55
II 4>(t, tQ) II.
< me
-a(t-tQ)
(4-23)
for some positive constants m and a. Hence, to show condition (2) of the
Poincare-Liapunov theorem and also to conclude stability we must have
sup HF*(t)
t>0 A
< L
where L is small enough to insure that
(4-24)
(mL a) < 0
(4-25)
Equation (4-24) can be verified by using the definitions for
k o
F^(t) and F^ to obtain the relationship
llF*(t) Fa"i < llF*(t) F"i + gVhiCKj, K2]ni (4-26)
Using (i) and (ii) then gives
sup llF*(t) Fll < ej + e2H[K1, K2]IIi (4-27)
If and e2 are small enough, then condition (2) of the Poincare-
Liapunov theorem is satisfied and the system NCT is asymptotically
stable.
The final condition which needs to be verified for a solution to
the servomechanism problem is the observability of the pair (A, K2).
This condition presents no problem, however, since it immediately
follows from Theorem 3.4 that (A, K2) is observable whenever the system