# Article

 Title: Delay-dependent robust stability conditions and decay estimates for systems with input delays (English) Author: Hrissagis, Kostas Author: Kosmidou, Olga I. Language: English Journal: Kybernetika ISSN: 0023-5954 Volume: 34 Issue: 6 Year: 1998 Pages: [681]-691 Summary lang: English . Category: math . Summary: The robust stabilization of uncertain systems with delays in the manipulated variables is considered in this paper. Sufficient conditions are derived that guarantee closed-loop stability under state-feedback control in the presence of nonlinear and/or time-varying perturbations. The stability conditions are given in terms of scalar inequalities and do not require the solution of Lyapunov or Riccati equations. Instead, induced norms and matrix measures are used to yield some easy to test robust stability criteria. The problem of constrained control is also discussed, and alternative stability tests for the case of saturation nonlinearities are presented. Estimates of the transient behavior of the controlled system are also obtained. Finally, an example illustrates the results. (English) Keyword: robust stability Keyword: state-feedback control Keyword: uncertain input delay Keyword: alternative stability tests Keyword: closed-loop stability Keyword: time-varying perturbations Keyword: decay estimates Keyword: transient behavior MSC: 34K35 MSC: 93C23 MSC: 93C41 MSC: 93D09 MSC: 93D15 MSC: 93D21 idZBL: Zbl 1274.93217 idMR: MR1695371 . Date available: 2009-09-24T19:21:46Z Last updated: 2015-03-28 Stable URL: http://hdl.handle.net/10338.dmlcz/135254 . Reference: [1] Bernstein D., Michel A. N.: A chronological bibliography on saturated actuators.Internat. J. Robust and Nonlinear Control 5 (1995), 375–380 MR 1346405, 10.1002/rnc.4590050502 Reference: [2] Bourlès H.: $\alpha$-stability of systems governed by functional differential equations – extensions of results for linear delay systems.Internat. J. Control 45 (1987), 2233–2238 MR 0891806, 10.1080/00207178708933878 Reference: [3] Bourlès H., Kosmidou O. I.: On the quadratic stability of uncertain systems in the presence of time-delays.In: Proceedings of the European Control Conference ECC’97, Brussels 1997 Reference: [4] Chen B. S., Wang S., Lu H. C.: Stabilization of time-delay systems containing saturating actuators.Internat. J. Control 47 (1988), 867–881 Zbl 0636.93063, MR 0935055, 10.1080/00207178808906058 Reference: [5] Choi H. H., Chung M. J.: Memoryless $H_{\infty }$ controller design for linear systems with delayed state and control.Automatica 31 (1995), 917–919 MR 1337341, 10.1016/0005-1098(95)00001-D Reference: [6] Halanay A.: Differential Equations.Academic Press, New York 1966 Zbl 0912.34002, MR 0216103 Reference: [7] Hale J. K., Lunel S. M. V.: Introduction to Functional Differential Equations.Springer–Verlag, New York 1993 Zbl 0787.34002, MR 1243878 Reference: [8] Hrissagis K., Crisalle O.: Robust stabilization of input constrained bilinear systems.In: Proceedings of the IFAC’96 World Congress, San Francisco 1996 Reference: [9] Hrissagis K., Crisalle O.: Simple robust stability tests for time-delay systems with nonlinearities: Application to CSTR with recycle.In: Workshop on Industrial Control Systems, Thessaloniki 1996 Reference: [10] Inamdar S. R., Kumar V. R., Kulkarni B. D.: Dynamics of reacting systems in the presence of time-delay.Chem. Engrg. Sci. 46 (1991), 901–908 10.1016/0009-2509(91)80197-7 Reference: [11] Kolmanovskii V. B., Nosov V. R.: Theory of Functional Differential Equations.Academic Press, New York 1985 Reference: [12] Krikelis N. J., Barkas S. K.: Design of tracking systems subject to actuator saturation and integrator wind-up.Internat. J. Control 39 (1984), 667–682 Zbl 0532.93023, 10.1080/00207178408933196 Reference: [13] Kwon W. H., Pearson A. E.: Feedback stabilization of linear systems with delayed control.IEEE Trans. Automat. Control 25 (1980), 266–269 Zbl 0438.93055, MR 0567387, 10.1109/TAC.1980.1102288 Reference: [14] Lee J. H., Lee Y. I., Kwon W. H.: $H_{\infty }$ controllers for input delayed systems.In: Proceedings of Conf. on Decision and Control, Lake Buena vista 1994 Reference: [15] Mori T. N.: Criteria for asymptotic stability of linear time-delay systems.IEEE Trans. Automat. Control 30 (1985), 158–161 Zbl 0557.93058, MR 0778636, 10.1109/TAC.1985.1103901 Reference: [16] Schell M., Ross J.: Effects of time-delay in rate processes.J. Chem. Phys. 85 (1986), 6489–6503 MR 0867203, 10.1063/1.451429 Reference: [17] Shen J.-C., Kung F.-C.: Stabilization of input delay systems with saturating actuator.Internat. J. Control 50 (1989), 1667–1680 Zbl 0687.93063, MR 1032426, 10.1080/00207178908953458 Reference: [18] Vidyasagar M.: Nonlinear Systems Analysis.Second edition. Prentice Hall, Englewood Cliffs, N. J. 1993 Zbl 1006.93001 .

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