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real axis

imag. axis

Ο

Ο

Ο

Ο

Ο

Ο

Ο

Ο

Ο

Ο

Ο

Ο

Ο

Ο

Ο

Ο

Ο

Ο

Ο

Ο

Ο

Ο

Ο

ZerosΟ

×××

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×××

Poles×

,! ?<9 Y

# 6%7 UYU

,1122

&

= <83- : <<4-4 =

= <83- <<4-4 =

= -- -. : .<<8 =

= -- -. .<<8 =

= 888-488 : 3<4.. =

= 888-488 3<4.. =

= 3.< : 4 < < =

= 3.< 4 < < =

= < .8<- : .443 =

= < .8<- .443 =

= .33833 : .3<< =

= .33833 .3<< =

.

= 4.333 : <4833 =

= 4.333 <4833 =

= ..43.. : 3-- 8 =

= ..43.. 3-- 8 =

= 4..- : -8<84 =

= 4..- -8<84 =

= 3 =

= 3-33-8 : 33<3<3 =

= 3-33-8 33<3<3 =

= 3 -4<38 : <<8<<8 =

= 3 -4<38 <<8<<8 =

2) *5B

D# ! 7 6%7 !Y !'9 K'HJK

F9

'H<M < N M<N M<N J

& 9

9 ! !

< 9 U! BYU

9 '

3$9

& !' ! 7

x∗ = 7πx + u

y = 10πx + u

3 6%79

,! HH

,&.; @9

,&39

,&3; @9

,&39

.=

,&01HHAAAA2

&

132 1 0'2

= D / N =

1-2 & &

-34433<4

12 & D &

3

1<2 & / &

3<384-.

182 & &

3

12 & N 1 2 &

1.2 & $ &

,1A3A329

!9

.?

-110

010

110

210

-8

-7

-6

-5

-4

-3

-2

-1

0Magnitude

Hz

db

.

-110

010

110

210

-180

-160

-140

-120

-100

-80

-60

-40

-20

0Phase

Hz

degrees

.

,! ? 9

2+ &!.&!!(> ! (!% !!

% !' !Y 6%7 # K(HJK !' D /! #

5 !' ! K- HJK /! !

! F 9

MNA- HJ

& 9 ! K(HJK

9 Y

3$9

&# !

A =[

1 −37 −2

] B =

[10

] C =

[4 −2

]1

.@

3 6%79

,&F3 9. -G9

,D&F39G9

,/&F< -G9

,&01HHAADA/2

&

132 1 0'2

= D / N =

1-2 & &

= 3 =

= . - =

12 & D &

= 3 =

= =

1<2 & / &

= < - =

182 & &

12 & N 1 2 &

= =

= =

1.2 & $ &

,

,&-12

&

: <

34 : : 2

2/ (!% !(

* 6%7 !

.A

s1 s2

s1

s1

s1

s1

s2

s2

s2

s2

+

+

+

s1*s2

s1 + s2

[s1 , s2]

[s1; s2 ]

s1 / . s2

,! ?=9 4

! 6%7 !Y ! !9

K -HJK ! !' !

F9

A -H J

&9

9 Y

9 + OP7&+-Q -

K HJK

F9

H M<NJ

&9

.E

9 !

9 !

3$9

S1 = 1s+2 ,

S2 = 13−s

& Y 9

. S1 + S2

S1 ∗ S2

<[

S1S2

]

[S1S2]

= S1S2

3 6%79

,&01AHH2

&

,3&31:-2

3 &

3

- :

,-&312

- &

3

,3&01HHA329

,-&01HHA-29

,-&-1-29

,3&-1329

.<

,1-2

& W W : W 3 W

0 & W3 W

,&-;39

,12

W 3 W W W

& W - W : W 3 W

0 & W3 W

,&-12

&

3

: 2

! ! !/! ! ! 9

,3;-

&

3

: 2

,3:-

&

8

: 2

,F39-G

&

= 3 =

= =

= - : =

= =

= 3 =

.<.

= =

= =

,F3 -G

&

= 3 3 =

= =

= - : =

,3-

&

. : 2

22 . %!

3 ! ' ! ' !' ! 7

&89 0&3'-;9 &3'9

&3'A

1A32&39

1A-;2&39

J &3'-;A

13AJ2&39

1AJ2&9

&39

+ ,3A

&9

&-'3A

J &-'-;3A

&-8;113AJ2:1AJ32:1:3AJ2:1AJ:3229

&11AJ229

,

&9

.<

A

1AJ2&9

1A0A29

3 ! 9

% #' ! /! D Y (nX2n) 5 /! ! ! ! :

% #' (V 1 = 1000, V 0 = 0)

% ' ! + DF .- ! 9 Vi,j = 1

4(V i−1,j + Vi,j−1 + Vi+1,jaVi,j+1) #$ V1 = 1000, V0 = 0

3 D! DF Y ! ! # +! DF .-

% ' D# !

!' D ! D# !'

