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Mathematical Control Theory I pp 349–362Cite as

Model Reduction by Generalized Differential Balancing

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Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 461))

Abstract

In this chapter, we give a generalization of differential balancing method for model reduction of nonlinear systems in the direction to computation. We generalize concepts of differential controllability and observability functions, then use them for model reduction. We show some stability properties are preserved under the model reduction and estimate the error bound by the model reduction.

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References

  1. D. Angeli, A Lyapunov approach to incremental stability properties. IEEE Trans. Autom. Control 47(3), 410–421 (2002)

    Article  MathSciNet  Google Scholar 

  2. A.C. Antoulas, Approximation of Large-Scale Dynamical Systems (SIAM, Philadelphia, 2005)

    Book  MATH  Google Scholar 

  3. A. Astolfi, Model reduction by moment matching for linear and nonlinear systems. IEEE Trans. Autom. Control 55(10), 2321–2336 (2010)

    Article  MathSciNet  Google Scholar 

  4. B. Besselink, N. Van De Wouw, J.M.A. Scherpen, H. Nijmeijer, Model reduction for nonlinear systems by incremental balanced truncation. IEEE Trans. Autom. Control 59(10), 2739–2753 (2014)

    Article  Google Scholar 

  5. F. Forni, R. Sepulchre. On Differentially Dissipative Dynamical Systems, in Proceedings of the 8th IFAC Symposium on Nonlinear Control Systems, pp. 21–25 (2013)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. K. Fujimoto, J.M.A. Scherpen, Nonlinear input-normal realizations based on the differential eigenstructure of hankel operators. IEEE Trans. Autom. Control 50(1), 2–18 (2005)

    Article  MathSciNet  Google Scholar 

  8. K. Fujimoto, J.M.A. Scherpen, Model reduction for nonlinear systems based on the balanced realization. SIAM J. Control Optim. 48(7), 4591–4623 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. T.C. Ionescu, A. Astolfi. Moment matching for nonlinear port Hamiltonian and gradient systems, in Preprints of the 9th IFAC Symposium on Nonlinear Control Systems (2013)

    Google Scholar 

  10. Y. Kawano, J.M.A. Scherpen. On Differential Balancing: Energy Functions and Balanced Realization, in Proceedings of European Control Conference, Linz, Austria, July 2015. To appear

    Google Scholar 

  11. W. Lohmiller, J.J.E. Slotine. On contraction analysis for non-linear systems. Automatica 34(6), 683–696 (1998)

    Google Scholar 

  12. I.R. Manchester, J.J.E. Slotine. Control contraction metrics and universal stabilizability, in Preprints of the 19th IFAC World Congress (2014)

    Google Scholar 

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

  14. M. Sassano, A. Astolfi, Dynamic generalized controllability and observability functions with applications to model reduction and sensor deployment. Automatica 50(5), 1349–1359 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. J.M.A. Scherpen, Balancing for nonlinear systems. Syst. Control Lett. 21(2), 143–153 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  16. A.J. van der Schaft, On DIfferential Passivity, in Proceedings of the 8th IFAC Symposium on Nonlinear Control Systems, pp. 15–20 (2013)

    Google Scholar 

  17. E.I. Verriest, W.S. Gray, Nonlinear balanced realizations, in 43rd IEEE Conference on Decision and Control, vol. 2, pp. 1164–1169 (2004)

    Google Scholar 

  18. E.I. Verriest, W.S. Gray, Flow balancing nonlinear systems, in 14th International Symposium on Mathematical Theory of Networks and Systems, Perpignan, France (2000)

    Google Scholar 

  19. K. Zhou, J.C. Doyle, K. Glover. Robust and Optimal Control, vol. 40. (Prentice Hall, New Jersey, 1996)

    Google Scholar 

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Acknowledgments

This work of Y. Kawano was partly supported by JST CREST and JSPS KAKENHI Grant Number 15K18087. The work was partly performed while Yu Kawano was visiting researcher at the University of Groningen.

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Authors and Affiliations

  1. Graduate School of Informatics, Kyoto University, Sakyo-ku, Kyoto, 606-8501, Japan

    Yu Kawano

  2. Engineering and Technology Institute Groningen, Jan C. Willems Center for Systems and Control, ENTEG-DTPA, University of Groningen, Nijenborgh, 4, 9747AG, Groningen, The Netherlands

    Jacquelien M.A. Scherpen

Authors
  1. Yu Kawano
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  2. Jacquelien M.A. Scherpen
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Corresponding author

Correspondence to Jacquelien M.A. Scherpen .

Editor information

Editors and Affiliations

  1. Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Groningen, The Netherlands

    M. Kanat Camlibel

  2. Department of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic Institute, Troy, New York, USA

    A. Agung Julius

  3. Department of Electrical Engineering, Indian Institute of Technology, Chennai, Tamil Nadu, India

    Ramkrishna Pasumarthy

  4. Engineering and Technology Institute Groningen, University of Groningen, Groningen, The Netherlands

    Jacquelien M.A. Scherpen

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Kawano, Y., Scherpen, J.M. (2015). Model Reduction by Generalized Differential Balancing. In: Camlibel, M., Julius, A., Pasumarthy, R., Scherpen, J. (eds) Mathematical Control Theory I. Lecture Notes in Control and Information Sciences, vol 461. Springer, Cham. https://doi.org/10.1007/978-3-319-20988-3_19

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  • DOI: https://doi.org/10.1007/978-3-319-20988-3_19

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20987-6

  • Online ISBN: 978-3-319-20988-3

  • eBook Packages: EngineeringEngineering (R0)

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