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.
Mots-clés : flatness, motion planning
@article{COCV_2003__9__419_0,
author = {Woittennek, Frank and Rudolph, Joachim},
title = {Motion planning for a class of boundary controlled linear hyperbolic {PDE's} involving finite distributed delays},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {419--435},
publisher = {EDP-Sciences},
volume = {9},
year = {2003},
doi = {10.1051/cocv:2003020},
zbl = {1075.93015},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2003020/}
}
TY - JOUR AU - Woittennek, Frank AU - Rudolph, Joachim TI - Motion planning for a class of boundary controlled linear hyperbolic PDE's involving finite distributed delays JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2003 SP - 419 EP - 435 VL - 9 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2003020/ DO - 10.1051/cocv:2003020 LA - en ID - COCV_2003__9__419_0 ER -
%0 Journal Article %A Woittennek, Frank %A Rudolph, Joachim %T Motion planning for a class of boundary controlled linear hyperbolic PDE's involving finite distributed delays %J ESAIM: Control, Optimisation and Calculus of Variations %D 2003 %P 419-435 %V 9 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2003020/ %R 10.1051/cocv:2003020 %G en %F COCV_2003__9__419_0
Woittennek, Frank; Rudolph, Joachim. Motion planning for a class of boundary controlled linear hyperbolic PDE's involving finite distributed delays. ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 419-435. doi : 10.1051/cocv:2003020. http://www.numdam.org/articles/10.1051/cocv:2003020/
[1] ,, and, Flatness and defect of non-linear systems: Introductory theory and examples. Internat. J. Control 61 (1995) 1327-1361. | MR | Zbl
[2] ,, and, Commande de l'équation des télégraphistes et restauration active d'un signal. Traitement du Signal 15 (1998) 619-625. | Zbl
[3] and, Controllability and observability of linear delay systems: An algebraic approach. ESAIM: COCV 3 (1998) 301-314. (URL: http://www.emath.fr/COCV/). | EuDML | Numdam | MR | Zbl
[4] and, Tracking control and -freeness of infinite dimensional linear systems, edited by G. Picci and D.S. Gilliam, Dynamical Systems, Control, Coding, Computer Vision. Birkhäuser (1999) 45-68. | MR | Zbl
[5] ,, and, 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.
[6] ,, and, Systèmes linéaires sur les opérateurs de Mikusiński et commande d'une poutre flexible. ESAIM Proc. 2 (1997) 183-193. (http://www.emath.fr/proc). | Zbl
[7] ,, and, 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).
[8] , Partial Differential Equations, 4th Edition. Springer-Verlag, New York (1991). | MR
[9] , and, Motion planning for the heat equation. Int. J. Robust Nonlinear Control 10 (2000) 629-643. | MR | Zbl
[10] and, Flachheitsbasierte Randsteuerung parabolischer Systeme mit verteilten Parametern. Automatisierungstechnik 48 (2000) 478-486.
[11] , Sur les équations différentielles du calcul opératoire et leurs applications aux équations aux dérivées partielles. Stud. Math. 12 (1951) 227-270. | MR | Zbl
[12] , Operational Calculus, Vol. 1. Pergamon, Oxford & PWN, Warszawa (1983). | MR | Zbl
[13] and, Operational Calculus, Vol. 2. Pergamon, Oxford & PWN, Warszawa (1987). | MR | Zbl
[14] ,, and, A flexible rod as a linear delay system, in Proc. 3rd European Control Conference. Rome, Italy (1995) 3676-3681.
[15] and, Motion planning for heavy chain systems. SIAM J. Control Optim. 40 (2001) 275-495. | MR | Zbl
[16] and, Dynamics and solutions to some control problems for water-tank systems. IEEE Trans. Automat. Control AC-47 (2002) 594-609. | MR
[17] , Über das Cauchysche Problem für Systeme von partiellen Differentialgleichungen. Mat. Sb. 2 (1937) 815-866. | JFM | Zbl
[18] , and, Flachheit: Ein neuer Zugang zur Steuerung und Regelung nichtlinearer Systeme. Automatisierungstechnik 45 (1997) 517-525.
[19] , Real and Complex Analysis, 3rd Edition. McGraw-Hill (1987). | MR | Zbl
[20] , Randsteuerung von Wärmetauschern mit örtlich verteilten Parametern: Ein flachheitsbasierter Zugang. Automatisierungstechnik 48 (2000) 399-406.
[21] and, Flachheitsbasierte Steuerung eines Timoshenko-Balkens. Z. Angew. Math. Mech. 83 (2003) 119-127. | MR | Zbl
[22] , A finite strain beam formulation. The three-dimensional dynamic problem. Part one. Comp. Meths. Appl. Mech. 49 (1985) 55-70. | Zbl
[23] , Operational Calculus. Springer-Verlag (1984). | MR | Zbl
[24] , 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. | MR
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