 6
7 8 9
>> eig(A)
ans =
15.9743 + 0.0000i
-0.4871 + 0.5711i
-0.4871 - 0.5711i
>> [V,D] = eig(A)
V =
-0.2155 0.0683 + 0.7215i 0.0683 - 0.7215i
-0.5277 -0.3613 - 0.0027i -0.3613 + 0.0027i
-0.8216 0.2851 - 0.5129i 0.2851 + 0.5129i
D =
15.9743 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i
0.0000 + 0.0000i -0.4871 + 0.5711i 0.0000 + 0.0000i
0.0000 + 0.0000i 0.0000 + 0.0000i -0.4871 - 0.5711i
>> A*V-V*D
ans =
27

1.0e-14 *
-0.0888 + 0.0000i 0.0777 - 0.1998i 0.0777 + 0.1998i
0.0000 + 0.0000i -0.0583 + 0.0666i -0.0583 - 0.0666i
0.0000 + 0.0000i -0.0555 + 0.2387i -0.0555 - 0.2387i
There are many other matrix functions in Matlab, many of them related to matrix factoriza-
tions. Some of the most useful are:
• lu computes the LU factorization of a matrix;
• chol computes the Cholesky factorization of a symmetric positive definite matrix;
• qr computes the QR factorization of a matrix;
• vd computes the singular values or singular value decomposition of a matrix;
• cond, condest, rcond computes or estimates various condition numbers;
• norm computes various matrix or vector norms.
9 Sumulink
Simulink is a toolbox of MATLAB that can be used for modeling, analyzing and simulating
dynamical systems. Here, we focus on the use of Simulink on the use with ordinary differential
equations.
Simulating the dynamical behavior of a system is important in engineering. Many systems
can be written as a differential equation, such as:
m¨x(t) + b ˙x(t) + kx(t) = u(t), (8)
describing the movement of a single mass, on which an actuation force u is applied. For some
simple differential equations, trajectories x(t) of the system can be computed analytically.
Many of these analytical solutions have been implemented in MATLAB and can be obtained
with the command dsolve.
However, many systems found in engineering or physics are described by more complex dif-
ferential equations, for which no analytical solution is known or even exists. Trajectories of
these systems can be approximated numerically. Hereto, solvers are developed that approx-
imate the change of the state variables over a small time step. Herewith, step by step, an
approximation of the trajectory is found. For example, the MATLAB functions ode23 and
ode45 are numerical solvers. Simulink is an other tool in MATLAB using numerical solvers.
Use of simulink has some advantages over the use of the functions ode23 and ode45. Simulink
has a graphical interface, such that the structure of the system can be clearly visible. Inputs,
such as a discontinuous signal u(t) for system (8) can be easily applied. Furthermore, real
time applications are possible.
28

9.1 Constructing a Simulink model
To compute trajectories with MATLAB by means of block diagrams, the tool Simulink should
be used. To open Simulink, type simulink in your command window or press the icon when in
the Home-tab in the MATLAB window. Now, the Simulink Library Browser should appear.
A new file can be opened by selecting File → New → Model or selecting the icon on the
toolbar. In the window appearing, one should build the model. Basically, you will have to
draw the block diagram. Hereto, blocks from the Library Browser should be dragged into the
model while holding your left mouse button.
In the library browser, you will find the integrator block in the category Continuous, the gain
and sum block are part of the category Math Operations. When a block is inserted in the
model, one can flip the block (press the right mouse button on block, select Format → Flip).
To connect two blocks, click on the >> exiting the first block, hold your right-mouse-button
and drag a line towards the >> entering the second block. You can give a name to a signal
by double-clicking on the connection.
Blocks and arrows can be moved in your model by using either the arrows on your keyboard
or by dragging them with your mouse.
Many blocks in Simulink have certain parameters. To edit these parameters, double click on
the block and change the value. The particular gain used is a parameter of the gain block.
The initial value of a signal exiting an integrator block is a parameter of this block. In the
parameters of the sum-block, one can make the block subtract signals instead of add them,
by changing the list of signs from |++ to |+-.
The integrator block has its initial condition as parameter. Here, one specifies the initial
output of the integrator block. By changing these initial conditions, a Simulink model can be
used to compute trajectories from predefined initial conditions.
Inputs in Simulink have to be defined as a function of time, for example u(t) = 3 sin(2t). You
will find many possible inputs in the Sources-category of your Library browser. Most used
are the sine wave, step or constant blocks. A sine wave block has the properties amplitude,
frequency and phase, the latter describing the initial phase of the signal. A step block has
parameters initial time, initial value and final value. A constant source has the constant value
as a parameter.
To handle outputs in Simulink, one needs to use the blocks in the category Sinks in the
Library browser. Most used are the Scope, To File, and To Workspace blocks. A scope will
visualize the signals entering it. When you have finished a simulation, double-click on the
scope to see the results. The To File block saves the entering signal to a .mat file. When you
select the block properties, you can change the file name, and the name of the stored variable.
The block To Workspace saves the signals to the workspace, such that one can use them later
in the command window. In this block’s properties, one can change the variable name, and
choose whether the data should be saved as structure, structure with time or array.
It is allowed to call variables that are present in your workspace in block parameters. For
example, a gain-block with gain k will work properly, when a scalar k is available in the
MATLAB workspace.
29

