Hostname: page-component-77f85d65b8-8wtlm Total loading time: 0 Render date: 2026-03-26T20:12:23.136Z Has data issue: false hasContentIssue false
  • English
  • Français

The Equivalence of Controlled Lagrangianand Controlled HamiltonianSystems

Published online by Cambridge University Press:  15 August 2002

Dong Eui Chang ,
Anthony M. Bloch ,
Naomi E. Leonard ,
Jerrold E. Marsden  and
Craig A. Woolsey
Dong Eui Chang
Affiliation:
Control and Dynamical Systems, California Institute of Technology, Pasadena, CA 91125, USA; dchang@cds.caltech.edu. Work partially supported by the California Institute of Technology and AFOSR grant ASOSR F49620-99-1-0190.
Anthony M. Bloch
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA; abloch@math.lsa.umich.edu. Work partially supported by NSF grants DMS 981283 and 0103895 and AFOSR.
Naomi E. Leonard
Affiliation:
Mechanical & Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA; naomi@princeton.edu. Work partially supported by NSF grant CCR-9980058, ONR grant N00014-98-1-0649 and AFOSR grant F49620-01-1-0382.
Jerrold E. Marsden
Affiliation:
Control and Dynamical Systems, California Institute of Technology, Pasadena, CA 91125, USA; marsden@cds.caltech.edu. Work partially supported by the California Institute of Technology and AFOSR grant ASOSR F49620-99-1-0190.
Craig A. Woolsey
Affiliation:
Aerospace & Ocean Engineering, Virginia Tech., Blacksburg, VA 24061, USA; cwoolsey@vt.edu.
Get access

Abstract

The purpose of this paper is to show that the method of controlledLagrangians and its Hamiltonian counterpart (based on the notionof passivity) are equivalent under rather general hypotheses. Westudy the particular case of simple mechanical control systems(where the underlying Lagrangian is kinetic minus potentialenergy) subject to controls and external forces in some detail.The equivalence makes use of almost Poisson structures (Poissonbrackets that may fail to satisfy the Jacobi identity) on theHamiltonian side, which is the Hamiltonian counterpart of a classof gyroscopic forces on the Lagrangian side.

Keywords

Controlled Lagrangiancontrolled HamiltonianenergyshapingLyapunov stabilitypassivityequivalence.

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 393 - 422
Copyright
© EDP Sciences, SMAI, 2002

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.)

