We have indicated that tracking occurs provided the initial state is correct. It is also necessary to show asymptotic stability of the system which models the transient dynamics. Since the formulation used
here is roughly the same as in previous derivations, the system NCT given in Chapter Two is still a valid model.
Feedback Gain Selection
In this section feedback gain calculation for the control scheme of
Figure 5-1 shall be discussed. Again, as in the previous chapter,
linearization procedures shall be employed so that Liapunov's indirect method can be used to determine stability.
Before proceeding, it is necessary to discuss the time-varying feedback law which has been chosen for the control scheme of Figure 5-1. Time-varying feedback is in contradiction with the requirements imposed to solve the servomechanism problem; however, it may be true that the time-varying gains vary slowly enough to treat them as constant for all practical purposes (the quasi-static approach). Here we assume this to be the case. Later is will become apparent that selecting timevarying feedback provides better compensation over the nominal trajectory. If the signals comprising the nominal trajectory vary
slowly enough then it is likely that the feedback gains K1(t) and K2(t) can be chosen to vary slowly.
Now consider the stabilization problem. Let us assume that all conditions for tracking and disturbance rejection have been met and the internal model system for the scheme of Figure 5-1 has been chosen in accordance with the theory of Chapter Two (using x*(t) and u*(t) in (A.2)). The linearization of the transient system NCT has been given previously but is repeated here for convenience.