49
Adhering to this more compact notation, we may write the linearized
approximation to NCT as
where
and
xA(t) FA(t)xA(t)
F*(t) G*(t)K1 -G*(t)K2
-BH A
(4-3)
(4-4)
F*(t)
3f(x,U,w)
3X
*
X (t)
u*(t)
w*(t)
G*(t)
_ 3f(x,U,w)
3U
*
X (t)
u*(t)
w*(t)
(4-5)
Note that (4-3) is the linearized system needed in conjunction with
Liapunov's indirect method (Theorem 2.3) and was originally given as
equation (2-40).
Although by applying Liapunov's indirect method we reduce the
problem from stabilizing a nonlinear system to stabilizing a linear
system, the time-dependency of this linear system can create compli
cations. Equations (4-4) and (4-5) show how this time-dependency enters
ic
into the linearized system due to the time-varying signals x (t), u (t),
and w*(t). To further complicate the stability problem, it is very
likely that the matrices F*(t) and G*(t) will not even be known. This
is because both F*(t) and G*(t) are implicitly dependent on the