Matlab Mech Eee Lectures 1

Motivation
How it is useful for:
Summary

Introduction to MATLAB
A Layman Approach

P Bharani Chandra Kumar
(bharani@aero.iitb.ac.in)

Department of Electrical Engineering
GMR Institute of Technology
Rajam, AP

Lecture series on MATLAB


Motivation
Summary

Outline

1 Motivation
History of MATLAB
Strengths of MATLAB
Weakness of MATLAB

2 How it is useful for:
Students
Engineers/ Scientists


Motivation History of MATLAB
How it is useful for: Strengths of MATLAB
Summary Weakness of MATLAB

Outline

1 Motivation
History of MATLAB
Strengths of MATLAB
Weakness of MATLAB

Students



MATLAB stands for MATrix LABoratory.
Developed primarily by Cleve Moler in the 1970s.
Need student access to Fortran subroutines for solving
linear (LINPACK) and eigenvalue (EISPACK) problems
without requiring knowledge of Fortran .
Developed as an interactive system to access LINPACK
and EISPACK.
Gained popularity primarily through word of mouth
In the 1980s, MATLAB was rewritten in C with more
functionality
Mathworks, Inc. was created in 1984 is now responsible for
development, sale, and support for MATLAB



MATLAB is relatively easy to learn
MATLAB code is optimized to be relatively quick when
performing matrix operations
MATLAB may behave like a calculator or as a programming
language
MATLAB is interpreted, errors are easier to ﬁx
Although primarily procedural, MATLAB does have some
object-oriented elements.



MATLAB is NOT a general purpose programming language
MATLAB is an interpreted language (making it for the most
part slower than a compiled language such as C++)
MATLAB is designed for scientiﬁc computation and is not
suitable for some things (such as parsing text)
MATLAB is an interpreted language, slower than a
compiled language such as C++
MATLAB commands are speciﬁc for MATLAB usage


Motivation
Students
Summary

Outline

1 Motivation
History of MATLAB
Strengths of MATLAB
Weakness of MATLAB

Students


Motivation
Students
Summary

How it is useful for students:
System dynamics can be analysed very easily
Messy equations can be solved very easily
Can enhance the skills required for present jobs
Can do the projects in a very easy way
Can be useful for analysis and study of multi-disciplinary
research areas


Motivation
Students
Summary

How it is useful for Engineer/Scientists:
Can do the research in various ﬁelds
Contains various inbuilt blocks like electric power systems,
IC engines, aircraft, process plant, etc .
Need not required to struggle alot to get output as
compared to other major packages
Can compare the existing literature results using wide
simulations
Can be useful for analysis and study of multi-disciplinary
research
Can publish papers based on the simulations obtained


Motivation
Summary

Summary
General intro to MATLAB.
History of MATLAB.
Who and how it can be utilized!


Appendix For Further Reading

For Further Reading I

Rudra Pratap.
Started with MATLAB : A Quick intro for Scientists and
Engineers.
Oxford University Press, 2006.
www.mathworks.com


Basics of MATLAB
(Lecture 2)


bharani@aero.iitb.ac.in

MATLAB GUI

Command Window

Workspace

Command History

Desktop Tools

Command Window
type commands

Workspace
view program variables
clear to clear
double click on a variable to see it in the Array Editor

Command History
view past commands
save a whole session using diary


Matrices

• A vector x = [1 2 5 1]

x =
1 2 5 1

• A matrix x = [1 2 3; 5 1 4; 3 2 -1]

x =
1 2 3
5 1 4
3 2 -1

• Transpose y = x.’ y =
1
2
5
1

Matrices (Contd…)
Let, x= [ 1 2 3
5 1 4
3 2 -1]
y = x(2,3)
• x(i,j) subscription
y =
4

• whole row y = x(3,:)
y =
3 2 -1
y = x(:,2)
• whole column
y = 2
1
2

Operators (Arithmetic)

+ addition
- subtraction
* multiplication .* element-by-element mult
/ division ./ element-by-element div
^ power .^ element-by-element power
‘ complex conjugate .‘ transpose
transpose


Operators (Relational, Logical)

== equal pi 3.14159265…
~= not equal j imaginary unit, −1
< less than i same as j
<= less than or equal
> greater than
>= greater than or equal

