The purpose of this paper is to show that the method of controlled lagrangians and its hamiltonian counterpart (based on the notion of passivity) are equivalent under rather general hypotheses. We study the particular case of simple mechanical control systems (where the underlying lagrangian is kinetic minus potential energy) subject to controls and external forces in some detail. The equivalence makes use of almost Poisson structures (Poisson brackets that may fail to satisfy the Jacobi identity) on the hamiltonian side, which is the hamiltonian counterpart of a class of gyroscopic forces on the lagrangian side.
Mots-clés : controlled lagrangian, controlled hamiltonian, energy shaping, Lyapunov stability, passivity, equivalence
@article{COCV_2002__8__393_0,
author = {Chang, Dong Eui and Bloch, Anthony M. and Leonard, Naomi E. and Marsden, Jerrold E. and Woolsey, Craig A.},
title = {The equivalence of controlled lagrangian and controlled hamiltonian systems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {393--422},
publisher = {EDP-Sciences},
volume = {8},
year = {2002},
doi = {10.1051/cocv:2002045},
mrnumber = {1932957},
zbl = {1070.70013},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2002045/}
}
TY - JOUR AU - Chang, Dong Eui AU - Bloch, Anthony M. AU - Leonard, Naomi E. AU - Marsden, Jerrold E. AU - Woolsey, Craig A. TI - The equivalence of controlled lagrangian and controlled hamiltonian systems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2002 SP - 393 EP - 422 VL - 8 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2002045/ DO - 10.1051/cocv:2002045 LA - en ID - COCV_2002__8__393_0 ER -
%0 Journal Article %A Chang, Dong Eui %A Bloch, Anthony M. %A Leonard, Naomi E. %A Marsden, Jerrold E. %A Woolsey, Craig A. %T The equivalence of controlled lagrangian and controlled hamiltonian systems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2002 %P 393-422 %V 8 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2002045/ %R 10.1051/cocv:2002045 %G en %F COCV_2002__8__393_0
Chang, Dong Eui; Bloch, Anthony M.; Leonard, Naomi E.; Marsden, Jerrold E.; Woolsey, Craig A. The equivalence of controlled lagrangian and controlled hamiltonian systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 393-422. doi : 10.1051/cocv:2002045. http://www.numdam.org/articles/10.1051/cocv:2002045/
[1] , and, Control of nonlinear underactuated systems. Comm. Pure Appl. Math. 53 (2000) 354-369. (See related papers at http://www.math.ksu.edu/~dav/). | MR | Zbl
[2] , and, The matching conditions of controlled Lagrangians and IDA passivity based control. Preprint (2001). | MR | Zbl
[3] ,, and, Controlled Lagrangians and the stabilization of mechanical systems II: Potential shaping. IEEE Trans. Automat. Control 46 (2001) 1556-1571. | MR | Zbl
[4] ,,, and, Stabilization of Mechanical Systems with Structure-Modifying Feedback
[5] and, Representation of Dirac structures on vector space and nonlinear L-C circuits, in Proc. Symp. on Appl. Math., AMS 66 (1998) 103-118. | MR
[6] and, Optimal control, optimization and analytical mechanics, in Mathematical Control Theory, edited by J. Baillieul and J. Willems Springer (1998) 268-321. | MR | Zbl
[7] ,, and, Stabilization of rigid body dynamics by internal and external torques. Automatica 28 (1992) 745-756. | MR | Zbl
[8] , and, Stabilization of mechanical systems using controlled Lagrangians, in Proc. IEEE CDC 36 (1997) 2356-2361.
[9] , and, Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem. IEEE Trans. Automat. Control 45 (2000) 2253-2270. | MR | Zbl
[10] , and, Controlled Lagrangians and the stabilization of Euler-Poincaré mechanical systems. Int. J. Robust Nonlinear Control 11 (2001) 191-214. | Zbl
[11] , Control theory and analytical mechanics, in 1976 Ames Research Center (NASA) Conference on Geometric Control Theory, edited by R. Hermann and C. Martin. Math Sci Press, Brookline, Massachusetts, Lie Groups: History, Frontiers, and Applications VII (1976) 1-46. | MR | Zbl
[12] and, Geometric Models for Noncommutative Algebras. Amer. Math. Soc., Berkeley Mathematics Lecture Notes (1999). | MR | Zbl
[13] , and, Lagrangian Reduction by Stages. Memoirs of the Amer. Math. Soc. 152 (2001). | MR
[14] , and, Geometric mechanics, Lagrangian reduction and nonholonomic systems, in Mathematics Unlimited-2001 and Beyond, edited by B. Enquist and W. Schmid. Springer-Verlag, New York (2001) 221-273. | MR | Zbl
[15] , Dirac manifolds. Trans. Amer. Math. Soc. 319 (1990) 631-661. | MR | Zbl
[16] and, Variational and Hamiltonian Control Systems. Springer-Verlag, Berlin, Lecture Notes in Control and Inform. Sci. 101 (1987). | Zbl
[17] ,Dirac Structures and Integrability of Nonlinear Evolution Equations. Chichester: John Wiley (1993). | MR
[18] , General matching conditions in the theory of controlled Lagrangians, in Proc. IEEE CDC (1999) 2519-2523.
