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Approximation of control laws with distributed delays: a necessary condition for stability. (English) Zbl 1265.93148
Summary: The implementation of control laws with distributed delays that assign the spectrum of unstable linear multivariable systems with delay in the input requires an approximation of the integral. A necessary condition for stability of the closed-loop system is shown to be the stability of the controller itself. An illustrative multivariable example is given.

93C35 Multivariable systems, multidimensional control systems
93D15 Stabilization of systems by feedback
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[1] Artstein Z.: Linear systems with delayed controls: a reduction. IEEE Trans. Automat. Control AC-27 (1982), 4, 869-879 · Zbl 0486.93011 · doi:10.1109/TAC.1982.1103023
[2] Brethé D., Loiseau J. J.: An effective algorithm for finite spectrum assignment of single-input systems with delays. Math. Comput. Simulation 45 (1998), 339-348 · Zbl 1017.93506 · doi:10.1016/S0378-4754(97)00113-4
[3] Bellman R., Cooke K. L.: Differential Difference Equations. Academic Press, London 1963 · Zbl 0163.10501
[4] Engelborghs K., Dambrine, M., Roose D.: Limitations of a class of stabilization methods for delay systems. IEEE Trans. Automat. Control AC-46 (2001), 2, 336-339 · Zbl 1056.93607 · doi:10.1109/9.905705
[5] Gantmacher F. R.: The Theory of Matrices. Vol. 1. AMS Chelsea Publishing, New York 1959 · Zbl 0927.15002
[6] Kamen E. W., Khargonekar P. P., Tannenbaum A.: Proper stable bezout factorizations and feedback control of linear time delay systems. Internat. J. Control 43 (1986), 3, 837-857 · Zbl 0599.93047 · doi:10.1080/00207178608933506
[7] Kolmanovski V. B., Nosov V. R.: Stability of Functional Differential Equations. Academic Press, New York 1986 · Zbl 0824.34081
[8] Manitius A. Z., Olbrot A. W.: Finite spectrum assignment problem for systems with delays. IEEE Trans. Automat. Control AC-24 (1979), 4, 541-553 · Zbl 0425.93029 · doi:10.1109/TAC.1979.1102124
[9] Mathews J. H.: Numerical Methods for Mathematics, Science, and Engineering. Prentice-Hall, Englewood Cliffs, N.J. 1992 · Zbl 0753.65002
[10] Mondié S., Santos O.: Une condition nécessaire pour l’implantation de lois de commande à retards distribués. Conférence Internationale Francophone d’Automatique, Lille 2000, pp. 201-206
[11] Morse A. S.: Ring models for delay-differential systems. Automatica 12 (1976), 529-531 · Zbl 0345.93023 · doi:10.1016/0005-1098(76)90013-3
[12] Palmor Z. J.: Modified predictors. The Control Handbook (W. Levine, CRC Press, Boca Raton 1996, Section 10.9
[13] Santos O., Mondié S.: Control laws involving distributed time delays: robustness of the implementation. American Control Conference, Chicago 2000, pp. 2479-2480
[14] Smith O. J. M.: Closer control of loops with dead time. Chem. Engrg. Prog. 53 (1959), 217-219
[15] Assche V. Van, Dambrine M., Lafay J. F., Richard J. P.: Some problems arising in the implementation of distributed-delay control laws. Proc. 38th IEEE Conference on Decision and Control, Phoenix 1999
[16] Vidyasagar M.: Control System Synthesis. MIT Press, Cambridge, MA 1985 · Zbl 0655.93001
[17] Watanabe K., Ito M.: A process model control for linear systems with delay. IEEE Trans. Automat. Control AC-26 (1981), 6, 1261-1268 · Zbl 0471.93035 · doi:10.1109/TAC.1981.1102802
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