α α
T
1 s + T (T s + 1) Gad (s (s) = = α 1 (αT s + 1) s + αT
0, 05 05 < < α < 1 ω→0 ω→∞
⇒ ⇒
|Gad ( jω )| → α
φ = 00
|Gad ( jω )| → 1
φ = 00
φm 1−α 2
Radio = Radio =
x =
sen (φm ) =
1−α 2
+α
1 α 1+α
−
α = 0, 1 1/T
10/T
0
) −5 B d ( d u t i −10 n g a M−15
20 lg
α
10 lg
α
10 lg
α
−20 −1
0
10
10 1/T
1
ω
m
10 10/T
ω
m
10/T 10
2
10
60 50
) s 40 o d a r g 30 ( e s a F 20 10 0
−1
0
10
10 1/T
1
2
10
Frecuencia (r/s)
ωm
1 lg ωm = 2
1 1 lg + lg T αT
⇒ω
m =
1 √ T α
φm 1−α 2
Radio = Radio =
x =
sen (φm ) =
1−α 2
+α
1 α 1+α
−
α = 0, 1 1/T
10/T
0
) −5 B d ( d u t i −10 n g a M−15
20 lg
α
10 lg
α
10 lg
α
−20 −1
0
10
10 1/T
1
ω
m
10 10/T
ω
m
10/T 10
2
10
60 50
) s 40 o d a r g 30 ( e s a F 20 10 0
−1
0
10
10 1/T
1
2
10
Frecuencia (r/s)
ωm
1 lg ωm = 2
1 1 lg + lg T αT
⇒ω
m =
1 √ T α
20lg Gad ( jω jω)) ωm = 20lg
|
|
√ α = 10lg α α
1/α
K c φm
α
α
φm = 65 6 50
α = 0,05
K c M F original original M Goriginal φm ∆φ 50 φm = M F deseado deseado φm
− M F
∆φ original original + ∆φ
α α
ωm
ωm
ωm
K v
≥ 4
α
T
αT
450
100
K v
K c
K V ım s V = l´ s→0
1 = 1 s (0, (0, 1s + 1) 1) (s + 1)
⇒ K = 4 c
40 20 0 ) B −20 d ( d −40 u t i −60 n g a −80 M −100 −120 −140
−1
10
0
1
10
2
10
3
10
10
−90
−135 ) s o d a r g −180 ( e s a F −225
−270
−1
10
0
1
10
2
10
3
10
10
Frecuencia (r/s)
G(s) =
4 s(0,1s+1)(s+1)
180
∆φ = 100
φm φm = 450
3 70 − 180 + 100 = 37
α sen (φm ) =
1 α 1+α
−
⇒ α = 0, 25 ωm
10log α = 10 log(0 log(0,, 25) =
−6, 045 dB
ωm 100
20log
4 jω (0, 1 jω + 1) ( jω + 1) ωm
ω=ωm
α
T
T =
−6, 045 dB ⇒
√
⇒ T = 0, 743
ωm α
(0, 743s + 1) (0, 185s + 1)
4 (0, 743 jω + 1) jω (0, 1 jω + 1) ( jω + 1) (0, 185 jω1)
MF comp = 1800
−
ωm =2,69
900 + 15, 060 + 69, 610 + 63, 420
T
Gat (s) =
s + 1/T Ts + 1 = β s + 1/βT βT s + 1 1
≥ β ≥ 15
K c
Gat (s) = K c
Ts + 1 βT s + 1
= 0 dB
− 26, 460
β β
ωm = 2, 69 r ad/s
αT
1
Gad (s) = 4
20log
=
= 45, 690
β = 10
K c = 1
0
) −5 B d ( d u −10 t i n g a M−15
20 lg β
−20 lg β −20 −2
10
−1
0
10 1/T
1
10 10/T
10
0 −10
) s −20 o d a r g −30 ( e s a F −40 −50 −60
−2
10
−1
0
10
1
10
10
Frecuencia (r/s)
α = 0, 1
β ma´x
15
23 dB
K c MF original MGoriginal ωcnueva φdeseada
MF deseado
φ
50
ωcnueva
φdeseado ω=ωcnueva =
|
β
−1800 + MF
deseado +
20log G(s) ωcnueva = 20logβ
|
φ
⇒ |G(s)|
|
ωcnueva
0 dB
= β
ωcnueva
1 ωc = nueva T 10
1
G(s) = MG
MF 450
≥ 10 dB
K v
≥
