with T5d (t) must be included in the internal model system (assuming they have not already been included). For example, if the nominal input
supplied to the system differs from the required nominal input by a constant, the internal model system must contain integrators.
Now consider the case when the nominal state is not generated correctly. Let us write
_d * . a
a (t) = R*(t) (t) (5-16)
where a(t) is the nominal state which is actually supplied to the system, x*(t) is the correct nominal state which should have been supplied, and . d(t) is the disturbance representing the difference between the correct and actual signals. Since the nominal state is fed through to the input via a linear feedback gain matrix (see Figure 5.1) it is apparent that d (t) can be modeled as an input disturbance. Hence, we may conclude that the dynamics associated with d(t) must
also be included in the internal model system.
To summarize, we have shown that robustness with respect to the open-loop signals R (t) and U (t) is obtained provided that any deviations from these signals are sucessfully modeled in the dynamics of the internal model system.
Digital Implementation
In the previous treatment of the servomechanism problem there has been an underlying assumption that the control will be implemented via continuous-time methods. Often it is desirable to implement the control using a digital computer and hence a discrete-time control law is