Relative gain array - Revision history

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	The '''relative gain array''' (RGA) is a classical widely-used{{Citation needed\|date=October 2018}} method for determining the best input-output pairings for multivariable [[process control]] systems.<ref>{{citation\|last=Bristol\|first=E.H.\|title=On a new measure of interaction for multivariable process control\|journal=IEEE Transactions on Automatic Control \|year=1966\|volume=1\|pages=133–134\|doi=10.1109/TAC.1966.1098266}}</ref> It has many practical open-loop and closed-loop control applications and is relevant to analyzing many fundamental steady-state closed-loop system properties such as stability and robustness.<ref>{{citation\|last1=Chen\|first1=Dan\|last2=Seborg\|first2=D.E.\|title=Relative Gain Array Analysis for Uncertain Process Models\|journal=AIChE Journal\|year=2002\|volume=48\|issue=2\|pages=302–310\|doi=10.1002/aic.690480214}}</ref>		The '''relative gain array''' (RGA) is a classical widely-used{{Citation needed\|date=October 2018}} method for determining the best input-output pairings for multivariable [[process control]] systems.<ref>{{citation\|last=Bristol\|first=E.H.\|title=On a new measure of interaction for multivariable process control\|journal=IEEE Transactions on Automatic Control \|year=1966\|volume=1\|pages=133–134\|doi=10.1109/TAC.1966.1098266}}</ref> It has many practical open-loop and closed-loop control applications and is relevant to analyzing many fundamental steady-state closed-loop system properties such as stability and robustness.<ref>{{citation\|last1=Chen\|first1=Dan\|last2=Seborg\|first2=D.E.\|title=Relative Gain Array Analysis for Uncertain Process Models\|journal=AIChE Journal\|year=2002\|volume=48\|issue=2\|pages=302–310\|doi=10.1002/aic.690480214\|bibcode=2002AIChE..48..302C }}</ref>

	==Definition==		==Definition==

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2024-09-23T19:38:30Z

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← Previous revision		Revision as of 19:38, 23 September 2024
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	The '''~~Relative~~ ~~Gain~~ ~~Array~~''' (RGA) is a classical widely-used{{Citation needed\|date=October 2018}} method for determining the best input-output pairings for multivariable [[process control]] systems.<ref>{{citation\|last=Bristol\|first=E.H.\|title=On a new measure of interaction for multivariable process control\|journal=IEEE Transactions on Automatic Control \|year=1966\|volume=1\|pages=133–134\|doi=10.1109/TAC.1966.1098266}}</ref> It has many practical open-loop and closed-loop control applications and is relevant to analyzing many fundamental steady-state closed-loop system properties such as stability and robustness.<ref>{{citation\|last1=Chen\|first1=Dan\|last2=Seborg\|first2=D.E.\|title=Relative Gain Array Analysis for Uncertain Process Models\|journal=AIChE Journal\|year=2002\|volume=48\|issue=2\|pages=302–310\|doi=10.1002/aic.690480214}}</ref>		The '''relative gain array''' (RGA) is a classical widely-used{{Citation needed\|date=October 2018}} method for determining the best input-output pairings for multivariable [[process control]] systems.<ref>{{citation\|last=Bristol\|first=E.H.\|title=On a new measure of interaction for multivariable process control\|journal=IEEE Transactions on Automatic Control \|year=1966\|volume=1\|pages=133–134\|doi=10.1109/TAC.1966.1098266}}</ref> It has many practical open-loop and closed-loop control applications and is relevant to analyzing many fundamental steady-state closed-loop system properties such as stability and robustness.<ref>{{citation\|last1=Chen\|first1=Dan\|last2=Seborg\|first2=D.E.\|title=Relative Gain Array Analysis for Uncertain Process Models\|journal=AIChE Journal\|year=2002\|volume=48\|issue=2\|pages=302–310\|doi=10.1002/aic.690480214}}</ref>

	==Definition==		==Definition==

NapoliRoma: NapoliRoma moved page Relative Gain Array to Relative gain array: Change to sentence case (MOS:AT)

2024-09-23T19:37:20Z

NapoliRoma moved page Relative Gain Array to Relative gain array: Change to sentence case (MOS:AT)

