Abstract
Transport-reaction processes with significant diffusive and dispersive mechanisms (e.g. packed-bed reactors, rapid thermal processing systems, chemical vapor deposition reactors, etc.) are typically characterized by strong nonlinearities and spatial variations, and are naturally modeled by nonlinear parabolic PDE systems. The main feature of parabolic PDEs is that the eigenspectrum of the spatial differential operator can be partitioned into a finite-dimensional slow one and an infinite-dimensional stable fast complement [18, 1, 29]. This implies that the dynamic behavior of such systems can be approximately described by finite-dimensional systems. Motivated by this, the standard approach to control parabolic PDEs involves the application of Galerkin’s method to the PDE system to derive ODE systems that describe the dynamics of the dominant (slow) modes of the PDE system, which are subsequently used as the basis for the synthesis of finite-dimensional controllers [20, 1, 26, 8]. However, there are two key controller implementation and closed-loop performance problems associated with this approach. First, the number of modes that should be retained to derive an ODE system that yields the desired degree of approximation may be very large, leading to high dimensionality of the resulting controllers [19]. Second, there is a lack of a systematic way to characterize the discrepancy between the solutions of the PDE system and the approximate ODE system in finite time, which is essential for characterizing the transient performance of the closed-loop PDE system.
Financial support for this work in part by UCLA through the SEAS Dean’s Fund, and National Science Foundation, CTS-9624725, is gratefully acknowledged.
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10. References
M. J. Balas. Feedback control of linear diffusion processes. Int. J. Contr., 29:523–533, 1979.
M. J. Balas. Stability of distributed parameter systems with finite-dimensional controller-compensators using singular perturbations. J. Math. Anal. Appi, 99:80–108, 1984.
M. J. Balas. Nonlinear finite-dimensional control of a class of nonlinear distributed parameter systems using residual mode filters: A proof of local exponential stability. J. Math. Anal. Appl., 162:63–70, 1991.
A. K. Bangia, P. F. Batcho, I. G. Kevrekidis, and G. E. Karniadakis. Unsteady 2-D flows in complex geometries: Comparative bifurcation studies with global eigenfunction expansion. submitted, 1996.
H. S. Brown, I. G. Kevrekidis, and M. S. Jolly. A minimal model for spatiotemporal patterns in thin film flow. In Pattern and Dynamics in Reactive Media, pages 11–31, R. Aris, D. G. Aronson, and H. L. Swinney, ed., Springer-Verlag, 1991.
C. A. Byrnes, D. S. Gilliam, and V. I. Shubov. On the dynamics of boundary controlled nonlinear distributed parameter systems. In Proc. of Symposium on Nonlinear Control Systems Design’95, pages 913–918, Tahoe City, CA, 1995.
J. Carr. Applications of Center Manifold Theory. Springer-Verlag, New York, 1981.
C. C. Chen and H. C. Chang. Accelerated disturbance damping of an unknown distributed system by nonlinear feedback. AIChE J., 38:1461–1476, 1992.
P. D. Christofides. Nonlinear Control of Two-Time-Scale and Distributed Parameter Systems. PhD thesis, Dept. of Chem. Eng. & Mat. Sci., University of Minnesota, Minneapolis, MN, 1996.
P. D. Christofides. Robust control of parabolic PDE systems. Chem. Eng. Sci. accepted; a version of this paper also appeared in the Proceedings of 36th Conference on Decision and Control, 1074–1081, San-Diego, CA, 1997.
P. D. Christofides and P. Daoutidis. Feedback control of hyperbolic PDE systems. AIChE J., 42:3063–3086, 1996.
P. D. Christofides and P. Daoutidis. Nonlinear control of diffusion-convection-reaction processes. Comp. Chem. Engng., 20(s): 1071–1076, 1996.
P. D. Christofides and P. Daoutidis. Distributed output feedback control of two-time-scale hyperbolic PDE systems. J. of Appl. Math. & Comp. Sci. in press, 1997.
P. D. Christofides and P. Daoutidis. Finite-dimensional control of parabolic PDE systems using approximate inertial manifolds. J. Math. Anal. Appl, in press; a version of this paper also appeared in the Proceedings of 36th Conference on Decision and Control, 1068–1073, San-Diego, California, 1997.
P. D. Christofides and P. Daoutidis. Robust control of hyperbolic PDE systems. Chem. Eng. Sci., in press, 1997.
C. Foias, M.S. Jolly, I.G. Kevrekidis, G.R. Sell, and E.S. Titi. On the computation of inertial manifolds. Phys. Lett A, 131:433–437, 1989.
C. Foias, G.R. Sell, and E.S. Titi. Exponential tracking and approximation of inertial manifolds for dissipative equations. J. Dynamics and Differential Equations, 1:199–244, 1989.
A. Friedman. Partial Differential Equations. Holt, Rinehart & Winston, New York, 1976.
D. H. Gay and W. H. Ray. Identification and control of distributed parameter systems by means of the singular value decomposition. Chem. Eng. Sci., 50:1519–1539, 1995.
G. Georgakis, R. Aris, and N. R. Amundson. Studies in the control of tubular reactors: Part I & II & III. Chem. Eng. Sci., 32:1359–1387, 1977.
E. M. Hanczyc and A. Palazoglu. Sliding mode control of nonlinear distributed parameter chemical processes. I & EC Res., 34:557–566, 1995.
I. Karafyllis, P. D. Christofides, and P. Daoutidis. Dynamical analysis of a reaction diffusion system with brusselator kinetics under feedback control. In Proceedings of American Control Conference, pages 2213–2217, Albuquerque, NM, 1997.
P. V. Kokotovic, H. K. Khalil, and J. O’Reilly. Singular Perturbations in Control: Analysis and Design. Academic Press, London, 1986.
W. Marquardt. Traveling waves in chemical processes. Int. Chem. Engng., 4:585–606, 1990.
H. M. Park and D. H. Cho. The use of the karhunen-loeve decomposition for the modeling of distributed parameter systems. Chem. Eng. Sci., 51:81–98, 1996.
W. H. Ray. Advanced Process Control. McGraw-Hill, New York, 1981.
A. Rigopoulos and Y. Arkun. Principal components analysis in estimation and control of paper machines. Comp. Chem. Engng., 20(s):1059–1064, 1996.
H. Sano and N. Kunimatsu. An application of inertial manifold theory to boundary stabilization of semilinear diffusion systems. J. Math. Anal. Appl., 196:18–42, 1995.
R. Temam. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag, New York, 1988.
A. N. Tikhonov. On the dependence of the solutions of differential equations on a small parameter. Mat. 56., 22: 193–204, 1948.
E. B. Ydstie and A. A. Alonso. Process systems and passivity via the Clausius-Planck inequality. Syst. & Contr. Lett., 30:253–264, 1997.
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Christofides, P.D., Daoutidis, P. (1998). Nonlinear Feedback Control of Parabolic PDE Systems. In: Berber, R., Kravaris, C. (eds) Nonlinear Model Based Process Control. NATO ASI Series, vol 353. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5094-1_13
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