Page 215 - Servo Motors and Industrial Control Theory -

Appendix C 213

Demand Position
Output Position

Loading
K + K i power Motor Mechanism
p
Unit
s

1+ K s
d
Position and velocity feedback

Note that there are two input variables. One is the demand position and the other
one is the external torque applied to the motor. Assume that the power unit is
capable of producing pure variable DC voltage with negligible time delay com-
pared to the overall response time of the system. The power unit produces a
voltage proportional to the control signal at high required current. You can set
the gain of the power unit to unity and assume that its gain is incorporated in the
controller. Define variables and parameters as required and determine the overall
transfer functions. You should find two transfer functions. One transfer function
that relates the output position to the demand position and one transfer function
which relates the output position to the input external torque. If any parameters
are not defined above, choose an engineering value for them and continue the
analysis. To find the transfer functions of the system you can use the principle
of superposition.
You should note that the characteristic equations of both transfer functions are
the same. Before deriving the transfer functions assume that both static and vis-
cous frictions are negligible which represent the worst conditions. For first ap-
proximation assume that the inductance is negligible. Having found the transfer
functions determine the values of proportional, integral and derivative feedback
gains so that there is at least a damping ratio 0.7 in the fundaments roots with
maximum speed of response. For this you choose a reasonable value for the
speed of response of fundamental root(s). They must not be too fast so that the
power unit may not be able to produce such a fast response. Predict a typical
response time of the system for small step input of demand position. Determine
the steady state errors for both inputs and show that both are zero for step input.
One important property of such position control systems is the dynamic position
drop when a step input of external torque is applied. Convert the transfer func-
tion which relates the output position to the external torque input, to state space
form and use numerical integration procedure to find the dynamic characteristic
of the output position when a step input of external torque is applied. Find the
maximum position drop when the external torque is applied.
Repeat the above analysis when the inductance of the motor cannot be ignored.
Discuss the effect of the inductance on the overall response characteristic.