Skip to main content
SpringerLink
Log in

Controlled Invariant Distributions and Differential Properties

  • Chapter
  • First Online:
Nonlinear Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 470))

  • 2030 Accesses

Abstract

The theory of (controlled) invariant (co-)distributions is reviewed, emphasizing the theory of liftings of vector fields, one-forms and (co-)distributions to the tangent and cotangent bundle. In particular, it is shown how invariant distributions can be equivalently described as invariant submanifolds of the tangent and cotangent bundle. This naturally leads to the notion of an invariant Lagrangian subbundle of the Whitney sum of tangent and cotangent bundle, which amounts to a special case of the central equation of contraction analysis. The interconnection of the prolongation of a nonlinear control system (living on the tangent bundle of the state space manifold) with its Hamiltonian extension (defined on the cotangent bundle) is shown to result in a differential Hamiltonian system. The invariant submanifolds of this differential Hamiltonian system corresponding to Lagrangian subbundles are seen to result in general differential Riccati and differential Lyapunov equations. The established framework thus yields a geometric underpinning of recent advances in contraction analysis and convergent dynamics.

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

Access this chapter

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Institutional subscriptions

References

  1. Abraham, R.A., Marsden, J.E.: Foundations of mechanics, 2nd edn. Benjamin / Cummings, Reading, MA (1978)

    MATH  Google Scholar 

  2. Cortés, J., van der Schaft, A.J., Crouch, P.E.: Characterization of gradient control systems. SIAM J. Control Optim. 44(4), 1192–1214 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Courant, T.J.: Dirac manifolds. Trans. Amer. Math. Soc. 319, 631–661 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Crouch, P.E., van der Schaft, A.J.: Variational and Hamiltonian control systems. Lecture Notes in Control and Information Sciences, vol. 101. Springer, New York (1987)

    MATH  Google Scholar 

  5. Dalsmo, M., van der Schaft, A.J.: On representations and integrability of mathematical structures in energy-conserving physical systems. SIAM J. Control and Optimization 37, 54–91 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dorfman, I.: Dirac Structures and Integrability of Nonlinear Evolution Equations. Wiley, Chichester (1993)

    MATH  Google Scholar 

  7. Forni, F., Sepulchre, R.: A differential Lyapunov framework for contraction analysis. IEEE Trans. Autom. Control 59(3), 614–628 (2014)

    Article  MathSciNet  Google Scholar 

  8. Forni, F., Sepulchre, R.: On differentially dissipative dynamical systems, Proceedings of 9th IFAC Symposium on Nonlinear Control Systems (NOLCOS2013), Toulouse, France, Sept. 4–6 (2013)

    Google Scholar 

  9. Forni, F., Sepulchre, R., van der Schaft, A.J.: On differential passivity of physical systems. In: Proceedings of 52nd IEEE Conference on Decision and Control (CDC), pp. 6580–6585. Florence, Italy (2013)

    Google Scholar 

  10. Jouffroy, J.: Some ancestors of contraction analysis. In: 44th IEEE Conference Decision and Control and European Control Conference (CDC-ECC ’05), pp. 5450–5455 (2005)

    Google Scholar 

  11. Kawano, Y., Ohtsuka, T.: Observability analysis of nonlinear systems using pseudo-linear transformation. In: Proceedings of 9th IFAC Symposium on Nonlinear Control Systems (NOLCOS2013), Toulouse, France, Sept. 4–6 (2013)

    Google Scholar 

  12. Kawano, Y., Ohtsuka, T.: Nonlinear eigenvalue approach to analysis of generalised differential Riccati equations (submitted, 2014)

    Google Scholar 

  13. Lohmiller, W., Slotine, J.-J.: On contraction analysis for non-linear systems. Automatica 34, 683–696 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Manchester, I.R., Slotine, J.J.: Control contraction metrics and universal stabilizability. 19th IFAC World Congress, Cape Town, South Africa (2014)

    Google Scholar 

  15. Manchester, I.R., Slotine, J.J.: Control contraction metrics: differential \(L^2\) gain and observer duality. arXiv:1403.5364v1 (2014)

