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┃ ┃(J r ┃
┣━━━╋━━━┫
┃x ┃ ┃
┗━━━┻━━━┛
a) Overdamped system
y(e)
┏━━━┓
┃~~ ┃
┃~/ ~/ ┃
┣━━━┫
┃ ┃
┗━━━┛
b) Oscillatory response
y(t)
c) Underdamped response
┏━━━┳━━━┓
┃ ┃\ ┃
┃ jco ┃ ┃
┃.A ┃ ┃
┃ ┃a - ┃
┣━━━╋━━━┫
┃ ┃ ┃
┗━━━┻━━━┛
y(r)
d) Critically damped response
Fig.5,3 Second-order system response.
i
tl
t:
;k'
::,
t$
t4 I
~
l4
.1
Fl .
r
. ~.
450 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
The unit-step input response of a second-order system ofEq. (5.60) is given by
t02
y(s) : S(S2 + 2<WnS
+ co:)
=ki+_ k2s+k3
s -s2+2_o S+~>2
(5.66)
(5.67)
where ki, k2, and k3 are constants. Expanding the partial fractions, taking the
inverse Laplace transforms, and simplifying, we obtain
y(t) = 1 -
-1-
where the phase angle ~ is grven by
e-<o.r COS(COdt - ~)
(5.68)
e-<eo., COS(COnt /Q:/ - ~) (5.69)
@ = tan-l
(5.70)
The typical response for various values of the damping parameter < are shown in
Fig. 5.4. Because the time appears as a product cont in Eq. (5.69), it is convenient
y(aht)
ant
Fig.5.4 Typicalsecond-order system response.
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW 451
y(t)
┏━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃ ┃\ ┃
┣━━━━━╋━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┫
┃ ┃ ┃~-.. M'ximun ium-Overshoa~ ┃
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