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5.3 Nonlinear Controllers 299

where

Then the error reaches the surface

in a finite time. In addition, once on the surface, q(t) will go to q d (t)
exponentially fast.

Proof:
Consider the Lyapunov function candidate

Then

Substituting for and using the skew-symmetric property of (q)-2V m (q, )
we obtain

Then, if we use the controller given by

we obtain

where

Copyright © 2004 by Marcel Dekker, Inc.