Editing Fractional calculus (section)

===PID controllers===
Generalizing [[PID controller]]s to use fractional orders can increase their degree of freedom. The new equation relating the ''control variable'' {{math|''u''(''t'')}} in terms of a measured ''error value'' {{math|''e''(''t'')}} can be written as
<math display="block">u(t) = K_\mathrm{p} e(t) + K_\mathrm{i} D_t^{-\alpha} e(t) + K_\mathrm{d} D_t^{\beta} e(t)</math>

where {{mvar|α}} and {{math|β}} are positive fractional orders and {{math|''K''<sub>p</sub>}}, {{math|''K''<sub>i</sub>}}, and {{math|''K''<sub>d</sub>}}, all non-negative, denote the coefficients for the [[proportional control|proportional]], [[integral]], and [[derivative]] terms, respectively (sometimes denoted {{mvar|P}}, {{mvar|I}}, and {{mvar|D}}).<ref>{{cite journal |last1=Tenreiro Machado |first1=J. A. |last2=Silva |first2=Manuel F. |last3=Barbosa |first3=Ramiro S. |last4=Jesus |first4=Isabel S. |last5=Reis |first5=Cecília M. |last6=Marcos |first6=Maria G. |last7=Galhano |first7=Alexandra F. |date=2010 |title=Some Applications of Fractional Calculus in Engineering |journal=[[Mathematical Problems in Engineering]] |language=en |volume=2010 |pages=1–34 |doi=10.1155/2010/639801 |doi-access=free|hdl=10400.22/13143 |hdl-access=free }}</ref>