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2.10 Useful Theorems and Lemmas 89

Then e 1 , y 2 , and y 1 , .
Basically, the small-gain theorem states that a feedback interconnection
of two systems is BIBO stable if the loop-gain is less than unity. In other
words, if a signal traverses the feedback-loop and decreases in magnitude
then the closed-loop system can not go unstable.

EXAMPLE 2.10–1: Applications of Small-Gain Theorem
Let the feedback connection of Figure 2.9.2 be such that

Then, we can show that

γ 1=0.5; ß 1=0

γ 2=1; ß 2=1.

Since γ 1 γ 2 =0.5<1, the feedback connection is BIBO stable.

The next theorem provides a test for the stability of a nonlinear system based
on the stability of its linear part. In fact, a certain degree of robustness is
achieved if the linear system is exponentially stable as described in [Anderson
et al. 1986].

Total Stability Theorem
THEOREM 2.10–2: Total Stability Consider the state-space system
described by

where

1. The system is exponentially stable i.e. there are some a>0
and K 1, such that

||x(t)|| Ke ; t 0
-at

Copyright © 2004 by Marcel Dekker, Inc.