
To illustrate the process of linearizing a nonlinear state variable model, consider the inverted
pendulum previously introduced in Section 5.4, redrawn in Figure 7.59.
The coupled nonlinear differential equations describing the system (Equations 5.50 and 5.51) can
be manipulated to read
€
x ¼
ml
_
u
2
sin u (mg= 2 ) sin 2u þ u
M þ m sin
2
u
(7:139)
€
u ¼
(ml=2)
_
u
2
sin 2u þ (m þ M)g sin u u cos u
l(M þ m sin
2
u)
(7:140)
State variables are x
1
, x
2
, x
3
, x
4
where x
1
¼ x, x
2
¼
_
x, x
3
¼ u, x
4
¼
_
u. The state derivatives are
given by
_
x
1
¼ f
1
( x, u ) ¼ x
2
(7:141)
_
x
2
¼ f
2
( x, u ) ¼
mlx
2
4
sin x
3
(mg= 2 ) sin 2x
3
þ u
M þ m sin
2
x
3
(7:142)
_
x
3
¼ f
3
( x, u ) ¼ x
4
(7:143)
_
x
4
¼ f
4
( x, u ) ¼