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228 Computed-Torque Control

and the derivative-filtering pole at

(4.5.15)

It is easy to implement this digital outer-loop filter in terms of difference
-1
equations on a DSP. First, write K(z) in terms of z , which is the unit delay in
the time domain (i.e., a delay of T seconds), as

(4.5.16)

-1
[Note: There is some abuse in notation in denoting (4.5.16) as K(z ); this we
shall accept.]
Now suppose that the control input u k is related to the tracking error as

(4.5.17)

Then u k may be computed from past and present values of e k using auxiliary
variables as follows:
(4.5.18)
(4.5.19)
(4.5.20)

The variables represent the integral and derivative portions of the
digital PID controller, respectively. These difference equations are easily
implemented in software.

Actuator Saturation and Integrator Antiwindup Compensation
Actuator saturation leading to integrator windup is a problem that can occur
in the outer PID loop of a robot controller. In Example 4.4.4 we saw the
deleterious effects of integrator windup. It is easy to implement antiwindup
protection digitally [Åström and Wittenmark 1984], [Lewis 1992]. The
antiwindup protection circuit would be placed into the outer linear feedback
loop of a robot control system, where the integrator (controller memory) is
located.
Suppose that the controller is given in transfer function form

(4.5.21)

Copyright © 2004 by Marcel Dekker, Inc.