Skip to main content
SpringerLink
Log in
Book cover

Model Order Reduction: Theory, Research Aspects and Applications pp 213–250Cite as

Time Variant Balancing and Nonlinear Balanced Realizations

  • Chapter

Part of the book series: Mathematics in Industry ((TECMI,volume 13))

Summary

Balancing for linear time varying systems and its application to model reduction via projection of dynamics (POD) are briefly reviewed. We argue that a generalization for balancing nonlinear systems may be expected to be based upon three sound principles: 1) Balancing should be defined with respect to a nominal flow; 2) Only Gramians defined over small time intervals should be used in order to preserve the accuracy of the linear perturbation model and; 3) Linearization should commute with balancing, in the sense that the linearization of a globally balanced model should correspond to the balanced linearized model in the original coordinates.

The first two principles lead to local balancing, which provides useful information about the dynamics of the system and the topology of the state space. It is shown that an integrability condition generically provides an obstruction towards a notion of a globally balanced realization in the strict sense. By relaxing the conditions of “strict balancing” in various ways useful system approximations may be obtained.

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   129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Fliess, M., Lamnahbi M. and Lamnabhi-Lagarrigue, F.: An algebraic approach to nonlinear functional expansions. IEEE Trans. Circ. Syst., 30,554–570 (1983)

    Article  MATH  Google Scholar 

  2. Gilbert, E.G.: Functional expansions for the response of nonlinear differential systems. IEEE Trans. Automatic Control, 22, 6 909–921 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  3. Gray, W.S. and Verriest, E.I.: Balanced realizations near stable invariant manifolds. Proc. MTNS-04, Leuven, Belgium, (2004).

    Google Scholar 

  4. Gray, W.S. and Verriest, E.I.: Balanced realizations near stable invariant manifolds. Automatica 42 653-659 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Gray, W.S. and Verriest, E.I.: Algebraically defined Gramians for nonlinear Systems. Proceedings of the 45-th IEEE Conference on Decision and Control, San Diego, CA 3730-3735 (2006).

    Google Scholar 

  6. Gucercin, S. and Antoulas, A.: A survey of model reduction by balanced truncation and some new results. Int. J. Control, 77 8 748–766 (2004)

    Article  Google Scholar 

  7. Isidori, A., Nonlinear Control Systems, 2-nd edition. Springer-Verlag (1989)

    Google Scholar 

  8. Jonckheere, E. and Silverman, L.: A new set of invariants for linear systems - Application to reduced order compensator design. IEEE Transactions on Automatic Control, 28 10 953–964 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  9. Khayatian, A. and Taylor, D.G.: Sampled-data modeling and control design for nonlinear pulse-width modulated systems. Proc. American Control Conference, 660–664 (1993)

    Google Scholar 

  10. Lall, S. and Beck, C.: Error-bounds for balanced reduction of linear time-varying systems. IEEE Trans. Automatic Control, 48 946–956 (2003)

    Article  MathSciNet  Google Scholar 

  11. Lall, L., Marsden, J.E. and Glavaški, S.: A subspace approach to balanced truncation for model reduction of nonlinear control systems. Int. J. Robust and Nonlinear Control, 12 6 519–535 (2002)

    Article  MATH  Google Scholar 

  12. Lesiak, C. and Krener, A.J.: The existence and uniqueness of Volterra series for nonlinear systems. IEEE Trans. Auto. Ctr. 23 6 1090–1095 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  13. Longhi, S. and Orlando, G.: Balanced reduction of linear periodic systems. Kybernetika, 35 737–751 (1999)

    MathSciNet  Google Scholar 

  14. Monaco, S. and Normand-Cyrot, D.: Input-output approximation of a nonlinear discrete time system from an equilibrium point. Proc. 23-th Conference on Decision and Control, Las Vegas, NV 90–96 (1984)

    Google Scholar 

  15. Monaco, S. and Normand-Cyrot, D.: On the sampling of a linear analytic control system. Proc. 24-th Conference on Decision and Control, Ft. Lauderdale, FL 1457–1462 (1985)

    Google Scholar 

  16. Moore, B.: Principal component analysis in linear systems: Controllability, observability and model reduction. IEEE Transactions on Automatic Contro;, 26 17–32 (1981)

    Article  MATH  Google Scholar 

  17. Morse, M.: What is analysis in the large? In Chern, S.S. (ed) Studies in Global Geometry and Analysis. Mathematical Association of America (1967)

    Google Scholar 

  18. Newman A. and Krishnaprasad, P.S.: Computation for nonlinear balancing. Proceedings of the 37th Conference on Decision and Control, Tampa, FL 4103–4104 (1998)

    Google Scholar 

  19. Opmeer. M.R. and Curtain, R.F.: Linear quadratic Gaussian balancing for discrete-time infinite-dimensional linear systems. SIAM Journal on Control and Optimization, 43 4 119–122 (2004)

    Article  MathSciNet  Google Scholar 

  20. Sandberg H. and Rantzer, A.: Error bounds for balanced truncation of linear time-varying systems. Proc. 39-th Conf. Decision and Control, Sydney, Australia 2892–2897 (2002)

    Google Scholar 

  21. Sandberg H. and Rantzer, A.: Balanced truncation of linear time-varying systems. IEEE Transactions on Automatic Control, 49 2 217–229 (2004)

    Article  MathSciNet  Google Scholar 

  22. Scherpen, J.M.A.: Balancing for nonlinear systems, Ph.D. Dissertation, University of Twente (1994)

    Google Scholar 

  23. Shokoohi, S., Silverman, L. and van Dooren, P.: Linear time-variable systems: Balancing and model reduction. IEEE Transactions on Automatic Control, 28, 833–844 (1983)

