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2001 | 11 | 1 | 165-188
Tytuł artykułu

Motion planning, equivalence, infinite dimensional systems

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Motion planning, i.e., steering a system from one state to another, is a basic question in automatic control. For a certain class of systems described by ordinary differential equations and called flat systems (Fliess et al. 1995; 1999a), motion planning admits simple and explicit solutions. This stems from an explicit description of the trajectories by an arbitrary time function, the flat output, and a finite number of its time derivatives. Such explicit descriptions are related to old problems on Monge equations and equivalence investigated by Hilbert and Cartan. The study of several examples (the car with -trailers and the non-holonomic snake, pendulums in series and the heavy chain, the heat equation and the Euler-Bernoulli flexible beam) indicates that the notion of flatness and its underlying explicit description can be extended to infinite-dimensional systems. As in the finite-dimensional case, this property yields simple motion planning algorithms via operators of compact support. For the non-holonomic snake, such operators involve non-linear delays. For the heavy chain, they are defined via distributed delays. For heat and Euler-Bernoulli systems, their supports are reduced to a point and their definition domain coincides with the set of Gevrey functions of order 2.
Rocznik
Tom
11
Numer
1
Strony
165-188
Opis fizyczny
Daty
wydano
2001
otrzymano
2000-09-01
poprawiono
2001-01-01
Twórcy
  • Ecole des Mines de Paris, Centre Automatique et Systemes, 60, bd. Saint-Michel, 75272 Paris, Cedex 06, France
Bibliografia
  • Aoustin Y., Fliess M., Mounier H., Rouchon P. and Rudolph J. (1997): Theory and practice in the motion planning and control of a flexible robot arm using Mikusiński operators. -Proc. 5th IFAC Symp. Robot Control, Nantes, France, pp.287-293.
  • Campion G., d'Andrea Novel G. and Bastin G. (1996): Structural properties and classification of kinematic and dynamic models of wheeled mobile robots. -IEEE Trans. Robot. Automat., Vol.12, No.1, pp.47-62.
  • Cartan E. (1914): Sur l'equivalence absolue de certains systèmes d'equations differentielles et sur certaines familles decourves. -Bull. Soc. Math. France, Vol.42, pp.12-48, also in: Ouvres Complètes, Part II, Vol.2, (1984), Paris: CNRS,pp.1133-1168.
  • Cartan E. (1915): Sur l'integration de certains systèmes indetermines d'equations differentielles. -J. fur reine und angew. Math., Vol.145, pp.86-91, also in: Ouvres Complètes, Part II, Vol.2, (1984), Paris: CNRS,pp.1164-1174.
  • Dubois F., Petit N. and Rouchon P. (1999): Motion planing and nonlinear simulations for a tank containing a fluid. - Proc.s European Control Conf., Karlsruhe, Germany, published on CD-ROM.
  • Fliess M., Levine J., Martin Ph. and Rouchon P. (1995): Flatness and defect of nonlinear systems: introductory theory and examples. -Int. J. Contr., Vol.61, No.6, pp.1327-1361.
  • Fliess M., Mounier H., Rouchon P. and Rudolph J. (1996): Systèmes lineaires sur les operateurs de Mikusiński et commande d'une poutre flexible. -ESAIM Proc. Conf. Elasticite, viscolelasticite et contrôle optimal, 8ème entretiens du centre Jacques Cartier, Lyon, France, pp.157-168.
  • Fliess M., Levine J., Martin P., Ollivier F. and Rouchon P. (1997): Controlling nonlinear systems by flatness, In: Systems and Control in the Twenty-First Century, (C.I. Byrnes, B.N. Datta, D.S. Gilliam and C.F. Martin, Eds.). - Boston: Birkhauser.
  • Fliess M., Mounier H., Rouchon P. and Rudolph J. (1998a): A distributed parameter approach to the control of a tubular reactor: A multi-variable case. - Proc. Conf. Decision and Control, Tampa, U.S.A., pp.439-442.
  • Fliess M., Mounier H., Rouchon P. and Rudolph J. (1998b): Tracking control of a vibrating string with an interior mass viewed as delay system. -ESAIM: Contr. Optim. Calc. Var, t www.eamth.frcocv, Vol.3, pp.315-321.
  • Fliess M., Levine J., Martin Ph. and Rouchon P. (1999a): A Lie-Backlund approach to equivalence and flatness of nonlinear systems. -IEEE Trans. Automat. Contr., Vol.44, No.5, pp.922-937.
  • Fliess M., Martin Ph., Petit N. and Rouchon P. (1999b): Active signal restoration for the telegraph equation. -Proc. Conf. Decision and Control, Phenix, U.S.A., pp.1107-1111.
  • Gelfand I.M. and Shilov G.E. (1964): Les Distributions, T.3. -Paris: Dunod.
  • Giaro A., Kumpera A. and Ruiz C. (1978): Sur la lecture d'un resultat d'Elie Cartan. -C.R. Acad. Sc. Paris, Serie A, Vol.287, pp.241-244.
  • Goursat E. (1923): Leccons sur le Problème de Pfaff. - Paris: Hermann.
  • Hilbert D. (1912): Uber den Begriff der Klasse von Differentialgleichungen. - Math. Ann., Vol.73, pp.95-108, also in: Gesammelte Abhandlungen, Vol.III, (1965) New York: Chelsea, pp.81-93.
  • Laroche B. and Martin Ph. (2000): Motion planning for a 1-d diffusion equation using a brunovsky-like decomposition. - Proc. MTNS 2000, Perpignan, France, published on CD-ROM.
