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and
2g(M þ m) sin
u
1
2M þ m u(t)
u ¼
) u
2
( sin
u
þ
u
(4
:
42)
(2M þ m)l cos
, and x 4 ¼ u
ne the state variables as x 1 ¼ x, x 2 ¼ x, x 3 ¼ u
De
, then we have
x 1 ¼ x 2
mg sin 2x 3
(2M þ m) cos x 3 þ
cos 2x 3
(2M þ m) cos x 3 u(t)
1
þ
x 2 ¼
(4
:
43)
x 3 ¼ x 4
2g(M þ m) sin x 3
(2M þ m)l cos x 3
1
2M þ m u(t)
x 4 ¼ x 4 sin x 3 þ
The above state equations are nonlinear. To obtain linear equations, we linearize
them about
u ¼
0. The results of linearization are
x 1 ¼ x 2
x 2 ¼
2mg
2M þ m x 3 þ
2
2M þ m u(t)
(4
:
44)
x 3 ¼ x 4
2g(M þ m)
(2M þ m)l x 3
1
2M þ m u(t)
x 4 ¼
The linear equation given by Equation 4.44 can be written in matrix form as
2
4
3
5
2
4
3
5
2
4
3
5
0 1
0
0
2
4
3
5
0
2
2M þ m
0
x 1 (t)
x 2 (t)
x 3 (t)
x 4 (t)
x 1 (t)
x 2 (t)
x 3 (t)
x 4 (t)
2mg
2M þ m
00
0
¼
þ
u(t)
(4
:
45)
0 0
0
1
00 2g(M þ m)
2M þ m
1
2M þ m
0
4.4 STATE-SPACE REPRESENTATION OF GENERAL
CONTINUOUS LTI SYSTEMS
Consider a SISO system described by the N th
order constant coef
cients differential
equation with input u ( t )
and output y ( t )
:
N y ( t )
d t N
N
1
N u ( t )
d t N
N
1
d
þ a 1 d
d t N 1 þþ a N y ( t ) ¼ b 0 d
y ( t )
þ b 1 d
u ( t )
d t N 1 þþ b N u ( t )
(
4
:
46
)
The state-space representation of the system de
ned by differential equation given in
Equation 4.46 is not unique. We de
ne two useful canonical forms that are suitable
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