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5.2 Feedback-Linearization Controllers 277

and using (5.2.8)–(5.2.11), we have that

(5.2.27)

The following theorem presents sufficient conditions for the boundedness of
the error that parallel those of Theorem 5.2.1.
THEOREM 5.2–2: Suppose that

and

(1)

Then the ∞ stability of the error is guaranteed if

(2)

Proof:
The condition above results from applying the small-gain theorem to the
closed-loop system, under the assumption that e(0)=0 so that the quadratic
2
term ||e|| is small. See [Craig 1988] for details.
Note that (2) reduces to

and further to

(5.2.28)

Let us study the inequality above to determine the effect of each term. The
following observations are made to help satisfy (5.2.28).

1. A large k v will help satisfy the stability condition. Note: That will also
imply a large k p.

Copyright © 2004 by Marcel Dekker, Inc.