& AND
| OR
~ NOT

Generating Vectors from functions

x = zeros(1,3)
• zeros(M,N) MxN matrix of zeros x =
0 0 0

• ones(M,N) MxN matrix of ones x = ones(1,3)
x =
1 1 1

• rand(M,N) MxN matrix of uniformly
x = rand(1,3)
distributed random numbers
on (0,1) x =
0.9501 0.2311 0.6068


Operators (in general)

[ ] concatenation x = [ zeros(1,3) ones(1,2) ]
x =
0 0 0 1 1

( ) subscription x = [ 1 3 5 7 9]
x =
1 3 5 7 9

y = x(2)
y =
3
y = x(2:4)
y =
3 5 7

MATRIX OPERATIONS
Let, A = eye(3)

>> A = [ 1 0 0
0 1 0
0 0 1 ]

>> eig(A) >> inv(A) >>A' >>A*A

ans = ans = ans = ans =

1 1 0 0 1 0 0 1 0 0
1 0 1 0 0 1 0 0 1 0
1 0 0 1 0 0 1 0 0 1

MATRIX OPERATIONS (Contd…)
Let, >> A = [1 2 3;4 5 6;7 8 9]

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

>> eig(A) >> B=A' >>C=A*B >> D=A.*B

ans = B= C= D=

16.1168 1 4 7 14 32 50 1 8 21
-1.1168 2 5 8 32 77 122 8 25 48
-0.0000 3 6 9 50 122 194 21 48 81

QUESTIONS ?


Applications of MATLAB
(Lecture 3)



Overview

Linear algebra
Solving a linear equation
Finding eigenvalues and eigenvectors

Curve fitting and interpolation

Data analysis and statistics

Nonlinear algebraic equations


Linear Algebra
Solving a linear system
Find the values of x, y and z for the following equations:

5x = 3y – 2z +10
8y +4z = 3x + 20
2x + 4y - 9z = 9
Step 1: Rearrange equations:

5x - 3y + 2z = 10
- 3x + 8y +4z = 20
2x + 4y - 9z = 9
Step 2: Write the equations in matrix form:
[A] x = b
 5 −3 2  10 
A = − 3 8
 4 b = 20
 
2
 4 − 9
 9
 

Linear Algebra (Contd…)
Step 3: Solve the matrix equation in MATLAB:

>> A = [ 5 -3 2; -3 8 4; 2 4 -9];
>> b = [10; 20; 9]

>> x = A b
x =
3.442
3.1982
1.1868
% Veification
>> c = A*x
>> c =
10.0000
20.0000
9.0000

Finding eigenvalues and eigenvectors
Eigenvalue problem in scientific computations shows up as:

Av=λv

The problem is to find ‘λ’ and ‘v’ for a given ‘A’ so that above eq.
is satisfied:

Method 1: Classical method by using pencil and paper:

a) Finding eigenvalues from the determinant eqn.

A − λI = 0
b) Sole for ‘n’ eigenvectors by substituting the corresponding
eigenvalues in above eqn.

Method 2: By using MATLAB:

Step 1: Enter matrix A and type [V, D] = eig(A)

>> A = [ 5 -3 2; -3 8 4; 2 4 -9];
>> [V, D] = eig(A)
V =

-0.1709 0.8729 0.4570
-0.2365 0.4139 -0.8791
0.9565 0.2583 -0.1357
D =

-10.3463 0 0
0 4.1693 0
0 0 10.1770



Step 2: Extract what you need:

‘V’ is an ‘n x n’ matrix whose columns are eigenvectors

D is an ‘n x n’ diagonal matrix that has the eigenvalues of
‘A’ on its diagonal.


Cross check:
Let us check 2nd eigenvalue and second eigenvector will
satisfy A v = λ v or not:

v2=V(:,2) % 2nd column of V
v2 =
0.8729
0.4139
0.2583

>> lam2=D(2,2) % 2nd eigevalue
lam2 =
4.1693
>> A*v2-lam2*v2
ans = 1.0e-014 *
0.0444
-0.1554
0.0888

Curve Fitting
What is curve fitting ?

It is a technique of finding an algebraic relationship that “best”
fits a given set of data.

There is no magical function that can give you this relationship.

You have to have an idea of what kind of relationship might exist
between the input data (x) and output data (y).

