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MATLAB

BEGINNING



MATLAB

Machine Code

MATLAB – High Level Programming Language

Assembly

Fortran C C++

BASIC JAVA

MATLAB

Easy Development

EfficientOperation



MATLAB

CHAPTER 1An Overview of MATLAB



MATLAB : Chapter 1. An Overview of MATLAB

References

1. Co So Matlab va Ung dung,Author : Pham Thi Ngoi Yen Publisher: Nha xuat ban Khoa hoc va Ky thuat, Ha Noi.Year : 2006

2. The MATLAB Help System

3. Essential MATLAB® for Engineers and Scientists Author : Brian D. Hahn and Daniel T. ValentinePublisher : Elsivier Year : 2007




Starting MatlabThe default MATLAB Desktop.




Operations (1/2)

Symbol Operation MATLAB form

^ exponentiation: a^b

* multiplication: a*b

/ right division: a/b

\ left division: a\b

+ addition: a+b

- subtraction: a-b

ba

ab

a/ b

b

b

b

a

b \aa

b

Scalar arithmetic operations




Operations (2/2)

Calculation result is savedto „ans ‟by default : ans = 9

6 is saved to ans by using ans one

more time

1.5 is saved to the variable „a‟byassign „a‟as the storing place

In the Matlab, „\ ‟= „ \‟




assignment , workspace, and command

The sign „=„ is called the„assignment or replacement‟operator.

e.g.x = 3 : assign the value 3 to the variable xx = x + 2 : add 2 to the current value of xWorkspace : the names and values of any variables in use in thecurrent work session

Commands for managing the worksession

clc clears the Command window

clear Removes all variables frommemory

exist(„name‟) Determines if a file or variableexists having the name ‟name‟

quit Stops MATLAB

who Lists the variable currently inmemory

whos Lists the current variables andsizes, and indicates if they haveimaginary parts

: Colon; generates and arrayhaving regularly spacedelements

, Comma; separates elements of an array

; Semicolon; suppresses screenprinting; also denotes a new rowin an array

… Ellipsis; continues a line




recall feature, and special variable

recall featurerecall a previously typed function or variablee.g.

up-arrow key( ↑) & down -arrow key(↓): move up and down through thepreviously typed lines one line at a time.Tab key: automatically completes the name of a function, variable, or file if youtype the first few letters of the name and press the Tab key.

Special variables and constants

ans Temporary variable containing the most recent answer

eps Specifies the accuracy of floating point precision

i , j The imaginary unit

Inf Infinity

NaN Indicates an undefined numerical result

pi The number π

1




Complex Number Operations

„;‟ is a command whichleads to a new line, alsoomits calculation results.




Formatting Commands

The Format command controls how numbers appear on the screen.

format short Four decimal digits (the default); 13.6745

format long 16 digits; 17.27484029463547

format short e Five digits (four decimals) plus exponent; 6.3793e+03

format long e 16 digits (15 decimals) plus exponent; 6.379243784781294e-04

format bank Two decimal digits; 126.73

format + Positive, negative, or zero; +

format rat Rational approximation; 43/7

format compact Suppresses some line feeds

format loose Resets to less compact display mode




Array(1/2)

One of the strengths of MATLAB is its ability to handle collections of numbers, called arrays . A numerical array is an ordered collection of numbers

We can use square bracketse.g .

>> x = [0, 1, 3, 6]

You need not type all the numbers in the array if they are regularlyspaced

e.g.

>> u = [0: 0.1: 10] u = [0, 0.1, 0.2, 0.3, … , 9.8, 9.9, 10]




Array(2/2)

You can compute ‘w=5*sin (u)’ for ‘u=[0: 0.1: 10]’

>> u=[0: 0.1: 10];>> w=5*sin(u);

computed the formula ‘w=5*sin(u)’ 101 times.

array index: points to a particular element in the array>>u(7)ans =

0.6000>>w(7)ans =

2.8232

length: determine how many values are in an array>>m = length(w)m =

101




Mathematical functions

Some commonly usedmathematical functions

Function MATLAB syntax ( )

exp(x)

sqrt(x)

log(x)log10(x)

cos(x)

sin(x)

tan(x)

acos(x)asin(x)

atan(x) The MATLAB trigonometric functionsuse radian measure

xe

x

xn xog 10

xosxinxan

xos - 1

xin- 1

xan - 1




Working with Files

M-file : MATLAB function files and program files are saved with theextension .m , and called M-files .

MAT-fi le : save the names and values of variables .

ASCII-file : files written in a specific format designed to make themusable to a wide variety of software.




System, directory, and file commands

system, directory, and file commands

addpath d i rname Adds the directory dirname to the search path.

cd d i rname Changes the current directory to dirname .

dir Lists all files in the current directory.

dir d i rname Lists all the files in the directory dirname .

path Displays the MATLAB search path.

pathtool Starts the Set Path tool.

pwd Displays the current directory.

rmpath d i rname Removes the directory dirname from the search path.

what Lists the MATLAB-specific files found in the current working directory. Most datafiles and other non-MATLAB files are not listed. Use di r to get a list of all files.

what d i rname Lists the MATLAB-specific files in directory dirname .




Plotting commands

Some MATLAB plotting commands

[x , y] = ginput(n) Enables the mouse to get n points from a plot, and returns the x andy coordinates in the vectors x and y, which have a length n .

grid puts grid lines on the plot.

gtext („text‟) Enables placement of text with the mouse.

plot (x , y) Generates a plot of the array y versus the array x on rectilinear axes.

title („text‟) Puts text in a title at the top of the plot.

xlabel („text‟) Adds a text label to the horizontal axis (the abscissa).

ylabel („text‟) Adds a text label to the vertical axis (the ordinate).




Plotting with MATLABVariable pairs must be written.(x,y) and (x,z) are pairs,so express „plot(x,y,x,z) ‟




Linear Algebra Equations

the left division operator(\)e.g.

64 zyx

5 zyx

7 0z2yx

Matlab ‟s basic unit is „matrix ‟




Statistics, Calculus, and Processing

Statisticsperform statistical calculations and other types of datamanipulation.

Numerical Calculus, Differential Equations, and SimulinkMATLAB can numerically compute the derivative andthe integral

Symbolic Processingobtain the derivative and the integral insymbolic form (a formula instead of as a set of numericalvalues)

dy/dx d x

dy/dx d x




Script Files and the Editor/Debugger(1/2)Two ways for performing operations in MATLAB

Interactive mode: directly enter the commands in the Command window

using sc r ip t files ( c o m m a n d s files): store the commands in script files M-files

sc r ip t files:when need to use many commands or arrays withmany elements

func t ion files:when need to repeat the operation of a set of commands




Script Files and the Editor/Debugger(2/2)

M-file make and save




Input/output commands

disp (A) Displays the contents, but not the name, of the array A.

disp („text‟) Displays the text string enclosed within single quotes.

format Controls the screen‟s output display format

fprintf Performs formatted writes to the screen or to a file

x = input („text‟) Displays the text in quotes, waits for user input from thekeyboard, and stores the value in x.

x = input („text‟ , ‟s‟) Displays the text in quotes, waits for user input from thekeyboard, and stores the input as a string in x

k = menu(„title‟,‟option1‟,‟option2‟,…)

Displays a menu whose title is in the string variable „title‟,and whose choices are „option1‟,‟option2‟, and so on.




The MATLAB Help System(1/4)

Help Browser Graphical user interfacefind information, view online documentation

Help Functionshelp, lookfor, docdisplay syntax information for specified function

Other Resourcesrun demos, contact technical support, participate in a newsgroup.The MathWorks Website

the home of MATLAB.http://www.mathworks.com




The MATLAB Help System(2/4)- Help Browser

The MATLAB Help Browser Contents: acontents listing tab

Index: a globalindex tab

Search: a searchtab having a findfunction and fulltext search features

Demos: abookmaking tab tostart built-indemonstrations




The MATLAB Help System(3/4)- Help Functions

MATLAB Help functions (help, lookfor, doc)

Function Use

doc Displays the start page of the documentation in the Help Browser

doc func t ion Displays the documentation for the MATLAB function func t ion

doc t o o l b o x /func t ion Displays the documentation for the specified toolbox function

doc t o o l b o x Displays the documentation road map page for the specified toolbox

help Displays a list all the function directories, with a description of thefunction category each represents.

help func t ion Displays in the Command window a description of the specified

function func t ion

helpwin top ic Displays the help text for the specified top ic inside the desktop HelpBrowser window

lookfor top ic Displays in the Command window a brief description for all functionswhose description includes the specified keyword top ic

type f i lename Displays the M-file f i lename without opening it with a text editor.




The MATLAB Help System(4/4)- Help Functions

Examples



28

MATLAB

CHAPTER 2Numeric, Cell, and Structure Arrays



MATLAB : Chapter 2. Numeric, Cell, and Structure Arrays

29

Arrays

Arrays : Collection of numbersthe basic building block in MATLABaddition, subtraction, multiplication, division, and exponentiationpolynomial algebra and root

Available classes of arrays in MATLAB 7

numeric arrays : contains only numeric values cell arrays : access data by its locationstructure arrays : access data by name

Array

numeric character logical cell structure functionhandle

Java




30

Arrays

Cartesian coordinates x, y, and zunit vector i, j, kexpress the vector p = x i + y j + z ke.g.

p = 5 i + 7 j + 2 kin MATLAB: write in a specific order,separate with a space,identify the group with brackets

[5 7 2]

vector row vector [5 7 2]column vector




31

Arrays

Creating Vectors in MATLABMATLAB displays row vectors horizontally and column vectors vertically.Create a row vector : space or commas separate elements

g = [3 7 9] = [3,7,9]Create a column vector : semicolon or Enter separate elements, you can usetranspose

g = [3;7;9] = [3,7,9]‟ = [3 (Enter)7 (Enter)9]

Create vectors by “appending” r = [2, 4, 20], w = [9, -6, 3]u = [r, w] = [2, 4, 20, 9, -6, 3]

MATLAB Ch 2 N i C ll d S A




32

Arrays

Creating Vectors in MATLABGenerate a large vector of regularly spaced elements using colon operator (:)

x = [m:q:n]m : the first value

q : the increment (default = 1)n : the last value <= nx = [0:2:8] = [0, 2, 4, 6, 8] x = [0:2:7] = [0, 2, 4, 6]y = [-3:2] = [-3, -2, -1, 0, 1, 2]u = [10:-2:4] = [10, 8, 6, 4]

The linspace command, logspace commandlinspace(x1, x2, n) : n is number of points between x1 and x2

linspace(5,8,31) = [5:0.1:8]logspace(a, b, n) : n is number of points between 10 a and 10 b

logspace(-1,1,4) = [0.1000, 0.4642, 2.1544, 10.000]

MATLAB Ch 2 N i C ll d S A




33

ArraysTwo-Dimensional Arrays

An array can have multiple rows, multiple columns, or bothcalled a matrix

.

Creating MatricesSpace or commas separate elements in different columnsSemicolons separate elements in different rows

You can also create a matrix from row or column vectors

<example (a = [1,3,5], b=[7,9,11]) >

7316

1042M

[1, 3, 5; 7, 9, 11]

[a; b]

[a b]

1197

531

1197531

MATLAB Ch t 2 N i C ll d St t A




34

ArraysArray Addressing

The row number is always listed firstThe colon operator selects individual elements, rows, columns,or ”subarrays” of arrays

v( : ) represents all the row or column elements of the vector v.v(2 : 5) represents the second through fifth elementsA( : , 3) denotes all the elements in the third column of the matrix A

C = B(2:3, 1:3)

948

7316C

1715123

25948

187316

131042

B

The general indexes of two dimensional array

MATLAB : Chapter 2 Numeric Cell and Structure Arrays




35

ArraysThe empty or null array

contains no elements, [ ]Rows and columns can be deleted by setting the selected row or column equal to the null array.

A(3, :) = [ ] deletes the third row in A

A(:, 2:4) = [ ] deletes the second through fourth columns In A A([1 4], : ) = [ ] deletes the first and fourth rows of A

< Other examples >

751

496A

A(1,5) = 300751

30496A

B = A(:,5:-1:1)

15700

69403B





36

ArraysSome Useful Array Functions

Command Description

cat(n, A, B, C, …) Creates a new array by concatenating the arrays A, B, C, and so on along the dimension n.

find(x) Computes an array containing the indices of the nonzero elements of the aray x.

[u, v, w] = find(A) Computes the arrays u and v, containing the row and column indices of the nonzero elements of thematrix A, and the array w, containing the values of the nonzero elements. The array w may be omitted.

length(A) Computes either the number of elements of A if A is a vector or the largest value of m or n if A is an m

× n matrix.

linspace(a,b,n) Creates a row vector of n regularly spaced values between a and b

logspace(a,b,n) Creates a row vector of n logarithmically spaced values between a and b

max(A) Returns the algebraically largest element in A if A is a vector. Returns a row vector containing thelargest elements in each column if A is a matrix. If any of the elements are complex, max(A) returns theelements that have the largest magnitudes.

