Observer for Linear Distributed-Parameter Systems with Application to Isothermal Plug-Flow Reactor

Nadia  Barje; Mohamed Elarbi  Achhab; Vincent  Wertz

doi:10.4236/ica.2013.44045

Nadia Barje, Mohamed Elarbi Achhab, Vincent Wertz

Abstract: This paper presents a conception of an exponential observer for a class of linear distributed-parameter systems (DPSs), in which the dynamics are partially unknown. The given distributed-parameter observer ensures asymptotic state estimator with exponentially decay error, based on the theory of C0-semigroups in a Hilbert space. The theoretical observer developed is applied to a chemical tubular reactor, namely the isothermal Plug-Flow reactor basic dynamical model for which measurements are available at the reactor output only. The process is described by Partial differential equations with unknown initial states. For this application, performance issues are illustrated in a simulation study.

Keywords: Distributed-Parameter Systems; C0-semigroup; Exponential Observers; Perturbed Systems; Tubular Reactor

Cite this paper: N. Barje, M. Achhab and V. Wertz, "Observer for Linear Distributed-Parameter Systems with Application to Isothermal Plug-Flow Reactor," Intelligent Control and Automation, Vol. 4 No. 4, 2013, pp. 379-384. doi: 10.4236/ica.2013.44045.

References

[1] M. Renardy and R. C. Rogers,” An Introduction to Partial Dif ferential Equations,” Springer, New York, 1993.

[2] A. Pazy, “Semigroups of Linear Operators and Applications to Partial Differential Equations,” Springer-Verlag, New York, 1983.
http://dx.doi.org/10.1007/978-1-4612-5561-1

[3] R. F. Curtain and J. Zwart, “An Introduction to Infinite Dimentional Linear Systems Theory,” Springer, New York, 1995. http://dx.doi.org/10.1007/978-1-4612-4224-6

[4] D. Luenberger, “An Introduction to Observers,” IEIEEE Transactions on Automatic Control, Vol. 16, No. 6, 1971, pp. 596-602.
http://dx.doi.org/10.1109/TAC.1971.1099826

[5] T. Kobayashi and S. Hitotsuya, “Observers and Parameter Determination for Distributed Parameter Systems,” International Journal of Control, Vol. 33, No. 1, 1981, pp. 31-50.
http://dx.doi.org/10.1080/00207178108922906

[6] A. VandeWouwer and M. Zeitz, “State Estimation in Distributed Parameter Systems,” In: Encyclopedia of Life Support Systems (EOLSS): Control Systems, Robotics and Automation. EOLSS Publishers, London, 2003.

[7] M. A. Demetriou, “Natural Second-Order Observers for Second-Order Distributed Parameter Systems,” Systems & Control Letters, Vol. 51, No. 3-4, 2004, pp. 225-234.
http://dx.doi.org/10.1016/j.sysconle.2003.08.005

[8] A. Smyshlyaev and M. Krstic, “Backstepping Observers for a Class of Parabolic PDEs,” Systems & Control Letters, Vol. 54, No. 7, 2005, pp. 613-625.
http://dx.doi.org/10.1016/j.sysconle.2004.11.001

[9] T. D. Nguyen, “Second-Order Observers for SecondOrder Distributed Parameter Systems in R2,” Systems & Control Letters, Vol. 57, No. 10, 2008, pp. 787-795.
http://dx.doi.org/10.1016/j.sysconle.2008.03.011

[10] R. Miranda, I. Chairez and J. Moreno, “Observer Design for a Class of Parabolic PDE via Sliding Modes and Backstepping,” Proceedings of International Workshop on Variable Structure Systems, Mexico City, Jun 2010, pp. 215-220.

[11] J. Winkin, D. Dochain and P. Ligarius, “Dynamical Analysis of Distributed Parameter Tubular Reactors,” Automatica, Vol. 36, No. 3, 2000, pp. 349-361.
http://dx.doi.org/10.1016/S0005-1098(99)00170-3

[12] I. Y. Smets, D. Dochain and J. F. Van Impe, “Optimal Temperature Control of a Steady-State Exothermic PlugFlow Reactor,” AIChE Journal, 2002, Vol. 48, No. 2, pp. 279-286.
http://dx.doi.org/10.1002/aic.690480212