27
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 'x(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 x(t) depends on 'ri(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 x(t) = 0,
ri(t) = 0 is
£(t)
F*(t) G*(t)K1
-G*(t)K2
x(t)
S(t)
-BH
A
(t)