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90 Introduction to Control Theory

2. f(t, 0)=0 i.e. the origin is an equilibrium point of f(t, x)

3. || f(t, x 1 )-f(t, x 2 )|| ß 1 ||x 1 -x 2 ||, for some ß 1 >0
4. ||g(t, x 1 )|| ß 2 r, for some ß 2 >0
5. ||g(t, x 1 )-g(t, x 2 ) ß 2 ||x 1 -x 2 ||

6.
Then, there exists a unique solution x(t) to 1.5.5 and

The total stability theorem will be used to design controllers that will
make the linear part of the system exponentially stable. In effect, this
theorem guarantees that if the linear part of the system is “very” stable
(exponentially stable), the destabilizing effect of the bounded
nonlinearities may not be sufficient to destabilize the system and the state
will remain bounded.

EXAMPLE 2.10–2: Total Stability
Consider the nonlinear system

Let |x 0|<1, and note the following

K=1; a=2; ß 1=0.5; ß 2=1.

Note first that all conditions of the theorem are satisfied, then there exists a
unique solution x(t) which is bounded by

A version of the Bellman-Gronwall lemma is proved in [Sastry and Bodson
1989] and is presented next.

Copyright © 2004 by Marcel Dekker, Inc.