Article | Proceedings of the 4th International Workshop on Equation-Based Object-Oriented Modeling Languages and Tools; Zurich; Switzerland; September 5; 2011 | LIEDRIVERS— A Toolbox for the Efficient Computation of Lie Derivatives Based on the Object-Oriented Algorithmic Differentiation Package ADOL-C

Title:
LIEDRIVERS— A Toolbox for the Efficient Computation of Lie Derivatives Based on the Object-Oriented Algorithmic Differentiation Package ADOL-C
Author:
Klaus Röbenack: Technische Universität Dresden. Faculty of Electrical and Computer Engineering. Institute of Control Theory, Germany Jan Winkler: Technische Universität Dresden. Faculty of Electrical and Computer Engineering. Institute of Control Theory, Germany Siqian Wang: Technische Universität Dresden. Faculty of Electrical and Computer Engineering. Institute of Control Theory, Germany
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Full text (pdf)
Year:
2011
Conference:
Proceedings of the 4th International Workshop on Equation-Based Object-Oriented Modeling Languages and Tools; Zurich; Switzerland; September 5; 2011
Issue:
056
Article no.:
007
Pages:
57-66
No. of pages:
10
Publication type:
Abstract and Fulltext
Published:
2011-11-03
ISBN:
978-91-7519-825-5
Series:
Linköping Electronic Conference Proceedings
ISSN (print):
1650-3686
ISSN (online):
1650-3740
Publisher:
Linköping University Electronic Press; Linköpings universitet


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Lie derivatives are widely used in mathematics and physics. They are usually computed symbolically using computer algebra software. This symbolic computation might fail for very complicated expressions. Moreover; symbolic differentiation becomesmore difficult if the function to be differentiated is not described explicitly as a function but by an algorithm. This is a situation occuring quite often in modeling languages. In this contribution we present an approach for calculating Lie derivatives based on algorithmic differentiation using the software package ADOL-C avoiding the drawbacks of symbolic differentiation.

Keywords: Lie derivatives; algorithmic differentiation

Proceedings of the 4th International Workshop on Equation-Based Object-Oriented Modeling Languages and Tools; Zurich; Switzerland; September 5; 2011

Author:
Klaus Röbenack, Jan Winkler, Siqian Wang
Title:
LIEDRIVERS— A Toolbox for the Efficient Computation of Lie Derivatives Based on the Object-Oriented Algorithmic Differentiation Package ADOL-C
References:

[1] CppAD: A Package for Differentiation of C++ Algorithms. http://www.coin-or.org/CppAD/.


[2] Maxima; a computer algebra system. http://maxima. sourceforge.net/.


[3] R. Abraham; J. E. Marsden; and T. Ratiu. Manifolds;Tensor Analysis; and Applications. Springer; New York; 2nd edition; 1983.


[4] www.autodiff.org. Web-Portal ĂĽber Automatisches Differenzieren.


[5] C. Bendtsen and O. Stauning. FADBAD; a flexible C++ package for automatic differentiation. Technical Report IMM-REP-1996-17; TU of Denmark; Dept. of Mathematical Modelling; Lungby; 1996.


[6] 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.


[7] G. L. Blankenship and H. G. Kwatny. Computational methods for control of dynamical systems. In N. Munro; editor; Symbolic methods in control system analysis and design; volume 56 of IEE Control Engineering Series; chapter 15. IEE Press; London; 1999.


[8] B. Christianson. Reverse accumulation and accurate rounding error estimates for Taylor series. Optimization Methods & Software; 1:81–94; 1992.


[9] G. Ciccarella; M. Dalla Mora; and A. Germani. A Luenberger-like observer for nonlinear systems. Int. J. Control; 57(3):537–556; 1993.


[10] J.-M. Cornil and P. Testud. An Introduction to Maple V. Springer; 2001.


[11] M. DallaMora; A. Germani; and C.Manes. A state observerfor nonlinear dynamical systems. Nonlinear Analysis; Theory; Methods & Applications; 30(7):4485–4496; 1997.


[12] B. de Jager. The use of symbolic computation in nonlinearcontrol: Is it viable? IEEE Trans. on Automatic Control; 40(1):84–89; 1995.


[13] S. A. Forth. An efficient overloaded implementation offorward mode automatic differentiation in MATLAB. ACM Transactions on Mathematical Software; 32(2):195–222; jun 2006.


[14] J. P. Gauthier; H. Hammouri; and S. Othman. A simple observer for nonlinear systems—application to bioreactors. IEEE Trans. on Automatic Control; 37(6):875–880; 1992.


[15] R. Giering and Th. Kaminski. Recomputations in reverse mode ad. In G. Corliss; Ch. Faure; A. Griewank; Hascoët; and U. Naumann; editors; Automatic Differentiation: From Simulation to Optimization; chapter 33; pages 283–290. Springer; 2002.


[16] A. Griewank; D. Juedes; and J. Utke. ADOL-C: Apackage for automatic differentiation of algorithms written in C/C++. ACM Trans. Math. Software; 22:131–167; 1996.


[17] A. Griewank and A. Walther. Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. SIAM; 2nd edition; 2008.


[18] D. D. Holm; T. Schmah; and C. Stoica. Geometric Mechanicas and Symmetry. Oxford University Press; 2009.


[19] C. J. Isham. Modern Differential Geometry for Physicists. World Scientific; 2 edition; 2001.


[20] A. Isidori. Nonlinear Control Systems: An Introduction. Springer-Verlag; London; 3. edition; 1995.


[21] A. Kugi; K. Schlacher; and R. Novaki. Symbolic Computation for the Analysis and Synthesis of Nonlinear Control Systems; volume 2 of Software Studies; pages 255–264. WIT-Press; Southampton; 1999.


