More on Matlab

MATLAB

MATLAB was originally developed as a MATrix LABoratory in the late seventies. Today, it is much more powerful and remains convenient and fast in numeric computations of matrices. Hence, it is also powerful for graphics. MATLAB has many special toolboxes such as optimization, signal processing and wavelet transforms. But its Symbolic Math Toolbox is based on Maple, and is not as convenient as Maple for symbolic computations. For details, refer to The Student Edition of MATLAB, The Language of Technical Computing (The Math Works, Inc., published by Prentice-Hall, Inc. 1997), or visit http://www.mathworks.com.

Matlab is an interactive program for numerical computation and data visualization; it is used extensively by control engineers for analysis and design. There are many different toolboxes available which extend the basic functions of Matlab into different application areas; in these tutorials, we will make extensive use of the Control Systems Toolbox. Matlab is supported on Unix, Macintosh, and Windows environments; a student version of Matlab is available for personal computers. For more information on Matlab, contact the Mathworks.

The idea behind these tutorials is that you can view them in one window while running Matlab in another window. You should be able to re-do all of the plots and calculations in the tutorials by cutting and pasting text from the tutorials into Matlab or an m-file.

Vectors

Let's start off by creating something simple, like a vector. Enter each element of the vector (separated by a space) between brackets, and set it equal to a variable. For example, to create the vector a, enter into the Matlab command window (you can "copy" and "paste" from your browser into Matlab to make it easy):

a = [1 2 3 4 5 6 9 8 7]

Matlab should return:

a =

         
                           1  2  3  4  5  6  9  8  7

Let's say you want to create a vector with elements between 0 and 20 evenly spaced in increments of 2 (this method is frequently used to create a time vector):

t = 0:2:20

t =

          0  2  4 
                           6  8  10  12  14  16  18  20

Manipulating vectors is almost as easy as creating them. First, suppose you would like to add 2 to each of the elements in vector 'a'. The equation for that looks like:

b = a + 2

b =

         
                           3  4  5  6  7  8  11  10  9

Now suppose, you would like to add two vectors together. If the two vectors are the same length, it is easy. Simply add the two as shown below:

c = a + b

     c
                           =

          4  6  8  10  12  14  20  18  16

Subtraction of vectors of the same length works exactly the same way.

Functions

To make life easier, Matlab includes many standard functions. Each function is a block of code that accomplishes a specific task. Matlab contains all of the standard functions such as sin, cos, log, exp, sqrt, as well as many others. Commonly used constants such as pi, and i or j for the square root of -1, are also incorporated into Matlab.

sin(pi/4)

ans =

     0.7071

To determine the usage of any function, type help [function name] at the Matlab command window.

Matlab even allows you to write your own functions with the function command; follow the link to learn how to write your own functions and see a listing of the functions we created for this tutorial.

Plotting

It is also easy to create plots in Matlab. Suppose you wanted to plot a sine wave as a function of time. First make a time vector (the semicolon after each statement tells Matlab we don't want to see all the values) and then compute the sin value at each time.

t=0:0.25:7;

y = sin(t);

plot(t,y)

The plot contains approximately one period of a sine wave. Basic plotting is very easy in Matlab, and the plot command has extensive add-on capabilities. I would recommend you visit the plotting page to learn more about it.

Polynomials

In Matlab, a polynomial is represented by a vector. To create a polynomial in Matlab, simply enter each coefficient of the polynomial into the vector in descending order. For instance, let's say you have the following polynomial:

To enter this into Matlab, just enter it as a vector in the following manner

x = [1 3 -15 -2 9]

x =

         
                           1  3  -15  -2  9

Matlab can interpret a vector of length n+1 as an nth order polynomial. Thus, if your polynomial is missing any coefficients, you must enter zeros in the appropriate place in the vector. For example,

would be represented in Matlab as:

y = [1 0 0 0 1]

You can find the value of a polynomial using the polyval function. For example, to find the value of the above polynomial at s=2,

z = polyval([1 0 0 0 1],2)

z =

You can also extract the roots of a polynomial. This is useful when you have a high-order polynomial such as

Finding the roots would be as easy as entering the following command;

roots([1 3 -15 -2 9])

     ans =

        -5.5745

        
                           2.5836

        -0.7951

         0.7860

Let's say you want to multiply two polynomials together. The product of two polynomials is found by taking the convolution of their coefficients. Matlab's function conv that will do this for you.

x = [1 2];

y = [1 4 8];

z = conv(x,y)

z =

          1  6  16  16

Dividing two polynomials is just as easy. The deconv function will return the remainder as well as the result. Let's divide z by y and see if we get x.

