Sampling Time

It is important to select a good sampling time, otherwise the stability of the system can be affected (get overshooting, never converges, does not respond quick enough, etc). When sample time T is too large, you get Aliased Signals or fake signals from noise where your controller would respond to these fake signals as if they were real which could make your controller worse and once again, affect stability.

Also if you choose the sampling time to be too large, the Poles will start to drift towards the edge of the unit disk which will cause numerical issues when solving Optimization Problems.

Choosing a Sampling Time

• Consider the time scales of all signals and systems in our closed-loop sampled-data systems:

  1. Bandwidth (${\omega }_{BW}$)
  2. Sampling Frequency (${\omega }_s$)
  3. Frequency of Signals (${\omega }_0$)
  4. We want ${\omega }_s>\max \left({\omega }_{BW},{\omega }_0\right)=>\text{small T}$

• The smaller we pick our sampling time, the more expensive our hardware becomes!

• If the sampling time is too fast, it might not be time for computing new control actions.

• You could also reach limits of numerical precision

Options

1. Choose $T$ small enough (${\omega }_s>\left(5-10\right)\text{ x }\max ({\omega }_{BW},{\omega }_0)$)
2. Add Continuous Time Low Pass Filter where the idea here is to filter out the high frequency signals to avoid aliasing

Conversion

Usually we solve for ${\omega }_{BW}$ where we actually want $T_s$, which we can get by $T_s=\frac{2\pi }{{\omega }_{BW}}$

Sampling → Stability

Given $x\left(t\right)=e^{\lambda t}x(0)$ and sampling at $t=kT$, you can prove that $\left|e^{\lambda }\right|$ is $e^{\mathrm{Re}\lambda }$ which is stable when $\mathrm{Re}\left(\lambda \right)<0$ which implies $e^{\mathrm{Re}\left(\lambda \right)}<1$. Therefore, when a system is stable in Continuous Time, it is also stable when sampled in Discrete Time