Sampled-Data Systems

Sampled-data systems mix a continuous plant with a discrete controller, requiring analysis of sampling, holds, and aliasing together.

Continuous Plant, Discrete Controller

A sampled-data system is a hybrid: a physical plant that evolves continuously in time, controlled by a digital device that acts only at discrete sample instants. Understanding it requires treating the continuous and discrete parts together, along with the converters that link them. Almost every modern control system is of this kind.

The signal chain

• Sampler: an analog-to-digital converter reads the continuous output at each sample instant.
• Digital controller: computes the command from sampled data, typically once per sample period.
• Hold: a digital-to-analog converter, usually a zero-order hold, keeps the command constant between updates.
• Plant: the continuous system driven by the held command.

The zero-order hold

The zero-order hold reconstructs a continuous signal from discrete samples by holding each value until the next update, producing a staircase. This introduces an effective delay of about half a sample period, adding phase lag that erodes stability margin. Discretizing a continuous plant together with its hold gives the equivalent discrete transfer function used for design.

Aliasing and the Nyquist rate

Sampling folds any signal content above half the sample rate, the Nyquist frequency, down into lower frequencies, where it appears as spurious low-frequency components indistinguishable from real ones. This aliasing corrupts measurements irreversibly. An analog anti-aliasing filter placed before the sampler removes the offending high-frequency content, and is a mandatory part of any sampled-data design.

Choosing the sample rate

The sample rate is a central design parameter. Too slow and the loop loses performance or goes unstable, since the controller sees and corrects the plant only intermittently; too fast and it wastes computation and can amplify quantization noise. A practical range is ten to thirty times the closed-loop bandwidth, balanced against processor and converter limits.

Intersample behavior

A subtlety unique to sampled-data systems is that the output between samples is not seen by the controller and can differ from what the samples suggest, including hidden oscillations. Rigorous sampled-data theory analyzes this intersample behavior directly, rather than assuming the discrete model tells the whole story, which matters for high-performance and safety-critical loops.