The stability of an irrigation canal system

Bounit, Hamid

International Journal of Applied Mathematics and Computer Science, Tome 13 (2003), p. 453-468 / Harvested from The Polish Digital Mathematics Library

Résumé

In this paper we examine the stability of an irrigation canal system. The system considered is a single reach of an irrigation canal which is derived from Saint-Venant's equations. It is modelled as a system of nonlinear partial differential equations which is then linearized. The linearized system consists of hyperbolic partial differential equations. Both the control and observation operators are unbounded but admissible. From the theory of symmetric hyperbolic systems, we derive the exponential (or internal) stability of the semigroup underlying the system. Next, we compute explicitly the transfer functions of the system and we show that the input-output (or external) stability holds. Finally, we prove that the system is regular in the sense of (Weiss, 1994) and give various properties related to its transfer functions.

Détail

Publié le : 2003-01-01

Zbl 1115.93333

EUDML-ID : urn:eudml:doc:207657

Bibliographie

[000] Baume J.P. and Sau H. (1997): Steady of irrigation canal dynamics for control purpose. - Proc. 1st Int. Workshop Regulation of Irrigation Canals, RIC'97, Marrakech, Morocco, pp. 3-12.

[001] Baume J.P. (1990): Regulation des cannaux d'irrgation: Etude du sous-systèmes bief avec vanne. - M.Sc. thesis, Universite des Scienceset Techniques du Languedoc (USTL), Montpellier, France.

[002] Bounit H. (2003a): Robust PI-controller for an irrigation canal system. - (In preparation). | Zbl 1115.93333

[003] Bounit H. (2003b): H^∞-controller for an irrigation canal system. - (In preparation). | Zbl 1115.93333

[004] Bounit H., Hammouri H. and Sau J. (1997): Regulation of an irrigation canal through the semigroup approach. - Proc. 1st Int. Workshops Regulation of Irrigation Canals, RIC'97, Marrakech, Morocco, pp. 261-267.

[005] Burt C.M., Clemmens A.J. and Streslkoff T.S. (1998): Influence of canal geometryand dynamics on controllability. - J.Irrig. Drain. Eng., ASCE, Vol. 124, No. 1, pp. 16-22.

[006] Chow V.T. (1985): Open Channels Hydraulics. - New York: Mac Graw Hill .

[007] Clemmens A.J., Streslkoff T.S. and Gooch R.S. (1995): Influence of canal geometryand dynamics on controllability. - Proc. 1st Int. Conf. Water Res. Eng., ASCE, Reston, VA, pp. 21-25.

[008] Curtain R.F. (1988): Equivalence of input-output stability and exponential stability for infinite dimensional systems. - Math. Syst. Theory, Vol. 21, pp. 19-48. | Zbl 0657.93050

[009] Curtain R.F. and Zwart H. (1995): An Introduction to Infinite Dimensional Linear System Theory. - New York: Springer. | Zbl 0839.93001

[010] Engel K.J. and Nagel R. (2000): One Parameter Semigroups for Linear Evolution Equations. - New York: Springer. | Zbl 0952.47036

[011] Francis B.A. and Zames G. (1984), : On H^∞-optimal sensitivity theory for SISO feedback systems. - IEEE Trans. Automat. Contr., Vol. AC-29, No. 1, pp. 9-16. | Zbl 0601.93015

[012] Gauthier J.P. and Xu C.Z. (1991): H^∞-control of a distributed parameter system withnon-minimun phase. - Int. J. Contr., Vol. 53, No. 1, pp. 45-79. | Zbl 0724.93028

[013] Mahmood M.A. and Yevjevich V. (1975): Unsteady Flow in Open Channels, Vols. 1 and 2. - Fort Collins USA: Water Resources Publications.

[014] Miller M.A. and V. Yevjevich V. (1975): Unsteady Flow in Open Channels, Vol. 3. - Fort Collins USA: Water Resources Publications

[015] Pazy A. (1983): Semigroups of Linear Operators and Applications to Partial Differential Equations. - New York: Springer. | Zbl 0516.47023

[016] Pohjolainen S.A. (1985a): Robust controller for infinite systems with exponential strongly continuous semigroups. - J.Math. Anal. Appl., Vol. 111, pp. 622-636. | Zbl 0577.93037

[017] Pohjolainen S.A. (1985b): Robust multivariable PI-controllers for infinite dimensional systems. - IEEE Trans. Automat. Contr., Vol. 27, pp. 17-30. | Zbl 0493.93029

[018] Rauch J. and Taylor M. (1974): Exponential decay of solution to hyperbolic equations in bounded domain. - Indiana Univ. Math. J., Vol. 24, pp. 79-86. | Zbl 0281.35012

[019] Rebarber R. (1993): Conditions for the equivalence of internal and external stability for distributed parameter systems. - IEEE Trans. Automat. Contr., Vol. 38, No. 6, pp. 994-998. | Zbl 0786.93087

[020] Russel D.L. (1978): Controllability and stabilizability theory for linear partial differential equation: Recent progressand open questions. - SIAM Review, Vol. 20, No. 4, pp. 639-739.

[021] Weiss G. (1989a): Admissible observation operators for linear semigroups. - Israel J. Math., Vol. 65, pp. 17-43. | Zbl 0696.47040

[022] Weiss G. (1989b): Admissibility of unbounded control operators. -SIAM. J. Contr. Optim., No. 27, pp. 527-545. | Zbl 0685.93043

[023] Weiss G. (1994): Transfert function of regular linear systems. Part I: Characterization of regularity. - Trans. Amer. Math. Soci., Vol. 342, pp. 827-854. | Zbl 0798.93036

[024] Xu C.Z. and Jerbi H. (1995): A robust PI-controller for infinite dimensional systems. - Int. J. Contr., Vol. 61, No. 1, pp. 33-45. | Zbl 0820.93036

[025] Yoon M.G. and Lee B.H. (1991): An approximation approach to H^∞ control problems for distributed parameter systems. - Automatica, Vol. 33, No. 11, pp. 2049-2052. | Zbl 0910.93028

[026] Zames G. and Francis B.A. (1983): Feedback, minimax sensitivity and optimal robustness. - IEEE Trans. Automat. Contr., Vol. AC-28, No. 5, pp. 585-600. | Zbl 0528.93026