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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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