If you do not have a firm idea but you have data that you trust,
MATLAB can help you to explore the best possible fit.


Curve Fitting (Contd…)
Example 1 : straight line (linear) fit:

x 5 10 20 50 100
Y 15 33 53 140 301

Step 1: Plot raw data:

Enter the data in MATLAB and plot it:
>> x = [ 5 10 20 50 100];
>> y = [15 33 53 140 301];
>> plot (x,y,’o’)
>> grid


Curve Fitting (Contd…)
Example 2 : Comparing different fits:

x = 0: pi/30 : pi/3
y = sin(x) + rand (size(x))/100

Step 1: Plot raw data:
>> plot (x,y,’o’)
>> grid

Step 2: Use basic fitting to do a quadratic and cubic fit

Step 3 : Choose the best fit based on the residuals


Interpolation

What is interpolation ?

It is a technique of finding a functional relationship between
variables such that a given set of discrete values of the variables
satisfy that relationship.

Usually, we get a finite set of data points from experiments.

When we want to pass a smooth curve through these points or
find some intermediate points, we use the technique of
interpolation.

Interpolation is NOT curve fitting, in that it requires the
interpolated curve to pass through all the data points.

Data can be interpolated using Splines or Hermite interpolants.

Interpolation (Contd…)
MATLAB provides the following functions to facilitate
interpolation:

interp1 : One data interpolation i.e. given yi and xi, finds yj at
desired xj from yj = f(xj).
ynew = interp1(x,y,xnew, method)

interp2 : Two dimensional data interpolation i.e. given zi at
(xi,yi) from z = f(x,y).
znew = interp2(x,y,z,xnew,ynew, method)

interp3 : Three dimensional data interpolation i.e. given vi at
(xi,yi,zi) from v = f(x,y,z).
vnew = interp2(x,y,z,v,xnew,ynew,znew,
method)

spline : ynew = spline(x,y,xnew, method)

Example:

x = [1 2 3 4 5 6 7 8
9]
y = [1 4 9 16 25 36 49
64 81]
Find the value of 5.5?

Method 1: Linear Interpolation

MATLAB Command :

>> yi=interp1(x,y,5.5,'linear')

yi = 30.5000


Method 2: Cubic Interpolation

MATLAB Command :

>> yi = interp1(x,y,5.5,’cubic')

yi =
30.2479

Method 3: Spline Interpolation

MATLAB Command :

>> yi = interp1(x,y,5.5,’spline') Note: 5.5*5.5=

yi =
30.2500

Data Analysis and statistics

It includes various tasks, such as finding mean, median,
standard deviation, etc.

MATLAB provides an easy graphical interface to do such type of
tasks.

As a first step, you should plot your data in the form you wish.

Then go to the figure window and select data statistics from the
tools menu.

Any of the statistical measures can be seen by checking the
appropriate box.


Data Analysis and statistics (Contd…)

Example:

x = [1 2 3 4 5 6 7 8
9]
y = [1 4 9 16 25 36 49
64 81]

Find the minimum value, maximum value, mean, median?


It can be performed directly by using MATLAB commands also:

Consider:

x = [1 2 3 4 5]

mean (x) : Gives arithmetic mean of ‘x’ or the avg. data.
MATLAB usage: mean (x) gives 3.

median (x) : gives the middle value or arithmetic mean of two middle
Numbers.
MATLAB usage: median (x) gives 3.

Std(x): gives the standard deviation

Max(x)/min(x): gives the largest/smallest value

Solving nonlinear algebraic equations
Step 1: Write the equation in the standard form:

f(x) = 0

Step 2: Write a function that computes f(x).

Step 3: Use the built-in function fzero to find the solution.

