Page 87 - Servo Motors and Industrial Control Theory

80 4 Electrical DC Servo Motors

positive to the input variable θi. The second method is to inject the state variables
immediately after the proportional control. Because already a position feedback
is used the second method will be used. A positive feedback is considered and the
gains might become positive or negative depending on the state matrix. The voltage
equation then becomes
K( θ − ) K1 θ o + θ + K2s θ + K3 θ m + K4s θ m + K5I:=RI+LsI+C ·s θ m
o
o
m
i
(4.24)
In Eq. (4.24), K1, K2,…K5 are the gains of the state variables that must be selected
to a value that all roots of characteristic equation are moved to the desired position
as defined above. Rearranging Eq. (4.24) and writing it in state space form, the volt-
age equation becomes

sx := K ⋅θ − K x ⋅ + K1 x ⋅ + K2 x ⋅ + K3 x ⋅ + K4 x ⋅ + K5 x ⋅ − R x ⋅ − C m x ⋅ (4.25)
1 L i L 1 L 1 L 2 L 3 L 4 L 5 L 5 L 4
The closed loop state equations, by introducing new gains as

K11 = K1/ LK22 = K2 / L K33 = K3/ LK44 = K4 / L K55 = K5 / L

can now be written in matrix form as

d x:=ACx+BU (4.26)
dt

where

0 1 0 0 0
− 1.282·10 4 0 1.282·10 3 0 0
AC:= 0 0 0 1 0
1.282·10 4 0 − 1.282·10 3 0 106.4
5
− 1.136·10 + K11 K22 K33 − 943.2 K44+ − 409.1 K55+

The values of gains must be selected in such way that the eigenvalues of matrix
AC be the same as desired values given above. For this, the characteristic equation
which is the determinant of the dynamic matrix,
AC sI− (4.27)

must be calculated. Using the symbolic expansion of calculating the determinant
with MathCad, gives

( 106.4K44+1.14·10 )s +−
s + (409.1 K55)s− 4 +− 5 3 ( 1.41·10 K55
5
4
( 1.36·10 K22 1.36·10 K44−
+ 5.77·10 − 106.4K33)s +− 5 6 (4.28)
6
2
+ 1.29·10 )s ( 1.36·10 K33 1.36·10 K11 1.55·10 )+− 6 − 5 + 10
9