Dasar control system dengan matlab

Bab 1 Pendahuluan

MATLAB berasal dari kata Matrix Laboratory. Versi pertamanya ditulis di University of New

Mexico dan Stanford University pada akhir tahun 70-an, dimaksudkan untuk memecahkan

permasalahan di teori matriks, aljabar linier, dan analisa numerik. Sebelumnya telah dikenal

program-program LINPACK DAN EISPACK, yang dikemas dalam FORTRAN untuk

perhitungan matriks.

MATLAB yang saat ini digunakan kemampuannya melampaui Matrix Laboratory yang

sebenarnya. MATLAB adalah bahasa pemrograman tingkat tinggi untuk perhitungan

teknik. Isinya mencakup perhitungan, visualisasi, dan pemrograman dalam lingkungan yang

mudah digunakan dimana masalah dan solusinya ditampilkan dalam notasi matematika yang

sudah dikenal. Fungsi-fungsinya meliputi :

Matematika dan komputasi

Pengembangan Algoritma

Modelling, simulasi, dan pembuatan proto-type

Analisa data, eksplorasi, dan visualisasi

Ilmu pengetahuan dan grafik teknik,

Pengembangan aplikasi, termasuk pembuatan antarmuka pengguna secara

grafis.

MATLAB terus berkembang dari tahun ke tahun dengan berbagai masukan dari para

pengguna MATLAB. Versi Matlab meliputi hal-hal yang terkait di setiap bagian MATLAB

dari deret matematika dan konstruksi bahasa pemrograman, sampai stuktur data dan

metoda-metoda orientasi objek, untuk visualisasi dan perlengkapan pembuatan GUI

MATLAB adalah suatu lingkungan dan bahasa pemrograman , dan satu dari kelebihannya

ada kenyataan bahwa bahasa MATLAB memungkinan kita untuk membuat peralatan yang

bisa dipakai ulang. Fungsi-fungsi dan program-program khusus dapat ditulis secara mudah

di dalam kode MATLAB. Sebagaimana fungsi-fungsi MATLAB terus ditulis untuk

memecahkan masalah-masalah tertentu, tentu diperlukan meng-kelompokan fungsi-fungsi

yang berhubungan menjadi satu direktori. Hal ini adalah konsep dasar dari Toolbox:

kumpulan khusus dari M-file untuk bekerja pada masalah-masalah tertentu.

Materi Pendukung Kuliah : Sistim Pengaturan (Perhitungan, Pemrograman dan Visualisasi dengan MATLAB)

2

Penggunaan M-files dalam Matlab

Ada perbedaan penggunaan Matlab di dalam platform yang berbeda :

Macintosh

Terdapat sebuah built-in editor untuk m-files; pilih "New M-file" dari menu File. Editor

lainnya juga dapat digunakan (tapi pastikan ahwa file telah disave dalam format text, dan

dapat diload bila akan mulai menggunakan Matlab).

Windows

Menggunakan Matlab dalam Windows serupa dengan cara menggunakannya di dalam

Macintosh. Akan tetapi, perlu diketahui bahwa m-file akan disimpan di dalam clipboard.

Karenanya, perlu dipastikan bahwa m-file telah disimpan sebagai namafile.m .

Unix

Di dalam unix, editor dijalankan secara terpisah dari Matlab. Cara terbaik adalah dengan

membuat sebuah direktori untuk semua m-files, kemudian cd ke direktori yang dimaksud

sebelum menjalankan Matlab maupun editor. Untuk mulai menggunakan Matlab dari

window Xterm hanya perlu dengan mengetikkan : matlab.

Perintah juga dapat diketikkan langsung ke dalam matlab, atau taruh semua dari perintah

yang diperlukan untuk bersama-sama di dalam sebuah an m-file, dan hanya jalankan file.

Bila semua m-files diletakkan dalam directory yang sama dengan direktori tempat matlab

dijalankan, maka matlab akan selalu menjalankannya.

Menu help dalam Matlab

Matlab memberikan menu help dengan cukup mengetikkan :

help commandname


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Pengertian m-file

Sebuah m-file atau file script, adalah file teks yang sederhana yang berisi perintah Matlab. Bila file

ini dijalankan, Matlab membaca command ini dan mengeksekusinya sebagaimana bila setiap

command diketik langsung pada prompt Matlab. Semua m-file memiliki ekstensi “. m”. Bila sebuah

m-file baru dibuat dengan nama yang sama dengan yang telah ada di m-file yang sudah ada, Matlab

akan memilih m-file yang lebih dahulu berada di urutan path. Untuk itu, buatlah m-file dengan

nama yang belum ada di default Matlab, bisa dicoba dengan diberi nama dalam bahasa Indonesia.

Untuk mengetahui file mana yang sudah ada, ketik help filename pada prompt Matlab

Alasan Menggunakan m-file

Untuk permasalahan yang sederhana, mengetikkan perintah langsung pada prompt Matlab adalah

hal yang mudah, akan tetapi, jumlah baris dari command makin lama makin banyak, dan bila sedang

melakukan suatu percobaan, maka bila harus mengetikkannya berkali-kali adalah hal yang sia-sia.

Dalam hal ini, m-file dapat membantu, bahkan penting untuk menghindar dari masalah ini.

Cara Menjalankan m-file

Setela m-file disimpan dengan nama *.m pada folder Matlab, atau direktori, file ini dapat dieksekusi

hanya dengan mengetikkan nama file pada prompt Matlab

Cara Membuat m-file

Untuk membuat M-file pada PC, pilih New dari File menu dan pilih M-file. Prosedur ini akan

menampilkan text editor window dimana perintah-perintah MATLAB dapat diberikan.. Tampilan

berikut menunjukkan perintah yang dikirimkan melalui file skrip.

% Contoh.m file skrip untuk masalah acid water bath initial_con=90 min_con=50 lost=0.01; n=floor(log2(initial_con/min_con)/log2(1+lost))


4

File ini disimpan sebagai M-file contoh.m pada disket dengan memilih Save dari menu File,

kemudian MATLAB akan menjalankan perintah dari contoh.m pada saat contoh diketikkan pada

prompt MATLAB:

Dikarenakan perlengkapan dari file skrip, MATLAB melengkapi beberapa fungsi yang berguna bila

digunakan di dalam M-file, yaitu :

Perintah Arti

disp(ans) menampilkan hasil tanpa mengindentifikasi nama

variabel

echo mengendalikan command window mengulang

perintah-perintah pada file skrip

input prompt buatan untuk masukan

keyboard memberikan kendali pada keyboard secara temporary

(ketik return untuk keluar)

pause berhenti sampai sembarang tombol keyboard ditekan

pause(n) berhenti untuk n detik

waitforbuttonp

ress

berhenti sampai pemakai menekan tombol mouse atau

key keyboard

Contoh :

% Contoh.m file skrip untuk masalah acid water bath initial_con=90 min_con=50 lost=input(‘Masukkan persentasi kehilangan dengan setiap dip >’)/100n=floor(log2(initial_con/min_con)/log2(1+lost))

EDU>>contoh

initial_con=

90

min_con=

50

n =

59


5

Hasil file skrip :

Contoh :

Hasil :

echo on % Contoh.m file skrip untuk masalah acid water bath initial_con=90 min_con=50 lost=input(‘Masukkan persentasi kehilangan dengan setiap dip >’)/100n=floor(log2(initial_con/min_con)/log2(1+lost)) echo off

EDU>> contoh

% Contoh.m file skrip untuk masalah acid water bath

initial_con=90

initial_con =

90

min_con=50

min_con =

50

lost=input(‘Masukkan persentasi kehilangan dengan setiap

dip >’)/100

Masukkan persentasi kehilangan dengan setiap dip > 20

lost =

0.2

n=floor(log2(initial_con/min_con)/log2(1+lost))

n =

3

echo off

Masukkan persentasi kehilangan dengan setiap dip > 5

lost =

0.05

n =

12


6

Tutorial Dasar Matlab

Matlab adalah program interaktif untuk komputasi numeric dan visualisasi data yang umum

digunakan oleh ilmuwan kendali untuk analisa dan perancangan. Ada berbagai toolbox yang

berbeda yang dapat digunakan untuk berbagai area aplikasi yang berbeda. Untuk diktat ini, hanya

penggunaan pada bidang sistem kendali yang akan dibahas.

