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On computation of the logarithm of the Chen-Fliess series for nonlinear systems

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Nonlinear and Adaptive Control

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 281))

Abstract

We obtain expressions for the logarithm of the Chen-Fliess series for a nonlinear control system. This logarithm provides an alternative to Sussmann’s exponential product expansion of the Chen-Fliess series. We also formulate a rule for generating the coefficient of any Lie brackets in the logarithm expansion.

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References

  1. Agrachev A.A., Gamkrelidze R.V. (1978) Exponential representation of flows and chronological calculus. Matem. Sbornik 107, 467–532. English transl. in: Math. USSR Sbornik 35 (1979), 727–785.

    MathSciNet  Google Scholar 

  2. Agrachev A.A., Gamkrelidze R.V. (1979) Chronological algebras and nonstationary vector fields. Journal Soviet Mathematics 17, 1650–1675 (in Russian).

    Article  Google Scholar 

  3. Agrachev A.A., Gamkrelidze R.V., Sarychev A.V. (1989) Local invariants of smooth control systems. Acta Applicandae Mathematicae 14, 191–237.

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen K.T. (1957) Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Annals of Mathematics 65, 163–178.

    Article  MathSciNet  Google Scholar 

  5. Fliess M. (1989) Fonctionelles causales nonlinéaires et indeterminées noncommutatives. Acta Appl. Math. 15, 3–40.

    Article  MathSciNet  Google Scholar 

  6. Kawski M. (2000) Calculating the logarithm of the Chen Fliess series. International Journal Control (submitted) (also: elec. proc. MTNS 2000).

    Google Scholar 

  7. Kawski M., Sussmann H.J. (1997) Noncommutative power series and formal Lie-algebraic techniques in nonlinear control theory. in: Helmke U., Prätzel-Wolters D., Zerz E. Eds. Systems and Linear Algebra. Teubner, Stuttgart, 111–128.

    Google Scholar 

  8. Loday J-L. (2001) Dialgebras. in: Dialgebras and related operads. Springer Lecture Notes in Math. 1763, 7–66.

    Article  Google Scholar 

  9. Reutenauer C. (1993) Free Lie algebras. London Mathematical Society Monographs-new series 7, Oxford Science Publications.

    Google Scholar 

  10. Rocha E.M. (2001) On computation of the logarithm of the Chen-Fliess series for nonlinear systems. CM01/I-08, Departamento de matemática, Universidade de Aveiro.

    Google Scholar 

  11. Sarychev A.V. (2000) Stability criteria for time-periodic systems via high-order averaging techniques. in: Isidori A., Lamnabhi-Lagarrique F., Respondek W. Eds., Nonlinear Control in the Year 2000, 2, Springer-Verlag, Berlin et al, 365–377.

    Google Scholar 

  12. Sussmann H.J. (1986) A product expansion of the Chen series. In Theory and Applications of Nonlinear Control Systems, Byrnes C.I., Lindquist A. eds., Elsevier, North-Holland.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Departamento de Matemática, University of Aveiro, 3810-193, Aveiro, Portugal

    Eugénio M. Rocha

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  1. Eugénio M. Rocha
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Editor information

Editors and Affiliations

  1. Departement of AppliedMathematics, The University of Sheffield, Western Bank, Sheffield, S10 2TN, UK

    Alan Zinober

  2. Dept. of Automatic Control and Systems Engineering, The University of Sheffield, Mappin Street, Sheffield, S1 3JD, UK

    David Owens

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© 2003 Springer-Verlag Berlin Heidelberg

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Rocha, E.M. (2003). On computation of the logarithm of the Chen-Fliess series for nonlinear systems. In: Zinober, A., Owens, D. (eds) Nonlinear and Adaptive Control. Lecture Notes in Control and Information Sciences, vol 281. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45802-6_25

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  • DOI: https://doi.org/10.1007/3-540-45802-6_25

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43240-1

  • Online ISBN: 978-3-540-45802-9

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