[xx, R] = deconv(z,y)

     xx =

         
                           1  2

R =

         
                           0  0  0  0

As you can see, this is just the polynomial/vector x from before. If y had not gone into z evenly, the remainder vector would have been something other than zero.

If you want to add two polynomials together which have the same order, a simple z=x+y will work (the vectors x and y must have the same length). In the general case, the user-defined function, polyadd can be used. To use polyadd, copy the function into an m-file, and then use it just as you would any other function in the Matlab toolbox. Assuming you had the polyadd function stored as a m-file, and you wanted to add the two uneven polynomials, x and y, you could accomplish this by entering the command:

z = polyadd(x,y)

x =

         
                           1  2

y =

         
                           1  4  8

z =

     
                               1  5  10

Matrices

Entering matrices into Matlab is the same as entering a vector, except each row of elements is separated by a semicolon (;) or a return:

B
                           = [1 2 3 4;5 6 7 8;9 10 11 12]

     B
                           =

          1    2    3     4

         
                           5    6    7     8

         
                           9   10   11    12

B = [ 1  2  3  4

     
                           5  6  7  8

      9 10 11 12]

B =

          1    2    3    
                           4

          5    6    7     8

         
                           9   10   11    12

Matrices in Matlab can be manipulated in many ways. For one, you can find the transpose of a matrix using the apostrophe key:

C = B'

     C
                           =

          1   5    9

          2   6   10

         
                           3   7   11

         
                           4   8   12

It should be noted that if C had been complex, the apostrophe would have actually given the complex conjugate transpose. To get the transpose, use .' (the two commands are the same if the matix is not complex).

Now you can multiply the two matrices B and C together. Remember that order matters when multiplying matrices.

D = B * C

D =

        
                           30    70   110

        
                           70   174   278

       
                           110   278   446

D  = C * B

D =

        107   122   137   152

        122  
                           140   158   176

       
                           137   158   179   200

        152  
                           176   200   224

Another option for matrix manipulation is that you can multiply the corresponding elements of two matrices using the .* operator (the matrices must be the same size to do this).

E = [1 2;3 4]

F = [2 3;4 5]

G = E .* F

E =

         
                           1   2

          3   4

F =

          2   3

          4  
                           5

     G
                           =

          2   6

        
                           12  20

If you have a square matrix, like E, you can also multiply it by itself as many times as you like by raising it to a given power.

E^3

     ans
                           =

         37    54

        
                           81   118

If wanted to cube each element in the matrix, just use the element-by-element cubing.

E.^3

     ans =

         
                           1     8

        
                           27    64

You can also find the inverse of a matrix:

X = inv(E)

X =

        -2.0000    1.0000

         1.5000   -0.5000

or its eigenvalues:

eig(E)

     ans
                           =

        -0.3723

         5.3723

There is even a function to find the coefficients of the characteristic polynomial of a matrix. The "poly" function creates a vector that includes the coefficients of the characteristic polynomial.

p = poly(E)

p =

    1.0000   -5.0000   -2.0000

Remember that the eigenvalues of a matrix are the same as the roots of its characteristic polynomial:

roots(p)

     ans =

        
                           5.3723

        -0.3723

Printing

Printing in Matlab is pretty easy. Just follow the steps illustrated below:

Macintosh

To print a plot or a m-file from a Macintosh, just click on the plot or m-file, select Print under the File menu, and hit return.

Windows

To print a plot or a m-file from a computer running Windows, just selct Print from the File menu in the window of the plot or m-file, and hit return.

Unix

To print a plot on a Unix workstation enter the command:

print
                           -P<printername>

If you want to save the plot and print it later, enter the command:

print plot.ps

Sometime later, you could print the plot using the command "lpr -P plot.ps" If you are using a HP workstation to print, you would instead use the command "lpr -d plot.ps"

To print a m-file, just print it the way you would any other file, using the command "lpr -P <name of m-file>.m" If you are using a HP workstation to print, you would instead use the command "lpr -d plot.ps<name of m-file>.m"

Using M-files in Matlab

There are slightly different things you need to know for each platform.

Macintosh

There is a built-in editor for m-files; choose "New M-file" from the File menu. You can also use any other editor you like (but be sure to save the files in text format and load them when you start Matlab).

Windows

Running Matlab from Windows is very similar to running it on a Macintosh. However, you need to know that your m-file will be saved in the clipboard. Therefore, you must make sure that it is saved as filename.m

Unix

You will need to run an editor separately from Matlab. The best strategy is to make a directory for all your m-files, then cd to that directory before running both Matlab and the editor. To start Matlab from your Xterm window, simply type: matlab.

You can either type commands directly into matlab, or put all of the commands that you will need together in an m-file, and just run the file. If you put all of your m-files in the same directory that you run matlab from, then matlab will always find them.