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Feedback stabilization of relative equilibria for mechanical systems with symmetry

Bloch, Anthony M. and Marsden, Jerrold E. and De Alvarez, G. Sanchez (1997) Feedback stabilization of relative equilibria for mechanical systems with symmetry. In: Current and future directions in applied mathematics. Birkhauser Boston Inc. , Cambridge, MA, USA, pp. 43-64. ISBN 081763956X.

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This paper is an outgrowth of the work of Bloch, Krishnaprasad, Marsden and Sánchez de Alvarez [1992], where a feedback control that stabilizes intermediate axis rigid body rotation using an internal rotor was found. Stabilization is determined by use of the energy-Casimir (Arnold) method. In the present paper we show that this feedback controlled system can be written as the Euler-Lagrange equations for a modified Lagrangian: a velocity shift associated with a change of connection turns the free (unforced) equations into the feedback controlled equations. We also show how stabilization of the inverted pendulum on a cart can be achieved in an analogous way. We provide a general systematic construction of such controlled Lagrangians. The basic idea is to modify the kinetic energy of the free Lagrangian using a generalization of the Kaluza-Klein construction in such a way that the extra terms obtained in the Euler-Lagrange equations can be identified with control forces. The fact that the controlled system is Lagrangian by construction enables one to make use of energy techniques for a stability analysis. Once stabilization is achieved in a mechanical context, one can establish asymptotic stabilization by the further addition of dissipative controls. The methods here can be combined with symmetry breaking controls obtained by modifying the potential energy and also can be used for tracking.

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Additional Information:© 1997. Received September, 1996. We thank John Ballieul, Francesco Bullo, Joel Burdick, Neil Getz, Dan Koditschek, P.S. Krishnaprasad, Naomi Leonard, Richard Murray and Tudor Ratiu for helpful comments. Some of the ideas here were directly inspired by our efforts to combine ideas in Bloch, Krishnaprasad Marsden and Sánchez de Alvarez [1992], those of Wang and Krishnaprasad [1992], and those in the PhD thesis of Neil Getz.
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)F49620-96-1-0100
Guggenheim FellowshipUNSPECIFIED
Department of EnergyDE-FG0395-ER25251
National Council for Science and Technology (CONICIT)-VenezuelaUNSPECIFIED
Record Number:CaltechAUTHORS:20100714-154512455
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ID Code:19056
Deposited By: Ruth Sustaita
Deposited On:04 Aug 2010 18:18
Last Modified:08 Nov 2021 23:49

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