# How to determine sampling time, acceptable dead-time to reject output disturbances in FO- and SOPDT models?

I have a few questions regarding sampling time, dead-time, and output disturbance rejection.

Given a typical first-order plus dead-time model:

$$P = {Kexp^{-sL} \over T_ps+1}.$$

I would like to calculate an acceptable dead-time, L, and appropriate sampling time, T, to reject (attenuate) output disturbances $\le \omega$.

In my past projects, I have followed the rule of thumb outlined here, that the sampling time should be 10 times per process time constant or faster, and taken it for granted

$$T \le 0.1Tp$$

However, I don't know how (or if) faster sampling can improve disturbance rejection. Modeling this in Simulink shows a diminishing return.

Furthermore, I am convinced that if T is fast enough, disturbance rejection performance is limited to dead-time. In the past, I have modeled closed-loop systems in Simulink and iteratively increased L until disturbance rejection was unacceptable. Is there a mathematical relationship between L and $\omega$ that clearly shows this?

Lastly, how do you apply the same exercise to second-order systems where Tp is not explicitly available?

Thank you,

• Is $P$ the disturbance attenuation transfer function? – fibonatic Apr 14 '18 at 16:01