American Institute of Mathematical Sciences

Advanced
March  2010, 3(1): 19-36. doi: 10.3934/dcdss.2010.3.19

Controlled Lagrangians and stabilization of discrete mechanical systems

Anthony M. Bloch 1, , Melvin Leok 2, , Jerrold E. Marsden 3, and  Dmitry V. Zenkov 4,

1. 

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, United States
2. 

Department of Mathematics, University of California, San Diego, 9500 Gilman Drive La Jolla, CA 92093-0112, United States
3. 

Control and Dynamical Systems, California Institute of Technology, Pasadena, CA 91125, United States
4. 

Department of Mathematics, North Carolina State University, Raleigh, NC 27695, United States

Received  September 2008 Revised  February 2009 Published  December 2009

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.
Keywords: digital control., discrete mechanics, Matching.
Mathematics Subject Classification: Primary: 70Q05, 93C55, 93D15; Secondary: 39A1.
Citation: Anthony M. Bloch, Melvin Leok, Jerrold E. Marsden, Dmitry V. Zenkov. Controlled Lagrangians and stabilization of discrete mechanical systems. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 19-36. doi: 10.3934/dcdss.2010.3.19
[1]

François Gay-Balmaz, Tudor S. Ratiu. Clebsch optimal control formulation in mechanics. Journal of Geometric Mechanics, 2011, 3 (1) : 41-79. doi: 10.3934/jgm.2011.3.41
[2]

Juan Carlos Marrero, David Martín de Diego, Ari Stern. Symplectic groupoids and discrete constrained Lagrangian mechanics. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 367-397. doi: 10.3934/dcds.2015.35.367
[3]

Angel Angelov, Marcus Wagner. Multimodal image registration by elastic matching of edge sketches via optimal control. Journal of Industrial & Management Optimization, 2014, 10 (2) : 567-590. doi: 10.3934/jimo.2014.10.567
[4]

Rama Ayoub, Aziz Hamdouni, Dina Razafindralandy. A new Hodge operator in discrete exterior calculus. Application to fluid mechanics. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2155-2185. doi: 10.3934/cpaa.2021062
[5]

Ismet Cinar, Ozgur Ege, Ismet Karaca. The digital smash product. Electronic Research Archive, 2020, 28 (1) : 459-469. doi: 10.3934/era.2020026
[6]

Hyung-Chun Lee. Efficient computations for linear feedback control problems for target velocity matching of Navier-Stokes flows via POD and LSTM-ROM. Electronic Research Archive, 2021, 29 (3) : 2533-2552. doi: 10.3934/era.2020128
[7]

Aleksandar Zatezalo, Dušan M. Stipanović. Control of dynamical systems with discrete and uncertain observations. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4665-4681. doi: 10.3934/dcds.2015.35.4665
[8]

V.N. Malozemov, A.V. Omelchenko. On a discrete optimal control problem with an explicit solution. Journal of Industrial & Management Optimization, 2006, 2 (1) : 55-62. doi: 10.3934/jimo.2006.2.55
[9]

Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571
[10]

Qiying Hu, Wuyi Yue. Optimal control for resource allocation in discrete event systems. Journal of Industrial & Management Optimization, 2006, 2 (1) : 63-80. doi: 10.3934/jimo.2006.2.63
[11]

Evelyn Herberg, Michael Hinze, Henrik Schumacher. Maximal discrete sparsity in parabolic optimal control with measures. Mathematical Control & Related Fields, 2020, 10 (4) : 735-759. doi: 10.3934/mcrf.2020018
[12]

Qiying Hu, Wuyi Yue. Optimal control for discrete event systems with arbitrary control pattern. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 535-558. doi: 10.3934/dcdsb.2006.6.535
[13]

Jean-Marie Souriau. On Geometric Mechanics. Discrete & Continuous Dynamical Systems, 2007, 19 (3) : 595-607. doi: 10.3934/dcds.2007.19.595
[14]

Jianjun Zhang, Yunyi Hu, James G. Nagy. A scaled gradient method for digital tomographic image reconstruction. Inverse Problems & Imaging, 2018, 12 (1) : 239-259. doi: 10.3934/ipi.2018010
[15]

Lizhong Peng, Shujun Dang, Bojin Zhuang. Localization operator and digital communication capacity of channel. Communications on Pure & Applied Analysis, 2007, 6 (3) : 819-827. doi: 10.3934/cpaa.2007.6.819
[16]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183
[17]

Galina Kurina, Sahlar Meherrem. Decomposition of discrete linear-quadratic optimal control problems for switching systems. Conference Publications, 2015, 2015 (special) : 764-774. doi: 10.3934/proc.2015.0764
[18]

Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control & Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195
[19]

Jean-Baptiste Caillau, Bilel Daoud, Joseph Gergaud. Discrete and differential homotopy in circular restricted three-body control. Conference Publications, 2011, 2011 (Special) : 229-239. doi: 10.3934/proc.2011.2011.229
[20]

Le Viet Cuong, Thai Son Doan. Assignability of dichotomy spectra for discrete time-varying linear control systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3597-3607. doi: 10.3934/dcdsb.2020074

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (72)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences