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period. This would correspond to eigenvalues at j2irF for a
continuous-time implementation. If F is zero then we wish to indicate
that the internal model system contains integrators (i.e., eigenvalues
at 1.0 for the discrete-time implementation).
The responses shown in Figure 7-2 demonstrate the ability of the
controller to reject a sinusoidal disturbance force. No reference is
used. Recall from Chapter Six that the controller only needs to contain
poles (eigenvalues) at the frequencies of the disturbance in order to
achieve perfect tracking. The poles in the internal model (I.M.) system
correspond to the 1.5 Hz disturbance and it is clearly seen that the
error does indeed go to zero. Figure 7-3 shows a similar test, however,
in addition to a sinusoidal disturbance a constant disturbance is also
introduced. The poles of the I.M. system are adjusted accordingly
(i.e., integrators are added) and the error once again goes to zero in
steady-state. Figure 7-4 is included to demonstrate that a simple
scheme employing just integrators in the I.M. system will not adequately
compensate for sinusoidal disturbances. Here we see that the steady-
state error to a sinusoidal disturbance is a sinusoid at the frequency
of the disturbance.
Figures 7-5 through 7-7 show the error which occurs when a
sinusoidal reference signal is introduced. A disturbance force in not
considered in these tests. First, Figure 7-5 shows the performance of
the design which results from using only linear servomechanism theory.
The I.M. system contains poles at the frequency of the reference signal
(1.5 Hz in this case). It is evident that the steady-state error is
sinusoidal in nature with a frequency of 3 Hz; twice that of the
reference signal. In order to improve the steady-state performance we