[x, k] = max(A) Similar to max(A) but stores the maximum values in the row vector x and their indices in the row vector k

min(A) Same as max(A) but returns minimum values.

[x, k] = min(A) Same as [x, k] =max(A) but returns minimum values.

size(A) Returns a row vector [m n] containing the sizes of the m × n array A.

sort(A) Sorts each column of the array A in ascending order and returns an array the same size as A.

sum(A) Sums the elements in each column of the array A and returns a row vector containing the sums.





37

ArraysMagnitude, Length, and Absolute Value of a Vector

length : the number of elements in the vector.

magnitude :absolute value : The absolute value of a vector x is a vector whose elements are theabsolute values of the elements of x.

<example>x = [2, -4, 5]length = 3, magnitude = 6.7082, absolute value = [2, 4, 5]

The Array Editor

A graphical interface for working with arrays.To open the Array Editor from the Workspace Browser, double-click on the variable youwant to open.

22

2

2

1 n x x x





38

Multidimensional ArraysMATLAB supports multidimensional arraysThe first two dimensions are the row and column, as with a matrix.The higher dimensions are called pages .

<example>Want to create a three dimensional array whose first page is

and whose second page is

>>A = [4, 6, 1;5, 8, 0;3, 9, 2];>>A( : , : , 2) = [6, 2, 9;0, 3, 1;4, 7, 5];

293

085

164

574

130

926





Multidimensional Arrays cat command

59

28A

37

64 B C = cat(3, A, B) produces a three-dimensional

array

<example>

>> A = [1 2;3 4];>> B = [5 6;7 8];>> C = cat(3, A, B) % concatenating the arrays A, B along the dimension three.C(:,:,1) =

1 23 4

C(:,:,2) =5 67 8

>> d = size(c)d =

2 2 2





40

Element-by-Element OperationsElement-by-element operations

Symbol Operation Form Example

+ Scalar-array addition A + b [6, 3] + 2 = [8 ,5]

- Scalar-array subtraction A – b [8, 3] - 5 = [3, -2]

+ Array addition A + B [6, 5] + [4, 8] = [10, 13]

- Array subtraction A – B [6, 5] - [4, 8] = [2, -3]

.* Array multiplication A.*B [3, 5] .* [4, 8] = [12, 40]

./ Array right division A./B [2, 5] ./ [4, 8] = [2/4, 5/8]

.\ Array left division A.\B [2, 5] .\ [4, 8] = [2\4, 5\8]

.^ Array exponentiation A.^B [3, 5] .^2 = [3^2, 5^2]

2 .^ [3, 5] = [2^3, 2^5]

[3, 5] .^ [2, 4] = [3^2, 5^4]




: p . , , y

Element-by-Element Operations<example1> <example2>

>> x = [ 1 2 3];>> y = [ 4 5 6];>> x.^yans=

1 32 729

>>y.^2ans=

16 25 36

>>2.^[x y]

ans=2 4 8 16 32 64

cf) ==> 2^[1 2 3 4 5 6] = [ 2^1 2^2 2^3 2^4 2^5 2^6 ]

>>x = 0:0.01:5>>y = sin(x^2);??? Error using ==> ^Matrix must be square.

>>y=sin(x.^2);

>>plot(x,y);




p , , y

42

Matrix OperationsMatrix Multiplication

Use the operator * to perform matrix multiplication in MATLAB<example>

>>A = [6, -2,;10,3;4,7];>>B = [9,8;-5,12];>>A*B

ans =64 2475 1161 116

Special Matrices

Command Description

eye(n) Creates an n × n identity matrix.

eye(size(A)) Creates an identity matrix the same size as the matrix A.

ones(n) Creates an n × n matrix of ones.ones(m,n) Creates an m × n array of ones.

ones(size(A)) Creates an array of ones the same size as the array A.

zeros(n) Creates an n × n matrix of zeros.

zeros(m,n) Creates an m × n array of zeros.

zeros(size(A)) Creates an array of zeros the same size as the array A.




p y

43

Matrix OperationsMatrix Division

Right and left operators, / and \Chapter 6 covers matrix division and matrix inverse.

Matrix Exponentiationmust be a square matrix

to find A 2, type A^2

Special products

Command Syntax

cross(A, B) Computes a 3 × n array whose columns are the crossproducts of the corresponding columns in the 3 × narrays A and B. Returns a three-element cross-product vector if A and B are three-element vectors.

dot(A, B) Computes a row vector of length n whose elementsare the dot products of the corresponding columnsof the m × n arrays A and B




44

Polynomial Operations Using Arrays

Type help polyfun for more information

We will use following notation

Polynomial Addition and Subtrctionadd the arrays that describe their coefficientsif the polynomials are of different degrees, add zeros to the coefficient array of theleower-degree polynomial.

<example>

coefficient array is f=[9, -5, 3, 7] and g = [6, -1, 2]g = [0 g] = [0, 6, -1, 2]h = f+g = [9, 1, 2, 9]

12

12

31

21)( nnnnnn a xa xa xa xa xa x f

7359)(23

x x x x f 26)(2

x x x g )()()( x g x f xh

929)( 23 x x x xh




45

Polynomial Operations Using ArraysPolynomial Multiplication and Division

Multiply polynomials : use conv function (it stands for “convolve”)

synthetic division : use deconv function (it stands for “deconvolve”)

Polynomial functions

Command Description

conv(a, b) Computes the product of the two polynomials described by the coefficient arrays a and b.

[q, r] = deconv(num,

den)

Computes the result of dividing a numerator polynomial, whose coefficient array is num, by a

denominator polynomial represented by the coefficient array den. The quotient polynomial isgiven by the coefficient array q, and the remainder polynomial is given by the coefficient arrayr.

poly(r) Computes the coefficients of the polynimial whose roots are specfied by the vector r.

polyval(a, x) Evaluates a polynomial at specified values of its independent variable x, which can be a matrixor a vector. The polynomial‟s coefficients of descending powers are stored in the array a. Theresult is the same size as x.

roots(a) Computes the roots of a polynomial specified by the coefficient array a. The result is a columnvector that contains the polynomial‟s roots.

1429413954

)26)(7359()()(2345

223

x x x x x

x x x x x x g x f

5833.05.126

7359)()(

2

23

x x x

x x x x g x f

>>f = [9, -5, 3, 7];

>>g = [6, -1, 2];

>>product = conv(f, g)

product =

54 -39 41 29 -1 14

>>quotient, remainder] = deconv(f,g)

quotient =

1.5 -0.5833

remainder =

0 0 -0.5833 8.1667

Acsl , Pos tech ATLAB : Chapter 2. Numeric, Cell, and Structure Arrays



46

Polynomial Operations Using Arrays

Plotting PolynomialsThe polyval(a,x) function is very useful for poltting polynomials.

<example>plot the polynomial

7359)( 23 x x x x f 52 x for

-2 -1 0 1 2 3 4 5-200

0

200

400

600

800

1000

1200

x

f ( x

)




47

Cell Arrays

Cell array : an array in which each element is a bin, or cell which can containan array.

• examplesB = [2, 4], [6,-9;3, 5]; [7;2], 10 ;

H = [2, 4, 8], [6, -8, 3]; [2:6], [9, 2, 5]; [1, 4, 5], [7, 5, 2] ;J = [ H1, 1; H1, 2; H(2, 2 ];

10

53

96 42

27

529

386

842

J




48

Cell Arrays

Cell array functions

Function Description

C = cell(n) Creates an n × n cell array C of empty matrices.

C = cell(n,m) Creates an n × m cell array C of empty matrices.celldisp(C) Display the contents of cell array C.

cellplot(C) Displays a graphical representation of the cell array C.

c = num2cell(A) Converts a numeric array A into a cell array C.

[X, Y, …] = deal(A,B, …) Matches up the input and output lists. Equivalent to X = A, Y = B, …

[X, Y, …] = deal(A) Matches up the input and output lists. Equivalent to X = A, Y = A, …

iscell(C) Returns a 1 if C is a cell array; otherwise, returns a 0.




49

Cell Arrays

Cell Array Function Examples




50

Structure Arrays

Structure ar rays : composed of s t ructures .enables you to store dissimilar arrays together accessed using namef ie lds

Creat ing St ruc tures using ass ign ment s ta tements

u s ing the s t ruc t func t ion

dot n ota t ion ( .) : to sp ec i fy and to access the f ie lds

<example>

student.name = ‘ John Smith ’ ; student.SSN = ‘ 392-77-1786 ’ ; student.email = ‘ [email protected] ’ ; student.tests = [67, 75, 84];

student(2).name = ‘ Mary Jones ’ ; student(2).SSN = ‘ 431-56-9832 ’ ; student(2).email = ‘ [email protected] ’ ; student(2).tests = [84, 78, 93];

Structure array“

student”

Student(1)

Name: John Smith

SSN: 392-77-1786

Email: [email protected]

Tests: 67, 75, 84

Student(2)

Name: Mary Jones

SSN: 431-56-9832

Email: [email protected]

Tests: 84, 78, 93




51

Structure Arrays

Structure functions

Function Description

names = fieldnames(S) Returns the field names associated with the structure array S asnames , a cell array of strings

F = getfield(S, „field‟) Returns the contents of the field „ field ‟ in the structure array S .Equivalent to F = S.field

isfield(S, „field‟) Returns 1 if „field‟ is the name of a field in the structure array S , and0 otherwise

isstruct(S) Returns 1 if the array S is a structure array, and 0 otherwise

S = rmfield(S, „field‟) Removes the field „ field ‟ from the structure array S .

S = setfield(S, „field‟, V) Sets the contents of the field „ field ‟ to the value V in the structurearray S

S = struct(„f1‟, „v1‟, „f2‟, „v2‟, …) Creates a structure array with the fields „f1‟, „f2, … having thevalues „v1‟, „v2‟, ….




52

Structure ArraysStructure Arrays Examples




MATLAB

CHAPTER 3Functions and Files




Elementary Mathematical Functionsl ookfo r : find functions that are relevant to your

application

help : when you know the correct spelling of the function




Exponential and Logarithmic Functions

Some commonmathematicalfunctions

Exponential

exp(x) Exponential;

sqrt(x) Square root;

Logarithmiclog(x) Natural logarithm; ln x

log10(x) Common(base10)logarithm;log(x) means ln x.

xe x

Complex

abs(x) Absolute value; xangle(x) Angle of a complex number x

conj(x) Complex conjugate

imag(x) Imaginary part of a complexnumber x

real(x) Real part of a complex number xNumeric

ceil(x) Round to the nearest integer toward

fix(x) Round to the nearest integer toward zero.

floor(x) Round to the nearest integer toward

round(x) Round toward nearest integer

sign(x) Signum function:+1 if x> 0; if x = 0; -1 if x < 0




Complex Number Functions(1/2)

rectangular representationa + ib

polar representation

abs (x ), ang le(x ) absolute value (magnitude)angle

conj(x) complex conjugate

θ




Complex Number Functions(2/2)




Numeric FunctionsMATLAB has been optimized to deal with arrays.

round(y) : rounds to the nearest integer ans = 2, 3, 4f ix(y) : truncates to the nearest integer toward zero ans = 2, 2, 3ceil(y) : rounds to the nearest integer toward ∞ ans = 3, 3, 4e.g. z = [-2.6 , -2.3 , 5.7]floor(z) : rounds to the nearest integer toward - ∞ ans = -3, -3, 5fix(z) = -2, -2, 5abs(z) = 2.6 , 2.3, 5.7




Trigonometric Functions

Table 3.1-2Trigonometric functions

Trigonometric*angle : radian

cos(x) Cosine; cos x.cot(x) Cotangent; cot x.

csc(x) Cosecant; csc x.

sec(x) Secant; sec x.

sin(x) Sine; sin x.

tan(x) Tangent; tan x.

