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Chapter 5
to get away with a simple gain multiplier. In these cases an interpolated look-up table or
more complex mathematical relationship may be useful.
The rabbit derivative and second derivative terms
The second relationship we can quickly identify has to do with the slope or deriva-
tive of the rabbit. For a position servo, the rabbit derivative is the rabbit velocity, and
nd
the rabbit 2 derivative is the rabbit acceleration.
If the rabbit is attacking positively, we will need to pour on the coals to heat the fur-
nace up or to accelerate the robot. This term is thus proportional to the derivative of
the rabbit, and we will call its gain the rabbit derivative gain or rabbit velocity gain.
If the rabbit is accelerating, then we will need even more power. This term is most
commonly found in position controls and its gain is thus called the rabbit acceleration
gain. If we perfectly guess these gains, then the servo will track the rabbit perfectly,
with the minimum possible delay.
The big advantage to the rabbit derivative terms as opposed to the error derivative
term is that, because they work from the nice clean rabbit command, they do not
amplify system response noise.
Combined reactive and predictive controls
Unfortunately, a predictive control of this type has no way of correcting for variables
such as ambient temperature or payload. It is also unlikely that our perfectly selected
gains will be as perfect for every unit we build. It is possible to learn these gains dur-
ing operation, but this will be discussed later.
It is therefore not practical to expect exact performance from a predictive control.
Yet, the prediction may buy us a guess within say 10% of the correct value, and its
output is instantaneous and stable. For this reason, it is often optimal to combine
terms of a conventional PID with those of a predictive control. Using our analog
control as a metaphor, consider the configuration shown in Figure 5.4.
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