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4.5 Digital Robot Control 239

Instead, it is necessary to use two derivative filters, one for estimating joint
velocities and one for providing an estimate for the velocity error k . A
simulation that attempts to use 1 makes the point quite well.
Note that the velocity estimates are also used instead of in computing
the nonlinear terms

EXAMPLE 4.5–2: Digital PI Controller with Anti-windup Compensation
A general digital PI controller with sampling period of T seconds is
given by

(1)

We have sampled the continuous PI controller by the modified matched-
polezero (MPZ) method [Lewis 1992] to obtain a delay of T seconds in the
integrator to allow for computation time. The proportional gain is k and
the reset time is T I ; both are fixed in the design stage. The tracking error is
e k .
-1
Multiply by (1-z ) to write
(2)

which is in the transfer function form (4.5.21). The corresponding difference
equation form for implementation is

(3)

This controller will experience windup problems since the autoregressive
-1
polynomial R=1- z has a root at z=1, making it marginally stable. Thus,
when u k is limited, the integrator will continue to integrate, “winding up”
beyond the saturation level.

a. Antiwindup Compensation
To correct this problem, select an observer polynomial of

(4)

Copyright © 2004 by Marcel Dekker, Inc.