Nonlinear Control Sy.. - Free

4.1. DEFINITIONS 109
All of these definitions are similar to their counterpart in Chapter 3. The difference is in
the inclusion of the initial time to. This dependence on the initial time is not desirable and
motivates the introduction of the several notions of uniform stability.
Definition 4.2 The equilibrium point x = 0 of the system (4.1) is said to be
Uniformly stable if any given e > 0, 16 = b(e) > 0 :
X(0)11 < 6 1x(t)jj < e Vt > to > 0 (4.6)
Uniformly convergent if there is 61 > 0, independent of to, such that
jxoIIoo.
Equivalently, x = 0 is uniformly convergent if for any given E1 > 0, IT = T(El) such
that
lx(0)II < 61 Ix(t)II < El Vt > to + T
Uniformly asymptotically stable if it is uniformly stable and uniformly convergent.
Globally uniformly asymptotically stable if it is uniformly asymptotically stable and
every motion converges to the origin.
As in the case of autonomous systems, it is often useful to restate the notions of uniform
stability and uniform asymptotic stability using class IC and class ICL functions. The following
lemmas outline the details. The proofs of both of these lemmas are almost identical
to their counterpart for autonomous systems and are omitted.
Lemma 4.1 The equilibrium point x = 0 of the system (4.1) is uniformly stable if and only
if there exists a class IC function a constant c > 0, independent of to such that
Ix(0)II < c => Ix(t)II to. (4.7)
Lemma 4.2 The equilibrium point x = 0 of the system (4.1) is uniformly asymptotically
stable if and only if there exists a class ICL and a constant c > 0, independent
of to such that
1x(0)11 < c => lx(t)II to. (4.8)
Definition 4.3 The equilibrium point x = 0 of the system (4.1) is (locally) exponentially
stable if there exist positive constants a and A such that
11x(t)II < a
whenever jx(0) I < 6. It is said to be globally exponentially stable if (4.9) is satisfied for
anyxER".
Ilxollke-at (4.9)