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.

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.

DOI : https://doi.org/10.1051/cocv:2003020
Classification : 35B37,  35L55,  44A40,  93C20
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
DA  - 2003///
SP  - 419
EP  - 435
VL  - 9
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2003020/
UR  - https://zbmath.org/?q=an%3A1075.93015
UR  - https://doi.org/10.1051/cocv:2003020
DO  - 10.1051/cocv:2003020
LA  - en
ID  - COCV_2003__9__419_0
ER  - 
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] M. Fliess, J. Lévine, Ph. Martin and P. Rouchon, Flatness and defect of non-linear systems: Introductory theory and examples. Internat. J. Control 61 (1995) 1327-1361. | MR 1613557 | Zbl 0838.93022

[2] M. Fliess, Ph. Martin, N. Petit and P. Rouchon, Commande de l'équation des télégraphistes et restauration active d'un signal. Traitement du Signal 15 (1998) 619-625. | Zbl 1001.93032

[3] M. Fliess and H. Mounier, Controllability and observability of linear delay systems: An algebraic approach. ESAIM: COCV 3 (1998) 301-314. (URL: http://www.emath.fr/COCV/). | EuDML 90525 | Numdam | MR 1644427 | Zbl 0908.93013

[4] M. Fliess and H. Mounier, Tracking control and π-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 1684835 | Zbl 0918.93010

[5] 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.

[6] M. Fliess, H. Mounier, P. Rouchon and J. Rudolph, 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 0898.93018

[7] 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).

[8] F. John, Partial Differential Equations, 4th Edition. Springer-Verlag, New York (1991). | MR 1185075

[9] B. Laroche, Ph. Martin and P. Rouchon, Motion planning for the heat equation. Int. J. Robust Nonlinear Control 10 (2000) 629-643. | MR 1776232 | Zbl 1022.93025

[10] A.F. Lynch and J. Rudolph, Flachheitsbasierte Randsteuerung parabolischer Systeme mit verteilten Parametern. Automatisierungstechnik 48 (2000) 478-486.

[11] J. Mikusiński, Sur les équations différentielles du calcul opératoire et leurs applications aux équations aux dérivées partielles. Stud. Math. 12 (1951) 227-270. | MR 46550 | Zbl 0044.12701

[12] J. Mikusiński, Operational Calculus, Vol. 1. Pergamon, Oxford & PWN, Warszawa (1983). | MR 737380 | Zbl 0532.44003

[13] J. Mikusiński and Th.K. Boehme, Operational Calculus, Vol. 2. Pergamon, Oxford & PWN, Warszawa (1987). | MR 902363 | Zbl 0643.44005

[14] 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.

[15] N. Petit and P. Rouchon, Motion planning for heavy chain systems. SIAM J. Control Optim. 40 (2001) 275-495. | MR 1806175 | Zbl 0967.93073

[16] 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. | MR 1893517

[17] I.G. Petrovskij, Über das Cauchysche Problem für Systeme von partiellen Differentialgleichungen. Mat. Sb. 2 (1937) 815-866. | JFM 63.0466.03 | Zbl 0018.40503

[18] R. Rothfuß, J. Rudolph and M. Zeitz, Flachheit: Ein neuer Zugang zur Steuerung und Regelung nichtlinearer Systeme. Automatisierungstechnik 45 (1997) 517-525.

[19] W. Rudin, Real and Complex Analysis, 3rd Edition. McGraw-Hill (1987). | MR 924157 | Zbl 0925.00005

[20] J. Rudolph, Randsteuerung von Wärmetauschern mit örtlich verteilten Parametern: Ein flachheitsbasierter Zugang. Automatisierungstechnik 48 (2000) 399-406.

[21] J. Rudolph and F. Woittennek, Flachheitsbasierte Steuerung eines Timoshenko-Balkens. Z. Angew. Math. Mech. 83 (2003) 119-127. | MR 1960115 | Zbl 1090.74033

[22] J.C. Simo, A finite strain beam formulation. The three-dimensional dynamic problem. Part one. Comp. Meths. Appl. Mech. 49 (1985) 55-70. | Zbl 0583.73037

[23] K. Yosida, Operational Calculus. Springer-Verlag (1984). | MR 752699 | Zbl 0542.44001

[24] 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. | MR 1607161

Cité par Sources :