[19] , Controlled Lagrangians, symmetries and conditions for strong matching, in Lagrangian and Hamiltonian Methods for Nonlinear Control: A Proc. Volume from the IFAC Workshop, edited by N.E. Leonard and R. Ortega. Pergamon (2000) 57-62.
[20] ,, and, Dirac brackets in constrained dynamics. Fortschr. Phys. 30 (1999) 459-492. | MR | Zbl
[21] and, Stabilization of relative equilibria II. Regul. Chaotic Dyn. 3 (1999) 161-179. | MR | Zbl
[22] and, Stabilization of relative equilibria. IEEE Trans. Automat. Control 45 (2000) 1483-1491. | MR | Zbl
[23] , Nonlinear Systems. Prentice-Hall, Inc. Second Edition (1996). | Zbl
[24] and, The Poisson reduction of nonholonomic mechanical systems. Reports on Math. Phys. 42 (1998) 101-134. | MR | Zbl
[25] , Lie-Poisson structures, dual-spin spacecraft and asymptotic stability. Nonl. Anal. Th. Meth. and Appl. 9 (1985) 1011-1035. | Zbl
[26] and, Introduction to Mechanics and Symmetry. Springer-Verlag, Texts in Appl. Math. 17 (1999) Second Edition. | MR | Zbl
[27] , and, An Intrinsic Hamiltonian Formulation of the Dynamics of LC-Circuits. IEEE Trans. Circuits and Systems 42 (1995) 73-82. | MR | Zbl
[28] ,, and, Passivity-based Control of Euler-Lagrange Systems. Springer-Verlag. Communication & Control Engineering Series (1998).
[29] ,, and, Stabilization of underactuated mechanical systems via interconnection and damping assignment. IEEE Trans. Aut. Control (to appear).
[30] , Controllability of Poisson control systems with symmetry. Amer. Math. Soc., Providence, RI., Contemp. Math. 97 (1989) 399-412. | MR | Zbl
[31] , Underactuated mechanical systems, in Control Problems in Robotics and Automation, edited by B. Siciliano and K.P. Valavanis. Spinger-Verlag, Lecture Notes in Control and Inform. Sci. 230. [Presented at the International Workshop on Control Problems in Robotics and Automation: Future Directions Hyatt Regency, San Diego, California (1997).]
[32] , Hamiltonian dynamics with external forces and observations. Math. Syst. Theory 15 (1982) 145-168. | MR | Zbl
[33] , System Theoretic Descriptions of Physical Systems, Doct. Dissertation, University of Groningen; also CWI Tract #3, CWI, Amsterdam (1983). | MR | Zbl
[34] , Stabilization of Hamiltonian systems. Nonlinear Anal. Theor. Meth. Appl. 10 (1986) 1021-1035. | MR | Zbl
[35] , -Gain and Passivity Techniques in Nonlinear Control. Springer-Verlag, Commun. Control Engrg. Ser. (2000). | Zbl
[36] and, On the Hamiltonian formulation of nonholonomic mechanical systems. Rep. Math. Phys. 34 (1994) 225-233. | MR | Zbl
[37] , System theoretic models for the analysis of physical systems. Ricerche di Automatica 10 (1979) 71-106. | MR | Zbl
[38] , Energy Shaping and Dissipation: Underwater Vehicle Stabilization Using Internal Rotors, Ph.D. Thesis. Princeton University (2001).
[39] ,, and, Physical dissipation and the method of controlled Lagrangians, in Proc. of the European Control Conference (2001) 2570-2575.
[40] ,, and, Dissipation and controlled Euler-Poincaré systems, in Proc. IEEE CDC (2001) 3378-3383.
[41] and, Modification of Hamiltonian structure to stabilize an underwater vehicle, in Lagrangian and Hamiltonian Methods for Nonlinear Control: A Proc. Volume from the IFAC Workshop edited by N.E. Leonard and R. Ortega. Pergamon (2000) 175-176.
[42] ,, and, Matching and stabilization of the unicycle with rider, Lagrangian and Hamiltonian Methods for Nonlinear Control: A Proc. Volume from the IFAC Workshop, edited by N.E. Leonard and R. Ortega. Pergamon (2000) 177-178.
[43] , and, Flat nonholonomic matching, Proc ACC 2002 (to appear). | MR
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