given by (4-21) is exponentially stable. This completes the proof.
In order to interpret Theorem 4.2 we need certain continuity
conditions to hold. That is, F*(t) and G*(t) should be continuous functions of the reference r*(t) and the disturbance w*(t). Then
assumptions (i) and (ii) are realistic since both iiF (t) - Fi and
viG*(t) - Goo i will be small whenever i r*(t)o and iw (t)I are small. Furthermore, because F*(t) and G*(t) are often periodic due to the periodicity of x*(t), u*(t), and w*(t), C I and 62 will be,
respectively, the maximum values that iiF (t) - F011i and iiG*(t) - iii assume over one period.
As a final point, note that if in condition (iii) of Theorem 4.2 the word "stabilizable" is replaced by "controllable" then the feedback gains K1 and K2 can be selected to arbitrarily assign the eigenvalues of
the system (4-21). This may, in turn, make it possible to obtain a large ratio of a/m with suitably chosen feedback gains. Then, provided that 11K1, K211ii does not become too large, L will increase and hence
larger reference and disturbance signals will be allowed. Note also
that if the input enters into the nonlinear system by a linear timeinvariant mapping, E2 will be zero so that increasing the ratio a/m will
always increase L.
We now give a rather lengthy example which makes use of many of the results obtained so far for the nonlinear servomechanism problem. In
this example, simulated test results are provided to show the performance of the control algorithm. Also, simulations are provided which show the consequence of using a controller based on linear servomechanism theory.