81
where
and
xA(t) F*(t)xA(t)
xA(t)
x(t) x*(t)
n(t) n*(t)
F*A(t)
F*(t) G*(t)K1(t)
-BH
-G*(t)K2(t)
A
(5-6)
(5-7)
(5-8)
The Jacobian matrices F*(t) and G*(t) are evaluated at the signals
x (t), u (t), and w (t). More precisely, we may write
F*(t)
x = x (t)
u = u*(t)
*
w = w (t)
G*(t)
3f(X,U,W)
3U
X = X (t)
U = U*(t)
*
w = w (t)
(5-9)
if
Usually these Jacobian matrices cannot be evaluated apriori since x (t),
fa ^
u (t), and w (t) are not known. Consequently, to show stability of the
linearized system we shall use a technique already presented in Chapter
Four; namely, the Poincare-Liapunov theorem. Let us write the
linearized equation given by (5-6) as
xA(t) FA(t)xA(t) + CFA(t) FA(t)]xA(t)
(5-10)