After watching a PI controller take an incredibly long time to cycle through a very small resolution limit, I figured it would be helpful to develop an equation that would provide an estimate of the period of the limit cycle (sustained equal amplitude oscillation) from a valve resolution limit (e.g. stick-slip). The attached equation shows that while dead band in itself does not cause a limit cycle in a single self-regulating loop, dead band can have a profound effect on the period of the oscillation. Since on-off rotary valves disguised as throttling valves can have dead bands more than 10 times the resolution limit, drawn out limit cycles is not an uncommon phenomenon. The following file tells the story.