Sample Rate
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## 1 Introduction to Sample Rate

Controls engineers use several rules of thumb. One of the most common rules of thumb is that the sample rate of the feedback sensor must be at least 10x (10 times) the closed loop bandwidth. Sample rates of 15x and 20x the closed loop bandwidth are even better. The benefits of feedback sensor sample rates much higher than 20x rapidly diminish.

## 2 Sample Rate: Nyquist

Recently I was asked why we need 10x instead of 2x as Nyquist would suggest. The Nyquist rule of thumb states that a signal can be reconstructed only to 1/2 the sample rate. Not all signals can be reconstructed to 1/2 the sample rate but no signal can be reconstructed to a frequency higher than 1/2 the sample rate.

## 3 Sample Rate: Controls

Figure 1: Sine Wave representing Error

For purposes of controls the Nyquist rate is irrelevant. The error at several points must be measured. A 2x sampling could be at the peak and trough or it could be at the 2 zero crossings. If the samples were near the zero crossings then the error measurements would be near zero. The control system would have little error signal to follow.

Since the closed loop system relies on sensor feedback for correction the error measurements must be accurate and often. If we model the error with a sine wave we can see (in Figure 1) that the 2 samples could line up in such a way that the error would be small or at least close to zero.

Figure 2 shows the worst case scenario for 10 samples per sine wave. Notice that the maximum error is approximately 5%. The higher the sampling rate, the smaller the time between samples, the smaller the maximum sampling error. This sampling error will move through the closed loop system just like sensor noise will. Sensor noise goes straight through the closed loop system unattenuated to become part of the residual error.

Figure 2: Sine Wave representing Error with Samples