Stability analysis of delay-differential equations by the method of steps and inverse Laplace transform

Differential Equations and Dynamical Systems

Volumn 17, Issue 1-2, 2009, Pages 185-200

  Kalmár Nagy, Tamás ^a  

^a Department of Electrical and Computer Engineering   (United States)

Author keywords

Stability; Time delay systems

Indexed keywords

EID: 84859707502     PISSN: 09713514     EISSN: 09746870     Source Type: Journal    
DOI: 10.1007/s12591-009-0014-x     Document Type: Article

References (31)

1. Asl F. M. and Ulsoy A. G., Analysis of a System of Linear Delay Differential Equations, Journal of Dynamic Systems, Measurement, and Control, 125, pp. 215-223, (2003).
2. Baker C. T. H., Paul C. A. H. and Wille D. R., Issues in the numerical solution of evolutionary delay differential equations, Advances in Comput. Math. 3, pp. 171-196, (1995).
3. Barnett S., Polynomials and Linear Control Systems, Marcel Dekker (1983).
4. Bellman R. and Cooke K., Differential Difference Equations, Academic Press, (1963).
5. Breda D., Maset S. and Vermiglio R., Computing the characteristic roots for delay differential equations, IMA Journal of Numerical Analysis, 24, pp. 1-19, (2004).
6. Butcher E. A., Ma H., Bueler E., Averina V. and Szabó Zs., Stability of time-periodic delay-differential equations via Chebyshev polynomials, International Journal for Numerical Methods in Engineering, 59, pp. 895-922, (2004).
7. Chen S.-G., Ulsoy A. G. and Koren Y., Computational stability analysis of chatter in turning, Journal of Manufacturing Science and Engineering, 119, pp. 457-460, (1997).
8. Driver R. D., Ordinary and Delay Differential Equations, vol. 20 of Applied Mathematical Sciences, Springer-Verlag, New York (1977).
9. Dubner H. and Abate J., Numerical Inversion of Laplace Transforms by Relating them to the Finite Fourier Cosine Transform, Journal of the ACM (JACM), 15(1), p. 115-123, (1968).
10. Elsgolts L. E. and Norkin S. B., Introduction to the theory and application of differential equations with deviating arguments, Academic Press. Translated from the Russian by John L. Casti, Mathematics in Science and Engineering, vol. 105, (1973).
11. 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).
12. Hale J. K. and Lunel S. M. V., Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99, Springer Verlag, New York, (1993).
13. Hassard B. D., Counting Roots of the Characteristic Equation for Linear Delay-Differential Systems, Journal of Differential Equations, 136, pp. 222-235, (1997).
14. Hayes N. D., Roots of the transcendental equation associated with a certain differential-difference equation, J. London Math. Soc., 25, pp. 226-232, (1950).
15. Insperger T. and Stépán G., Semidiscretization method for general delayed systems, International Journal for Numerical Methods in Engineering, 55, 503-518 (2002).
16. Ito K., Kappel F. and Salamon D., A Variational Approach to Approximation of Delay Systems, Differential and Integral Equations, 4, pp. 51-72, (1991).
17. Kalmár-Nagy T., A Novel Method for Efficient Numerical Stability Analysis of Delay-Differential Equations, in Proceedings of the American Control Conference, pp. 2823-2826, (2005).
18. Kalmár-Nagy T., Stépán G., Moon F. C., Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations, Nonlinear Dynamics, 26, pp. 121-142 (2001).
19. Kappel F., Galerkin type approximation schemes for delay systems, Annals of Differential Equations, 1(1), 57-82, (1985).
20. Kappel, F., Spline approximation for autonomous nonlinear functional differential equations, Nonlinear Analysis, Theory, Methods & Applications, 10(5), pp. 503-513, (1986).
21. Mäkilä P. M. and Partington J. R., Shift operator induced approximations of delay systems, SIAM Journal on Control and Optimization, 37, pp. 1897-1912, (1999).
22. Olgac N., and Sipahi R., An Exact Method for the Stability Analysis of Time-Delayed LTI Systems, IEEE Trans. Autom. Control, 47(5), pp. 793-797, (2002).
23. Partington J. R., Some frequency-domain approaches to the model reduction of delay systems, Annual Reviews in Control, 28, pp. 65-73 (2004).
24. Pinney E., Ordinary difference-differential equations, University of California Press, Berkeley (1958).
25. Richard J-P., Time-delay systems: an overview of some recent advances and open problems, Automatica, 39, pp. 1667-1694, (2003).
26. 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).
27. Smith H., MAT598 Applied Delay Differential Equations, Lecture Notes, (2004).
28. Stépán G., Retarded Dynamical Systems. Longman: Harlow, UK, (1989).
29. Tošić D. V. and Lutovac M. D., Symbolic computation of impulse, step and sine responses of linear time-invariant systems, Symp. Theor. El. Engineering, ISTET, Magdeburg, Germany, pp. 653-657, (1999).
30. Valkó P. P. and Vojta B. L., The List, http://pumpjack. tamu. edu/~valko.
31. Wahi P., Chatterjee A. Galerkin projections for delay differential equations, Journal of Dynamic Systems, Measurement, and Control, 127, pp. 80-, (2005).