    Article  Google Scholar 

  24. Varga, A.: Balancing related methods for minimal realization of periodic systems. Systems & Control Lett., 36 339–349 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  25. Varga, A.: Balanced truncation model reduction for periodic systems. Proc. 39-th Conf. Dec. and Contr., Sydney, Australia 2379–2484 (2000)

    Google Scholar 

  26. Verriest, E.I.: On generalized balanced realizations. PhD Dissertation, Stanford University (1980)

    Google Scholar 

  27. Verriest, E.I.: Low sensitivity design and optimal order reduction for the LQG-Problem. Proceedings of the 1981 Midwest Symposium on Stochastic Systems, Albuquerque, NM 365–369 (1981)

    Google Scholar 

  28. Verriest, E.I.: Suboptimal LQG-design via balanced realizations. Proc. 20th IEEE Conf. on Dec. and Contr., San Diego, CA 686–687 (1981)

    Google Scholar 

  29. Verriest, E.I.: Approximation and order reduction in nonlinear models using an RKHS approach. Proc. 18th Conf. on Information Sciences and Systems, Princeton, NJ 197–201 (1984)

    Google Scholar 

  30. Verriest, E.I.: Stochastic reduced order modeling of deterministic systems. Proc. 1985 American Control Conf., Boston, MA 1003–1004 (1985)

    Google Scholar 

  31. Verriest, E.I.: Model reduction via balancing and connections with other methods. In Desai, U.B. (ed) Modeling and Application of Stochastic Processes. Kluwer Academic Publishers (1986)

    Google Scholar 

  32. Verriest, E.I.: Reduced-order LQG design: Conditions for feasibility. Proceedings of the 1986 IEEE Conference on Decision and Control, Athens, Greece 1765–1769 (1986)

    Google Scholar 

  33. Verriest, E.I.: Balancing for robust modeling and uncertainty equivalence. In Helmke, U., Mennicken, R. and Saurer, J. (eds) Systems and Networks: Mathematical Theory and Applications, Uni. Regensburg, 543–546 (1994)

    Google Scholar 

  34. Verriest, E.I.: Balancing for discrete periodic nonlinear systems. IFAC Workshop Prepr.: Periodic Control Systems, Como, Italy 253–258 (2001)

    Google Scholar 

  35. Verriest, E.I.: Pseudo balancing for discrete nonlinear systems. Proceedings of the MTNS 2002, Notre Dame (2002)

    Google Scholar 

  36. Verriest, E.I.: Nonlinear balancing and Mayer-Lie interpolation. Proc. SSST-04, Atlanta, GA 180–184 (2004)

    Google Scholar 

  37. Verriest, E.I.: Uncertainty equivalent model reduction. Proceedings of the MTNS 2006, Kyoto, Japan 1987–1992 (2006)

    Google Scholar 

  38. Verriest, E.I., and Gray, W.S.: Flow balancing nonlinear systems. Proceedings of the MTNS 2000, Perpignan, France (2000)

    Google Scholar 

  39. Verriest, E.I. and Gray, W.S.: Discrete time nonlinear balancing. Proceedings IFAC NOLCOS 2001, St Petersburg, Russia (2001)

    Google Scholar 

  40. Verriest, E.I. and Gray, W.S.: Nonlinear balanced realizations. Proc. 40-th IEEE Conference on Decision and Control, Orlando, 3250–3251 (2001)

    Google Scholar 

  41. Verriest, E.I. and Gray, W.S.: Nonlinear balanced realizations. Proc. 43d IEEE Conf. Dec. and Contr., Paradise Island, Bahamas, 1164-1169 (2004)

    Google Scholar 

  42. Verriest, E.I. and Gray, W.S.: Geometry and topology of the state space via balancing. Proceedings of the MTNS 2006, Kyoto, Japan 840–848 (2006)

    Google Scholar 

  43. Verriest, E.I. and Helmke, U.: Periodic balanced realizations. Proceedings of the IFAC Conference on System Structure and Control, Nantes, France 519–524 (1998)

    Google Scholar 

  44. Verriest, E.I. and Kailath, T.: On generalized balanced realizations. Proceedings of the 19th Conference on Decision and Control, Albuquerque, NM, 504–505 (1980)

    Google Scholar 

  45. Verriest, E.I. and Kailath, T.: On generalized balanced realizations. IEEE Transactions on Automatic Control, 28 8 833–844 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  46. Verriest, E.I. and Pajunen, G.A.: Robust control of systems with input and state constraints via LQG balancing. Proceedings of the 1992 IEEE Conference on Decision and Control, Tucson, AZ 2607–2610 (1992)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, 30332-0250, USA

    E. I. Verriest

Authors
  1. E. I. Verriest
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Physical Design Methods – Mathematics, NXP Semiconductors Corporate I&T/DTF/Design Methods, High Tech Campus 37, 5656, AE Eindhoven, The Netherlands

    Wilhelmus H. A. Schilders

  2. Physical Design Mothods, NXP Semiconductors Corporate I&T/DTF/Design Methods, High Tech Campus 37, 5656, AE Eindhoven, The Netherlands

    Henk A. van der Vorst

  3. Mathematical Institute, Utrecht University, Budapestlaan 6, 3508, TA Utrecht, The Netherlands

    Joost Rommes

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Verriest, E.I. (2008). Time Variant Balancing and Nonlinear Balanced Realizations. In: Schilders, W.H.A., van der Vorst, H.A., Rommes, J. (eds) Model Order Reduction: Theory, Research Aspects and Applications. Mathematics in Industry, vol 13. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78841-6_11

Download citation

Publish with us

Policies and ethics

Search

Navigation