Vektor

Sebagai permulaan, matlab akan digunakan untuk suatu permasalahan yang sederhana seperti

vector. Setiap elemen dari vector akan disusun dengan dipisahkan oleh space di dalam tanda

kurung. Sebagai contoh, dapat diketik seperti berikut ini pada command window dari program

matlab :

Matlab akan kembali menuliskan :

Untuk menyusun deret bilangan dari 0 sampai dengan 20 dengan jarak 2 maka diketikkan :

Untuk memanipulasi vector dengan menambahkan dua kepada setiap elemen dari vector a, maka

persamaannya adalah :

a =

1 2 3 4 5 6 9 8 7

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

t = 0:2:20

t =

0 2 4 6 8 10 12 14 16 18 20

b = a + 2

b =

3 4 5 6 7 8 11 10 9


7

Bila kedua vector, yaitu a dan b akan ditambahkan,

dan seterusnya

c = a + b

c =

4 6 8 10 12 14 20 18 16


8

Fungsi-Fungsi Matlab

Matlab telah menyediakan beberapa fungsi standar. Setiap fungsi adalah satu blok kode yang

mengerjakan tugas tertentu. Pada matlab telah meliputi fungsi-fungsi standar yaitu : sin, cos, log,

exp, sqrt, dan sebagainya. Biasanya, konstanta seperti pi, and i or j untuk akar dari -1, juga telah

termasuk dalam Matlab.

Untuk mengetahui fungsi dari fungsi-fungsi yang ada, ketikkan :

help [function name] pada Matlab command window.

Saat menuliskan perintah seperti roots, plot, atau step ke dalam Matlab, sebenarnya yang sedang

dilakukan adalah menjalankan sebuah m-file dengan masukan dan keluaran yang telah ditulis untuk

memenuhi tugas tertentu. Jenis m-file ini serupa dengan subrutin dalam bahasa pemrograman yang

memiliki masukkan (parameter yang telah dimasukkan ke dalam m-file), keluaran (nilai yang

diperoleh dari m-file), dan satu badan perintah-perintah yang berisi variabel local. Matlab menyebu

m-file ini sebagai function (fungsi). Untuk itu, sebuah perintah function dapat ditulis baru, sesuai

tujuan, artinya, sebuah toolbox baru dapat disusun.

Fungsi baru adalah sebuah m-file dan memiliki ekstensi file “. m”. File ini disimpan pada direktori

yang sama dengan software Matlab, atau di dalam direktori yang berisi path yang diacu oleh Matlab.

Baris pertama sebuah function haruslah berikut ini :

function [output1,output2] = filename(input1,input2,input3)

Sebuah fungsi dapat menerima input dan mengeluarkan output dengan jumlah variabel sebanyak

yang diperlukan. Baris yang merupakan sebuah keterangan untuk memudahkan pembacaan

program biasa diberikan tanda “%” pada awal tiap baris.

sin(pi/4)

ans =

0.7071


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Fungsi dapat ditulis dengan trik, dan praktis diperlukan untuk keberhasilan menuliskan sebuah

fungsi untuk mendapatkan hasil yang diperlukan. Contoh berikut adalah m-file yaitu add.m yang

berisi :

Bila ketiga baris perintah ini diberi nama add.m pada direktori Matlab, maka pada file ini dapat

dieksekusi pada command line :

Polynomials

Pada Matlab, sebuah polynomial ditampilkan dengan sebuah vector. Untuk menuliskan sebuah

polynomial di dalam matlab dapat dilihat contoh berikut ini :

Persamaan :

Dalam Matlab ditulis :

Matlab dapat mengartikan sebuah vector sepanjang n+1 sebagai deret polynomial sepanjang n.

Untuk itu, dapat dilakukan dengan :

dipresentasikan dalam Matlab sebagai :

y = [1 0 0 0 1]

function [var3] = add(var1,var2)

%add is a function that adds two numbers

var3 = var1+var2;

y = add(3,8)

x = [1 3 -15 -2 9]

x =

1 3 -15 -2 9


10

Pada polynomial dengan s=2,

z = polyval([1 0 0 0 1],2)

z =

17

Untuk mencari akar persamaan dari :

adalah dengan menuliskan :

Untuk mengkalikan dua polynomial :

Contoh :

Untuk membagi dua polynomial :

roots([1 3 -15 -2 9])

ans =

-5.5745

2.5836

-0.7951

0.7860

x = [1 2];

y = [1 4 8];

z = conv(x,y)

z =

1 6 16 16

[xx, R] = deconv(z,y)

xx =

1 2

R =

0 0 0 0


11

Sebagaimana dapat dilihat, bila untuk menjumlahkan dua polynomial, dengan panjang yang sama,

dapat digunakan perintah polyadd, yang sudah dijadikan fungsi di dalam mfile :

Matrik

Entering matrices into Matlab is the same as entering a vector, except each row of elements is

separated by a semicolon (;) or a return:

B = [1 2 3 4;5 6 7 8;9 10 11 12]

B =

1 2 3 4

5 6 7 8

9 10 11 12

B = [ 1 2 3 4

5 6 7 8

9 10 11 12]

B =

1 2 3 4

5 6 7 8

9 10 11 12

Matrices in Matlab can be manipulated in many ways. For one, you can find the

transpose of a matrix using the apostrophe key:

C = B'

z = polyadd(x,y)

x =

1 2

y =

1 4 8


12

C =

1 5 9

2 6 10

3 7 11

4 8 12

It should be noted that if C had been complex, the apostrophe would have actually

given the complex conjugate transpose. To get the transpose, use .' (the two

commands are the same if the matix is not complex).

Now you can multiply the two matrices B and C together. Remember that order

matters when multiplying matrices.

D = B * C

D =

30 70 110

70 174 278

110 278 446

D = C * B

D =

107 122 137 152

122 140 158 176

137 158 179 200

152 176 200 224

Another option for matrix manipulation is that you can multiply the corresponding

elements of two matrices using the .* operator (the matrices must be the same size

to do this).

E = [1 2;3 4]

F = [2 3;4 5]

G = E .* F

E =

1 2

3 4


13

F =

2 3

4 5

G =

2 6

12 20

If you have a square matrix, like E, you can also multiply it by itself as many

times as you like by raising it to a given power.

E^3

ans =

37 54

81 118

If wanted to cube each element in the matrix, just use the element-by-element

cubing.

E.^3

ans =

1 8

27 64

You can also find the inverse of a matrix:

X = inv(E)

X =

-2.0000 1.0000

1.5000 -0.5000

or its eigenvalues:


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eig(E)

ans =

-0.3723

5.3723

There is even a function to find the coefficients of the characteristic polynomial of a

matrix. The "poly" function creates a vector that includes the coefficients of the

characteristic polynomial.

p = poly(E)

p =

1.0000 -5.0000 -2.0000

Remember that the eigenvalues of a matrix are the same as the roots of its

characteristic polynomial:

roots(p)

ans =

5.3723

-0.3723


15

Plotting in Matlab

Plotting

Menggambarkan plotting pada matlab dapat dilakukan dengan mudah. Bila sebuah gelombang sinus

akan diplot sesuai fungsi waktu, maka pertama-tama akan dibuat vector waktu (dengan memberikan

tanda titik koma, sebagai tanda bahwa tidak semua nilai ditampilkan), maka matlab akan

menggambarkan keluarannya.

Contoh :

One of the most important functions in Matlab is the plot function. Plot also happens

to be one of the easiest functions to learn how to use. The basic format of the

function is to enter the following command in the Matlab command window or into a

m-file.

plot(x,y)

t=0:0.25:7;

y = sin(t);

plot(t,y)


16

This command will plot the elements of vector x on the horizontal axis of a figure,

and the elements of the vector y on the vertical axis of the figure. The default is that

each time the plot command is issued, the current figure will be erased; we will

discuss how to override this below. If we wanted to plot the simple, linear formula:

y=3x

We could create a m-file with the following lines of code:

x = 0:0.1:100;

y = 3*x;

plot(x,y)

which will generate the following plot,

One thing to keep in mind when using the plot command is that the vectors x and y

must be the same length. The other dimension can vary. Matlab can plot a 1 x n

vector versus a n x 1 vector, or a 1 x n vector versus a 2 x n matrix (you will get two

lines), as long as n is the same for both vectors.

The plot command can also be used with just one input vector. In that case the

vector columns are plotted versus their indices (the vector 1:1:n will be used for the

horizontal axis). If the input vector contains complex numbers, Matlab plots the real

part of each element (on the x-axis) versus the imaginary part (on the y-axis).


