Abstract
In this paper we describe various aspects of Jerry Marsden’s work and influence on control theory and its connections with mechanics. In particular we trace the role of his key ideas on reduction and symmetries in the setting of nonlinear control.
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References
Åströom, K. J. and K. Furuta [1996], Swinging up a pendulum by energy control, IFAC World Congress, San Francisco 13.
Abraham, R. and J. E. Marsden [1978], Foundations of Mechanics, Second Edition, Addison-Wesley.
Auckly, D., L. Kapitanski, and W. White [2000], Control of nonlinear underactuated systems, Comm. Pure Appl. Math., 53, 354–369.
Baillieul, J. [1999], Matching conditions and geometric invariants for second-order control systems, in Proc. IEEE Conf. Decision and Control 38, 1664–1670.
Bloch, A. M., D. E. Chang, N. E. Leonard, and J. E. Marsden [2001], Controlled Lagrangians and the stabilization of mechanical systems II: Potential shaping, IEEE Trans. Automatic Control 46, 1556–1571.
Bloch, A. M., D. E. Chang, N. E. Leonard and J. E. Marsden [2000], Potential and kinetic shaping for control of underactuated mechanical systems, in Proc. American Control Conf., 3913–3917.
Bloch, A. M., D. E. Chang, N. E. Leonard, J. E. Marsden and C. A. Woolsey [2000], 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, 51–56.
Bloch, A. M. and P. E. Crouch [1999], Optimal control, optimization and analytical mechanics, in Mathematical Control Theory (J. Baillieul and J. Willems eds.), 268–321, Springer.
Bloch, A. M., P. S. Krishnaprasad, J. E. Marsden, and R. Murray [1996], Nonholonomic mechanical systems with symmetry, Arch. Rat. Mech. An., 136, 21–99.
Bloch, A. M., P. S. Krishnaprasad, J. E. Marsden, and G. Sánchez de Alvarez [1992], Stabilization of rigid body dynamics by internal and external torques, Automatica 28, 745–756.
Bloch, A. M., N. E. Leonard and J. E. Marsden [1997], Stabilization of mechanical systems using controlled Lagrangians, Proc. IEEE Conf. Decision and Control 36, 2356–2361.
Bloch, A. M., N. E. Leonard and J. E. Marsden [1998], Matching and stabilization by the method of controlled Lagrangians, Proc. IEEE Conf. Decision and Control 37, 1446–1451.
Bloch, A. M., N. E. Leonard and J. E. Marsden [1999a], Stabilization of the pendulum on a rotor arm by the method of controlled Lagrangians, Proc. IEEE Int. Conf. Robotics and Automation, 500–505.
Bloch, A. M., N. E. Leonard and J. E. Marsden [1999b], Potential shaping and the method of controlled Lagrangians, Proc. IEEE Conf. Decision and Control 38, 1653–1657.
Bloch, A. M., N. E. Leonard and J. E. Marsden [2000], Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem, IEEE Trans. Automatic Control 45, 2253–2270
Bloch, A. M., N. E. Leonard and J. E. Marsden [2001], Controlled Lagrangians and the stabilization of Euler-Poincaré mechanical systems, Int. J. Nonlinear and Robust Control 11, 191–214.
Bloch, A. M., J. E. Marsden and G. Sánchez de Alvarez [1997], Stabilization of relative equilibria of mechanical systems with symmetry, Current and Future Directions in Applied Mathematics, (M. Alber, B. Hu, and J. Rosenthal, eds.), Birkhäuser, 43–64.
Bloch, A. M. and J. E. Marsden [1989], Controlling homoclinic orbits, Theoretical and Computational Fluid Dynamics 1, 179–190.
Brockett, R. W. [1976], Control theory and analytical mechanics, in 1976 Ames Research Center NASA Conference on Geometric Control Theory, (R. Hermann and C. Martin, eds.), Lie Groups: History Frontiers and Applications, 7, Math. Sci. Press, Brookline, Mass., USA.
Grizzle, J. W. and S. Marcus [1985], The structure of nonlinear control systems posessing symmetries, IEEE Trans. Automatic Control 30, 248–258.
Hamberg, J. [1999], General matching conditions in the theory of controlled Lagrangians. Proc. IEEE Conf. Decision and Control 38, 2519–2523.
Holmes, P., J. Jenkins, and N. E. Leonard [1998], Dynamics of the Kirchhoff equations I: Coincident centers of gravity and buoyancy, Physica D, 118, 311–342.
Jalnapurkar, S. M. and J. E. Marsden [1999], Stabilization of relative equilibria II, Reg. and Chaotic Dyn. 3, 161–179.
Jalnapurkar, S. M. and J. E. Marsden [2000], Stabilization of relative equilibria, IEEE Trans. Automatic Control 45, 1483–1491.
Koon, W. S. and J. E. Marsden [1997], Optimal control for holonomic and nonholonomic mechanical systems with symmetry and Lagrangian reduction, SIAM J. Control and Optim., 35, 901–929.
Koon, W-S., M. Lo, J. E. Marsden and S. D. Ross [2000], Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics, Chaos, 10, 427–469.
