Choosing Sampling Time

To choose the sampling time, we consider the time scales of all signals and systems in our closed-loop sampled-data system:

1. Plant bandwith ${\omega }_{bw}$
2. Sampling frequency ${\omega }_s:=\frac{2\pi }{T}$
3. Frequency of reference and/or disturbance and/or settling time (${\omega }_0$)

Taking these into account, we have some options:

1. Choose $T$ small enough. Rule of thumb: ${\omega }_s>(5-10)\times \max \{{\omega }_{bw},{\omega }_0\}$.
2. Add continuous-time low pass filter The idea is to filter out the high frequency signals to avoid aliasing

Plant bandwidth

At high enough frequencies, the gains of all (physical) plans approach zero. At some point, if input is high enough frequency, you simply can’t respond fast enough. ${\omega }_{bw}$ serves as an estimate for the fastest time scales for the plant.

Two ways to find ${\omega }_{bw}$:

1. Find $\omega$ where $|P(j\omega )|=-3\text{ dB}$
2. ${\omega }_{bw}\approx {\max }_{\text{poles }{p}_i\in P(s)}\text{Re}(p_i)$
   • For example, $P(s)=\frac{(s-2)(s+3)}{(s+4)(s+5)}$ would have ${\omega }_{bw}\approx 5$.

Sampling frequency

Sampling frequency ${\omega }_s:=\frac{2\pi }{T}$. We desire the sampling frequency to be high such that ${\omega }_s>\text{max}({\omega }_{bw},{\omega }_0)$.

• This means that we want a small sampling time $T$. However, this typically requires more expensive hardware, and may not even be feasible due to hardware limitations.
• If we sample too fast, there might not be time for computing new control actions.
• We might also reach the limits of our numerical precision

Frequency of reference/disturbance/settling time

Frequency of reference and/or disturbance and/or settling time (${\omega }_0$):

• Ex. $r(t)=\cos ({\omega }_{0}t)$
• Ex. ${\omega }_0=\frac{2\pi }{\text{settling time}}$