A Reduction Method for the Boundary Control of the Heat Conduction Equation

[+] Author and Article Information
H. M. Park, O. Y. Kim

Department of Chemical Engineering, Sogang University, Seoul, Korea

J. Dyn. Sys., Meas., Control 122(3), 435-444 (Nov 18, 1998) (10 pages) doi:10.1115/1.1286365 History: Received November 18, 1998
Copyright © 2000 by ASME
Your Session has timed out. Please sign back in to continue.


Lausterer,  G. K., and Ray,  W. H., 1979, “Distributed parameter state estimation and optimal feedback control—An experimental study in two space dimensions,” IEEE Trans. Autom. Control, AC-24, p. 179.
Nulman,  J. , and Krusius,  J. P, and Gat,  A., 1985, “Rapid thermal processing of thin gate dielectrics-oxidation of silicon,” IEEE Electron Device Lett., EDL-6, p. 205.
Park,  H. M., and Cho,  D. H., 1996, “The use of the Karhunen–Loève decomposition for the modeling of distributed parameter systems,” Chem. Eng. Sci., 51, pp. 81–98.
Park,  H. M., and Cho,  D. H., 1996, “Low dimensional modeling of flow reactors” Int. J. Heat Mass Transf., 39, pp. 3311–3323.
Loève, M., 1995, Probability Theory, Van Nostrand, Princeton, NJ.
Park,  H. M., and Sirovich,  L., 1990, “Turbulent thermal convection in a finite domain: Numerical results,” Phys. Fluids A, 2, pp. 1659–1668.
Moin,  P., and Moser,  R. D., 1989, “Characteristic-eddy decomposition of turbulence in a channel,” J. Fluid Mech., 200, pp. 471–506.
Aubry,  N., Holmes,  P., Lumley,  J. L., and Stone,  E., 1988, “The dynamics of coherent structures in the wall region of a turbulent boundary layer,” J. Fluid Mech., 192, pp. 115–173.
Deane,  A. E., Kevrekidis,  I. G., Karniadakis,  G. E., and Orszag,  S. A., 1991, “Low-dimensional models for complex geometry flows: Application to grooved channels and circular cylinders,” Phys. Fluids, 3, pp. 23–37.
Park,  H. M., and Lee,  J. H., 1998, “A method of solving inverse convection problems by means of mode reduction,” Chem. Eng. Sci., 53, pp. 1731–1744.
Baker,  J., and Christofides,  P. D., 1999, “Nonlinear control of rapid thermal chemical vapor deposition under uncertainty,” Comput. Chem. Eng., 23, pp. 233–236.
Christofides,  P. D., 1998, “Robust control of parabolic PDE systems,” Chem. Eng. Sci., 53, pp. 2449–2465.
Courant, R., and Hilbert, D., 1953, Methods of Mathematical Phyiscs, Vol. 1, Interscience Publishers, New York.
Fletcher,  R., and Reeves,  R. M., 1964, “Function minimization by conjugate gradients,” Comput. J. (UK), 7, pp. 149–154.


Grahic Jump Location
The system and relevant boundary conditions
Grahic Jump Location
Definition of shape functions. (a) Temporal shape functions; (b) spatial shape functions.
Grahic Jump Location
Some dominant empirical eigenfunctions with large eigenvalues. (a) The first eigenfunction (λ1=0.741); (b) the second eigenfunction (λ2=0.156); (c) the third eigenfunction (λ3=4.969×10−2); (d) the fourth eigenfunction (λ4=2.127×10−2).
Grahic Jump Location
Some typical empirical eigenfunctions with small eigenvalues. (a) The 11th eigenfunction (λ11=1.136×10−3); (b) the 12th eigenfunction (λ12=8.807×10−4); (c) the 13th eigenfunction (λ13=5.832×10−4); (d) the 14th eigenfunction (λ14=4.251×10−4).
Grahic Jump Location
Relative errors of the low dimensional dynamic model with respect to the finite difference solution for various heat flux functions q(x,t)
Grahic Jump Location
The space-time contours of the optimal control obtained either by the FDM-CG or by the KLG-CG
Grahic Jump Location
The temporal profiles of the optimal control at x=0.5 and x=0.8 obtained either by the FDM-CG or by the KLG-CG. The original control, Eqs. (24a24b) is also displayed for comparison. (a) x=0.5; (b) x=0.8.
Grahic Jump Location
Convergence of the KLG-CG for various initial approximations of q(x,t). (a) Profiles of various initial approximations at x=0.5; (b) converged profiles of optimal control q(x,t) at x=0.5; (c) profiles of various initial approximations at x=0.8; (d) converged profiles of optimal control q(x,t) at x=0.8.




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In