s s s +1)( 25 +1)( 40 +1) s( 10
≥ 20
K v K voriginal = lim s s→0
K c
1 = 1 s s s s( 10 + 1)( 25 + 1)( 40 + 1)
K c = 20
⇒
40 20 0 ) B d ( −20 d u t i −40 n g −60 a M −80 −100 −120 0
10
1
2
10
3
10
10
−90 −135
) s o −180 d a r g −225 ( e s a F −270 −315 −360 0
10
1
2
10
3
10
10
Frecuencia (r/s)
G(s) =
20
s s s +1)( 25 +1)( 40 +1) s( 10
ωcnueva MF deseado φdeseado ω=ωcnueva =
|
β
−1800 + 450 + 50 = −130
|G(s)|
ωcnueva
= 4, 06
−
0 dB
T = 2, 27
Ts + 1 = 20 βT s + 1
jω
2, 27s + 1 9, 22s + 1
20 jω( 10 + 1)( 25 + 1)( jω 40 + 1)
20log
jω
2, 27 jω + 1 9, 22 jω + 1
900 + 23, 750 + 9, 980 + 6, 280 + 84, 280 MG MF
= 0, 06 dB
ω =4,4
− 88, 590
+ 1800 = 45, 690
φ=
MG ω = 11, 3 rad/s
MG =
= 4, 4 rad/s
β Gat (s) = K c
MF =
cnueva
⇒ β = 4, 06
1 ωc = nueva = 0, 44 T 10 T
⇒ω
−20log
⇒
20 jω( 10 + 1)( 25 + 1)( jω 40 + 1) jω
jω
φ =
−180
−1800
2, 27 jω + 1 9, 22 jω + 1
= ω =11,3
− (−12 dB) = 12 dB
K c
Gad
at (s)
−
= K c
s+ s+
1 T 1
1 αT 1
s+ s+
1 T 2
1 βT 2
T 1 s + 1 = K c αβ αT 1 s + 1
T 2 s + 1 βT 2 s + 1
0, 05 < α < 1
1
≥ β ≥ 15
α = 1/β K c = 1
α = 1/β = 0, 1
0
) −5 B d ( d u t i −10 n g a M−15
Aproximación asintótica
−20 −4 10
−3
10
−2
10T 0,1/ 2
−1
1/10 T2
0
0
1/10 T1
0
1/10 T1
ω
1
10T 10/ 1
2
α
10
3
10
60 40
) s 20 o d a r g 0 ( e s a F −20 −40 −60 −4 10
−3
10
−2
10T 2 0,1/
−1
1/10 T2
ω
0
1
10T 10/ 1
2
10
3
10
Frecuencia (r/s)
α = 1/β α
β
K c MF MGoriginal ωcnueva
ωcnueva 1 ωc = nueva T 2 10
φm
φm
α
β
βT 2
dB
−20logβ X dB
ωcnueva
0 dB dB 20logβ
20 dB/dc
−
α
G(s) =
1
K v
s(s+1)(s+4)
≥ 10
MF
≥ 500
MG
≥ 10 dB
K v K voriginal = lim s s→0
K c
1 = 0, 25 s(s + 1)(s + 4)
⇒
K c = 40
−150 0, 5 r/s
40 20
) B d ( d u t i n g a
0 −20 −40
M −60 −80 −100 −1 10
0
1
10
2
10
10
−90
−135 ) s o d a r g −180 ( e s a F −225
−270 −1
0
10
1
10
2
10
10
Frecuencia (r/s)
G(s) =
40
s(s+1)(s+4)
ωcnueva ωcoriginal
2, 8 r/s
ωcnueva = 2 r/s ωcnueva
MF ωcnueva
1 2 = = 0, 2 T 2 10 φm = 500 φm senφm =
α 1 α 1+α
−
β
⇒ T 2 = 5
− 00 + 100 = 600
⇒ α = 0, 07
βT 2
⇒ β = 13, 93
⇒ βT 2 = 69, 64
−20logβ = −22, 87 dB ≈ − 23 dB
1
≈ 0, 31
T 1
1
⇒ T 1 = 3, 23
αT 1
= 4, 42
⇒ αT 1 = 0, 23 ωcnueva
Gad
−
20log
MF comp =
−
40 jω( jω + 1)( jω + 4)
3, 23 jω + 1 0, 23 jω + 1
900 + 60, 950 + 24, 230 + 80, 240
MG φ =
MG =
−20log
20 jω( 10 + 1)( 25 + 1)( jω 40 + 1) jω
jω
5s + 1 69, 64s + 1
5 jω + 1 69, 64 jω + 1
