s
s
1 + KG K G(s)H (s) = 0 K K s
|KG (s)H (s)| ∠KG (s)H (s) = −1 + j 0 |KG (s)H (s)| = 1
∠KG (s)H (s)
= 1800 ± k 3600
k = 0, ±1, ±2, . . . . . .
m
(s + zi )
KG (s)H (s) = K
i=1 n
(s + p j )
j =1
m
|s + zi |
K
i=1 n
=1
|s + p j |
j =1
m
n
[∠ (s + p j )] = 1800 ± k3600
[∠ (s + zi )] −
i=1
i=1
K
K n
m
(s + p j ) + K
j =1
i=1
n
(s + zi )
n
=0 ⇒
(s + p j )
m
(s + p j ) + K
j =1
j =1
K
K
K
G(s)H (s) =
(s + 3) s(s + 1)(s + 2)(s + 4)
i=1
(s + zi )
=0
s
G(s)H (s) =
p1
s1
p2
(s+3) s(s+1)(s+2)(s+4)
z1
p1
s1
∠ (s + z1 )
− [∠ (s + p1 ) + ∠ (s + p2 )] = 00 − 00 + 00 = 00 = −1800 s2
p1
z1
s2
∠ (s + z1 )
− [∠ (s + p1 ) + ∠ (s + p2 )] = 00 − 1800 + 00 = −1800
s3
s4
s3
s4
s3
s4
∠ (s + z1 ) ∠ (s + z1 )
− [∠ (s + p1 ) + ∠ (s + p2 )] = 00 − 1800 + 00 = −00 = −1800
− [∠ (s + p1 ) + ∠ (s + p2 )] = −1800 − 1800 + 1800 = −3(1800 )
G(s)H (s) =
n
(s+3) s(s+1)(s+2)(s+4)
m
n>m N A σA φA N A = n − m n
σA =
polos de G(s)H (s) − N A
ceros de G(s)H (s)
φA =
(2q − 1) N A
=
m
(− p j ) −
j =1
1800
q = 0, 1, 2, · · · , (N A − 1)
N A σA
φA
N A = 4 − 1 = 3 σA =
(−1 − 2 − 4) − 3
(−3)
4 = − = −1, 33 3
φA1 = 600 φA2 = 1800 φA3 = 3000
i=1
N A
(−zi )
σA G(s)H (s) =
(s+3) s(s+1)(s+2)(s+4)
s = −2 s = −3 s = −4 K
K K K K
K sR
K = p (s) = −
dK ds
s = 0
s = −1
sR
1 G(s)H (s)
dp(s) = ds
=0 sR
K
K = −
s(s + 1)(s + 2)(s + 4) s4 + 7s3 + 14s2 + 8s =− (s + 3) (s + 3)
dK ds sR1 = −0, 43
sR
3s4 + 26s3 + 77s2 + 84s + 24 =− (s + 3)2
sR2 = −1, 6
sR3 = −3, 3 + 0 , 68 j
=0 sR
sR4 = −3, 3 + 0 , 68 j
sR1 = −0, 43
G(s)H (s) =
(s+3) s(s+1)(s+2)(s+4)
s4 + 7s3 + 14s2 + 8s + K (s + 3) = 0
90−K 7 −K 2 −65K +720
b1 =
c1 =
b1
> 0
s4 s3 s2 s1 s0
14 3 K 8 + K 0 b1 3K 0 0 0 c1 3K 0
⇒ ⇒
b1 > 0 K 1 < −74, 64 K 2 < 9 , 65
1 7
⇒ ⇒
K < 90 K < 9 , 65
s1 = −4, 37
s2 = −2, 62
s3 = +1 , 58 j
s4 = −1, 58 j
ω = 1, 58 K = 9, 65 s = j ω K
ω
( jω )4 + 7 ( jω )3 + 14 ( jω )2 + 8 ( jω ) + K ( jω + 3) = 0
ω 4 − 7ω 3 j − 14ω 2 + (8 + K ) ωj + 3 = 0
ω4 − 14ω 2 + 3K = 0
ω = 1, 5877
⇒ 2
−7ω − 8 − K = 0
K = 9, 65
s
N A
σA
φA
K
8
6 System: sys Gain: 9.38 Pole: −0.00784 + 1.57i Damping: 0.00499 Overshoot (%): 98.4 Frequency (rad/sec): 1.57
4 System: sys Gain: 0.533 Pole: −0.458 Damping: 1 Overshoot (%): 0 Frequency (rad/sec): 0.458
o i r 2 a n i g a 0 m I e j E −2 −4
−6
−8 −12
−10
−8
−6
−4
−2
0
2
4
Eje Real
G(s)H (s) =
(s+3) s(s+1)(s+2)(s+4)
K G(s)H (s) =
(s+3)(s+8) s(s+1)(s+5)(s+6)
G(s)H (s) =
(s+8) (s+1)(s+6)2 (s+3)2
