Computing $\mu$-values for LTI Systems

Mutti-Ur Rehman; Jehad Alzabut; Javed Hussain Brohi; Mutti-Ur Rehman; Jehad Alzabut; Javed Hussain Brohi

doi:10.3934/math.2021019

AIMS Mathematics

2021, Volume 6, Issue 1: 304-313. doi: 10.3934/math.2021019

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Research article

Computing $\mu$-values for LTI Systems

1.
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
2.
Department of Mathematics, Sukkur IBA University, 65200, Sukkur-Pakistan
3.
Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 12435, Saudi Arabia

Received: 13 August 2020 Accepted: 23 September 2020 Published: 12 October 2020
MSC : 15A03, 15A18, 80M50

In this article we consider certain linear time-varying control systems and investigate their stability using structured singular values ($\mu$-values). We use the low rank ordinary differential equations based methodology to compute the lower bounds for $\mu$-values. The inner-outer algorithm computes the local extremizer of an admissible perturbation and adjusts the desired perturbation level. Further, we present a comparison of our results via the well-known MATLAB routine mussv which is available in MATLAB control toolbox.
- eigenvalues,
- singular values,
- structured singular values,
- low rank ODE's
Citation: Mutti-Ur Rehman, Jehad Alzabut, Javed Hussain Brohi. Computing $\mu$-values for LTI Systems[J]. AIMS Mathematics, 2021, 6(1): 304-313. doi: 10.3934/math.2021019

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Abstract

In this article we consider certain linear time-varying control systems and investigate their stability using structured singular values ($\mu$-values). We use the low rank ordinary differential equations based methodology to compute the lower bounds for $\mu$-values. The inner-outer algorithm computes the local extremizer of an admissible perturbation and adjusts the desired perturbation level. Further, we present a comparison of our results via the well-known MATLAB routine mussv which is available in MATLAB control toolbox.

References

[1]	B. Zhou, G. B. Cai, G. R. Duan, Stabilisation of time-varying linear systems via Lyapunov differential equations, Int. J. Control, 86 (2013), 332-347. doi: 10.1080/00207179.2012.728008
[2]	Z. D. Sun, A robust stabilizing law for switched linear systems, Int. J. Control, 77 (2004), 389-398. doi: 10.1080/00207170410001667468
[3]	C. J. Harris, J. F. Miles, Stability of linear systems: some aspects of kinematic similarity, Switzerland: Academic Press, 1980.
[4]	J. C. Doyle, B. A. Francis, A. R. Tannenbaum, Feedback control theory, New York: Dover Publications, 2009.
[5]	J. Doyle, Analysis of feedback systems with structured uncertainties, IET. Control Theory A., 129 (1982), 242-250.
[6]	Ph. Mullhaupt, D. Buccieri, D. Bonvin, A numerical sufficiency test for the asymptotic stability of linear time-varying systems, Automatica, 43 (2007), 631-638. doi: 10.1016/j.automatica.2006.10.014
[7]	B. Zhou, On asymptotic stability of linear time-varying systems, Automatica, 68 (2016). 266-276.
[8]	N. P. Van Der Aa, H. G. Ter Morsche, R. R. M. Mattheij, Computation of eigenvalue and eigenvector derivatives for a general complex-valued eigensystem, Electron J. Linear Al., 16 (2007), 1-10.
[9]	A. Packard, J. Doyle, The complex structured singular value, Automatica, 29 (1993), 71-109. doi: 10.1016/0005-1098(93)90175-S
[10]	T. G. Wright, L. N. Trefethen, 2002. Available from: Software available at http://www.comlab.ox.ac.uk/pseudospectra/eigtool.
[11]	N. Guglielmi, M. Rehman, D. Kressner, A novel iterative method to approximate structured singular values, SIAM J. Matrix Anal. A., 38 (1993), 361-386.

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