On-Off, Discrete-Time Control of a Stable First-Order Linear Plant

This paper discusses the on-off sampled-data regulation of a stable first-order linear plant. Two control laws are considered:(1) A linear switching law, which leads to a discrete map already studied by Nagumo and Sato and others; and (2) Optimal control which minimizes mean squared error from the command at sampling instants. The latter is found by showing equivalence to an Ising spin problem. The two controls lead to the same class of limit cycles, but their average outputs are different Devil´s staircase functions of set-point or bias input. An interesting mathematical relationship between the lengths o corresponding steps in the two staircases is found.