It is apparent that '(t) and %(t) are simply the deviations of the true state and input from the nominal state and input.
As indicated, the tracking error will be zero whenever the state and input of the plant are x*(t) and u*(t) respectively. Assuming there
exists a state trajectory n (t) for the internal model system which allows this to happen, we must then have
K2(t)n*(t) = -K1(t)R*(t) - %*(t) (5-5)
The above equation is a mere consequence of definition (5-3) and the structure of the controller.
Although K1(t) and K2(t) are shown to be functions of time, for the
present, assume that they are constant. Also, assume that x (t) and
S*(t) satisfy a linear differential equation of the form given in assumption (A.2). It then readily follows (see Chapter Two) that if the internal model system is chosen to contain the modes of x*(t) and * (t) and if the pair (A, K2(t)) is observable, tracking will occur for some
initial state [x*(O), n (0)]. Consequently, we shall require that x Wt) and u*(t) satisfy the differential equation given in (A.2). The internal model system is designed accordingly.
One advantage of the new requirement is that, effectively, the class of signals for which x*(t) and u*(t) are allowed to belong is increased. For example, Figure 5-2 shows a state trajectory x*(t) which
consists of a sinusoidal trajectory Z *t) superimposed on some nominal trajectory R*Ct). The trajectory - (t) is not restricted by assumption (A.2).