17

Plot aesthetics

The color and point marker can be changed on a plot by adding a third parameter (in

single quotes) to the plot command. For example, to plot the above function as a

red, dotted line, the m-file should be changed to:

x = 0:0.1:100;

y = 3*x;

plot(x,y,'r:')

The plot now looks like:

The third input consists of one to three characters which specify a color and/or a

point marker type. The list of colors and point markers is as follows:

y yellow . point

m magenta o circle

c cyan x x-mark

r red + plus

g green - solid

b blue * star

w white : dotted

k black -. dashdot

-- dashed

You can plot more than one function on the same figure. Let's say you want to plot a

sine wave and cosine wave on the same set of axes, using a different color and point

marker for each. The following m-file could be used to do this:


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x = linspace(0,2*pi,50);

y = sin(x);

z = cos(x);

plot(x,y,'r', x,z,'gx')

You will get the following plot of a sine wave and cosine wave, with the sine wave in a

solid red line and the cosine wave in a green line made up of x's:

By adding more sets of parameters to plot, you can plot as many different functions on

the same figure as you want. When plotting many things on the same graph it is useful

to differentiate the different functions based on color and point marker. This same effect

can also be achieved using the hold on and hold off commands. The same plot

shown above could be generated using the following m-file:


y = sin(x);

plot(x,y,'r')

z = cos(x);

hold on

plot(x,z,'gx')

hold off

Always remember that if you use the hold on command, all plots from then on will be

generated on one set of axes, without erasing the previous plot, until the hold off

command is issued.


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Subplotting

More than one plot can be put on the same figure using the subplot command. The

subplot command allows you to separate the figure into as many plots as desired, and

put them all in one figure. To use this command, the following line of code is entered

into the Matlab command window or an m-file:

subplot(m,n,p)

This command splits the figure into a matrix of m rows and n columns, thereby creating

m*n plots on one figure. The p'th plot is selected as the currently active plot. For

instance, suppose you want to see a sine wave, cosine wave, and tangent wave plotted

on the same figure, but not on the same axis. The following m-file will accomplish this:


y = sin(x);

z = cos(x);

w = tan(x);

subplot(2,2,1)

plot(x,y)

subplot(2,2,2)

plot(x,z)

subplot(2,2,3)

plot(x,w)

As you can see, there are only three plots, even though I created a 2 x 2 matrix of 4

subplots. I did this to show that you do not have to fill all of the subplots you have


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created, but Matlab will leave a spot for every position in the matrix. I could have easily

made another plot using the line subplot(2,2,4) command. The subplots are arranged

in the same manner as you would read a book. The first subplot is in the top left corner,

the next is to its right. When all the columns in that row are filled, the left-most column

on the next row down is filled (all of this assuming you fill your subplots in order i.e. 1,

2, 3,..).

One thing to note about the subplot command is that every plot command issued later

will place the plot in whichever subplot position was last used, erasing the plot that was

previously in it. For example, in the m-file above, if a plot command was issued later in

the m-file, it would be plotted in the third position in the subplot, erasing the tangent

plot. To solve this problem, the figure should be cleared (using clf), or a new figure

should be specified (using figure).

Changing the axis

Now that you have found different ways to plot functions, you can customize your plots

to meet your needs. The most important way to do this is with the axis command. The

axis command changes the axis of the plot shown, so only the part of the axis that is

desirable is displayed. The axis command is used by entering the following command

right after the plot command (or any command that has a plot as an output):

axis([xmin, xmax, ymin, ymax])

For instance, suppose want to look at a plot of the function y=exp(5t)-1. If you enter the

following into Matlab t=0:0.01:5;

y=exp(5*t)-1;

plot(t,y)


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you should have the following plot:

As you can see, the plot goes to infinity. Looking at the y-axis (scale: 8e10), it is

apparent that not much can be seen from this plot. To get a better idea of what is going

on in this plot, let's look at the first second of this function. Enter the following

command into the Matlab command window.

axis([0, 1, 0, 50])

and you should get the following plot:

Now this plot is much more useful. You can see more clearly what is going on as the

function moves toward infinity. You can customize the axis to your needs. When using

the subplot command, the axis can be changed for each subplot by issuing an axis

command before the next subplot command. There are more uses of the axis command

which you can see if you type help axis in the Matlab command window.


22

Adding text

Another thing that may be important for your plots is labeling. You can give your plot a

title (with the title command), x-axis label (with the xlabel command), y-axis label

(with the ylabel command), and put text on the actual plot. All of the above commands

are issued after the actual plot command has been issued.

A title will be placed, centered, above the plot with the command: title('title

string'). The x-axis label is issued with the following command: xlabel('x-axis

string'). The y-axis label is issued with the following command: ylabel('y-axis

string').

Furthermore, text can be put on the plot itself in one of two ways: the text command

and the gtext command. The first command involves knowing the coordinates of

where you want the text string. The command is text(xcor,ycor,'textstring'). To

use the other command, you do not need to know the exact coordinates. The command

is gtext('textstring'), and then you just move the cross-hair to the desired location

with the mouse, and click on the position you want the text placed.

To further demonstrate labeling, take the step response plot from above. Assuming that

you have already changed the axis, copying the following lines of text after the axis

command will put all the labels on the plot:

title('step response of something')

xlabel('time (sec)')

ylabel('position, velocity, or something like that')

gtext('unnecessary labeling')

The text "unnecessary labeling" was placed right above the position, I clicked on. The

plot should look like the following:


23

Other commands that can be used with the plot command are:

• clf (clears the current plot, so it is blank)

• figure (opens a new figure to plot on, so the previous figure is saved)

• close (closes the current figure window)

• loglog (same as plot, except both axes are log base 10 scale)

• semilogx (same as plot, except x-axis is log base 10 scale)

• semilogy (same as plot, except y-axis is log base 10 scale)

• grid (adds grid line to your plot)

Of course this is not a complete account of plotting with Matlab, but it should give you

a nice start

Printing

Printing in Matlab is pretty easy. Just follow the steps illustrated below:

Macintosh

To print a plot or a m-file from a Macintosh, just click on the plot or m-file, select

Print under the File menu, and hit return.

Windows


24

To print a plot or a m-file from a computer running Windows, just selct Print from

the File menu in the window of the plot or m-file, and hit return.

Unix

To print a plot on a Unix workstation enter the command:

print -P<printername>

If you want to save the plot and print it later, enter the command:

print plot.ps

Sometime later, you could print the plot using the command "lpr -P plot.ps" If

you are using a HP workstation to print, you would instead use the command

"lpr -d plot.ps"

To print a m-file, just print it the way you would any other file, using the

command "lpr -P <name of m-file>.m" If you are using a HP workstation to

print, you would instead use the command "lpr -d plot.ps<name of m-file>.m"


25

Matlab Commands List

The following list of commands can be very useful for future reference. Use "help" in

Matlab for more information on how to use the commands.

In these tutorials, we use commands both from Matlab and from the Control Systems

Toolbox, as well as some commands/functions which we wrote ourselves. For those

commands/functions which are not standard in Matlab, we give links to their descriptions.

For more information on writing Matlab functions, see the function page.

Note:Matlab commands from the control system toolbox are highlighted in red.

Non-standard Matlab commands are highlighted in green.

Command Description

abs Absolute value

acker Compute the K matrix to place the poles of A-BK, see also place

axis Set the scale of the current plot, see also plot, figure

bode Draw the Bode plot, see also logspace, margin, nyquist1

c2dm Continuous system to discrete system

clf Clear figure (use clg in Matlab 3.5)

conv Convolution (useful for multiplying polynomials), see also deconv

ctrb The controllability matrix, see also obsv

deconv Deconvolution and polynomial division, see also conv

det Find the determinant of a matrix

dimpulse Impulse response of discrete-time linear systems, see also dstep

dlqr Linear-quadratic requlator design for discrete-time systems, see also

lqr

dlsim Simulation of discrete-time linear systems, see also lsim

dstep Step response of discrete-time linear systems, see also stairs

eig Compute the eigenvalues of a matrix

eps Matlab's numerical tolerance

feedback Feedback connection of two systems.


