4. Background
MATLAB = Matrix Laboratory (developed by The MathWorks).
Opening MATLAB
Working
Memory
Command
History
Command
Window
Working
Path
5. Variables
Have not to be previously declared
Variable names can contain up to 63 characters
Variable names must start with a letter followed by letters, digits,
and underscores( Valid: x1 = 5, Invalid: 1x = 5)
Variable names are case sensitive (a1 does NOT equal A1)
EX: Using command window
>> x = 5; % this is used as a comment
>> y = 3;
>> z = x + y
z = 8
Semicolon (;) : Suppresses output
Percentage (%) : Commenting. Only good for that line
Operators
6. MATLAB Special Variables
ans Default variable name for results
pi Value of pi number (3.14)
eps Smallest incremental number
inf Infinity
NaN Not a number e.g. 0/0
realmin The smallest usable positive real number
realmax The largest usable positive real number
i,j Imaginary unit
7. MATLAB Assignment & Operators
Assignment = a = b (assign b to a)
Addition + a + b
Subtraction - a - b
Multiplication * or.* a*b or a.*b
Division / or ./ a/b or a./b
Power ^ .^ a^b or a.^b
8. MATLAB Matrices
MATLAB treats all variables as matrices. For our purposes a
matrix can be thought of as an array, in fact, that is how it is
stored.
Vectors are special forms of matrices and contain only one
row OR one column.
Scalars are matrices with only one row AND one column
9. MATLAB Matrices
A matrix with only one row is called a row vector. A row vector can be
created in MATLAB as follows (note the commas):
» rowvec = [12 , 14 , 63]
rowvec =
12 14 63
A matrix with only one column is called a column vector. A column
vector can be created in MATLAB as follows (note the semicolons):
» colvec = [13 ; 45; -2]
colvec =
13
45
-2
10. MATLAB Matrices
A matrix can be created in MATLAB as follows (note the commas
AND semicolons):
» matrix = [1 , 2 , 3 ; 4 , 5 ,6 ; 7 , 8 , 9]
matrix =
1 2 3
4 5 6
7 8 9
11. Index
MATLAB index starts with 1, NOT 0!
Vector Index:
◦ a = [22 17 7 4 42]
a(1) = 22
a(3) = 7
Matrix Index:
◦ a = [7 12 42; 5 1 23; 4 9 10];
a(1, 3) = 42
a(3, 2) = 9
12. Some matrix functions in MATLAB
X = ones(r,c) Creates matrix full with ones
X = zeros(r,c) Creates matrix full with zeros
A = diag(x) Creates squared matrix with vector x in diagonal
[r,c] = size(A) Return dimensions of matrix A
X = A’ Transposed matrix
X = inv(A) Inverse matrix squared matrix
X = pinv(A) Pseudo inverse
d = det(A) Determinant
[X,D] = eig(A) Eigenvalues and eigenvectors
[U,D,V] = svd(A) singular value decomposition
13. Dot Operator
For scalar operations, nothing new is needed. Example:
a = 5; b = 3; ==> c = a*b %c = 15
For element operations, a dot must be used before the operator.
Note: Dot operator not the same as dot product!
Example:
◦ a = [1 2 3 4];
◦ b = [5 6 7 8];
◦ c = a*b
◦ Result: ??? Error using ==> mtimes
inner matrix dimensions must agree
Now, try:
◦ c = a.*b %notice the dot!
◦ Result: c=[5 12 21 32]
Notice what it is doing: a(1)*b(1), a(2)*b(2), etc.
14. Dot Product
Check Dimensions
◦ A*B
◦ A*B’
◦ A’*B
Use MATLAB command
>> C=dot(A,B)
Cross Product
By definition
Use MATLAB command
>> C=cross(A,B)
Vector Products
Consider & generate two random 3*1 vectors, A & B
A=rand(3,1) , B=rand(3,1)
A(2)*B(3)-A(3)*B(2)
A(3)*B(1)-A(1)*B(3)
A(1)*B(2)-A(2)*B(1)
15. Loops
Loop statements do not need parenthesis. The statements
are recognized via tabs.
3 general types of loops:
◦ If/else/elseif loops
◦ For loops
◦ While loops
There is a 4th type, called nested loop, that can be any of the
above 3 (and any combinations of them).
17. For Loops
Counter variable does not have to be i; it can be any variable
Iterations can be tightly controlled with min:stepsize:max.
No need to pre-define the counter because you are declaring in
the for loop itself!
for i=1:10
statement
end
18. While Loops
As long as output of condition is a logic true, it will continue
looping until the condition becomes false.
