fA(t, XA) := fA(t, XA) - FA(t)xA (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 rsup 1fifA(t' XA)11
11 XsA 1+0 tO x ) = 0 (2-47)
F 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