26

figure Create a new figure or redefine the current figure, see also subplot,

axis

for For, next loop

format Number format (significant digits, exponents)

function Creates function m-files

grid Draw the grid lines on the current plot

gtext Add a piece of text to the current plot, see also text

help HELP!

hold Hold the current graph, see also figure

if Conditionally execute statements

imag Returns the imaginary part of a complex number, see also real

impulse Impulse response of continuous-time linear systems, see also step,

lsim, dlsim

input Prompt for user input

inv Find the inverse of a matrix

jgrid Generate grid lines of constant damping ratio (zeta) and settling time

(sigma), see also sgrid, sigrid, zgrid

legend Graph legend

length Length of a vector, see also size

linspace Returns a linearly spaced vector

lnyquist1 Produce a Nyquist plot on a logarithmic scale, see also nyquist1

log natural logarithm, also log10: common logarithm

loglog Plot using log-log scale, also semilogx/semilogy

logspace Returns a logarithmically spaced vector

lqr Linear quadratic regulator design for continuous systems, see also

dlqr

lsim Simulate a linear system, see also step, impulse, dlsim.

margin Returns the gain margin, phase margin, and crossover frequencies,

see also bode

norm Norm of a vector

nyquist1 Draw the Nyquist plot, see also lnyquist1. Note this command was


27

written to replace the Matlab standard command nyquist to get more

accurate Nyquist plots.

obsv The observability matrix, see also ctrb

ones Returns a vector or matrix of ones, see also zeros

place Compute the K matrix to place the poles of A-BK, see also acker

plot Draw a plot, see also figure, axis, subplot.

poly Returns the characteristic polynomial

polyadd Add two different polynomials

polyval Polynomial evaluation

print Print the current plot (to a printer or postscript file)

pzmap Pole-zero map of linear systems

rank Find the number of linearly independent rows or columns of a

matrix

real Returns the real part of a complex number, see also imag

rlocfind Find the value of k and the poles at the selected point

rlocus Draw the root locus

roots Find the roots of a polynomial

rscale Find the scale factor for a full-state feedback system

set Set(gca,'Xtick',xticks,'Ytick',yticks) to control the number and

spacing of tick marks on the axes

series Series interconnection of Linear time-independent systems

sgrid Generate grid lines of constant damping ratio (zeta) and natural

frequency (Wn), see also jgrid, sigrid, zgrid

sigrid Generate grid lines of constant settling time (sigma), see also jgrid,

sgrid, zgrid

size Gives the dimension of a vector or matrix, see also length

sqrt Square root

ss Create state-space models or convert LTI model to state space, see

also tf

ss2tf State-space to transfer function representation, see also tf2ss

ss2zp State-space to pole-zero representation, see also zp2ss


28

stairs Stairstep plot for discreste response, see also dstep

step Plot the step response, see also impulse, lsim, dlsim.

subplot Divide the plot window up into pieces, see also plot, figure

text Add a piece of text to the current plot, see also title, xlabel, ylabel,

gtext

tf Creation of transfer functions or conversion to transfer function, see

also ss

tf2ss Transfer function to state-space representation, see also ss2tf

tf2zp Transfer function to Pole-zero representation, see also zp2tf

title Add a title to the current plot

wbw Returns the bandwidth frequency given the damping ratio and the

rise or settling time.

xlabel/ylabel Add a label to the horizontal/vertical axis of the current plot, see

also title, text, gtext

zeros Returns a vector or matrix of zeros

zgrid Generates grid lines of constant damping ratio (zeta) and natural

frequency (Wn), see also sgrid, jgrid, sigrid

zp2ss Pole-zero to state-space representation, see also ss2zp

zp2tf Pole-zero to transfer function representation, see also tf2zp


29

Modeling Tutorial

Matlab can be used to represent a physical system or a model. To begin with, let's start

with a review of how to represent a physical system as a set of differential equations.

Train system

In this example, we will consider a toy train consisting of an engine and a car.

Assuming that the train only travels in one direction, we want to apply control to the

train so that it has a smooth start-up and stop, along with a constant-speed ride.

The mass of the engine and the car will be represented by M1 and M2, respectively. The

two are held together by a spring, which has the stiffness coefficient of k. F represents

the force applied by the engine, and the Greek letter, mu (which will also be represented

by the letter u), represents the coefficient of rolling friction.

Photo courtesy: Dr. Howard Blackburn

Free body diagram and Newton's law

The system can be represented by following Free Body Diagrams.

From Newton's law, you know that the sum of forces acting on a mass equals the mass

times its acceleration. In this case, the forces acting on M1 are the spring, the friction

and the force applied by the engine. The forces acting on M2 are the spring and the

friction. In the vertical direction, the gravitational force is canceled by the normal force

applied by the ground, so that there will be no acceleration in the vertical direction. The

equations of motion in the horizontal direction are the followings:


30

State-variable and output equations

This set of system equations can now be manipulated into state-variable form. Knowing

state-variables are X1 and X2 and the input is F, state-variable equations will look like

the following:

Let the output of the system be the velocity of the engine. Then the output equation will

become:

1. Transfer function

To find the transfer funciton of the system, first, take Laplace transforms of above state-

variable and output equations.

Using these equations, derive the transfer function Y(s)/F(s) in terms of constants.

When finding the transfer function, zero initial conditions must be assumed. The

transfer function should look like the one shown below.


31

2. State-space

Another method to solve the problem is to use the state-space form. Four matrices A, B,

C, and D characterize the system behavior, and will be used to solve the problem. The

state-space form that were manipulated from the state-variable and the output equations

is shown below.

Matlab representation

Now we will show you how to enter the equations derived above into an m-file for

Matlab. Since Matlab can not manipulate symbolic variables, let's assign numerical

values to each of the variables. Let

• M1 = 1 kg

• M2 = 0.5 kg

• k = 1 N/sec

• F= 1 N

• u = 0.002 sec/m

• g = 9.8 m/s^2

Create an new m-file and enter the following commands.

M1=1;

M2=0.5;


32

k=1;

F=1;

u=0.002;

g=9.8;

Now you have one of two choices: 1) Use the transfer function, or 2) Use the state-

space form to solve the problem. If you choose to use the transfer function, add the

following commands onto the end of the m-file which you have just created.

num=[M2 M2*u*g 1];

den=[M1*M2 2*M1*M2*u*g M1*k+M1*M2*u*u*g*g+M2*k M1*k*u*g+M2*k*u*g];

If you choose to use the state-space form, add the following commands at the end of the

m-file, instead of num and den matrices shown above.

A=[ 0 1 0 0;

-k/M1 -u*g k/M1 0;

0 0 0 1;

k/M2 0 -k/M2 -u*g];

B=[ 0;

1/M1;

0;

0];

C=[0 1 0 0];

D=[0];

See the Matlab basics tutorial to learn more about entering matrices.

Continue solving the problem

Now, you are ready to obtain the system output (with an addition of few more commands). It should be

noted that many operations can be done using either the transfer function or the state-space model.

Furthermore, it is simple to transfer between the two if the other form of representation is required. If you

need to learn how to convert from one representation to the other, click Conversion.


33

Perhitungan dan Pemrograman dengan MATLAB

2.1. Matematika Sederhana

Seperti kalkulator, MATLAB dapat melakukan perhitungan matematika

sederhana.

Contoh : Lakukan perhitungan di bawah ini dengan menggunakan MATLAB

!!!

“Wati berbelanja ke toko buku membeli 10 buku tulis @Rp 1500, 1 buah

penghapus pinsil Rp 1000, dan 2 isi bolpoint @ Rp 800,-“

Maka yang harus dibayarkan oleh Wati adalah :

EDU>>10*1500+1000+2*800

ans =

17600

Dapat pula dihitung dengan menuliskan program :

EDU>> buku=10

buku =

10

EDU>> penghapus=1

penghapus =

1

EDU>> isi = 2;

EDU>> bayar = buku*1500 + penghapus * 1000 + isi * 800

bayar =

17600

MATLAB tidak memperhatikan spasi, untuk sebagian besar, dan perkalian

diprioritaskan terlebih dahulu dari penjumlahan. Juga perlu diketahui,

MATLAB menyebut hasil sebagai ans (singkatan dari answer).

Pada penjumlahan dan perkalian, MATLAB menggunakan operasi aritmatik

dasar berikut ini :


34

Operasi Rumus Aritmatik Simbol Penulisan

Penjumlahan a + b + a + b

Pengurangan a – b - a – b

Perkalian a x b * a * b

Pembagian a : b / atau \ a / b atau b \ a

Pangkat ab ^ a^b

Urutan operasi yang akan dilakukan adalah dengan menggunakan aturan

yang biasa digunakan, yaitu perhitungan dihitung dari kiri ke kanan,

operasi pangkat merupakan urutan pertama, diikuti dengan perkalian dan

pembagian, dan terakhir adalah penjumlah dan pengurangan. Untuk

operasi yang dilakukan dalam tanda kurung, maka operasi itu yang

pertama akan dilakukan dari operasi yang ada di luarnya.


35

2.2. Komentar dan Pemberian Tanda Baca

Semua huruf di belakang tanda % dibaca sebagai pernyataan bebas (tidak

mempengaruhi program)

Contoh : EDU>> buku = 10 % jumlah dari buku yang dibeli

buku =

10

Beberapa perintah dapat diletakkan pada satu baris bila dipisah dengan

koma, atau titik koma.

Contoh : EDU>> buku = 10, penghapus = 1; isi = 2

buku =

10

isi =

2

2.3. Bilangan Komplek

Salah satu dari kelebihan MATLAB adalah tidak diperlukannya penanganan

khusus untuk bilangan komplek. Bilangan komplek ditampilkan oleh

MATLAB dengan beberapa cara.

