Matlab: Matrix Laboratory

 

Getting started:  Menu / Help / Examples and demos

 

 

X = 12

X is a 1*1 matrix

 

Math functions:

 

 

log(X)

log(33)

sin(X)

cos(X)

cosh(X)

sqrt(X)

…

 

 

Basic stuff:

 

 

a = [1 9 –8.9 0 5]

a = [1 9 –8.9 0 5];

 

sin(a)

 

b = a + 2

 

a = a * 2

 

d = a .* a

 

 

plot(d)

grid on

gird off

 

bar(b)

pie(b)

 

A = zeros(5)

B = ones(8)

B = ones(2,3)

C = rand(2)

 

 

 

 

a = 32:50

 

a = 0:0.1:2*pi;

b = sin(a).*sin(a);

 

plot(b)

plot(a,b)

 

hold off

hold on

 

 

c = cos(a)

plot(a, c);

 

 

hold off

plot(a, b, ':r', a, c, '-b')    

 

 

 

 

 

 

 

 

 

 

s = size(a)

str = sprintf('Sin^2(t) and Cos(t) from 0.0 to %4.1f', a(s(2)) );

str = title(str)

 

Doesn’t echo the result

 

Sin() of each element

 

Each element +2

 

Each element time 2

 

Element by element

 

 

 

 

 

 

 

 

 

5*5 matrix all elements 0

8*8 matrix all elements 1

2*3 matrix all elements 1

2*2 matrix all elements random numbers between 0 and 1

 

(all numbers are in long format)

 

an array

 

 

 

 

x axis: b’s index

x axis: a’s elements

 

 

 

 

 

 

Hold is on Ž both plots

 

 

 

A nicer way to have both plots.

:r means dotted red

-b means solid blue

 

See help (icon: ?) on plot

 

 

Use plot in 2D. For 3D use mesh, surf, plot3, …

 

 

 

formatted print into a string

 

 

NOTE: single quotations unlike C

 

 

legend('sin(t)^2', 'cos(t)')

 

 

 

 

 

 

xlabel('t')

ylabel('');

gtext('Blah blah')    

 

 

axis ([ -1 8 -1.5 1.5])

 

 

 

 

 

 

 

 

 

a = 1:0.1:50;

b = a.^2 + 3*a;

plot(a, b);

semilogx(a, b);

grid

 

Can drag with mouse

 

 

(notice how nice ^2 is shown in the title and in the legend!)

 

 

 

 

put text with mouse

 

 

[x_min  x_max  y_min y_max ]

 

 

logarithmic charts: use semilogx, semilogy, and loglog  (instead of plot)

 

polar for polar plots

 

 

Matrix calculations:

 

 

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

B = A'

C = A*B  

CC = A .* A

det(A)

D = inv(A)

 

 

A * A

 

V = eig(A)       

 

 

 

 

 

Svd(A)

 

[v, d] = eig(A)

 

B = A * v

 

B ./ v

 

 

 

A(:, 1)

A(2, ::)

A(:, 2:3)

 

B = [ A(1,:)  2 4 6]

3*3 matrix

Transpose

Matrix multiplication

Element by element

determinant

Inverse of A (inf if non det(A)==0)

 

 

is I

 

Eigen values of matrix A

( A.V = l.V )

 

if det(A)==0 => at least one 0.0 eigenvalue

 

singular value decomposition

 

eigenvalue decomposition

 

 

 

 

 

 

 

The first column

The second row

The 2nd and 3rd rows

 

 

 

 

 

VERY IMPORTANT NOTICE FOR C PROGRAMMERS: index from 1 (NOT 0)

 

 

Example (convolution):

 

 

clear

 

a (1, 1:100) = 0;

a(1, 30:70) = 2;

plot(a)

axis([0 100 -1 5])    

 

 

b(1, 1:50) = 0;

b(1, 30:40) = 3;

plot(b)

axis([0 60 -1 5])

 

c = conv(a, b);

plot(c)

axis([0 200 -10 100])

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Trapezoid

(if the two pulses have equal length Ž triangle)

 

 

Solving equations:

 

 

A = [1 0 1]

 

Roots(A)

X2 + 1 = 0

 

1 and –1

 

 

step response of an LTI system defined by state-space equations

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Or simply:

 

 

 

 

 

 

 

 

 

 

Impulse response:

 

 

 

 

 

 

 

clear

A = [0 1; -1 -2]

B = [0; 1]

C = [1 0]

D = 1

t = 0:0.01:10;

 

y = step(A, B, C, D, 1, t);

plot(t,y)

 

 

y = step(A, B, C, D)

grid

 

 

 

A = [0 1 ; -2 -0.5]

y = step(A, B, C, D, 1, t);

plot(t,y)

grid on

 

 

 

 

y = impulse(A, B, C, D, 1, t);

plot(t,y)

 

Y  = C X + D U

 

 

 

 

 

Step response

 

 

 

 

 

 

 

 

 

 

 

Has imaginary poles left side of the jw axis Ž damping oscillation

 

 

 

Step response of an LTI system defined by its transfer function

 

 

 

 

 

 

num = [1 1]

den1 = [0 1 2]

den2 = [1 1 1]

den = conv(den1, den2)

 

G = tf(num, den)

 

 

 

Damp(G)

 

 

Pole(G)

Tzero(G)

 

 

pzmap(G)

 

 

 

 

 

rlocus(G)

[k, poles ] = rlocfind (G)

 

 

step(G)

impulse(G)

 

bode(G)

 

 

 

 

 

 

Transfer function

 

 

 

Eigenvalues, damping and natural frequencies

 

Poles and zeros

 

 

 

Pole & zero map

 

 

 

 

 

 

Root locus

 

 

 

Step and impulse responses

 

 

Bode diagram

 

 

change into space-space representation:

 

Question: What would the dimension of 'A' be?

 

 

 

Pss = ss(G)

 

step(Pss)  

 

[a, b, c, d] = ssdata(Pss)

 

 

Same as step(G)

 

 

Note to C/C++ programmers: notice that in Matlab functions can have multiple return values

 

 

State-space Ž transfer function

 

 

[num, den] = ss2tf(a, b, c, d, 1)

 

 

 

M files:

 

Apart from very basic things, everything else is an M file.

Example: lti\impulse.m

 

Interpreter (not a compiler) Ž in case of error stops at the error

 

Save your files with .m extension and add your folder to Matlab’s path

 

 

A simple m file that calculates the square root of a number

 

Another simple m file that has a function that returns the sinus and cosinus of a number

 

 

Example: an m file that shows the step response of the following system for 3 different values for k: 2, 20 and 100

 

 

A hint for having faster m files: for loops are very slow. Avoid them as much as you can!

 

Exmple:

 

Instead of:

 

 

For  i =0:100

     A(i+1) = sin(i);

end

 

 

Use this:

 

 

A = sin([0:100])

 

 

 

Want to learn more:

1)      Use the help and demos

2)      Google “Matlab Tutorial” and you’ll fine a zillion! (there are especially some good tutorials on how to avoid for loops)