[22] H. G. Kwatny and G. L. Blankenship. Nonlinear Control and Analytical Mechanics: A Computational Approach. Birkhäuser; Boston; 2000.


[23] J. M. Lee. Introduction to Smooth Manifolds; volume 218 of Graduate Texts in Mathematics. Springer; New York; 2006.


[24] M. Lemmen; T. Wey; and M. Jelali. NSAS – ein Computer- Algebra-Packet zur Analyse und Synthese nichtlinearer systeme. Forschungsbericht Nr. 20/95; Gerhard-Mercator- Universität-GH Duisburg; Meß-; Steuer- und Regelungstechnik; 1995.


[25] H. Nijmeijer and A. J. van der Schaft. Nonlinear Dynamical Control systems. Springer; New York; 1990.


[26] R. Oloff. Geometrie der Raumzeit. Vieweg Verlag; Wiesbaden; 3. edition; 2004.


[27] V. Polyakov; R. Ghanadan; and G. L. Blankenship. Symbolic numerical computational tools for nonlinear and adaptive control. In Proc. IEEE/IFAC Joint Symposium on Computer-Aided Control System Design; pages 117–122; Tucson; Arizona; 1994.


[28] K. Röbenack. On the efficient computation of higher order maps adk fg(x) using Taylor arithmetic and the Campbell- Baker-Hausdorff formula. In Alan Zinober and David Owens; editors; Nonlinear and Adaptive Control; volume 281 of Lecture Notes in Control and Information Science; pages 327–336. Springer; 2002.


[29] K. Röbenack. Automatic differentiation and nonlinear controller design by exact linearization. Future Generation Computer Systems; 21(8):1372–1379; 2005.


[30] K. Röbenack. Computation of high gain observers for nonlinear systems using automatic differentiation. Journal Dynamic Systems; Measurement; and Control; 127(1):160– 162; 2005.


[31] K. Röbenack. Computation of Lie derivatives of tensorfields required for nonlinear controller and observer design employing automatic differentiation. Proc. in Applied Mathematics and Mechanics; 5(1):181–184; 2005.


[32] K. Röbenack. Nonlinear controller design based on algorithmic plant description. Mathematical and Computer Modelling of Dynamical Systems; 13(2):193–209; 2007.


[33] 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; pages 625–628; 2000.


[34] K. Röbenack and K. J. Reinschke. Reglerentwurf mit Hilfe des Automatischen Differenzierens. Automatisierungstechnik; 48(2):60–66; 2000.


[35] K. Röbenack and K. J. Reinschke. The computation of Lie derivatives and Lie brackets based on automatic differentiation. Z. Angew. Math. Mech.; 84(2):114–123; 2004.


[36] R. Rothfuss and M. Zeitz. Einführung in die Analysenichtlinearer Systeme. In S. Engell; editor; Entwurf nichtlinearer Regelungen; pages 3–22. Oldenbourg-Verlag; München; 1995.


[37] C. Rui; I. V. Kolmanovsky; and N. H. McClamroch. Symboliccomputation in nonlinear control of multibody space systems. In ACC’1998 (American Control Conference); Philadelphia; 1998.


[38] J. Schaffner andM. Zeitz. Variants of nonlinear normal form observer design. In H. Hijmeijer and T. I. Fossen; editors; New Direction in Nonlinear Observer Design; volume 244 of Lecture Notes in Control and Information Science; pages 161–180. Springer-Verlag; London; 1999.


[39] H. Skaug and D Fournier. Automatic approximation of the marginal likelihood in nonlinear hierarchical models. Computational Statistics and Data Analysis; 51(2):699– 709; 2006.


[40] S. Stamatiadis; R. Prosmiti; and S. C. Farantos.AUTO_DERIV: Tool for automatic differentiation of a FORTRAN code. Comput. Phys. Commun.; 127(2&3):343–355; may 2000. Catalog number: ADLS.


[41] O. Stauning and C. Bendtsen. FADBAD++: Flexibleautomatic differentiation using templates and operator overloading in ANSI C++. http://www2.imm.dtu. dk/~km/FADBAD/.


[42] V. Toth. Tensor manipulation in GPL Maxima. arXiv:cs/0503073v2 [cs.SC]; 2005.


[43] V. S. Varadarajan. Lie Groups; Lie Algebras; and Their Representation. Springer-Verlag; 1984.


[44] A. Verma. ADMAT: Automatic differentiation for MATLAB using object oriented methods. In SIAM workshop on object oriented methods; pages 174–183; 1999.


[45] G. M. von Hippel. Taylur; an arbitrary-order automatic differentiation package for fortran 95. Comput.Phys.Commun.; 174:569–576; 2006.


[46] A. Walther; A. Griewank; and O. Vogel. ADOL-C: Automatic differentiation using operator overloading in C++. Proc. in Applied Mathematics and Mechanics; 2(1):41–44; 2003.


[47] P. J. Werbos. The Roots of Backpropagation: From Ordered Derivatives to Neural Networks and Political Forecasting. Wiley; 1994.


[48] S. Wolfram. The MATHEMATICA Book. Cambridge University Press; 1999.


[49] M. Zeitz. The extended Luenberger observer for nonlinear systems. Systems & Control Letters; 9:149–156; 1987.

Proceedings of the 4th International Workshop on Equation-Based Object-Oriented Modeling Languages and Tools; Zurich; Switzerland; September 5; 2011

Author:
Klaus Röbenack, Jan Winkler, Siqian Wang
Title:
LIEDRIVERS— A Toolbox for the Efficient Computation of Lie Derivatives Based on the Object-Oriented Algorithmic Differentiation Package ADOL-C
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