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(5.56)
(5.57)
(5.5 8)
(5.59)
,j
j
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ei
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~
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N
s
:~
r'
7j\
' J.
y(s) = G(s)r (s)
= ( ::b)(l)
PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
A second-order system has two poles.ln general, the response of a second-order
system can be any one of the four types of responses as shown in Fig. 5.3. Suppose
the system poles, which depend on the values ofa and b, are both real and negative
as shown in Fig. 5.3a; then the corresponding response is a steady rise, without
any overshoot, to the final value. This type of response is called an "overdamped"
response.lf the poles are purely imaginary, then the response is a constant ampli-
tude sinusoid that will continue forever because there is no damping in the system
(Fig. 5.3b). This type of response is called "oscillatory response." The frequency
of this undamped oscillation is called the natural frequencPyuof the system. If the
system poles are a pair of complex conjugate numbers with negative real parts,
then ffie transient response will be oscillatory and is. charactaned by overshoots
as shown in Fig. 5.3c. This type of response is caried i{underdampeYd response-
and the frequency of this oscillation is called the cxponential decay frequency or
the damped frequency.lf the poles are real, negative, and equal to each other, then
the response is said to be critically damped as shown in Fig. 5.3d.
The transfer function of the second-order system given by the Eq. (5.57) can be
expressed in the standard fonn as follows:
co: (5.60)
G(s) = S2 + 2 cons +(D2
COn = \~
(5.61)
a (5.62)
<=2 =2-~ff/
Here, COn iS the natural frequency of the system, and 4 is the damping ratio of
the system. The damping ratio is defined as the ratio of the existing damping to
that required for critical damping. For C > 1, the second-order system has two
real, negative, and unequal roots, and the system has an overdamped response
as in Fig. 5.3a. When C :1, the two real negative roots become equal, and the
motion associated with this case is called critically damped motion as shown in
Fig. 5.3d. When < < 1.0, the second-order system has a pair of complex roots
with negative real parts, and the system displays a damped oscillatory motion
as in Fig. 5.3c. Thus, the condition g' = 1 represents the boundary between the
overdamped exponential motion and the damped oscill,ator}r motion. .
The damping ratio ; and the natural frequency COn are two unportant parameters
that characterize a second-order system. The response of a second-order system
depends on the values of these two parameters. The system poles, in terms of
frequency and damping ratio, are given by
S1,2 = -ad + jcod (5.63)
We have
tTd-.<tOn COd=tOnA (5.64)
so that
S1,2 = -<COrt :+: jtDn
Here, COd iS called the damped frequency.
(5.65)
LINEAR SYSTEMS, THEORY, AND DESIGN:A BRIEF REVtEW 449
x Poles of G(s = ~
┏━━━━━━━━━┳━┓
┃ ┃b ┃
┃ j(D ┃ ┃
┃. . . -- u ┃ ┃
┗━━━━━━━━━┻━┛
z
J L
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