Skip to main content
SpringerLink
Log in
Book cover

Mathematics of Complexity and Dynamical Systems pp 143–159Cite as

Discrete Control Systems

  • Reference work entry

Article Outline

Glossary

Definition of the Subject

Introduction

Discrete Lagrangian and Hamiltonian Mechanics

Optimal Control of Discrete Lagrangian and Hamiltonian Systems

Controlled Lagrangian Method for Discrete Lagrangian Systems

Future Directions

Acknowledgments

Bibliography

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   600.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Abbreviations

Discrete variational mechanics:

A formulation of mechanics in discrete‐time that is based on a discrete analogue of Hamilton's principle, which states that the system takes a trajectory for which the action integral is stationary.

Geometric integrator:

A numerical method for obtaining numerical solutions of differential equations that preserves geometric properties of the continuous flow, such as symplecticity, momentum preservation, and the structure of the configuration space.

Lie group:

A differentiable manifold with a group structure where the composition is differentiable. The corresponding Lie algebra is the tangent space to the Lie group based at the identity element.

Symplectic:

A map is said to be symplectic if given any initial volume in phase space, the sum of the signed projected volumes onto each position‐momentum subspace is invariant under the map. One consequence of symplecticity is that the map is volume‐preserving as well.

Bibliography

Primary Literature

  1. Auckly D, Kapitanski L, White W (2000) Control of nonlinear underactuated systems. Commun Pure Appl Math 53:354–369

    Article  MathSciNet  MATH  Google Scholar 

  2. Benettin G, Giorgilli A (1994) On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms. J Stat Phys 74:1117–1143

    Article  MathSciNet  MATH  Google Scholar 

  3. Betts JT (2001) Practical Methods for Optimal Control Using Nonlinear Programming. SIAM, Philadelphia, PA

    MATH  Google Scholar 

  4. Bloch AM (2003) Nonholonomic Mechanics and Control. In: Interdisciplinary Applied Mathematics, vol 24. Springer, New York

    Google Scholar 

  5. Bloch AM, Leonard N, Marsden JE (1997) Matching and stabilization using controlled Lagrangians. In: Proceedings of the IEEE Conference on Decision and Control. Hyatt Regency San Diego, San Diego, CA, 10–12 December 1997, pp 2356–2361

    Google Scholar 

  6. Bloch AM, Leonard N, Marsden JE (1998) Matching and stabilization by the method of controlled Lagrangians. In: Proceedings of the IEEE Conference on Decision and Control. Hyatt Regency Westshore, Tampa, FL, 16–18 December 1998, pp 1446–1451

    Google Scholar 

  7. Bloch AM, Leonard N, Marsden JE (1999) Potential shaping and the method of controlled Lagrangians. In: Proceedings of the IEEE Conference on Decision and Control. Crowne Plaza Hotel and Resort, Phoenix, AZ, 7–10 December 1999, pp 1652–1657

    Google Scholar 

  8. Bloch AM, Leonard NE, Marsden JE (2000) Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem. IEEE Trans Syst Control 45:2253–2270

    Article  MathSciNet  MATH  Google Scholar 

  9. Bloch AM, Chang DE, Leonard NE, Marsden JE (2001) Controlled Lagrangians and the stabilization of mechanical systems II: Potential shaping. IEEE Trans Autom Contr 46:1556–1571

    Article  MathSciNet  MATH  Google Scholar 

  10. Bloch AM, Leok M, Marsden JE, Zenkov DV (2005) Controlled Lagrangians and stabilization of the discrete cart‐pendulum system. In: Proceedings of the IEEE Conference on Decision and Control. Melia Seville, Seville, Spain, 12–15 December 2005, pp 6579–6584

    Google Scholar 

  11. Bloch AM, Leok M, Marsden JE, Zenkov DV (2006) Controlled Lagrangians and potential shaping for stabilization of discrete mechanical systems. In: Proceedings of the IEEE Conference on Decision and Control. Manchester Grand Hyatt, San Diego, CA, 13–15 December 2006, pp 3333–3338

    Google Scholar 

  12. Bryson AE, Ho Y (1975) Applied Optimal Control. Hemisphere, Washington, D.C.

    Google Scholar 

  13. Chang D-E, Bloch AM, Leonard NE, Marsden JE, Woolsey C (2002) The equivalence of controlled Lagrangian and controlled Hamiltonian systems. Control Calc Var (special issue dedicated to Lions JL) 8:393–422

