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2003 | 13 | 4 | 453-468
Tytuł artykułu

The stability of an irrigation canal system

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Rocznik
Tom
13
Numer
4
Strony
453-468
Opis fizyczny
Daty
wydano
2003
otrzymano
2002-12-18
poprawiono
2003-09-12
Twórcy
autor
  • Department of Mathematics, Ibnou Zohr University Faculty of Sciences, BP. 8106, 80000 Agadir, Morocco
Bibliografia
  • 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.
  • 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.
  • Bounit H. (2003a): Robust PI-controller for an irrigation canal system. - (In preparation).
  • Bounit H. (2003b): H^∞-controller for an irrigation canal system. - (In preparation).
  • 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.
  • 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.
  • Chow V.T. (1985): Open Channels Hydraulics. - New York: Mac Graw Hill .
  • 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.
  • Curtain R.F. (1988): Equivalence of input-output stability and exponential stability for infinite dimensional systems. - Math. Syst. Theory, Vol. 21, pp. 19-48.
  • Curtain R.F. and Zwart H. (1995): An Introduction to Infinite Dimensional Linear System Theory. - New York: Springer.
  • Engel K.J. and Nagel R. (2000): One Parameter Semigroups for Linear Evolution Equations. - New York: Springer.
  • 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.
  • 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.
  • Mahmood M.A. and Yevjevich V. (1975): Unsteady Flow in Open Channels, Vols. 1 and 2. - Fort Collins USA: Water Resources Publications.
  • Miller M.A. and V. Yevjevich V. (1975): Unsteady Flow in Open Channels, Vol. 3. - Fort Collins USA: Water Resources Publications
  • Pazy A. (1983): Semigroups of Linear Operators and Applications to Partial Differential Equations. - New York: Springer.
  • Pohjolainen S.A. (1985a): Robust controller for infinite systems with exponential strongly continuous semigroups. - J.Math. Anal. Appl., Vol. 111, pp. 622-636.
  • Pohjolainen S.A. (1985b): Robust multivariable PI-controllers for infinite dimensional systems. - IEEE Trans. Automat. Contr., Vol. 27, pp. 17-30.
  • 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.
  • 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.
  • 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.
  • Weiss G. (1989a): Admissible observation operators for linear semigroups. - Israel J. Math., Vol. 65, pp. 17-43.
  • Weiss G. (1989b): Admissibility of unbounded control operators. -SIAM. J. Contr. Optim., No. 27, pp. 527-545.
  • Weiss G. (1994): Transfert function of regular linear systems. Part I: Characterization of regularity. - Trans. Amer. Math. Soci., Vol. 342, pp. 827-854.
  • 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.
  • 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.
  • 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.
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Bibliografia
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