Example 1: Solve

sin x = e x − 5
Solution:

x= fzero('sin(x)-exp(x)+5',1)

x =
1.7878

Solving nonlinear algebraic equations
(contd…)

• Example 2: Solve

x2 − 2x + 4 = 0
Solution:

x= fzero(‘x*x-2*x+4',1)

x x=e
sin
= x
−5
1.7878


Optimization Techniques
through MATLAB
(Lecture 4)



Overview

Unconstrained Optimization

Constrained Optimization

Constrained Optimization through gradients


Why Optimize!
Engineers are always interested in finding the
‘best’ solution to the problem at hand
Fastest
Fuel Efficient

Optimization theory allows engineers to
accomplish this
Often the solution may not be easily obtained
In the past, it has been surrounded by certain
mistakes


The Greeks started it!
Queen Dido of Carthage (7 century
BC)
– Daughter of the king of Tyre
– Agreed to buy as much land as
she could “enclose with one
bull’s hide”
– Set out to choose the largest
amount of land possible, with
one border along the sea
• A semi-circle with side
touching the ocean
• Founded Carthage
– Fell in love with Aeneas but
committed suicide when he left.


The Italians Countered
Joseph Louis Lagrange (1736-1813)
His work Mécanique Analytique (Analytical
Mechanics) (1788) was a mathematical
masterpiece
Invented the method of ‘variations’ which
impressed Euler and became ‘calculus of
variations’
Invented the method of multipliers
(Lagrange multipliers)
Sensitivities of the performance index to
changes in states/constraints
Became the ‘father’ of ‘Lagrangian’
Dynamics
Euler-Lagrange Equations

The Multi-Talented Mr. Euler
Euler (1707-1783)
Friend of Lagrange
Published a treatise which became
the de facto standard of the
‘calculus of variations’
The Method of Finding Curves
that Show Some Property of
Maximum or Minimum
He solved the brachistachrone
(brachistos = shortest, chronos =
time) problem very easily
Minimum time path for a bead
on a string, Cycloid


Hamilton and Jacobi
William Hamilton (1805-1865)
Inventor of the quaternion

Karl Gustav Jacob Jacobi (1804-
1851)
Discovered ‘conjugate points’ in
the fields of extremals
Gave an insightful treatment to
the second variation

Jacobi criticized Hamilton’s work
Hamilton-Jacobi equation
Became the basis of Bellman’s
work 100 years later


What to Optimize?
Engineers intuitively know what they are
interested in optimizing

Straightforward problems
Fuel
Time
Power
Effort
More complex
Maximum margin
Minimum risk
The mathematical quantity we optimize is called
a cost function or performance index

Optimization through MATLAB
Consider initially the problem of finding a minimum
to the function:

MATLAB function FMINCON solves problems of the form:

min F(X)

subject to: A*X <= B, Aeq*X = Beq (linear constraints)

C(X) <= 0, Ceq(X) = 0 (nonlinear constraints)

LB <= X <= UB


(Contd…)

X = FMINCON(FUN,X0,A,B) starts at X0 and finds a minimum
X to the function FUN, subject to the linear inequalities A*X <= B.

X=FMINCON(FUN,X0,A,B,Aeq,Beq) minimizes FUN subject to
the linear equalities:

Aeq*X = Beq as well as A*X <= B.

(Set A=[ ] and B=[ ] if no inequalities exist.)


(Contd…)
X=FMINCON(FUN,X0,A,B,Aeq,Beq,LB,UB) defines a set of
lower and upper bounds on the design variables, X, so that the
solution is in the range LB <= X <= UB.

Use empty matrices for LB and UB if no bounds exist.
Set LB(i) = -Inf if X(i) is unbounded below; and set UB(i) = Inf if X(i)
is unbounded above.

X=FMINCON(FUN,X0,A,B,Aeq,Beq,LB,UB,NONLCON) subjects
the minimization to the constraints defined in NONLCON.

The function NONLCON accepts X and returns the vectors C and
Ceq, representing the nonlinear inequalities and equalities
respectively.


Unonstrained Optimization : Example
Consider the above problem with no constraints:

f ( x) = e x (4 x 2 + 2 y 2 + 4 xy + 2 y + 1)

Solution by MATLAB:

Step 1: Create an inline object of the function to be minimized
fun = inline('exp(x(1)) * (4*x(1)^2 + 2*x(2)^2 +
4*x(1)*x(2) + 2*x(2) + 1)');

Step 2: Take a guess at the solution:
x0 = [-1 1];

Step 3: Solve using fminunc function:
[x, fval] = fminunc(fun, x0);


Unconstrained Optimization : Example
(Contd…)
>> x =

0.5000 -1.0000

>> fval =

1.3028e-010


Constrained Optimization : Example
Consider initially the problem of finding a minimum
to the function:
f ( x) = e x (4 x 2 + 2 y 2 + 4 xy + 2 y + 1)

Subjected to:

1.5 + x(1).x(2) - x(1) - x(2) < = 0

- x(1).x(2) < = 10


(contd…)
Solution using MATLAB:

Step 1: Write the m-file for objective function:

function f = objfun(x)
% objective function
f=exp(x(1)) * (4*x(1)^2 + 2*x(2)^2 + 4*x(1)*x(2) +
2*x(2) + 1);

Step 2: Write the m-file for constraints:
function [c, ceq] = confun(x)
% Nonlinear inequality constraints:
c = [1.5 + x(1)*x(2) - x(1) - x(2);
-x(1)*x(2) - 10];
% no nonlinear equality constraints:
ceq = [];

(contd…)
Step 3: Take a guess at the solution
x0 = [-1 1];

options =
optimset('LargeScale','off','Display','iter');

% We have no linear equalities or inequalities or
bounds,
% so pass [] for those arguments

[x,fval,exitflag,output] =
fmincon('objfun',x0,[],[],[],[],[],[],'confun',options)
;


max Directional
Iter F-count f(x) constraint Step-size derivative
Procedure
1 3 1.8394 0.5 1 0.0486
2 7 1.85127 -0.09197 1 -0.556 Hessian
modified twice
3 11 0.300167 9.33 1 0.17
4 15 0.529834 0.9209 1 -0.965
5 20 0.186965 -1.517 0.5 -0.168
6 24 0.0729085 0.3313 1 -0.0518
7 28 0.0353322 -0.03303 1 -0.0142
8 32 0.0235566 0.003184 1 -6.22e-006
9 36 0.0235504 9.032e-008 1 1.76e-010 Hessian
modified
Optimization terminated successfully:

% A solution to this problem has been found at:
x

x =

-9.5474 1.0474


% The function value at the solution is:
fval

fval =

0.0236

% Both the constraints are active at the solution:
[c, ceq] = confun(x)

c =

1.0e-014 *

0.1110
-0.1776

ceq =

[]


System Identification
(Lecture 5)



Overview

Introduction

Basic questions about system identification

Common terms used in system identification

Basic information about dynamical systems

Basic steps for system identification

An exciting example


Introduction

What is system identification?

Water Can you Say
Heater What is this!

Cold Water Hot Water



Determining system dynamics from input-output
data

Generate enough data for estimation and
validation

Select range for estimation and validation

Select order of the system

Check for best fit and determine the system
dynamics

Basic questions about system
identification


It enables you to build mathematical models of a
dynamic system based on measured data.

You adjust the parameters of a given model until its
output coincides as well as possible with the
measured output.

How do you know if the model is any good?
A good test is to compare the output of the model to
measured data that was not used for the fit.

identification (contd…)

What models are most common?
The most common models are difference-equation
descriptions, such as ARX and ARMAX models, as
well as all types of linear state-space models.

• Do you have to assume a model of a particular
type?
For parametric models, you specify the model
structure. This can be as easy as selecting a single
integer -- the model order -- or it can involve several
choices.



What does the System Identification Toolbox
contain?
It contains all the common techniques used to
adjust parameters in all kinds of linear models.

• How do I get started?
If you are a beginner, browse through The
Graphical User Interface. Use the graphical user
interface (GUI) and check out the built-in help
functions.



Is this really all there is to system identification?

There is a great deal written on the subject of
system identification.

However, the best way to explore system
identification is by working with real data.

It is important to remember that any estimated
model, no matter how good it looks on your screen,
is only a simplified reflection of reality.


Common terms used in system
identification

Estimation data: The data set that is used to create
a model of the data.

Validation data: The data set (different from
estimation data) that is used to validate the model.

Model views: The various ways of inspecting the
properties of a model, such as zeros and poles, as
well as transient and frequency responses.



Model sets or model structures are families of
models with adjustable parameters.

Parameter estimation is the process of finding the
"best" values of these adjustable parameters.

The system identification problem is to find both
the model structure and good numerical values of
the model parameters.



• This is a matter of using numerical search to find
those numerical values of the parameters that give
the best agreement between the model's (simulated
or predicted) output and the measured output.

