Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-20T03:36:24.013Z Has data issue: false hasContentIssue false
  • English
  • Français

Motion planning for a class of boundary controlled linear hyperbolic PDE's involving finite distributed delays

Published online by Cambridge University Press:  15 September 2003

Frank Woittennek  and
Joachim Rudolph
Frank Woittennek
Affiliation:
Technische Universität Dresden, Germany; [email protected].,
Joachim Rudolph
Affiliation:
Technische Universität Dresden, Germany; [email protected].,
Get access

Abstract

Motion planning and boundary control for a class of linear PDEs with constant coefficients is presented. With the proposed method transitions from rest to rest can be achieved in a prescribed finite time. When parameterizing the system by a flat output, the system trajectories can be calculated from the flat output trajectory by evaluating definite convolution integrals. The compact kernels of the integrals can be calculated using infinite series. Explicit formulae are derived employing Mikusiński's operational calculus. The method is illustrated through an application to a model of a Timoshenko beam, which is clamped on a rotating disk and carries a load at its free end.

Keywords

Flatnessmotion planning.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 419 - 435
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Fliess, M., Lévine, J., Martin, Ph. and Rouchon, P., Flatness and defect of non-linear systems: Introductory theory and examples. Internat. J. Control 61 (1995) 1327-1361. CrossRef
Fliess, M., Martin, Ph., Petit, N. and Rouchon, P., Commande de l'équation des télégraphistes et restauration active d'un signal. Traitement du Signal 15 (1998) 619-625.
Fliess, M. and Mounier, H., Controllability and observability of linear delay systems: An algebraic approach. ESAIM: COCV 3 (1998) 301-314. (URL: http://www.emath.fr/COCV/). CrossRef
M. Fliess and H. Mounier, Tracking control and $\pi$ -freeness of infinite dimensional linear systems, edited by G. Picci and D.S. Gilliam, Dynamical Systems, Control, Coding, Computer Vision. Birkhäuser (1999) 45-68.
M. Fliess, H. Mounier, P. Rouchon and J. Rudolph, Controllability and motion planning for linear delay systems with an application to a flexible rod, in Proc. 34th IEEE Conference on Decision and Control. New Orleans (1995) 2046-2051.
Fliess, M., Mounier, H., Rouchon, P. and Rudolph, J., Systèmes linéaires sur les opérateurs de Mikusinski et commande d'une poutre flexible. ESAIM Proc. 2 (1997) 183-193. (http://www.emath.fr/proc). CrossRef
M. Fliess, H. Mounier, P. Rouchon and J. Rudolph, Controlling the transient of a chemical reactor: A distributed parameter approach, in Proc. Computational Engineering in Systems Application IMACS Multiconference, (CESA'98). Hammamet, Tunisia (1998).
F. John, Partial Differential Equations, 4th Edition. Springer-Verlag, New York (1991).
Laroche, B., Martin, Ph. and Rouchon, P., Motion planning for the heat equation. Int. J. Robust Nonlinear Control 10 (2000) 629-643. 3.0.CO;2-N>CrossRef
Lynch, A.F. and Rudolph, J., Flachheitsbasierte Randsteuerung parabolischer Systeme mit verteilten Parametern. Automatisierungstechnik 48 (2000) 478-486. CrossRef
Mikusinski, J., Sur les équations différentielles du calcul opératoire et leurs applications aux équations aux dérivées partielles. Stud. Math. 12 (1951) 227-270.
J. Mikusinski, Operational Calculus, Vol. 1. Pergamon, Oxford & PWN, Warszawa (1983).
J. Mikusinski and Th.K. Boehme, Operational Calculus, Vol. 2. Pergamon, Oxford & PWN, Warszawa (1987).
H. Mounier, J. Rudolph, M. Petitot and M. Fliess, A flexible rod as a linear delay system, in Proc. 3rd European Control Conference. Rome, Italy (1995) 3676-3681.
Petit, N. and Rouchon, P., Motion planning for heavy chain systems. SIAM J. Control Optim. 40 (2001) 275-495. CrossRef
N. Petit and P. Rouchon, Dynamics and solutions to some control problems for water-tank systems. IEEE Trans. Automat. Control AC-47 (2002) 594-609.
Petrovskij, I.G., Über das Cauchysche Problem für Systeme von partiellen Differentialgleichungen. Mat. Sb. 2 (1937) 815-866.
Rothfuß, R., Rudolph, J. and Zeitz, M., Flachheit: Ein neuer Zugang zur Steuerung und Regelung nichtlinearer Systeme. Automatisierungstechnik 45 (1997) 517-525. CrossRef
W. Rudin, Real and Complex Analysis, 3rd Edition. McGraw-Hill (1987).
Rudolph, J., Randsteuerung von Wärmetauschern mit örtlich verteilten Parametern: Ein flachheitsbasierter Zugang. Automatisierungstechnik 48 (2000) 399-406.
Rudolph, J. and Woittennek, F., Flachheitsbasierte Steuerung eines Timoshenko-Balkens. Z. Angew. Math. Mech. 83 (2003) 119-127. CrossRef
Simo, J.C., A finite strain beam formulation. The three-dimensional dynamic problem. Part one. Comp. Meths. Appl. Mech. 49 (1985) 55-70. CrossRef
K. Yosida, Operational Calculus. Springer-Verlag (1984).
K. Yuan, Control of slew maneuver of a flexible beam mounted non-radially on a rigid hub: A geometrically exact modelling approach, Vol. 204 (1997) 795-806.