Matlab Tutorial June15 2005

MATLAB Tutorial

Jitkomut Songsiri

Department of Electrical Engineering,

Faculty of Engineering,

Chulalongkorn University

Room 404, EE Bldg.

June 15, 2005

Lecture 1

Introduction

• MATLAB is a computer program that can be veryhelpful in solving the kinds of mathematical problemsyou will frequently encounter.

• MATLAB can solve a wide variety of numerical prob-lems, from the very basic to complex problems.

• MATLAB can be used to plot several kinds of graphs

• MATLAB is supported to Windows, Mac OS X, andLinux platforms.

Introduction 1 – 2Other Scientific Computing Softwares

Software Windows Linux Free SoftwareMathematicahttp://www.wolfram.com/products/mathematica/index.html 4 4 7

SciLabhttp://scilabsoft.inria.fr/ 4 4 4

Octavehttp://www.octave.org 7 4 4

Maplehttp://www.maplesoft.com 4 4 7

Matlab tutorial, June 15, 2005

http://www.wolfram.com/products/mathematica/index.html

http://scilabsoft.inria.fr/

http://www.octave.org

http://www.maplesoft.com

Lecture 2

Variables

• Double array

• Char array

Variables 2 – 1Double array

Generate at the command line.

>> x = 1

x =

1

>> y = [1 2]

y =

1 2

>> z = [1 -1 0; 0 1 -1; 0 0 1]

z =1 -1 0

0 1 -1

0 0 1>> whos x y z

Name Size Bytes Class

x 1x1 8 double array

y 1x2 16 double array

z 3x3 72 double array

A scalar, a vector , or a matrix is stored asa double array with different sizes.

A polynomial is also stored as a row vector.

q(s) = s2 + 5s + 2

>> q = [1 5 2]


Variables 2 – 2Char array


>> str = ’hello’

str =

hello

>> whos str


str 1x5 10 char array

Grand total is 5 elements using 10 bytes

str is stored in a char array whose size is equal to the number of characters


Lecture 3

Help and Matlab environment

• The help command

• The lookfor command

• The help desk and link to the mathworks

• The workspace

• The save command

• The search path

• Disk file manipulation

Help and Matlab environment 3 – 1The help command

To get syntax and instructions for inv,

� help inv

INV Matrix inverse.

INV(X) is the inverse of the square

matrix X. A warning message is printed

if X is badly scaled or nearly singular.

See also SLASH, PINV, COND, CONDEST,

LSQNONNEG, LSCOV.

All functions are organized into directories, e.g., all linear algebra functions are in matfun.To list the name and description of functions in matfun,

� help matfun


Help and Matlab environment 3 – 2The help command (cont.)

Alternatively, to list the name only,

� what matfun

The command help alone lists all the directories,

� help

HELP topics:

matlab\general - General purpose commands

matlab\ops - Operators and special . . .matlab\lang - Programming language . . .. . .


Help and Matlab environment 3 – 3The help command (cont.)

The help window

• Select Help Window under the Help menu (on PCs)

• Click the question mark on the menu bar (on PCs)

• Type helpwin

Help window gives the same information as help,


Help and Matlab environment 3 – 4The lookfor command

• Search for functions based on a keyword.

• Search through the first line of help text.

• Return the first line containing a keyword.

For example,

� help inverse

inverse.m not found

One should type

� lookfor inverse


Help and Matlab environment 3 – 5The lookfor command

Results:

INVHILB Inverse Hilbert matrix.

ACOS Inverse cosine.

ASIN Inverse sine.

ATAN Inverse tangent.

ERFINV Inverse error function.

INV Matrix inverse.

PINV Pseudoinverse.

. . .

Adding the option -all,

� lookfor -all inverse

searches the entire help text (not just the first line).


Help and Matlab environment 3 – 6The help desk

The Matlab Help Desk

• Provide access to helps and references stored in your system.

• Many documents in HTML format.

• Access via internet web browser.

• Select the Help Desk under the Help menu (on PCs).

• Type helpdesk on the workspace.

Note This gives more details than help.


Help and Matlab environment 3 – 7The workspace

• The area of memory.

• Accessible from the command line.

