82
where
and
-*
pA(t)
F*(t) G^tiK^t)
-BH
-G*(t)K2(t)
A
(5-11)
F*(t)
x = x*(t)
u = u*(t)
"k
w = w (t)
G*(t)
9f(x,U,w)
9U
x*(t)
u*(t)
_*
W (t)
(5-12)
Now (see Theorem 4.1) if the system
xA(t) = F*(t)xA(t) (5-13)
k w,^k
is the exponentially asymptotically stable and sup nF.(t) F.(t)n. is
t>0 at
suitably small then (5-10) is exponentially asymptotically stable as
desired.
First consider conditions under which the quantity
k ^~k
sup iiF.(t) F. (t) II. is sufficiently small. When the true reference
t>0 at
and disturbance signals are precisely equal to the nominal reference and
disturbance signals then obviously this quantity is zero. If the true
k i
signals deviate from the nominal signals then sup nF.(t) F.(t)n. is
t>0 at
k
not zero, however, it is generally small (assuming ^(t) depends
continuously on the reference and disturbance) whenever the deviations
from the nominal signals are small. Hence, the control law developed
here will be effective when the true reference and disturbance signals
are close to the nominal reference and disturbance signals.