Commenced in January 2007
Paper Count: 31100
Tracking Control of a Linear Parabolic PDE with In-domain Point Actuators
Abstract:This paper addresses the problem of asymptotic tracking control of a linear parabolic partial differential equation with indomain point actuation. As the considered model is a non-standard partial differential equation, we firstly developed a map that allows transforming this problem into a standard boundary control problem to which existing infinite-dimensional system control methods can be applied. Then, a combination of energy multiplier and differential flatness methods is used to design an asymptotic tracking controller. This control scheme consists of stabilizing state-feedback derived from the energy multiplier method and feed-forward control based on the flatness property of the system. This approach represents a systematic procedure to design tracking control laws for a class of partial differential equations with in-domain point actuation. The applicability and system performance are assessed by simulation studies.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1082899Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1781
 M. Krsti'c and A. Smyshyaev, Boundry Control of PDEs: A Course on Backstepping Designs. New York: SIAM, 2008.
 I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Cambridge, UK: Cambridge Universioty Press, 2000.
 A. Bensoussan, G. Da Prato, M. Delfor, and S. K. Mitter, Representation and Control of Infinite-Dimensional Systems. Boston: Birkhauser, 2006.
 Z. Luo, B. Guo, and O. Morgul, Stability and Stabilization of Infinite Dimentional Systems with Applications. London: Springer, 1999.
 R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimentional Linar System Theory, ser. Text in Applied Mathematic. NY: Springer- Verlag, 1995, vol. 12.
 I. Lasiecka, Mathematical Control Theory of Coupled PDEs. Philadelpia: SIAM, 2002.
 T. Meurer and A. Kugi, "Tracking control for boundary controlled parabolic pdes with varying parameters: Combining backstepping and differential flatness," Automatica, vol. 45, pp. 1182-1194, 2009.
 T. Meurer, D. Thull, and A. Kugi, "Flatness-based tracking control of a piezoactuated euler-bernoulli beam with non-collocated output feedback: Theory and experinebts," International Journal of Control, vol. 81, no. 3, pp. 473-491, 2008.
 F. Ollivier and A. Sedoglavic, "A generalization of flatness to nonlinear system of partial differential equations. application to the command of a flexible rod," in Proc. of the 5th IFAC Symposium, vol. 1, Saint Petersburg, Russia, 2001, pp. 196-200.
 A. Kharitonov and O. Sawodny, "Optimal flatness based control for heating processes in the glass industry," in Proc. of the 43rd IEEE Conference on Decision and Control, Atlantis, Bahamas, 2004, pp. 2435-2440.
 J. Corriou, Process Control- Theory and Applications. London: Springer-Verlag, 2004.
 M. Fliess, J. L'evine, P. Martin, and P. Rouchon, "Flatness and defect of nonlinear system introductory theory and examples," International Journal of Control, vol. 61, pp. 1327-1361, 1995.
 J. L'evine, Analysis and Control of Nonlinear Systems: A Flatness-based Approach. Berlin: Springer-Verlag, 2009.
 R. Rothfur, J. Rudolph, and M. Zeitz, "Flatness-based control of a nonlinear chemical reactor model," Automatica, vol. 32, pp. 1433-1439, 1996.
 H. Sira-Ramirez and S. K. Agrawal, Differential Flat Systems. NY: Marcel Dekker Inc., 2004.
 B. Laroche, P. Martin, and P. Rouchon, "Motion planning for the heat equation," International Journal of Robust Nonlinear Control, vol. 10, pp. 629-643, 2000.
 A. F. Lynch and J. Rudolph, "Flatness-based boundary control of a class of quasilinear parabolic distributed parameter systems," International Journal of Control, vol. 75, no. 15, 2005.
 N. Petit, P. Rouchon, J. M. Boueih, F. Guerin, and P. Pinvidic, "Control of an industrial polymerization reactor using flatness," International Journal of Control, vol. 12, no. 5, pp. 659-665, 2002.
 J. Rudolph, Flatness Based Control of Distributed Parameter Systems. Aachen: Shaker-Verlag, 2003.
 R. A. C. Georgakis and N. Amundson, "Studies in the control of tubular reactors- i, ii, iii," Chemical Engineering Science, vol. 32, pp. 1359- 1387, 1977.
 K. Ammari and M. Tucsnak, "Stabilization of Bernoulli−Euler Beams by means of a pointwise feedback force," SIAM J. Control Optimal, vol. 39, no. 4, pp. 1160-1181, 2000.
 W. Han and B. D. Reddy, Plasticity: Mathematical Theory and Numerical Analysis. New York: Springer, 1999.
 K. Yosida, Functional Analysis, reprint of 6th edition ed. Spring- Verlage, 1995.
 W. Suli and D. Mayers, An Introduction to Numerical Analysis. UK: Cambridge University Press, 2006.
 J. V. Egorov and Kondratev, "The oblique derivative problem," Mathematics of the Ussr-Sbornik, vol. 7, pp. 139-169, 1969.
 P. Martin, R. M. Murray, and P. Rouchon, "Flat systems, equivalence and trajectory generation," CDS, Caltech, CA, CDS Technical Report, 2003.
 B. Laroche, P. Martin, and P. Rouchon, "Motion planning for a class of partial differential equations with boundary control," in Proc. of the 37th IEEE Conference on Decision and Control, Tampa, FL, USA, 1998, pp. 3494-3497.