Applied and Computational Mathematics

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The Relationship between the Condition Number, RGA and Interaction in Multivariable Systems

Received: 17 June 2014    Accepted: 08 July 2014    Published: 20 July 2014
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Abstract

One of the most widely used input and output controllability measure is relative gain array (RGA). RGA measures input-output interaction in multi input multi output (MIMO) systems. The other significant measure in use is the smallest singular value of frequency subordinate. The condition number is defined as the ratio between the largest and smallest singular values of a system. In this paper, the relationship of relative gain array (RGA) with condition number and interaction as well as condition number in relation to interaction will be investigated respectively. The results indicate that the parameters under investigation are not always correlated, that is, the two-way relationship is not established between them all the time.

DOI 10.11648/j.acm.20140304.12
Published in Applied and Computational Mathematics (Volume 3, Issue 4, August 2014)
Page(s) 121-124
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Relative Gain Array, Condition Number, Interaction, MIMO Systems

References
[1] D. Kincaid, W. Cheney, Numerical Analysis. Mathematics of Scientific Computing, Third Edition, 2008.
[2] S. Skogestad and K. Havre., The use of RGA and Condition Number as Robustness Measures,European Symposium on Computer Aided Process Engineering -Part B, vol. 20, pp. 1005- 1010, 1996.
[3] D.S. Lubinsky, Condition numbers of Hankel matrices for exponential weights, Mathematical Analysis and Applications 314, 266–285, 2006.
[4] E. Bristol,On a new measure of interaction for multivariable process control, IEEE Transactions on Automatic Control, vol. 11, no. 1, pp. 133-134, 1966.
[5] P. Grosdidier, M. Morari, B.R. Holt, Closed-loop properties from steady-state gain information, Industrial & engineering chemistry. Fundamentals - ACS, vol. 24, no. 2, pp. 221-235, 1985.
[6] S. Skogestad, M. Morari, Implications of large RGA elements on control performance, Industrial & engineering chemistry research 26 (11), 2323-2330,1987.
[7] C. N. Nett, V.Manousiouthakis, Euclidean Condition and Block Relative Gain, Connections, Conjectures, and Clarifications, IEEE Transactions on Automatic Control, vol. 32, no. 5, pp. 405-407, 1987.
[8] K. Razzaghi, F. Shahraki, A Survey for the Selection of Control Structure for Distillation Columns Based on Steady State Controllability Indexes, Iranian Journal of Chemical Engineering Vol. 6, No. 2 (Spring), 2009.
[9] Q.Liang, Is the relative gain array a sensitivity measure?, IFAC workshop on interactions between process design and process control, London,UK,pp.133-138,1992.
[10] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: Analysis and Design (2nd Edition). UK: Wiley, 2005.
[11] A. Khaki-Sedigh, B. Moaveni, Control Configuration Selection for Multivariable Plants, LNCIS 391, Springer Verlag, 2009.
[12] B. Halvarsson, Interaction Analysis in Multivariable Control Systems Applications to Bioreactors for Nitrogen Removal, Phd Thesis, Uppsala University, Sweden,2010.
[13] M.Mousavi, M.Haeri, Welding current and arc voltage control in a GMAW process using ARMarkov based MPC, Control Engineering Practice, Vol. 19, PP.1408–1422, 2011.
[14] J.M.Maciejowski, Multivariable Feedback Design, Addison-Wesley, 1989.
Author Information
  • EE Department, Imam Khomeini International University, Qazvin, Iran

  • EE Department, Imam Khomeini International University, Qazvin, Iran

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    Aref Shahmansoorian, Sahar Jamebozorg. (2014). The Relationship between the Condition Number, RGA and Interaction in Multivariable Systems. Applied and Computational Mathematics, 3(4), 121-124. https://doi.org/10.11648/j.acm.20140304.12

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    Aref Shahmansoorian; Sahar Jamebozorg. The Relationship between the Condition Number, RGA and Interaction in Multivariable Systems. Appl. Comput. Math. 2014, 3(4), 121-124. doi: 10.11648/j.acm.20140304.12

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    AMA Style

    Aref Shahmansoorian, Sahar Jamebozorg. The Relationship between the Condition Number, RGA and Interaction in Multivariable Systems. Appl Comput Math. 2014;3(4):121-124. doi: 10.11648/j.acm.20140304.12

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  • @article{10.11648/j.acm.20140304.12,
      author = {Aref Shahmansoorian and Sahar Jamebozorg},
      title = {The Relationship between the Condition Number, RGA and Interaction in Multivariable Systems},
      journal = {Applied and Computational Mathematics},
      volume = {3},
      number = {4},
      pages = {121-124},
      doi = {10.11648/j.acm.20140304.12},
      url = {https://doi.org/10.11648/j.acm.20140304.12},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.acm.20140304.12},
      abstract = {One of the most widely used input and output controllability measure is relative gain array (RGA). RGA measures input-output interaction in multi input multi output (MIMO) systems. The other significant measure in use is the smallest singular value of frequency subordinate. The condition number is defined as the ratio between the largest and smallest singular values of a system. In this paper, the relationship of relative gain array (RGA) with condition number and interaction as well as condition number in relation to interaction will be investigated respectively. The results indicate that the parameters under investigation are not always correlated, that is, the two-way relationship is not established between them all the time.},
     year = {2014}
    }
    

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    AB  - One of the most widely used input and output controllability measure is relative gain array (RGA). RGA measures input-output interaction in multi input multi output (MIMO) systems. The other significant measure in use is the smallest singular value of frequency subordinate. The condition number is defined as the ratio between the largest and smallest singular values of a system. In this paper, the relationship of relative gain array (RGA) with condition number and interaction as well as condition number in relation to interaction will be investigated respectively. The results indicate that the parameters under investigation are not always correlated, that is, the two-way relationship is not established between them all the time.
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