We see that F 0 is simply the matrix resulting from a linearization of
A
the system NCT about the origin. In addition, because the original
system is time-invariant F°, Go, and F are constant matrices.
A
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.1) 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) - F0ni 6
t)O
(ii) sup iIG*(t) - Gi=2
t)O 1
where s 1 and c2 are positive constants.
(iii) The pair [FO, GO] is stabilizable
(iv) rank F0 = n + p
-H 0
for all xi which are characteristic roots of the differential
equation given in assumption (A.2).