  • Laroche B., Martin Ph. and Rouchon P. (1998): Motion planning for a class of partial differential equations with boundary control. -Proc. Conf. Control and Decision, Tampa, U.S.A., pp.3494-3497.
  • Lemon C. and Hause J.E. (1994): Design and initial flight test of the Champagne Flyer. -Proc. 33nd IEEE Conf. Decision and Control, Lake Buena Vista, U.S.A., pp.3852-3853.
  • Lenoir Y., Martin Ph. and Rouchon P. (1998): 2kπ, the juggling robot. -Proc. Conf. Control and Decision, Tampa, U.S.A., pp.1995-2000.
  • Martin Ph. (1992): Contribution à l'etude des systèmes diffèrentiellement plats. - Ph.D. Thesis, Ecole des Mines de Paris.
  • Martin Ph., Devasia S. and Paden B. (1996): A different look at output tracking: Control of a VTOL aircraft. -Automatica, Vol.32, No.1, pp.101-107.
  • Martin Ph., Murray R. and Rouchon P. (1997): Flat systems. -Proc. 4th European Control Conf., Brussels, Belgium, Plenary Lectures and Mini-courses, pp.211-264.
  • Martin Ph. and Rouchon P. (1994): Feedback linearization and driftless systems. -Math. Contr. Signal Syst., Vol.7, No.7, pp.235-254.
  • Mounier H. (1995): Proprietes structurelles des systèmes lineaires à retards: Aspects theoriques et pratiques. - Ph.D. Thesis, Universite Paris Sud, Orsay.
  • Mounier H. and Rudolph J. (1998): Flatness based control of nonlinear delay systems: A chemical reactor example. -Int. J. Contr., Vol.71, pp.871-890.
  • Mounier H., Rudolph J., Petitot M. and Fliess M. (1995): A flexible rod as a linear delay system. -Proc. 3rd European Control Conf., Rome, Italy, pp.3676-3681.
  • Mounier H., Rouchon P. and Rudolph J. (1996): π-freeness of a long electric line. -Proc. CESA '96 IMACS Multiconf., Lille, France, pp.28-29.
  • Murray R.M. (1994): Nilpotent bases for a class on nonintegrable distributions with applications to trajectory generation for nonholonomic systems. -Math. Contr. Signal Syst., Vol.7, No.1, pp.58-75.
  • Murray R.M. (1995): Caltech ducted fan reference manual. - Techn. Rep., California Institute of Technology, Division of Engineering and Applied Science.
  • Murray R.M. (1996): Trajectory generation for a towed cable flight control system. -Proc. IFAC World Congress, San Francisco, U.S.A., pp.395-400.
  • Murray R.M. and Sastry S.S. (1993): Nonholonomic motion planning: Steering using sinusoids. -IEEE Trans. Automat. Contr., Vol.38, No.5, pp.700-716.
  • Pasillas-Lepine W. (2000): Systèmes de contact et structures de Goursat: Theorie et application au contrôle des systèmes mecaniques nonholonomes. - Ph.D. Thesis, Universite de Rouen, France.
  • Petit N., Creff Y. and Rouchon P. (1998): Motion planning for two classes of nonlinear systems with delays depending on the control. - Proc. Conf. Decision and Control, Tampa, U.S.A., pp.1007-1011.
  • Petit N. and Rouchon P. (2001): Motion planning for heavy chain systems. -Wave 2000, (to appear).
  • Pommaret J.F. (1978): Systems of Partial Differential Equations and Lie Pseudogroups. -New York: Gordon and Breach.
  • Pommaret J.F. (1995): Dualite differentielle et applications. -C.R. Acad. Sci. Paris, Serie I, Vol.320, pp.1225-1230.
  • Ramis J.P. (1979): Devissage Gevrey. -Asterisque, Vol.59-60, pp.173-204.
  • Rothfus R., Becker U. and Rudolph R. (2000): Controlling a solenoid valve: A distributed parameter approach. -Proc. MTNS 2000, Perpignan, France, published on CD-ROM.
  • Rouchon P., Fliess M., Levine J. and Martin Ph. (1993a): Flatness and motion planning: the car with n-trailers. -Proc. European Control Conf., Groningen, the Netherlands, pp.1518-1522.
  • Rouchon P., Fliess M., Levine J. and Martin Ph. (1993b): Flatness, motion planning and trailer systems. -Proc. 32nd IEEE Conf. Decision and Control, San Antonio, U.S.A., pp.2700-2705.
  • Rouchon P. and Rudolph J. (2000): Reacteurs chimiques differentiellement plats: Planification et suivi de trajectoires. - Part II, Chapter III.f of the book Automatique et procedes chimiques (J.P. Corriou, Ed.), Paris: Hermès, (in print).
  • Rouchon P. and Rudolph J. (1999): Invariant tracking and stabilization: Problem formulation and examples, In: Stability and Stabilization of Nonlinear Systems(D. Acyels, F. Lamnabhi-Lagarrigve, A. van der Schaft, Eds.). - Berlin: Springer, pp.261-273.
  • Tilbury D.M. (1994): Exterior differential systems and nonholonomic motion planning. - Ph.D. Thesis, University of California, Berkeley, Memorandum No. UCBERL M9490.
  • Valiron G. (1950): Equations Fonctionnelles, 2nd Ed. -Paris: Masson.
  • von Weber E. (1898): Zur Invarianten Theorie der Systeme Pfaff'scher Gleigungen. -Berichte Verhandlungen der Koniglich Sachsichen Gesellshaft der Wissenschaften Mathematische-Physicalische Klasse, Leipzig, Vol.50, pp.207-229.
Typ dokumentu
Bibliografia
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