f(x,u,w) = 3x + x2 + (2x2 + 4)w + u
so that
F (t)=
X=alsi nt
* ( f(x,u,w) G*(t)= . au
= (3+2x+4wx)
X=alsint
w: a2
= 3+(2a 1+412)sint
(4-33)
X=alsi nt w= a2
(4-34)
The linearization of NCT may be written as
xA(t)
[F (t)-K1] [
0( 0C 0( 0( -1(
1
0
0
0
-4
* -K2
0 1 0 0 0
0
0
1
0
-5
x A(t)
(4-35)
X (t)
where xA(t) := (4-36)
It is not difficult to show that all conditions needed to apply Theorem 4.2 are satisfied. Evaluation of the linearized system about the origin gives
F° = 3 4 Go = 1
(4-32)
(4-37)