[Martin Taylor 2003.12.07.2329 EST]
Too many separate threads are being discussed under the no longer
descriptive title of "Lova and Hate".
[From Bruce Gregory (2003.12.07.2255)]
[Martin Taylor 2003.12.07.1945 EST]
The "getting to work on time" control system experiences little
increase in its (currently near zero) error if you brake, but the
"distance to car in front" control system experiences a rapid and
large increase in error if you don't. If two systems have equal gain,
the one with greater error will have greater output. The foot goes on
I understand, but I don't agree. Why do you say that the 'getting to
work on time' control system experiences little or no error when I act
in a way that insures that I will not get to work on time? Suppose no
car cuts in front of me. What happens then if my car brakes due to a
malfunction in the system? Does the 'getting to work on time' control
system experience little error?
Yes, assuming two things. (1) the time lost is a small portion of the
time left for getting to work, so that a small (and easily
controlled) speedup late in the trip will correct the problem, and
(2) the error in the "getting to work on time" system was initially
small or zero, meaning that you were not at that point already highly
concerned that you were going to be late.
In the situation you originally described, the car cutting in front
will make a minuscule difference to your arrival time, anyway,
compared to the loss of time that would occur if you crashed.
If you are subjected to a breakdown or a traffic jam, the time lost
can be a good proportion of the time needed to get to work on time,
and then the error in that system can become large. Likewise, if you
are already late, you may not be so ready to brake--you may take the
risk of trying to pass the guy who cut in, so as to maintain your
speed. The outputs of the two control systems will be more nearly
equal. And another degree of freedom is available that allows the
"don't hit the other guy" system which influences the "put the brakes
on" action to control without lowering the speed reference.
Howsoever that may be, "overwhelming" is a function of output, not of
gain alone. Output is a function of the gain function operating on
the error magnitude.
As you are driving, leaving enough space between you and the car in
front that you experience no error, that control system has no output
(i.e. you don't slow down to increase the gap, and you don't speed up
to shorten it). At the same time, you are controlling your speed
through another control system, one based on the reference value for
arrival and your perception of your expected arival time based on
your current speed and location, and your expectation of the speed
you are likely to be able to maintain on the rest of the journey
(imagination comes in, here, but let's worry only about the resulting
perception). That control system also has no output at the moment,
because your reference "time to arrival" is no earlier than your
perceived "expected time of arrival." So your speed is satisfactory,
and you don't change your reference for its value.
Now a car cuts in front and puts its brakes on. Suddenly, the "space
to car in front" perception is much shorter than its reference value,
which creates output that affects the reference for perception of
your speed. The "time to arrival" control system hasn't experienced
much, if any, change in its perception, so it still has zero output.
And it won't experience much error until you have been slowed or
stopped for an appreciable time, because it isn't only your current
speed that enters into its perception, but also your (imagined)
perception of speed attainable on the rest of the journey, distance
to go, and time available.
I suppose the net result of this way of looking at it is that the
hypothetical situation isn't actually an example of irresolvable
conflict in which one system overwhelms another, though one system in
practice will overwhelm the other in respect of controlling the
instantaneous speed right now. Both systems may well be able to
control their perceptions, despite the disturbance. The "gap to car
in front" dominates "speed now", whereas "time of arrival" can
exercise control by shifting reference speeds for late parts of the
journey. Lots of degrees of freedom!
I suppose in all this, we should be careful to keep in mind that none
of these systems are linear. That's fairly important.
Does this help?