Contoh :

EDU>> c1=1-2i % i menunjukkan bagian imajiner

c1 =

1.0000 – 2.0000i

EDU>> c1=1-2j % j juga bisa digunakan

c1 =

1.0000 – 2.0000i

EDU>> c2=3*(2-sqrt(-1)*3)

c2 =

6.0000 – 9.0000i

EDU>> c3=sqrt(-2)

c3 =

0 + 1.4142 i

dan seterusnya.


36

2.4. Fungsi-fungsi Matematika yang umum digunakan

Pernyataan Fungsi

abs(x) nilai absolut atau besar bilangan komplek

acos(x) invers cosinus

acosh(x) invers cosinus hyperbolis

angle(x) sudut 4 kuadrant dari komplek

asin(x) invers sinus

asinh(x) invers sinus hyperbolis

atan(x) invers tangen

atan2(x,y) invers tangen 4 kuadrant

atanh(x) invers tangen hyperbolis

ceil(x) gerak siklus plus tak hingga

conj(x) konjugasi komplek

cos(x) cosinus

cosh(x) cosinus hyperbolis

exp(x) eksponensial (ex)

fix(x) gerak siklus nol

floor(x) gerak siklus minus tak hingga

gcd(x,y) pembagi terbesar dari bilangan bulat x dan y

imag(x) bagian bilangan imajiner

lcm(x,y) perkalian terkecil dari bilangan bulat x dan y

log(x) logaritma alami

log10(x) logaritma biasa

real(x) bagian bilangan real

rem(x,y) sisa dari pembagian (rem (x,y) memberikan sisa

dari x/y)

round(x) bilangan bulat terdekat dari siklus lingkar

sign(x) fungsi signum

sin(x) sinus

sinh(x) sinus hyperbolis

sqrt(x) akar persamaan

tan(x) tangen

tanh(x) tangen hyperbolic


37

Latihan-latihan Soal dengan MATLAB

Latihan 1 : Perhitungan dengan Matrik Didalam menulis suatu bentuk matrik, setiap kolom dipisahkan dengan

matrik, dan setiap baris dipisahkan dengan tanda titik koma.

Contoh :

Matrik G =

1 2 3

4 5 6

7 8 9

Maka ditulis :

EDU>> g=[1 2 3;4 5 6;7 8 9]

g =

1 2 3

4 5 6

7 8 9

atau :

EDU>> g=[1 2 3

4 5 6

7 8 9]

g =

1 2 3

4 5 6

7 8 9

EDU>>size(g) % ordo dari matrik a

ans =

2 2

Fungsi lain matriks yang harus diketahui :

inv(a) ∗ inverse dari matrik a

transpose(a) ∗ transpose dari matrik a


38

det(a) ∗ determinan dari matrik a

Soal Latihan :

1. A =

1 2

3 4

a. Tulislah perintah dari MATLAB prompt untuk matrik A

b. Hitunglah transpose dari matrik A

c. Hitunglah determinan dari matrik A

d. Hitunglah invers dari matrik A

e. Buktikan bahwa A-1=1/det * A?

f. Buktikan bahwa A x A-1 = I


39

Latihan 2 : Transfer Function

Untuk menuliskan fungsi dari diagram blok di atas dan kemudian

menuliskannya setelah diberi umpan balik H(s)=1 adalah :

Bila suatu transfer function (fungsi Laplace) akan diubah ke dalam bentuk

perhitungan ruang keadaan (state space equation) untuk diagram blok di

atas di mana

x(t) = ax(t) + bu(t)

y(t) = cx(t) + du(t)

maka ditulis program sebagai berikut :

Buatlah m.file dari kedua program di atas dan lihatlah hasilnya

S + 2

S2 + 2S + 1

num=[1 2] den=[1 2 1] printsys(num,den) [a,b,c,d]=ss2tf(num,den)

num=[1 2] den=[1 2 1] printsys(num,den) [numcl,dencl]=cloop(num,den) printsys(numcl,dencl)


40

Fungsi – fungsi lain yang harus diketahui :

[num,den]=ss2tf(a,b,c,d)

∗ untuk mengubah dari state space menjadi

transfer function

[nump,denp]=parallel(num1,den1,num2,den2)

∗ menghitung paralel dari dua fungsi

[nums,dens]=series(num1,den1,num2,den2)

∗ menghitung seri dari dua fungsi

[numf,denf]=feedback(num1,den1,numh,denh)

∗ menghitung fungsi alih dengan umpan balik

fungsi H(s)

step(num,den) ∗ melukis tanggapan waktu dari fungsi alih

loop tertutup

figure ∗ untuk membuat window gambar baru

Tugas :

Fungsi 1 :

S + 1

( S + 3 )( S + 5 )

Fungsi 2 :

S + 6

S + 10

a. Hitunglah fungsi alih lingkar tertutup dari fungsi 1 ( H(s) = 1 )

b. Hitunglah fungsi alih dari fungsi 1 dan fungsi 2 yang dihubung paralel.

c. Hitunglah fungsi alih dari fungsi 1 dan fungsi 2 yang dihubung seri.

d. Hitunglah fungsi alih dengan fungsi 2 sebagai umpan balik.

e. Hitung state space equation dari fungsi 1

f. Gambarkan fungsi tanggapan waktu dari fungsi 1

g. Gambarkan fungsi tanggapan waktu dari fungsi 2

Save dengan nama “Latih1.m”


41

Latihan 3 : Fungsi Tanggapan Waktu & Tempat Kedudukan Akar

4.3.1. Fungsi Tanggapan Waktu

4.3.1.1. Sistem Orde Satu

Dengan masukan R(s)=1/2, sistem orde satu dapat dinyatakan dengan :

C(s) = R(s)G(s) = a .

s(s+a)

Invers sistem di atas memberikan tanggapan step :

c(t) = cf(t) + cn(t) = 1 – e-at

Dari persamaan di atas terlihat pentingnya parameter a, satu-satunya

parameter yang berpengaruh terhadap tanggapan transien. Jika t = 1/a

e-at⏐t=1/1 = e-1 = 0.37

atau

x(t) = 1-e-at⏐t=1/a = 1- 0.37 = 0.63

☺ Konstanta Waktu (Time Constant)

Konstanta waktu adalah waktu yang dibutuhkan sistem untuk mencapai

harga 63% dari harga akhir.

☺ Rise Time, Tr

Rise Time didefinisikan sebgai waktu yang dibutuhkan untuk beranjak dari

titik 10% hingga 90% dari harga akhir. Rumus Tr =

Tr = 2,2

2ζωn

☺ Settling Time, Ts

Settling time didefinisikan sebagai waktu yang dibutuhkan tanggapan

untuk mencapai 98% dari harga akhir. Rumus settling time :

Ts = 4

2ζωn

4.3.1.2. Sistem Orde Dua

Berbeda dengan sistem orde pertama, sistem orde dua mempunyai tingkat

kompleksitas yang lebih besar, lebih banyak parameter yang harus

dianalisa dan diamati untuk dapat menentukan tanggapan sistem.


42

Tanggapan pada sistem orde dua ada empat macam :

Tanggapan teredam lebih

Tanggapan kurang teredam

Tanggapan osilasi

Tanggapan teredam kritis

Ada beberapa istilah yang biasa digunakan untuk menyatakan sistem-

sistem berorde dua ini, yang pertama adalah frekuensi alami. Frekuensi

alami ωn adalah frekuensi osilasi sistem tanpa peredaman.

