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Technology / Technical guide
Discrete Control & Zero-Order Hold: Between Samples
A discrete calculation produces values at selected instants, while the mechanical or electrical process continues between them. A zero-order hold describes one way those values become a piecewise-constant signal. The hold is a representation of the output interval; it is neither ideal interpolation nor automatically an extra sample of delay.
A constant command does not freeze the physical process.
Distinguish the sequence, the held waveform and the continuously evolving model that receives it.
01 / Held intervals
Each value persists until the next update.
For ideal updates at kTs, a zero-order hold sets the output to u[k] from kTs up to, but excluding, (k+1)Ts.
For ideal updates at kTs, a zero-order hold sets the output to u[k] from kTs up to, but excluding, (k+1)Ts. MathWorks describes holding the input for the sample period. The sequence value and continuous-time waveform have different domains.
A hypothetical sequence of 3, 5 and 5 arbitrary units with Ts = 0.1 seconds gives levels 3 then 5 then 5 across successive intervals. Equal adjacent values leave no jump at that update; the absence of a jump does not imply the update never occurred.
02 / Continuous response
The receiving system still evolves between updates.
A continuous dynamic system can accelerate, settle or store energy while its input remains constant.
A continuous dynamic system can accelerate, settle or store energy while its input remains constant. Its state at the next update depends on the whole intervening response, not just a list of input values without an interval.
For a declared linear model, that evolution can be integrated under the constant-input assumption to obtain a discrete state relationship. This mathematical correspondence depends on the model and Ts; it does not establish how an actual controller or actuator updates.
03 / Not a unit delay
Holding and postponing are different operations.
A one-sample delay uses an earlier sequence value u[k−1] at an update.
A one-sample delay uses an earlier sequence value u[k−1] at an update. The ideal hold defined here uses the current u[k] during its assigned interval. Computation or transmission can add a separate delay, but it is not inherent in this indexing definition.
Confusing those operations can shift the model by a full interval. The pure-delay guide explains an explicit time shift, while a hold determines the shape of the inter-update signal.
04 / Pulse response
The ideal hold has a frequency-dependent envelope.
Normalising the rectangular hold pulse by its duration gives frequency response exp(−jωTs/2) multiplied by sin(ωTs/2)/(ωTs/2), with the zero-frequency limit defined as one.
Normalising the rectangular hold pulse by its duration gives frequency response exp(−jωTs/2) multiplied by sin(ωTs/2)/(ωTs/2), with the zero-frequency limit defined as one. This follows by integrating a constant pulse over its stated interval.
Below the first zero, the sinc factor is positive and the phase is −ωTs/2, while magnitude falls from its low-frequency value. The hold therefore is not a pure delay at every frequency, even though a half-interval phase term appears in this model.
05 / Illustrative value
A response factor is not an actual controller setting.
At invented Ts = 0.1 seconds and f = 1 Hz, ωTs/2 = π/10.
At invented Ts = 0.1 seconds and f = 1 Hz, ωTs/2 = π/10. The normalised hold magnitude is approximately 0.984 and its phase −18°. These numbers describe the ideal rectangular-pulse relationship, not a turbine feedback result.
The sampling representation and later reconstruction remain separate from this hold calculation. The frequency-grid guide addresses a different question about analysing retained records rather than producing a held command.
06 / Scope
Ideal intervals omit physical update details.
Nonuniform update times, actuator dynamics and constrained output can alter the physical waveform.
Nonuniform update times, actuator dynamics and constrained output can alter the physical waveform. A model using a zero-order hold must retain the boundary where that approximation applies rather than assuming every digital signal has the same behaviour.
No MARSTON hold, update period or discretisation method is declared. The page explains inter-update representation without implementing software, selecting Ts, or changing controller or interface settings.
Related technical guides: Feedback Delay: Phase Lag and Stability Boundaries · Control Transfer Functions: Poles, Zeros & Initial State · Sampling Jitter: Timing Error, Signal Slope & SNR.
General background: MathWorks: Hold-period behaviour and distinction from a unit delay. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| u[k] | Discrete sequence value | Its index refers to a declared update. |
| Ts | Uniform update interval, seconds | Not inferred from a display refresh. |
| Held u(t) | Piecewise-constant waveform | Uses current u[k] on its stated interval. |
| u[k−1] | Explicit one-sample delayed value | Different from holding u[k]. |
| Normalised pulse response | Sinc magnitude with a half-interval phase term | An ideal model, not a pure delay at all frequencies. |
Engineering questions.
Does holding a command stop plant motion?
No. The continuous system can evolve under a constant input.
Is every hold an extra one-sample delay?
No. Indexing and explicit delays must be distinguished.
Is the held waveform ideal interpolation?
No. It is piecewise constant.
Does half-interval phase mean a pure delay everywhere?
No. The hold also has a frequency-dependent magnitude.
Is 0.1 seconds a product update interval?
No. It is an invented illustration.
Does this page select a sampling period?
No actual control setting is prescribed.
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