Does this article's appendix contain an error (is the heating system described integral control)?

Question

I am confused by the primer on control theory found at the end of the following paper, which concerns an instance of integral control in a biological system. Since it concerns integral control, I thought electrical engineers might be better able to answer. I apologize if the article is behind a paywall but it is a very popular paper evidently, so you may be able to google for a pdf.

Yi et al. (2000). Robust perfect adaptation in bacterial chemotaxis through integral feedback control. http://www.pnas.org/content/97/9/4649.abstract

Within the primer, the authors use as an example of integral control a heating system where the power setting on a furnace is proportional to the difference between a thermostat's set point and the ambient temperature. The claim is that this is not proportional control because the temperature in the room is the integral of the heat released, and the system is responding to error in temperature. I thought an integral control system would need to keep track of past temperature deviations from the set point. Is their argument accurate, and if so, could someone explain?

To be slightly more specific, one reason I have trouble grasping their argument is that, I imagine, if a window were left open (so that heat was constantly being lost from the room) then the heating system would never adapt but would operate with droop just as a proportional heating system would...right?

Thanks for your help!

Answer 1

Without looking at the paper, and responding only to your question- the heating system described is called proportional control in the industry (assuming the setpoint is temperature, the measured variable is temperature and the output is power-- as is typical). In fact in many heating systems the gain can be high enough that the droop due to demand changes is negligible (and the system can be correct either by changing the setpoint or by applying a correction factor called 'manual reset' that the error is nulled a given setpoint with nominal demand.

So, you are correct that if there was a large demand change from nominal there would be a persistent error with proportional control, but it's not necessarily of any practical importance.

The thermal capacity of the object being heated does indeed introduce a pole in the response, but the heat loss increases with temperature difference (at least proportionally, but often much faster with convection or radiation losses) so the result is not an integral control response.

If you had a block of material that was sufficiently isolated from the environment (almost no conductive, convective or radiation losses) then you could consider it to be an integral controller, but that does not represent even a rough approximation to reality in any of the thousands of systems I've worked with.

Answer 2

First of all, it's a very poor analogy because pretty much any heating system in the world uses a bang-bang, on-off control, operating the heater either at "full throttle" or not at all, since this is the most efficient way to do it.

But even if the heat output was proportional to the temperature error, the temperature would only qualify as "integral" if the thing being heated was perfectly insulated, with no heat loss whatsoever. In that case, the total heat in the system — and therefore the temperature — would be the integral of the heat input. But since that's impossible in real life, the temperature ends up being a balance between heat power in and heat power out. In other words, the temperature rises until the power out (which is generally proportional to temperature) just balances the power in.

As such, the system would require a constant error to run at all, which means that I agree with you — it's effectively a proportional-only system. A true integral system would be able to drive a constant error to zero.

Answer 3

Check to see if they are trying to control the heat in the room or the temperature.

If you want to control the quantity of heat in the room, and the temperature is the measurement you are taking, then it is an integral control system. The integration is done physically by the nature of the temperature vs. heat relationship.

However, if you are trying to control the temperature in a room, then it is a simple proportional control system.

Usually thermostats are trying to control heat, so the argument would be invalid in the typical case, but since I haven't seen the paper, I'm not sure exactly what they are saying they are trying to control.

In any case, it's a bad example for a primer.