Tracking Control of a Linear Parabolic PDE with In-domain Point Actuators
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.
[1] M. Krsti'c and A. Smyshyaev, Boundry Control of PDEs: A Course on
Backstepping Designs. New York: SIAM, 2008.
[2] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential
Equations: Continuous and Approximation Theories. Cambridge, UK:
Cambridge Universioty Press, 2000.
[3] A. Bensoussan, G. Da Prato, M. Delfor, and S. K. Mitter, Representation
and Control of Infinite-Dimensional Systems. Boston: Birkhauser, 2006.
[4] Z. Luo, B. Guo, and O. Morgul, Stability and Stabilization of Infinite
Dimentional Systems with Applications. London: Springer, 1999.
[5] 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.
[6] I. Lasiecka, Mathematical Control Theory of Coupled PDEs.
Philadelpia: SIAM, 2002.
[7] 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.
[8] 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.
[9] 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.
[10] 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.
[11] J. Corriou, Process Control- Theory and Applications. London:
Springer-Verlag, 2004.
[12] 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.
[13] J. L'evine, Analysis and Control of Nonlinear Systems: A Flatness-based
Approach. Berlin: Springer-Verlag, 2009.
[14] R. Rothfur, J. Rudolph, and M. Zeitz, "Flatness-based control of a
nonlinear chemical reactor model," Automatica, vol. 32, pp. 1433-1439,
1996.
[15] H. Sira-Ramirez and S. K. Agrawal, Differential Flat Systems. NY:
Marcel Dekker Inc., 2004.
[16] 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.
[17] 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.
[18] 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.
[19] J. Rudolph, Flatness Based Control of Distributed Parameter Systems.
Aachen: Shaker-Verlag, 2003.
[20] 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.
[21] 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.
[22] W. Han and B. D. Reddy, Plasticity: Mathematical Theory and Numerical
Analysis. New York: Springer, 1999.
[23] K. Yosida, Functional Analysis, reprint of 6th edition ed. Spring-
Verlage, 1995.
[24] W. Suli and D. Mayers, An Introduction to Numerical Analysis. UK:
Cambridge University Press, 2006.
[25] J. V. Egorov and Kondratev, "The oblique derivative problem," Mathematics
of the Ussr-Sbornik, vol. 7, pp. 139-169, 1969.
[26] P. Martin, R. M. Murray, and P. Rouchon, "Flat systems, equivalence
and trajectory generation," CDS, Caltech, CA, CDS Technical Report,
2003.
[27] 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.
[1] M. Krsti'c and A. Smyshyaev, Boundry Control of PDEs: A Course on
Backstepping Designs. New York: SIAM, 2008.
[2] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential
Equations: Continuous and Approximation Theories. Cambridge, UK:
Cambridge Universioty Press, 2000.
[3] A. Bensoussan, G. Da Prato, M. Delfor, and S. K. Mitter, Representation
and Control of Infinite-Dimensional Systems. Boston: Birkhauser, 2006.
[4] Z. Luo, B. Guo, and O. Morgul, Stability and Stabilization of Infinite
Dimentional Systems with Applications. London: Springer, 1999.
[5] 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.
[6] I. Lasiecka, Mathematical Control Theory of Coupled PDEs.
Philadelpia: SIAM, 2002.
[7] 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.
[8] 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.
[9] 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.
[10] 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.
[11] J. Corriou, Process Control- Theory and Applications. London:
Springer-Verlag, 2004.
[12] 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.
[13] J. L'evine, Analysis and Control of Nonlinear Systems: A Flatness-based
Approach. Berlin: Springer-Verlag, 2009.
[14] R. Rothfur, J. Rudolph, and M. Zeitz, "Flatness-based control of a
nonlinear chemical reactor model," Automatica, vol. 32, pp. 1433-1439,
1996.
[15] H. Sira-Ramirez and S. K. Agrawal, Differential Flat Systems. NY:
Marcel Dekker Inc., 2004.
[16] 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.
[17] 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.
[18] 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.
[19] J. Rudolph, Flatness Based Control of Distributed Parameter Systems.
Aachen: Shaker-Verlag, 2003.
[20] 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.
[21] 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.
[22] W. Han and B. D. Reddy, Plasticity: Mathematical Theory and Numerical
Analysis. New York: Springer, 1999.
[23] K. Yosida, Functional Analysis, reprint of 6th edition ed. Spring-
Verlage, 1995.
[24] W. Suli and D. Mayers, An Introduction to Numerical Analysis. UK:
Cambridge University Press, 2006.
[25] J. V. Egorov and Kondratev, "The oblique derivative problem," Mathematics
of the Ussr-Sbornik, vol. 7, pp. 139-169, 1969.
[26] P. Martin, R. M. Murray, and P. Rouchon, "Flat systems, equivalence
and trajectory generation," CDS, Caltech, CA, CDS Technical Report,
2003.
[27] 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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:63395", author = "Amir Badkoubeh and Guchuan Zhu", title = "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.", keywords = "Tracking Control, In-domain point actuation, PartialDifferential Equations.", volume = "5", number = "11", pages = "1758-6", }
