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2001 | 11 | 2 | 287-317

Tytuł artykułu

Finite-dimensional control of nonlinear parabolic PDE systems with time-dependent spatial domains using empirical eigenfunctions

Treść / Zawartość

Warianty tytułu

Języki publikacji

EN

Abstrakty

EN
This article presents a methodology for the synthesis of finite-dimensional nonlinear output feedback controllers for nonlinear parabolic partial differential equation (PDE) systems with time-dependent spatial domains. Initially, the nonlinear parabolic PDE system is expressed with respect to an appropriate time-invariant spatial coordinate, and a representative (with respect to different initial conditions and input perturbations) ensemble of solutions of the resulting time-varying PDE system is constructed by computing and solving a high-order discretization of the PDE. Then, the Karhunen-Loaève expansion is directly applied to the ensemble of solutions to derive a small set of empirical eigenfunctions (dominant spatial patterns) that capture almost all the energy of the ensemble of solutions. The empirical eigenfunctions are subsequently used as basis functions within a Galerkin model reduction framework to derive low-order ordinary differential equation (ODE) systems that accurately describe the dominant dynamics of the PDE system. The ODE systems are subsequently used for the synthesis of nonlinear output feedback controllers using geometric control methods. The proposed control method is used to stabilize an unstable steady-state of a diffusion-reaction process with nonlinearities, spatially-varying coefficients and time-dependent spatial domain, and is shown to lead to the construction of accurate low-order models and the synthesis of low-order controllers. The performance of the low-order models and controllers is successfully tested through computer simulations.

Rocznik

Tom

11

Numer

2

Strony

287-317

Opis fizyczny

Daty

wydano
2001
otrzymano
2000-11-24
poprawiono
2001-02-10

Twórcy

  • Department of Chemical Engineering, University of California, Los Angeles, CA 90095-1592, USA
  • Department of Chemical Engineering, University of California, Los Angeles, CA 90095-1592, USA