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← Previous revision		Revision as of 12:34, 6 June 2022
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	==Generalizations==		==Generalizations==
	The RGA is often generalized in practice to be used when <math>\mathrm{G}</math> is singular, e.g., non-square, by replacing the inverse of <math>\mathrm{G}</math> with its [[Moore–Penrose inverse]] (pseudoinverse).<ref>{{citation\|last1=van de Wal\|first1=M.\|last2=de Jager\|first2=B.\|title=A review of methods for input/output selection\|journal=Automatica\|year=2001\|volume=37\|issue=4\|pages=487–510\|doi=10.1016/S0005-1098(00)00181-3}}</ref> However, it has been shown that the Moore–Penrose pseudoinverse fails to preserve the critical scale-invariance property of the RGA (#2 above) and that the unit-consistent (UC) generalized inverse must therefore be used.<ref>{{citation\|last1=Uhlmann\|first1=Jeffrey\|title=On the Relative Gain Array (RGA) with Singular and Rectangular Matrices\|journal=Applied Mathematics Letters\|year=2019\|volume=93\|pages=52–57\|bibcode=2018arXiv180510312U\|arxiv=1805.10312\|doi=10.1016/j.aml.2019.01.031\|s2cid=44092817 }}</ref> <ref>{{citation\|last1=Qasim Al Yousuf\|first1=Rafal\|last2=Uhlmann\|first2=Jeffrey\|title=On Use of the Moore-Penrose Pseudoinverse for Evaluating the RGA of Non-Square Systems\|journal=Iraqi Journal of Computers, Communications, Control & Systems Engineering \|volume=3\|issue=21\|doi=10.33103/uot.ijccce.21.3.8\|year=2021\|~~doi-broken-date~~=~~2022-06-05~~ \|arxiv=2106.09766 }}</ref>		The RGA is often generalized in practice to be used when <math>\mathrm{G}</math> is singular, e.g., non-square, by replacing the inverse of <math>\mathrm{G}</math> with its [[Moore–Penrose inverse]] (pseudoinverse).<ref>{{citation\|last1=van de Wal\|first1=M.\|last2=de Jager\|first2=B.\|title=A review of methods for input/output selection\|journal=Automatica\|year=2001\|volume=37\|issue=4\|pages=487–510\|doi=10.1016/S0005-1098(00)00181-3}}</ref> However, it has been shown that the Moore–Penrose pseudoinverse fails to preserve the critical scale-invariance property of the RGA (#2 above) and that the unit-consistent (UC) generalized inverse must therefore be used.<ref>{{citation\|last1=Uhlmann\|first1=Jeffrey\|title=On the Relative Gain Array (RGA) with Singular and Rectangular Matrices\|journal=Applied Mathematics Letters\|year=2019\|volume=93\|pages=52–57\|bibcode=2018arXiv180510312U\|arxiv=1805.10312\|doi=10.1016/j.aml.2019.01.031\|s2cid=44092817 }}</ref> <ref>{{citation\|last1=Qasim Al Yousuf\|first1=Rafal\|last2=Uhlmann\|first2=Jeffrey\|title=On Use of the Moore-Penrose Pseudoinverse for Evaluating the RGA of Non-Square Systems\|journal=Iraqi Journal of Computers, Communications, Control & Systems Engineering \|volume=3\|issue=21\|doi=10.33103/uot.ijccce.21.3.8\|year=2021\|pages=89–97 \|arxiv=2106.09766 \|s2cid=235485155 }}</ref>