    Google Scholar 

  14. Hairer E, Lubich C, Wanner G (2006) Geometric Numerical Integration, 2nd edn. Springer Series in Computational Mathematics, vol 31. Springer, Berlin

    Google Scholar 

  15. Hamberg J (1999) General matching conditions in the theory of controlled Lagrangians. In: Proceedings of the IEEE Conference on Decision and Control. Crowne Plaza Hotel and Resort, Phoenix, AZ, 7–10 December 1999, pp 2519–2523

    Google Scholar 

  16. Hamberg J (2000) Controlled Lagrangians, symmetries and conditions for strong matching. In: Lagrangian and Hamiltonian Methods for Nonlinear Control. Elsevier, Oxford

    Google Scholar 

  17. Hussein II, Leok M, Sanyal AK, Bloch AM (2006) A discrete variational integrator for optimal control problems in SO(3). In: Proceedings of the IEEE Conference on Decision and Control. Manchester Grand Hyatt, San Diego, CA, 13–15 December 2006, pp 6636–6641

    Google Scholar 

  18. Iserles A, Munthe-Kaas H, Nørsett SP, Zanna A (2000) Lie-group methods. In: Acta Numerica, vol 9. Cambridge University Press, Cambridge, pp 215–365

    Google Scholar 

  19. Junge D, Marsden JE, Ober-Blöbaum S (2005) Discrete mechanics and optimal control. In: IFAC Congress, Praha, Prague, 3–8 July 2005

    Google Scholar 

  20. Kane C, Marsden JE, Ortiz M (1999) Symplectic‐energy‐momentum preserving variational integrators. J Math Phys 40(7):3353–3371

    Article  MathSciNet  MATH  Google Scholar 

  21. Kane C, Marsden JE, Ortiz M, West M (2000) Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. Int J Numer Meth Eng 49(10):1295–1325

    Article  MathSciNet  MATH  Google Scholar 

  22. Kelley CT (1995) Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia, PA

    Book  MATH  Google Scholar 

  23. Lee T, Leok M, McClamroch NH (2005) Attitude maneuvers of a rigid spacecraft in a circular orbit. In: Proceedings of the American Control Conference. Portland Hilton, Portland, OR, 8–10 June 2005, pp 1742–1747

    Google Scholar 

  24. Lee T, Leok M, McClamroch NH (2005) A Lie group variational integrator for the attitude dynamics of a rigid body with applications to the 3D pendulum. In: Proceedings of the IEEE Conference on Control Applications. Toronto, Canada, 28–31 August 2005, pp 962–967

    Google Scholar 

  25. Lee T, Leok M, McClamroch NH (2006) Optimal control of a rigid body using geometrically exact computations on SE(3). In: Proceedings of the IEEE Conference on Decision and Control. Manchester Grand Hyatt, San Diego, CA, 13–15 December 2006, pp 2710–2715

    Google Scholar 

  26. Lee T, Leok M, McClamroch NH (2007) Lie group variational integrators for the full body problem. Comput Method Appl Mech Eng 196:2907–2924

    Article  MathSciNet  MATH  Google Scholar 

  27. Lee T, Leok M, McClamroch NH (2007) Lie group variational integrators for the full body problem in orbital mechanics. Celest Mech Dyn Astron 98(2):121–144

    Article  MathSciNet  MATH  Google Scholar 

  28. Leimkuhler B, Reich S (2004) Simulating Hamiltonian Dynamics. Cambridge Monographs on Applied and Computational Mathematics, vol 14. Cambridge University Press, Cambridge

    Google Scholar 

  29. Leok M (2004) Foundations of Computational Geometric Mechanics. Ph D thesis, California Instittute of Technology

    Google Scholar 

  30. Marsden JE, Ratiu TS (1999) Introduction to Mechanics and Symmetry, 2nd edn. Texts in Applied Mathematics, vol 17. Springer, New York

    Book  Google Scholar 

  31. Marsden JE, West M (2001) Discrete mechanics and variational integrators. In: Acta Numerica, vol 10. Cambridge University Press, Cambridge, pp 317–514

    Google Scholar 

  32. Marsden JE, Pekarsky S, Shkoller S (1999) Discrete Euler–Poincarée and Lie–Poisson equations. Nonlinearity 12(6):1647–1662

    Article  MathSciNet  MATH  Google Scholar 

  33. Maschke B, Ortega R, van der Schaft A (2001) Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. IEEE Trans Autom Contr 45:1498–1502

