Matlab solved problems

MotivationHow it is useful for:

Summary

A Layman Approach

bharani@aero.iitb.ac.in

Command Window

Command History

Workspace

MATLAB GUI

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]

1 2 5 1

• A matrix x = [1 2 3; 5 1 4; 3 2 -1]

3 2 -1

• Transpose y = x.’ y =

Matrices (Contd…)

• x(i,j) subscription

• whole row

• whole column

y = x(2,3)

y = x(3,:)

3 2 -1

y = x(:,2)

Let, x = [ 1 2 3

3 2 -1]

Operators (Arithmetic)

+ addition

- subtraction

* multiplication

/ division

^ power

‘ complex conjugate

transpose

.* element-by-element mult

./ element-by-element div

.^ element-by-element power

.‘ transpose

Operators (Relational, Logical)

== equal

~= not equal

< less than

<= less than or equal

> greater than

>= greater than or equal

pi 3.14159265…

j imaginary unit,

i same as j

Generating Vectors from functions

• zeros(M,N) MxN matrix of zeros

• ones(M,N) MxN matrix of ones

• rand(M,N) MxN matrix of uniformly

distributed random numbers

on (0,1)

x = zeros(1,3)

x = ones(1,3)

x = rand(1,3)

0.9501 0.2311 0.6068

Operators (in general)

[ ] concatenation

( ) subscription

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

0 0 0 1 1

x = [ 1 3 5 7 9]

1 3 5 7 9

y = x(2)

y = x(2:4)

MATRIX OPERATIONS

Let, A = eye(3)

>> A = [ 1 0 0

0 0 1 ]

>> eig(A)

>> inv(A)

MATRIX OPERATIONS (Contd…)

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

A = [ 1 2 3

7 8 9 ]

>> eig(A)

16.1168

-1.1168

-0.0000

>> B=A'

>>C=A*B

14 32 50

32 77 122

50 122 194

>> D=A.*B

1 8 21

8 25 48

21 48 81

QUESTIONS ?

Applications of MATLAB(Lecture 3)

P Bharani Chandra Kumar

Overview

Linear algebra

Solving a linear equation

Finding eigenvalues and eigenvectors

Curve fitting and interpolation

Data analysis and statistics

Nonlinear algebraic equations

Linear AlgebraSolving 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

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

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:

A v = λ v

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.

b) Sole for ‘n’ eigenvectors by substituting the corresponding eigenvalues in above eqn.

0=− IA λ

Linear Algebra (Contd…)

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)

-0.1709 0.8729 0.4570

-0.2365 0.4139 -0.8791

0.9565 0.2583 -0.1357

-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

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:

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

301140533315Y

1005020105x

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

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')

30.2479

Method 3: Spline Interpolation

MATLAB Command :

>> yi = interp1(x,y,5.5,’spline')

30.2500

Note: 5.5*5.5=

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

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

Solution:

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

1.7878

5sin −=

Solving nonlinear algebraic equations

(contd…)

5sin −=

• Example 2: Solve

Solution:

x= fzero(‘x*x-2*x+4',1)

1.7878

0422=+− xx

QUESTIONS ?

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

– 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-

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

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

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.

(Contd…)

Unonstrained Optimization : Example

Consider the above problem with no constraints:

Solution by MATLAB:

Step 1: Create an inline object of the function to be minimizedfun = 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);

)12424()( 22++++= yxyyxexf

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:

Subjected to:

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

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

)12424()( 22++++= yxyyxexf

(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:

-9.5474 1.0474

% The function value at the solution is:

fval =

0.0236

% Both the constraints are active at the solution:

[c, ceq] = confun(x)

1.0e-014 *

0.1110

-0.1776

QUESTIONS ?

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?

Can you Say

What is this!

Cold Water Hot Water

Heater

Determining system dynamics from input-output

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

� 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

Validate models using Models Views.

Export models to the MATLAB workspace for

further processing .

QUESTIONS ?

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

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!’)

x = 1:n;

term = sqrt(x);

y = sum(term)

Jump to here if TRUE

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

∆ < 0 (no real roots)

∆ = 0 (one real root)

∆ > 0 (two real roots)

MATLAB needs to make

decisions (based on ∆)

42−±−

acb 42−=∆

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

Print message ‘Two real roots’

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:\n\n')

disp(['discriminant = ', num2str(delta)])

elseif delta == 0

fprintf('\nEquation has one real root:\n')

xone = -b/(2*a)

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 = %10.2e\n\t Second root = %10.2f', x(1),x(2))

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

QUESTIONS ?

Graphics

G Satish

Outline

� 2-D plots

� 3-D plots

� Handle objects

2-D plots

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’)

Overlay plots

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 -. dashdotc cyan + plus -- dashed

m magenta * stary yellow s square

k black d diamond

v triangle (down)^ triangle (up)

< triangle (left)> triangle (right)

p pentagramh 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);

3-D plots

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

>> 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);

View (contd…)

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)

Mesh plots(cont…)

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)

Surface plots (contd…)

Handle Graphics

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])

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