| Title:
             | 
Approximation of control laws with distributed delays: a necessary condition for stability (English) | 
| Author:
             | 
Mondié, Sabine | 
| Author:
             | 
Dambrine, Michel | 
| Author:
             | 
Santos, Omar | 
| Language:
             | 
English | 
| Journal:
             | 
Kybernetika | 
| ISSN:
             | 
0023-5954 | 
| Volume:
             | 
38 | 
| Issue:
             | 
5 | 
| Year:
             | 
2002 | 
| Pages:
             | 
[541]-551 | 
| Summary lang:
             | 
English | 
| . | 
| Category:
             | 
math | 
| . | 
| 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. (English) | 
| Keyword:
             | 
linear multivariable system with delay | 
| Keyword:
             | 
stability | 
| MSC:
             | 
93C23 | 
| MSC:
             | 
93C35 | 
| MSC:
             | 
93D05 | 
| MSC:
             | 
93D15 | 
| idZBL:
             | 
Zbl 1265.93148 | 
| idMR:
             | 
MR1966944 | 
| . | 
| Date available:
             | 
2009-09-24T19:48:35Z | 
| Last updated:
             | 
2015-03-25 | 
| Stable URL:
             | 
http://hdl.handle.net/10338.dmlcz/135485 | 
| . | 
| Reference:
             | 
[1] Artstein Z.: Linear systems with delayed controls: a reduction.IEEE Trans. Automat. Control AC-27 (1982), 4, 869–879 Zbl 0486.93011, MR 0680488, 10.1109/TAC.1982.1103023 | 
| Reference:
             | 
[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, MR 1622412, 10.1016/S0378-4754(97)00113-4 | 
| Reference:
             | 
[3] Bellman R., Cooke K. L.: Differential Difference Equations.Academic Press, London 1963 Zbl 0163.10501, MR 0147745 | 
| Reference:
             | 
[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, MR 1814584, 10.1109/9.905705 | 
| Reference:
             | 
[5] Gantmacher F. R.: The Theory of Matrices.Vol. 1. AMS Chelsea Publishing, New York 1959 Zbl 0927.15002, MR 0107649 | 
| Reference:
             | 
[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, MR 0828360, 10.1080/00207178608933506 | 
| Reference:
             | 
[7] Kolmanovski V. B., Nosov V. R.: Stability of Functional Differential Equations.Academic Press, New York 1986 MR 0860947 | 
| Reference:
             | 
[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, MR 0538808, 10.1109/TAC.1979.1102124 | 
| Reference:
             | 
[9] Mathews J. H.: Numerical Methods for Mathematics, Science, and Engineering.Prentice–Hall, Englewood Cliffs, N.J. 1992 Zbl 0753.65002 | 
| Reference:
             | 
[10] Mondié S., Santos O.: Une condition nécessaire pour l’implantation de lois de commande à retards distribués.In: Conférence Internationale Francophone d’Automatique, Lille 2000, pp. 201–206 | 
| Reference:
             | 
[11] Morse A. S.: Ring models for delay-differential systems.Automatica 12 (1976), 529–531 Zbl 0345.93023, MR 0437162, 10.1016/0005-1098(76)90013-3 | 
| Reference:
             | 
[12] Palmor Z. J.: Modified predictors.In: The Control Handbook (W. Levine, ed.), CRC Press, Boca Raton 1996, Section 10.9 | 
| Reference:
             | 
[13] Santos O., Mondié S.: Control laws involving distributed time delays: robustness of the implementation.In: American Control Conference, Chicago 2000, pp. 2479–2480 | 
| Reference:
             | 
[14] Smith O. J. M.: Closer control of loops with dead time.Chem. Engrg. Prog. 53 (1959), 217–219 | 
| Reference:
             | 
[15] Assche V. Van, Dambrine M., Lafay J. F., Richard J. P.: Some problems arising in the implementation of distributed-delay control laws.In: Proc. 38th IEEE Conference on Decision and Control, Phoenix 1999 | 
| Reference:
             | 
[16] Vidyasagar M.: Control System Synthesis.MIT Press, Cambridge, MA 1985 Zbl 0655.93001, MR 0787045 | 
| Reference:
             | 
[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, 10.1109/TAC.1981.1102802 | 
| . |