control engineering lab 2

8/19/2019 control engineering lab 2

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MTS-362

Control Engineering Lab-2

Introduction to MATLAB its functions and applications

Plotting Cur!e "itting #Part-II$

2%& Plotting'

The simplest graphs to create are plots of points in the cartesian plane. For example:

>> x = [1;2;3;4;5];

>> y = [0;.25;3;1.5;2];

>> plot(x,y)

The resulting graph is displayed in Figure

"igure 2%(' A simple Matlab graph

Notice that, by default, Matlab connects the points with straight line segments. An alternative

is the following !:

>> plot(x,y,'o')

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"igure 2%2' Another simple Matlab graph

All Matlab variables must have numerical values:

>> x = -10:.1:10;

The basic plot command:

>> plot(sin(x))

"igure 2%3' (raph of )ine

Note that the hori*ontal axis is mar+ed according to the index, notthe value of x. Fix this as follows:

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>> plot( x, sin(x) )

"igure 2%)' Another )ine curve

This is the same thing as plotting -parametric curves.-

e can plot the -inverse relationship- for example, the s/uaring function and 0&1 s/uare root!easily:

>> plot( x, x.^2 ) >> plot( x.^2, x )

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or a spiral in two or in three dimensions:

>> t = 0:.1:10; plot( t .*cos(t), t .* sin(t) )

>> plot3( t .* cos(t), t .*sin(t), t )

3lot several curves simultaneously with plot(x1, y1, x2,

y2, ...):

>> plot( x, cos(x), x, 1 - x.^2./2, x, 1 -x.^2./2 + x.^4./24 )



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"igure 2%*'

2%( Putting se!eral grap+s in one ,indo,

The subplot command creates several plots in a single window. To be precise,

subplot(m,n,i) creates mn plots, arranged in an array with m rows and n columns. 5talso sets the next plot command to go to the ith coordinate system counting across the rows!.6ere is an example

>> t = (0:.1:2*pi)';

>> subplot(2,2,1)

>> plot(t,sin(t))

>> subplot(2,2,2)

>> plot(t,os(t))

>> subplot(2,2,3)

>> plot(t,!xp(t))

>> subplot(2,2,4)>> plot(t,1."(1#t.$2))

2%2 o, to plot in .I""E/E0T C1L1/S

%& you ! + tim! s!!in som! o& t! plots tt you +o in

mtlb on t! olo -osttions, you soul+ pobbly n!

t! olos.

/o &in+ out o- to +o tt, you n typ! t t! mtlb pompt:

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!lp plot

%n sot, olo n b! sp!i&i!+ insi+! t! plot ommn+. ou

n typ!:

plot(x,y,'-')

to plot -it! lin!. ou n lso us! ot! olos t!y

! ll list!+ in t! plot !lp in&omtion.

2%3 Soe ore about plotting

6ere are other ways to graph multiple curves, using matrices plotted by

columns! and using -hold.-

"igure 2%6'

Functions of two variables may be plotted, as well, but some -setup- is re/uired7

>> [x y] = !s"#$i%(-3:.1:3, -3:.1:3);>> & = x.^2 - y.^2;

6ere are two options for plotting the surface. $oo+ at the help page for details.



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"igure 2%'

2%) 3. Plots'

5n order to create a graph of a surface in 21space or a contour plot of a surface!, it isnecessary to evaluate the function on a regular rectangular grid. This can be done using the

meshgrid command. First, create %9 vectors describing the grids in the x1 and y1directions:

>> x = (0:2*pi"20:2*pi)';

>> y = (0:4*pi"40:4*pi)';

Next, spread;; these grids into two dimensions using m!si+:

>> [,] = m!si+(x,y);

>> -os

m! i! 6yt!s 7lss

41x21 8999 +oubl! y

41x21 8999 +oubl! y

x 21x1 189 +oubl! y

y 41x1 329 +oubl! y

n+ totl is 194 !l!m!nts usin 1422 byt!sThe effect of m!si+ is to create a vector with the x1grid along each row, and a vector

with the y1grid along each column. Then, using vectori*ed functions and&or operators, it is

easy to evaluate a function z < f x, y! of two variables on the rectangular grid:>> = os().*os(2*);

6aving created the matrix containing the samples of the function, the surface can be graphed

using either the m!s or the su& commands :

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>> m!s(x,y,)

>> su&(x,y,)

"igure 2%4' >sing the mesh command

"igure 2%5' >sing the surf command

The difference is that su& shades the surface, while m!s does not.! 5n addition, a contour

plot can be created :

>> ontou(x,y,)

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2.' o! o$! ot lottin# n% $p"s

plot(x,y) !t!s 7t!sin plot o& t! !tos x < y

plot(y) !t!s plot o& y s. t! num!il lu!s o& t!

