24
It is seen that the system NCT has an equilibrium point at
x(t) = n(t) = 0
Theorem 2.2: Suppose that the hypotheses of Theorem 2.1 are satisfied
and that for some choice of Kj and the system NCT is locally
asymptotically stable. Then, with the control scheme defined by system
NC, local tracking of r (t) with disturbance w (t) will occur. If in
addition, system NCT is globally asymptotically stable then global
tracking of r*(t) with disturbance w*(t) will occur.
Proof: Obvious since x(t) + 0 as t +  and e(t) = r (t) Hx(t) =
H[x*(t) x(t)] = -Hx(t).
Since global stability is often difficult to obtain in many
practical systems using constant-gain feedback, the local stability
result of Theorem 2.2 will most often apply. Consequently, success of
the control scheme will depend on the initial state of the original
system and of the internal model system. This will usually mean that
tracking and disturbance rejection can be achieved only if the reference
signal and the disturbance signal are not excessively large.
Stability of the system NCT will be a major topic of the next
section as well as subsequent chapters. At this point, however, it is
appropriate to generalize the results obtained so far. This is
important since previous results have been developed with the assumption
that only one particular reference signal r*(t) and one particular
"if
disturbance signal w (t) will be applied to the system.
The