Editing Proportional–integral–derivative controller (section)

===Standard versus parallel (ideal) form===
The form of the PID controller most often encountered in industry, and the one most relevant to tuning algorithms is the ''standard form''. In this form the <math>K_p</math> gain is applied to the <math>I_{\mathrm{out}}</math>, and <math>D_{\mathrm{out}}</math> terms, yielding:

:<math>u(t) = K_p \left( e(t) + \frac{1}{T_i}\int_0^t e(\tau)\,d\tau + T_d\frac{d}{dt}e(t) \right)</math>
where
:<math>T_i</math> is the ''integral time''
:<math>T_d</math> is the ''derivative time''

In this standard form, the parameters have a clear physical meaning. In particular, the inner summation produces a new single error value which is compensated for future and past errors. The proportional error term is the current error. The derivative components term attempts to predict the error value at <math>T_d</math> seconds (or samples) in the future, assuming that the loop control remains unchanged. The integral component adjusts the error value to compensate for the sum of all past errors, with the intention of completely eliminating them in <math>T_i</math> seconds (or samples). The resulting compensated single error value is then scaled by the single gain <math>K_p</math> to compute the control variable.

In the parallel form, shown in the controller theory section
:<math>u(t) = K_p e(t) + K_i \int_0^t e(\tau)\,d\tau + K_d\frac{d}{dt}e(t)</math>

the gain parameters are related to the parameters of the standard form through <math>K_i = K_p/T_i</math> and <math>K_d = K_p T_d</math>. This parallel form, where the parameters are treated as simple gains, is the most general and flexible form. However, it is also the form where the parameters have the weakest relationship to physical behaviors and is generally reserved for theoretical treatment of the PID controller. The standard form, despite being slightly more complex mathematically, is more common in industry.