• The commands who and whos show the current contents.

who −→ a short listwhos −→ size and storage information

For example, (variables from the Variables Lecture)

>> who

Your variables are:

A P sysss syszpkz

B Z sysssz x

C den systf y

D num systfz z

K str syszpk


Help and Matlab environment 3 – 8The workspace (cont.)

Results from whos

>> whos


A 2x2 32 double array

B 2x1 16 double array

C 1x2 16 double array

D 1x1 8 double array

K 1x1 8 double array

P 1x2 16 double array

Z 1x1 8 double array

den 1x3 24 double array

num 1x2 16 double array

str 1x5 10 char array

sysss 1x1 2824 ss object

sysssz 1x1 2824 ss object

...


Help and Matlab environment 3 – 9The workspace (cont.)

To delete all the existing variables from the workspace, enter the command clear

� clear

To delete specific variables use the commend clear followed by variables

� clear A B C D


Help and Matlab environment 3 – 10The save command

• Save the contents of the workspace as a MAT-file

• Read the MAT-file with command load

Save all contents to var_data.mat:� save var data

Save specific contents:� save var data systf sysss syszpk

Append the data to var_data.mat:� save -append var data systf sysss syszpk

Restore the saved data to workspace:� load var data


Help and Matlab environment 3 – 11The search path

To view the search path,

• Type path.

• Choose Set Path from the File menu (on PCs).

Disk file manipulation

Generic system commands: dir, type, delete, and cd.

MATLAB MS-DOS UNIX

dir dir ls

type type cat

delete del rm

cd chdir cd


Lecture 4

The command window

• The format command

• Suppressing output

• Long command lines

• Command line editing

The command window 4 – 1The format command

• specify the displayed numeric format

• does not affect computation

Example. Display x in different formats

� x = [4/3 1.2345e-6]

format short

1.3333 0.0000

format short e

1.3333e+000 1.2345e-006

format short g

1.3333 1.2345e-006

format long

1.33333333333333 0.00000123450000


The command window 4 – 2The format command (con.)

format long e

1.333333333333333e+000 1.234500000000000e-006

format long g

1.33333333333333 1.2345e-006

format bank

1.33 0.00

format rat

4/3 1/810045

format hex

3ff5555555555555 3eb4b6231abfd271

Note For proper spacing, use Fixedsys or Courier.


The command window 4 – 3Suppressing output

•Matlab automatically displays results.

• Need no display. −→ End the line with “;”.

Useful for:

– Large outputs,

– Scripts and functions.

• Multiple expressions in one line. −→ Separate each with “;”.

� a = 1; b = a+1;

• Force to display results. −→ Use “,” instead.

� a = 1, b = a+1,

a =

1

b =

2


The command window 4 – 4Long command lines

Statement does not fit in one line. −→ Use three periods, then Enter.

� s = 1 -1/2 + 1/3 ...

-1/4 + 1/5 - 1/6 ...

+ 1/7 - 1/8 + 1/9 ...

- 1/10 + 1/11 - 1/12

s =

0.6532

Note

• Blank spaces around =, +, and - are optional.

• Frequently used inside scripts and functions


The command window 4 – 5Command line editing

Suppose you mistakenly enter

� x = (1 + sqt(5))/2

Undefined function or variable ’sqt’

The error is due to the misspell of sqrt.

Solutions:

• Press the ↑ key to correct;

• Guide Matlab with a few characters, and press ↑;

• Use command history window.


The command window 4 – 6Command line editing (con.)

Command line editing keys.

↑ Ctrl + p Recall previous line

↓ Ctrl + n Recall next line

← Ctrl + b Move back one character

→ Ctrl + f Move forward one character

Ctrl + ← Ctrl + r Move right one word

Ctrl + → Ctrl + l Move left one word


The command window 4 – 7Command line editing (con.)

Home Ctrl + a Move to beginning of line

End Ctrl + e Move to end of line

Esc Ctrl + u Clear line

Del Ctrl + d Delete character at cursor

Backspace Ctrl + h Delete character before cursor

Ctrl + k Delete to end of line


Lecture 5

Elementary Mathematical Functions

The commands are listed in elfun category

>> help elfun

Elementary math functions.

