M. De La Sen, "Stabilization criteria for continuous linear time-invariant systems with constant lags", Discrete Dynamics in Nature and Society, vol. 2006, Article ID 087062, 19 pages, 2006. https://doi.org/10.1155/DDNS/2006/87062
Stabilization criteria for continuous linear time-invariant systems with constant lags
Some criteria for asymptotic stability of linear and time-invariant systems with constant point delays are derived. Such criteria are concerned with the properties of robust stability related to two relevant auxiliary delay-free systems which are built by deleting the delayed dynamics or considering that the delay is zero. Explicit asymptotic stability results, easy to test, are given for both the unforced and closed-loop systems when the stabilizing controller for one of the auxiliary delay-free systems is used for the current time-delay system. The proposed techniques include frequency domain analysis techniques including the use of norms.
- H. Bourlès, “-stability of systems governed by a functional-differential equation—extension of results concerning linear delay systems,” International Journal of Control, vol. 45, no. 6, pp. 2233–2234, 1987.
- S. D. Brierley, J. N. Chiasson, E. B. Lee, and S. H. Żak, “On stability independent of delay for linear systems,” IEEE Transactions on Automatic Control, vol. 27, no. 1, pp. 252–254, 1982.
- J. Chen, “On computing the maximal delay intervals for stability of linear delay systems,” IEEE Transactions on Automatic Control, vol. 40, no. 6, pp. 1087–1093, 1995.
- C. Corduneanu and N. Luca, “The stability of some feedback systems with delay,” Journal of Mathematical Analysis and Applications, vol. 51, no. 2, pp. 377–393, 1975.
- R. F. Datko, “Remarks concerning the asymptotic stability and stabilization of linear delay differential equations,” Journal of Mathematical Analysis and Applications, vol. 111, no. 2, pp. 571–584, 1985.
- R. F. Datko, “Time-delayed perturbations and robust stability,” in Differential Equations, Dynamical Systems, and Control Science, K. D. Elworthy, W. N. Everitt, and E. B. Lee, Eds., vol. 152 of Lecture Notes in Pure and Applied Mathematics Series, pp. 457–468, Marcel Dekker, New York, 1994.
- M. De La Sen, “On some structures of stabilizing control laws for linear and time-invariant systems with bounded point delays and unmeasurable states,” International Journal of Control, vol. 59, no. 2, pp. 529–541, 1994.
- M. De La Sen, “Allocation of poles of delayed systems related to those associated with their undelayed counterparts,” Electronics Letters, vol. 36, no. 4, pp. 373–374, 2000.
- M. De La Sen, “Preserving positive realness through discretization,” Positivity, vol. 6, no. 1, pp. 31–45, 2002.
- P. Dorato, L. Fortuna, and G. Muscato, Robust Control for Unstructured Perturbations—An Introduction, vol. 168 of Lecture Notes in Control and Information Sciences, Springer, Berlin, 1992, series editors M. Thoma and A. Wyner.
- G. F. Franklin and J. D. Powell, Digital Control of Dynamic Systems, Addison-Wesley, Massachusetts, 1980.
- K. Gu, “Discretized LMI set in the stability problem of linear uncertain time-delay systems,” International Journal of Control, vol. 68, no. 4, pp. 923–934, 1997.
- J. K. Hale, E. F. Infante, and F. S. P. Tsen, “Stability in linear delay equations,” Journal of Mathematical Analysis and Applications, vol. 105, no. 2, pp. 533–555, 1985.
- T. Kailath, Linear Systems, Prentice-Hall Information and System Sciences Series, Prentice-Hall, New Jersey, 1980.
- E. W. Kamen, “On the relationship between zero criteria for two-variable polynomials and asymptotic stability of delay differential equations,” IEEE Transactions on Automatic Control, vol. 25, no. 5, pp. 983–984, 1980.
- D. Kincaid and W. Cheney, Numerical Analysis. Mathematics of Scientific Computing, Brooks/Cole, California, 1991.
- H. Kwakernaak, Uncertainty Models and the Design of Robust Control Systems, edited by J. Ackerman, Lecture Notes in Control and Information Sciences Series, no. 70, Springer, Berlin, 1992, series editor M. Thoma.
- J. S. Luo and P. P. J. van den Bosch, “Independent of delay stability criteria for uncertain linear state space models,” Automatica, vol. 33, no. 2, pp. 171–179, 1997.
- J. M. Ortega, Numerical Analysis. A Second Course, Computer Science and Applied Mathematics, Academic Press, New York, 1972.
- A. D. Wunsch, Complex Variables with Applications, Addison-Wesley, Massachusetts, 2nd edition, 1994.
- B. Xu, “Stability robustness bounds for linear systems with multiple time-varying delayed perturbations,” International Journal of Systems Science, vol. 28, no. 12, pp. 1311–1317, 1997.
- B. Xu, “Stability criteria for linear time-invariant systems with multiple delays,” Journal of Mathematical Analysis and Applications, vol. 252, no. 1, pp. 484–494, 2000.
Copyright © 2006 M. De La Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.