necessary. In this section we give a brief discussion as to how to devise a satisfactory discrete-time algorithm. It is assumed that the discrete-time control algorithm will closely approximate the performance of the already developed continuous-time algorithm. Consequently,
essentially no new theory will be needed. In addition, since discretetime control is a well known subject area [22], [23] much unnecessary detail shall be omitted from this discussion.
Discrete-time control requires sampling of the various outputs (or states) of the plant and if T denotes the spacing between samples, then
sampling occurs at the times t = kT, k = 0,1,2. It is necessary
that the sample rate (l/T) be chosen high enough so that, for all practical purposes, the resulting control algorithm will behave as a continuous-time control law. For example, if the state x*(t) and the input u*(t) consist of sinusoidal signals then obviously the sample rate should be higher than the highest frequency in the sinusoidal signals.
With the above comments in mind we give a typical digital
implementation of the control scheme shown in Figure 5-1. This is shown in Figure 5-4. Notice that the needed continuous-time signals from the plant are converted into discrete-time signals by sampling so that they may be processed digitally. Additionally, the control u(t) to the plant
is produced by converting the discrete-time signal u(k) into an analog signal. This is accomplished by means of a zero-order hold (z.o.h.) which can be thought of simply as a digital-to-analog convertor
providing a piecewise constant version of u(k).
The remainder of Figure 5-4 is basically self-explanatory, however,
we shall discuss two issues in more detail; namely, construction of the internal model system and selection of the feedback gains.