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Z 1 , Z 1 =
− b ± √ b
2
− 4 ac
2 a
Z 1 , Z 1 =
Z 1 , Z 1 =
=−0.25 ± 1.56 i
Numerador 2 ∗( S +0.25+1.56 i )( S +0.25−1.56 i )
Denominador (s+0.4786) (s^2 + 5.521s + 8.357)
(S+0.4786)(S+2.7607 + 0.8579i)(S+2.7607 - 0.8579i)
Y ( S )
U ( S )
2 ( S + 0.25+1.56 i )( S +0.25−1.56 i )
( S +0.4786)( S +2.7607+0.8579 i )( S +2.7607−0.8579 i )
Y ( S )
U ( S )
K ( S + a + ib )( S + a − ib )
( S + f )( S + c + id )( S + c − id )
X 2 ( S )
X 1 ( S )
S + 2 +1.56 i
S +2.76+ 0.86 i
X 2 ( S )∗ S +2.76+ 0.86 i = X 1 ( S )( S + 2 +1.56 i )
x 2 ( t )+ x 2 ( t ) ( 2.76+ 0.86 i ) =
x 1 ( t ) + x 1 ( t )( 2 +1.56 i )
A =
[
]
B =
[
]
C =[ 4 5 1 ]
Estabilidad
det ( SI − A )= 0
det
([
S 0 0
0 S 0
0 0 S
]
[
])
det
([
S − 1 0
0 S − 1
6 11 S + 6
])
( S (^ S
2
S
3
2
( S + 1 ) ( S + 2 ) ( S + 3 )= 0
Polos S=-1,-2,-
Estable
Observabilidad
θ o
[
C
C ∗ A
C ∗ A ∗ A
]
[
]
Rango 2, sistema no es totalmente observable, una variable de estado no se puede observar
Controlabilidad
θ c
=[ B A ∗ B A ∗ A ∗ B ]
X 3 ( S )
X 2 ( S )
S + 1
S + 3
X 3 ( S ) ( S + 3 )= X 2 ( S ) ( S + 1 )
x 3 ( t )=− 3 x 3 ( t ) +
x 2 ( t )+ x 2 ( t )
x 3 (^ t )=− 3 x 3 ( t )^ + 2 x 2 (^ t )^ + u ( t )+ x 1 ( t )+ x 2 ( t )
x 3 (^ t )=− 3 x 3 ( t )^ + 3 x 2 (^ t )+ u ( t )+ x 1 ( t )
x 1 (^ t )=− x 1 (^ t )^ + u ( t )
x 2 (^ t )= 2 x 2 (^ t )^ + u ( t )+ x 1 ( t )
x 3 (^ t )=− 3 x 3 ( t )^ + 3 x 2 (^ t )+ u ( t )+ x 1 ( t )
y ( t ) = x 3 ( t )
x 1
x 2
x 3
[
]
x 1
x 2
x 3
[
]
u ( t )
y ( t ) =[ 0 0 1 ]
x 1
x 2
x 3
V.E
ic ( t )= C vc ´( t )
vL ( t )= L
d il
d t
VE: Voltaje en el condensador y corriente en la bobina.
Vs ( t )= i ( t ) R 1 + vc ( t ) + R 1 ic ( t )
i ( t )= ic ( t ) + il ( t )
Vs ( t )=( ic ( t ) + il ( t )) R 1 + vc ( t ) + R 1 ic ( t )
Vs ( t )= ic ( t ) R 1 + il ( t ) R 1 + vc ( t ) + R 1 ic ( t )
Vs ( t )= 2 R 1 ic ( t ) + il ( t ) R 1 + vc ( t )
Vs ( t )= 2 R 1 C vc ´ ( t )+ il ( t ) R 1 + vc ( t )
vc ´ ( t )=
2 R 1 C
Vs ( t )−
2 C
il ( t )−
2 R 1 C
vc ( t )
Vs ( t )= VR 1 + VR 2 + Vl ( t )
Vs ( t )= VR 1 + R 2 iL ( t )+ L
d il
d t
i ( t )= ic ( t ) + il ( t )
Vs ( t )= R 1 ( ic ( t ) + il ( t ))+ R 2 iL ( t )+ L
d il
d t
Vs ( t )= R 1 ic ( t ) + R 1 il ( t )+ R 2 iL ( t )+ L
d il
d t
Vs ( t )= R 1 ic ( t ) +( R 1 + R 2 ) iL ( t )+ L
d il
d t
Vs ( t )= 2 R 1 ic ( t ) + il ( t ) R 1 + vc ( t )
y (^ t )^ =[ 1 0 ]^
vc ( t )
il ( t )
θo =[
C
CA
]
CA =[ 1 0 ]
[
]
=[−0.5 − 500 ]
θo =
[
]
θc =[ B A ∗ B ]
A ∗ B =
[
][
]
[
]
θc =[
]
Rango 2.
det ( SI − A )= 0
det
([
S 0
0 S
]
[
])
det
([
S + 0.5 500
0 .5 S + 550
])
( S +0.5 ) ( S + 550 ) −( 0.5∗ 500 )= 0
S
2
+550.5 S + 25 = 0
Z 1 , Z 2 =
2
− 100
Z 1 , Z 2 =
Z 1 , Z 2 =
Z 1 , Z 2 =
Z 1 =−0.
Z 1 =−550.