BE CAREFUL OF INFINITE LOOPS.
while condition
statement
end
19. Nested Loops
Can use a mix of the different types of loops.
Very useful for performing algorithm/operations on vectors
and matrices.
20. Examples
Write a code to print out integer numbers from 5 to 36
Write a code to print out integers between 45 to 109 which are divisible
to 3
Write a code to print out integers between 45 to 109 which are divisible
to 5
Write a code :
◦ If n is divisible to 3 print “divisible to 3”
◦ elseIf n is divisible to 5 print “divisible to 5”
◦ Elseif n is divisible by 5 and 3 print “divisible by 15”
21. Section II
Feb 6st, 2016
◦ Graphing & Plots
◦ Scripts and Functions
◦ Linear & System of Equations Solving
22. Plots
plot –plot(x,y,linespec)
◦ x and y vectors must be same length!
subplot –subplot(m,n,p) where mxn matrix,
p = current graph
figure –creates new window for graph
title –creates text label at top of graph
xlabel/ylabel–horizontal and vertical labels
for graph
26. Axis-specific
Helps focus on what is important on the graph.
Change only the x or y axis limits:
◦ xlim([xminxmax]) or ylim([yminymax])
◦ min and max can be positive or negative.
example
27. Some useful commands
Grid on
Hold on
Legend
Plot3 : example
◦ t=0:pi/50:10*pi;
◦ st=sin(t);
◦ ct=cos(t)
◦ figure;
◦ plot3(st,ct,t)
28. plotyy
2D line plots with y-axis on both left and right side
◦ [AX,H1,H2]=plotyy(x1,y1,x2,y2)
29. Scripts & functions
Two kind of M-files:
◦ Scripts
◦ Functions:
With parameters and returning values
Only visible variables defined inside the
function or parameters
Usually one file for each function defined
Structure:
30. Function example
In function:
◦ function y=average(x)
◦ y=sum(x)/length(x);
◦ end
In script:
◦ z=1:99;
◦ average(z)
31. Polynomials
We can use an array to represent a polynomial. To do so we
use list the coefficients in decreasing order of the powers. For
example x3+4x+15 will look like [1 0 4 15]
To find roots of this polynomial we use roots command. roots
([1 0 4 15])
To create a polynomial from its roots poly command is used.
poly([1 2 3]) where r1=1, r2=2, r3=3
To evaluate the new polynomial at x =5 we can use polyval
command. polyval([ 1 -6 11 -6], 5)
32. Systems of Equations
Consider the following system of equations
◦ x+5y+15z=7
◦ x-3y+13z=3
◦ 3x-4y-15z=11
One way to solve this system of equations is to use matrices.
First, define matrix A:
◦ A=[1 5 15; 1 -3 13; 3 -4 15];
Second, matrix b:
◦ b=[7;3;11];
Third, we solve the equation Ax=b for x, taking the inverse of A
and multiply it by b:
◦ x=inv(A)*b
Note that we cannot solve equation Ax=b by dividing b by A
because vectors A and b have different dimensions!
34. Systems of Equations
Consider the following system of equations
◦ x+5y+15z=7
◦ x-3y+13z=3
◦ 3x-4y-15z=11
One way to solve this system of equations is to use matrices.
First, define matrix A:
◦ A=[1 5 15; 1 -3 13; 3 -4 15];
Second, matrix b:
◦ b=[7;3;11];
Third, we solve the equation Ax=b for x, taking the inverse of A
and multiply it by b:
◦ x=inv(A)*b
Note that we cannot solve equation Ax=b by dividing b by A
because vectors A and b have different dimensions!
36. Symbolic toolbox example
Define functions
F1=6x3-4x2+bx-5, F2= Sin (y),
and
F3= Sqrt(x).
Use int() function to determine:
37. Solving ODE with dsolve, 1st
order
First order equations
◦ y’=xy
y=dsolve(‘Dy=y*x’,’x’)
y=dsolve(‘Dy=y*x’,’y(1)=1’,’x’)
◦ Or
eq1=‘Dy=y*x’;
y=dsolve(eq1,’x’);
◦ Issues
our expression for y(x) isn’t suited for array operations:
vectorize()
y, as MATLAB returns it, is actually a symbol : eval()
◦ Plotting
x = linspace(0,1,20);
z = eval(vectorize(y));
plot(x,z)
38. Solving ODE with dsolve, 2nd
order
Second order equations
◦ EQ:
eqn2 = ’D2y + 8*Dy + 2*y =cos(x)’;
inits2 = ’y(0)=0 , Dy(0)=1’;
y=dsolve(eqn2,inits2,’x’)
◦ Plotting
x = linspace(0,1,20);
z = eval(vectorize(y));
plot(x,z)
39. Solving ODE with dsolve,
systems
Systems of equations
◦ EQs:
[x,y,z]=dsolve(’Dx=x+2*y-z’,’Dy=x+z’,’Dz=4*x-4*y+5*z’)
inits=’x(0)=1,y(0)=2,z(0)=3’;
[x,y,z]=dsolve(’Dx=x+2*y-z’,’Dy=x+z’,’Dz=4*x-4*y+5*z’,inits)
Notice that since no independent variable was specified,
MATLAB used its default, t.