Trigonometric.

sin - Sine.

sinh - Hyperbolic sine.

asin - Inverse sine.

asinh - Inverse hyperbolic sine.

cos - Cosine.

cosh - Hyperbolic cosine.

acos - Inverse cosine.

acosh - Inverse hyperbolic cosine.

tan - Tangent.

tanh - Hyperbolic tangent

...

Elementary Mathematical Functions 5 – 1Example Usages

Example 1 Check the validation of Euler formula

ejθ = cos θ + j sin θ

by using the command

exp - Exponential

sin - Sine

cos - Cosine

>> theta= pi/3;

>> exp(j*theta)

ans =

0.5000 + 0.8660i

>> cos(theta)+j*sin(theta)

ans =

0.5000 + 0.8660i


Elementary Mathematical Functions 5 – 2Example Usages (con.)

Example 2 Find the absolute value of a complex number, z = a + jb

|z| =√

a2 + b2

by using the command

abs - Absolute value

real - Complex real part

imag - Complex imaginary part

sqrt - Square root

>> z = 3+j*4 ;

>> abs(z)

ans =

5

>> sqrt(real(z)^2+imag(z)^2)

ans =

5


Lecture 6

Matrix operations

• Generating matrices

• Loading matrices

• Concatenation

• Manipulating rows and columns

• Matrix relating function

Matrix operations 6 – 1Generating matrices


� A = [1 -1 0; 0 1 -1; 0 0 1]

A =1 -1 0

0 1 -1

0 0 1

The numeric format of the command window is specified by the format command:

Several interesting functions,

• ones(n,m),• zeros(n,m),• eye(n).


Matrix operations 6 – 2Generating matrices (cont.)

Generate via M-file editor.

A = [ 1 -1 0

0 1 -1

0 0 1 ]

Save as an M-file, say mat_gen.m, and run at the command line,

� mat gen.m

A =1 -1 0

0 1 -1

0 0 1

The extension .m can be omitted.


Matrix operations 6 – 3Loading matrices

The load command

• Read binary files (MAT files)

• Read text files

Text files must be a rectangular table of numbers, separated by blanks. For example, createoutside Matlab :

16 3 2

5 10 11

8 4 7

Save as matrix.txt, and run

� load matrix.txt


Matrix operations 6 – 4Concatenation

Concatenate matrices to a bigger matrix:

� A = [B C; D]

• # of rows of B = C,

• # of columns of D = B + C.

For example, we can get A =

1 −1 00 1 −10 0 1

from

B =

[1 −10 1

], C =

[0−1

], D =

[0 0 1

].


Matrix operations 6 – 5Manipulating rows and columns

Access a row.

� X = A(3,:)

X =

0 0 1

Access a column.

� Y = A(:,2)

Y =-1

1

0

Replace a row

� A(2,:) = [1 2 3]

A =1 -1 0

1 2 3

0 0 1


Matrix operations 6 – 6Manipulating rows and columns (cont.)

Delete a row.

� A(2,:) = [ ]

A =1 -1 0

0 0 1

Obtain a part of matrix. −→ Apply vector subscript.

� X = A([1 2],[2 3])

X =-1 0

1 -1


Matrix operations 6 – 7Matrix relating functions

Basic arithmetic functions:

• plus(A,B) or A + B,

• minus(A,B) or A - B,

• mtimes(A,B) or A*B,

• A^b.

A scalar is applied to any matrix elementwisely.

Other linear algebraic functions:

• det(A),• trace(A),• A’,• inv(A).


Matrix operations 6 – 8Matrix relating functions

Examples for A =

1 −1 00 1 −10 0 1

.

� det(A)

ans =

1

� trace(A)

ans =

3

� A’

ans =1 0 0

-1 1 0

0 -1 1

� inv(A)

ans =1 1 1

0 1 1

0 0 1


Matrix operations 6 – 9Matrix relating functions (cont.)

size and length:

• size of matrices; length of vectors;

• outputs of size: Suppose B =

[1 −1 00 1 −1

].

� X = size(B)

X =

2 3

� [Y, Z] = size(B)

Y =

2

Z =

3

To get # of rows or columns, specify 1 or 2 as the second argument, respectively.

� X = size(B,1)

X =

2

� Y = size(B,2)

Y =

3


Lecture 7

Graphics in Matlab

• Plotting

• Figure windows

• Add plots to an existing figure

• Subplots

• Print graphics

Graphics in Matlab 7 – 1Plotting

Different forms of plot.

