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┏━━━━┳━━━━┓
┃jco     ┃        ┃
┃        ┃{       ┃
┣━━━━╋━━━━┫
┃        ┃     a  ┃
┃N       ┃ c      ┃
┗━━━━┻━━━━┛
                 
┏━━━┳━━━┓
┃      ┃k     ┃
┃  jto ┃      ┃
┃x     ┃      ┃
┃      ┃(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)
                                                                                                 
                                                                                                 
                                                                                                 
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