XA(t) FA(t)xA(t) (5-6)
where
[x(t)- x*(t
xA(t) [n(t) - *(t)] (5-7)
and
F F * (t) - G*(t)KI(t) -G*(t)2(t
FA(t) -BH (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
Ft* ( - f(x,u,w) ( G (t) = af(xuw)*
ax x x (t) au x x (t)
u = (t) u u* (t) (5-9)
w (t) w w (t)
Usually these Jacobian matrices cannot be evaluated apriori since x*(t), 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
A(t) s FA(t)xA(t) + [FA(t) - FA(t)]xA(t) (5-10)