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270 Robust Control of Robotic Manipulators

Consider the closed-loop system given by (5.2.8), with the controller
(5.2.12), and choose the following Lyapunov function candidate:

(1)

where is the Lyapunov function corresponding to the SPR system
(5.2.14). Then if ≥0, we have that V>0. This condition is satisfied for ≥µ 2 I.
Then differentiate to find

(2)

To guarantee that V < 0 recall the bounds (5.2.8)–(5.2.11), and write

(3)

where
. Note that ||e|| may be factored out of (3) without affecting
the sign definiteness of the equation. The uniform boundedness of the error
is then guaranteed using Lemma 2.10.3 and Theorem 2.10.3 if

(4)

which is guaranteed if
(5)

or
(6)

The error will be bounded by a term that goes to zero as a increases (see
Theorem 2.10.3 and its proof in [Dawson et al. 1990] for details). This
analysis then allows to be arbitrarily large as long as ≥µ 2I, as shown in
the next example. In fact, if N were known, global asymptotic stability is
assured from the passivity theorem since in that case =0. The controller is
summarized in Table 5.2.1.
It is instructive to study (6) and try to understand the contribution of
each of its terms. The following choices will help satisfy (6).

1. Large gains K p and K v which correspond to a large a.

Copyright © 2004 by Marcel Dekker, Inc.