
The natural modes from Equation 7.166 are s
1
, s
2
¼(13.5022)
1=2
¼j 3.6745, and the frequency
of undamped oscillations is v
n
¼3.6745 rad=s (see Figures 7.63 and 7.64). Furthermore, the two
remaining characteristic roots are, from Equation 7.165, both zero. Laplace transforming Equations
7.165 and 7.166 and solving for Du(s) and DX(s) result in
Du(s) ¼
0:444
s
2
þ 13:5022
DU(s)(7:167)
DX( s) ¼ 0:333
s
2
þ 13:0666
s
2
(s
2
þ 13:5022)
DU( s)(7:168)
The inverted pendulum is often used as an example of an inherently (open-loop) unstable system,
and numerous linear controls texts demonstrate techniques for designing linear controllers to
balance the pendulum in the upright ...