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Symplectic feedback using Hamiltonian Lie algebra and its applications to an inverted pendulum

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Abstract

From the symplectic representation of an autonomous nonlinear dynamical system with holonomic constraints, i.e., those that can be represented through a symplectic form derived from a Hamiltonian, we present a new proof on the realization of the symplectic feedback action, which has several theoretical advantages in demonstrating the uniqueness and existence of this type of solution. Also, we propose a technique based on the interpretation, construction and characterization of the pull-back differential on the symplectic manifold as a member of a one-parameter Lie group. This allows one to synthesize the control law that governs a certain system to achieve a desired behavior; and the method developed from this is applied to a classical system such as the inverted pendulum.

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Authors and Affiliations

  1. Departamento de Ciencias Fisicas, Facultad de Ingenieria, Ciencias y Administracion, Universidad de La Frontera, Avenida Francisco Salazar 01145, Casilla 54-D, Temuco, Chile

    J. Lorca Espiro

  2. Departamento de Ingenieria Electrica, Facultad de Ingenieria, Ciencias y Administracion, Universidad de La Frontera, Avenida Francisco Salazar 01145, Casilla 54-D, Temuco, Chile

    C. Muñoz Poblete

Authors
  1. J. Lorca Espiro
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  2. C. Muñoz Poblete
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Correspondence to J. Lorca Espiro.

Additional information

J. Lorca ESPIRO was born in Temuco, Chile, in 1984. He received his Electronics Engineering title in 2008, he also holds a B.S. degree in Physics and a M.S. degree in Engineering Sciences from Universidad de La Frontera. He is currently active in areas of theoretical and mathematical physics, such as quantum field theory and gravitation and cosmology, as well as in classical mechanics and symplectic geometry.

C. Muñoz POBLETE was born in Collipulli, Chile, in 1966. He graduated in Electronic Engineering in 1990 at the Department of Electrical Engineering (DIE) of the Universidad de La Frontera (Chile), and received his Ph.D. degree in Engineering Sciences at the Pontificia Universidad Catlica (Chile) in 2001. He is currently an associate professor of Automatic Control at DIE. His research interest include nonlinear and predictive control, and wireless sensor networks.

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Espiro, J.L., Poblete, C.M. Symplectic feedback using Hamiltonian Lie algebra and its applications to an inverted pendulum. J. Control Theory Appl. 11, 275–281 (2013). https://doi.org/10.1007/s11768-013-1073-7

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  • DOI: https://doi.org/10.1007/s11768-013-1073-7

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