Bibliografia

  • Armaou A. and Christofides P.D. (1999): Nonlinear feedback control of parabolic PDE systems with time-dependent spatial domains. - J. Math. Anal. Appl., Vol.239, No.1, pp.124-157.
  • Armaou A. and Christofides P.D. (2001): Robust control of parabolic PDE systems with time-dependent spatial domains. - Automatica, Vol.37, No.1, pp.61-69.
  • Baker J. and Christofides P.D. (2000): Finite dimensional approximation and control of nonlinear parabolic PDE systems. -Int. J. Contr., Vol.73, No.5, pp.439-456.
  • Balas M.J. (1979): Feedback control of linear diffusion processes. -Int. J. Contr., Vol.29, No.3, pp.523-533.
  • Balas M.J. (1982): Trends in large scale structure control theory: Fondest hopes, wildest dreams. -IEEE Trans. Automat. Contr., Vol.27, No.3, pp.522-535.
  • Bangia A.K., Batcho P.F., Kevrekidis I.G. and Karniadakis G.E. (1997): Unsteady 2-D flows in complex geometries: Comparative bifurcation studies with global eigenfunction expansion. -SIAM J. Sci. Comp., Vol.18, No.3, pp.775-805.
  • Byrnes C.A., Gilliam D.S. and Shubov V.I. (1994): Global lyapunov stabilization of a nonlinear distributed parameter system. -Proc. 33rd IEEE Conf. Decision and Control,Orlando, FL, pp.1769-1774.
  • Byrnes C.A., Gilliam D.S. and Shubov V.I. (1995): On the dynamics of boundary controlled nonlinear distributed parameter systems. -Proc. Symp. Nonlinear Control Systems Design'95, Tahoe City, CA, pp.913-918.
  • Chen C.C. and Chang H.C. (1992): Accelerated disturbance damping of an unknown distributed system by nonlinear feedback. -AIChE J., Vol.38, No.9, pp.1461-1476.
  • Christofides P.D. (2001): Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes. - Boston: Birkhauser.
  • Curtain R.F. (1982): Finite-dimensional compensator design for parabolic distributed systems with point sensors and boundary input. -IEEE Trans. Automat. Contr., Vol.27, No.1, pp.98-104.
  • Curtain R.F. and Zwart H.J. (1995): An Introduction to Infinite Dimensional Linear Systems. - New York: Springer.
  • Friedman A. (1976): Partial Differential Equations. - New York: Holt, Rinehart and Winston.
  • Fukunaga K. (1990): Introduction to Statistical Pattern Recognition. - New York: Academic Press.
  • Graham M.D. and Kevrekidis I.G. (1996): Alternative approaches to the Karhunen-Loève decomposition for model reduction and data analysis. -Comp. Chem. Eng., Vol.20, No.5, pp.495-506.
  • Holmes P., Lumley J.L. and Berkooz G. (1996): Turbulence, Coherent Structures, Dynamical Systems and Symmetry. - New York: Cambridge University Press.
  • Isidori A. (1989): Nonlinear Control Systems: An Introduction, 2nd Ed. - Berlin-Heidelberg: Springer, second edition.
  • Kokotovic P.V., Khalil H.K. and O'Reilly J. (1986): Singular Perturbations in Control: Analysis and Design. - London: Academic Press.
  • Kowalewski A. (1998): Optimal control of a distributed hyperbolic system with multiple time-varying lags. -Int. J. Contr., Vol.71, No.3, pp.419-435.
  • Kowalewski A. (2000): Optimal control of distributed hyperbolic systems with deviating arguments. -Int. J. Contr., Vol.73, No.11, pp.1026-1041.
  • Lasiecka I. (1995): Control of systems governed by partial differential equations: A historical perspective. -Proc. 34th IEEE Conf. Decision and Control,New Orleans, LA, pp.2792-2797.
  • Palanki S. and Kravaris C. (1997): Controller synthesis for time-varying systems by inputoutput linearization. -Comp. Chem. Eng., Vol.21, No.8, pp.891-903.
  • Park H.M. and Cho D. (1996): The use of the Karhunen-Loeve decomposition for the modeling of distributed parameter systems. -Chem. Eng. Sci., Vol.51, No.1, pp.81-98.
  • Pazy A. (1983): Semigroups of Linear Operators and Applications to Partial Differential Equations. - New York: Springer.
  • Ray W.H. (1981): Advanced Process Control. - New York: McGraw-Hill.
  • Ray W.H. and Seinfeld J.H. (1975): Filtering in distributed parameter systems with moving boundaries. -Automatica, Vol.11, No.5, pp.509-515.
  • Rowley C.W. and Marsden J.E. (2000): Reconstruction equations and the Karhunen-Loève expansion for systems with symmetry. -Physica D, Vol.142, No.1-2, pp.1-19.
  • Sano H. and Kunimatsu N. (1995): An application of inertial manifold theory to boundary stabilization of semilinear diffusion systems. -J. Math. Anal. Appl., Vol.196, No.1, pp.18-42.
  • Shvartsman S.Y. and Kevrekidis I.G. (1998): Nonlinear model reduction for control of distributed parameter systems: A computer assisted study. -AIChE J., Vol.44, No.7, pp.1579-1595.
  • Sirovich L. (1987): Turbulence and the dynamics of coherent structures: I-III. -Quart. Appl. Math., Vol.XLV, No.3, pp.561-590.
  • Sirovich L., Knight B.W. and Rodriguez J.D. (1990): Optimal low-dimensional dynamical approximations. -Quart. Appl. Math., Vol.XLVIII, No.3, pp.535-548.
  • Theodoropoulou A., Adomaitis R.A. and Zafiriou E. (1999): Inverse model based real-time control for temperature uniformity of RTCVD. -IEEE Trans. Sem. Manuf., Vol.12, No.1, pp.87-101.
  • van Keulen B. (1993): H_∞-Control for Distributed Parameter Systems: A State-Space Approach. - Boston: Birkhauser.
  • Wang P.K.C. (1967): Control of a distributed parameter system with a free boundary. -Int. J. Contr., Vol.5, No.3, pp.317-329.
  • Wang P.K.C. (1990): Stabilization and control of distributed systems with time-dependent spatial domains. -J. Optim. Theor. Appl., Vol.65, No.2, pp.331-362.

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Bibliografia

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