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Analytical methods for thermostatic on-off controls

Traditionally, automatic control theory has mainly been concerned with the behaviour of linear controllers, where the controller output is some continuous function of the error signal. These use combinations of Proportional, Integral and Derivative actions. In practice, however, the cheapest and most commonly used type of controller is on-off, which is non-linear. In an ordinary room thermostat the control is produced by switching the heat on or off: the control is discontinuous. Methods for the analysis of non-linear systems are rather less well-known than those suitable for linear systems, and it would be worthwhile reviewing and illustrating some of the methods available. Three methods are described and each of them is applied to a range of simple transfer functions. The results are compared and conclusions are drawn as to the relative accuracy and ease of application of the methods.