often specified in advance, there is little difficulty associated with introducing the dynamics of the state trajectory into the internal model system. For example, if the desired trajectory consists of sinusoidal signals then the internal model system must have poles (eigenvalues of the A matrix) at the frequencies of the sinusoidal signals.
The dynamics which must be included in the internal model system in order to generate the appropriate input signal are investigated here. The input needed to produce a certain state trajectory can be calculated from (6-7). This input is seen to be a function of the state trajectory (i.e., e*(t), ;*(t) and o (t)) as well as the disturbance w(t).
Unfortunately, the disturbance is not generally known in advance so that determining an exact input may not be possible. In many situations,
however, one has an idea of the frequencies and also the ranges of amplitudes which might occur in the disturbance signal. Consequently, the input TA(t) can be evaluated for various combinations of the
anticipated disturbance signals. Using Fourier analysis, the dominant frequencies in the resulting inputs can then be identified and the
internal model system can be designed by placing eigenvalues in locations corresponding to these anticipated frequencies.
Now consider a very important situation which does not require Fourier analysis; namely tracking a constant (or step) reference signal while sinusoidal disturbances are being applied. When e*(t) is constant then 6*(t) and U*(t) are both zero. Consequently, equation (6-7) can be written in the following form
TA(t) = -Tg(e*) - D(O*)w(t) (6-9)