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2.8 Input/Output Stability 81

EXAMPLE 2.8–1: Input/Output Versus Lyapunov Stability

Consider the time-varying system

The system is asymptotically stable with a single equilibrium point at y e =0.
On the other hand, a unit step input (which is definitely bounded) starting at
t=0 will lead to the response

which grows unbounded as t increases.

Therefore, we need to discuss the conditions under which a bounded input
will result in a bounded output [Boyd and Barratt], [Desoer and Vidyasagar
1975]. This was actually presented when discussing the system-induced
norms (see Definition 2.5.7) and the current discussion should serve to
contrast these concepts with Lyapunov stability. Consider the nonlinear
system

(2.8.1)
y(t)=g[x(t), t, u(t)] (2.8.2)

DEFINITION 2.8–1 The dynamical system (2.8.1) is bounded-input-
bounded-output (BIBO) stable if for any

||u(t)|| M<8

there exist finite γ>0 and b such that

||y(t)|| γM+b

Note that BIBO stability implies the uniform boundedness of all equilibrium
states.

Copyright © 2004 by Marcel Dekker, Inc.