Stability of Linear Delay Differential Equations using Modified Algebraic Approach

Authors

  • K. H. Mohammedali School of Mathematical Sciences, Universiti Sains Malaysia, 11800, Penang, Malaysia.
  • N. A. Ahmad School of Mathematical Sciences, Universiti Sains Malaysia, 11800, Penang, Malaysia.

Keywords:

Stability of Delay Differential Equations, Characteristic Equation, Linear Delay Differential Equations.

Abstract

An algebraic approach have been developed to study the stability of delay differential equations with m-retarded arguments, each of them is a multiple of fixed unknown time lag. The method has its basis upon transforming the characteristic equation related to the delay differential equation into an equivalent system of two algebraic equations in order to evaluate the value of the time lag which ensures the stability of the delay differential equation.

References

Arivo O. et al. (eds.), “Delay Differential Equations and Applications”, Springer, Berline, 477-517, 2006.

Asl F. M. and Ulsoy A. G., “Analysis of a System of Linear Delay Differential Equations”, Journal of Dynamic Systems, Measurement, and Control, Vol.125, 215-223, 2003.

Breda D., Maset S. and Vermiglio R., “Computing the Characteristic Roots for Delay Differential Equations”, IMA Journal of Numerical Analysis, 24, 1-19, 2004.

Butcher E. A., Ma H., Bueler E., Averina V. and SzabóZs., “Stability of Time-Periodic Delay Differential Equations Via Chebyshev Polynomials”, International Journal for Numerical Methods in Engineering, 59, 895-922, 2004.

Campbell S. A., “Introduction to Delay Differential Equations”, Department of Applied Mathematics, University of Waterloo, 2007.

Chen S. G., Ulsoy A. G. and Koren Y., “Computational Stability Analysis of Chatter in Turning”, Journal of Manufacturing Science and Engineering, 119, 457-460, 1997.

Engelborghs K. and Roose D., “On Stability of LMS Methods and Characteristic Roots of Delay Differential Equations”, SIAM Journal of Numerical Analysis, 40, 629-650, 2002.

Hale J. K. and Lunel S. M. V., Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99, Springer Verlag, New York, 1993.

Hassard B. D., “Counting Roots of the Characteristic Equation for Linear Delay-Differential Systems”, Journal of Differential Equations, Vol.136, 222-235, 1997.

Hsu C. S., “Application of the Tau-Decomposition Method to Dynamical Systems Subjected to Retarded Follower Forces”, J. Applied Mechanics, 37, 259-266, 1970.

Insperger T. and Stépán G., “Semidiscretization Method for General Delayed Systems”, International Journal for Numerical Methods in Engineering, 55, 503-518, 2002.

Jury E. I. and Zeheb E., An Algebraic Algorithm for Determining the Desired Gain of Multivariable Feedback Systems”, Int. J. Control, Vol.37, No.5, 1081-1094, 1983.

Nagy T. K., “Stability Analysis of Delay Differential Equations by the Method of Steps and Inverse Laplace Transform”, Differential Equations and Dynamical Systems, Vol.17, No. 1 & 2, 185-200, 2009.

Olgac N., and Sipahi R., “An Exact Method for the Stability Analysis of Time-Delayed LTI Systems”, IEEE Trans. Autom. Control, 47(5), 793-797, 2002.

Postlethwaite I. and Macfarlane A. G. J., A complex Variable Approach to the Analysis of Linear Multivariable Feedback Systems, Lecture Notes in Control and Information Science, 12, New York, Springer Verlag, 1979.

Sipahi R. and Olgac N., “A Novel Stability Study on Multiple Time-Delay Systems (MTDS) Using the Root Clustering Paradigm”, Proceedings of the ACC, 2004.

Stépán G., Retarded Dynamical Systems. Longman: Harlow, UK, 1989.

Verriest E. I., New Qualitative Methods for Stability of Delay Systems, Kybernetika, Vol.37, No.3, 229-238, 2001

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Published

2016-10-01

How to Cite

Mohammedali, K. H., & Ahmad, N. A. (2016). Stability of Linear Delay Differential Equations using Modified Algebraic Approach. Journal of Telecommunication, Electronic and Computer Engineering (JTEC), 8(7), 157–163. Retrieved from https://jtec.utem.edu.my/jtec/article/view/1299