53
We see that F is simply the matrix resulting from a linearization of
the system NCT about the origin. In addition, because the original
system is time-invariant F, G, and F are constant matrices.
Let us investigate the stability of NCT using the Poincare-Liapunov
theorem. To do this we give the following result.
Theorem 4.2: Suppose that for a particular time-invariant system
assumptions (A.l) and (A.2) are satisfied and that the conditions needed
to apply Liaponov's indirect method to the system NCT (see eq. (2-47)
and (2-48)) hold true. In addition, assume that the following
conditions are satisfied.
(i)sup iiF*(t) Fii. = e
t>0 1
(ii)sup llG*(t) GII. = e,
t>0 1
where and e^ are positive constants.
(iii)The pair [F, G] is stabilizable
(iv)
rank
xiI F
-H
n + p
for all which are characteristic roots of the differential
equation given in assumption (A.2).