Inversetrigonometric

acos(x) Inverse cosine;arccos x =

acot(x) Inverse cotangent;arccot x =

acsc(x) Inverse cosecant;arccsc x =

asec(x) Inverse secant;arcsec x =

asin(x) Inverse sine;arcsin x =

atan(x) Inverse tangent;arctan x =

atan2(y,x) four-quadrantinverse tangent.

xos 1

xo t 1

xsc 1

xec 1

xin 1

xan 1




Hyperbolic Functions(1/2)

Hyperbolicfunctions

Hyperbolic

cosh(x) Hyperbolic cosine; cosh x =coth(x) Hyperbolic cotangent; cosh x / sinh x

csch(x) Hyperbolic cosecant; 1 / sinh x

sech(x) Hyperbolic secant; 1 / cosh x

sinh(x) Hyperbolic sine; sinh x =

tanh(x) Hyperbolic tangent; sinh x / cosh x

)/2ex

)/2e x




Hyperbolic Functions(2/2)

Hyperbolicfunctions

Inversehyperbolic

acosh (x) Inverse hyperbolic cosine;acoth (x) Inverse hyperbolic cotangent;

acsch (x) Inverse hyperbolic cosecant;

asech (x) Inverse hyperbolic secant;asinh (x) Inverse hyperbolic sine;

atanh (x) Inverse hyperbolic tangent;

1,n(xosh 2

121

x x,1

1ln(o th or

0x,x1

x1

ln(s c h 21

1 ,x

1

x

1ln(ec h

2

1

x ,n(xinh 2

1 ,x

xln(

2

1xanh 1




User-Defined Functions(1/4)Funct ion f i l e : when need to repeat a set of commands several times.

variables : localsyntax

function [output variables] = function_name (input variables)function_name = saved file name ( with the .m extension)

function_name = drop : file name = drop.m

e.g.




User-Defined Functions(2/4)

e.g.




User-Defined Functions(3/4)

the order of arguments is important, no t the names of thearguments

use arrays as input arguments




User-Defined Functions(4/4)more than one output




Variations in the Function Line

Funct io n def ini t ion l ine Fi le name

1. function [area_square] = square(side); square.m

2. function area_square = square(side); square.m

3. function [volume_box] = box(height,width,length); box.m

4. function [area_circle, circumf] = circle(radius); circle.m

5. function sqplot(side); sqplot.m




Local Variables, Global Variables

Local variablesvariables created by a function file are local to that function.local : their values are not available outside the function.

Global variablestheir values are available to the basic workspace and to other functions that declare these variables global.

variables in script file : global




Minimization and root-finding functionsTable 3.2-1 Minimization and root-finding functions

Function Descriptionfminbnd(„function‟,x1

,x2)

Returns a value of x in the interval x1 ≤x ≤x2

that corresponds to a minimum of the single-variable function described by the string‘function’.

fminsearch(„function‟,x0)

Uses the starting vector x0 to find a minimumof the multivariable function described by the

string ‘function’ .fzero(„function‟,x0) Uses the starting value x0 to find a zero of

the single-variable function described by thestring ‘function’ .




Advanced Function Programming

Function Handlesusing the at sign : @

e.g.

>> sine_handle = @ sin;

>> plot([0:0.01:6], sine_handle(0: 0.01: 6))function x=gen_plot(fun_handle, interval)plot(interval, fun_handle, interval)

>>gen_plot(sine_handle, [0: 0.01: 6]) or >>gen_plot(@sin, [0: 0.01: 6])

advantagesspeed of execution and providing access to subfunctions.a standard MATLAB data type, and thus can be used in thesame manner as other data types.




Methods for Calling Functionsfour ways to invoke, or “call,” a function into action.

1. As a character string identifying the appropriate function M-file.function y = fun1(x)y = x.^2-4;

>>[x, value] = fzero(„fun1‟,[0,3]) 2. As a function handle.

>>[x, value] = fzero(@fun1,[0,3])3. As an “inline” function object, or

>>fun1 = „x.^2 -4‟; >>fun_inline = inline(fun1);>>[x, value] = fzero(fun_inline, [0,3])

4. As a string expression.>>fun1 = „x.^2 -4‟; >>[x, value] = fzero(fun1, [0,3]) or as>>[x, value] = fzero(„x.^2 -4‟, [0,3])




Types of Functions(1/2)

primary functionscontains the main program.the only function that you can call from the MATLAB command line or fromanother M-file function

anonymous functionscreate a simple function without needing to create an M-file for it.

provide a quick way of making a function from any MATLAB expression

subfunctionsplaced in the primary functionuse multiple functions within a single primary function M-file




Types of Functions(2/2)

nested functionsdefined within another functionhelp to improve the readability of your program

difference between nested functions and subfuntions: subfunctions normallycannot be accessed outside of their primary function fle

overloaded functionsfunctions that respond differently to different types of input arguments.created to treat integer inputs differently than inputs of class double.

private functionsrestrict access to a functioncalled only from an M-file function in the parent directory




Anonymous Functionscreate a simple function without needing to create an M-filefor it.

MATLAB command linefrom within another function or script

syntaxfhan dle = @(arglis t) expr

arglist: a comma-separated list of input arguments to be passed to thefunctionexpr: any single, valid MATLAB expressione.g.

sq = @(x) x.^2;

>>sq(5)>>sq([5,7])

be useful for more complicated functions involvingnumerous keystrokes.




Variables and Anonymous Functions

variables can appear in anonymousfunctions in two ways

as variables specified in the argument liste.g.

f = @(x) x.^3;

as variables specified in the body of the expressione.g.

plane = @(x, y) A*x + B *y;




Subfunctions(1/2)

all other functions in the pr imary function are called sub func t i ons .

the order for checking functions inMATLAB

1. checks to see if the function is a built-in function such s sin.2. checks to see if the function is subfunction in the file.3. checks to see if the function is private function.

may use subfunctions with the samename as another existing M-file.

allow you to name subfunctions without being concerned about whether another function exists with the same nameprotects you from using another function unintentionally




Subfunctions(2/2)

e.g.function y = subfun_demo(a)y = a – mean(a);

%function w = mean(x)w = sqrt(sum(x.^2))/length(x);a sample session follows.>> y =subfun_demo([4,-4])y =

1.1716 -6.8284if had used the MATLAB M-function mean different answer

>> a= [4, -4];>> b =a – mean(a)b =

4 -4

primary function

subfunction




Nested Functions(1/2)

Functions that are defined within the main function.contains the usual components of an M-file function.must always terminate with an en d statement.

e.g.function f = parabola(a, b, c)f =@p;

function y = p(x)y = a*x^2 + b*x + c;

endendIn the command window type>>f = parabola(4, -50, 5);>>fminbnd(f, -10, 10)ans =

6.2500

6.2500




Nested Functions(2/2)

two unique properties1. can access the workspace of all functions inside of which it is nested.2. function handle

stores the information needed to access the nested functionstores the values of all variables shared between the nested function andthose functions that contain it

call a nested function1. from the level immediately above it2. from a function nested at the same level within the same parent function3. from a function at any lower level




Private Functions

reside in subdirectories with the

special name pr ivate .visible only to functions in the parentdirectory.

invisible outside the parent directory.MATLAB looks for private functionsbefore standard M-file functions.


W ki g ith D t Fil



Working with Data Files

header : a comment that describe whatthe data represent, the date it wascreated, and who created the date.Im po r t ing da ta : bring data created byother applications into the MATLABworkspace.Expor t ing da ta : package workspace

variables so that they can be used byother applications.Im po r t ing Wizard : a graphical user

interface




Using Import Functions with Text Data

• To import text data from the command line or in an M-file, choice of functiondepends on how the data in the text file is formatted.

• The text data must be formatted in a uniform pattern of rows and columns, using atext character, called a delimiter or column separator, to separate each data item.

•The delimiter can be a space, comma, semicolon, tab, or any other character. Theindividual data items can be alphabetic or numeric characters or a mix of both.

• The text file can also contain one or more lines of text, called header lines, or canuse text headers to label each column or row.

•To find out how the data is formatted, view it in a text editor. After determine theformat, find the sample in the table that most closely resembles the format of data.

•Then read the topic referred to in the table for information on how to import thatformat.


T bl 6 1 ASCII D Fil F



Data Format Sample File Extension

Description

1 2 3 4 56 7 8 9 10

.txt.dat or other

See Importing Numeric Text Data or Using the Import Wizard with Text Data for information.

1; 2; 3; 4; 56; 7; 8; 9; 10or

1, 2, 3, 4, 56, 7, 8, 9, 10

.txt

.dat

.csv

or other

See Importing Delimited ASCII Data Files or Using the Import Wizard with Text Data for information.

Ann Type1 12.34 45 YesJoe Type2 45.67 67 No

.txt

.dator other

See Importing Numeric Data with Text Headersfor information.

Grade1 Grade2 Grade391.5 89.2 77.388.0 67.8 91.067.3 78.1 92.5.

txt.dator other

• See Importing Numeric Data with Text Headersor Using the Import Wizard with Text Data for information.

If you are familiar with MATLAB import functions but are not sure when to use them,

see the following table, which compares the features of each function.

Table 6-1: ASCII Data File Formats


T bl 6 2 ASCII D I F i F



Function Data Type Delimiters

Number of Return Val

ues

Notes

Csvread Numeric data Commasonly

One Primarily used with spreadsheet data.See Working with Spreadsheets.

Dlmread Numeric data Anycharacter

One Flexible and easy to use.

Fscanf Alphabetic and num

eric; however, both types returned in a single return variable

Any

character

One Part of low-level file I/O routines. Requi

res use of fopen to obtain file identifier and fclose after read.

load Numeric data Spaces only

One Easy to use. Use the functional form of load to specify the name of the output variable.

textread Alphabetic and numeric

Anycharacter

Multiple values in cell arrays

Flexible, powerful, and easy to use. Use format string to specify conversions

textscan Alphabetic and numeric

Any character

Multiple values returned to one cellarray

More flexible than textread. Also more f ormat options

Table 6-2: ASCII Data Import Function Features




MATLAB

CHAPTER 4Programming with MATLAB




OUTLINEProgram Design and DevelopmentRelational Operators and Logical VariablesLogical Operators and Functions

Conditional StatementsLoopsThe „switch‟ structure

Input and OutputDebugging MATLAB ProgramsSummary


Program design and development(1/3)



Program design and development(1/3)Algo r i thm s and co n t ro l s t ruc tu res

Sequential operationsConditional operationsInterative operations (loops)

Struc tu red p rog ram m ing Table 4.1-1 Steps for developing a computer solution1. State the problem concisely

2. Specify the data to be used by the program. This is the “input.” 3. Specify the information to be generated by the program, this is the “output.”

4. Work through the solution steps by hand or with a calculator; use a simpler set of data if necessary

5. Write and run the program

6. Check the output of the program with your hand solution

7. Run the program with your input data and perform a reality check on theoutput

8. If you will use the program as a general tool in the future, test is by runningit for a range of reasonable data values; perform a reality check on theresults






St ruc tu re char t & F low ch art

P s e d o c o d e Examp le 1 . sequent ia l op erat ions

p=a+b+c

a= inp ut ( ˙ Enter the value of s id e a: );

b= inpu t ( ˙ Enter the value of s ide b : );

c= inpu t ( ˙ Enter the value of s ide c : );

p=a+b+c;

s=p/2; A=sqrt(s*(s-a)*(s-b)*(s-c));

disp(˙ The perimeter is: ˙ )

p

disp(˙ The area is: ˙ )

A

))()(( c sb sa s s A2 p

s

Example 2 . co ndi t ion al op erat ions

x=input( ˙ Enter the va lue of s ide x : ); y=input ( ˙ Enter the va lue of s ide y : ); r=sqr t (x^2+y^2) ; If x >=0

theta=atan(y/x); else

theta=atan(y/x)+pi; en d

d i sp (˙ The hypoteneuse is: ˙ ) disp(r ) theta=theta*(180/pi); d i sp (˙ The angle is degrees is: ˙ ) disp(theta)

22 y xr )(tan 1

x y


P d i d d l (3/3)




Examp le 3 . i te rat ive o pera t ions

Determine how many terms arerequired for the sum of the series10k 2 -4k+2 , k=1,2,3,… to exceed 20,000

Tot al = 0; K = 0; Wh ile total < 2e+4

k = k +1;

Total = 10*k^ 2 – 4*k +2 + to tal; en d disp ( ‘ The number of terms is : ‘ ) d i sp ( k ) disp ( ‘ The sum is : ‘ ) d isp ( to ta l )

Total > 20,000 ?

Total = 0k = 0

Total =10k 2 -4k+2

k = k + 1

START

END

DISP k, Total

True

False




Relational operators and logical variables(1/2)

Table 4.2-1 Relat ion al o perators

Relationaloperator

Meaning

< Less than

<= Less than or equalto

> Greater than

>= Greater than or

equal to== Equal to

~= Not equal to

Example


R l i l d l i l i bl (2/2)



Relational operators and logical variables(2/2)

The logical class

example

The logica l funct ion

example




Logical operators and functions(1/6)

Tab le 4.3-1 log ical o p erators Operator Name Definition

~ NOT ~A returns an array the same dimension as A; the newarray has ones where A is zero and zeros where A isnonzero.

& AND A&B returns an array the same dimension as A and B;the new array has ones where both A and B havenonzero elements and zeros where either A or B is zero.

| OR A|B returns an array the same dimension as A and B; thenew array has ones where at least one element in A or Bis nonzero and zeros where A and B are both zero.

