Page 60 - Servo Motors and Industrial Control Theory -

52 3 State Variable Feedback Control Theory

Transfer function (3.8) is similar to the previous example and we can define three
state variables as

x := w
1
d
x := w (3.9)
2
dt
d
x := x
3 2
dt
and the state equation becomes
x 1 0 1 0 x 1 0
d x:= 0 0 1 x⋅ +⋅ (3.10)
0 u
dt 2 2
x 3 − 3 − 9 − 6x 3 1
The output is then given by

2
y :(= 5 s + 2 s + 2 ) X
Hence,
x
1
y:= (2 2 5 ) x⋅ 2
x
3
It can be seen that in this case, all state variables contribute to the output. Although
the above form of equations were derived from the transfer function, the state vari-
ables as will be shown in the proceeding chapter, can be defined from the governing
differential equation for each element.
In the above example, the system matrix A, the input vector B, and output vector,
C are as follows:
0 1 0
A:= 0 0 1
− 3 − 9 − 6
0
B:0= C :(2 2 5)=
1
When the system matrix A, the input matrix B, and the output matrix C are defined,
the dynamic behavior of the system can completely be studied. In some applica-
tions, although very rare in servo control systems, there might be direct contribution
from the input vector U to the output vector Y. In this case, another term should be
added to the output equation.