gives a delay-free system by using approximations of order ord of the time delays in system sys.

Details and Options

  • The system can be either a StateSpaceModel or a TransferFunctionModel.
  • For continuous-time systems, delays are approximated using PadeApproximant with order ord.
  • For discrete-time systems, delays are approximated poles for the integer part of the delay and a Thiran all-pass filter of order ord for the fractional part of the delay.


open allclose all

Basic Examples  (2)

Approximate a StateSpaceModel with an input delay:

A TransferFunctionModel with delay:

Scope  (4)

A continuous-time StateSpaceModel with symbolic delay:

A discrete-time StateSpaceModel:

A continuous-time TransferFunctionModel:

A discrete-time TransferFunctionModel with integer delay:

With fractional delay:

Use only a first-order approximation:

Generalizations & Extensions  (1)

For discrete-time systems, if the order is omitted, it will be chosen automatically:

Applications  (1)

The model of the lathe below uses an internal delay to account for a varying chip size:

The delay causes an instability:

Approximate the delays to allow for controller design:

Apply the controller to the original time-delay system:

The resulting closed-loop system is stable:

Properties & Relations  (1)

Zero-order approximations are equivalent to setting the delay equal to zero:

Possible Issues  (1)

Approximating a discrete-time system with too large an order creates an unstable system:

Instead, allow the order to be chosen automatically:

Wolfram Research (2012), SystemsModelDelayApproximate, Wolfram Language function,


Wolfram Research (2012), SystemsModelDelayApproximate, Wolfram Language function,


@misc{reference.wolfram_2020_systemsmodeldelayapproximate, author="Wolfram Research", title="{SystemsModelDelayApproximate}", year="2012", howpublished="\url{}", note=[Accessed: 16-April-2021 ]}


@online{reference.wolfram_2020_systemsmodeldelayapproximate, organization={Wolfram Research}, title={SystemsModelDelayApproximate}, year={2012}, url={}, note=[Accessed: 16-April-2021 ]}


Wolfram Language. 2012. "SystemsModelDelayApproximate." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2012). SystemsModelDelayApproximate. Wolfram Language & System Documentation Center. Retrieved from