&& Short-CircuitAND

Operator for scalar logical expressions. A && B returnstrue if both A and B evaluate to true, and false if they donot.

|| Short-CircuitOR

Operator for scalar logical expressions. A || B returnstrue if either A or B or both evaluate to true, and false if they do not.





Example 1. Not

2. An d

3. Or





Table 4.3-2 order of preced ence for op erator typ es

Precedence Operator type

First Parentheses „ () „; evaluated starting with theinnermost pair.

Second Arithmetic operators and logical NOT(~); evaluatedfrom left to right.

Third Relational operators ; evaluated from left to right

Fourth Logical AND.

Fifth Logical OR.






x y ~x x|y x&y xor(x,y)

True True False True True False

True False False True False True

False True True True False True

False False True False False False

Tabl e 4.3-3 Tru th tab le

example

Shor t -c i rcu i t op erators A && B Returns true (logical 1) if both A and B evaluate to true, and false (logical 0) if they donot.

A | | B Returns true (logical 1) if either A or B, or both, evaluate to true, and false (logical 0) if they do not.





Table 4.3-4 logic al func tion s Logical function Definition

all(x) Returns a scalar, which is 1 if all the elements in the vector x are nonzero and 0 otherwise.

all(A) Returns a row vector having the same number of columns as the matrix A and containing ones and zeros,depending on whether or not the corresponding column of A has all nonzero elements.

any(x) Returns a scalar, which is 1 if any of the elements in the vector x is nonzero and 0 otherwise.

any(A) Returns a row vector having the same number of columns as A and containing ones and zeros, depending onwhether or not the corresponding column of the matrix A contains any nonzero elements.

find(A) Computes an array containing the indices of the nonzero elements of the array A.

[u,v,w]=find(A)Computes the arrays u and v containing the row and column indices of the nonzero elements of the array Aand computes the array w containing the values of the nonzero elements. The array w may be omitted.

finite(A) Returns an array of the same dimension as A with ones where the elements of A are finite and zeroselsewhere.

ischar(A) Returns a 1 if A is a character array and 0 otherwise.

isempty(A) Returns a 1 if A is an empty matrix and 0 otherwise.

isinf(A) Returns an array of the same dimension as A, with ones where A has „inf‟ and zeros elsewhere.

isnan(A) Returns an array of the same dimension as A with ones where A has „NaN‟ and zeros elsewhere. („NaN‟stands for “ not a munber,” which means an undefined result.)

isnumeric(A) Returns a 1 if A is a numeric array and 0 otherwise.

isreal(A) Returns a 1 if A has no elements with imaginary parts and 0 otherwise.

logical(A) Converts the elements of the array A into logical values.

xor(A,B) Returns an array the same dimension as A and B; the new array has ones where either A or B is nonzero, but

not both, and zeros where A and B are either both nonzero or both zero.





Logical operators and functions(6/6)Logical operators and the f ind function

Find(x): computes an array containing the indices of the nonzero elements of the array x.example


Conditional statements(1/5)




the i f statement

The i f statement‟s basic form is

i f logical expressionstatements

en d

the nested i f statement

i f logical expression 1statements group 1i f logical expression 2

statement group 2en d

en d

Start

LogicalExpression 1

StatementGroup 1

True

False

FalseLogicalExpression 2

StatementGroup 2

Statement

Group 3

True

End

Figure 4.4-3 flowchart for

the general if structure





Conditional statements(2/5)Example






the else statement

i f logical expressionstatements group 1

else statements group 2

end

example

Start

StatementGroup2

StatementGroup1

LogicalExpression

False

End

True

Figure 4.4-2 flowchart forthe general else structure






the elseif statement

if logical expressionstatements group 1

elseif logicalexpression2

statements group 2else

statements group 3en d

Checking the number of input andoutput arguments

nargin -number of input arguments

Function z=sqrtfun(x,y)if (nargin==1)

z=sqrt(x);elseif (nargin==2)

z=sqrt((x+y)/2);

end





Strings and conditional statementsNameNumber findstr Input


Loops (1/4)



Loops (1/4)fo r Loops

fo r loop variable=m:s:nstatements

en d

Example

for k=5:10:35x=k^2

end

Figure 4.5-1 flowchartof a for loop.

Start

Set k = m

Incrementk by s

k > n ?

False

End

True

Statements

Statementsfollowing the

End statement


Loops (2/4)



Loops (2/4)

break – terminates the loopbut does not stop the entire program

example

for k=1:10x=50-k^2;if x<0

breakendy=sqrt(x)

end

Cont inue- passes control to thenext iteration of the for or whileloop in which it appears

example

x=[10,1000,-10,100];y=NaN*x;

for k=1:length(x)if x(k)<0

continueendy(k)=log10(x(k));

endy

The result is y=1,3,NaN,2.


Loops (3/4)



Loops (3/4)Using an array as a loop index

example A=[1,2,3;4,5,6];for v=A

disp(v)end

Imp l ied L oops

Use of logical ar rays as m asks

example


Loops (4/4)



Loops (4/4)whi l e Loops

whi l e loop variable=m:s:nstatements

En d

Example

x=5;while x<25

disp(x)x=2*x-1;

end

1. The loop variable must have a valuebefore the while statement is executed.

2. The loop variable must be changedsomehow by the statements.

Figure 4.5-3 flowchartof the while loop.

Statementsfollowing the

End statement

End

Start

LogicalExpression

True

False

Statements(which incrementthe loop variable)




The swi tch structure

sw i tch inp u t express ion (sca lar o r s t r ing)

case value 1statement group 1

case value 2statement group 2

.

.

.

Otherwisestatement group n

end

example

switch anglecase 45

disp(„Northeast‟)

case 135disp(„Southeast‟)

case 225disp(„Southwest‟)

case 315disp(„Northwest‟)

otherwisedisp(„Direction Unknown‟)

end



107

MATLAB

CHAPTER 5Advanced Plotting and Model Building

MATLAB : Chapter 5. Advanced Plotting and Model Building



108

xy Plotting Functions

0 1 2 3 4 5 6 7 8 9 10 0

200

400

600

800

1000

1200

1400

1600

Time (seconds)

H e

i g h t ( f e e

t )

Height of a Falling Object Versus Time

Zero Drag Model

Data

LEGEND

TICK MARK

TICK-MARK LABEL

AXIS LABEL

DATA SYMBOL

PLOT TITLE

The Anatomy of a Plot

SPaC, Postech ATLAB : Chapter 5. Advanced Plotting and Model Building



109


Requirements for a Correct Plot1. Each axis must be labeled with the name of the quantity being plotted and its

units.2. Each axis should have regularly spaced tick marks at convenient intervals with a

spacing that is easy to interpret and interpolate.

3. If you are plotting more than one curve or data set, label each on its plot or use alegend to distinguish them.

4. If you are preparing multiple plots of similar type or if the axes‟ labels cannotconvey enough information, use a title.

5. If you are plotting measured data, plot each data point with a symbol.6. Sometimes data symbols are connected by lines to help the viewer visualize the

data. You should be careful to prevent the misinterpretation when connecting thedata points, especially with a solid line.

7. If you are plotting points generated by evaluating a function, do not use a symbolto plot the points. Instead, be sure to generate many points, and connect thepoints with solid lines.




110


Plot, Label, and Title Commandsxlabel and ylabel commands put labelson the abscissa and the ordinate.title command puts a title at the top of theplot.

<example>

>>x = [0:0.1:52] ;>>y = 0.4*sqrt(1.8*x) ;>>plot (x,y)>>xlabel ( „ Distance (miles)‟ ) >>ylabel ( „ Height (miles)‟ ) >>title ( „ Rocket Height as a Function of

Downrange Distance‟ )

0 10 20 30 40 50 60 0

0.5

1

1.5

2

2.5

3

3.5

4

Distance (miles)

H e

i g h t ( m i l e s

)

Rocket Height as a Function of Downrange Distance




111


grid and axis Commandsgrid : displays gridlines at the tick marks corresponding to the tick labels.axis : override the MATLAB selections for the axis limits.

Basic syntax: axis ( [xmin xmax ymin ymax] )




112


Plots of Complex Numbersplot (y) : Plots the imaginary parts of y versus the real parts if y is a vector having

complex values.





113

y gThe Function Plot Command fplot

fplot : automatically analyzes the function to be plotted and decideshow many plotting points to use so that the plot will show all thefeatures of the function.Basic syntax: fplot ( „string‟ , [xmin xmax] )

where „string‟ is a text string that describes the function to be plottedand [xmin xmax] specifies the minimum and maximum values of the independent variable.





114


Plotting PolynomialsWe can plot polynomials more easily by using the polyval function

<example>Plot the polynomial

>>x = [-6:0.01:6];>>p = [3,2,-100,2,-7,90];

>>plot (x, polyval (p,x)), xlabel („x‟),ylabel („p‟)

)( x f

-6 -4 -2 0 2 4 6-3000

-2000

-1000

0

1000

2000

3000

4000

5000

907210023)( 2345 x x x x x x f )66( x


Subplots and Overlay Plots



115

Subplots and Overlay PlotsBasic xy plotting commands

Command Description

axis([xmin, xmax yminymax])fplot(„string‟, [xmin xmax])

Gridplot(x,y)plot(y)

print title („text‟) xlabel („text‟) ylabel („text‟)

Sets the minimum and maximum limits of the x- and y-axes.Performs intelligent plotting of functions, where „string‟ is atext string that described the function to be plotted and[xmin xmax] specifies the minimum and maximum valuesof the independent variable. The range of the dependentvariable can also be specified. In this case the syntax isfplot(„string‟, [xmin xmax ymin ymax]).Display gridlines at the tick marks corresponding to thetick labels.Generates a plot of the array y versus the array x onrectilinear axes.Plots the values of y versus their indices if y is a vector.Plots the imaginary parts of y versus the real parts if y is avector having complex values.Prints the plot in the Figure window.Puts text in a title at the top of a plot.Adds a text label to the x-axis(the abscissa).Adds a text label to the y-axis(the ordinate).





116

p ySubplots

Basic syntax: subplot (m, n, p)<example1> subplot (3, 2, 5) : This command creates an array of six pane, three panes deep and twopanes across, and directs the next plot to appear in the fifth pane (in the bottom-left corner).

<example2> X = [0:0.01:5];

y = exp(-1.2*x).*sin(10*x+5);

subplot(1,2,1)

plot(x,y),xlabel('x'),ylabel('y'),…

axis([0 5 -1 1])

x = [-6:0.01:6];y = abs(x.^3-100);

subplot(1,2,2)

plot(x,y),xlabel('x'),ylabel('y'),…

axis([-6 6 0 350])





117

p yOverlay plots

Need to plot more than one curve or data set on a single plot.

Data Markers and Line Types

Data markers Line types Colors

Dot (.) .Asterisk (*) *Cross (x) xCircle (o) oPlus sign (+) +Square ( ) s

Diamond ( ◊ ) dFive-pointed star ( ) p

Solid line -Dashed line - -Dash-dotted line -.Dotted line :

Black kBlue bCyan cGreen gMagenta mRed r

White w Yellow y





118

p yData Markers and Line Types

plot(x,y,'*',x,y,':‘)

plot(x,y,x,y,'o‘)

plot(x,y,'o‘)





119

Labeling Curves and Datalegend command: legend („string1‟, „string2‟) where string1 and string2 are text strings of your choice.gtext command: gtext („string‟) add a text string to the plot.text command: text (x, y, „string‟)





120

The hold Commandhold command create a plot that needs two or more plotcommands.be useful with some of the advanced MATLAB toolboxcommands that generate specialized plots.

<example>

-1 0 1 2 3 4 5 6-1

0

1

2

3

4

5

6

7

y2 versus y1

Imag(z) versus Real(z)




121

Subplots and Overlay PlotsPlot enhancement commands

Command Description

axesgtext („text‟)

holdlegend („leg1‟, ‟leg2‟, …)

plot (x, y, u, v)plot (x, y, „type‟)

plot (A)

plot (P, Q)

refreshsubplot (m, n, p)

text (x, y, „text‟)

Creates axes objects.Places the string text in the Figure window at a point specified by the mouse.

Freezes the current plot for subsequent graphics commands.Creates a legend using the strings leg1, leg2, and so on and specifies itsplacement with the mouse.Plots, on rectilinear axes, four arrays: y versus x and v versus u.Plots the array y versus the array x on rectilinear axes, using the line type,data marker, and colors specified in the string type.Plots the columns of the m × n array A versus their indices and generates ncurves.Plots array Q versus array P. See the text for a description of the possiblevariants involving vectors and/or matrices: plot (x, A), plot (A, x), and plot (A,

B).Redraws the current Figure window.Splits the Figure window into an array of subwindows with m rows and ncolumns and directs the subsequent plotting commands to the p-thsubwindow.Places the string text in the Figure window at a point specified bycoordinates x, y.





