Which sampling type takes samples at equal intervals t = kT and is the most common in digital controllers?

Periodic (conventional) sampling

What describes multiple-rate sampling?

Different loops or signals are sampled at different periods, such as a fast inner loop and a slow outer loop

Samples are taken at random times

All signals sampled at the same period

Why is random sampling the hardest to design for?

Instants are random, so it needs statistical analysis

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Types of sampling operations compared

Short answer

Periodic sampling uses equal intervals and is easiest to analyze with the z-transform. Multiple-order sampling has a repeating pattern with unequal spacing within the pattern. Multiple-rate sampling uses different periods for different control loops. Random sampling occurs at irregular instants and is hard to analyze. Periodic flat-top sampling is most common in digital control.

Compare different types of sampling operations used for sampling of continuous time signal.

Sampling converts a continuous-time signal into a sequence of values (pulses) taken at certain instants. Sampling operations are classified by how the sampling instants are chosen.

Types

Periodic (conventional) sampling: samples are taken at equal intervals, $t = kT$ . This is the most common type; it gives simple difference equations and allows use of the z-transform.
Multiple-order sampling: the pattern of sampling instants repeats periodically, i.e. $t_{k+r} - t_k$ is constant, but the spacing within a pattern is not equal.
Multiple-rate sampling: different loops or signals are sampled at different periods, e.g. a fast inner current loop and a slow outer speed loop.
Random sampling: the instants are random, e.g. in event-driven systems or networked control with random delays. It is hard to analyse.

Comparison

Feature	Periodic	Multiple-order	Multiple-rate	Random
Sampling instants	$kT$ , equal spacing	Pattern repeats with period	Different $T$ for different signals	Random
Analysis tool	z-transform	Modified methods	Multirate z-transform	Statistical
Hardware	Simple, one clock	Moderate	Several clocks	Event logic
Use	Most digital controllers	Special systems	Cascade loops with fast and slow parts	Networks, events
Ease of design	Easiest	Harder	Harder	Hardest

Sampling by pulse shape

Ideal (impulse) sampling	Natural sampling	Flat-top sampling
Train of impulses of strength $x(kT)$	Pulses of width $\tau$ follow the signal shape	Pulses of width $\tau$ with constant height $x(kT)$
Mathematical model	Switch closed for $\tau$	Practical S/H output

In digital control, periodic flat-top sampling (S/H) is used in practice, and it is modelled as ideal impulse sampling followed by a zero order hold.

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Types of sampling operations compared

Types

Comparison

Sampling by pulse shape

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