Nonlinear Control Sy.. - Free

4.5. ANALYSIS OF LINEAR TIME-VARYING SYSTEMS 121
where
A(t) = of f of (o, t)
l==o
ax ax
since the higher-order terms tend to be negligible "near" the equilibrium point, (4.31) seems
to imply that "near" x = 0, the behavior of the nonlinear system (4.30) is similar to that
of the so-called linear approximation (4.31). We now investigate this idea in more detail.
More explicitly, we study under what conditions stability of the nonlinear system (4.30) can
be inferred from the linear approximation (4.31). We will denote
Thus
g(x, t) de f f (x, t) - A(t)x.
x = f (x, t) = A(t)x(t) + g(x, t).
Theorem 4.7 The equilibrium point x = 0 of the nonlinear system
is uniformly asymptotically stable if
x = A(t)x(t) + g(x, t)
(i) The linear system a = A(t)x is exponentially stable.
(ii) The function g(x, t) satisfies the following condition. Given e > 0, there exists 5 > 0,
independent of t, such that
This means that
uniformly with respect to t.
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