− 22, 490
−180
ω = 4, 65 rad/s
3, 23s + 1 at (s) = 40 0, 23s + 1
+ 83, 660
= ω =1,8
−0, 28 dB
− 89, 540
+ 1800 = 56, 70
MG
⇒
φ =
−1800
2, 27 jω + 1 9, 22 jω + 1
= ω =4,65
− (−13 dB) = 13 dB
P
K c 20logK c K c > 1 60 40 20 0 ) B −20 d ( −40 d u −60 t i n −80 g a −100 M−120 −140 −160 −180 −3 10
Sin controlador Proporcional
−2
10
−1
0
10
1
10
10
2
10
−90
−135 ) s o d a r g −180 ( e s a F −225
−270 −3 10
−2
10
−1
0
10
1
10
10
2
10
Frecuencia (r/s)
PD
K c
T d
Gc (s) = K c (T d s + 1) = K c T d s +
1 = K c T d (s + ω0 ) T d
PD
K c = dB 20 /dc
PD 900 ω0
40
) 30 B d ( d u t i 20 n g a M10 Aproximación Asintótica 0 −1 10
0
10
ω
0
=
1/T
1
d
2
10
10
90 75
) s 60 o d a r g 45 ( e s a F 30 15 0 −1 10
0
1
10
2
10
10
Frecuencia (r/s)
PD PD
K c
ω0 T d
◦
450
ω0
◦
ω0
◦
ω0 450
1
K c
T i
K c
Gc (s) = K c
1 K c 1+ = T i s T i
|G ( jω)| c
ω →∞
T i s + 1 s
= K c P I
20logK c
K c
−900
30 20
) B d ( 10 d u t i n g 0 a M
20 lg K
−10
c
Aproximación Asintótica −20 −1 10
0
10
1
2
10 i 1/T
10
1 1/T 10 i
10
3
10
0 −15
) s o −30 d a r g −45 ( e s a F −60 −75 −90 −1 10
0
10
2
3
10
Frecuencia (r/s)
P I
00
ωcnueva
MF deseado
φdeseada
φ 50
ωcnueva
1 ωc = nueva T i 10 K c 0 dB 20logK c =
−20log |G( jω)|
ωcnueva
K c/T i
K 0 = K c/T i
ωcnueva φdeseada
MF deseado
−
20 dB/dc
−20log |G( jω)|
ωcnueva
ω=1
ω0
T i = 1/ω
0
Gc (s) = K 0
1 ω0
s+1 s
K c = T i
T i s + 1 s
P ID
K c
T d
T i
Gc (s) = K c
1 K c 1 + T d s + = T i s T i
T d T i s2 + T i s + 1 s P ID
30
20
) B d ( 10 d u t i n 0 g a M −10
−20 −1 10
0
1
10
10
2
10
3
4
10
5
10
10
90
) s o d a r g ( e s a F
45
0
−45
−90 −1 10
0
1
10
10
2
10
3
4
10
5
10
10
Frecuencia (r/s)
P ID
P I PD
G(s) =
4
s(s+1)(s+2)
60 40 20
) B 0 d ( d −20 u t i n −40 g a −60 M −80
−100 −120
−2
10
−1
10
0
10
1
2
10
10
−90
−135
) s o d a r g −180 ( e s a F −225
−270 −2
10
−1
10
0
10
1
2
10
10
Frecuencia (r/s)
G(s) =
4
s(s+1)(s+2)
500
0, 1 500
120 ,
φan˜ adir = M F deseado
− MF
original =
50
− 120 = 380 ω0 450
ωc = 1, 15
ω0 =
20log
4 jω( jω + 1)( jω + 2)
MF cont = 1800
1 = 1, 15 T d
(0, 87 jω + 1)
−
⇒ T = 0, 87 d
= 0 dB ω =ωcnueva
⇒
ωcnueva = 1, 65 r ad/s
900 + 58, 780 + 39, 520 + 62, 210 = 53, 910
P I P ID
ωcnueva φdeseado
ωcnueva φdeseado ωcnueva =
|
−1800 + 500 + 50 = −1250
⇒ω
cnueva
= 0, 43 rad/s
K c 20log G( jω) ωcnueva = 12, 41 dB
|
⇒ 20log (K ) = −12, 41 dB
|
⇒ K = 0, 24
c
c
T i 1 = 0, 043 T i
20log
4 jω( jω + 1)( jω + 2)
MF cont = 1800 +
−
⇒ T = 23, 25 i
0, 24 23, 25
23, 25 jω + 1 jω
900 + 21, 800 + 11, 310 + 84, 290
= 0 dB
ω =0,43
− 900
= 48, 890
50
G(s) =
50
1000 s(s+10)
20 r/seg
35 r/seg
P P I P D
P ID
MG
≥ 6 dB
≥ 450
MF
60
40
) B 20 d ( d u t i 0 n g a M−20 −40
−60
−1
0
10
1
10
2
10
3
10
10
−90
−105
) s −120 o d a r g −135 ( e s a F−150 −165
−180 −1
0
10
1
10
2
10
3
10
10
Frecuencia (r/s)
G(s) =
1000 s(s+10)
MG
−1350
17 dB
K c
20logK c =
−17 dB
⇒ K = 0, 141 c
20 r/seg ωc = 30, 7 r/seg MF =
180 PD
PD ω0 = 40 r/seg ω0 =
20log
1 = 40 T d
1000 (0, 025 jω + 1) jω ( jω + 10)
d
= 0 dB
⇒
ω =ωcnueva
MF cont = 1800 + 41, 990
⇒ T = 0, 025
−
ωcnueva = 36 r ad/s
900 + 74, 480 = 57, 510
P P D P I K p
≥ 40 K ≥ 120 K ≥ 120 K ≥ 120 p v
v
P ID
≥ 450 MF ≥ 450 MF ≥ 200 MF ≥ 500
MF
G(s) =
20 (s + 3) (s + 0, 1) (s + 5) (s + 10)2
10 0 −10
) B−20 d ( d −30 u t i n −40 g a M−50 −60 −70 −80 −3 10
−2
10
−1
10
0
1
10
2
10
10
0 −30 −60
) s −90 o d a r −120 g ( −150 e s a F−180 −210 −240 −270 −3
−2
10
10
−1
10
0
1
10
2
10
10
Frecuencia (r/s)
G(s) =
20(s+3) (s+0,1)(s+5)(s+10)2
K p K poriginal = lim s→0
K c
20 (s + 3) = 1, 2 (s + 0, 1) (s + 5) (s + 10)2
K c =
20logKc = 30, 5 dB
4 rad/s
600
40 = 33, 33 1, 2
K p K c = 100
1, 2 20log100 = 40 dB 0 dB
≈ 150
9, 5 rad/s
MF
PD ω0 =
20log 100(0, 105 jω + 1)
1 = 9, 5 T d
⇒ T
20 ( jω + 3) ( jω + 0, 1) ( jω + 5) ( jω + 10)2
MF cont = 1800 + 76, 290 + 52, 320
−
= 0, 105
d
= 0 dB
⇒
ω =ωcnueva
ωcnueva = 12, 3 r ad/s
89, 530 + 67, 880 + 50, 890 + 50, 890 = 49, 420
K v P I K v = 120 K v K 0 = K c/T i = 100
0 dB
ωcnueva ωcnueva
φdeseado K 0 φdeseado
|
ωcnueva
−2, 88 dB ≈ −3 dB
=
−1800 + 200 + 50 = −1500
⇒ω
cnueva
= 7, 95 rad/s
−20log |G( jω)|
ωcnueva
=
10 0
Ubicación del cero en w = 1,3
) B−10 d ( d −20 u t i n −30 g a M−40
Nuevo 0 dB
−50
dB a atenuar
polo en el origen sin ganancia
−60 −3 10
−2
10
10
−1
10
0
10
1
2
10
0 −30 −60
) s −90 o d a −120 r g ( e −150 s a −180 F
a esta frecuencia el MF será aproximadamente 25
−210 −240 −270 −3 10
−2
10
10
−1
10
0
10
1
2
10
Frecuencia (r/s)
P I
ω
≈ 1, 3
Gc (s) = 100
0, 77 jω + 1 20log 100 s
1 1,3 s + 1
s
=
20 ( jω + 3) ( jω + 0, 1) ( jω + 5) ( jω + 10)2
MF cont = 1800 + 69, 440 + 80, 770
K v
−
(0, 77s + 1) s
