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4.5 Digital Robot Control 241

(9)

with corresponding difference equation implementation

(10)

If u k is not in saturation, this amounts to updating the plant control by
adding the second and third terms on the right-hand side of (10) to u k-1.
These terms are, therefore, nothing but u k-u k-1. The compensator with α=0
is thus called the velocity form of the PI controller.

b. Digital Control of DC Motor
Consider the simplified model for a do motor given by

(11)

with the angular velocity. A motor speed controller has the form

(12)

where e=r-w is the tracking error, with r the desired command angular velocity.
Taking a=1 and b=1, with k 1=k 2=-3.318, we obtain poles at s=-1, -3.318. The
slower pole is canceled by a zero, so that the step response is fast, having only
-3.318t .
a mode like e
Writing the PI controller as
(13)

we see that

(14)

The digital controller obtained using the modified MPZ is given by (1).
The time constant of the closed-loop system is so
that a sampling period of T=0.05 s is reasonable. The sampling period
should be about one-tenth the time constant.

Copyright © 2004 by Marcel Dekker, Inc.