58
f(x,u,w)
3x +
x2
(2x2 +
4)w + u
(4-32)
so that
F*(t)
G*(t)
3f(x,u,w)
3X
3f(x,U,w)
3U
x=ajSint
w=cu
x=c^sint
w=a0
= (3+2x+4wx)
x=a^sint
W=a0
= 1
3+ (2a^+4a j otg) s i n t
(4-33)
(4-34)
The linearization of NCT may be written as
xA(t)
[F (t)Kj3 [
0
0
0
0
-1
0
0
0
0
0
1
0
0
0
-4
-K2
0
1
0
0
0
0
0
1
0
-5
0
0
0
1
0
Xfl(i)
(4-35)
where
xA(t)
x(t)
(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
9
G = 1
(4-37)