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SmithDelayCompensator
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SmithDelayCompensator

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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

open allclose all

Basic Examples  (1)Summary of the most common use cases

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

In[1]:=1
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https://wolfram.com/xid/0dbh01ehk6gshiriua-6jueqr
Out[1]=1

Scope  (2)Survey of the scope of standard use cases

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

In[1]:=1
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https://wolfram.com/xid/0dbh01ehk6gshiriua-co8a7
Out[1]=1

A compensator for a StateSpaceModel with internal delays:

In[1]:=1
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https://wolfram.com/xid/0dbh01ehk6gshiriua-fiw3v3
Out[1]=1

Applications  (2)Sample problems that can be solved with this function

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

In[1]:=1
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https://wolfram.com/xid/0dbh01ehk6gshiriua-b65aiy
In[2]:=2
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https://wolfram.com/xid/0dbh01ehk6gshiriua-fk9gqy
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dbh01ehk6gshiriua-e5kog8
Out[3]=3
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0dbh01ehk6gshiriua-d7s0t
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0dbh01ehk6gshiriua-jl96o8
Out[6]=6

The closed-loop system has no internal delays:

In[7]:=7
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https://wolfram.com/xid/0dbh01ehk6gshiriua-jdl7t2
Out[7]=7

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

In[1]:=1
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https://wolfram.com/xid/0dbh01ehk6gshiriua-l0skk7

A Smith predictor for a diagonal feedback matrix:

In[2]:=2
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https://wolfram.com/xid/0dbh01ehk6gshiriua-gg9prh
In[3]:=3
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https://wolfram.com/xid/0dbh01ehk6gshiriua-6recp
Out[3]=3

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

In[4]:=4
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https://wolfram.com/xid/0dbh01ehk6gshiriua-e05ond
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0dbh01ehk6gshiriua-h5xscj
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0dbh01ehk6gshiriua-n3dfqm
In[7]:=7
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https://wolfram.com/xid/0dbh01ehk6gshiriua-hhs5za
Out[7]=7
In[8]:=8
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https://wolfram.com/xid/0dbh01ehk6gshiriua-fzqxq4
Out[8]=8

Properties & Relations  (1)Properties of the function, and connections to other functions

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

In[1]:=1
✖
https://wolfram.com/xid/0dbh01ehk6gshiriua-cl6s8i
Out[1]=1

The resulting closed-loop system has no internal delays:

In[2]:=2
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https://wolfram.com/xid/0dbh01ehk6gshiriua-i18fpm
Out[2]=2
Wolfram Research (2012), SmithDelayCompensator, Wolfram Language function, https://reference.wolfram.com/language/ref/SmithDelayCompensator.html.
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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.

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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.

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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

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

BibTeX

@misc{reference.wolfram_2025_smithdelaycompensator, author="Wolfram Research", title="{SmithDelayCompensator}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/SmithDelayCompensator.html}", note=[Accessed: 29-March-2025 ]}

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@misc{reference.wolfram_2025_smithdelaycompensator, author="Wolfram Research", title="{SmithDelayCompensator}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/SmithDelayCompensator.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_smithdelaycompensator, organization={Wolfram Research}, title={SmithDelayCompensator}, year={2012}, url={https://reference.wolfram.com/language/ref/SmithDelayCompensator.html}, note=[Accessed: 29-March-2025 ]}

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@online{reference.wolfram_2025_smithdelaycompensator, organization={Wolfram Research}, title={SmithDelayCompensator}, year={2012}, url={https://reference.wolfram.com/language/ref/SmithDelayCompensator.html}, note=[Accessed: 29-March-2025 ]}