• Nonparametric identification methods: Techniques
to estimate model behavior without necessarily
using a given parameterized model set.

• Model validation is the process of gaining
confidence in a model.

Basic information about dynamical
systems


Basic Steps for System
Identification

Import data from the MATLAB workspace.

Plot the data using Data Views.

Preprocess the data using commands in the
Preprocess menu.

For example, you can remove constant offsets or
linear trends (for linear models only), filter data, or
select regions of interest.


Basic Steps for System
Identification (contd…)

Select estimation and validation data.

Estimate models using commands in the Estimate
menu.

Validate models using Models Views.

Export models to the MATLAB workspace for
further processing .


Solution of Ordinary Differential
Equations and Engineering Computing

(Lecture 6)



Solution of Ordinary Differential
Equaions


Overview

Introduction

Examples


Introduction
DSOLVE Symbolic solution of ordinary differential
equations.

DSOLVE('eqn1','eqn2', ...) accepts symbolic
equations representing ordinary differential
equations and initial conditions.

Several equations or initial conditions may be
grouped together, separated by commas, in a single
input argument.

By default, the independent variable is 't'.

The independent variable may be changed from 't'
to some other symbolic variable by including
that variable as the last input argument.

Introduction (contd…)

The letter 'D' denotes differentiation with respect
to the independent variable, i.e. usually d/dt.

A "D" followed by a digit denotes repeated
differentiation; e.g., D2 is d^2/dt^2.

Any characters immediately following these
differentiation operators are taken to be the
dependent variables; e.g., D3y denotes the third
derivative of y(t).

Note that the names of symbolic variables should
not contain the letter "D".



Initial conditions are specified by equations like
'y(a)=b' or 'Dy(a) = b' where y is one of the
dependent variables and a and b are
constants.

If the number of initial conditions given is less
than the number of dependent variables, the
resulting solutions will obtain arbitrary
constants, C1, C2, etc.

Three different types of output are possible. For
one equation and one output, the resulting
solution is returned, with multiple solutions to
a nonlinear equation in a symbolic vector.



If no closed-form (explicit) solution is found, an
implicit solution is attempted. When an implicit
solution is returned, a warning is given.

If neither an explicit nor implicit solution can be
computed, then a warning is given and the
empty sym is returned.

In some cases concerning nonlinear equations,
the output will be an equivalent lower order
differential equation or an integral.


Examples
1) dsolve('Dx = -a*x') returns

ans: exp(-a*t)*C1

2) x = dsolve('Dx = -a*x','x(0) = 1','s') returns

ans: x = exp(-a*s)

3) y = dsolve('(Dy)^2 + y^2 = 1','y(0) = 0') returns
ans: y=
[ sin(t)]
[ -sin(t)]


Examples (contd…)

4) S = dsolve('Df = f + g','Dg = -f + g','f(0) = 1','g(0) = 2')

Ans: S.f = exp(t)*cos(t)+2*exp(t)*sin(t)
S.g = -exp(t)*sin(t)+2*exp(t)*cos(t)

5) dsolve('Df = f + sin(t)', 'f(pi/2) = 0')

Ans: -1/2*cos(t)- 1/2*sin(t)+1/2*exp(t)/(cosh(1/2*pi)+
sinh(1/2*pi))

Engineering Computing


Outline of lecture

MATLAB as a calculator revisited

Concept of M-files

Decision making in MATLAB

Use of IF and ELSEIF commands

Example: Real roots of a quadratic


MATLAB as a calculator

MATLAB can be used as a ‘clever’ calculator
This has very limited value in engineering

Real value of MATLAB is in programming
Want to store a set of instructions
Want to run these instructions sequentially
Want the ability to input data and output
results
Want to be able to plot results
Want to be able to ‘make decisions’

Example

n
1 1 1 1
y=∑ = + + + ...
i =1 i 1 2 3

Can do using MATLAB as a calculator
>> x = 1:10;
>> term = 1./sqrt(x);
>> y = sum(term);

Far easier to write as an M-file


How to write an M-file

File → New → M-file

Takes you into the file editor

Enter lines of code (nothing happens)

Save file (we will call ours bharani.m)

Run file

Edit (ie modify) file if necessary


Bharani.m Version 1

n = input(‘Enter the upper limit: ‘);
x = 1:n; % Matlab is case sensitive
term = sqrt(x);
y = sum(term)

What happens if n < 1 ?


