[From Bill Powers (981207.0146 MDT)]
Attached: contin1.mdl
I guess everyone who is going to do the thermostat model has done it by
now; I'm starting to get inquiries about when we can to on to the next thing.
So here it is:
The next project is to take the thermostat model and turn it into a
continuous-control model. All the relationships in the model are already
continuous, except for two parts of the control system, the comparator and
the output function. We have seen that when you use an on-off comparator
and an output function that is either on or off, the entire system goes
into oscillation, cycling continuously with the temperature either too low
or too high. However, I hope you are now convinced that even in this case
we get continuous temperature control, with the temperature variations
being small. Looked at from a distance, we see that the output is actually
continuously variable if we define it in terms of the on/off _ratio_. As
the outside temperature falls, we see the proportion of furnace on-time
increasing smoothly. Only when we look in detail at each on-off cycle does
the digital or logical character of the system stand out.
Now, however, we will look at a more sophisticated temperature control
system, the kind that's used in a laboratory experiment in which
temperature has to be kept constant within a hundredth of a degree or
better. You can't have the furnace cycling between full on and completely
off in this kind of system. What is needed is a furnace that can be turned
partly on, and in fact which can vary its heat output between zero and
maximum smoothly as an actuating signal goes smoothly from zero to maximum.
Attached is a model made of thermo1.mdl (furnace, house, and heat losses)
with a control system below it. The equations for the control system and
its constants have not been filled in -- that's the problem for this
exercize. The only difference from the previous models is that the
perceptual delay has been removed -- you can put it back in later if you
want to see its effect. For the precise temperature control we want here,
the air has to be vigourously stirred, not only to keep the air temperature
uniform, but to eliminate the perceptual lag as much as possible.
Not so obvious are the Model ... Time Bounds changes. The computing
interval is now reduced to 0.01 hour, to allow high loop gain. We still
plot only every 0.2 hours, so 500 output values are plotted.
The outside temperature is set to 0 degrees, with a pulse of warming by 25
degrees that lasts for 10 hours.
The assignment is to explain the details of the plot once you have filled
in the equations for the model.
Best,
Bill P.
contin1.mdl (60 Bytes)