represents a time delay of δ in a StateSpaceModel or TransferFunctionModel .
SystemsModelDelay
represents a time delay of δ in a StateSpaceModel or TransferFunctionModel .
Details
- SystemsModelDelay [δ] makes it possible to represent systems involving time delays and efficiently manipulate, approximate, and simulate these systems.
- SystemsModelDelay [δ] is typeset in StandardForm as δ and can be entered using delay.
- For a StateSpaceModel , SystemsModelDelay [δ] can occur linearly in any of the system matrices. For a signal , SystemsModelDelay [δ]w[t] is taken to be .
- For a TransferFunctionModel , SystemsModelDelay [δ] can occur linearly in the coefficients of the polynomials. It is taken to represent a transformed time delay:
-
-δ s in a continuous-time systemz-δ in a discrete-time system
- Here, s is the Laplace-transform variable, and z is the z-transform variable.
- For discrete-time systems, the delay δ is taken to be a multiple of the SamplingPeriod .
Examples
open all close allBasic Examples (3)
A state-space model with an input delay:
A transfer-function model with delay:
A time-delay state-space model created from delay differential equations:
Scope (7)
A continuous-time state-space model with a state delay:
A discrete-time state-space model with an output delay:
A continuous-time transfer-function model with delay:
Or represent the delay as an exponential:
A discrete-time transfer-function model with delay:
A state-space model with a delay in the descriptor matrix:
A continuous-time state-space model created from delay differential equations:
A discrete-time system from a difference equation including SystemsModelDelay :
Applications (1)
The delay differential equation below describes the motion of the cutting tool in a lathe where the force on the tool depends on the position of the tool from the previous rotation:
The delay creates peaks in the frequency response:
Properties & Relations (5)
A zero delay reduces to 1:
Delays in continuous-time transfer functions are equivalent to exponentials:
Delays in discrete-time transfer-function models are equivalent to additional poles:
State-space systems with neutral delays have delays in the descriptor matrix:
When converting to a transfer function, state delays appear in the denominator:
Possible Issues (1)
In discrete-time systems, delays should have integer values:
When necessary, zero-order approximations are used:
Approximating delays before simulation can give a more accurate result:
Related Guides
History
Text
Wolfram Research (2012), SystemsModelDelay, Wolfram Language function, https://reference.wolfram.com/language/ref/SystemsModelDelay.html.
CMS
Wolfram Language. 2012. "SystemsModelDelay." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SystemsModelDelay.html.
APA
Wolfram Language. (2012). SystemsModelDelay. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SystemsModelDelay.html
BibTeX
@misc{reference.wolfram_2025_systemsmodeldelay, author="Wolfram Research", title="{SystemsModelDelay}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/SystemsModelDelay.html}", note=[Accessed: 17-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_systemsmodeldelay, organization={Wolfram Research}, title={SystemsModelDelay}, year={2012}, url={https://reference.wolfram.com/language/ref/SystemsModelDelay.html}, note=[Accessed: 17-November-2025]}