78
It is apparent that 'X'(t) and ui(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) = -K^tj/ft) lT*(t) (5-5)
The above equation is a mere consequence of definition (5-3) and the
structure of the controller.
Although K-^(t) and ^(t) are shown to be functions of time, for the
present, assume that they are constant. Also, assume that x (t) and
ii*(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 u (t)
and if the pair (A, ^(t)) is observable, tracking will occur for some
initial state [x (0), n (0)]. Consequently, we shall require that
x*(t) and D*(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 "x*(t) superimposed on some nominal
trajectory x*(t). The trajectory x*(t) is not restricted by assumption
(A.2).