Reduction of controlled lagrangian and hamiltonian systems with symmetry

SIAM Journal on Control and Optimization

Volumn 43, Issue 1, 2005, Pages 277-300

  Chang, Dong Eui ^a   Marsden, Jerrold E ^b  

^a PSL RESEARCH UNIVERSITY   (France)

^b CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Controlled Hamiltonian systems; Controlled Lagrangian systems; Equivalence; Reduction

Indexed keywords

CONTROLLED HAMILTONIAN SYSTEMS; CONTROLLED LAGRANGIAN SYSTEMS; EQUIVALENCE; REDUCTION THEORY;

ASYMPTOTIC STABILITY; CONTROL SYSTEMS; HAMILTONIANS; LAGRANGE MULTIPLIERS; POISSON DISTRIBUTION; ROTORS; SPACECRAFT;

CONTROL THEORY;

EID: 14244253962     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012902412951     Document Type: Article

References (36)

1. R. ABRAHAM AND J. E. MARSDEN, Foundations of Mechanics, 2nd ed., Addison-Wesley, Reading, MA, 1978.
2. R. ABRAHAM, J. E. MARSDEN, AND T. S. RATIU, Manifolds, Tensor Analysis, and Applications, Appl. Math. 75, 2nd ed., Springer-Verlag, New York, 1988.
3. D. AUCKLY, L. KAPITANSKI, AND W. WHITE, Control of nonlinear underactuated systems, Comm. Pure Appl. Math., 53 (2000), pp. 354-369.
4. D. AUCKLY AND L. KAPITANSKI, On the λ-equations for matching control laws, SIAM J. Control Optim., 41 (2002), pp. 1372-1388.
5. G. BLANKENSTEIN, R. ORTEGA, AND A. J. VAN DER SCHAFT, The matching conditions of controlled Lagrangians and interconnection and damping assignment passivity based control, Internat. J. Control, 75 (2002), pp. 645-665.
6. G. BLANKENSTEIN AND A. J. VAN DER SCHAFT, Symmetry and reduction in implicit generalized Hamiltonian systems, Rep. Math. Phys., 47 (2001), pp. 57-100.
7. A. M. BLOCH, D. E. CHANG, N. E. LEONARD, AND J. E. MARSDEN, Controlled Lagrangians and the stabilization of mechanical systems II: Potential shaping, IEEE Trans. Automat. Control, 46 (2001), pp. 1556-1571.
8. A. M. BLOCH, D. E. CHANG, N. E. LEONARD, J. E. MARSDEN, AND C. WOOLSEY, Asymptotic stabilization of Euler-Poincaré mechanical systems, in Lagrangian and Hamiltonian Methods for Nonlinear Control: A Proceedings Volume from the IFAC Workshop, N. E. Leonard and R. Ortega, eds., Pergamon, Elmsford, NY, 2000, pp. 51-56.
9. A. M. BLOCH, P. S. KRISHNAPRASAD, J. E. MARSDEN, AND G. SÁNCHEZ DE ALVAREZ, Stabilization of rigid body dynamics by internal and external torques, Automatica, 28 (1992), pp. 745-756.
10. A. M. BLOCH, P. S. KRISHNAPRASAD, J. E. MARSDEN, AND T. RATIU, The Euler-Poincaré Equations and double bracket dissipation, Comm. Math. Phys., 175 (1996), pp. 1-42.
11. A. M. BLOCH, N. E. LEONARD, AND J. E. MARSDEN, Stabilization of mechanical systems using controlled Lagrangians, in Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, CA, 1997, pp. 2356-2361.
12. A. M. BLOCH, N. E. LEONARD, AND J. E. MARSDEN, Matching and stabilization by the method of controlled Lagrangians, in Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, FL, 1998, pp. 1446-1451.
13. A. M. BLOCH, N. E. LEONARD, AND J. E. MARSDEN, Stabilization of the pendulum on a rotor arm by the method of controlled Lagrangians, in Proceedings of the International Conference on Robotics and Automation, Detroit, MI, IEEE, Piscataway, NJ, 1999, pp. 500-505.
14. A. M. BLOCH, N. E. LEONARD, AND J. E. MARSDEN, Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem, IEEE Trans. Automat. Control, 45 (2000), pp. 2253-2270.