= 0 dB ωcnueva
ωcnueva = 8 r/seg
⇒
89, 280 + 57, 990 + 38, 660 + 38, 660 + 900 = 15, 620 150
50 MF
≥ 500
≥ 120
P ID P I P I
ω = 8 rad/seg ω = 10 rad/seg Gc (s) = 100
1 1,3 s
+1
s
1 10 s + 1
(0, 77s + 1) (0, 1s + 1) = 100 s
20log 100
(0,77 jω +1)(0,1 jω +1) s
20( jω +3) ( jω +0,1)( jω +5)( jω +10)2
ω =ωcnueva
= 0 dB
ωcnueva = 10, 1 r/seg MF cont = 1800 + 73, 460 + 82, 670 + 45, 280 89, 430 + 63, 660 + 45, 280 + 45, 280 = 47, 740
−
≥ 450
MF
MG
G(s) =
≥ 10 dB
K v
≥ 10
(0, 2s + 1) s (s + 1) (s + 2)
K v K c K V = l´ım s s→0
MG
(0, 2s + 1) = 1 s (s + 1) (s + 2)
⇒ K = 100 c
≈ −30
≈ −3 dB
MF
40 20
) B d ( d u t i n g a
0 −20 −40
M −60 −80 −100 −1
10
0
1
10
10
2
10
−90 −105
) s −120 o d a −135 r g ( −150 e s a F−165 −180 −195 −1
10
0
1
10
10
2
10
Frecuencia (r/s)
G(s) =
(0,2s+1) s(s+1)(s+2)
−1800 ∆φ = 100
φm φm = 450 α sen (φm ) =
− −30
+ 100 = 580
1 α = 0, 84 1+α
−
⇒ α = 0, 082 ωm
10log α = 10 log(0, 082) =
−10, 85 dB
ωm
−100 20log
100(0, 2 jω + 1) jω ( jω + 1) ( jω + 2) ωm T =
α
=
ω =ωm
T
1
√
ωm α
⇒ T = 0, 8
20log
100(0, 2 jω + 1) (0, 8 jω + 1) jω ( jω + 1) ( jω + 2) (0, 066 jω1)
MF comp = 1800 + 41, 020 MG
−
⇒ αT = 0, 066
(0, 8s + 1) (0, 066s + 1)
= 0 dB
ω =ωc
900 + 77, 05 + 65, 310
⇒
+ 73, 970
−1800
ωcnueva MF deseado
ωm = 4, 35 rad/s
αT
Gad (s) = 100
−10, 85 dB ⇒
ωc = 4, 35 r/seg
− 16, 020
= 46, 610
φdeseado ω=ωcnueva = β
|
−1800 + 450 + 50 = −130
|G(s)|
ωcnueva
= 13, 77
cnueva
= 0, 6 r ad/s 0 dB
⇒ β = 13, 77
1 ωc = nueva = 0, 066 T 10 T
⇒ω
T = 16, 66
β Gat (s) = K c
Ts + 1 16, 66s + 1 = 100 βT s + 1 229, 5s + 1
50
MF 20log
100(0, 2 jω + 1) jω ( jω + 1) ( jω + 2)
MF comp = 1800 + 6, 840
MG MF
−
16, 66 jω + 1 229, 5 jω + 1
= 0, 05 dB ω =0,6
900 + 30, 960 + 16, 700 + 84, 290
− 89, 580 φ =
MG ω = 2, 2 rad/s MG =
−20log
100(0, 2 jω + 1) jω ( jω + 1) ( jω + 2)
φ =
⇒
16, 66 jω + 1 229, 5 jω + 1
= 43, 890
−180
−1800
= ω =2,2
− (−20 dB) = 20 dB
ωcnueva Kc 2 r/s
MF =
1 2 = = 0, 2 T 2 10 φm = 450 φm senφm
α 1 α = 1+α
−
ωcnueva =
40
⇒ T 2 = 5
− 40 + 100 = 590
β
⇒ α = 0, 077
βT 2
⇒ β = 13
⇒ βT 2 = 65
−20logβ = −22, 28 dB ≈ − 23 dB
1 T 1
≈ 0, 37
1
⇒ T 1 = 2, 7
αT 1
= 4, 81
⇒ αT 1 = 0, 207 ωcnueva
Gad
20log
100(0, 2 jω + 1) jω ( jω + 1) ( jω + 2)
MF comp = 1800 + 16, 170 MG
at (s)
−
−
= 40
2, 7s + 1 0, 208s + 1
2, 7 jω + 1 0, 208 jω + 1
