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Transcript of 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.
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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
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