15. A. M. BLOCH, N. E. LEONARD, AND J. E. MARSDEN, Controlled Lagrangians and the stabilization of Euler-Poincaré mechanical systems, Internat. J. Robust Nonlinear Control, 11 (2001), pp. 191-214.
16. W. M. BOOTHBY, An Introduction to Differentiate Manifolds and Riemannian Geometry, Academic Press, New York, 1986.
17. H. CENDRA, J. E. MARSDEN, AND T. S. RATIU, Lagrangian reduction by stages, Mem. Amer. Math. Soc., 152 (2001), number 722.
18. H. CENDRA, J. E. MARSDEN, AND T. S. RATIU, Geometric mechanics, Lagrangian reduction and nonholonomic systems, in Mathematics Unlimited - 2001 and Beyond, B. Enquist and W. Schmid, eds., Springer-Verlag, New York, 2001, pp. 221-273.
19. D. E. CHANG, Controlled Lagrangian and Hamiltonian Systems, Ph.D. Thesis, Control and Dynamical Systems Department, California Institute of Technology, Pasadena, CA, 2002.
20. D. E. CHANG, A. M. BLOCH, N. E. LEONARD, J. E. MARSDEN, AND C. WOOLSEY, Equivalence of controlled Lagrangian and controlled Hamiltonian systems, ESAIM Control Optim. Calc. Var., 8 (2002), pp. 393-422.
21. D. E. CHANG, AND J. E. MARSDEN, Asymptotic stabilization of the heavy top using controlled Lagrangians, in Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, 2000, pp. 269-273.
22. J. HAMBERG, General matching conditions in the theory of controlled Lagrangians, in Proceedings of the 39th IEEE Conference on Decision and Control, Phoenix, AZ, 1999, pp. 2519-2523.
23. J. HAMBERG, Controlled Lagrangians, symmetries and conditions for strong matching, in Lagrangian and Hamiltonian Methods for Nonlinear Control: A Proceedings Volume from the IFAC Workshop, N. E. Leonard and R. Ortega, eds., Pergamon, Elmsford, NY, 2000, pp. 62-67.
24. D. D. HOLM, J. E. MARSDEN, AND T. S. RATIU, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), pp. 1-81.
25. S. KOBAYASHI AND K. NOMIZU, Foundations of Differential Geometry, John Wiley & Sons, New York, 1963.
26. P. S. KRISHNAPRASAD, Lie-Poisson structures, dual-spin spacecraft and asymptotic stability, Nonlinear Anal., 9 (1985), pp. 1011-1035.
27. J. E. MARSDEN, Lectures on Mechanics, London Math. Soc. Lecture Note Ser. 174, Cambridge University Press, Cambridge, UK, 1992.
28. J. E. MARSDEN AND T. S. RATIU, Introduction to Mechanics and Symmetry, 2nd ed., Texts in Appl. Math. 17, Springer-Verlag, New York, 1999.
29. J. E. MARSDEN AND J. SCHEURLE, Lagrangian reduction and the double spherical pendulum, Z. Angew. Math. Phys., 44 (1993), pp. 17-43.
30. J. E. MARSDEN AND J. SCHEURLE, The reduced Euler-Lagrange equations, Fields Inst. Commun., 1 (1993), pp. 139-164.
31. R. MONTGOMERY, J. E. MARSDEN, AND T. S. RATIU, Gauged Lie-Poisson structures, Contemp. Math., 28 (1984), pp. 101-114.
32. 2-Gain and Passivity Techniques in Nonlinear Control, Communication & Control Engineering Series, Springer-Verlag, New York, 2000.
33. L. S. WANG AND P. S. KRISHNAPRASAD, Gyroscopic control and stabilization, J. Nonlinear Sci., 2 (1992), pp. 367-415.
34. C. WOOLSEY AND N. E. LEONARD, Underwater vehicle stabilization by internal rotors, in Proceedings of the American Control Conference, San Diego, CA, 1999.
35. H. YOSHIMURA AND J. E. MARSDEN, Variational Principles, Dirac Structure and Implicit Lagrangian Systems, preprint.
36. D. V. ZENKOV, A. M. BLOCK, AND J. E. MARSDEN, The Lyapunov-Malkin theorem and stabilization of the unicycle with rider, Systems Control Lett., 45 (2002), pp. 293-302.