Nonlinear Control Sy.. - Free

Chapter 7
Input-to-State Stability
So far we have seen two different notions of stability: (1) stability in the sense of Lyapunov
and (2) input-output stability. These two concepts are at opposite ends of the spectrum.
On one hand, Lyapunov stability applies to the equilibrium points of unforced state space
realizations. On the other hand, input-output stability deals with systems as mappings
between inputs and outputs, and ignores the internal system description, which may or
may not be given by a state space realization.
In this chapter we begin to close the gap between these two notions and introduce the
concept of input-to-state-stability. We assume that systems are described by a state space
realization that includes a variable input function, and discuss stability of these systems in
a way to be defined.
7.1 Motivation
Throughout this chapter we consider the nonlinear system
x = f(x,u)
where f : D x Du -+ 1R' is locally Lipschitz in x and u. The sets D and Du are defined by
D = {x E 1R : jxjj < r}, Du = {u E Ift"` : supt>0 IIu(t)JI = 1jullc_ < ru}.
These assumptions guarantee the local existence and uniqueness of the solutions of
the differential equation (7.1). We also assume that the unforced system
x = f(x,0)
has a uniformly asymptotically stable equilibrium point at the origin x = 0.
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(7.1)