Bharani.m Version 2

n = input(‘Enter the upper limit: ‘);
if n < 1
disp (‘Your answer is meaningless!’)
end
x = 1:n;
term = sqrt(x); Jump to here if TRUE
y = sum(term)

Jump to here if FALSE


Decision making in MATLAB

For ‘simple’ decisions?
IF … END (as in last example)
More complex decisions?
IF … ELSEIF … ELSE ... END

Example: Real roots of a quadratic equation


Roots of ax2+bx+c=0

Roots set by discriminant − b ± b 2 − 4ac
∆ < 0 (no real roots)
x=
2a
∆ = 0 (one real root)
∆ > 0 (two real roots)
2
MATLAB needs to make
∆ = b − 4ac
decisions (based on ∆)


One possible M-file
Read in values of a, b, c
Calculate ∆
IF ∆ < 0
Print message ‘ No real roots’→ Go END
ELSEIF ∆ = 0
Print message ‘One real root’→ Go END
ELSE
Print message ‘Two real roots’
END


M-file (bharani.m)
%================================================
% Demonstration of an m-file
% Calculate the real roots of a quadratic equation
%================================================

clear all; % clear all variables
clc; % clear screen

coeffts = input('Enter values for a,b,c (as a vector): ');

% Read in equation coefficients
a = coeffts(1);
b = coeffts(2);
c = coeffts(3);

delta = b^2 - 4*a*c; % Calculate discriminant


M-file (bharani.m) (contd…)
% Calculate number (and value) of real roots

if delta < 0

fprintf('nEquation has no real roots:nn')
disp(['discriminant = ', num2str(delta)])

elseif delta == 0
fprintf('nEquation has one real root:n')
xone = -b/(2*a)
else
fprintf('nEquation has two real roots:n')
x(1) = (-b + sqrt(delta))/(2*a);
x(2) = (-b – sqrt(delta))/(2*a);
fprintf('n First root = %10.2ent Second root = %10.2f',
x(1),x(2))
end

Conclusions

MATLAB is more than a calculator
its a powerful programming environment

• Have reviewed:
– Concept of an M-file
– Decision making in MATLAB
– IF … END and IF … ELSEIF … ELSE … END
– Example of real roots for quadratic equation


Outline
2-D plots
3-D plots
Handle objects

Simple plot
>> a=[1 2 3 4]
>> plot(a)

Overlay plots
using plot command
>> t=0:pi/10:2*pi;
>> y=sin(t); z=cos(t);
>> plot(t,y,t,z)

Overlay plots(contd…)
using hold command
>> plot(t,sin(t));
>> grid
>> hold
>> plot(t,sin(t))

Overlay plots(contd..)
using line command
>> t=linspace(0,2*pi,10)
>> y=sin(t); y2=t; y3=(t.^3)/6+(t.^5)/120;
>> plot(t,y1)
>> line(t,y2,’linestyle’,’--’)
>> line(t,y3,’marker’,’o’)

Style options
>> t=0:pi/10:2*pi;
>> y=sin(t); z=cos(t);
>>plot(t,y,’go-’,t,z,’b*--’)

Style options
b blue . point - solid
g green o circle : dotted
r red x x-mark -. dashdot
c cyan + plus -- dashed
m magenta * star
y yellow s square
k black d diamond
v triangle (down)
^ triangle (up)
< triangle (left)
> triangle (right)
p pentagram
h hexagram

Labels, title, legend and other text
objects

>> a=[1 2 3 4]; b=[ 1 2 3 4]; c=[4 5 6 7]
>> plot(a,b,a,c)
>> grid
>> xlabel(‘xaxis’)
>> ylabel(‘yaxis’)
>> title(‘example’)
>> legend(‘first’,’second’)
>> text(2,6,’plot’)

Modifying plots with the plot editor
To activate this tool go to
figure window and click on the
left-leaning arrow
Now you can select and
double click on any object in
the current plot to edit it.
Double clicking on the selected
object brings up a property
editor window where you can
select and modify the current
properties of the object

Subplots
>> t=linspace(0,2*pi,10); y=sin(t); z=cos(t);
>> figure
>> subplot(2,1,1)
>> subplot(2,1,2)
>> subplot(2,1,1); plot(t,y)
>> subplot(2,1,2); plot(t,z)