G(s)H (s) =
(s+3)(s+8) s(s+1)(s+5)(s+6)
G(s)H (s) =
(s+3)(s+8) s(s+1)(s+5)(s+6)
N A
σA
φA N A = 4 − 2 = 2 σA =
s = 0
(−1 − 5 − 6) − 2
(−3 − 8)
1 = − = −0, 5 2
φA1 = 900 φA2 = 2700
s = −1 s= 0
s = −1 K = −
dK ds sR1 = −9, 88
sR
s(s + 1)(s + 5)(s + 6) s4 + 12s3 + 41s2 + 30s =− s2 + 11s + 24 (s + 3)(s + 8)
2s5 + 45s4 + 360s3 + 1285s2 + 1968s + 720 =− (s2 + 11s + 24)2
sR2 = −5, 55
sR3 = −3, 27 + 1, 4 j
=0 sR
sR4 = −3, 27 + 1, 4 j sR1 = −0, 52
G(s)H (s) =
K
(s+3)(s+8) s(s+1)(s+5)(s+6)
sR5 = −0, 52
8
6
4 System: sys Gain: 0.33 Pole: −0.529 Damping: 1 Overshoot (%): 0 Frequency (rad/sec): 0.529
o i r 2 a n i g a 0 m I e j E −2 −4
−6
−8 −9
−8
−7
−6
−5
−4
−3
−2
−1
0
1
Eje Real
G(s)H (s) = G(s)H (s) =
(s+3)(s+8) s(s+1)(s+5)(s+6)
(s+8) (s+1)(s+6)2 (s+3)2
G(s)H (s) =
(s+8) (s+1)(s+6)2 (s+3)2
N A
σA
φA N A = 5 − 1 = 4
σA =
(−1 − 3 − 3 − 6 − 6) − 4
(−8)
=−
11 = −2, 75 4
φA1 = 450 φA2 = 1350 φA3 = 2250 φA4 = 3150
s = −1
s = −3
s = −3
s = −6 s = −6
s = −8
K = −
(s + 1)(s + 6)2 (s + 3)2 s5 + 19s4 + 135s3 + 441s2 + 648s + 324 =− (s + 8) (s + 8)
dK ds
sR
sR1 = −8, 85
4s5 + 97s4 + 878s3 + 3681s2 + 7056s + 4860 =− (s + 8)2 sR2 = −6
sR3 = −4, 8
sR4 = −3
=0 sR
sR5 = −1, 58
sR3 = −4, 8
G(s)H (s) =
sR5 = −1, 58
(s+8) (s+1)(s+6)2 (s+3)2
s5 + 19s4 + 135s3 + 441s2 + 648s + 324 + K (s + 8) = 0
( jω )5 + 19 ( jω )4 + 135 ( jω )3 + 441 ( jω )2 + (648 + K ) ( jω ) + 324 + 8K = 0
jω 5 + 19ω 4 − 135 jω 3 − 441ω 2 + (648 + K ) jω + 324 + 8K = 0
19ω 4 − 441ω 2 + 324 + 8K = 0
ω = 2, 6
⇒ ω4
− 135ω 2
+ 648 + K = 0
K = 223
10
8 System: sys Gain: 221 Pole: −0.00689 + 2.59i Damping: 0.00266 Overshoot (%): 99.2 Frequency (rad/sec): 2.59
6
4
o i r a n i g a m I e j E
System: sys Gain: 5.53 Pole: −4.86 Damping: 1 Overshoot (%): 0 Frequency (rad/sec): 4.86
2
0 System: sys Gain: 3.55 Pole: −1.62 Damping: 1 Overshoot (%): 0 Frequency (rad/sec): 1.62
−2
−4
−6
−8
−10 −12
−10
−8
−6
−4
−2
0
2
4
6
Eje Real
G(s)H (s) =
(s+8) (s+1)(s+6)2 (s+3)2
K
K
G(s)H (s) =
1 (s + 2)
K K K K
K G(s)H (s) =
1 (s + 2)(s + 4)
1.5
1
1.5
o i r a n i g a m I e j E
0.5 1
o i r a n i g a m I e j E
0
−0.5
0.5
0
−0.5
−1 −1
−1.5 −40
−1.5 −4.5
−35
−30
−25
−20
−15
−10
−5
0
−4
−3.5
5
K K
G(s)H (s) =
−3
−2.5
−2
Eje Real
Eje Real
1 s(s + 2)(s + 4)
−1.5
−1
−0.5
0
G(s)H (s) =
(s + 3) s(s + 2)(s + 4)
6
10
8 4 6
o i r a n i g a m I e j E
4
o i r a n i g a m I e j E
2
0
2
0
−2
−2 −4
−6 −4 −8
−6 −5
−4
−3
−2