    Article  Google Scholar 

  34. Moser J, Veselov AP (1991) Discrete versions of some classical integrable systems and factorization of matrix polynomials. Commun Math Phys 139:217–243

    Article  MathSciNet  MATH  Google Scholar 

  35. Ortega R, Spong MW, Gómez-Estern F, Blankenstein G (2002) Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment. IEEE Trans Autom Contr 47:1218–1233

    Google Scholar 

  36. Sanz-Serna JM (1992) Symplectic integrators for Hamiltonian problems: an overview. In: Acta Numerica, vol 1. Cambridge University Press, Cambridge, pp 243–286

    Google Scholar 

  37. Scheeres DJ, Fahnestock EG, Ostro SJ, Margot JL, Benner LAM, Broschart SB, Bellerose J, Giorgini JD, Nolan MC, Magri C, Pravec P, Scheirich P, Rose R, Jurgens RF, De Jong EM, Suzuki S (2006) Dynamical configuration of binary near-Earth asteroid (66391) 1999 KW4. Science 314:1280–1283

    Article  Google Scholar 

  38. Scheeres DJ, Hsiao F-Y, Park RS, Villac BF, Maruskin JM (2006) Fundamental limits on spacecraft orbit uncertainty and distribution propagation. J Astronaut Sci 54:505–523

    Google Scholar 

  39. Xiu D (2007) Efficient collocational approach for parametric uncertainty analysis. Comm Comput Phys 2:293–309

    MathSciNet  MATH  Google Scholar 

  40. Zenkov DV, Bloch AM, Leonard NE, Marsden JE (2000) Matching and stabilization of low‐dimensional nonholonomic systems. In: Proceedings of the IEEE Conference on Decision and Control. Sydney Convention and Exhibition Centre, Sydney, NSW Australia; 12–15 December 2000, pp 1289–1295

    Google Scholar 

  41. Zenkov DV, Bloch AM, Leonard NE, Marsden JE (2002) Flat nonholonomic matching. In: Proceedings of the American Control Conference. Hilton Anchorage, Anchorage, AK, 8–10 May 2002, pp 2812–2817

    Google Scholar 

Books and Reviews

  1. Bloch AM (2003) Nonholonomic Mechanics and Control. Interdisciplinary Appl Math, vol 24. Springer

    Google Scholar 

  2. Bullo F, Lewis AD (2005) Geometric control of mechanical systems. Texts in Applied Mathematics, vol 49. Springer, New York

    Google Scholar 

  3. Hairer E, Lubich C, Wanner G (2006) Geometric Numerical Integration, 2nd edn. Springer Series in Computational Mathematics, vol 31. Springer, Berlin

    Google Scholar 

  4. Iserles A, Munthe-Kaas H, Nørsett SP, Zanna A (2000) Lie-group methods. In: Acta Numerica, vol 9. Cambridge University Press, Cambridge, pp 215–365

    Google Scholar 

  5. Leimkuhler B, Reich S (2004) Simulating Hamiltonian Dynamics. Cambridge Monographs on Applied and Computational Mathematics, vol 14. Cambridge University Press, Cambridge

    Google Scholar 

  6. Marsden JE, Ratiu TS (1999) Introduction to Mechanics and Symmetry, 2nd edn. Texts in Applied Mathematics, vol 17. Springer

    Google Scholar 

  7. Marsden JE, West M (2001) Discrete mechanics and variational integrators. In Acta Numerica, vol 10. Cambridge University Press, Cambridge, pp 317–514

    Google Scholar 

  8. Sanz-Serna JM (1992) Symplectic integrators for Hamiltonian problems: an overview. In: Acta Numerica, vol 1. Cambridge University Press, Cambridge, pp 243–286

    Google Scholar 

Download references

Acknowledgments

TL and ML have been supported in part by NSF Grant DMS‐0504747 and DMS-0726263. TL and NHM have been supported in part by NSF Grant ECS-0244977 and CMS-0555797.

Author information

Authors and Affiliations

  1. Department of Aerospace Engineering, University of Michigan, Ann Arbor, USA

    Taeyoung Lee & Harris McClamroch

  2. Department of Mathematics, Purdue University, West Lafayette, USA

    Melvin Leok

Authors
  1. Taeyoung Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Melvin Leok
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Harris McClamroch
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag

About this entry

Cite this entry

Lee, T., Leok, M., McClamroch, H. (2012). Discrete Control Systems. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_10

Download citation

Publish with us

Policies and ethics

Search

Navigation