Kedua, adalah damping ratio ζ. Damping ratio ini didefinisikan sebagai

perbandingan antara frekuensi pengurangan eksponensial dengan

frekuensi alaminya. Untuk memperjelas konsep, perhatikan sistem umum

berikut :

G(s) = ωn2 .

s2 + 2ζωns + ωn2

☺ Peak Time, Tp

Peak time adalah waktu yang dibutuhkan untuk mencapai puncak pertama

atau puncak maksimum. Nilai peak time dirumuskan dengan :

21 ζω

π

−=

n

Tp

☺ Percent Overshoot, %OS

Percent overshoot adalah perbandingan selisih harga maksimum dengan

harga akhir sistem (dalam persen). Rumusan untuk mendapatkan nilainya

adalah :

)100/(%ln100/ln(%

22 OSOS

+

−=

πζ

☺ Settling Time, Ts

Settling time adalah waktu yang dibutuhkan transien osilasi teredam

untuk berada pada ± 2% harga keadaan tunak. Rumus Ts :

nsT

ζω4

=


43

☺ Rise Time, Tr

Rise time adalah waktu yang dibutuhkan bentuk gelombang untuk

beranjak dari harga 10% ke 90% harga akhir. Nilai rise time secara analitis

tidak dapat didapat.

n

d

d

tgTrζωω

ω −= −11

dimana

21 ζωω −= nd

Penulisan perhitungan-perhitungan di atas beberapa diantaranya bila

ditulis dalam MATLAB adalah sebagai berikut :

4.3.2. Tempat Kedudukan Akar

Tempat Kedudukan Akar adalah metoda yang cukup efektif dalam

perancangan dan analisis stabilitas dan tanggapan transien. TKA dapat

digunakan secara kualitatif menerangkan unjuk kerja sebuah sistem

dengan berbagai variasi perubahan parameter. Sebagai contoh, efek

num = den = finalvalue = polyval(num,0)/polyval(den,0) [y,x,t]=step(num,den); [Y,k]=max(y); %menghitung time to peak (tp) timetopeak=t(k) %menghitung percentovershoot percentovershoot=100*(Y-finalvalue)/finalvalue %menghitung rise time (tr) n=1; while y(n)<0.1*finalvalue,n=n+1;end m=1; while y(m)<0.9*finalvalue,m=m+1;end risetime=t(m)-t(n) %menghitung settling time (ts) l=length(t); while(y(l)>0.98*finalvalue)&(y(l)<1.02*finalvalue) l=l-1; end settlingtime=t(l)


44

variasi penguatan percent overshoot, settling time, dan peak time.

Disamping tanggapan transien, TKA dapat memberikan informasi grafis

tentang stabilitas sebuah sistem. Dapat terlihat secara jelas range

stabilitas, range ketidakstabilan dan kondisi yang menyebabkan sistem

menuju osilasi.

☺ Aturan Perancangan TKA

Beberapa istilah yang berpengaruh dalam perancangan TKA :

1. Jumlah percabangan

2. Simetri

3. Segmen sumbu real

4. Titik awal dan titik akhir

5. Beberapa aturan tambahan.

Untuk melukiskan tempat kedudukan akar dan menentukan k dan poles

dimana :

k = -den(p)/num(p)

dan poles adalah kutub-kutub dari persamaan sistem lingkar terbuka, di

dalam MATLAB adalah :

Tugas :

Sebuah sistem memiliki persamaan sebagai berikut :

num = den = wn = damping = rlocus(num,den) sgrid(damping,wn) axis([-2 2 –6 6]) [k,poles]=rlocfind(num,den)

1

S2 + 2S + 1

C(s) R(s)


45

a. Tentukan Tr, Ts, Td, %OS, ωn

b. Gambarkan Fungsi Tanggapan waktunya

c. Tentukan k dan kutub-kutub dari persamaan di atas

d. Lukiskan Tempat Kedudukan akarnya


46

Latihan 4 : Transfer Fungsi Kontinyu – Diskrit, dan Fungsi

Tanggapan Waktu Diskrit dengan metode ZOH dan FOH

4.4.1. Sinyal Waktu Diskrit

Sinyal waktu diskrit timbul bila sistim melibatkan suatu operasi

pencuplikan dari sinyal waktu kontinyu, atau bila sistim melibatkan suatu

proses iteratif yang dibawa oleh komputer digital.

Urutan dari nilai atau angka biasanya ditulis sebagai x(k), dimana k

menunjukkan urutan nilai atau angka muncul pada deret, contoh x(0),

x(1), x(2), …

Bila sinyal waktu kontinyu dicuplik, periode pencuplikan T menjadi

parameter yang penting

4.4.2. Impulse Sampling

Suatu sampling mempunyai periode T, dan duration sampling sangat

singkat.

Sampling ditimbulkan oleh sampler dari fungsi yang kontinyu, menjadi

pulsa train, misal :

t=0, T, 2T, … dengan T, perioda sampling.

Data Hold Circuit, mengkonversikan sample data ke dalam sinyal yang

kontinyu.

Data hold adalah proses pembangkitan sinyal waktu kontinyu h(t) dari

deret waktu diskrit x(kT)

Sinyal h(t) selama interval waktu kT ≤ t ≤ (k+1)T

dapat didekati oleh polinomial dalam τ :

n(kT+τ) = anτn + an-1τn-1 + … + a1τ + a0

untuk 0≤τ≤t, h(kT) = x(kT)

4.4.3. Fungsi Alih Rangkaian Hold

Fungsi alih ZOH : Gh0(s) = s

sT−−ε1

dengan ZOH, integrator reset ke nol setelah satu periode sampling.

Fungsi alih FOH : Gh1(s) = T

Tss

sT 112

+⎟⎟⎠

⎞⎜⎜⎝

⎛ − −ε


47

Keluaran h(t) dari FOH adalah garis lurus yang merupakan extrapolasi dari

2 nilai sampling.

Dengan Matlab, semua ketentuan di atas dapat diperoleh dengan cara

yang jauh lebih mudah. Seperti contoh berikut ini :


48

Tugas :

Sebuah sistem memiliki persamaan sebagai berikut :

a. Tentukan transformasi Z dengan (Ts = 0.3) loop tertutup dari fungsi di

atas

Ts= 0.5; %periode pencuplikan num=[1]; den=[1 8 4]; printsys(num,den) %sistem kontinyu loop terbuka [numc,denc]=cloop(num,den); printsys(numc,denc) %sistim kontinyu loop tertutup [numd,dend]=c2dm(numc,denc,Ts); %transformasi kontinyu-diskrit printsys(numd,dend,'z') %fungsi diskrit subplot(121), step(numc,denc) subplot(122), dstep(numd,dend) figure [ndf,ddf]=c2dm(numc,denc,Ts,'foh'); %pencuplikan dengan first order hold printsys(ndf,ddf,'z') [ndz,ddz]=c2dm(numc,denc,Ts,'zoh'); %pencuplikan dengan zero order hold printsys(ndz,ddz,'z') subplot(121), dstep(ndf,ddf) subplot(122), dstep(ndz,ddz)

1

S2 + 2S + 1

C(s) R(s)


49

b. Buatlah grafik tanggapan waktu kontinyu

c. Buatlah grafik tanggapan waktu diskritnya

d. Buatlah fungsi pencuplikan dengan cara ZOH dan grafiknya

e. Buatlah fungsi pencuplikan dengan cara FOH dan grafiknya.


50

Latihan 5 : Kriteria Kestabilan Menurut Jury-Test Suatu sistem dengan persamaan karakteristik

P(z) = aoZn + a1Zn-1 + … an-1z + an

Sistem stabil bila memenuhi semua kondisi di bawah ini :

1. ⏐an⏐ < ⏐a0⏐

2. P(z)⏐z=1 > 0

3. P(z)⏐z=-1 untukngenp

iluntuknganj00

><⟨

4. ⏐bn-1⏐ >⏐b0⏐

⏐cn-2⏐ >⏐c0⏐

⏐dn-3⏐ >⏐d0⏐

…

⏐q2⏐ > ⏐q0⏐

dimana :

bk = 10

1

+

−−

k

knn

aaaa

k = 0,1,2,3,…,n-1

ck = 10

21

+

−−−

k

knn

bbbb

k = 0,1,2,3,…,n-2

dk = 10

22

+

−−−

k

knn

cccc

k = 0,1,2,3,…,n-3

…

qk = 10

23

+

−

k

k

pppp

k = 0,1,2

Baris terakhir dari tabel terdiri dari 3 elemen (untuk sistem order-2, 2n-3

= 1 dan tabel Jury hanya berisi satu baris yang terdiri dari tiga elemen.

Elemen-elemen dari baris genap adalah reverse dari peletakan elemen-

elemen baris ganjil.

