Understanding oversampling in control system

Oversampling means to sample at significantly more than the Nyquist Rate. I can well understand it in signal re-construction: more samples can improve the ADC's resolution, signal-to-error ratio, etc.

However, when in a control system, I have some difficulities understanding it. Assume that the frequency of the process variable is $$f$$, so the Nyquist Rate is $$2f$$, and we sample at $$2f$$ (the controller will take action upon each sample). Maybe we don't get a stable process variable, so we decide to improve the sampling rate.

Next, we sample at $$20f$$, a sampling frequency 10 times higher (also the controller takes action upon each sample), and this time we obtain a stable process variable. However, because the controller action is also 10 times faster, the frequency of the process variable is no longer $$f$$, but much higher, say $$f'$$. Now the Nyquist Rate becomes $$2f'$$.

Comparing sampling frequency $$20f$$ and new Nyquist rate $$2f'$$, they may be very close, so we cannot call it "oversampling".

Is my understanding towards oversampling and Nyquist rate correct?

• Why do you assume that you change the controller action? – Solar Mike Jan 29 '20 at 15:32
• @SolarMike Since I have more samples, if the controller action remain the same, the more samples will be wasted. This is my understanding: more samples --> more controller action. Is this incorrect? – Bloodmoon Jan 29 '20 at 15:36

First, you have your process, which takes place at a frequency $$f_p$$. If you sample by $$f_s = 2 f_p$$ you are able to observe the behavior of process up to the frequency $$f_p$$. The Nyquist frequency is the minimum frequency at which a finite bandwidth signal needs to be sampled to retain all of the information.
However, if the controller needs to to react on the behavior observed at $$f_p$$, the controller bandwidth should be at least $$f_p$$, while the sampling frequency needs to be much higher. According to (Franklin, 2015, p.613), the sampling rate should be at least 20 times the bandwidth of the controller. However, best performance is obtained when the sample rate is greater than 25 times the controller bandwidth.