MATLAB/Simulink
Basic MATLAB matrices
operators
script and function files
flow control
plotting
Basics of Simulink state-space models (1st order ordinary diff eqns)
setting integration properties
setting initial conditions
input types
getting data to the workspace
Basic MATLAB
optional windows
workspace
current directory
type commands here
command window
screen shot of the Matlab window
Matlab’s help features
type “help” at the command prompt
and Matlab returns a list of help topics
Matlab’s help features
>> help lang
Matlab’s language constructs
Matlab’s help features
>> help for
how to use Matlab’s “for” statement
Matlab’s help features
you can also access “on-line” help by clicking the
question mark in the toolbar
separate window
MATLAB Variables
all variables are stored in 32bit floating point format
no distinction between real and integer
>>a = 3;
same assignment for “a”
>>a = 3.0;
Matlab is case sensitive
>>A=3;
Aa
>>a=2;
MATLAB Variables
can use numbers and underscore in variable names
>>case34=6.45;
OK
>>case_34=6.45;
names must start with a letter
>>34case=23.45;
results in a syntax error
string (text) variables enclosed in single quotes.
The variable is stored as array of characters
>>title=‘This is the title’;
MATLAB Variables
if a variable is defined,
typing the variable name returns its value
>>a=45.57;
>>a
a=
45.57
Matlab returns the value
to clear a variable from memory
>>a=4
>>clear a
MATLAB Variables
Matlab will “echo” commands unless a semi-colon is used
>>a=23.2;
>>
>>a=23.2
a=
23.2
>>
Matlab echoes the command
MATLAB Variables
Vectors
column vectors
1 
 
a  2 
3 
 
>>a=[1;2;3];
>>a
a=
1
2
3
use semi-colon
to separate rows
row vectors
a  1 2 3
>>a=[1,2,3];
>>a
a=
1 2 3
use comma
to separate columns
MATLAB Variables
Matrices
2-dimensional matrices
1 2 3
a

4
5
6


>>a=[1,2,3;4,5,6];
>>a
a=
1 2 3
4 5 6
again, separate columns with commas and rows with semi-colons
MATLAB Variables
Indexing Matrix elements
A vector is a special type of matrix
row vector is a 1 x n matrix, 1 row n columns
column vector is a n x 1 matrix, n rows 1 column
>>a=[1,2,3];
>>a(2)
ans =
2
could also reference by a(1,2)
note, a(2,1) would produce an error
because “a” only has one row
MATLAB Variables
Indexing Matrix elements
more examples
1 2 3
a

4
5
6


addressing
>>a(2,3)
ans =
6
>>a=[1,2,3;4,5,6];
assigning
>>a(2,2)=9;
>>a
a=
1 2 3
4 9 6
MATLAB Variables
complex-valued numbers
Typically, the variable “i” or “j” is used to represent the
complex variable; e.g.
i  1
Then, a complex number is represented as
z = a + ib
Re(z) = a
Im(z) = b
MATLAB Variables
complex-valued numbers
Unless i or j has been previously defined, Matlab assigns
i and j the complex variable value
In Matlab, a complex variable is represented in the
following format
(assuming all variables are cleared)
>>z=23+i*56;
>>z
z=
23.00 + 56.00i
>>z=23+j*56;
>>z
z=
23.00 + 56.00i
Matlab always uses the symbol “i” to represent a complex number
MATLAB Variables
complex-valued numbers
What happens in this case?
>>i=3;
>> z=23+i*56;
>>z
z=
What happens in this case?
>>a=sqrt(-1);
>>z=23+a*56;
>>z
z=
MATLAB Variables
complex-valued numbers
Note, a real-valued number is a special case of a
complex-valued number
assigning any element of a matrix as complex-valued
makes the entire matrix complex-valued
>>a=[1,2];
>>a
a=
1 2
>>a(1)=1+i*5;
>>a
a=
1.00+5.00i
2.00+0.00i
MATLAB Variables
Advanced data types
n-dimensional arrays
structures
cell arrays
MATLAB Operations
Basic operations
addition
subtraction
multiplication
division
right division
left division
>>a=3;b=4;
>>c1=a/b;
>>c2=a\b;
+
*
/
\
c1=0.75
c2=1.3333….
?
so, be careful!
MATLAB Operations
Mixed Real and Complex valued Variables
if both variables are real-valued, a real-valued result is obtained
if one variable is complex-valued, Matlab recasts the real
variable as complex and then performs the operation. The
result is complex-valued
however, the type casting is done internally, the real-valued
variable remains real after the operation
MATLAB Operations
Other (Scalar) Operations
Math representation
z  yx
Matlab interpretation
>>z=y^x;
y  ex
>>y=exp(x);
y  ln(x)
>>y=log(x);
y  log(x)
>>y=log10(x)
y  sin(x) y  sin 1 (x)
>>y=sin(x);
>>y=asin(x);
y  cos(x) y  cos 1 (x)
>>y=cos(x);
>>y=acos(x);
y  tan(x) y  tan 1 (x)
>>y=tan(x);
>>y=atan(x);
MATLAB Operations
Examples
y x
>>y=x^0.5;
>>y=x^(1/2);
>>y=sqrt(x);
All variables in the preceding operations can be
real or complex, negative or positive
for x < 0, y is complex. Matlab assumes you allow complex
valued numbers. If y is not to be complex, you must
provide error checking.
MATLAB Operations
Matrices
Only matrices of the same dimension can be added and subtracted
For multiplication, the inner dimensions must be the same
1 2 3
A

4
5
6


 2 3 4
B

5
6
7


No error
>>D=A+B;
>>D=A-B;
>>D=A*C;
>>D=C*A;
Matrix multiplication
not commutative
4 5
C  6 7 