	==References==		==References==

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← Previous revision		Revision as of 21:49, 5 June 2022
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	==Generalizations==		==Generalizations==
	The RGA is often generalized in practice to be used when <math>\mathrm{G}</math> is singular, e.g., non-square, by replacing the inverse of <math>\mathrm{G}</math> with its [[Moore–Penrose inverse]] (pseudoinverse).<ref>{{citation\|last1=van de Wal\|first1=M.\|last2=de Jager\|first2=B.\|title=A review of methods for input/output selection\|journal=Automatica\|year=2001\|volume=37\|issue=4\|pages=487–510\|doi=10.1016/S0005-1098(00)00181-3}}</ref> However, it has been shown that the Moore–Penrose pseudoinverse fails to preserve the critical scale-invariance property of the RGA (#2 above) and that the unit-consistent (UC) generalized inverse must therefore be used.<ref>{{citation\|last1=Uhlmann\|first1=Jeffrey\|title=On the Relative Gain Array (RGA) with Singular and Rectangular Matrices\|journal=Applied Mathematics Letters\|year=2019\|volume=93\|pages=52–57\|bibcode=2018arXiv180510312U\|arxiv=1805.10312\|doi=10.1016/j.aml.2019.01.031}}</ref> <ref>{{citation\|last1=Qasim Al Yousuf\|first1=Rafal\|last2=Uhlmann\|first2=Jeffrey\|title=On Use of the Moore-Penrose Pseudoinverse for Evaluating the RGA of Non-Square Systems\|journal=Iraqi Journal of Computers, Communications, Control & Systems Engineering \|volume=3\|issue=21\|doi=10.33103/uot.ijccce.21.3.8\|year=2021}}</ref>		The RGA is often generalized in practice to be used when <math>\mathrm{G}</math> is singular, e.g., non-square, by replacing the inverse of <math>\mathrm{G}</math> with its [[Moore–Penrose inverse]] (pseudoinverse).<ref>{{citation\|last1=van de Wal\|first1=M.\|last2=de Jager\|first2=B.\|title=A review of methods for input/output selection\|journal=Automatica\|year=2001\|volume=37\|issue=4\|pages=487–510\|doi=10.1016/S0005-1098(00)00181-3}}</ref> However, it has been shown that the Moore–Penrose pseudoinverse fails to preserve the critical scale-invariance property of the RGA (#2 above) and that the unit-consistent (UC) generalized inverse must therefore be used.<ref>{{citation\|last1=Uhlmann\|first1=Jeffrey\|title=On the Relative Gain Array (RGA) with Singular and Rectangular Matrices\|journal=Applied Mathematics Letters\|year=2019\|volume=93\|pages=52–57\|bibcode=2018arXiv180510312U\|arxiv=1805.10312\|doi=10.1016/j.aml.2019.01.031\|s2cid=44092817 }}</ref> <ref>{{citation\|last1=Qasim Al Yousuf\|first1=Rafal\|last2=Uhlmann\|first2=Jeffrey\|title=On Use of the Moore-Penrose Pseudoinverse for Evaluating the RGA of Non-Square Systems\|journal=Iraqi Journal of Computers, Communications, Control & Systems Engineering \|volume=3\|issue=21\|doi=10.33103/uot.ijccce.21.3.8\|year=2021\|doi-broken-date=2022-06-05 \|arxiv=2106.09766 }}</ref>

