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

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

             1       Ts  
  Kp * (1 + ---- * ------)
             Ti      z-1

Thanks!

Answer 1

[Christoph F.]

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.

Answer 2

[Christoph F.]

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]

In the formula above it's "z-1" not "z^-1". Does that make difference?