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Controlled Lagrangians and stabilization of discrete mechanical systems

Bloch, Anthony M. and Leok, Melvin and Marsden, Jerrold E. and Zenkov, Dmitry V. (2010) Controlled Lagrangians and stabilization of discrete mechanical systems. Discrete and Continuous Dynamical Systems Series S, 3 (1). pp. 19-36. ISSN 1937-1632.

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Controlled Lagrangian and matching techniques are developed for the stabilization of relative equilibria of discrete mechanical systems with symmetry and equilibria of discrete mechanical systems with broken symmetry. Unexpected phenomena arise in the controlled Lagrangian approach in the discrete context that are not present in the continuous theory. In particular, to make the discrete theory effective, one can make an appropriate selection of momentum levels or, alternatively, introduce a new parameter into the controlled Lagrangian to complete the kinetic shaping procedure. New terms in the controlled shape equation that are necessary for potential shaping in the discrete setting are introduced. The theory is illustrated with the problem of stabilization of the cart-pendulum system on an incline, and the application of the theory to the construction of digital feedback controllers is also discussed.

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Additional Information:© 2009 AIMS. Received September 2008; revised February 2009; published December 2009. The authors would like to thank the reviewers for helpful remarks. The research of AMB was supported by NSF grants DMS-0604307, CMS- 0408542 and DMS-0907949. The research of ML was partially supported by NSF grants DMS-0504747, DMS-0726263, and CAREER Award DMS-0747659. The research of JEM was partially supported by AFOSR Contract FA9550-08-1-0173. The research of DVZ was partially supported by NSF grants DMS-0306017, DMS- 0604108, and DMS-0908995.
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)FA9550-08-1-0173
Subject Keywords:Matching, discrete mechanics, digital control.
Classification Code:2000 Mathematics Subject Classification: Primary: 70Q05, 93C55, 93D15; Secondary: 39A11.
Record Number:CaltechAUTHORS:20100601-130852433
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:18507
Deposited By: Ruth Sustaita
Deposited On:29 Jun 2010 15:50
Last Modified:26 Dec 2012 12:05

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