Adhering to this more compact notation, we may write the linearized approximation to NCT as
A(t) FA(t)xA(t) (4-3)
where
F*t) LF*(t) - G*(t)K1 -G*(t)K2(
A(-BH A
and
* af(x,u,w) G * (t): *
F(t) ax x= x (t) au x= x (t)
u u (t) u u (t) (45)
w w(t) w w(t)
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 complications. Equations (4-4) and (4-5) show how this time-dependency enters 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