122

p yAnnotating Plots

\ tau & \ omega represent the Greek letters τ and ω .

^ : superscript, _ : subscript.<example>X_13 x 13

Hints for Improving Plots

1. Start scales from zero whenever possible.2. Use sensible tick-mark spacing.3. Minimize the number of zeros in the data being plotted.4. Determine the minimum and maximum data values for each axis before

plotting the data.5. Use a different line type for each curve when several are plotted on a single

plot and they cross each other 6. Do not put many curves on one plot, particularly if they will be close to each

other or cross one another at several points.7. Use the same scale limits and tick spacing on each plot if you need to

compare information on more than one plot.


Special Plot Types



123

Logarithmic plotslog-log plot: log scales on both axes.

semilog plot: a log scale on only one axis.


Special Plot Types



124

Specialized plot commands

Command Description

bar (x, y)

loglog (x, y)

plotyy (x1, y1, x2, y2)

polar (theta, r, „type‟)

semilogx (x, y)

semilogy (x, y)

stairs (x, y)

stem (x, y)

Creates a bar chart of y versus x.

Produces a log-log plot of y versus x.

Produces a plot with two y-axes, y1 on the left and y2 onthe right.

Produces a polar plot from the polar coordinates thetaand r, using the line type, data marker, and colorsspecified in the string type.

Produces a semilog plot of y versus x with logarithmicabscissa scale.

Produces a semilog plot of x versus y with logarithmicabscissa scale.

Produces a stairs plot of y versus x.

Produces a stem plot of y versus x.


Special Plot Types



125

Frequency-Response Plots and Filter Circuits

<example> Frequency-Response Plot of a Low-Pass Filter

, where s = ω i and RC = 0.1 second1

1 RCs A

A

i

o

R

C

+

-vi vo

100

101

102

10-1

100

Frequency (rad/s)

O u

t p u

t / I n p u

t R a

t i o

Frequency Response of a Low-Pass RC Circuit (RC = 0.1 s)


Th Di i l Pl



126

Three-Dimensional PlotsExample

[X, Y] = meshgrid(-2:0.1:2);Z = X.*exp(-((X-Y.^2).^2+Y.^2));

mesh(X,Y,Z) meshc(X,Y,Z) surf(X,Y,Z)

surfc(X,Y,Z) meshz(X,Y,Z) waterfall(X,Y,Z)

http://c/Documents%20and%20Settings/123/Local%20Settings/Temp/Rar$DI00.750/mesh(X,Y,Z)

http://c/Documents%20and%20Settings/123/Local%20Settings/Temp/Rar$DI00.750/meshc(X,Y,Z)

http://c/Documents%20and%20Settings/123/Local%20Settings/Temp/Rar$DI00.750/surf(X,Y,Z)

http://c/Documents%20and%20Settings/123/Local%20Settings/Temp/Rar$DI00.750/surfc(X,Y,Z)

http://c/Documents%20and%20Settings/123/Local%20Settings/Temp/Rar$DI00.750/meshz(X,Y,Z)

http://c/Documents%20and%20Settings/123/Local%20Settings/Temp/Rar$DI00.750/waterfall(X,Y,Z)

http://c/Documents%20and%20Settings/123/Local%20Settings/Temp/Rar$DI00.750/waterfall(X,Y,Z)

http://c/Documents%20and%20Settings/123/Local%20Settings/Temp/Rar$DI00.750/meshz(X,Y,Z)

http://c/Documents%20and%20Settings/123/Local%20Settings/Temp/Rar$DI00.750/surfc(X,Y,Z)

http://c/Documents%20and%20Settings/123/Local%20Settings/Temp/Rar$DI00.750/surf(X,Y,Z)

http://c/Documents%20and%20Settings/123/Local%20Settings/Temp/Rar$DI00.750/meshc(X,Y,Z)

http://c/Documents%20and%20Settings/123/Local%20Settings/Temp/Rar$DI00.750/mesh(X,Y,Z)



MATLAB

CHAPTER 6Numerical calculus and differential equations

MATLAB : Chapter 8. Numerical calculus and differential equations

OUTLINE



OUTLINE

Numerical integrationI(2)

Numerical differentiation(3)

Numerical Methods for Differential Equations(7)

Extension to Higher-Order Equations(6)ODE Solvers in the Control System Toolbox(7)

Advanced Solver Syntax(2)


Numerical integrationI(1/2)



Rectangular integration trapezoidal integration

Numerical integration functions

Command Description

quad („function‟,a,b,tol) Uses an adaptive Simpson‟s rule to compute the integral of the function

‘function ’ with a as the lower integration limit and b as the upper limit. Theparameter tol is optional. tol indicates the specified error tolerance.

quadl („function‟, a,b,tol) Uses Lobatto quadrature to compute the integral of the function ‘function’. Therest of the syntax is identical to quad.

trapz (x,y) Uses trapezoidal integration to compute the integral of y with respect to x ,where the array y contains the function values at the points contained in the

array x.

y

xa b

y=f(x) y=f(x)

y

xa b··· ···

0sin dx

2cos0coscossin00

xdx

(exact solution)

(use of the trapzfunction)


Numerical integrationI(2/2)



Quadratu re func t ions

quad („function‟,a,b)

quadl („function‟,a,b)

Slope func t ion s ingu la ri t i es

example

xdxdy

x y

/5.0/(The slope has a singularity at x=0)

The answer is A1=A2=0.6667, A3=0.6665

Compared to Trapz, exactness

is high, but it is impossible tointegrate discretized data (onlypossible for fixed to function)


Numerical differentiation(1/3)



x y y

x x y y

m B

23

23

23

x1 x2 x3

y1

y2

y3

Δ x Δ x

y=f(x)

True slope

A

B

C

x y y

x y y

x y ymm

mB A

C

221

2132312

x

y y

x x

y ym A

12

12

12

x y

dxdy

limΔx 0

: backward difference

: central difference

: forward difference

: Difinition





The d i ff Func t ion

d= di ff (x) ->( d=[x(2)-x(1),x(3)- x(2), … ] )

Backward difference & centraldifference method

example :

example : x=[5, 7, 12, -20];d= diff(x)

d = 2 5 -32





b = polyder (p)p = [a 1,a 2,…,a n]b = [b 1,b 2,…,b n-1 ]b = polyder (p1,p2)[num, den] = polyder(p2,p1)

nnnnn a xa xa xa xa x f 1

23

121)(

12

21

1 )1( nnn a xan xna

dxdf

12

21

1 nnn b xb xb

Example p1= 5x +2

p2=10x 2+4x-3

Result: der2 = [10, 4, -3]

prod = [150, 80, -7]

num = [50, 40, 23]

den = [25, 20, 4]

Command Description

d=diff(x) Returns a vector d containing the differences between adjacent elements in the vector x .

b=polyder(p) Returns a vector b containing the coefficients of the derivative of the polynomial representedby the vector p .

b=polyder(p1,p2) Returns a vector b containing the coefficients of the polynomial that is the derivative of theproduct of the polynomials represented by p1 and p2 . cf. equivalent comment:b=polyder(conv(p1,p2))

[num, den]=polyder(p2,p1)

Returns the vector num and d en containing the coefficients of the numerator and denominator polynomials of the derivative of the quotient p 2 /p 1, where p1 and p 2 are polynomials.

Numerical differentiation functions

po lyno m ial Derivat ives


Numerical Methods for Differential Equations(1/7)



Runge-Kutta MethodsThe second-order Runge-Kutta methods:

(8.5-17) , where : constant weighting factors

(8.5-18)(8.5-19)

To duplicate the Taylor series through the h 2 term, these coefficientsmust satisfy the following:

(8.5-19)

(8.5-19)

(8.5-19)

22111 g w g w y y k k

),(

),(

2

1

k k k

k k

hf yht hf g

yt hf g

21, ww

2

12

1

1

2

1

21

w

w

ww





The fourth-order Runge-Kutta methods:

(8.5-23)

(8.5-24)

443322111 g w g w g w g w y y k k

])(,[

])(,[

),(

),(

1333332334

1222223

112

1

g g g yht hf g

g g yht hf g

g yht hf g

yt hf g

k k

k k

k k

k k

0

1

21

21

31

61

3

33

2

21

32

41

ww

ww





MATLAB ODE Solvers ode23 and ode45MATLAB provides functions, called solvers , that implement Runge-Kutta

methods with variable step size.The ode23 function uses a combination of second- and third-order Runge-Kutta methods, whereas ode45 uses a combination of fourth- and fifth-order methods.

<Table 8.5-1> ODE solvers

Solver name Description

ode23 Nonstiff, low-order solver.ode45 Nonstiff, medium-order solver.ode113 Nonstiff, variable-order solver.

ode23s Stiff, low-order solver.ode23t Moderately stiff, trapezoidal-rule solver.ode23tb Stiff, low-order solver.ode15s Stiff, variable-order solver.





Solver Syntax

<Table 8.5-2> Basic syntax of ODE solvers Command Description

[t, y] = ode23 ( „ydot‟, tspan, y0) Solves the vector differential equation specifiedin function file ydot, whose inputs must be t and y andwhose output must be a column vector representing; that is, . The number of rows in this columnvector must equal the order of the equation. The vector tspan contains the starting and ending values of theindependent variable t, and optionally, any intermediatevalues of t where the solution is desired. The vector y0contains. The function file must have two inputarguments t and y even for equations where is nota function of t. The syntax is identical for the other solvers.

),(.

yt f y

dt dy

),(.

yt f y

)( 0t y

),( yt f



Effect of Step Size

The spacing used by ode23 is smaller than that used by ode45 becauseode45 has less truncation error than ode23 and thus can use a larger stepsize.ode23 is sometimes more useful for plotting the solution because it oftengives a smoother curve.

Numerical Methods and Linear EquationsIt is sometimes more convenient to use a numerical method to find thesolution.Examples of such situations are when the forcing function is a complicatedfunction or when the order of the differential equation is higher than two.

Use of Global ParametersThe global x y z command allows all functions and files using that command to sharethe values of the variables x, y, and z.





Use of Global Parameters( Example ) The circuit‟s equation was given earlier. It is

Solving for the derivative and using RC=0.1, we obtain

( )dy RC y v t dt

10[ ( ) ]dy

v t ydt

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2

0

2

4

Time (s)

A p p

l i e d V o

l t a g e

( V

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

0

1

2

3

Time (s)

C a p a c

i t o r

V o

l t a g e

( V )

< Solution >

< Result >


Extension to Higher-Order Equations(1/6)h h f h bl f



The Cauchy form or the state-variable formConsider the second-order equation

(8.6-1)

(8.6-2)

If , , then

5 7 4 ( ) y y y f t 1 4 7

( )5 5 5

y f t y y

y x1

2 x y

21

.

2

2

.

1

57

54

)(51

x xt f x

x xThe Cauchy form

or state-variable form


Extension to Higher-Order Equations(2/6).



<example> Solve (8.6-1) for with the initial conditions .

( Suppose that and use ode45 .)60 t 9)0(,3)0(

.

y y)sin()( t t f

xdot(1)

xdot(2)

x(1)

x(2)

.

1 x.

2 x

1 x

2 x

The in i t ial condi t ion for

the vec tor x .

0 1 2 3 4 5 6-4

-2

0

2

4

6

8

10


Extension to Higher-Order Equations(3/6)



Matrix Methods

< a mass and spring with viscous surface friction > y

f(t)m

k

c

)(...

t f ky yc ym

21

. x x

212 )(1. xmc x

mk t f

m x

( Letting , ) y x1

.2 y x

)(1010

2

1

2

1..

t f m x

x

mc

mk

x

x

(Matrix form)

)(

.t f

mc

mk

10

m10

2

1

x

x

(Compact form)

(8.6-6)

(8.6-7)





Characteristic Roots from the eig Function

Substituting ,

Cancel the terms

Its roots are s = -6.7321 and s = -3.2679.

211 3.

x x x212 7

. x x x

(8.6-8)

(8.6-8)

st st st

st st st

e Ae Ae sA

e Ae Ae sA

212

211

7

3

st e At x 11 )( st e At x 22 )(

st e

0)7(

0)3(

21

21

A s A

A A s 02210

71

13 2 s s

s

s

A nonzero solution will exist for A1 and A2 if

and only if the determinant is zero





MATLAB provides the eig function to compute the characteristic roots.Its syntax is eig(A)

<example> The matrix A for the equations (8.6-8) and (8.6-9) is

To find the time constants, which are the negative reciprocals of thereal parts of the roots, you type tau = -1./real (r). The time constantsare 0.1485 and 0.3060.