To save your model, press the save button in the toolbar of your model window. Simulink
models will be saved with the extension .mdl. Later, you can open these models by selecting
the ﬁle saved.
Please note, that the Library browser contains many more blocks as mentioned in this section.
When you double click on a certain block in the library browser, you will see the purpose and
parameters of the block.
9.2 Creating a block diagram
Most dynamical systems can be represented in a block diagram. A diﬀerential equation is
splitted in simple blocks. Arrows connecting blocks are signals. The most used blocks are
given in the following table:
Integrator y(t) =
t
0 x(s) ds
Sum z(t) = x(t) + y(t)
Gain y(t) = 2x(t)
Table 7: Popular Simulink blocks
9.3 Example of a block diagram
To clarify, a block diagram is constructed for the system (8), i.e.
m¨x(t) + b ˙x(t) + kx(t) = u(t).
The highest state derivatives is ¨x. Integrating this state once, we obtain ˙x. Integration of ˙x
yields x. ¨x is represented by ddx and ˙x by dx in Figure 7. Assume that m = 1, b = 2 and
k = 3 and rewriting (8), yields an expression for ¨x:
¨x = u − 2 ˙x − 3x (9)
and we get the Simulink model shown in Figure 7.
9.4 Running a simulation
To run a simulation of your Simulink model, you can select the button “Start Simulation”
on the toolbar of the model. Take care, that the correct initial conditions are used for the
30

Figure 7: The Simulink block diagram.
integrator blocks. The simulation will finish when the maximum simulation time is reached.
If this takes to long, you can terminate your simulation in the meantime by pressing the “Stop
Simulation” button on the toolbar.
When the simulation is stopped, you can double-click on a scope to see the signal over time.
Signals send to “To Workspace” or “To file” blocks are created in your workspace, or in the
file in the current directory, respectively.
9.4.1 Simulation parameters
To edit the parameters used by Simulink to simulate the behavior of the system, click on
Simulation → Configuration Simulation Parameters. A screen similar to the one in Figure 8
will be shown.
Figure 8: Simulink setup window
In this window, you will be able to set a start and stop time for your simulation. Note, that
these times represent the simulated time, which usually differs from the computation time.
31

9.4.2 Type of solver
Many solver options can be changed. The most important one is the solver type. The solver
type determines the time steps Simulink is taking. Though very small time steps are usually
more accurate, they require longer computation time.
One can choose Variable-Step or Fixed-Step solvers. In a fixed step solver, the solver takes
fixed time steps and computes at each time step, how the states should be updated. The
most important option for this solver is the fixed-step size.
More complex solvers are variable-step solvers. They try to take their step size as large as
possible, but will reduce the step size when the computation becomes less accurate. Usually,
variable-step solvers are faster. By default, one should select the ode45 solver. However, for
certain problems, ode45 will need very long computation times. In that case, one can choose
a different solver, such as ode15s.
Important parameters that should be chosen for a variable step solver are the Relative Tol-
erance and the Absolute Tolerance. The relative tolerance measures the error relative to the
size of each state. The relative tolerance represents a percentage of the state’s value. The
default, 1e-3, means that the computed state will be accurate to within 0.1%. The Absolute
Tolerance is a threshold error value. This tolerance represents the acceptable error as the
value of the measured state approaches zero. A variable step solver chooses its time steps
such, that both absolute and relative tolerance are satisfied.
32

References
F. B. L. Martens, P. van Zutven, J. C. v. d. Meer, H. A. v. Essen, and J. J. B. Biemond,
“Interactive matlab course,” Eindhoven University of Technology, Tech. Rep., 2012.
M. S. Gockenbach, “A practical introduction to matlab,” Michigan Tech, Tech. Rep., 1999.
33

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