	==References==		==References==

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2022-06-05T21:48:20Z

← Previous revision		Revision as of 21:48, 5 June 2022
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	==Generalizations==		==Generalizations==
	The RGA is often generalized in practice to be used when <math>\mathrm{G}</math> is singular, e.g., non-square, by replacing the inverse of <math>\mathrm{G}</math> with its [[Moore–Penrose inverse]] (pseudoinverse).<ref>{{citation\|last1=van de Wal\|first1=M.\|last2=de Jager\|first2=B.\|title=A review of methods for input/output selection\|journal=Automatica\|year=2001\|volume=37\|issue=4\|pages=487–510\|doi=10.1016/S0005-1098(00)00181-3}}</ref> However, it has been shown that the Moore–Penrose pseudoinverse fails to preserve the critical scale-invariance property of the RGA (#2 above) and that the unit-consistent (UC) generalized inverse must therefore be used.<ref>{{citation\|last1=Uhlmann\|first1=Jeffrey\|title=On the Relative Gain Array (RGA) with Singular and Rectangular Matrices\|journal=Applied Mathematics Letters\|year=2019\|volume=93\|pages=52–57\|bibcode=2018arXiv180510312U\|arxiv=1805.10312\|doi=10.1016/j.aml.2019.01.031}}</ref> <ref>{{citation\|last1=Qasim Al Yousuf\|first1=Rafal\|last2=Uhlmann\|first2=Jeffrey\|title=On Use of the Moore-Penrose Pseudoinverse for Evaluating the RGA of Non-Square Systems\|journal=Iraqi Journal of Computers, Communications, Control & Systems Engineering ~~(IJCCCE)~~\|volume=3\|issue=21\|doi=10.33103/uot.ijccce.21.3.8\|year=2021}}</ref>		The RGA is often generalized in practice to be used when <math>\mathrm{G}</math> is singular, e.g., non-square, by replacing the inverse of <math>\mathrm{G}</math> with its [[Moore–Penrose inverse]] (pseudoinverse).<ref>{{citation\|last1=van de Wal\|first1=M.\|last2=de Jager\|first2=B.\|title=A review of methods for input/output selection\|journal=Automatica\|year=2001\|volume=37\|issue=4\|pages=487–510\|doi=10.1016/S0005-1098(00)00181-3}}</ref> However, it has been shown that the Moore–Penrose pseudoinverse fails to preserve the critical scale-invariance property of the RGA (#2 above) and that the unit-consistent (UC) generalized inverse must therefore be used.<ref>{{citation\|last1=Uhlmann\|first1=Jeffrey\|title=On the Relative Gain Array (RGA) with Singular and Rectangular Matrices\|journal=Applied Mathematics Letters\|year=2019\|volume=93\|pages=52–57\|bibcode=2018arXiv180510312U\|arxiv=1805.10312\|doi=10.1016/j.aml.2019.01.031}}</ref> <ref>{{citation\|last1=Qasim Al Yousuf\|first1=Rafal\|last2=Uhlmann\|first2=Jeffrey\|title=On Use of the Moore-Penrose Pseudoinverse for Evaluating the RGA of Non-Square Systems\|journal=Iraqi Journal of Computers, Communications, Control & Systems Engineering \|volume=3\|issue=21\|doi=10.33103/uot.ijccce.21.3.8\|year=2021}}</ref>

	==References==		==References==

216.131.22.42 at 00:31, 28 May 2022

2022-05-28T00:31:05Z

← Previous revision		Revision as of 00:31, 28 May 2022
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	==Generalizations==		==Generalizations==
	The RGA is often generalized in practice to be used when <math>\mathrm{G}</math> is singular, e.g., non-square, by replacing the inverse of <math>\mathrm{G}</math> with its [[Moore–Penrose inverse]] (pseudoinverse).<ref>{{citation\|last1=van de Wal\|first1=M.\|last2=de Jager\|first2=B.\|title=A review of methods for input/output selection\|journal=Automatica\|year=2001\|volume=37\|issue=4\|pages=487–510\|doi=10.1016/S0005-1098(00)00181-3}}</ref> However, it has been shown that the Moore–Penrose pseudoinverse fails to preserve the critical scale-invariance property of the RGA (#2 above) and that the unit-consistent (UC) generalized inverse must therefore be used.<ref>{{citation\|last1=Uhlmann\|first1=Jeffrey\|title=On the Relative Gain Array (RGA) with Singular and Rectangular Matrices\|journal=Applied Mathematics Letters\|year=2019\|volume=93\|pages=52–57\|bibcode=2018arXiv180510312U\|arxiv=1805.10312\|doi=10.1016/j.aml.2019.01.031}}</ref>		The RGA is often generalized in practice to be used when <math>\mathrm{G}</math> is singular, e.g., non-square, by replacing the inverse of <math>\mathrm{G}</math> with its [[Moore–Penrose inverse]] (pseudoinverse).<ref>{{citation\|last1=van de Wal\|first1=M.\|last2=de Jager\|first2=B.\|title=A review of methods for input/output selection\|journal=Automatica\|year=2001\|volume=37\|issue=4\|pages=487–510\|doi=10.1016/S0005-1098(00)00181-3}}</ref> However, it has been shown that the Moore–Penrose pseudoinverse fails to preserve the critical scale-invariance property of the RGA (#2 above) and that the unit-consistent (UC) generalized inverse must therefore be used.<ref>{{citation\|last1=Uhlmann\|first1=Jeffrey\|title=On the Relative Gain Array (RGA) with Singular and Rectangular Matrices\|journal=Applied Mathematics Letters\|year=2019\|volume=93\|pages=52–57\|bibcode=2018arXiv180510312U\|arxiv=1805.10312\|doi=10.1016/j.aml.2019.01.031}}</ref> <ref>{{citation\|last1=Qasim Al Yousuf\|first1=Rafal\|last2=Uhlmann\|first2=Jeffrey\|title=On Use of the Moore-Penrose Pseudoinverse for Evaluating the RGA of Non-Square Systems\|journal=Iraqi Journal of Computers, Communications, Control & Systems Engineering (IJCCCE)\|volume=3\|issue=21\|doi=10.33103/uot.ijccce.21.3.8\|year=2021}}</ref>