• plot(y) −→ y vs indices of elements.

• plot(x,y) −→ y vs x.

For example, to plot sin(x) from 0 to 2π:

� x = 0:pi/100:2*pi;

� y = sin(x);

� plot(x,y)

Note The ”;” is used to suppressed the output.


Graphics in Matlab 7 – 2Plotting: Axis title and label

� xlabel(’x (rad)’)

� ylabel(’Sine of x’)

� title(’Sine Function’,’FontSize’,12)

0 1 2 3 4 5 6 7−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x (rad)

Sin

e of

x

Sine Function


Graphics in Matlab 7 – 3Plotting: Multiple data sets in single graph

� y1= sin(x);

� y2 = sin(x-.25);

� y3 = sin(x-.5);

� plot(x,y1,x,y2,x,y3)

The legend command help identify each plot.

� legend(’sin(x)’,’sin(x-.25)’,’sin(x-.5)’)


Graphics in Matlab 7 – 4Plotting (cont.)

0 1 2 3 4 5 6 7−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1sin(x)sin(x−.25)sin(x−.5)


Graphics in Matlab 7 – 5Plotting: Line styles, markers, and colors

� plot(x,y,’color style marker’)

color_style_marker is a string defining a color, a line style, and a marker type:

• Colors:

’c’ cyan, ’g’ green’m’ magenta, ’b’ blue’y’ yellow, ’w’ white’r’ red, ’k’ black


Graphics in Matlab 7 – 6Plotting: Line styles and marker types

• Line styles:

’-’ solid ’:’ dotted’--’ dashed ’-.’ dash-dot

• Marker types:

’+’ plus sign, ’s’ square ’p’ pentagram’o’ o sign, ’d’ diamond ’h’ hexagram’*’ star sign, ’^’ up triangle ’>’ right triangle’x’ x sign, ’v’ down triangle ’<’ left triangle

Note again that you can edit colors, line styles, and markers interactively.


Graphics in Matlab 7 – 7Plotting examples

Example 1: Red dotted line, plus sign.

� plot(x,y,’r:+’)

Example 2: Black squares, no line.

� plot(x,y,’ks’)

Example 3: # of markers < # plotting points.

� x1 = 0:pi/100:2*pi;

� x2 = 0:pi/10:2*pi;

� plot(x1,sin(x1),’r:’,x2,sin(x2),’r+’)


Graphics in Matlab 7 – 8Plotting examples (cont.)

0 1 2 3 4 5 6 7−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1


Graphics in Matlab 7 – 9Figure windows

For any graphing functions,

• No figure window. −→ open a new one.

• One figure window. −→ use the existing one.

• Many figure windows. −→ chooses the current figure

Make figure #n a current figure.

� figure(n)

Open blank figure.

� figure


Graphics in Matlab 7 – 10Add plots to an existing figure

The command hold prepares to add a new graph to the existing axis.

� sys = tf(1,[1 2 3])

� [y1,t1] = step(sys); plot(t1,y1,’--’)

� hold on

� [y2,t2] = impulse(sys); plot(t2,y2,’:’)

� hold off

Toggle holding states:

� sys = tf(1,[1 2 3])

� [y1,t1] = step(sys); plot(t1,y1,’--’)

� hold

Current plot held

� [y2,t2] = impulse(sys); plot(t2,y2,’:’)

� hold

Current plot released


Graphics in Matlab 7 – 11Adding plots to an existing graph (cont.)

0 1 2 3 4 5 6−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


Graphics in Matlab 7 – 12Subplots

The subplot command:

• Display multiple plots in one window.

• Print multiple plots on one piece of paper.

Type subplot(m,n,p):

• Partition a window into an m× n table of subplots.

• Select the pth subplot.

Figure # runs from −→ and from ↓.


Graphics in Matlab 7 – 13Subplots (cont.)

� t = 0:pi/10:2*pi;

� [X,Y,Z] = cylinder(4*cos(t));

� subplot(2,2,1); mesh(X)

� subplot(2,2,2); mesh(Y)

� subplot(2,2,3); mesh(Z)

� subplot(2,2,4); mesh(X,Y,Z)

010

2030

0

20

40−5

0

5

010

2030

0

20

40−5

0

5

010

2030

0

20

400

0.5

1

−50

5

−5

0

50

0.5

1


Graphics in Matlab 7 – 14Saving and Exporting

Saving a figure

• Select Save from the File menu.