7113





Programming Detailed Forcing Functions< An armature-controlled dc motor>

v

R L

I

+

-

w K e

i K T T w

cwi

cwi K dt dw

I

t vw K Ridt di

L

T

e )(

)(0

1

2

1

2

1..

t v L x x

I c

I K L

K

L

R

x x

T

e

Apply Kirchhoff‟s voltage law and Newton‟s law

(8.6-10)

(8.6-11)

( matrix form )

: motor‟s current , : rotational velocity i w

L: inductance, R: resistance, I: inertia, : torque constant, : back emf constant,

c: viscous damping constant, : applied voltage

T K e K

)(t v

Letting w xi x 21 ,


ODE Solvers in the Control System Toolbox(1/7)



Model FormsThe reduced form

The state-model of the same system

)(532 ... t f x x x (8.7-1)

2

1

210

23

25

10

.

x

x

u

(8.7-5)

(8.7-3)

(8.7-4)

(8.7-2)

Both model forms contain the sameinformation. However, each formhas its own advantages, dependingon the purpose of the analysis .

The specification of theoutput:

DuC y


ODE Solvers in the Control System Toolbox(2/7)< Table 8 7 1> LTI object functions



< Table 8.7-1> LTI object functions

Command Description

sys = ss (A, B, C, D) Creates an LTI object in state-space form, where thematrices A, B, C, and D correspond to those in themodel .

[A, B, C, D] = ssdata (sys) Extracts the matrices A, B, C, and D corresponding tothose in the model

.

sys = tf (right, left) Creates an LTI object in transfer-function form, wherethe vector right is the vector of coefficients of the right-hand side of the equation, arranged in descending

derivative order, and left is the vector of coefficients of the left-hand side of the equation, also arranged indescending derivative order.

[right, left] = tfdata (sys) Extracts the coefficients on the right- and left-handsides of the reduced-form model.

DuC yu,.

DuC yu,.





Example


ODE Solvers in the Control System Toolbox(4/7)ODE Solvers



ODE Solvers<Table 8.7-2> Basic syntax of the LTI ODE solvers

LTI Viewers : It provides an interactive user interface that allows you toswitch between different types of response plots and between the analysisof different systems. The viewer is invoked by typing ltiview.

Command Description

Impulse (sys) Computes and plots the unit-impulse response of the LTIobject sys.

Initial (sys, x0) Computes and plots the free response of the LTI object sys

given in state-model form, for the initial conditions specified inthe vector x0.

lsim (sys, u, t) Computes and plots the response of the LTI object sys to theinput specified by the vector u, at the times specified by thevector t.

step (sys) Computes and plots the unit-step response of the LTI objectsys.



Example

0 1 2 3 4 5 6 7 8 9 10-0.5

0

0.5

1

1.5

Time

R e s p o n s e


Advanced Solver Syntax(1/2)The odeset Function< Table 8 8 1 > Complete syntax of ODE solvers



< Table 8.8-1 > Complete syntax of ODE solversCommand Description

[t, y] = ode23 („ydot‟,tspan, y0, options, p1,p2, …)

Solves the differential equation specified in the function the ydot,whose inputs must be t and y and whose output must be a column vector representing dy/dt; that is, f(t,y). The number of rows in this column vector mustequal the order of the equation. The vector tspan contains the starting andending values of the independent variable t, and optionally, any intermediatevalues of t where the solution is desired. The vector y0 contains y(t 0). Thefunction file must have two input arguments t and y even for equations wheref(t,y) is not a function of t.The options argument is created with the odeset function, and p1, p2, … areoptional parameters that can be passed to the function file ydot every time it iscalled. If these optional parameters are used, but no options are set, useoptions = [ ] as a placeholder.The function file ydot can take additional input arguments. It has the form ydot (t,y, flag, p1, p2, …), where flag is an argument that notifies the function ydot thatthe solver is expecting a specific kind of information. The syntax for all thesolvers is identical to that of ode23.

Options = odeset(„name1‟, „value1‟,„name2‟, „value2‟, …)

Creates an integrator options structure options to be used with the ODE solver,in which the named properties have the specified values, where name is thename of a property and value is the value to be assigned to the property. Anyunspecified properties have default values. odeset with no input argumentsdisplays all property names and their possible values.

),(.

yt f y


Advanced Solver Syntax(2/2)



Stiff Differential Equations

A stiff differential equation is one whose response changes rapidly over atime scale that is short compared to the time scale over which we areinterested in the solution.

A small step size is needed to solve for the rapid changes, but many steps

are needed to obtain the solution over the longer time interval, and thus alarge error might accumulate.

The four solvers specifically designed to handle stiff equations: ode15s (avariable-order method), ode23s (a low-order method), ode23tb (another low-order method), ode23t (a trapezoidal method).



SPaC, POSTECH 156

MATLAB

CHAPTER 7Simulink

MATLAB : Chapter 9. Simulink

Simulation Diagram



157

Simulation diagram (block diagram)Consider the equation

Simulation diagrams for

)(10. t f y

dt t f t y )(10)(

)(10)( t f t xdt t xt y )()(

10f(t) y(t)x(t) f yx

10 s1

)(10.

t f y


Introduction to Simulink



158

Type Simulink

in the MATLABCommand windowto start window.

The Simulink Library Browser

Type „simulink‟ In the Command

Window





159

Create a new modelClick on the icon that resembles aclean sheet of paper, or select Newfrom the File menu in the Browser.

Click

this iconFile New

Or





160

① Double-click appropriate library.② See a list of blocks within that library.③ Click on the block name or icon.④ Hold the mouse button, and drag

it to the new model window.⑤ release the button.

click, hold,drag, andrelease





161

⑥ Use the File menu in the model window to Open, Close, and Save model files. ⑦ To print, File print⑧ Edit : to copy, cut and paste blocks.⑨ you can also use mouse for these operations





162

ExampleSimulink Solution of

Use Simulink to solve the following problem for 0 ≤ t ≤ 13.

The exact solution is

)sin(10.

t y

)sin(10 t dt dy 0)0( y

)cos1(10)( t t y





163

① Start Simulink and open a new model window.② Select and place the Sine Wave block from the Source library .





164

③ Double-click to open the Block Parameterswindow .Make sure that Amplitude= 1, Frequency= 1,Phase= 0, Sample time= 0Then click OK .





165

④ Select and place the Gain block from the Math Operations library .

the Block Parameters window: the Gain value = 10.Note that the value ‟10‟ then appears in the triangle.

⑤ Select and place the Integrator block from the Continuous library .Initial condition=0 (because y(0)=0)





166

⑥ Select and place the Scope block from the Sinks library .

⑦ Connect each input and output port.





167

⑧ Click on the Simulation menu, and click the Configuration Parameters item.(If you use MATLAB 6.5 or earlier, click Simulation parameters instead of

Configuration Parameters .)Click on the solver tab, and enter 13 for the stop time.Make sure the Start time is 0.





168

⑨ Run the simulation by clicking on the Simulation menu,and clicking the Start item.

OR





169

⑩ After the simulation, double-click on the Scope block .

Double-click!


Linear State-Variable Models



170

State-variable models can havemore than one input and morethan on output.Simulink has the State-spaceblock that represents the linear state-variable model

DuCx y Bu Ax x ,.





171

ExampleSimulink Model of the Two-Mass System

m1=5, m2=3, c1=4, c2=8, k1=1, and k2=4The equations of motion are

These equations can be expressed in state-variableform as

)(48483

0485125

11222

22111

....

....

t f x x x x x

x x x x x

43

21

.

.

z z

z z

)4(31.

)5(51.

)(848

8412

43214

43212

t f z z z z z

z z z z z





172

Vector matrix form

)(.

t Bf Az z

A =

0 1 0 0

-1 -12/5 4/5 8/50 0 0 14/3 8/3 -4/3 -8/3

0

00

1/3

B =

2

2

1

1

.

.

x

x

x

xz1

z2z3

z4

z = =





173

Initinal conditions

Output equation

)(: t Df Cz youtput

1 0 0 0

0 0 1 0C =

0

0D =

0)0(2,5.0)0(2,0)0(1,2.0)0(1..x x x x



175

③Select and place the State-Spacelock . Enter A,B,C,D. Then enter initial condition.

0.2; 0; 0.5; 0





176

④Select and place the Scope block.

⑤connect each port.


Linear State-Variable Models⑥experiment with different values of the Stop time until the Scope shows thath d h b h d



177

the steady-state response has been reached.Ex) when stop time = 25

steady-state


Piecewise-Linear Models



178

Closed-form solutions are not available for most nonlinear differentialequations, We must solve such equations numerically.

Ex)

Piecewise-l inear models are actually nonlinear, although they mayappear to be linear.

Ex) a mass attached to a spring and sliding on a horizontal surface withCoulomb friction.

0.

5..

y y y y 0)sin(.

y y 0.

y y

0 )(

0 )(

...

...

x x x

x x x

if mg t f k m

if mg t f k m


Piecewise-Linear Models



179

These two equations can be expressed as the single, nonlinear equation.

Solutions of models that contain piecewise-linear functions are very tediousto program. However, Simulink has built-in blocks that represent many of thecommonly-found functions such as Coulomb friction. Therefore Simulink isespecially useful for such applications. One such block is the Saturationblock in the Discontinuities library.

0 1

0 1)sing( )sing()(

......

x

x x x x x

if

if wheremg t f k m


Simulink Model of a Rocket-Propelledsled



180

sled

A rocket propelled sled

Compute the sled‟s velocity v for 0 ≤t≤6 if v(0) = 0 The rocket thrust is 4000N and the sled mass is 450kg.The sled‟s equation of motion is

.

450 4000cos( )v

θ f

b

v

)/( 50..

srad



183

s ed

④ Set the Stop time 10, run thesimulation, and examine the resultin Scope.





184

(b)①modify the model as follows.② the Saturation block from theDiscontinuities library .Upper limit= 60*pi/180,

Lower limit=0.③mux

Generate thesolution when theengine angle θ=0.

mux





185

Scope window

Θ=0

Θ≠ 0


The Relay Block



186

The Simulink Relay block is an example of something that is tedious toprogram in MATLAB but is easy to implement in Simulink.

A graph of the logic of a relay.

The relay switches the output between two specified values, named Onand Off in the figure.

On

Off

SwOnSwOff

Off

OnSwOff SwOn

The relay function. (a) The case whereOn>Off. (b) The case where On<Off

(a) (b)




Model of a Relay-Controlled Motor



188

x1=i, x2= ω

R=0.6 Ω, L=0.002H, K T=0.04N·m/A, K e=0.04V·s/rad, c=0.01N·m·s/rad, andI=6·10^(-5)kg·m^2Suppose we have a sensor that measure the motor speed, and we use thesensor‟s signal to activate a relay to switch the applied voltage v(t) between0 and 100V to keep the speed between 250 and 350rad/s.

SwOff=250, SwOn=350, Off=100, On=0

2

1

.

.

x

x=

2

1

x x-R/L -K e /L

KT/I -c/I+

1/L 0

0 -1/I

V(t)

Td(t)

On

Off

SwOnSwOff



190

①Create a new Simulink model.

②Select and place a Step block from the Sources library . Label itDisturbance Step .

Step time=0.05, Initial and Finaltime=0 and 3, Sample time=0

③Select and place a Relay blockfrom the Discontinuities library.Switch-on and Switch-off points=350 and 250,Output when on and Output whenoff=0 and 100.





191

④Select and place the Mux block from the signal Routing library .Display option to Signals.number of input=2.

⑤Select and place the State-Space block from theContinuous library .enter the A,B,C,D.enter [0;0] for the initial conditoin.B tells 2 input, C and D tells 1output.





192

⑥Select and place the Scopeblock from the Sinks library.

⑦connect each port.

⑧Stop time=0.1 and run thesimulation. (the plot of ω (t) in thescope.)

Note :

Connect signal1(first input) to the outputof the Relay block

Connect signa2(second input) to theoutput of the Disturbance Step



194

Speed becomes constantV=100, the system achieves a steady-state condition in which the motor torque equals the sum of thedisturbance torque and the viscousdamping torque. Thus the accelerationis zero.





195

⑨If you want examine the current i(t), change thematrix C to [1,0], and run the simulation again.




Transfer-Function Models



197

mass-spring-damper system

As with the control System toolbox, Simulink can accepts a system description in transfer-function form and in state-variable formif force f(t) is sinusoidal function, it is easy to use the MATLAB commands presented.However, suppose that the force f(t) is created by applying a sinusoidal input voltage to ahydraulic piston that has a dead-zone nonlinearity. This means that the piston does notgenerate a force until the input voltage exceeds a certain magnitude.

A graph of a particular dead-zone nonlinearity is shown below.