	==References==		==References==

Comp.arch at 16:51, 31 March 2021

2021-03-31T16:51:07Z

← Previous revision		Revision as of 16:51, 31 March 2021
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	==Definition==		==Definition==

	Given a linear time-invariant (LTI) system represented by a nonsingular matrix <math>\mathrm{G}</math>, the relative gain array (RGA) is defined as		Given a linear time-invariant (LTI) system represented by a nonsingular matrix <math>\mathrm{G}</math>, the relative gain array (RGA) is defined as

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	==Properties==		==Properties==

	The following are some of the linear-algebra properties of the RGA:<ref>{{citation\|last1=Johnson\|first1=C.R.\|title=Mathematical aspects of the relative gain array (A◦A<sup>−T</sup>)\|last2=Shapiro\|first2=H.M.\|journal=SIAM Journal on Algebraic and Discrete Methods \|year=1986\|volume=7\|issue=4\|pages=627–644\|doi=10.1137/0607069}}</ref>		The following are some of the linear-algebra properties of the RGA:<ref>{{citation\|last1=Johnson\|first1=C.R.\|title=Mathematical aspects of the relative gain array (A◦A<sup>−T</sup>)\|last2=Shapiro\|first2=H.M.\|journal=SIAM Journal on Algebraic and Discrete Methods \|year=1986\|volume=7\|issue=4\|pages=627–644\|doi=10.1137/0607069}}</ref>

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	==Generalizations==		==Generalizations==
		⚫	The RGA is often generalized in practice to be used when <math>\mathrm{G}</math> is singular, e.g., non-square, by replacing the inverse of <math>\mathrm{G}</math> with its [[Moore–Penrose inverse]] (pseudoinverse).<ref>{{citation\|last1=van de Wal\|first1=M.\|last2=de Jager\|first2=B.\|title=A review of methods for input/output selection\|journal=Automatica\|year=2001\|volume=37\|issue=4\|pages=487–510\|doi=10.1016/S0005-1098(00)00181-3}}</ref> However, it has been shown that the Moore–Penrose pseudoinverse fails to preserve the critical scale-invariance property of the RGA (#2 above) and that the unit-consistent (UC) generalized inverse must therefore be used.<ref>{{citation\|last1=Uhlmann\|first1=Jeffrey\|title=On the Relative Gain Array (RGA) with Singular and Rectangular Matrices\|journal=Applied Mathematics Letters\|year=2019\|volume=93\|pages=52–57\|bibcode=2018arXiv180510312U\|arxiv=1805.10312\|doi=10.1016/j.aml.2019.01.031}}</ref>

⚫	The RGA is often generalized in practice to be used when <math>\mathrm{G}</math> is singular, e.g., non-square, by replacing the inverse of <math>\mathrm{G}</math> with its [[~~Moore-Penrose~~ inverse]] (pseudoinverse).<ref>{{citation\|last1=van de Wal\|first1=M.\|last2=de Jager\|first2=B.\|title=A review of methods for input/output selection\|journal=Automatica\|year=2001\|volume=37\|issue=4\|pages=487–510\|doi=10.1016/S0005-1098(00)00181-3}}</ref> However, it has been shown that the ~~Moore-Penrose~~ pseudoinverse fails to preserve the critical scale-invariance property of the RGA (#2 above) and that the unit-consistent (UC) generalized inverse must therefore be used.<ref>{{citation\|last1=Uhlmann\|first1=Jeffrey\|title=On the Relative Gain Array (RGA) with Singular and Rectangular Matrices\|journal=Applied Mathematics Letters\|year=2019\|volume=93\|pages=52–57\|bibcode=2018arXiv180510312U\|arxiv=1805.10312\|doi=10.1016/j.aml.2019.01.031}}</ref>

