Digital Control Systems [MEE 4003] - Kckong.info

2.1. STATE SPACE REALIZATION OF DYNAMIC SYSTEMS 17
[Example 2-3] The governing equation of the two-mass-spring system
[ ][ ] [ ][ ] [ ][ ]
m1 0 ẍ1 0 0 ẋ1 k −k x1
+ + =
0 m 2 ẍ 2 0 0 ẋ 2 −k k x 2
where the outputs are x 1 andx 2 , is to be represented in a state space.
i.e.,
[
u1
u 2
]
Note that the original equation of motion consists of two differential equations,
ẍ 1 = 1 m 1
[−kx 1 +kx 2 ]+ 1 m 1
u 1 (2.2)
ẍ 2 = 1 m 2
[kx 1 −kx 2 ]+ 1 m 2
u 2 (2.3)
The variables with the lowest order in the equation of motion and the output arex 1 and
x 2 . Let them be y 1 and z 1 . New variables, y 2 and z 2 , are defined as the derivatives of
y 1 and z 1 , respectively, i.e.
d
dt y 1 = y 2
d
dt z 1 = z 2
Differentiating once more, the original differential equations in (2.2)–(2.3) appear, i.e.
d
dt y 2 = 1 m 1
[−ky 1 +kz 1 ]+ 1 m 1
u 1
d
dt z 2 = 1 m 2
[ky 1 −kz 1 ]+ 1 m 2
u 2
Arranging the new first-order differential equations,
⎡ ⎤ ⎡ ⎤⎡
⎤ ⎡
y 1 0 1 0 0 y 1
d
⎢ y 2
⎥
dt ⎣ z 1
⎦ = −k k
⎢ m 1
0
m 1
0
⎥⎢
y 2
⎥
⎣ 0 0 0 1 ⎦⎣
z 1
⎦ + ⎢
⎣
z 2 0 −k
m 2
0 z 2
k
m 2
0 0
1
m 1
0
0 0
0
1
⎤
[ ⎥ u1
⎦
m 2
u 2
]
Since the outputs of the system arey 1 = x 1 and z 1 = x 2 ,
⎡ ⎤
[ ]
x 1
1 0 0 0
y = ⎢ x 2
⎥
0 1 0 0 ⎣ x˙
1
⎦ ∈ R2
x˙
2
Digital Control Systems, Sogang University
Kyoungchul Kong