Editing Proportional–integral–derivative controller (section)

===Stability===

If the PID controller parameters (the gains of the proportional, integral and derivative terms) are chosen incorrectly, the controlled process input can be unstable; i.e., its output [[divergence (computer science)|diverges]], with or without [[oscillation]], and is limited only by saturation or mechanical breakage. Instability is caused by ''excess'' gain, particularly in the presence of significant lag.

Generally, stabilization of response is required and the process must not oscillate for any combination of process conditions and setpoints, though sometimes [[marginal stability]] (bounded oscillation) is acceptable or desired.{{Citation needed|date=March 2011}}

Mathematically, the origins of instability can be seen in the [[Laplace domain]].<ref name=Bechhoefer>{{cite journal | last = Bechhoefer | first = John | title = Feedback for Physicists: A Tutorial Essay On Control | journal = Reviews of Modern Physics | volume = 77 | issue = 3 | pages = 783–835 | doi=10.1103/revmodphys.77.783| citeseerx = 10.1.1.124.7043 | year = 2005 | bibcode = 2005RvMP...77..783B }}</ref>

The closed-loop transfer function is
: <math>H(s) = \frac{K(s)G(s)}{1 + K(s)G(s)},</math>
where <math>K(s)</math> is the PID transfer function, and <math>G(s)</math> is the plant transfer function. A system is ''unstable'' where the closed-loop transfer function diverges for some <math>s</math>.<ref name="Bechhoefer"/> This happens in situations where <math>K(s)G(s) = -1</math>. In other words, this happens when <math>|K(s)G(s)| = 1</math> with a 180° phase shift. Stability is guaranteed when <math>K(s)G(s) < 1</math> for frequencies that suffer high phase shifts. A more general formalism of this effect is known as the [[Nyquist stability criterion]].