Page 88 - Servo Motors and Industrial Control Theory

4.6 DC Servo Motors for Very High Performance Requirements 81

Equating the characteristic Eq. (4.28) and the desired characteristic Eq. (4.23) the
gains can be calculated as

K11: 1.8·10 K22 := 6 =− 2083 K33:=− 2.3·10 K44 :=− 902 K55:=− 241
5
The actual gains are obtained by multiplying the above gains by the inductance L,
which gives
K1:=1584 K2:= 1.83 K3:= 202.4 K4:= 0.79 K5:= 0.21− − − − (4.29)

It can be seen that the gains are negative except K1. The positive gain compensates
for any deflection that may have caused by external torque. To check that the gains
(4.29) are correct they must be substituted in matrix AC and to calculate the eigen-
values again.

0 1 0 0 0
− 1.282·10 4 0 1.282·10 3 0 0
AC:= 0 0 0 1 0 (4.30)
1.282·10 4 0 − 1.282·10 3 0 106.4
− 1.136·10 + K11 K22 K33 − 943.2 K44+ − 409.1 K55+
5
Matrix AC is now the closed loop system matrix and the values of the gains are now
known.
Substituting the gains in the matrix (4.30) and calculating the eigenvalues again
gives

w := eigenvals AC( )
+
−189 .767 201 .423 i
−
−189 .767 201 .423 i (4.31)
w= −106 .2446 86 78+ . i
− 106 246 86 78− . i
.
− 58 075
.

It can be seen from the eigenvalues (4.31) that indeed the eigenvalues have been
moved to the required value. There are slight differences, which are due to the
rounding off error.
It can be seen that the roots of the characteristic equation all have a damping
ratio of greater than 0.7 because the real part of the roots have moved to the left of
− 45° line on the complex plane. The real root has moved to the required position
on the complex plane. There has been a small change in the required values of the
eigenvalues due to a rounding off error in the calculation. This indicates that the
eigenvalues are not very sensitive to the values of gains. This is useful for practical
applications where the exact gains might be achievable. The reader is encouraged to
change each gain by approximately 10 % to observe how sensitive the eigenvalues
are when the gain is changed.