)( t f yk yc ym

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5-0.4-0.3-0.2-0.1

00.1

0.20.30.40.5

Input

O u t p u t





198

[EXAMPLE 9.5-1] Response with a Dead ZoneCreate and run a Simulink simulation of a mass-spring-damper model

m=1, c=2 and k=4f(t) = sin(1.4t)The system has the dead-zone nonlinearity shown in Figure 9.5-1

SolutionStart Simulink and construct

fallowing Simulink model

)( t f yk yc ym





199

Solution (cont’d)

Details of Blocks

drag

double-click

& changeparameters

Simulink librarybrowser





200

solution (cont’d)

Modification of the dead-zone model to include aMux blockTo plot both the input and the output of theTransfer Fcn block versus time on the same graph.

(More informative)

dragdouble-click& changeparameters




Nonlinear State-Variable Models



202

Solution

let

)(sin108.025.0)(sin1 t M t M mgLc I

Integrate both sides of each equation over time to obtain

dt t M

dt

)(sin108.025.0





203

Solution (cont‟d) Construct following Simulink model

Select blocks and place them.Double-click blocks and change parameters.

θ θ

initial = 0

initial = pi/4





204

Solution (cont‟d) Details of new blocks

drag

double-click& change

parameters





205

Solution (cont‟d) Simulation result

NoteTo flip the box left to right, right-click on it, select Filp“Fnc” block : General expression block. Use “u” as input variable name.


Subsystems



206

To simulate a complex system, the diagram can become rather largeand therefore some what cumbersome.Simulink, however, provides for the creation of s u b s y s t e m b l o c k s

S u b s y s t em b l o c k play a role analogous to that of subprograms in a programming languagea Simulink program represented by a single block.can be used in other Simulink programs.

Example – A Hydraul ic System


Subsystems



207

Example – A Hydraul ic System liquid mass density ρ cylindrical tank with a bottom area Amass flow rate q mi (t)total mass in the tank m = ρ Ah

If the outlet is a pipe that discharges to atmospheric pressure p a and provides a resistanceto flow that is proportional to the pressure difference across its ends, then the outlet flow rate is

where R is called the fluid resistance.

The transfer function is

momi qqdt

dh A

dt

dm

h R g

qdt dh

A mo

R gh

p p gh R

q aamo

])[(1

R g As sQ s H

mi /1

)()(


SubsystemsExample – A Hydraulic System (cont’d)



208

Example A Hydraulic System (cont d)

On the other hand, the outlet may be a valve or other restriction that provides nonlinear resistance to the flow.In such cases, a common model is the signed-square-root relation

where q mo is the outlet mass flow rate, R is the resistance, Δ p is the pressure differenceacross the resistance, and

in Simulink : sgn(u)*sqrt(abs(u)) (note: in Matlab~sign(u))Consider the slightly different system

flow source q and two pumps that supply liquid at the pressures p l and p r .

Suppose the resistances are nonlinear and obey the signed-square-root relationthen the model of the system is

)(1

pSSR R

qmo

00)(

pif p pif p pSSR

)(1

)(1

r r

l l

p pSSR R

p pSSR R

qdt dh

A


Subsystems



209

Create a subsystem blockdrag the Subsystem block from the libraryor first creation a Simulink model and then “encapsulating”

Create s subsystem block for the liquid-level system shown in Figure 9.7-2.First construct the Simulink model as fallow

input port(Ports and

Subsystemslibrary)

output port(Ports and

Subsystemslibrary)


SubsystemsCreate s subsystem block (Cont‟d)



210

Create s subsystem block (Cont d) create a “bounding box” surrounding the diagram. (holding the mouse button down, anddragging)choose Create Subsystem from the Edit menu. (or in the Right-click menu)

The result


Subsystems



211

Connecting Subsystem Blockscreate a simulation of the system shown below

where the mass inflow rate q is a step function

Simulink model of the systemDouble-click subsystem block Tank 1 and changeparameters.

1/R_l=0; 1/R_r=1/R_1; 1/rho*A=1/rho*A_1initial condition of the integrator to h10

in Tank 21/R_l = 1/R_1; 1/R_r = 1/R_2; 1/rho*A=1/rho*A_2initial condition of the integrator to h20




Dead Time in Models



213

Dead Time (transport delay)a time delay between an action and its effect.The transfer function for a dead-time process is(from the shifting property of the Laplace transform)Some systems have an unavoidable time delay in the interaction between components.

The delay often results from the physical separation of the components and typicallyoccurs as a delay between a change in the actuator signal and its effect on thesystem being controlled, or as delay in the measurement of the output

Another, source of dead time is the computation time required for digital control computer to calculate the control algorithm.There are an infinite number of characteristic roots for a system with dean time.

The analysis of dead-time processes is difficult, and often simulation is the only practicalway to study such processes.Easily simulated in Simulink. Using “Transport Delay” Block.

Tse

2/111

22 sT Tsee Ts

Ts


Dead Time in Models



214

Consider the model of the height h of liquid in a tank.input is a mass flow rate qiit takes a time T for the change in input flow to reach thetank following a change in the valve opening.the transfer function

Simulink model for this system

15 2)( )( se sQ s H Ts

i


Dead Time in ModelsTransport Delay block



215

Transport Delay block

in Continuous libraryDouble-click the block and set the delay to 1.25

Transfer Fcn (with initial outputs) block

Double-click the block and set the initial output to 0.2The Saturation and Rate Limiter Blocks

suppose that the minimum and maximum flow rates available from the input flow valve are0 and 2In additional, some actuators have limits on how fast they can react. (to avoid damage tothe unit)Saturation block

in Discontinuity libraryDouble-click the block and set the Upper limit to 2, the Lower limit to 0

Rate Limiter blockin Discontinuity library Double-click the block and set the Rising slew rate to 2, the Falling slew rate to 0


Dead Time in Models

A C t l S t



216

A Control SystemThe Simulink model shown in Figure 9.8-1 is for a specific type of control system called a PIcontroller.whose response f(t) to the error signal e(t) is the sum of a term proportional to the error signaland a term proportional to the integral of the error signal.

where K P and K I are called the proportional and integral gains.The error signal e(t) is the difference between the unit-step command representing the desiredheight and the actual height.In transform notation

In Figure 9.8-1, we used the values K P=4 and K I=5/4.

Simulation ResultStop time to 30

t

I P dt t e K t e K t f 0

)()()(

)()()()( s E s

K K s E

s K

s E K s F I p

I P


Simulation of a Vehicle Suspension



217

Often in the design of an engineering system, we must eventually dealwith nonlinearities in the system and with more complicated inputs suchas trapezoidal functions, and this must often be dome with simulation.In this section we introduce four additional Simulink elements that enableus to model a wide range of nonlinearities and input functions,

the Derivative block,the Signal Builder block,the Look-Up Table block, andthe MATLAB Fcn block

As our example, we will use the

single-mass suspension modelshown in Figure 9.9-1


Simulation of a Vehicle SuspensionSingle-mass model of a vehicle suspension



218

g p

the spring and damper forces fs and fd have the nonlinear models shown inFigures 9.9-1 and 9.9-3

The bump is represented by the trapezoidal function y(t) shown in Figure 9.9-4





219

The system model from Newton‟s law is

where m = 400 kg, is the nonlinear spring fuction shown in Figure 9.9-2and is the nonlinear damper function shown n Figure 9.9-3

The corresponding simulation diagram

)..

()(..

x y f x y f xm d s

)( x y f s

)..

( x y f d


Simulation of a Vehicle SuspensionSimulink model of a vehicle suspension system.



220

The Derivative BlockSimulink uses numerical methods,it computes derivatives only approximately

in Continuous libraryThe Signal Builder Blocks

in Source librarydouble-click and create the functionshown in Figure 9.9-4





221

The look-Up Table BlockThe spring function fs is created with the Look-Up Table block.In Lookup Tables librarydouble-click on it and enter [-0.5, -0.1, 0, 0.1, 0.5] for the Vector of input valuesand [-4500, -500, 0, 500, 4500] for the Vector of output values.

To Workspace block and the Clockenable us to plot x(t) and y(t)-x(t) versus t in the MATLAB Command window.

The MATLAB Fcn Blockwe must write a user-definedfunction to describe the damper function.

double-click on the block andchange function name.





222

Simulation Result

You can plot the response x(t) in the Command window as follows

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

t (s)

x ( m )



MATLAB

CHAPTER 10Symbolic processing with MATLAB

MATLAB : Chapter 10. Symbolic processing with MATLAB

OUTLINE



Symbolic expressions and algebra(10)

Algebraic and transcendental equations(3)Calculus (9)Differential Equations (6)


Symbolic expressions and algebra(1/10)

The s y m function Symbolic expressions



The s y m function

Ex >>x = sym ('x')>>x = sym ('x', 'real')>>syms x y u v>>pi=sym('pi')>>fraction=sym('1/3')>>sqroot2=sym('sqrt(2)')

ex >>syms x y>>s = x + y>>r = sqrt(x^2+y^2)

ex >>symbolic matrix A>>n=3

>>syms x;>>A=x.^((0:n)'*(0:n))

A=[1,1,1,1][1,x,x^2,x^3][1,x^2,x^4,x^6][1,x^3,x^6,x^9]


Symbolic expressions and algebra(2/10)expand (E)Manipulating expressions



ex >>syms x y>>expand((x+y)^2)

% applies algebra rulesans =

x^2+2*x*y+y^2

>>expand(sin(x+y))% applies trig identities

ans =

>>sin(x)*cos(y)+cos(x)*sin(y)

>>simplify(6*((sin(x))^2+(cos(x))^2))% applies another trig identity

ans =6

collect (E)

ex >>syms x y>>E=(x-5)^2+(y-3)^2;>>collect (E)

ans =

x^2-10*x+25+(y-3)^2

>>collect(E,y)ans =

y^2-6*y+(x-5)^2+9



factor (E) Ex> >>syms x y



factor (E)ex >>syms x y

>>factor(x^2-1)ans =

(x-1)*(x+1)

simplify (E)ex >>syms x y

>>simplify(x*sqrt(x^8*y^2))ans =

x*(x^8*y^2)^(1/2)

simple (E)[r, how] = simple (E)

y y

>>E1=x^2+5; % define two expressions>>E2=y^3-2;>>E3=x^3+2*x^2+5*x+10; % define a third

expressions >>S1=E1+E2 % add the expressions

S1 =x^2+3+y^3

>>S2=E1*E2 % multiply the expressionsS2 =(x^2+5)*(y^3-2)

>>expand(S2) % expand the productans =

x^2*y^3-2*x^2+5*y^3-10>>S3=E3/E1 % divide two expressions

S3 =(x^3+2*x^2+5*x+10)/(x^2+5)

>>simplify(S3) % see if some terms cancelans =

x+2





[num den] = numden (E)ex >> syms x>> E1=x^2+5;>> E4=1/(x+6);>> [num, den]=numden(E1+E4)

num =x^3+6*x^2+5*x+31

den =x+6

double (E)ex >> sqroot2=sym('sqrt(2)');

>> y=6*sqroot2y =

6*2^(1/2)>> z=double(y)

z =8.4853



poly2sym (p)



poly2sym (p)poly2sym (p, 'v')

ex >> poly2sym([2,6,4])

ans =2*x^2+6*x+4

>> poly2sym([5,-3,7],'y')ans =

5*y^2-3*y+7

sym2poly (E)

ex >> syms x

>> sym2poly (9*x^2+4*x+6)ans =

9 4 6

pretty (E)

ex >> E5=1/(x+6);>> pretty(E5)

1-----

x + 6

subs (E,old,new)ex >> syms x y

>> E=x^2+6*x+7;>> F=subs(E,x,y)

F =y^2+6*y+7

ex >> syms t>> f=sym('f(t)');>> g=subs(f,t,t+2)-f

g =f(t+2)-f(t)



ex (n-1)!



ex (n-1)!>> kfac=sym('k!');>> syms k n>> E=subs(kfac,k,n-1)

E =(n-1)!

>> expand(E)ans =

n!/n

To compute a numeric factorial ex >> 5!

>> factorial(5)>> prod(1:5)

ex >> syms a b x>> E=a*sin(b);>> F=subs(E,a,b,x,2)

F =x*sin(2)





Evaluating expressionsex >> syms x

>> E=x^2+6*x+7;>> G=subs(E,x,2)

G =23

>> class(G)

ans =double

digit (d) to change the number of digits MATLAB uses for calculating and evaluating expressions

vpa (E) to compute the expression E to the number of digits specified by the default value of 32 or thecurrent setting of digits

vpa (E,d) to compute the expression E using d digits





Plotting expressionsezplot (E)explot ( E, [xmin xmax] )

ex >> syms x

>> E=x^2-6*x+7;>> ezplot(E,[-2 6])


Symbolic expressions and algebra(9/10)Order of precedenceex syms x

E=x^2-6*x+7;F=-E/3

F = -1/3*x^2+2*x-7/3normally F= - (x2-6x+7)/3



Functions for creating and evaluating symbolic expressions

Command Description

class (E) Returns the class of the expression E.

digits (d) Sets the number of decimal digits used to do variable precision arithmetic. The default is 32 digits.

double (E) Converts the expression E to numeric form.

ezplot (E) Generates a plot of a symbolic expression E, which is a function of one variable. The default range of theindependent variable is the interval [-2 π , 2 π ] unless this interval contains a singularity. The optional formezplot (E,[xmin xmax]) generates a plot over the range from xmin to xmax .

findsym(E) Finds the symbolic variables in a symbolic expression or matrix, where E is a scalar or matrix symbolicexpression, and returns a string containing all the symbolic variables appearing in E. the variables arereturned in alphabetical order and are separated by commas. If no symbolic variables are found, findsym returns the empty string.

findsym(E,n) Returns the n symbolic variables in E closest to x, with the tie breaker going to the variable closer to z.