	==References==		==References==

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	==Generalizations==		==Generalizations==

	The RGA is often generalized in practice to be used when <math>\mathrm{G}</math> is singular, e.g., non-square, by replacing the inverse of <math>\mathrm{G}</math> with its [[Moore-Penrose inverse]] (pseudoinverse).<ref>{{citation\|last1=van de Wal\|first1=M.\|last2=de Jager\|first2=B.\|title=A review of methods for input/output selection\|~~series~~=Automatica\|year=2001\|volume=37\|pages=487–510}}</ref> However, it has been shown that the Moore-Penrose pseudoinverse fails to preserve the critical scale-invariance property of the RGA (#2 above) and that the unit-consistent (UC) generalized inverse must therefore be used.<ref>{{citation\|last1=Uhlmann\|first1=Jeffrey\|title=On the Relative Gain Array (RGA) with Singular and Rectangular Matrices\|~~series~~=Applied Mathematics Letters\|year=2019\|volume=93\|pages=52–57\|bibcode=2018arXiv180510312U\|arxiv=1805.10312}}</ref>		The RGA is often generalized in practice to be used when <math>\mathrm{G}</math> is singular, e.g., non-square, by replacing the inverse of <math>\mathrm{G}</math> with its [[Moore-Penrose inverse]] (pseudoinverse).<ref>{{citation\|last1=van de Wal\|first1=M.\|last2=de Jager\|first2=B.\|title=A review of methods for input/output selection\|journal=Automatica\|year=2001\|volume=37\|issue=4\|pages=487–510\|doi=10.1016/S0005-1098(00)00181-3}}</ref> However, it has been shown that the Moore-Penrose pseudoinverse fails to preserve the critical scale-invariance property of the RGA (#2 above) and that the unit-consistent (UC) generalized inverse must therefore be used.<ref>{{citation\|last1=Uhlmann\|first1=Jeffrey\|title=On the Relative Gain Array (RGA) with Singular and Rectangular Matrices\|journal=Applied Mathematics Letters\|year=2019\|volume=93\|pages=52–57\|bibcode=2018arXiv180510312U\|arxiv=1805.10312\|doi=10.1016/j.aml.2019.01.031}}</ref>

	==References==		==References==

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← Previous revision		Revision as of 18:13, 1 December 2019
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	The '''Relative Gain Array''' (RGA) is a classical widely-used{{Citation needed\|date=October 2018}} method for determining the best input-output pairings for multivariable [[process control]] systems.<ref>{{citation\|last=Bristol\|first=E.H.\|title=On a new measure of interaction for multivariable process control\|~~series~~=IEEE Transactions on Automatic Control \|year=1966\|volume=1\|pages=133–134}}</ref> It has many practical open-loop and closed-loop control applications and is relevant to analyzing many fundamental steady-state closed-loop system properties such as stability and robustness.<ref>{{citation\|last1=Chen\|first1=Dan\|last2=Seborg\|first2=D.E.\|title=Relative Gain Array Analysis for Uncertain Process Models\|~~series~~=AIChE Journal\|year=2002\|volume=48\|pages=302–310}}</ref>		The '''Relative Gain Array''' (RGA) is a classical widely-used{{Citation needed\|date=October 2018}} method for determining the best input-output pairings for multivariable [[process control]] systems.<ref>{{citation\|last=Bristol\|first=E.H.\|title=On a new measure of interaction for multivariable process control\|journal=IEEE Transactions on Automatic Control \|year=1966\|volume=1\|pages=133–134\|doi=10.1109/TAC.1966.1098266}}</ref> It has many practical open-loop and closed-loop control applications and is relevant to analyzing many fundamental steady-state closed-loop system properties such as stability and robustness.<ref>{{citation\|last1=Chen\|first1=Dan\|last2=Seborg\|first2=D.E.\|title=Relative Gain Array Analysis for Uncertain Process Models\|journal=AIChE Journal\|year=2002\|volume=48\|issue=2\|pages=302–310\|doi=10.1002/aic.690480214}}</ref>

	==Definition==		==Definition==