reference and disturbance signals and these signals (especially the disturbance) are usually not known in advance.
At this point we consider one method of dealing with the stability problem which will be applicable when the reference and disturbance signals are small. This assumption, although restrictive, is necessary to show stability of the system NCT when the feedback gains are kept constant. We first give what is known as the Poincare-Liapunov theorem [18].
Theorem 4.1: Consider the system
= Fl(t)x + fl(t,x) , x(to) = xo (4-6)
where
fl(t, 0) = 0 (4-7)
Assume that the following conditions are also satisfied
(1) Fl(t) is such that the system
: Fl(t)x (4-8)
is exponentially asymptotically stable for the equilibrium point x : 0. In otherwords, the state transition matrix D(t, to) associated with (4-8) is such that
S110(t, t 0 )I i< me- a(t-t0) (4-9)
for some positive constants m and a.