8 9 
Error
>>D=A+C;
>>D=A*B;
>>D=B*A;
MATLAB Operations
Left(\) and Right(/) Matrix “division”
Math representation
Matlab interpretation
C  A1B
>>C=A\B;
C  BA1
>>C=B/A;
Remember, A must be square and full rank
(linearly independent rows/columns)
MATLAB Operations
Matrix Transpose
Math representation
C  AT
Matlab interpretation
>>C=A’;
For complex-valued matrices, complex conjugate transpose
1 2 3
A

4
5
6


a  1  j2 3  j4
>>B=A’;
>>b=a’;
1 4
B  2 5


 3 6 
1  j2 
b

3

j4


MATLAB m-files
Two types of m-files
script files
collection of commands that Matlab executes
when the script is “run”
function files
collection of commands which together
represent a function, a procedure or a method
Both types are separate files with a “.m” extension
MATLAB m-files
To create an m-file, open the Matlab text editor
Click on the “page” icon
The Matlab text editor window will open
MATLAB m-files
Script Files
On the command line
In the script file named test.m
>>x=3.0;
>>y=x^2;
>>y
y =
9.0
>>
On the command line
>>test
y =
9.0
>>
MATLAB m-files
Script Files
script files share the workspace memory
>>x=5.0;
>>test
>>y
y =
25.0
>>
test.m script
MATLAB m-files
Script Files
script files can call other script files
inner.m script
>>outter
y =
36.0
>>
outter.m script
MATLAB m-files
Function Files
Matlab identifies function files from script files by
using the “function” and “return” keywords
the name of the function file must be
the same name as the function
MATLAB m-files
Function Files
The function file x2.m
>>r=3;
>>d=x2(r);
>>d
d =
9.0
>>
>>h=x2(4.2);
>>h
h =
17.64
>>
MATLAB m-files
Function Files
Multiple Inputs and Outputs
outputs in square brackets, [ ]
inputs in parentheses ( )
MATLAB m-files
Function Files
variables created in the function are not retained
in the workspace, except for the output variables
the function does not have access to workspace
variables, except for the inputs
variables passed to the function are “copies” of the
workspace variables. Changing their value inside the
function has no effect on their value in the workspace.
MATLAB Flow Control
The “while” and “if” statements
while expression
statements
end
if expression
statements
end
if expression
statements1
else
statements2
end
Matlab evaluates expression as logical “true” or “false”
“false” equivalent to zero
“true” equivalent to any non-zero number
statements, any valid Matlab command
MATLAB Flow Control
evaluating expression
any valid equation
a=4;
b=5;
c=5;
if a+b “True”
if b-c “False”
watch out for round-off
and word length error
if sin(0) “False”
if sin(pi) “True”
sin(pi) = 1.22e-16
conditional operators
==
equal to
<
less than
>
greater than
<= less than or equal to
>= greater than or equal to
~= not equal to
logical operators
& and
| or
while(3<=a)&(a<=5)
MATLAB Flow Control
The “for” statement
for index = start : [increment :] end
statements
end
index, start, increment, and end do not need to be integer valued
increment is optional, if increment is not specified
increment defaults to 1
index can be incremented positive (increment > 0) or
negative (increment < 0)
loop stops when index > end (or index < end)
MATLAB Flow Control
example
script file to cycle through x values
function file to generate the y values
MATLAB Plotting
Basic 2D plotting functions
plot(x1,y1[,x2,y2,x3,y3.....])
xlabel(‘x axis name’)
ylabel(‘y axis name’)
title(‘graph name’)
Additional functions
grid on
grid off
axis([xmin,xmax,ymin,ymax])
MATLAB Plotting
example y = sin(t)
the “plot” function alone
MATLAB Plotting
example y = sin(t)
script file to generate
a graph of y = sin(t)
MATLAB Plotting
example y = sin(t)
function file to generate
a graph of y = sin(t)
>>graphsin
>>
MATLAB Plotting
Adding a Legend for multiple graphs
“legend” remembers
the order the graphs
were plotted
Simulink Basics
click the Simulink button
the Simulink window
Simulink Basics
click the “new” button
create a new model or
open an existing one
the simulink model window
Simulink Example
Best thing to do is to go through an example
2nd order, constant coefficient, linear differential equation
y  c1y  c0 y  b0f (t)
Response to a “step” command
Simulink Example
Get an equivalent block diagram for the system
use mouse to drag blocks into
the model window and to
connect blocks with arrows
use integrators to get dy/dt and y
Simulink Example
add gain and summer blocks
Simulink Example
add the step input block
Simulink Example
add the output block
Simulink Example
Now, double click the blocks to open and set the block’s parameters
set gain value
set initial condition
set variable name
set output format to “array”
Simulink Example
To set the simulation parameters….
select Simulation -> Simulation Parameters
set Start and Stop time (in seconds)
set numerical integration type
Simulink Example
Time to run the simulation
click the “run” button to begin the simulation
when the simulation is complete, “Ready” appears at the bottom
Simulink Example
Simulink will automatically save a variable named “tout” to
the workspace.
This variable contains the time values used in the simulation,
important for variable time integration types
Simulink also will create the output variable(s) you specified
Simulink Example
>>plot(tout,yoft)
graph of the step response
Simulink Example
Another approach to solving the 2nd order single DOF
problem, is to cast it as a 1st order 2 DOF problem
x1  y
x1  x 2
x2  y
x 2  bo f  c1x 2  co x1
In Matrix (or State Space) form….
 x1 
x 
x 2 
 0
A
 co
uf
C  1 0
1 
c1 
x  Ax  Bu
y  Cx
0
B 
 bo 
Simulink Example
1st Order State-Space Models
Simulink Example
Multi Input Multi Output Systems
use Mux and Demux blocks to combine and extract vector signals
specify number of signals