Contoh dari penulisan program untuk menguji suatu sistim dengan orde 4

adalah :


51

%PENGUJIAN KESTABILAN SISTIM DISKRIT ORDE 4 %DENGAN METODE JURY'S TEST clear N=5; disp('Orde sistem = ') for k=1:5 kt=k-1 a(k)=input ('Masukkan harga a =') end pst=polyval(a,1) pmt=polyval(a,-1) for k=4:-1:1 matrikb=[a(N) a(N-k);a(1) a(k+1)] b(k)=det(matrikb) end for k=3:-1:1 matrikc=[b(N-1) b(N-1-k);b(1) b(k+1)] c(k)=det(matrikc) end if abs(a(5))<abs(a(1)) s1='Y' else s1='N' end if pst>0 s2='Y' else s2='N' end if pmt>0 s3='Y' else s3='N' end if abs(b(4))>abs(b(1)) s4a='Y' else s4a='N' end if abs(c(3))>abs(c(1)) s4b='Y' else s4b='N' end if s1&s2&s3&s4a&s4b=='Y' sistim='Stabil' else sistim='Tidak stabil' end


52

Contoh : Untuk menguji suatu sistim orde 4 yang memiliki persamaan karakteristik =

P(z)=z4 - 1.2z3 + 0.07z2 + 0.3z - 0.08

adalah :

EDU» Tan_Jury4

Orde sistem =

kt =

0

Masukkan harga a =1

a =

1


53

kt =

1

Masukkan harga a =-1.2

a =

1.0000 -1.2000

kt =

2

Masukkan harga a =.07

a =

1.0000 -1.2000 0.0700

kt =

3

Masukkan harga a =.3

a =

1.0000 -1.2000 0.0700 0.3000

kt =


54

4

Masukkan harga a =-0.08

a =

1.0000 -1.2000 0.0700 0.3000 -0.0800

P(z)=z4 - 1.2z3 + 0.07z2 + 0.3z - 0.08

pst =

0.0900

P(1) = 0.09

pmt =

1.8900

P(-1) = 1.89

matrikb =

-0.0800 1.0000

1.0000 -0.0800

b =

0 0 0 -0.9936


55

matrikb =

-0.0800 -1.2000

1.0000 0.3000

b =

0 0 1.1760 -0.9936

matrikb =

-0.0800 0.0700

1.0000 0.0700

b =

0 -0.0756 1.1760 -0.9936

matrikb =

-0.0800 0.3000

1.0000 -1.2000

b =

-0.2040 -0.0756 1.1760 -0.9936

b0 b1 b2 b3

matrikc =

-0.9936 -0.2040


56

-0.2040 -0.9936

c =

0 0 0.9456

matrikc =

-0.9936 -0.0756

-0.2040 1.1760

c =

0 -1.1839 0.9456

matrikc =

-0.9936 1.1760

-0.2040 -0.0756

c =

0.3150 -1.1839 0.9456

c0 c1 c2

s1 =

Y


57

s2 =

Y

s3 =

Y

s4a =

Y

s4b =

Y

sistim =

Stabil


58

Latihan 6 : Fungsi Tanggapan Frekuensi

Bode Plot Gambar-gambar Bode melukiskan kestabilan relatif sebuah sistem. Dalam

kenyataannya, margin fasa dan margin gain seringkali didefinisikan dalam

istilah-istilah gambar Bode. Ukuran-ukuran kestabilan relatif ini dapat

ditentukan untuk sebuah sistem tertentu dengan suatu usaha menghitung

yang minimum dengan menggunakan gambar-gambar Bode, khususnya

untuk hal-hal di mana tersedia data tanggapan frekuensi dari percobaan.

Magnituda ⏐P(jω)⏐ dari setiap fungsi P(jω) untuk setiap harga ω

digambarkan pada suatu skala logaritmik dalam satuan-satuan desibel

dimana,

dB ≡ 20 log10 ⏐P(jω)⏐

Karena desibel merupakan suatu satuan logaritmik, magnituda db dari

sebuah fungsi tanggapan frekuensi yang terdiri dari suatu hasilkali suku-

suku adalah sama dengan jumlah magnituda db dari masing-masing suku.

Jadi bila digunakan skala logaritmik, gambar magnituda dari sebuah

fungsi tanggapan frekuensi yang dapat dinyatakan sebagai hasil kali lebih

dari satu suku dapat diperoleh dengan menjumlahkan masing-masing

gambar magnituda db untuk setiap suku hasilkali.

Gambar magnituda db versus logω disebut gambar magnituda Bode, dan

gambar sudut fasa versus logω adalah gambar sudut fasa Bode. Dalam

kepustakaan kadang-kadang gambar magnituda Bode disebut gambar Log-

modulus.

Bentuk Bode untuk fungsi :

))...()(())...()((

)(21

21

nm

n

pspspsszszszsK

sG++++++

=

dimana m adalah bilangan bulat tak negatif, diperoleh dengan

menguraikan semua z dan p dan menyusunnya kembali dalam bentuk :

ωjspspspss

zszszsKsG

nm

n→

+++

+++=

)(...)()(

(...)()()(

21

)21

Pengubahan tanggapan magnitudo ke dalam dB menghasilkan


59

20 log ⏐G(jω)⏐= 20 log K + 20 log ⏐(s+z1)⏐ + 20 log ⏐(s+z2)⏐ +…

-20 log ⏐(sm)⏐ + 20 log ⏐(s+p1)⏐- … ⏐s → jω

Sementara fasa dari fungsi adalah 1/sm dimana s= jω adalah :

[ ])(arg ωjP

Contoh program dengan Matlab untuk membuat Bode Plot dari suatu

sistim dengan persamaan karakteristik :

)10)(15.0()22.0(*10)( 2

2

+++++

=ssssssF

adalah :

Bode representation

The Bode plots take each one of the points on the above plot and breaks it down into

magnitude and phase. The magnitude is then plotted as gain in decibels and the

phase is plotted in degrees. The frequency (on the independent axis) is plotted on a

logarithmic scale. Let's take a look at the Bode plots for this function and see if our

answers match.

bode(num,den)

% Sistem yang diberikan num=10*[1 0.2 2]; den=conv([1 0.5 1],[1 10]);%(s^2+0.5s+1)(s+10) disp('Persamaan sistem =') printsys(num,den,'s') % Diskrit dari Sistem dengan Zero Order Hold Ts=0.3 [numd,dend]=c2dm(num,den,Ts,'zoh'); %Contoh membuat Bode dari Sistem bode(num,den) dbode(numd,dend,Ts) Title('Contoh Bode Plot') xlabel('Frekuensi (rad/sec)')


60

If you look at the plots at a frequency of 3rad/s you will see the magnitude is a little

over 2.5 dB and the phase is around 140 degrees. This agrees with our previous

results.

Tugas :

Sebuah sistem memiliki persamaan karakteristik sebagai berikut :

)5)(12()13.0(*15)( 2

2

+++++

=ssssssF

Buatlah Bode Plot dari fungsi tersebut

Nyquist

Plotting Frequency Response

In order to illustrate the plotting of G(jw), we will generate our own plot of

G(j*w)and compare it with the Bode plots and the Nyquist diagram. If we run the

following m-file, we will see the points that comprise G(jw), in particular the point

corresponding to G(j*3).

%define the transfer function

num = conv([1,4],[1 6]);

den = conv([1,-3],[1 -5]);


61

%create a frequency vector

clf

w = logspace(-1,2,100);

hold on

%plot a '+' for each point corresponding to G(jw)

for i = 1:100

plot(polyval(num,j*w(1,i))/polyval(den, j*w (1,i)), '+')

end

%plot a green 'X' for frequency of 3rad/s

plot(polyval(num,j*3)/polyval(den, j*3), 'xg')

%plot vector from origin to this point

n= polyval(num,j*3)/polyval(den, j*3);

x = [0,real(n)];

y = [0,imag(n)];

plot(x,y, '-g')

%find magnitude and phase

Mag = abs(n);

Phase = 180 +360*atan(imag(n)/real(n))/(2*pi);

%find gain

Gain = 20*log10(Mag);

grid

Mag, Phase, Gain

The image should look similar to this:

Please note that each yellow cross represents the transfer function evaluated at a

specific frequency and that only positive frequencies are used. Also note the point


62

and vector associated with a frequency of 3 rad/s (green). In the Matlab command

window, you should see the Magnitude, Phase, and Gain (in dB) associated with this

vector:

Note: a decibel is defined as 20*log10 ( |G(j*w| )

Mag =

1.3558

Phase =

139.3987

Gain =

2.6440

Now that we know where Bode plots come form, let's see where Nyquist plots come from. The

Nyquist diagram is basically a plot of G(j*w). However, both positive and negative frequencies

of the contour we showed previously are taken into account. Let's plot the Nyquist diagram on

top of our G(j*w) plot (this time, we will include the negative frequencies as well).

%define the transfer function

num = conv([1,4],[1 6]);

den = conv([1,-3],[1 -5]);

%create a frequency vector similar to the one described above

w = logspace(-1,2,100);

negw = -1*w;

clf

hold on

%plot a '+' for each point corresponding to G(jw)

for i = 1:100

plot(polyval(num,j*w(1,i))/polyval(den, j*w (1,i)), '+')

plot(polyval(num,j*negw(1,i))/polyval(den, j*negw (1,i)), '+')

end

%draw Nyquist diagram on top to compare

nyquist(num,den)

hold off


63

A perfect match! Also note that the negative frequencies produce a mirror image

across the real axis of the positive frequency response.


64

Gain and Phase Margin

Let's say that we have the following system:

where K is a variable (constant) gain and G(s) is the plant under consideration. The

gain margin is defined as the change in open loop gain required to make the system

unstable. Systems with greater gain margins can withstand greater changes in system

parameters before becoming unstable in closed loop.

