zbMATH — the first resource for mathematics

Sufficiency-type stability and stabilization criteria for linear time-invariant systems with constant point delays. (English) Zbl 1067.34078
The following linear system is considered \[ \dot{x}(t)=A_{0}x(t)+\sum_{k=1}^{r}A_{k}x(t-\tau_{k}),\;t\geq 0\;, \] where \(\tau_{k},k=1,\dots,r\), are unknown delays.
Criteria for \(\alpha\)-stability (anti-stability) local in the delays and \(\varepsilon\)-stability independent of the delays are established. Moreover, questions on robust stability and closed-loop stabilization are discussed.

34K20 Stability theory of functional-differential equations
93D09 Robust stability
93D15 Stabilization of systems by feedback
Full Text: DOI
[1] Bourles, H., a-stability of systems governed by a functional differential equation — Extension of results concerning linear systems, Int. J. Control, 45, 2233-2234, (1987) · Zbl 0637.93058
[2] Brierley, S. D.; Chiasson, J. N.; Lee, E. B.; Zak, S. H., On stability independent of delay for linear systems, IEEE Trans. Automat. Control, 27, 252-254, (1982) · Zbl 0469.93065
[3] Chen, J., On computing the maximal delay intervals for stability of linear delay systems, IEEE Trans. Automat. Control, 40, 1087-1093, (1995) · Zbl 0840.93074
[4] Corduneanu, C.; Luca, N., The stability of some feedback systems with delay, J. Math. Anal. Appl., 51, 377, (1975) · Zbl 0312.34051
[5] Datko, R. F., Remarks concerning the asymptotic stability and stabilization of linear delay differential equations, J. Math. Anal. Appl., 111, 571-584, (1985) · Zbl 0579.34052
[6] Datko, R. F.; Elworthy, K. D. (ed.); Everitt, W. N. (ed.); Lee, E. B. (ed.), Time-delayed perturbations and robust stability, 457-468, (1994), New York · Zbl 0792.93098
[7] De la Sen, M., On some structures of stabilizing control laws for linear and time-invariant systems with bounded point delays, Int. J. Control, 59, 529-541, (1994) · Zbl 0799.93048
[8] De la Sen, M., Allocation of poles of delayed systems related to those associated with their undelayed counterparts, Electron. Lett., 36, 373-374, (2000)
[9] Franklin, G. F. and Powell, J. D.: Digital Control of Dynamic Systems, Addison-Wesley, Reading, MA, 1980.
[10] Gu, J., Discretized LMI set in the stability problem of linear uncertain time-delay systems, Int. J. Control, 68, 923-934, (1977) · Zbl 0986.93061
[11] Hale, J. K.; Infante, E. F.; Tsen, F. S. P., Stability in linear delay equations, J. Math. Anal. Appl., 105, 533-555, (1985) · Zbl 0569.34061
[12] Kamen, E. W., On the relationship between zero criteria for two-variable polynomials and asymptotic stability of delay-differential equations, IEEE Trans. Automat. Control, 25, 983-992, (1980) · Zbl 0458.93046
[13] Kincaid, D. and Cheney, W.: Numerical Analysis. Mathematics of Scientific Computing, Brooks/Cole Publishing Co. (Wadsworth, Inc.), Pacific Grove, CA, 1991. · Zbl 0745.65001
[14] Luo, J. S.; van den Bosch, P. P. J., Independent of delay stability criteria for uncertain linear state space models, Automatica, 33, 171-179, (1997) · Zbl 0887.93048
[15] Ortega, J. M.: Numerical Analysis, Academic Press, New York, 1972. · Zbl 0248.65001
[16] Xu, B., Stability robustness bounds for linear systems with multiple time-varying delayed perturbations, Int. J. Systems Sci., 28, 1311-1317, (1997) · Zbl 0899.93029
[17] Xu, B., Stability criteria for linear time-invariant systems with multiple delays, J. Math. Anal. Appl., 252, 484-494, (2000) · Zbl 0982.34064
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.