# What is 'z' in the formula for a discrete PID controller?

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Nikoloz Jandieri on 8 Dec 2016
Answered: Christoph F. on 8 Dec 2016
What is 'z' in the formula for a discrete PID?
1 Ts
Kp * (1 + ---- * ------)
Ti z-1
Thanks!

Christoph F. on 8 Dec 2016
1/(z-1) is equal to the infinite sum (z^-1) + (z^-2) + (z^-3) + ...; applied to a signal, this means the sum of all input samples of the past. This is how the digital PID controller approximates the integral over the input signal.
An in-depth explanation of the z-transform would probably be beyond the scope of this forum, and there is plenty of literature about the topic. Basically, the z-transform is as fundamental to signal processing and control in the discrete domain (where signals are time-discrete) as the Fourier- and LaPlace-transformations are for the continuous domain.

Christoph F. on 8 Dec 2016
z is the variable of a transfer function in the z-domain.
(Basically, z^-1 means a time shift of one sample backwards in time. For the mathematical background, refer to an explanation of the z-transformation).
Nikoloz Jandieri on 8 Dec 2016
In the formula above it's "z-1" not "z^-1". Does that make difference?

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