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SmithDelayCompensator[sys,con]

gives the Smith compensator for the time-delay system sys and the delay-free controller con.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
See Also
Related Guides
History
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SmithDelayCompensator

SmithDelayCompensator[sys,con]

gives the Smith compensator for the time-delay system sys and the delay-free controller con.

Details and Options

  • The system can be a StateSpaceModel or TransferFunctionModel object.
  • A Smith delay compensator is a controller for time-delay systems that eliminates internal delays in the closed-loop system.
  • Block diagram of a Smith predictor with controller con and system model sys, where sys0 denotes all delays being set to zero:

Examples

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Basic Examples  (1)

A compensator for a time-delay TransferFunctionModel with feedback gain k:

Wolfram Language code: SmithDelayCompensator[TransferFunctionModel[{{{SystemsModelDelay[4]}}, 2 + s}, s], TransferFunctionModel[{{{k}}, 1}, s]]

Scope  (2)

A symbolic controller for a time-delay TransferFunctionModel with a PI controller:

Wolfram Language code: SmithDelayCompensator[TransferFunctionModel[{{{k/E^s}}, 1 + s*τ}, s], TransferFunctionModel[{{{Subscript[k, i] + s*Subscript[k, p]}}, s}, s]]

A compensator for a StateSpaceModel with internal delays:

Wolfram Language code: SmithDelayCompensator[StateSpaceModel[{{{1, (1/2) + Subscript[, 3] - 2 Subscript[, T]}, {0, -1}}, {{0}, {3}}, {{1, 0}, {0, 1}}, {{0}, {0}}}, SamplingPeriod -> None, SystemsModelLabels -> None], TransferFunctionModel[{{{-16/3, 8/3}}, 1}, s]]

Applications  (2)

Use a Smith predictor to design a pole-placement controller for a time-delay system:

Wolfram Language code: ssm = StateSpaceModel[{{{1, -2 Subscript[, T]}, {0, -1}}, {{0}, {3}}, {{1, 0}, {0, 1}}, {{0}, {0}}}, SamplingPeriod -> None, SystemsModelLabels -> None];
Wolfram Language code: delayfree = SystemsModelDelayApproximate[ssm, 0]
Wolfram Language code: l = EstimatorGains[delayfree, {-1, -4}] k = StateFeedbackGains[delayfree, {-3, -5}]
Wolfram Language code: con = EstimatorRegulator[delayfree, {l, k}]
Wolfram Language code: sp = SmithDelayCompensator[ssm, con]

The closed-loop system has no internal delays:

Wolfram Language code: Simplify@TransferFunctionModel[SystemsModelFeedbackConnect[SystemsModelSeriesConnect[sp, ssm]]]

A chemical reactor train (Ogunnaike and Ray, 1979):

Wolfram Language code: reactor = StateSpaceModel[{{{-2, 0.5 Subscript[, 1]}, {0.5, -2}}, {{0.4, 0}, {0, 0.5}}, {{Subscript[, 3], 0}, {0, Subscript[, 2]}}, {{0, 0}, {0, 0}}}, SamplingPeriod -> None, SystemsModelLabels -> {{Subscript[c, f][t], Subscript[c, 2 f][t]}, {Overscript[Subscript[c, 1], ^ ][-3 + t], Overscript[Subscript[c, 2], ^ ][-2 + t]}, {Subscript[c, 1][t], Subscript[c, 2][t]}}];

A Smith predictor for a diagonal feedback matrix:

Wolfram Language code: con = TransferFunctionModel[{{Subscript[k, 1c], 0}, {0, Subscript[k, 2c]}}, s];
Wolfram Language code: sp = SmithDelayCompensator[reactor, con]

Compare the closed-loop response with and without the Smith predictor:

Wolfram Language code: cl1 = SystemsModelFeedbackConnect[SystemsModelSeriesConnect[sp, reactor]] /. {Subscript[k, 1c] -> 20, Subscript[k, 2c] -> 20}//Simplify
Wolfram Language code: cl2 = SystemsModelFeedbackConnect[SystemsModelSeriesConnect[con, reactor]] /. {Subscript[k, 1c] -> 3.0, Subscript[k, 2c] -> 3.5}//Simplify
Wolfram Language code: or1 = OutputResponse[cl1, {.5, 1}UnitStep[t], {t, 0, 20}]; or2 = OutputResponse[cl2, {.5, 1}UnitStep[t], {t, 0, 20}];
Wolfram Language code: Plot[or1, {t, 0, 20}, PlotRange -> All]
Wolfram Language code: Plot[or2, {t, 0, 20}, PlotRange -> All]

Properties & Relations  (1)

A Smith predictor for a time-delay TransferFunctionModel and PID controller:

Wolfram Language code: sp = SmithDelayCompensator[tf = TransferFunctionModel[{{{Subscript[ω, n]^2/E^(del*s)}}, s^2 + ((2*s)*ζ)*Subscript[ω, n] + Subscript[ω, n]^2}, s], TransferFunctionModel[{{{s^2*Subscript[k, d] + Subscript[k, i] + s*Subscript[k, p]}}, s}, s]]

The resulting closed-loop system has no internal delays:

Wolfram Language code: SystemsModelFeedbackConnect[SystemsModelSeriesConnect[sp, tf]]//Simplify
Wolfram Research (2012), SmithDelayCompensator, Wolfram Language function, https://reference.wolfram.com/language/ref/SmithDelayCompensator.html.

Text

Wolfram Research (2012), SmithDelayCompensator, Wolfram Language function, https://reference.wolfram.com/language/ref/SmithDelayCompensator.html.

CMS

Wolfram Language. 2012. "SmithDelayCompensator." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SmithDelayCompensator.html.

APA

Wolfram Language. (2012). SmithDelayCompensator. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SmithDelayCompensator.html

BibTeX

@misc{reference.wolfram_2026_smithdelaycompensator, author="Wolfram Research", title="{SmithDelayCompensator}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/SmithDelayCompensator.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_smithdelaycompensator, organization={Wolfram Research}, title={SmithDelayCompensator}, year={2012}, url={https://reference.wolfram.com/language/ref/SmithDelayCompensator.html}, note=[Accessed: 13-June-2026]}