5 jω + 1 69, 64 jω + 1
900 + 55, 410 + 35, 940 + 82, 150
5s + 1 69, 64s + 1
− 89, 390
−1800
MF 46, 410 43, 890 66, 490
MG
∞ 20 dB ∞
ωc (rad/s) 4, 35 0, 6 1, 45
ω =1,45
=
−0, 06 dB
+ 75, 680
− 16, 770
= 66, 490
10
G(s) =
s(s2 +6s+12)
≥ 600
ess
≤ 0, 1
MF
K c ess =
K V = l´ım s s→0
MG
1 K v
≤ 0, 1
⇒ K ≥ 10 v
10 10 = s(s2 + 6s + 12) 12
⇒ K = 12 c
≈ −140
≈ −5 dB
MF
40 20
) B 0 d ( d u −20 t i n g a −40
M
−60 −80 −1
0
10
1
10
2
10
10
−90
−135 ) s o d a r g −180 ( e s a F−225
−270 −1
0
10
1
10
2
10
10
Frecuencia (r/s)
G(s) =
120
s(s2 +6s+12)
650 ∆φ = 100
φm φm = 600
140 + 100 = 840
− −
420 sen (φm ) =
α
1 α = 0, 19 1+α
−
⇒ α = 0, 66
ωm 2(10log α) = 2 (10 log(0, 27)) =
−14 dB
ωm 20log
120 jω(( jω) + 6 jω + 12) 2
=
ω =ωm
−14 dB ⇒
ωm = 8, 2 rad/s
−450 α 2(10log (0, 05)) =
ωm
−26 dB 1300
20log
120 2 jω(( jω) + 6 jω + 12) ωm
α
ωm
T
T =
20log
G(s) =
−26 dB ⇒
√
⇒ T = 0, 34
ωm α
(0, 34s + 1) (0, 0167s + 1)
2
120 jω(( jω)2 + 6 jω + 12)
MF comp = 1800 +
900 + 78, 680 + 75, 400 + 2 77, 50
−
ωm = 13, 25 rad/s
αT
1
Gad (s) = 12
=
(0, 34 jω + 1) (0, 0167 jω1)
= 0 dB
ω =ωc
(s+1)(s+2) s2 (s+0,5)
K a
≥ 4
− 12, 50 MF
K a = l´ım s s→0
(s + 1) (s + 2) = 4 s2 (s + 0, 5)
⇒ K = 1 c
= 65, 920
≥ 500
K c 2
ωc = 13, 25 r ad/s
⇒
MG
≈ −17 dB
≈ 250
MF
( 1, 0)
−
60
) 40 B d ( 20 d u t i n 0 g a M
−20
−40 −1
0
10
1
10
2
10
10
−90 −105
) s −120 o d a r −135 g ( −150 e s a F−165 −180
−1
0
10
1
10
2
10
10
Frecuencia (r/s)
G(s) =
(s+1)(s+2) s2 (s+0,5)
G(s) =
MF φm φm φm = 500
φ
∆φ =
− −250 − 100 = 150
α sen (φm ) =
(s+1)(s+2) s2 (s+0,5)
1 α = 0, 43 1+α
−
⇒ α = 0, 4 ωm
10log α = 10log(0, 4) =
−3, 9 dB
−100
ωm
−100 20log
( jω + 1) ( jω + 2) jω 2 ( jω + 0, 5) ωm T =
α 1
ω =ωm
T
√
20log
( jω + 1) ( jω + 2) (0, 69 jω + 1) jω 2 ( jω + 0, 5) (0, 38 jω1)
MF comp = 1800 + 66, 040 + 48, 370
P I
−3, 9 dB ⇒
−
⇒ αT = 0, 28
(0, 69s + 1) (0, 28s + 1)
= 0 dB
ωc = 2, 25 rad/s
⇒
ω=ωc
900 + 900 + 77, 470
ωm = 2, 25 rad/s
αT
⇒ T = 0, 69
ωm α
Gad (s) =
=
+ 57, 500
P ID
− 32, 50
P
= 61, 930
PD
MF MF
φω=
|
20log
( jω + 1) ( jω + 2) jω 2 ( jω + 0, 5)
−1200
=
ω =4,5
ω = 4, 5 rad/s
⇒
−12, 13 dB ⇒
20logK c = 12, 13 dB
⇒ 350
ω0
ωcoriginal
≈ 1, 7 ω0 =
ω0 = 3 1 =2 T d
⇒ T = 0, 33 d
K c = 4
20log
( jω + 1) ( jω + 2) jω 2 ( jω + 0, 5)
MF cont = 1800
G(s) = 8 dB
−
(0, 33 jω + 1)
61, 610 + 42, 770
= 0 dB ω =ωcnueva
