Abstract
The computation of ad kf g requires derivatives of f and g up to order k. For small dimensions, the Lie brackets can be computed with computer algebra packages. The application to non-trivial systems is limited due to a burden of symbolic computations involved. The author proposes a method to compute function values of ad kf g using automatic differentiation.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
C. Bendtsen and O. Stauning. TADIFF, a flexible C++ package for automatic differentiation. Technical Report IMM-REP-1997-07, TU of Denmark, Dept. of Mathematical Modelling, Lungby, 1997.
A. Bensoussan and J. L. Lions, editors. Analysis and Optimization of Systems, Part 2, volume 63 of Lecture Notes in Control and Information Science. Springer, 1984.
Y. F. Chang. Automatic solution of differential equations. In D. L. Colton and R. P. Gilbert, editors, Constructive and Computational Methods for Differential and Integral Equations, volume 430 of Lecture Notes in Mathematics, pp. 61–94. Springer Verlag, New York, 1974.
Y. F. Chang. The ATOMCC toolbox. BYTE, 11(4):215–224, 1986.
B. Christianson. Reverse accumulation and accurate rounding error estimates for Taylor series. Optimization Methods and Software, 1:81–94, 1992.
G. F. Corliss and Y. F. Chang. Solving ordinary differential equations using Taylor series. ACM Trans. Math. Software, 8:114–144, 1982.
J.-M. Cornil and P. Testud. An Introduction to Maple V. Springer, 2001.
B. de Jager. The use of symbolic computation in nonlinear control: is it viable? IEEE Trans. on Automatic Control, AC-40(1):84–89, 1995.
A. Griewank. ODE solving via automatic differentiation and rational prediction. In D. F. Griffiths and G. A. Watson, editors, Numerical Analysis 1995, volume 344 of Pitman Research Notes in Mathematics Series. Addison-Wesley, 1995.
A. Griewank. Evaluating Derivatives — Principles and Techniques of Algorithmic Differentiation, volume 19 of Frontiers in Applied Mathematics. SIAM, 2000.
A. Griewank, D. Juedes, and J. Utke. A package for automatic differentiation of algorithms written in C/C++. ACM Trans. Math. Software, 22:131–167, 1996.
R. Hermann and A. J. Krener. Nonlinear controllability and observability. IEEE Trans. on Automatic Control, AC-22(5):728–740, 1977.
A. Isidori. Nonlinear Control Systems: An Introduction. Springer, 3rd edition, 1995.
B. Jakubczyk, W. Respondek, and K. Tchon, editors. Geomatric Theory of Nonlinear Control Systems. Technical University of Wroclaw, 1985.
D. Juedes and K. Balakrishnan. Generalized neuronal networks, computational differentiation, and evolution. In M. Berz, editor, Proc. of the Second International Workshop, pp. 273–285. SIAM, 1996.
A. J. Krener. (ad f,g), (ad f,g) and locally (ad f,g) invariant and controllability distributions. SIAM J. Control and Optimization, 23(4):523–524, 1985.
A. J. Krener and A. Isidori. Linearization by output injection and nonlinear observers. Systems & Control Letters, 3:47–52, 1983.
A. Kugi, K. Schlacher, and R. Novaki. Symbolic Computation for the Analysis and Sythesis of Nonlinear Control Systems, volume IV of Software for Electrical Engineering, Analysis and Design, pp. 255–264. WIT-Press, 1999.
R. Marino and G. Cesareo. Nonlinear control theory and symbolic algebraic manipulation. In Mathematical Theory of Networks and Systems, Proc. of MTNS’83, Beer Sheva, Israel, June 20–24, 1983, volume 58 of Lecture Notes in Control and Information Science, pp. 725–740. Springer, 1984.
R. Marino and G. Cesareo. The use of symbolic computation for power system stabilization: An example of computer aided design. In J. L. Lions, editors. Analysis and Optimization of Systems, Part 2, volume 63 of Lecture Notes in Control and Information Science. Springer, 1984 Bensoussan and Lions [2], pp. 598–611.
Neil Munro, editor. Symbolic methods in control system analysis and design. IEE, 1999.
H. Nijmeijer and A. J. van der Schaft. Nonlinear Dynamical Control systems. Springer, 1990.
W. Oevel, F. Postel, G. Rüscher, and St. Wehrmeier. Das MuPAD Tutorium. Springer, 1999. Deutsche Ausgabe.
K. Röbenack and K. J. Reinschke. A efficient method to compute Lie derivatives and the observability matrix for nonlinear systems. In Proc. 2000 International Symposium on Nonlinear Theory and its Applications (NOLTA’2000), Dresden, Sept. 17–21, volume 2, pp. 625–628, 2000.
K. Röbenack and K. J. Reinschke. Reglerentwurf mit Hilfe des Automatischen Differenzierens. Automatisierungstechnik, 48(2):60–66, 2000.
R. Rothfuß, J. Schaffner, and M. Schaffner, and M. Zeitz. Rechnergestützte Analyse und Synthese nichtlinearer Systeme. In S. Engell, editor, Nichtlineare Regelungen: Methoden, Werkzeuge, Anwendungen, volume 1026 of VDI-Berichte, p. 267–291. VDI-Verlag, Düsseldorf, 1993.
E. D. Sontag. Mathematical Control Theory, volume 6 of Texts in Applied Mathematics. Springer-Verlag, 2nd edition, 1998.
S. Wolfram. The MATHEMATICA Book. Cambridge University Press, 1999.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Röbenack, K. (2003). On the efficient computation of higher order maps ad kf g(x) using Taylor arithmetic and the Campbell-Baker-Hausdorff formula. 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_26
Download citation
DOI: https://doi.org/10.1007/3-540-45802-6_26
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43240-1
Online ISBN: 978-3-540-45802-9
eBook Packages: Springer Book Archive