<Dl> <Dd> u (t) = K p e (t) + K i ∫ 0 t e (t ′) d t ′ + K d d e (t) d t, (\ displaystyle u (t) = K_ (\ text (p)) e (t) + K_ (\ text (i)) \ int _ (0) ^ (t) e (t') \, dt'+ K_ (\ text (d)) (\ frac (de (t)) (dt)),) </Dd> </Dl> <Dd> u (t) = K p e (t) + K i ∫ 0 t e (t ′) d t ′ + K d d e (t) d t, (\ displaystyle u (t) = K_ (\ text (p)) e (t) + K_ (\ text (i)) \ int _ (0) ^ (t) e (t') \, dt'+ K_ (\ text (d)) (\ frac (de (t)) (dt)),) </Dd> <P> where K p (\ displaystyle K_ (\ text (p))), K i (\ displaystyle K_ (\ text (i))), and K d (\ displaystyle K_ (\ text (d))), all non-negative, denote the coefficients for the proportional, integral, and derivative terms respectively (sometimes denoted P, I, and D). </P> <P> In the standard form of the equation (see later in article), K i (\ displaystyle K_ (\ text (i))) and K d (\ displaystyle K_ (\ text (d))) are respectively replaced by K p / T i (\ displaystyle K_ (\ text (p)) / T_ (\ text (i))) and K p T d (\ displaystyle K_ (\ text (p)) T_ (\ text (d))); the advantage of this being that T i (\ displaystyle T_ (\ text (i))) and T d (\ displaystyle T_ (\ text (d))) have some understandable physical meaning, as they represent the integration time and the derivative time respectively . </P>

In a pid controller the values of proportional integral and derivative are dependent on