29
(2-46)
We now present a theorem based on Liapunov's indirect method which can
be used to show local tracking. In order to apply Liapunov's indirect
method, the following two technical conditions are required
lim ,sup
ixAi+0 lt>0
(2-47)
Fa(*) is bounded
(2-48)
It is mentioned that the above conditions are almost always satisfied in
practical systems.
Theorem 2.3: Suppose that the hypotheses of Theorem 2.1 are satisfied
and also assume that conditions (2-47) and (2-48) hold true. If in
addition, the system (2-40) is asymptotically stable then, with the
control scheme defined by system NC, local tracking of r*(t) with
disturbance w*(t) will occur.
Extensive use will be made of Theorem 2.3 in later chapters.
The Relation Between the Dimension of the
Internal Model System and the Input/Output Dimensions
In the previous sections, the servomechanism problem was treated
where it was assumed that the number of inputs to the plant was the same