◦ Plotting
t=linspace(0,.5,25);
xx=eval(vectorize(x));
yy=eval(vectorize(y));
zz=eval(vectorize(z));
plot(t, xx, t, yy, t, zz)
40. Solving ODE with dsolve,
systems
solve the given differential equations symbolically.
41. ODE Solving Numerically
Defining an ODE function in an M-file
Solving First-Order ODEs
Solving Systems of First-Order ODEs
Solving higher order ODEs
42. Numerical methods are used to solve initial value
problems where it is difficult to obtain exact solutions
MATLAB has several different built-ins
functions for numerical solution of
ODEs
44. Solving first-order ODEs
Define the function M-file
function y_dot=EOM(y,t)
global alpha gamma
y_dot=alpha * y - gamma * y^2;
end
Make a script M-file
global alpha gamma
alpha=0.15;
gamma=3.5;
y0=1;
[y t]=ode45(@EOM,[0 10],y0);
plot(t,y);
45. Solving systems of ODEs
Example
Create function containing the equations
Change the error tolerance using odeset
Plotting the columns of returned vector
46. Solving a Stiff system
Equations
Create the function
Ode and plotting
47. Example
Use “ode45” to solve the following differential
equation and plot y(x) in the interval of [0,6π]. Put
your name in the plot title.
48. Solving higher-order ODE
Project: Simple pendulum
Equation of motion is given by:
2nd order Nonlinear ODE
Convert the 2nd order ODE to standard form
49. Simple Pendulum
Initial conditions and constants
Coding
◦ Make a function M-file for equation of motion
function z_dot=EOM_pendulum(t,z)
global G L
theta=z(1);
theta_dot=z(2);
theta_dot2=-(G/L)*sin(theta);
z_dot=[theta_dot;theta_dot2];
end
◦ Make a script M-file to run the code
global G L
G=9.8;
L=2;
tspan=[0 2*pi];
inits=[pi/3 0];
[t, y]=ode45(@EOM_pendulum,tspan,inits);
52. Simulink
>> simulink
◦ Continuous and discrete dynamics blocks, such as
Integration, Transfer functions, Transport Delay, etc.
◦ Math blocks, such as Sum, Product, Add, etc
◦ Sources, such as Ramp, Random Generator, Step, etc
54. Example
Open a new window and drag the following blocks
into that file
Run the simulation and double click on the scope
to see the results
55. Example
Open a new window and drag the following blocks
into that file
Run the simulation and double click on the scope
to see the results
56. Example
Open a new window and drag the following blocks
into that file
Run the simulation and double click on the scope
to see the results
57. simple model
Build a Simulink model that solves the differential
equation
Initial condition
First, sketch a simulation diagram of this mathematical
model (equation)
Input is the forcing function 3sin(2t)
Output is the solution of the differential equation x(t)
t
x 2
sin
3
58. simple model
Build a Simulink model that solves the differential
equation
Double-click on
the Sine Wave
block to set
amplitude = 3
and freq = 2.
This produces the
desired input of
3sin(2t)
59. simple model
It should look like this, change the initial condition
in integrator to -1
60. Example – Toy Train
consisting of an engine and a car. Assuming that
the train only travels in one direction, we want to
apply control to the train so that it has a smooth
start-up and stop, along with a constant-speed ride
The mass of the engine and the car will be
represented by M1 and M2, respectively. The two
are held together by a spring, which has the
stiffness coefficient of k. F represents the force
applied by the engine, and the Greek letter, mu,
represents the coefficient of rolling friction.
64. Example – Toy Train
Before running the model, we need to assign numerical
values to each of the variables used in the model.
Create an new m-file and enter the following
commands.
M1=1;
M2=0.5;
k=1;
F=1;
mu=0.002;
g=9.8;
Execute your m-file to define these values. Simulink will
recognize MATLAB variables for use in the model.