[num den]=numden(E) Returns two symbolic expressions that represent the numerator expression num and denominator expression den for the rational representation of the expression E.

x=sym („x‟) Creates the symbolic variable with name x. typing x=sym („x‟,‟real‟) tells MATLAB to assume that x is real.Typing x=sym („x‟,‟unreal‟) tells MATLAB to assume that x is not real.

syms x y u v Creates the symbolic variables x,y,u ,and v. when used without arguments, syms lists the symbolic objectsin the workspace.

vpa (E,d) Sets the number of digits used to evaluate the expression E to d. typing vpa(E) causes E to be evaluated



Functions for creating and evaluating symbolic expressions



g g y p

Command Description

collect (E) Collects coefficients of like powers in the expression E.

expand (E) Expands the expression E by carrying out powers.

factor (E) Factors the expression E.

poly2sym (p) Converts a polynomial coefficient vector p to a symbolic polynomial. The formpoly2sym(p,‟v‟) generates the poly

pretty (E) Displays the expression E on the screen in a form that resembles typeset mathematics.

simple (E) Searches for the shortest form of the expression E in terms of number of characters. Whencalled, the function displays the results of each step of its search. When called without theargument, simple acts on the previous expression. The form [ r, how] = simple (E) does notdisplay intermediate steps, but saves those steps in the string how. The shortest form foundis stored in r .

simplify (E) Simplify the expression E using Maple‟s simplification rules.

subs (E,old,new) Substitutes new for old in the expression E, where old can be a symbolic variable or expression, new can be a symbolic variable ,expression ,or matrix, or a numeric value or matrix.

sym2poly (E) Converts the expression E to a polynomial coefficient vector.




Algebraic and transcendental equations(2/3)



solve ( E, 'v')

ex >> solve('b^2+8*c+2*b=0') % solves for c becauseit is closer to xans =

-1/8*b^2-1/4*b

ex >> solve('b^2+8*c+2*b=0','b') % solves for bans =

[ -1+(1-8*c)^(1/2)][ -1-(1-8*c)^(1/2)]

[x ,y] = solve (eq1,eq2)

ex >> eq4='6*x+2*y=14';>> eq5='3*x+7*y=31';>> solve(eq4,eq5)ans =

x : [1x1 sym]y : [1x1 sym]

>> x=ans.xx =

1>> y=ans.yy =

4ex >> [x ,y] = solve(eq4,eq5)

>> S=solve(eq4,eq5)S=

x : [1x1 sym]y : [1x1 sym]

>> S.xans=

1>> S.yans =

4


Algebraic and transcendental equations(3/3)



Command Description

solve (E) Solves a symbolic expression or equation represented by the

expression E. if E represents an equation, the equation‟s expression

must be enclosed in single quotes. If E represents an expression,

then the solution obtained will be the roots of the expression E; that is,

the solution of the equation E=0. You need not declare the symbolic

variable with the sym or syms function before using solve .

solve (E1,…,En) Solves multiple expressions or equations

S= solve (E) Saves the solution in the structure S .

Functions for solving algebraic and transcendental equations


Calculus (differentiation) (1/9)



diff (E)1n

n

nxdx

dxex

>> syms n x y>> diff(x^n)ans =

x^n*n/x>> simplify(ans)ans =

x^(n-1)*n

xdx xd 1lnex

>> diff(log(x))

ans =1/x

x xdx

xd cossin2sin2ex

>> diff((sin(x) 2)

ans =2*sin(x)*cos(x)

ydy

yd cos

sinex

>> diff((sin(y))

ans =cos(y)


Calculus (differentiation) (2/9))sin(),( xy y x f ex



)cos( xy ydxdf

2 x

>> diff(sin(x*y))

ans =cos(x*y)*y

ex

1) >> diff(sin(x*y))

ans =cos(x*y)*y

2) >> E=„x^2‟;

>> diff(E)ans=

2*x

3) >> syms x>> diff(x^2)ans =

2*x

)cos()]sin([ 2 xy x y

xy xex

>> syms x y>> diff(x*sin(x*y),y)

ans =x^2*cos(x*y)

diff (E, v)

diff (E, n)

diff (E, v, n)

xdx

xd 6

)(2

32

ex

>> syms x>> diff(x^3,2)ans =

6*x

)sin()]sin([ 32

2

xy x y

xy xex

>> syms x y>> diff(x*sin(x*y),y,2)ans =

-x^3*sin(x*y)


Calculus (integration) (3/9)



int (E)

ex >> syms x>> int(2*x)ans =

x^2

ex

>> syms n x y>> int(x^n)ans =

x^(n+1)/(n+1)>> int(1/x)ans =

log(x)>> int(cos(x))ans=

sin(x)>> int(sin(y))ans=

-cos(y)

1

1

n x

dx xn

n

xdx x

ln1

x xdx sincos

y ydy cossin



xn



xdn xnlnex

>> syms n x>> int(x^n,n)

ans =1/log(x)*x^n

int (E, v)

int (E, a, b) 393

52

5

2

32 xdx xex

>> syms x

>> int(x^2,2,5)

ans =39





33

332 abdx x

b

a

int (E, v, a, b)

x y

xdy xy3

1253

50

35

0

2ex

>> syms x y>> int(x*y^2,y,0,5)

ans = 125/3*x

int (E, m, n)

ex

>> syms a b x>> int(x^2,a,b)

ans =1/3*b^3-1/3*a^3

t e x xdx

t x

dx x

t e

t

et

t t

t t

cos)cos(cossin

21

21

22

1

2

1ex

>> syms t x>> int(x,1, t)

ans = 1/2*t^2-1/2>> int(sin(x),t,exp(t))ans =

-cos(exp(t))+cos(t)

1ln1

1xdx

xex

>> syms x>> int(1/(x-1))ans =

log(x-1)>> int(1/(x-1),0,2)ans =

NaN :singularity at x=1


Calculus (Taylor series) (6/9)

Taylor‟s theorem



taylor (f, n, a)

ex x x x x

x x ,!7!5!3

sin753

x x x x

x ,!6!4!2

1cos642

x x x x

xe x ,!4!3!2

1432

nb xn

n

n a xdx

f d n

R )(!

1

nk

a xk

k

a xa x Ra xdx

f d k

a xdx

f d a x

dxdf

a f x f

)(!

1)(

21

)()()( 22

2

,

>> syms x>> f=exp(x);>> taylor(f,4)ans =

1+x+1/2*x^2+1/6*x^3>> taylor(f,3,2)ans =

exp(2)+exp(2)*(x-2)+1/2*exp(2)*(x-2)^2

ex

])2(21

)2(1[ 22 x xe=( )


Calculus (sums) (7/9)symsum (E)



S = symsum (E, a, b) >> syms k n>> symsum(k,0,10)ans =

55

>> symsum(k,0,n-1)ans =

1/2*n^2-1/2*n

>> factor(ans)ans =

1/2*n*(n-1)

>> symsum(k^2,1,4)ans =

30

)()2()1()()( b E a E a E a E x E b

a x

)1()2()1()0()(1

0

x E E E E x E x

x

ex 55109321010

0k

k

nnnk n

k 21

21

13210 21

0

30169414

1

2

k

k

ex


Calculus (limits) (8/9)



limit (E, a) ( )

limit (E) : limit as

limit (E, v, a) : limit as

>> syms h x>> limit((x-3)/(x 2-9),3)ans =

1/6>> limit((sin(x+h)-sin(x))/h,h,0)ans =

cos(x)

limit (E, v, a, 'right ')limit (E, v, a, „left ')

0 x

)(lim x E a x

ex

>> syms x

>> limit(1/x,x,0,'left')ans =-Inf

>> limit(1/x,x,0,'right')ans =

Inf

x1

lim

x1

lim

0 x

0 x

ex

>> syms a x>> limit(sin(a*x)/x)ans =

a

a x

ax)sin(lim

0 x

av

ex 6

1

9

3

lim 2 x

x

h xh x )sin()sin(

lim

3 x

0h


Calculus (9/9)

Command Description

diff (E) Returns the derivative of the expression E with respect to the default independent variable.

Symbolic calculus functions



diff (E,v) Returns the derivative of the expression E with respect to the variable v.diff (E,n) Returns the n th derivative of the expression E with respect to the default independent variable.

diff (E,v,n) Returns the nth derivative of the expression E with respect to the variable v.

int (E) Returns the integral of the expression E with respect to the default independent variable.

int (E,v) Returns the integral of the expression E with respect to the variable v.

int (E,a,b) Returns the integral of the expression E with respect to the default independent variable over the interval [a,b] ,

where a and b are numeric quantities.int (E,v,a,b) Returns the integral of the expression E with respect to the variable v over the interval [a,b] , where a and b are

numeric quantities.

int (E,m,n) Returns the integral of the expression E with respect to the default independent variable over the interval [m,n] ,where m and n are symbolic expressions.

limit (E) Returns the limit of the expression E as the default independent variable goes to 0.

limit (E,a) Returns the limit of the expression E as the default independent variable goes to a .

limit (E,v,a) Returns the limit of the expression E as the variable v goes to a .

limit (E,v,a,‟d‟) Returns the limit of the expression E as the variable v goes to a from the direction specified by d , which may beright or left.

symsum (E) Returns the symbolic summation of the expression E.

taylor (f,n,a) Gives the first n-1 terms in the Taylor series for the function defined in the expression f , evaluated at the pointx=a . If the parameter a is omitted, the function returns the series evaluated at x=0 .


Differential Equations (1/6)



Solving a Single Differential EquationThe dsolve function‟s syntax: dsolve („eqn‟) → returns a symbolic solution of theODE specified by the symbolic expression eqn .The uppercase letter D → the first derivative, D2 → the second derivative. (ex) Dw → dw/dt.

Cannot use uppercase D as symbolic variable when using the dsolve function.

(example)

122 ydt dy t eC t y 2

16)(Ana ly t ic so lu t ion :

A symbolic solution using the dsolve function .





Solving Sets of EquationsThe appropriate syntax: dsolve („eqn1‟, „eqn2‟,…) → returns a symbolic solutionof the set of equations specified by the symbolic expressions eqn1 and eqn2 .

(example)

y xdt dy

y xdt dx

34

43Ana ly t ic so lu t ion :

t eC t eC t y

t eC t eC t xt t

t t

4cos4sin)(

4sin4cos)(3

23

1

32

31



f l d d d



Specifying Initial and Boundary ConditionsThe appropriate syntax: dsolve(„eqn‟,‟cond1‟,‟cond2‟) → returns a symbolicsolution of the ODE specified by the symbolic expression eqn , subject to theconditions specified in the expressions cond1 , cond2 , and so on.

(example)

22

2 , (0) 1, (0) 0d y

c y y ydt

2/)()( ct ct eet yAna ly t ic so lu t ion :



Pl i h S l i (1)



Plotting the Solution (1)The ezplot function can be used to plot the solution

(example)

0)0(),4sin(41010 yt ydt

dy

Ana ly t ic so lu t ion :

t et t t y 10

2925

)4sin(2910

)4cos(294

1)(

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t

1-4/29 cos(4 t)+10/29 sin(4 t)-25/29 exp(-10 t)



Pl i h S l i (2)



Plotting the Solution (2)Sometimes the ezplot function uses too few values of the independent variableand thus does not produce a smooth plot.To override the spacing chosen by the ezplot function, you can use the subs function to substitute an array of values for the independent variable.

Define t to be a sy mbo l ic var iable.



(T bl 10 4 1) Th d l f ti



(Table 10.4-1) The dsolve function

Command Description

dsolve(„eqn‟) Returns a symbolic solution of the ODE specified by the symbolicexpression eqn . Use the uppercase letter D to represent the firstderivative; use D2 to represent the second derivative, and so on.

Any character immediately following the differentiation operator istaken to be the dependent variable.

dsolve(„eqn1‟,‟eqn2‟,…) Returns a symbolic solution of the set of equations specified bythe symbolic expressions eqn1 , eqn2 , and so on.

dsolve(„eqn‟,‟cond1‟,‟cond2‟,…) Returns a symbolic solution of the ODE specified by the symbolic