!l!m!nts in t! y!to.

s!milox(x,y) plots lo(x) s y

s!miloy(x,y) plots x s lo(y)

lolo(x,y) plots lo(x) s lo(y)

i+ !t!s i+ on t! pis plot

titl!('t!xt') pl!s titl! t top o& pis plot

xlb!l('t!xt') -it!s 't!xt' b!n!t t! xxis o& plot

ylb!l('t!xt') -it!s 't!xt' b!si+! t! yxis o& plot

t!xt(x,y,'t!xt') -it!s 't!xt' t t! lotion (x,y)

t!xt(x,y,'t!xt','s') -it!s 't!xt' t point x,y ssumin

lo-! l!&t on! is (0,0) n+ upp! it on! is

(1,1).

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pol(t!t,) !t!s pol plot o& t! !tos <

t!t -!! t!t is in +ins.

b(x) !t!s b p o& t! !to x. (ot! lso t!

ommn+ stis(x).)

b(x,y) !t!s bp o& t! !l!m!nts o& t! !to

y, lotin t! bs o+in to t! !to

!l!m!nts o& 'x'. (ot! lso t! ommn+

stis(x,y).)

ol$ plots:n#l! = 0:.1*pi:3*pi;$%is = !xp(n#l!/20); pol$(n#l!,$%is),...titl!(n xpl! ol$ lot),...#$i%

$ #$p":

b(x,y) n+ b(y)

sti (x,y) n+ sti(y)

ltipl! plots:x1=0:.0'*pi:pi;y1=sin(x1); plot(x1,y1)"ol%

y2=cos(x1); plot(x1,y2)

ou n !t! multipl! ps by usin multipl! um!nts.

%n ++ition to t! !tos x,y !t!+ !li!, !t! t!

!tos ,b n+ plot bot !to s!ts simultn!ously s

&ollo-s.

= 1 : .1 : 3; = 10*!xp(-); plot(x,y,,)

ultipl! plots n b! omplis!+ lso by usin mti!s

t! tn simpl! !tos in t! um!nt. %& t! um!nts

o& t! 'plot' ommn+ ! mti!s, t! 7?@ o& y !

plott!+ on t! o+int! inst t! 7?@ o& x on t!

bsiss.


http://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.5.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.5.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.5.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.5.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.5.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.6.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.6.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.6.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.6.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.6.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.6.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.7.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.7.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.7.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.5.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.5.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.5.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.5.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.5.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.6.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.6.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.6.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.6.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.6.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.6.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.7.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.7.htmlhttp://web.ew.usna.edu/~mecheng/DESIGN/CAD/MATLAB/matlab/examples/example3.7.html


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2. 56789 76679

ny osions is! -!n -! ! s!t o& (x,y) pis

n+ -! +!si! to &in+ n !Aution o &untion tt B&itsB

t!s! +t. /! po!+u! -! &ollo- n b! !n!lly

lssi&i!+ in on! o& t-o t!oi!s, int!poltin &untions

o l!st sAu!s &untions.

/! l!stsAu!s &untion is on! tt obtins t! b!st

&it, -!! t! m!nin !! o& Bb!stB is bs!+ on minimiin

t! sum o& t! sAu!s o& t! +i&&!!n!s b!t-!!n t! &untion

n+ t! +t pis. 7l!ly t! i+! o& Bl!st sAu!sB is to

i!! som! sot o& !! o m!n lu! &untion -os!

pil u! +o!s not typilly pss tou ny o& t!

(x,y) +t pis. %t is

ppopit! &o !xp!im!ntl +t in -i !!y +t pi is

obtin!+ un+! on+itions o& un!tinty n+ &o -i ny

oll!t!+ +t, s ons!Au!n! o& !xp!im!ntl !o, my b!!t! o l!ss tn t! tu! lu!.

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t! unno-n o!&&ii!nts n on!ni!ntly &ollo- 7m!'s

Dul!. %n !n!l, o-!!, it is ommon to !ly on mtix

l!b to mnipult! t!s! !Autions into &omt &o -i

n op!tions o& b substitution -ill obtin t! unno-n

o!&&ii!nts. E!! -! -ill simply !ly on t! m!to+s o& it

(o l!&t) +iision in 'mtlb' to obtin t! unno-n

o!&&ii!nts, l!in t! +!tils o& t! num!il m!to+ &o

b substitution &o lt! +isussion.

ot! tt t! s!t o& !Autions n b! on!ni!ntly

!xp!ss!+ by t! mtix !Aution

[y] = [][]

-!!

y1 1 x1 x12.....x1n#1 1

y2 1 x2 x22.....x2n#1 2[y] = [] = [] =

y3 1 x3 x32.....x3n#1 3

. . . . . .