• The standard format is FIG.

Exporting figure

• Select Export from the File menu.

• EPS format −→ LATEX

• EMF format −→ MS Powerpoint


Lecture 8

M-Files

An M-File is used to

• Store the set of commands to be executed at once

• Create your own function

M-Files 8 – 1Script M-file

For example, save the commands from ”Graphics in Matlab” Lecture to a file namedplotsine.m

The command

>> plotsine

gives the same results as we did and all variables are shown in the workspace.


M-Files 8 – 2My function

For example, to calculate age from a given birthdate, we develop a function namedcal age.m in Matlab editor.

% CAL_AGE(MONTH,YEAR) give the age of a person by assigning month

% and year of the birthdate. MONTH and YEAR are specified in

% numeric format and your age will be returned.

%

% [yr,mo] = CAL_AGE(7,1978);

function[yr,mo] = cal_age(month,year)

today = clock; % get the current date from clock command

yr = today(1)-year- (month > today(2)) ; % year difference

mo = today(2)-month + 12*(today(2) < month); % month difference

fprintf(’Your age is %d years %d months’,yr,mo) % print the result

end


M-Files 8 – 3My function (cont.)

First, type the information of this function and comment it. This message will be displayedwhen we run help command.

>> help cal age

CAL_AGE(MONTH,YEAR) give the age of a person by assigning month

and year of the birthdate. MONTH and YEAR are specified in numeric

format and your age will be returned.

[yr,mo] = CAL_AGE(7,1978);

Second, define input arguments and returned values. Terminate the function by using end

function[yr,mo] = cal age(month,year)

...

end


M-Files 8 – 4My function (cont.)

Third, insert your codes and save this file with the extension .m

Your function will be executed by using the filename as a command

>> [yr,mo] = CAL_AGE(7,1978)

Your age is 26 years 10 months

yr =

26

mo =

10

Note:

• Variables defined in a function are local. They will not be shown in the workspace.

• The function can be run only if you are in the same path as the file is kept.


Lecture 9

Control Systems Toolbox

• TF, SS, ZPK object

• Frequently Used Commands

• Classical Design

• Time-domain analysis

• Frequency-domain analysis

• Simulink

Control Systems Toolbox 9 – 1TF Object

A tf object is a continuous-time or a discrete-time transfer function created by tf command.

Define the numerator and denominator of a transfer function.>> num = [1 2]; den = [1 2 1] ;

num and den which are stored as double arrays represent the coefficients of the numeratorand denominator, respectively.

Create a continuous-time transfer function.

>> systf = tf(num,den)

Transfer function:

s + 2

-------------

s^2 + 2 s + 1

Create a discrete-time transfer function andspecify the sampling time (1 sec).

>> systfz = tf(num,den,1)

Transfer function:

z + 2

-------------

z^2 + 2 z + 1

Sampling time: 1


Control Systems Toolbox 9 – 2SS Object

An SS Object is a continuous-time or a discrete-time state-space model.

Define state-space system matrices.>> A = [-0.5 0; 0 -0.2] ; B = [1; 0] ; C =[1 0 ] ; D = 0 ;

Create a continuous-time state-space model.

>> sysss = ss(A,B,C,D)

a = b =

x1 x2 u1

x1 -0.5 0 x1 1

x2 0 -0.2 x2 0

c = d =

x1 x2 u1

y1 1 0 y1 0

Continuous-time model.


Control Systems Toolbox 9 – 3SS Object (cont.)

Create a discrete-time state-space model and specify the sampling time (1 sec).

>> sysssz = ss(A,B,C,D,1)

a = b =

x1 x2 u1

x1 -0.5 0 x1 1

x2 0 -0.2 x2 0

c = d =

x1 x2 u1

y1 1 0 y1 0

Sampling time: 1

Discrete-time model.


Control Systems Toolbox 9 – 4ZPK Object

A ZPK Object is a zero-pole-gain model.

Define zeros (Z), poles (P), and gain (K) of a transfer function.>> Z = -0.5 ; P = [-0.7 -1] ; K = 1 ;

Create a continuous-time ZPK model.

