6.003: Signals and Systems
Lecture 9
October 8, 2009
1
6.003: Signals and Systems
SecondOrder Systems
October 8, 2009
Last Time
We analyzed a mass and spring system.
x
(
t
)
y
(
t
)
F
=
K
(
x
(
t
)
−
y
(
t
)
)
=
M
¨
y
(
t
)
+
K
M
A
A
−
1
x
(
t
)
y
(
t
)
˙
y
(
t
)
¨
y
(
t
)
Y
X
=
K
M
A
2
1 +
K
M
A
2
Last Time
We also analyzed a leaky tanks system.
r
0
(
t
)
r
1
(
t
)
r
2
(
t
)
h
1
(
t
)
h
2
(
t
)
τ
1
˙
r
1
(
t
) =
r
0
(
t
)
−
r
1
(
t
)
τ
2
˙
r
2
(
t
) =
r
1
(
t
)
−
r
2
(
t
)
+
1
τ
1
A
+
1
τ
2
A
r
0
(
t
)
r
2
(
t
)
˙
r
1
(
t
)
r
1
(
t
)
˙
r
2
(
t
)
−
−
R
2
R
0
=
A
/τ
1
1 +
A
/τ
1
×
A
/τ
2
1 +
A
/τ
2
SecondOrder Systems
Today:
Look more carefully at growth and decay of oscillatory re
sponses by studying an analogous electrical circuit.
v
i
v
o
R
L
C
But First ...
The canonical forms for CT and DT differ.
+
A
s
0
X
Y
+
Delay
z
0
X
Y
H
=
Y
X
=
A
1
−
s
0
A
H
=
Y
X
=
1
1
−
z
0
R
h
(
t
) =
e
s
0
t
u
(
t
)
h
[
n
] =
z
n
0
u
[
n
]
A →
1
s
R →
1
z
s
plane
s
plane
z
plane
z
plane
Check Yourself
What if we had used the DT canonical form for CT?
+
A
s
0
X
Y
What is the impulse response of this system?
1.
s
0
e
s
0
t
u
(
t
)
2.
s
t
0
u
(
t
)
3.
1 +
s
0
e
s
0
t
u
(
t
)
4.
δ
(
t
) +
s
0
e
s
0
t
u
(
t
)
5.
none of the above
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6.003: Signals and Systems
Lecture 9
October 8, 2009
2
SecondOrder Systems
Today:
Look more carefully at growth and decay of oscillatory re
sponses by studying an analogous electrical circuit.
v
i
v
o
R
L
C
SecondOrder Systems
Solve with state variable approach.
State variables represent the minimum knowledge of the past (
t < t
0
)
needed to propagate the output into the future (
t > t
0
).
Check Yourself
v
C
i
C
v
L
i
L
i
C
=
C
dv
C
dt
v
L
=
L
di
L
dt
Which of the following can be state variables?
1.
v
C
and
v
L
2.
i
C
and
v
L
3.
i
C
and
i
L
4.
v
C
and
i
L
5.
i
C
and
v
C
and
i
L
and
v
L
6. none of above
SecondOrder Systems
State variable approach:
determine expressions for derivatives of
state variables in terms of (undifferentiated) state variables.
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 Spring '11
 DennisM.Freeman
 Trigraph, iL, secondorder systems, Exponential decay

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