  16. Manchester, I.R., Slotine, J.J.: Control contraction metrics: Convex and intrinsic criteria for nonlinear feedback design. arXiv:1503.03144v1 (2015)

  17. Nijmeijer, H., van der Schaft, A.J.: Controlled invariance for nonlinear systems. IEEE Trans. Autom. Control AC–27(4), 904–914 (1982)

    Google Scholar 

  18. Nijmeijer, H., van der Schaft, A.J.: Nonlinear Dynamical Control Systems. Springer, New York (1990). corrected printing 2016

    Book  MATH  Google Scholar 

  19. Pavlov, A., Pogromsky, A., van de Wouw, N., Nijmeijer, H.: Convergent dynamics: a tribute to Boris Pavlovich Demidovich. Syst. Control Lett. 52, 257–261 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pavlov, A., van de Wouw, N., Nijmeijer, H.: Uniform Output Regulation of Nonlinear Systems: A Convergent Dynamics Approach. Springer, Berlin (2005)

    MATH  Google Scholar 

  21. Simpson, J.W., Bullo, F.: Contraction theory on Riemannian manifolds. Syst. Control Lett. 65, 74–80 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. van der Schaft, A.J.: \(L_2\)-Gain and Passivity Techniques in Nonlinear Control, Lecture Notes in Control and Information Sciences, Vol. 218, 2nd edn. Springer, Berlin (1996), Springer, London (2000) (Springer Communications and Control Engineering series)

    Google Scholar 

  23. van der Schaft, A.J.: Implicit Hamiltonian systems with symmetry. Rep. Math. Phys. 41, 203–221 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. van der Schaft, A.J.: On differential passivity. In: Proceedings of 9th IFAC Symposium on Nonlinear Control Systems (NOLCOS2013), pp. 21–25. Toulouse, France, Sept. 4–6 (2013)

    Google Scholar 

  25. van der Schaft, A.J.: A geometric approach to differential Hamiltonian and differential Riccati equations. In: Proceedings of 54th IEEE Conference on Decision and Control, (CDC), pp. 7151–7156. Osaka, Japan (2015)

    Google Scholar 

  26. Tyner, D.R., Lewis, A.D.: Geometric Jacobian linearization and LQR theory. J. Geom. Mech. 2(4), 397–440 (2010)

    MathSciNet  MATH  Google Scholar 

  27. Yano, K., Ishihara, S.: Tangent and cotangent bundles. Marcel Dekker, New York (1973)

    Google Scholar 

Download references

Acknowledgments

I would like to thank the Dutch Nonlinear Systems Group for continuing inspiration and support.

Author information

Authors and Affiliations

  1. Johann Bernoulli Institute for Mathematics and Computer Science, Jan C. Willems Center for Systems and Control, University of Groningen, Groningen, The Netherlands

    Arjan van der Schaft

Authors
  1. Arjan van der Schaft
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arjan van der Schaft .

Editor information

Editors and Affiliations

  1. Department of Mechanical Engineering, Eindhoven University of Technology Department of Mechanical Engineering, Eindhoven, Noord-Holland, The Netherlands

    Nathan van de Wouw

  2. Department of Mechanical Engineering, Eindhoven University of Technology Department of Mechanical Engineering, Eindhoven, Noord-Brabant, The Netherlands

    Erjen Lefeber

  3. Department of Mechanical Engineering, Eindhoven University of Technology Department of Mechanical Engineering, Eindhoven, Noord-Brabant, The Netherlands

    Ines Lopez Arteaga

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing Switzerland

About this chapter

Cite this chapter

van der Schaft, A. (2017). Controlled Invariant Distributions and Differential Properties. In: van de Wouw, N., Lefeber, E., Lopez Arteaga, I. (eds) Nonlinear Systems. Lecture Notes in Control and Information Sciences, vol 470. Springer, Cham. https://doi.org/10.1007/978-3-319-30357-4_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-30357-4_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-30356-7

  • Online ISBN: 978-3-319-30357-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation