Page 83 - Servo Motors and Industrial Control Theory -

76 4 Electrical DC Servo Motors

state variables that are measurable directly by high performance transducers. By
eliminating T T from Eqs. (4.9)–(4.13) the governing equations can be converted
s r
to three sets of differential equations; that is,
K K R C
sI := θ− θ − I − m s θ (4.14)
L i L o L L m

s θ := K t I − C s θ − K s θ + K s θ (4.15)
2
m
J J m J ·N 2 m J ·N o
m m m m
K K C T
s θ o := s θ m − s θ− 1 s θ− 1 (4.16)
2
o
o
J ·N J J J
1 1 1 1
Equations (4.14)–(4.16) are now in a form that it is possible to define state vari-
ables. The current motor position and velocity, output position, and velocity are all
the state variables that are directly measurable with high performance transducers.
The following state variables are defined as
x: =θ o
1
x : s =θ o
2
x: =θ m (4.17)
3
x : s =θ m
4
x: I=
5
Substituting the state variables in the above equations, the state equation becomes

d x := Ax+ BU
dt (4.18)

For simplicity the following variables are defined and the system matrices become
as
A := JN A := − J 1 K s A := − J 1 C 1
K s
3
2
1
1 •
A := − 1 A := K t A := − C
4 J 1 5 J m 6 J m (4.19)
K
A := J − m N• K ss 2 A := J • s A := K L
m N
7
8
9
A := − K A := R A := − C m
10 L 11 L 12 L