## Basic properties of thermostatic temperature controllers

Temperature control systems called proportional controllers come a little closer to the complexity of the temperature regulation system in our bodies.

They do not have a simple thermostat with its obvious set point. They have the advantage over a thermostat that they do not oscillate, at least if they are working properly.

Stripped to essentials, a temperature regulation system with a proportional controller includes an effector device and a controller. The heating or cooling power of the effector device is adjustable in a linear, i.e., proportional relationship to the control signal applied to it, within the limits of its minimal and maximal output levels. The controller generates this control signal according to the temperature of the controlled medium, transduced by some sort of temperature sensor. This control system interacts with a the physical system whose temperature it is intended to defend against changes in the surrounding medium. What the control system responds to, i.e., what is really really controlled, is the temperature of the sensor, so care must be taken to assure that the sensor is located in an appropriate place within the physical system.

In the upper panel of the graphic is an example of properties of a controller that regulates heater output. Once temperature falls below a level labeled Tc-zero, the heater output increases progressively up to a maximal level in proportion to the temperature that is sensed by the controller. This sloped segment between the physical limits of the effector is the region of proportional control.

In the upper panel are two equations that each describe the sloped segment, just two different ways of describing the same straight line. The expression with the H-zero and Tc-zero is a convenient one because it focuses on that first point at which the upslope begins as temperature falls.

It's perfectly all right with me if you call this first point, Tc-zero a set point. I would rather call it a threshold, but I can live with set point. At least it can be defined operationally -- that is you can conceive of an experiment in which you can reproducibly identify this point.

One misconcept that accompanies the set point idea is that systems equilibrate at their set point. But they do not. They equilibrate where the effector output matches the thermal load and that usually requires an error signal.

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