Krishnaprasad, P. S. [1985], Lie-Poisson structures, dual-spins pacecraft and asymptotic stability, Nonl. Anal. Th. Meth. and Appl. 9, 1011–1035.
Leonard, N. E. [1997a], Stability of a bottom-heavy underwater vehicle, Automatica 33, 331–346.
Leonard, N. E. [1997b], Stabilization of underwater vehicle dynamics with symmetry-breaking potentials, Systems and Control Letters 32, 35–42.
Leonard, N. E. and J. E. Marsden [1997], Stability and drift of underwater vehicle dynamics: Mechanical systems with rigid motion symmetry, Physica D 105, 130–162.
Leonard, N. E. and C. A. Woolsey [1998], Internal actuation for intelligent underwater vehicle control, Proc. 10th Yale Workshop on Adaptive and Learning Systems, 295–300.
Lewis, A. and R. Murray [1997], Configuration controllability of simple mechanical systems, SIAM. J. Control and Optimization 41 555–574.
Marsden, J. E. [1992], Lectures on Mechanics London Mathematical Society Lecture note series. 174, Cambridge University Press.
Marsden, J. E. and T. S. Ratiu [1999], Introduction to Mechanics and Symmetry. Texts in Applied Mathematics, 17, Springer-Verlag, 1994. 2nd Ed.
Marsden, J. E. and A. Weinstein [1974], Reduction of symplectic manfolds with symmetry, Rep. Math. Phys 5, 121–130.
Nijmeijer, H. and A. van der Schaft [1990], Nonlinear dynamical control systems, Springer-Verlag, New York.
Ortega, R., A. Loria, P. J. Nicklasson and H. Sira-Ramirez [1998], Passivity-based Control of Euler-Lagrange Systems, Springer-Verlag, Communication & Control Engineering Series.
Ortega, R., A. van der Schaft and B. Maschke [1999], Stabilization of portcontrolled Hamiltonian systems via energy balancing, in Stability and Stabilization of Nonlinear Systems, Springer, London, 239–260.
Ortega, R., A. van der Schaft, B. Maschke, and G. Escobar [1999], Energy-shaping of port-controlled Hamiltonian systems by interconnection, in Proc. IEEE Conf. Decision and Control 38, 1646–1651.
Sánchez de Alvarez, G. [1986], Ph.D Thesis, Berkeley
Simó, J. C., D. K. Lewis and J. E. Marsden [1991], Stability of relative equilibria I: The reduced energy momentum method, Arch. Rat. Mech. Anal. 115, 15–59.
van der Schaft, A. J. [1981], Symmetries and conservation laws for Hamiltonian systems with inputs and outputs: A generalization of Noethe’s theorem, Syst. Contr. Letters 1, 108–115.
van der Schaft, A. J. [1983], System theoretic descriptions of physical systems, Doct. Dissertation, University of Groningen, also CWI Tract # 3, CWI, Amsterdam.
van der Schaft, A. J. [1986], Stabilization of Hamiltonian systems, Nonlinear Analysis, Theory, Methods and Applications, 10, 1021–1035.
Wang, L. S. and P. S. Krishnaprasad [1992], Gyroscopic control and stabilization, J. Nonlinear Sci. 2, 367–415.
Willems, J. C. [1972], Dissipative dynamical systems, Arch. Rat. Mech. and Anal. 45, 321–351.
Willems, J. C. [1979], System theoretic models for the analysis of physical systems, Ricerche di Automatica 10, 71–106.
Woolsey, C. A. and N. E. Leonard [1999a], Underwater vehicle stabilization by internal rotors, Proc. American Control Conf., 3417–3421.
Woolsey, C. A. and N. E. Leonard [1999b], Global asymptotic stabilization of an underwater vehicle using internal rotors, Proc. IEEE Conf. Decision and Control 38, 2527–2532.
Woolsey, C. A. and N.E. Leonard [2002], Stabilizing underwater vehicle motion using internal rotors, Automatica, (to appear).
Zenkov, D. V., A. M. Bloch and J. E. Marsden [1998], The energy momentum method for the stability of nonholonomic systems, Dyn. Stab. of Systems, 13, 123–166.
Zenkov, D. V., A. M. Bloch, and J. E. Marsden [1999], Stabilization of the unicycle with rider, Proc. IEEE Conf. Decision and Control 38, 3470–3471.
Zenkov, D. V., A. M. Bloch, N. E. Leonard and J. E. Marsden [2000], Matching and stabilization of the unicycle with rider, in Lagrangian and Hamiltonian Methods for Nonlinear Control: A Proceedings Volume from the IFAC Workshop (N. E. Leonard and R. Ortega eds.), Pergamon, 177–178
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To Jerry Marsden on the occasion of his 60th birthday
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Bloch, A.M., Leonard, N.E. (2002). Symmetries, Conservation Laws, and Control. In: Newton, P., Holmes, P., Weinstein, A. (eds) Geometry, Mechanics, and Dynamics. Springer, New York, NY. https://doi.org/10.1007/0-387-21791-6_14
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DOI: https://doi.org/10.1007/0-387-21791-6_14
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-95518-6
Online ISBN: 978-0-387-21791-8
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