where
_. F*(t) - G*(t)Kl(t) -G*(t)K2(t)
A -BH A
and
T*(t) := f(x,u,w) G*t = I f(x,u,w)
ax x au() x M
U () uu U*(t) (5-12)
w *t w
Now (see Theorem 4.1) if the system
xA(t) = FA(t)xA(t) (5-13)
is the exponentially asymptotically stable and sup IIF*(t) - FA(t)li. is
suitably small then (5-10) is exponentially asymptotically stable as desired.
First consider conditions under which the quantity sup 1iFA(t) - FA(t)ii is sufficiently small. When the true reference t>O
and disturbance signals are precisely equal to the nominal reference and
disturbance signals then obviously this quantity is zero. If the true signals deviate from the nominal signals then sup IIF*(t) -FA(t)i is t)O
not zero, however, it is generally small (assuming FA(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.