. . . . . .

. . . . . .

yn#1 1 xn#1 xn#12... xn#1n#1 n#1

ot! tt t! it si+! o& t! mtix !Aution must b! t!

po+ut o& 'n#1' x 'n#1' sAu! mtix tim!s 'n#1' x 1

olumn mtixF /! solution usin uss !limintion lls &o

l!&t +iision in 'mtlb' o

[] = []G[y]

@sin tis m!to+, -! n &in+ t! o!&&ii!nts &o n

+!!! polynomil tt pss!s !xtly tou 'n#1' +t

points.

%& -! ! l! +t s!t, ! +t pi inlu+in

som! !xp!im!ntl !o, t! n+!!! polynomil is / H I

7E%7J. Kolynomils o& +!!! l! tn &i! o six o&t!n

! t!ibly un!listi b!io 6J/CJJ t! +t points!!n tou t! polynomil u! pss!s tou !!y +t

point F Hs n !xmpl!, onsi+! t!s! +t.

x y

2 4

3 3

4 5



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5 4

8

5

9

L 10

10 L

/!! ! L +t points in tis s!t. %t is l! tt s x

in!s!s, so lso +o!s y in!s!; o-!!, it pp!s to b!

+oin so in nonlin! -y. ?!t's s!! -t t! +t loos

li!. %n 'mtlb', !t! t-o !tos &o t!s! +t.

xL = [2:1:10];

yL = [ 4 3 5 4 5 10 L ];

/o obs!! t!s! +t plott!+ s points, !x!ut!

plot(xL,yL,'o')

6!us! -! ! nin! +t pis, it is possibl! to onstut

n !it+!!! int!poltin polynomil.

y = 0 # 1x # 2x2 # 3x3 # ....... # 9x9

/o &in+ t! unno-n o!&&ii!nts, +!&in! t! olumn !to y

y = yL'

n+ t! mtix

=

[on!s(1,L);xL;xL.$2;xL.$3;xL.$4;xL.$5;xL.$8;xL.$;xL.$9]'

ot! tt is +!&in!+ usin t! tnspos!, t! on!s()

&untion n+ t! y op!to ' .$ '. Cit n+ y so

+!&in!+, t!y stis&y t! !Aution

[][] = [y]

ol! &o t! o!&&ii!nts, mtix in t! bo!!Aution, by

!nt!in t! ommn+

= Gy

-i !sults in

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= [ 1.0!#003*

3.9140

8.8204

4.931

1.995L

.445

.084L

.005

.0003

.0000 ]

ot! tt t! nint o!&&ii!nt (9) pp!s to b! !o.

Htully, it is &init!, it only pp!s to b! !o b!us!

'mtlb' is pintin only 4 sini&int &iu!s to t! l!&t o&

t! +!iml point. /o obs!! t! o!&&ii!nts -it mo!

sini&int &iu!s, !nt! t! ommn+s

&omt lon

n+ t! &omt is n!+ to on! -it 15 sini&int +iits.

7l!ly (9) is not !o, it is smll numb! b!us! it is

multiplyin numb!, x, is!+ to t! !it po-!.

o- tt -! ! t! o!&&ii!nts, l!t's !n!t! su&&ii!nt

numb! o& points to !t! smoot u!. Mo x, &om

!to o! t! n! 2 N= x N= 10 in in!m!nts o& 0.1.

x = [ 2:.1:10 ];

Mo y, lult! t! lu! o& t! !it +!!! polynomil &o

! x.

y =(1)# (2).*x # (3).*x.$2 # (4).*x.$3 #

(5).*x.$4...

# (8).*x.$5 # ().*x.$8 # (9).*x.$ # (L).*x.$9;

o- plot (x,y) n+ t! +t points (xL,yL).

plot(x,y,xL,yL,'o')

/! polynomil !sults pp! to pss !xtly tou !!y

+t point, but l!ly t! polynomil is us!l!ss &o

!p!s!ntin ou imp!ssion o& t! +t t ny ot! point

-itin t! n! o& xF /is is ommon b!io &o i

o+! int!poltin polynomils. Mo tis !son, on! soul+

n!! tt!mpt to us! i o+! int!poltin polynomils to

!p!s!nt !xp!im!ntl +t.

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?b /s

Klot ll tionom!ty, !xpon!ntil n+ ny polynomil &untion

usin subplot, m!s, su&, plot

"ontrol #ngineering $ab %'&%'

control engineering lab 2

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