Keep in mind that unity gain in magnitude is equal to a gain of zero in dB.

The phase margin is defined as the change in open loop phase shift required to make

a closed loop system unstable.

The phase margin also measures the system's tolerance to time delay. If there is a

time delay greater than 180/Wpc in the loop (where Wpc is the frequency where the

phase shift is 180 deg), the system will become unstable in closed loop. The time

delay can be thought of as an extra block in the forward path of the block diagram

that adds phase to the system but has no effect the gain. That is, a time delay can

be represented as a block with magnitude of 1 and phase w*time_delay (in

radians/second).

For now, we won't worry about where all this comes from and will concentrate on

identifying the gain and phase margins on a Bode plot:

The phase margin is the difference in phase between the phase curve and -180 deg at

the point corresponding to the frequency that gives us a gain of 0dB (the gain cross

over frequency, Wgc). Likewise, the gain margin is the difference between the

magnitude curve and 0dB at the point corresponding to the frequency that gives us a

phase of -180 deg (the phase cross over frequency, Wpc).


65

One nice thing about the phase margin is that you don't need to replot the Bode in

order to find the new phase margin when changing the gains. If you recall, adding

gain only shifts the magnitude plot up. This is the equivalent of changing the y-axis

on the magnitude plot. Finding the phase margin is simply the matter of finding the

new cross-over frequency and reading off the phase margin. For example, suppose

you entered the command bode(50,[1 9 30 40]). You will get the following bode plot:

You should see that the phase margin is about 100 degrees. Now suppose you added a

gain of 100, by entering the command bode(100*50,[1 9 30 40]). You should get the

following plot (note I changed the axis so the scale would be the same as the plot

above, your bode plot may not be exactly the same shape, depending on the scale

used):


66

As you can see the phase plot is exactly the same as before, and the magnitude plot

is shifted up by 40dB (gain of 100). The phase margin is now about -60 degrees. This

same result could be achieved if the y-axis of the magnitude plot was shifted down

40dB. Try this, look at the first Bode plot, find where the curve crosses the -40dB

line, and read off the phase margin. It should be about -60 degrees, the same as the

second Bode plot.

We can find the gain and phase margins for a system directly, by using Matlab. Just

enter the margin command. This command returns the gain and phase margins, the

gain and phase cross over frequencies, and a graphical representation of these on the

Bode plot. Let's check it out:

margin(50,[1 9 30 40])


67

Latihan : 7 Designing Lead and Lag Compensators

Lead and lag compensators are used quite extensively in control. A lead compensator

can increase the stability or speed of response of a system; a lag compensator can

reduce (but not eliminate) the steady state error. Depending on the effect desired,

one or more lead and lag compensators may be used in various combinations.

Lead, lag, and lead/lag compensators are usually designed for a system in transfer

function form. The conversions page explains how to convert a state-space model

into transfer function form.

Lead or phase-lead compensator using root locus

A first-order lead compensator can be designed using the root locus. A lead

compensator in root locus form is given by

where the magnitude of zo is less than the magnitude of po. A phase-lead

compensator tends to shift the root locus toward the left half plane. This results in

an improvement in the system's stability and an increase in the response speed.

How is this accomplished? If you recall finding the asymptotes of the root locus that

lead to the zeros at infinity, the equation to determine the intersection of the

asymptotes along the real axis is:

When a lead compensator is added to a system, the value of this intersection will be

a larger negative number than it was before. The net number of zeros and poles will

be the same (one zero and one pole are added), but the added pole is a larger

negative number than the added zero. Thus, the result of a lead compensator is that

the asymptotes' intersection is moved further into the left half plane, and the entire

root locus will be shifted to the left. This can increase the region of stability as well

as the response speed.

In Matlab a phase lead compensator in root locus form is implemented by using the

transfer function in the form


68

numlead=kc*[1 z];

denlead=[1 p];

and using the conv() function to implement it with the numerator and denominator

of the plant

newnum=conv(num,numlead);

newden=conv(den,denlead);


69

Lead or phase-lead compensator using frequency

response

A first-order phase-lead compensator can be designed using the frequency response.

A lead compensator in frequency response form is given by

Note that this is equivalent to the root locus form

with p = 1/T, z = 1/aT, and Kc = a. In frequency response design, the phase-lead

compensator adds positive phase to the system over the frequency range 1/aT to

1/T. A bode plot of a phase-lead compensator looks like the following

The two corner frequencies are at 1/aT and 1/T; note the positive phase that is

added to the system between these two frequencies. Depending on the value of a,

the maximum added phase can be up to 90 degrees; if you need more than 90

degrees of phase, two lead compensators can be used. The maximum amount of

phase is added at the center frequency, which is located at

The equation which determines the maximum phase is


70

Additional positive phase increases the phase margin and thus increases the stability

of the system. This type of compensator is designed by determining a from the

amount of phase needed to satisfy the phase margin requirements, and determining

T to place the added phase at the new gain-crossover frequency.

Another effect of the lead compensator can be seen in the magnitude plot.

The lead compensator increases the gain of the system at high frequencies

(the amount of this gain is equal to a). This can increase the crossover frequency,

which will help to decrease the rise time and settling time of the system.

In Matlab, a phase lead compensator in frequency response form is implemented by

using the transfer function in the form

numlead=[aT 1];

denlead=[T 1];

and using the conv() function to multiply it by the numerator and denominator of the

plant



Lag or Phase-Lag Compensator using Root Locus

A first-order lag compensator can be designed using the root locus. A lag

compensator in root locus form is given by

where the magnitude of zo is greater than the magnitude of po. A phase-lag

compensator tends to shift the root locus to the right, which is undesirable. For this

reason, the pole and zero of a lag compensator must be placed close together

(usually near the origin) so they do not appreciably change the transient response or

stability characteristics of the system.

How does the lag controller shift the root locus to the right? If you recall finding the

asymptotes of the root locus that lead to the zeros at infinity, the equation to

determine the intersection of the asymptotes along the real axis is:


71

When a lag compensator is added to a system, the value of this intersection will be a

smaller negative number than it was before. The net number of zeros and poles will

be the same (one zero and one pole are added), but the added pole is a smaller

negative number than the added zero. Thus, the result of a lag compensator is that

the asymptotes' intersection is moved closer to the right half plane, and the entire

root locus will be shifted to the right.

It was previously stated that that lag controller should only minimally change the

transient response because of its negative effect. If the phase-lag compensator is not

supposed to change the transient response noticeably, what is it good for? The

answer is that a phase-lag compensator can improve the system's steady-state

response. It works in the following manner. At high frequencies, the lag controller

will have unity gain. At low frequencies, the gain will be z0/p0 which is greater than

1. This factor z0/p0 will multiply the position, velocity, or acceleration constant (Kp,

Kv, or Ka), and the steady-state error will thus decrease by the factor z0/p0.

In Matlab, a phase lead compensator in root locus form is implemented by using the

transfer function in the form

numlag=[1 z];

denlag=[1 p];


of the plant

newnum=conv(num,numlag);

newden=conv(den,denlag);

Lag or Phase-Lag Compensator using Frequency

Response

A first-order phase-lag compensator can be designed using the frequency response. A

lag compensator in frequency response form is given by


72

The phase-lag compensator looks similar to a phase-lead compensator, except that a

is now less than 1. The main difference is that the lag compensator adds negative

phase to the system over the specified frequency range, while a lead compensator

adds positive phase over the specified frequency. A bode plot of a phase-lag

compensator looks like the following

The two corner frequencies are at 1/T and 1/aT. The main effect of the lag

compensator is shown in the magnitude plot. The lag compensator adds gain at low

frequencies; the magnitude of this gain is equal to a. The effect of this gain is to

cause the steady-state error of the closed-loop system to be decreased by a factor of

a. Because the gain of the lag compensator is unity at middle and high frequencies,

the transient response and stability are not impacted too much.

The side effect of the lag compensator is the negative phase that is added to the

system between the two corner frequencies. Depending on the value a, up to -90

degrees of phase can be added. Care must be taken that the phase margin of the

system with lag compensation is still satisfactory.

In Matlab, a phase-lag compensator in frequency response form is implemented by

using the transfer function in the form

numlead=[a*T 1];

denlead=a*[T 1];


73


of the plant



Lead-lag Compensator using either Root Locus or Frequency

Response A lead-lag compensator combines the effects of a lead compensator with those of a

lag compensator. The result is a system with improved transient response, stability

and steady-state error. To implement a lead-lag compensator, first design the lead

compensator to achieve the desired transient response and stability, and then add on

a lag compensator to improve the steady-state response.

Dasar control system dengan matlab

Self Improvement

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