half-plane, all modes associated with A must be forced to zero or NCT will not be asymptotically stable. From (2-38) we see that ' (t) is the only signal which can accomplish this task. Consequently, X(t) should be a function of the modes in 'n(t) induced by the eigenvalues of A. Since '(t) depends on 'n(t) only through the feedback coupling from the
gain K2 and (A, K2) is not observable, it is impossible for 'X(t) to depend on the unobservable modes. Thus, we can conclude that
observability of the pair (A, K2) is necessary for asymptotic stability of the system NCT. We therefore have the following result.
Proposition 2.3: Controllability of (A, B) and observability of (A, K2) are necessary conditions for asymptotic stability (either local or
global) of NCT.
Although the above result is important, it is even more important that a practical method is available which allows one to ascertain directly whether or not NCT is asymptotically stable. It has already
been indicated that the local stability results of Theorem 2.2 will most often apply. One convenient method for showing local stability is Liapunov's indirect method which can be found in standard texts on nonlinear systems (e.g., see [17]). The required linearization of the closed-loop transient system NCT about the equilibrium point (t) = 0,
( 0 is
LF t -BH -G (t)K2j [ (2-40)
n BH A rl Mt