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ADVANCED AND APPLIED SCIENCES

EISSN: 2313-3724, Print ISSN:2313-626X

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 Volume 5, Issue 11 (November 2018), Pages: 67-74

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 Original Research Paper

 Title: Approximating structured singular values for Chebyshev spectral differentiation matrices

 Author(s): Mutti-Ur Rehman 1, *, Shams Ul Islam 2, Sadaqat Ali 3

 Affiliation(s):

 1Department of Mathematics, Sukkur IBA University, 65200 Sukkur, Pakistan
 2Department of Mathematics, COMSATS University Islamabad, 44000 Islamabad, Pakistan
 3Department of Mathematics, University of Lahore, Gujrat campus, 50700 Gujrat, Pakistan

 https://doi.org/10.21833/ijaas.2018.11.009

 Full Text - PDF          XML

 Abstract:

In this article, we present the numerical computation of lower bounds of structured singular value known as the µ-value for a family of Chebyshev spectral differentiation matrices. The µ-value is a versatile tool used in control in order to analyze the robustness, performance, stability, and instability of feedback systems in system theory. The purposed methodology is based upon low-rank ordinary differential equations based technique and provides tight lower bounds of µ-value once compared with the well-known MATLAB routine mussv available in the MATLAB control toolbox. 

 © 2018 The Authors. Published by IASE.

 This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

 Keywords: Structured singular values, Singular values, Spectral radius, Low rank ODE’s

 Article History: Received 21 May 2018, Received in revised form 29 August 2018, Accepted 18 September 2018

 Digital Object Identifier: 

 https://doi.org/10.21833/ijaas.2018.11.009

 Citation:

 Rehman MU, Islam SU, and Ali S (2018). Approximating structured singular values for Chebyshev spectral differentiation matrices. International Journal of Advanced and Applied Sciences, 5(11): 67-74

 Permanent Link:

 http://www.science-gate.com/IJAAS/2018/V5I11/Rehman.html

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 References (14) 

  1. Bernhardsson B, Rantzer A, and Qiu L (1998). Real perturbation values and real quadratic forms in a complex vector space. Linear Algebra and its Applications, 270(1-3): 131-154. https://doi.org/10.1016/S0024-3795(97)00232-2   [Google Scholar]
  1. Braatz RP, Young PM, Doyle JC, and Morari M (1994). Computational complexity of/spl mu/calculation. IEEE Transactions on Automatic Control, 39(5): 1000-1002. https://doi.org/10.1109/9.284879   [Google Scholar]
  1. Chen J, Fan MK, and Nett CN (1996). Structured singular values with nondiagonal structures. I. Characterizations. IEEE Transactions on Automatic Control, 41(10): 1507-1511. https://doi.org/10.1109/9.539434   [Google Scholar]
  1. Fan MK, Tits AL, and Doyle JC (1991). Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics. IEEE Transactions on Automatic Control, 36(1): 25-38. https://doi.org/10.1109/9.62265   [Google Scholar]
  1. Hinrichsen D and Pritchard AJ (2005). Mathematical systems theory I: modelling, state space analysis, stability and robustness. Springer, Berlin, Germany. https://doi.org/10.1007/b137541   [Google Scholar]
  1. Karow M, Hinrichsen D, and Pritchard AJ (2006). Interconnected systems with uncertain couplings: Explicit formulae for mu-values, spectral value sets, and stability radii. SIAM Journal on Control and Optimization, 45(3): 856-884. https://doi.org/10.1137/050626508   [Google Scholar]
  1. Kato T (1980). Perturbation theory for linear operators, classics in mathematics. Springer‐Verlag, Berlin, Germany.   [Google Scholar]
  1. Packard A and Doyle J (1993). The complex structured singular value. Automatica, 29(1): 71-109. https://doi.org/10.1016/0005-1098(93)90175-S   [Google Scholar]
  1. Packard A, Fan MK, and Doyle J (1988). A power method for the structured singular value. In the 27th IEEE Conference on Decision and Control, IEEE, Austin, TX, USA: 2132-2137.   [Google Scholar]
  1. Qiu L, Bernhardsson B, Rantzer A, Davison EJ, Young PM, and Doyle JC (1995). A formula for computation of the real stability radius. Automatica, 31(6): 879–890. https://doi.org/10.1016/0005-1098(95)00024-Q   [Google Scholar]
  1. Rehman MU and Tabassum S (2017). Numerical computation of structured singular values for companion matrices. Journal of Applied Mathematics and Physics, 5(05): 1057-1072.   [Google Scholar]
  1. Young PM, Doyle JC, and Packard A (1994). Theoretical and computational aspects of the structured singular value. Systems, Control and Information, 38(3): 129-138.   [Google Scholar]
  1. Young PM, Newlin MP, and Doyle JC (1992). Practical computation of the mixed μ problem. In the American Control Conference, IEEE, Chicago, IL, USA: 2190-2194.   [Google Scholar]
  1. Zhou K, Doyle JC, and Glover K (1996). Robust and optimal control. Prentice Hall, Upper Saddle River, NJ, USA.   [Google Scholar]