Abstract
We discuss here the problem of analyzing which classes of control systems may be put in a closed-loop form which is Lagrangian or Hamiltonian and ask when such controls are useful.
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References
Aström, K.J. and K. Furuta [1996] Swinging up a pendulum by energy control. IFAC, 13.
Bloch, A.M., and R Crouch [1998] Optimal control, optimization and analytical mechanics, to appear in the Brockettfest.
Bloch, A.M., PS. Krishnaprasad, J.E. Marsden, and T.S. Ratiu [1996] The Euler-Poincaré equations and double bracket dissipation. Comm. Math. Phys. 175, 1–42.
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. and J. Marsden [1989] Controlling homoclinic orbits, Theoretical and Computational Fluid Dynamics 1 (1989), 179–190.
Bloch, A.M. and J. Marsden [1990] Stabilization of the rigid body equations and the Energy-Casimir method, Systems and Controls Letters 14, 341–346.
Bloch, A.M., J.E. Marsden and G. Sánchez [1997] Stabilization of relative equilibria of mechanical systems with symmetry, Current and Future Directions in Applied Mathematics, Edited by M. Alber, B. Hu, and J. Rosenthal, Birkháuser, 43–64.
Bloch, A.M., N.E. Leonard and J.E. Marsden [1997] Stabilization of mechanical systems using controlled Lagrangians. Proc. IEEE Conf. Dec. Contr., San Diego, CA, 2356–2361.
Bloch, A.M., N.E. Leonard and J.E. Marsden [1998] Controlled Lagrangians and the stabilization of mechanical systems. Preprint.
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, Vol. 7, Math. Sci. Press, Brookline, Mass., USA.
Brockett, R.W. [1978] Lie algebras and rational functions: some control theoretic connections, in Lie Theories and Their Applications ( W. Rossman, ed.). Kingston, Ontario: Queen’s University, Dept. of Mathematics, 268–280.
Brockett, R.W. and A. Rahimi [1972] Lie algebras and linear differential equations, in Ordinary Differential Equations (L. Weiss, ed. ), Academic Press, 379–386.
Crouch, P. and A.J. van der Schaft [1987] Variational and Hamiltonian Control Systems, Lecture Notes in Control and Informations Sciences 10, Springer Verlag.
Hermann, R. [1977] Differential Geometry and the Calculus of Variations, 2nd Edition, Interdisciplinary Mathematics Volume, XVII, Math. Sci. Press, Brookline, Mass., USA (First Edition 1968, Academic Press).
Holmes, P., J. Jenkins and N.E. Leonard [1998] Dynamics of Kirchhoff Equations I: Coincident Centers of Gravity and Buoyancy, Physica D. To appear.
Koditschek, D.E. [1989] The application of total energy as a Lyapunov function for mechanical control systems, in Dynamics and control of multibody systems (Brunswick, ME, 1988), 131–157, Contemp. Math., 97, Amer. Math. Soc., Providence, RI.
Krishnaprasad P.S. [1985] Lie-Poisson structures, dual-spin spacecraft and asymptotic stability, Nonl Anal Th. Meth. and Appl. 9, 1011–1035.
Leonard, N.E. [1997] Stabilization of underwater vehicle dynamics with symmetry-breaking potentials, Systems and Control Letters 32, 35–42.
Lewis, A.D. and R. Murray [1997] Configuration controllability of simple mechanical control systems, SIAM Journal on Control and Optimization 35, 766–790.
Marsden, J.E. [1992] Lectures on Mechanics, London Mathematical Society Lecture note series. 174, Cambridge University Press.
Marsden, J.E. and T.S. Ratiu [1994] Introduction to Mechanics and Symmetry. Texts in Applied Mathematics, 17, Springer-Verlag.
Ortega, R. A. Loria, R. Kelly, and L. Praly [1995], On passivity-based output feedback global stabilization of Euler-Lagrange systems, Int. J. Robust and Nonlinear Control, special issue on Control of mechanical systems, 5 no 4, 313–325.
Slotine, J.-J. [1988] Putting physics in control: the example of robotics IEEE Control Systems Magazine 8, 12–18.
Van der Schaft, A.J. [1982] Hamiltonian dynamics with external forces and observations, Mathematical Systems Theory 15, 145–168.
Van der Schaft, A. J. [1986] Stabilization of Hamiltonian systems, Non-linear Analysis, Theory, Methods and Applications, 10, 1021–1035.
Wang, L.S. and P.S. Krishnaprasad [1992] Gyroscopic control and stabilization, J. Nonlinear Science 2, 367–415.
Willems, J.C. [1979] System theoretic models for the analysis of physical systems, in Ricerche di Automatica 10, Special Issue on Systems Theory and Physics (R.W. Brockett ed. ) 71–106.
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Bloch, A.M., Leonard, N.E., Marsden, J.E. (1999). Mechanical feedback control systems. In: Blondel, V., Sontag, E.D., Vidyasagar, M., Willems, J.C. (eds) Open Problems in Mathematical Systems and Control Theory. Communications and Control Engineering. Springer, London. https://doi.org/10.1007/978-1-4471-0807-8_10
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DOI: https://doi.org/10.1007/978-1-4471-0807-8_10
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