>> syszpk = zpk(Z,P,K)

Zero/pole/gain:

(s+0.5)

-------------

(s+0.7) (s+1)

Create a discrete-time ZPK model with thesampling time = 1 sec.

>> syszpkz = zpk(Z,P,K,1)

Zero/pole/gain:

(z+0.5)

-------------

(z+0.7) (z+1)

Sampling time: 1


Control Systems Toolbox 9 – 5Frequently Used Commands

• Three combinations of conversions among TF, SS, ZPK.

• Conversion between continuous and discrete models.

• System poles, zeros, and eigenvalues.


Control Systems Toolbox 9 – 6Conversion among TF, SS, ZPK

TF : Transfer FunctionSS : State Space ModelZPK : Zeros, Poles, and Gain

• ss2tf• ss2zp• tf2ss

• tf2zp• zp2tf• zp2ss

For example, convert G(s) =(s + 1)

(s2 + 4s + 5)to the state-space model.

>> num = [1 1]; den = [1 4 5];

>> [a,b,c,d] = tf2ss(num,den)

a = b =

-4 -5 1

1 0 0

c = d =

1 1 0


Control Systems Toolbox 9 – 7Conversion betweenContinuous-Time and Discrete-Time Models

• c2d : Conversion of continuous-time models to discrete time.

• d2c : Conversion of discrete-time models to continuous time.

For example, convert G(s) = (s+1)(s2+4s+5)

to a discrete model by assuming Zero-Order hold on

the inputs with the sampling time = 0.1 sec.

>> systf = tf(num,den)

Transfer function:

s + 1

-------------

s^2 + 4 s + 5

Create a TF object

>> sysd = c2d(systf,0.1,’zoh’)

Transfer function:

0.08611 z - 0.07791

----------------------

z^2 - 1.629 z + 0.6703

Sampling time: 0.1

Convert to the discrete-time model.


Control Systems Toolbox 9 – 8Poles, Zeros, and Eigenvalues

According to systf created in the previous slide, we calculate poles, zeros and eigenvalues ofthe system by the following command.

>> pole(systf)

ans =

-2.0000 + 1.0000i

-2.0000 - 1.0000i

>> zero(systf)

ans =

-1

>> eig(systf)

ans =

-2.0000 + 1.0000i

-2.0000 - 1.0000i

• The input arguments of pole and zero are LTI models (SS or TF object)

• The input argument of eig could be:

1. An LTI model (SS or TF object)

2. A square matrix


Control Systems Toolbox 9 – 9Classical Design

• rlocus computes and plots the root locus of a single-input, single-output LTI model

• sisotool A Graphical User Interface that allows you to design single-input/single-output(SISO) compensators.

−10 −8 −6 −4 −2 0 2 4−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Root Locus

Real Axis

Imag

inar

y A

xis

rlocus sisotool


Control Systems Toolbox 9 – 10Root Loci

In plotting root loci with MATLAB, we deal with the system equation in the form of

1 + KG(s) = 0

For example, let G(s) =(s + 1)

(s2 + 4s + 5).

>> num=[1 1];den = [1 4 5];

>> [a,b,c,d]=tf2ss(num,den);

>> systf = tf(num,den);

>> sysss=ss(a,b,c,d);

Create LTI models in various types

>> rlocus(num,den)

>> rlocus(systf)

>> rlocus(a,b,c,d)

>> rlocus(sysss)

The command can be used with all modeltypes.


Control Systems Toolbox 9 – 11Root Loci (cont.)

−4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0−1.5

−1

−0.5

0

0.5

1

1.5

Root Locus

Real Axis

Imag

inar

y A

xis

−4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0−1.5

−1

−0.5

0

0.5

1

1.5

System: systf2 Gain: 1.66 Pole: −3.99 Damping: 1 Overshoot (%): 0 Frequency (rad/sec): 3.99

Root Locus

Real Axis

Imag

inar

y A

xis

• Clicking at a point on root locus returns the closed-loop poles, feedback gain (K), andother parameters.

• The user is able to design the gain (K) by dragging mouse to the appropriate pointcorresponding to the desired closed-loop pole and time response specification.


Control Systems Toolbox 9 – 12SISOTOOL

This GUI lets you design single-input/single-output (SISO) compensators by interacting withthe root locus, Bode, and Nichols plots of the open-loop system.