Logarthmic plots
>> x=0:0.1:2*pi;
>>subplot(2,2,1)
>> semilogx(x,exp(-x));
>> grid

>> subplot(2,2,2)
>> semilogy(a,b);
>> grid

>> subplot(2,2,3)
>> loglog(x,exp(-x));
>> grid

Zoom in and zoom out
>> figure
>> plot(t,sin(t))
>> axis([0 2 -2 2])

Specialised plotting routines
>> stem(t,sin(t));
>> bar(t,sin(t))
>> a=[1 2 3 4]; stairs(a);

Simple 3-D plot
Plot of a parametric space curve
>> t=linspace(0,1,100);
>> x=t; y=t; z=t.^3;
>> plot3(x,y,z); grid

View
>> t=linspace(0,6*pi,100);
>> x=cos(t); y=sin(t); z=t;
>> subplot(2,2,1); plot3(x,y,z);
>> subplot(2,2,2); plot3(x,y,z); view(0,90);

Mesh plots
>> x=linspace(-3,3,50);
>> [X,Y]= meshgrid(x,y);
>> Z=X.*Y.*(X.^2-Y.^2)./(X.^2+Y.^2);
>> mesh(X,Y,Z);
>> figure(2)

Surface plots
>> u = -5 : 0.2 : 5;
>> [X,Y] = meshgrid(u,u);
>> Z=cos(X).*cos(Y).*exp(-sqrt(X.^2+Y.^2)/4);
>> surf(X,Y,Z)

Handle Graphics Objects
Handle Graphics is an object-oriented
structure for creating, manipulating and
displaying graphics
Graphics in Matlab consist of objects
Every graphics objects has:
– a unique identifier, called handle
– a set of characteristics, called properties

Getting object handles
There are two ways for getting object handles
• By creating handles explicitly at the object-
creation level commands
• By using explicit handle return functions

By creating handles explicitly at the object-
creation level commands
>> hfig=figure
>> haxes=axes(‘position’,[0.1 0.1 0.4 0.4])
>> t=linspace(0,pi,10);
>> hL = line(t,sin(t))
>> hx1 = xlabel(‘Angle’)

By using explicit handle return functions
>> gcf gets the handle of the current
figure
>> gca gets handle of current axes
>> gco returns the current object in the
current figure

Example
>> figure
>> axes
>> line([1 2 3 4],[1 2 3 4])
>> hfig = gcf
>> haxes = gca
Click on the line in figure
>>hL=gco

Getting properties of objects
The function ‘get’ is used to get a property
value of an object specified by its handle
get(handle,’PropertyName’)
The following command will get a list of all
property names and their current values of an
object with handle h
get(h)

Getting properties of objects
Example
>> h1=plot([ 1 2 3 4]);returns a line
object
>> get(h1)
>> get(h1,’type’)
>> get(h1,’linestyle’)

Setting properties of objects
The properties of the objects can be set by
using ‘set’ command which has the following
command form
Set(handle, ‘PropertyName’,Propertyvalue’)
By using following command you can see the
the list of properties and their values
Set(handle)

example
>> t=linspace(0,pi,50);
>> x=t.*sin(t);
>> hL=line(t,x);

>> set(hL,’linestyle’,’--’);

>> set(hL,’linewidth’,3,’marker’,’o’)

>> yvec = get(hL,’ydata’);
>> yvec(15:20) = 0;
>> yvec(40:45) = 0;
>> set(hL, ’ydata’, yvec)

Creating subplots using axes
command
>> hfig=figure
>> ha1=axes(‘position’,[0.1 0.5 0.3 0.3])
>> line([1 2 3 4],[1 2 3 4])

>> ha2=axes(‘position’[0.5 0.5 0.3 0.3])
>> line([1 2 3 4],[ 1 10 100 1000])

>> ha3=axes(‘position’,[0.1 0.1 0.3 0.3])
>> line([1 2 3 4],[0.1 0.2 0.3 0.4])

>> ha4=axes(‘position’,[0.5 0.1 0.3 0.3])
>> line([1 2 3 4],[10 10 10 10])

Matlab Mech Eee Lectures 1

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