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88 Introduction to Control Theory

Figure 2.9.2: Feedback interconnection of two nonlinear systems.

2.10 Useful Theorems and Lemmas

Consider the block diagram shown in Figure 2.9.2. The blocks labelled H 1
and H 2 represent two systems (linear or nonlinear) which operate on the
inputs e 1 and e 2 as follows

y 1 =H 1 e 1 =H 1 (u 1 -y 2 )

y 2 =H 2 e 2 -H 2 (u 2 +y 1 )

Let H 1 be an m×p matrix function, and H 2 be a p×m matrix function.
Therefore, u 1 and y 2 are p×11 vectors, while u 2 and y 1 are m×11 vectors. The
first theorem gives sufficient conditions to guarantee that the closed-loop
system shown in Figure 2.9.2 is BIBO stable and is given in [Desoer and
Vidyasagar 1975].

Small-Gain Theorem
THEOREM 2.10–1: Small gain Theorem Let H 1 : and H 2 :
. Therefore H 1 and H 2 satisfy the inequalities

for all T [0,∞) and suppose that . If

Copyright © 2004 by Marcel Dekker, Inc.