1. Current parameters of the Compensator

2. Configuration of Feedback System

3. Root Locus

4. Open-Loop Bode Plot


Control Systems Toolbox 9 – 13SISOTOOL (cont.)

The structure of the compensator can becustomized.

Step response, Closed-Loop Bode plot,and other loop responses are provided inanalysis menu.


Control Systems Toolbox 9 – 14Time-domain analysis

• step Step response

• impulse Impulse response

• initial Response of state-space system with given initial conditions.


Control Systems Toolbox 9 – 15Step response

The step response of G(s) from t = 0 to t = 5 can be obtained by:

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Step Response

Time (sec)

Am

plitu

de

>> step(systf,5)


Control Systems Toolbox 9 – 16Impulse response

The impulse response of G(s) from t = 0 to t = 5 can be obtained by:

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Impulse Response

Time (sec)

Am

plitu

de

>> impulse(systf,5)


Control Systems Toolbox 9 – 17Initial response

The inital response of G(s) from t = 0 to t = 5 with a given initial condition (x0 = [−1 1])can be obtained by:

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Response to Initial Conditions

Time (sec)

Am

plitu

de

>> inital(sysss,[-1 1],5)

Note: The first argument of command initial must be an SS object or state-space model.


Control Systems Toolbox 9 – 18Frequency-domain analysis

• bode Bode diagrams of the frequency response

• nyquist Nyquist plot

• freqresp Frequency response over a frequency grid

• ltiview Response analysis GUI (LTI Viewer)


Control Systems Toolbox 9 – 19Bode

Draw a Bode plot of open-loop system G(s).

−40

−35

−30

−25

−20

−15

−10

Mag

nitu

de (

dB)

10−1

100

101

102

−90

−45

0

45

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

>> bode(systf)

The following command return the responsemagnitudes and phases in degrees along withthe frequency data.

>> [MAG,PHASE,W] = bode(systf);

>> MAG

MAG(:,:,1) =

0.2008

>> PHASE

PHASE(:,:,1) =

1.1275

Note: The argument of bode is any kind of LTI models.


Control Systems Toolbox 9 – 20Nyquist

The command nyquist draws the Nyquist plot of G(s).

>> nyquist(systf)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Nyquist Diagram

Real Axis

Imag

inar

y A

xis

Note: The argument of nyquist is any kind of LTI models.


Control Systems Toolbox 9 – 21Frequency Response

The frequency response of G(s)

G(s) =s + 1

s2 + 4s + 5

s=jw⇐⇒ G(jw) =(jw + 1)

(−w2 + 4jw + 5)

can be computed by :

1. Specifying a vector of frequency grid

>> w = logspace(-2,3,500)

0.0100

0.0102

...

2. Generating the command

>> Gjw = freqresp(systf,w)

Gjw(:,:,1) =

0.2000 + 0.0004i

Gjw(:,:,2) =

0.2000 + 0.0004i

...

• The commands mag(Gjw) andphase(Gjw) give the same resultsas that obtained from bode command.


Control Systems Toolbox 9 – 22Response analysis GUI: LTI Viewer

The LTI Viewer is an interactive graphical user interface (GUI) for analyzing the time andfrequency responses of linear systems and comparing many systems in the same time.A blank viewer is run by

>> ltiview


Control Systems Toolbox 9 – 23Response analysis GUI: LTI Viewer (cont.)

1. Import the data to the viewer

Choose the variable systf available in the workspace.


Control Systems Toolbox 9 – 24Response analysis GUI: LTI Viewer (cont.)

2. Select the option of graphs to be displayed. For example, step and bode response will beshown in the same window.


Control Systems Toolbox 9 – 25Simulink

Simulink is a tool for creating a diagramof a particular system and simulating itstime response.

See an example in the class.


Control Systems Toolbox 9 – 26References

[1] Matlab Help [Computer Software], The Mathworks Inc., version 6.5.0.180913a,Release 13, 2002.

[2] W. Khaisongkram, MATLAB Tutorial, CSRL, Dept. of Electrical Engineering,Chulalongkorn University, July 2004.


Matlab Tutorial June15 2005

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Transcript of Matlab Tutorial June15 2005