−
900 + 900 + 74, 880
15(s+1) s(s+10)(s2 +10s+50)
ωcnueva = 1, 85 rad/s
⇒
+ 31, 660 = 61, 160
K v
≥ 450
≥ 3
MF
0
) −20 B d ( −40 d u t i n −60 g a M −80
−100 −1
0
10
1
10
2
10
10
−45
−90
) s o d −135 a r g ( e s −180 a F −225
−270 −1
0
10
1
10
2
10
10
Frecuencia (r/s)
G(s) =
15(s+1) s(s+10)(s2 +10s+50)
K c K a = l´ım s s→0
2
15 (s + 1) = 0, 03 s (s + 10) (s2 + 10s + 50) 0 dB
MG
≈ 4 dB
≈ 200
MF
⇒ K = 100 c
−40 dB
MG
≥
∆φ = 100
φm φm = 450
200 + 100 = 350
−
α sen (φm ) =
1 α = 0, 57 1+α
−
⇒ α = 0, 27 ωm
10log α = 10 log(0, 27) =
−5, 67 dB
ωm
20log
15 ( jω + 1)
2
jω ( jω + 10) ( jω) + 10 jω + 50 650
10log (0, 05) = ωm 20log
=
−5, 67 dB ⇒
ωm = 13 rad/s
ω =ωm
α
−13 dB
15 ( jω + 1)
2
jω ( jω + 10) ( jω) + 10 jω + 50
=
−13 dB ⇒
ωm = 18 rad/s
ω =ωm
φ = 2100 MF ma´ximo = 350
650
−
ωcnueva MF deseado φdeseado ω=ωcnueva =
|
−1800 + 450 + 50 = −130
⇒ω
cnueva
β
0 dB
|G(s)|
ωcnueva
= 1, 82
1 ωc = nueva = 0, 684 T 10 T
= 6, 84 rad/s
⇒ β = 1, 82 T = 1, 46
β Gat (s) = K c
MF
Ts + 1 1, 46s + 1 = 100 βT s + 1 2, 66s + 1
50
20log
15 ( jω + 1)
100
2
jω ( jω + 10) ( jω) + 10 jω + 50
MF comp = 1800 + 81, 680
−
MG MF
16, 66 jω + 1 229, 5 jω + 1
= 0, 06 dB ω =6,84
900 + 34, 370 + 67, 110 + 20, 20 + 84, 290
− 86, 850
φ =
MG ω = 11, 4 rad/s MG =
−20log
φ =
⇒
15( jω +1) jω ( jω +10)(( jω )2 +10 jω +50)
=
−1800
100
16,66 jω +1 229,5 jω +1
− (−8, 13 dB) = 8, 13 dB
= 47, 440
−180 =
ω =11,4
ωcnueva Kc 8 rad/s
MF =
1 8 = = 0, 8 T 2 10
⇒ T 2 = 1, 25
MF original
ωcnueva φm = 450
φm
α 1 α senφm = 1+α
−
ωcnueva =
450
β
− 350 + 150 = 250
⇒ α = 0, 406
φ = 150 MF ≈ 350
βT 2
⇒ β = 2, 46
⇒ βT 2 = 3, 08
−20logβ = −7, 8 dB ≈ −8 dB
1 T 1
≈ 3, 7
1
⇒ T 1 = 0, 27
αT 1
= 9, 12
⇒ αT 1 = 0, 11 ωcnueva
Gad
20log
− − −
at (s)
−
= 100
0, 27s + 1 0, 11s + 1
15 ( jω + 1)
2
1, 25s + 1 3, 08s + 1
0, 27s + 1 0, 11s + 1
100
jω ( jω + 10) ( jω) + 10 jω + 50
1, 25s + 1 3, 08s + 1
−0, 09 dB
ω =8,3
MF comp = 1800 + 83, 130 900 + 39, 690 + 69, 400 + 33, 420 + 84, 490 87, 760 + 65, 970 42, 320 = 51, 010
MG
−1800
MG MF
φ=
−180
MG ω = 14, 25 rad/s
MG =
=
−20log
15 ( jω + 1)
100
2
jω ( jω + 10) ( jω) + 10 jω + 50
P I
φ =
⇒
−1800
0, 27s + 1 0, 11s + 1
1, 25s + 1 3, 08s + 1
= 8, 87dB ω =14,25
P ID
ωc
K c
100
MG ω0
PD 250 ωcoriginal 9, 8 1 ω0 = = 12 T d
≈
ω0 = 10, 5
⇒ T = 0, 083 d
≈ 4 dB
≈ 200
