How much additional positive phase should we request from a

How much additional positive phase should we request from a DT PID controller with a standard selection of the sampling rate at 6 samples-per-rise-time (assuming ZOH introduces a half-sample delay, and bandwidth is 1.5 times the cutoff frequency):

A. About 6 deg.

C. No additional phase

D. Depends on PI vs PID

E. None of the above

Solution

ample rate for digital signal processing
& fast sampling approximations of digital control systems.

fs > 2B allows exact reconstruction of a signal with bandwidth B (Nyquist sampling theorem)

Continuous controllers may be approximated in discrete time by dierence equations e.g. via Euler’s approximation

Reconstruction by a ZOH introduces a delay of T/2, equivalent to a phase delay of !T/2 at frequency !

ZOH phase delay implies a more oscillatory response but negligible eect if fs > 30B

We can design discrete controllers for lower sample rates;
this requires more accurate analysis of discrete signals and systems

The ZOH delay of T/2 (sec) causes

1 - 22

phase lag = !T /2 (rad) phase lag = /2 = 90 phase lag = /30 = 6

at ! rad s 1
at ! = /T [= Nyquist rate] at ! = /(15T)

? 90 phase lag could be catastrophic ? If !samp > 30 !max,

then system bandwidth: !max < /(15T ), so the maximum phase lag is less than 6

usu\"ally safe to ignore

About 6 deg(A)