Article purchase

Temporarily unavailable

References

Auckly, D., Kapitanski, L. and White, W., Control of nonlinear underactuated systems. Comm. Pure Appl. Math. 53 (2000) 354-369. (See related papers at http://www.math.ksu.edu/dav/). 3.0.CO;2-U>CrossRef
G. Blankenstein, R. Ortega and A. van Der Schaft, The matching conditions of controlled Lagrangians and IDA passivity based control. Preprint (2001).
Bloch, A.M., Chang, D.E., Leonard, N.E. and Marsden, J.E., Controlled Lagrangians and the stabilization of mechanical systems II: Potential shaping. IEEE Trans. Automat. Control 46 (2001) 1556-1571. CrossRef
A.M. Bloch, D.E. Chang, N.E. Leonard, J.E. Marsden and C.A. Woolsey, Stabilization of Mechanical Systems with Structure-Modifying Feedback. Presented at the 2001 SIAM Conf. on Control and its Applications, http://www.aoe.vt.edu/~cwoolsey/Lectures/SIAM.7.01.html
A.M. Bloch and P.E. Crouch, Representation of Dirac structures on vector space and nonlinear L-C circuits, in Proc. Symp. on Appl. Math., AMS 66 (1998) 103-118.
A.M. Bloch and P.E. Crouch, Optimal control, optimization and analytical mechanics, in Mathematical Control Theory, edited by J. Baillieul and J. Willems Springer (1998) 268-321.
Bloch, A.M., Krishnaprasad, P.S., Marsden, J.E. and Sánchez De, G. Alvarez, Stabilization of rigid body dynamics by internal and external torques. Automatica 28 (1992) 745-756. CrossRef
A.M. Bloch, N.E. Leonard and J.E. Marsden, Stabilization of mechanical systems using controlled Lagrangians, in Proc. IEEE CDC 36 (1997) 2356-2361.
Bloch, A.M., Leonard, N.E. and Marsden, J.E., Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem. IEEE Trans. Automat. Control 45 (2000) 2253-2270. CrossRef
Bloch, A.M., Leonard, N.E. and Marsden, J.E., Controlled Lagrangians and the stabilization of Euler-Poincaré mechanical systems. Int. J. Robust Nonlinear Control 11 (2001) 191-214. CrossRef
R.W. Brockett, 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.
A. Cannas Da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras. Amer. Math. Soc., Berkeley Mathematics Lecture Notes (1999).
H. Cendra, J.E. Marsden and T.S. Ratiu, Lagrangian Reduction by Stages. Memoirs of the Amer. Math. Soc. 152 (2001).
H. Cendra, J.E. Marsden and T.S. Ratiu, 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.
Courant, T., Dirac manifolds. Trans. Amer. Math. Soc. 319 (1990) 631-661. CrossRef
P.E. Crouch and A. J. van der Schaft, Variational and Hamiltonian Control Systems. Springer-Verlag, Berlin, Lecture Notes in Control and Inform. Sci. 101 (1987).
I. Dorfman,Dirac Structures and Integrability of Nonlinear Evolution Equations. Chichester: John Wiley (1993).
J. Hamberg, General matching conditions in the theory of controlled Lagrangians, in Proc. IEEE CDC (1999) 2519-2523.
J. Hamberg, 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.
Ibort, A., De Leon, M., Marrero, J.C. and Martin De, D. Diego, Dirac brackets in constrained dynamics. Fortschr. Phys. 30 (1999) 459-492. 3.0.CO;2-E>CrossRef
Jalnapurkar, S.M. and Marsden, J.E., Stabilization of relative equilibria II. Regul. Chaotic Dyn. 3 (1999) 161-179. CrossRef
Jalnapurkar, S.M. and Marsden, J.E., Stabilization of relative equilibria. IEEE Trans. Automat. Control 45 (2000) 1483-1491. CrossRef
H.K. Khalil, Nonlinear Systems. Prentice-Hall, Inc. Second Edition (1996).
Koon, W.S. and Marsden, J.E., The Poisson reduction of nonholonomic mechanical systems. Reports on Math. Phys. 42 (1998) 101-134. CrossRef
Krishnaprasad, P.S., Lie-Poisson structures, dual-spin spacecraft and asymptotic stability. Nonl. Anal. Th. Meth. and Appl. 9 (1985) 1011-1035. CrossRef
J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry. Springer-Verlag, Texts in Appl. Math. 17 (1999) Second Edition.
Maschke, B.M., van der Schaft, A.J. and Breedveld, P.C., Intrinsic Hamiltonian Formulation, An of the Dynamics of LC-Circuits. IEEE Trans. Circuits and Systems 42 (1995) 73-82. CrossRef
R. Ortega, A. Loria, P.J. Nicklasson and H. Sira-Ramirez, Passivity-based Control of Euler-Lagrange Systems. Springer-Verlag. Communication & Control Engineering Series (1998).
R. Ortega, M.W. Spong, F. Gómez-Estern and G. Blankenstein, Stabilization of underactuated mechanical systems via interconnection and damping assignment. IEEE Trans. Aut. Control (to appear).
Sánchez De, G. Alvarez, Controllability of Poisson control systems with symmetry. Amer. Math. Soc., Providence, RI., Contemp. Math. 97 (1989) 399-412. CrossRef
M.W. Spong, 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).]
van der Schaft, A.J., Hamiltonian dynamics with external forces and observations. Math. Syst. Theory 15 (1982) 145-168. CrossRef
A.J. van der Schaft, System Theoretic Descriptions of Physical Systems, Doct. Dissertation, University of Groningen; also CWI Tract #3, CWI, Amsterdam (1983).
van der Schaft, A.J., Stabilization of Hamiltonian systems. Nonlinear Anal. Theor. Meth. Appl. 10 (1986) 1021-1035. CrossRef
A.J. van der Schaft, L 2 -Gain and Passivity Techniques in Nonlinear Control. Springer-Verlag, Commun. Control Engrg. Ser. (2000).
van der Schaft, A.J. and Maschke, B., On the Hamiltonian formulation of nonholonomic mechanical systems. Rep. Math. Phys. 34 (1994) 225-233. CrossRef
Willems, J.C., System theoretic models for the analysis of physical systems. Ricerche di Automatica 10 (1979) 71-106.
C.A. Woolsey, Energy Shaping and Dissipation: Underwater Vehicle Stabilization Using Internal Rotors, Ph.D. Thesis. Princeton University (2001).
C.A. Woolsey, A.M. Bloch, N.E. Leonard and J.E. Marsden, Physical dissipation and the method of controlled Lagrangians, in Proc. of the European Control Conference (2001) 2570-2575.
C.A. Woolsey, A.M. Bloch, N.E. Leonard and J.E. Marsden, Dissipation and controlled Euler-Poincaré systems, in Proc. IEEE CDC (2001) 3378-3383.
C.A. Woolsey and N.E. Leonard, 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.
D.V. Zenkov, A.M. Bloch, N.E. Leonard and J.E. Marsden, 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.
D.V. Zenkov, A.M. Bloch and J.E. Marsden, Flat nonholonomic matching, Proc ACC 2002 (to appear).