MF
20log
15( jω +1) jω ( jω +10)(( jω )2 +10 jω +50)
100(0, 083 jω + 1)
ωcnueva = 11, 2 rad/s
⇒
= 0 dB ω =ωcnueva
MG MF cont = 1800
−
−
84, 90
1800
−
900 + 48, 240 + 72, 850 + 51, 120
G(s) = MF
≥ 450
MG
≥ 10 dB
Kp
+ 43, 020 = 45, 720
10(s+5) (s+50)(s2 +8s+100)
≥ 10
Kv
≥ 10
0
) B d ( d u t i n g a
−20
−40
−60
M −80
−100 0
10
1
2
10
10
3
10
45
0
) s o d a r g ( e s a F
−45
−90
−135
−180 0
10
1
2
10
10
3
10
Frecuencia (r/s)
G(s) =
10(s+5) (s+50)(s2 +8s+100)
G(s) = 1 s(s+1)(s+5)
≥ 550
MF
MG
≥ 8 dB
Kv
≥ 10/3
40 20 0 ) B d ( −20 d u −40 t i n g −60 a M −80 −100 −120 −2
10
−1
0
10
1
10
2
10
10
−90
−135 ) s o d a r g −180 ( e s a F −225
−270 −2
10
−1
0
10
1
10
2
10
10
Frecuencia (r/s)
G(s) =
G(s) = MF
50
1
s(s+1)(s+5)
10 3 (s+1)( 13 s+1)
0 dB 40 20
) B d ( 0 d u −20 t i n g a −40 M −60 −80 −1
10
0
1
10
2
10
10
0 −45
) s o −90 d a r g −135 ( e s a −180 F −225 −270
−1
10
0
1
10
2
10
10
Frecuencia (r/s)
G(s) =
G(s) =
10 3 (s+1)( 13 s+1)
4 s(s+2)
≥ 500
MF
MG
≥ 8 dB
Kv
≥ 20
40 20
) B 0 d ( d u t i −20 n g a −40 M −60 −80 −1
0
10
1
10
2
10
10
−90 −105
) s −120 o d a r g −135 ( e s a F −150 −165 −180 −1
0
10
1
10
2
10
10
Frecuencia (r/s)
G(s) =
4
s(s+2)
G(s) = MG MG
≥
MF 600 20 rad/s
6 dB
≥
MF 60 40
) B d ( 20 d u t i 0 n g a −20 M −40 −60 −1
10
0
10
1
10
2
3
10
10
−90 −105
) s o −120 d a r g −135 ( e s a −150 F −165 −180
−1
10
0
10
1
10
2
3
10
10
Frecuencia (r/s)
G(s) =
1000
s(s+10)
G(s) =
25
s2 +5s+25
1000 s(s+10)
0,1
MF 450 MF
Kv = 100
≥
10 rad/s
20 10
) 0 B d ( −10 d u t i −20 n g a −30 M−40 −50 −60 0
10
10
1
2
10
0 −15 −30 −45 ) s −60 o d a −75 r g −90 ( e s −105 a F −120 −135 −150 −165 −180 0
10
10
1
2
10
Frecuencia (r/s)
G(s) =
25
s2 +5s+25
G(s) = K = 1
≥ 450
MF K = 0, 1;10;100
50 40
) B d ( d u t i n g a
30 20 10 0
−10
M−20 −30 −40 −2
10
−1
0
10
1
10
10
−90 −105
) s −120 o d a r g −135 ( e s a −150 F −165 −180 −2
10
−1
0
10
1
10
10
Frecuencia (r/s)
G(s) =
1
s(s+0,5)
K s(s+0,5)
ess
≤ 0, 1
10(s+5) G(s) = (s+50)( s +8s+100) 8 dB K p 10 2
MF
Kv
≥
450
MG
≥
≥
≥ 10
−30 −40
) B −50 d ( d −60 u t i n −70 g a M −80 −90 −100 −1
10
0
1
10
2
10
10
3
10
30 15 0 −15 ) −30 s o d −45 a r −60 g −75 ( e −90 s a −105 F−120 −135 −150 −165 −180 −1
10
0
1
10
2
10
10
3
10
Frecuencia (r/s)
G(s) =
10(s+5) (s+50)(s2 +8s+100)
MF = 500 0 dB G(s) =
40 (s+1)(s+3)2
