1
Lecture 9: Time-Response Characteristics (II)
•
Time Specifications of Continuous-Time Systems
•
Mapping the
s
-Plane to the
z
-Plane
•
Time Specifications of Sampled-Data System
•
Reading: Chapter 6.4
Sampled-Data Systems
C
(
s
)
H
(
s
)
G
(
s
)
R
(
s
)
+
¡
T
Transfer function from
R
(
z
) to
C
(
z
):
C
(
z
)
R
(
z
)
=
G
(
z
)
1+
GH
(
z
)
Characteristic equation:
1 +
GH
(
z
) = 0
whose roots are the closed-loop poles:
p
1
,
p
2
,...,
p
n
Stable systems: all closed-loop poles within the unit circle
Dominant closed-loop poles: the closed-loop pole(s) furthest away from 0
Transient response largely determined by the dominant closed-loop pole(s)
Problem
: quantify the response characteristics based on dominant pole locations

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2
Continuous-Time Feedback Systems
1 +
G
(
s
)
H
(
s
) = 0
C
(
s
)
H
(
s
)
G
(
s
)
R
(
s
)
+
¡
Transfer function from
R
(
s
) to
C
(
s
):
C
(
s
)
R
(
s
)
=
G
(
s
)
1+
G
(
s
)
H
(
s
)
Characteristic equation:
q
1
,
q
2
,...,
q
n
Stable systems: all closed-loop poles strictly on the left half of the complex plane
Dominant closed-loop poles: the closed-loop pole(s) furthest to the right
Transient response largely determined by the dominant closed-loop pole(s)
whose
roots are the closed-loop poles:
Step Response
Time (sec)
Amplitude
0
5
10
15
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time Specifications of Continuous-Time Systems
•
t
d
:
delay time
, time for
s
(
t
)
to reach half of
s
(
1
)
•
t
r
:
rise time
, time for
s
(
t
) to
reach
s
(
1
)
•
t
p
:
peak time
, time for
s
(
t
) to
reach first peak
•
M
p
:
maximum overshoot
•
t
s
:
settling time
, time for
s
(
t
)
to settle within a range (2%
or 5%) of
s
(
1
)
Step response
s
(
t
)
t
d
t
r
t
p
M
p
=
s
(
t
p
)
¡
s
(
1
)
s
(
1
)
s
(
1
)
0
:
5
s
(
1
)
t
s