Page 296 -

5.2 Feedback-Linearization Controllers 281

(5.2.31)

and let the operator gains of P i , i=1,2, be given by

(5.2.32)

Note that 1 =max 11 , 12 . See Lemma 2.5.2 and Example 2.5.8. Consider
then

(5.2.33)

The next theorem gives sufficient conditions for the BIBO stability of the
trajectory error.

THEOREM 5.2–3: The BIBO stability of the closed-loop system (5.2.30)
will be guaranteed if

(1)

In fact, the trajectory error is bounded by

(2)

Proof:
By the small-gain theorem. See [Spong and Vidyasagar 1987] for
details.

If we study (1) carefully we note that it will be satisfied if

1. µ 1 is large or M(q) is large.

2. Good knowledge of N, resulting in a small 1 .
3. Small 1 due to a large gain of the compensator C.

Copyright © 2004 by Marcel Dekker, Inc.