0

100

200

300

400

500

600

700

800

900

1000

Z

01020304050

X

0 10 20 30 40 50 60 70 80 90 100Y

,! ??9 !' F

.<<

0

100

200

300

400

500

600

700

800

900

1000

Z

050100Y0 10 20 30 40 50X

,! ?@9

0

10

20

30

40

50

X

0 10 20 30 40 50 60 70 80 90 100Y

,! ?A9 ! <.+-

.<

29 (& &! .

3 6%7 ! ! ! 6& ! W ' !Y ! L(HJL LHJL

LHJL ! ! !

! F 9

AH<<J

&9

9 !' +'-

9 /! ! !'

9 ! !'

9 !

3$9

6%79

,# 1# 2

,&01AHH29

,&'3'9

,#

,&31;;-:329

,/

,O&39

, &39

,&O;13: ;29

,&;13:;29

,0&01HHA2

,&1HHAA029

,-1AA-29

,1H# 131;;-:322 O&3 &3HAHHAH012H29

!9

# .<=

0 1 2 3 4 5 60.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Proporcional.Derivativo (1/(s**2+1)) K=1 D=1

tiempo

y(t)

,! ?E9 &

' Z D# !Y L HJ L

39

,&01AHH29

,&'3'9

,#

,&31;;-:329

,/

, &3'8

,O&9

, &39

,&O;13: ;29

,&;13:;29

,0&01HHA29

,&1HHAA029

,-1AA29

.<?

,

,1H# 131;;-:322 O&F3 - < 8G &3HAHHAH012H29

>!9

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

1.2

Proporcional.Derivativo (1/(s**2+1)) K=[1 2 3 4 5] D=1

tiempo

y(t)

,! ?.9 [O .< =

&9

,&01AHH29

,&'3'9

,#

,&31;;-:329

,/

,&9

, &8'8'-89

,&:39

,O&9

, &9

,&O;13: ;29

# .<@

,&;13:;29

,0&01HHA29

,&1HHAA029

,-1AA29

,

,1H# 131;;-:322 O& &F8A 3A 38A -A -8GHAH

>!9

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

1.2

Proporcional.Derivativo (1/(s**2+1)) K=3 D=[0.5, 1, 1.5, 2, 2.5]

tiempo

y(t)

,! ?..9 &O = . .= =

V

,129

,&01AHH29

,&'3'89

,#

,&31<;:29

,/ 1#52

, &3'89

.<A

,O&9

,5&-9

,&O;13:315;229

,&;13:;29

,0&01HHA29

,&1HHAA029

,-1AA29

,9

,1H# 5! #&31<:2A O&F3 - < 8G 5&-HAHHAH012H29

>!9

0 10 20 30 40 500.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Proporcional Integrador Planta=1/(4s+6), K=[1 2 3 4 5] I=2

tiempo

y(t)

,! ?.9 6 O . < = 6O

6

,129

,&01AHH29

,&'3'89

,#

# .<E

,&31<;:29

,/ 1#52

,&9

, &<'<'-9

,&:39

,O&9

,5&9

,&O;13:315;229

,&;13:;29

,0&01HHA29

,&1HHAA029

,-1AA29

,9

,1H# 5! #&31<:2A O&F3 - < 8G 5&-HAHHAH01

!9

0 4 8 12 16 20 24 28 320.0

0.2

0.4

0.6

0.8

1.0

1.2

Proporcional Integrador Planta=1/(4s+6), K=3 I=[0.4, 0.8, 1.2, 1.6, 2]

tiempo

y(t)

,! ?.<9 6 [O< 6OP A . .? Q

.

.!& C! ? (& !(!.D(

6 %! (! ('Y %! ' *' ) 7D !'9 %%6)5 76&623)V65)%

!"#

I % =

FS @E

.

.A ...! A . @@# .E A <. < ..=$! ?< A .< 6& .<

& A& =<& <& 7 . &' .

.<3! =3&5 ?A3$ E. ..= .<3/! E<F .AF ..F .. .

. S . ...! !' .<!' .<!' .<!' .<!' .<!' .<!' .<,*)65)3 ==,! ..?,! ..E! .! ..?

AE(>I,65 @.(D# .E

B A

..= .

% .<. @E AA @E =%6)*: . = .A ... .. ..S @E

2 .@ <

. .

. )

2 <2Y 2Y 2Y <?2Y ! <2Y .2Y ! <=2Y ! < 2Y ! < @

) $ =@ .

5 E5 ' ...

. .< .E @. .E @A E?< E@ EA< .< ! =. .A E ' = . .E .! .' .@0 .A

>X .E>XY .EXY ..> => @E A?> .>2 .

@= .@ ..=

/ .. .< .? .A @E A! . . .?

. .A ..=4! .@

. A ? ! .= ? # .= = ! @ ? . = .. < @ ! !Y E ! A ! ! E ! @ ! <. ! <

0# ..E0 ...;0 A;6)&5; < ?

F @@F @@F E<