−1
0
1
−10 −4.5
2
Eje Real
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
Eje Real
K
K K K K
0
0.5
K
Lugar Geométrico 4
3
2
1
o i r a n i g a m 0 I e j E
S3
S4
S1
S2
−1
Polo Dominante −2
−3
−4 −5
−4
−3
−2 Eje Real
−1
0
1
2
K ts(2%) = 4τ ⇒ ts(2%) = 4
s1
1
s1
⇒ ts(2%) ≈ 4
1 K = G(s)H (s)
1 0, 2
⇒ ts(2%) ≈ 5
s2
ts(2%) = 4τ ⇒ ts(2%) ≈ 4
s=−0,43
|s| |s + 1| |s + 2| |s + 4| = |s + 3|
1 0, 43
⇒ ts(2%) ≈ 9, 3
⇒ K = s=−0,43
(0, 43)(0, 57)(1, 57)(3, 57) ⇒ K = 0, 5345 (2, 57)
ζ K K
ζ ζ = 0, 5
1
0.8
seda = 0,5
0.6
0.4
o i r a0.2 n i g a 0 m I e j −0.2 E −0.4
−0.6
−0.8
−1 −5 −4.8−4.6−4.4−4.2−4 −3.8−3.6−3.4−3.2 −3 −2.8−2.6−2.4−2.2 −2 −1.8−1.6−1.4−1.2−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
Eje Real
ζ = 0, 5 s = −0, 35 + 0, 6 j K
|s| |s + 1| |s + 2| |s + 4| K = |s + 3|
⇒ K = s=−0,35+0,6 j
(0, 69)(0, 88)(1, 76)(3, 70) ⇒ K = 1, 47 (2, 72)
ts(2%) ≤ 10
s1 = −0, 4
s2,3 = −0, 4 ± 0, 4 j
K s3 s2 |s| |s + 1| |s + 2| |s + 4| K 1 = |s + 3| K 2 =
|s| |s + 1| |s + 2| |s + 4| |s + 3|
⇒ K 1 =
(0, 4)(0, 6)(1, 6)(3, 6) ⇒ K 1 = 0, 53 (2, 6)
⇒ K 2 =
(0, 57)(0, 72)(1, 65)(3, 62) ⇒ K 2 = 0, 93 (2, 63)
s1 =−0,4
s2 =−0,4+0,4 j
K = K 2
K = K 1 K = K 2 ζ
1
ts = 10
0.8
0.6
S2, S3
S1
0.4
o i r a0.2 n i g a 0 m I e j −0.2 E −0.4
−0.6
−0.8
−1 −5 −4.8−4.6−4.4−4.2−4 −3.8−3.6−3.4−3.2−3 −2.8−2.6−2.4−2.2−2 −1.8−1.6−1.4−1.2−1 −0.8−0.6−0.4−0.2 0 0.2 0.4 0.6 0.8 1
Eje Real
ts(2%) = 10
5
4
3
2
o i r a n i g a m I e j E
1
0
−1
−2
−3
−4
−5 −14
−12
−10
−8
−6
−4
−2
0
2
Eje Real
(ωd = 4)
(s + 5, 8)
ωd K
5
4
3
S2
S1
2
o i r a 1 n i g a 0 m I e j E −1 −2
−3
−4
−5 −14
−12
−10
−8
−6
−4
−2
0
2
Eje Real
ωd = 4
GLA (s) =
|s| |s + 4| |s + 6| K 1 = |s + 5| |s + 8| K 2 =
K
|s| |s + 4| |s + 6| |s + 5| |s + 8|
(s + 5) (s + 8) s (s + 4) (s + 6) ⇒ K 1 =
s1 =−5,25+4 j
⇒ K 2 =
s2 =−10,5+4 j
(6, 6)(4, 19)(4, 07) ⇒ K 1 = 5, 79 (4, 01)(4, 85)
(11, 24)(7, 63)(6, 02) ⇒ K 2 = 16, 09 (6, 80)(4, 72)
K 1 K
K 2
ωd s1 s3
|s| |s + 4| |s + 6| K 3 = |s + 5| |s + 8|
K
⇒ K 3 = s3 =−5,25
(5, 25)(1, 25)(2, 25) ⇒ K 3 = 18, 17 (0, 25)(3, 25)
5
4
3
S1 2
o i r a n i g a m I e j E
1
S3 0
−1
−2
−3
−4
−5 −14
−12
−10
−8
−6
−4
−2
0
2
Eje Real
ts K 3 K 1 s3
5
1.5
4
Raíces para altos valores de K
3
Raíces para altos valores de K
1
2
o i r a 1 n i g a 0 m I e−1 j E
o i r a n i g a m I e j E
0.5
0
−0.5 −2
−3 −1 −4
−5 −14
−12
−10
−8
−6 Eje Real
−4
−2
0
2
−1.5 −9
−8
−7
−6
−5
−4
Eje Real
K
−3
−2
−1
0
1
