Computation of multiple Lie derivatives by algorithmic differentiation. (English) Zbl 1136.93017

Summary: Lie derivatives are often used in nonlinear control and system theory. In general, these Lie derivatives are computed symbolically using computer algebra software. Although this approach is well-suited for small and medium-size problems, it is difficult to apply this technique to very complicated systems. We suggest an alternative method to compute the values of iterated and mixed Lie derivatives by algorithmic differentiation.


93B29 Differential-geometric methods in systems theory (MSC2000)
93B40 Computational methods in systems theory (MSC2010)
93C10 Nonlinear systems in control theory


Full Text: DOI


[1] C. Bendtsen, O. Stauning, TADIFF, a flexible \(\operatorname{C} ++\) package for automatic differentiation, Technical Report IMM-REP-1997-07, TU of Denmark, Deparment of Mathematical Modelling, Lungby, 1997.
[2] M. Berz, C. Bischof, G. Corliss, A. Griewank (Eds.), Computational Differentiation: Techniques, Applications, and Tools, SIAM, Philadelphia, PA, 1996. · Zbl 0857.00033
[3] J. Birk, M. Zeitz, Computer-aided design of nonlinear observers, in: Proceedings of the IFAC-Symposium Nonlinear Control System Design, Capri, Italy, 1989.
[4] H. Chen, A. Kremling, F. Allgöwer, Nonlinear predictive control of a CSTR benchmark problem, in: Proceedings of the Third European Control Conference ECC’95, Roma, Italy, 1995, pp. 3247-3252.
[5] Chen, K.-T., Integration of paths, geometric invariants and a generalized baker – hausdorff formula, Ann. math., 65, 1, 163-178, (1957) · Zbl 0077.25301
[6] G. Corliss, C. Faure, A. Griewank, L. Hascoët, U. Naumann (Eds.), Automatic Differentiation: From Simulation to Optimization, Springer, New York, 2002. · Zbl 0983.68001
[7] Faa de Bruno, F., Note sur une nouvelle formule de calcule differentiel, Quart. J. math., 1, 359-360, (1856)
[8] Fliess, M.; Lamnabhi, M.; Lannabhi-Lagarrigue, F., An algebraic approach to nonlinear functional expansion, IEEE trans. circuits and systems, 30, 8, 554-570, (1983) · Zbl 0529.34002
[9] J. C. Gómez, Using symbolic computation for the computer aided design of nonlinear (adaptive) control systems, in: 14th IMACS World Congress on Computational and Applied Mathematics, Atlanta, USA, July 11-15, 1994.
[10] A. Griewank, Evaluating Derivatives—Principles and Techniques of Algorithmic Differentiation, Frontiers in Applied Mathematics, vol. 19, SIAM, Philadelphia, PA, 2000. · Zbl 0958.65028
[11] Griewank, A.; Juedes, D.; Utke, J., ADOL-C: a package for automatic differentiation of algorithms written in \(\operatorname{C} / \operatorname{C} ++\), ACM trans. math. software, 22, 131-167, (1996) · Zbl 0884.65015
[12] Griewank, A.; Utke, J.; Walther, A., Evaluating higher derivative tensors by forward propagation of univariate Taylor series, Math. comput., 69, 1117-1130, (2000) · Zbl 0952.65028
[13] Gröbner, W., Die Lie-reihen und ihre anwendung, (1967), Deutscher Verlag der Wissenschaften Berlin
[14] Hardy, M., Combinatorics of partial derivatives, Electron. J. combin., 13, 1-13, (2006) · Zbl 1080.05006
[15] Isham, C.J., Modern differential geometry for physicists, (2001), World Scientific Singapore · Zbl 0746.53001
[16] Isidori, A., Nonlinear control systems: an introduction, (1995), Springer London · Zbl 0569.93034
[17] de Jager, B., The use of symbolic computation in nonlinear control: is it viable?, IEEE trans. automat. control, 40, 1, 84-89, (1995) · Zbl 0925.93215
[18] B. de Jager, Symbolic computation in nonlinear control system modelling and analysis, in: 10th IEEE International Symposium on Computer Aided Control System Design jointly with the 1999 Conference on Control Applications (1999 IEEE CCA/CACSD), Kohala Cost, Island of Hawai, USA, August 22-27, 1999.
[19] K.-U. Klatt, S. Engell, Rührkesselreaktor mit Parallel- und Folgereaktion, in: S. Engell (Ed.), Nichtlineare Regelungen: Methoden, Werkzeuge, Anwendungen, VDI-Berichte, vol. 1026, VDI-Verlag, Düsseldorf, 1993, pp. 101-108.
[20] A. Kugi, K. Schlacher, R. Novaki, Symbolic Computation for the Analysis and Synthesis of Nonlinear Control Systems, Software Studies, vol. 2, WIT-Press, Southampton, 1999, pp. 255-264.
[21] Kwatny, H.G.; Blankenship, G.L., Nonlinear control and analytical mechanics: A computational approach, (2000), Birkhäuser Boston · Zbl 0963.93007
[22] Mishkov, R.L., Generalization of the formula of faa di bruno for a composite function with a vector argument, Internat. J. math. math. sci., 24, 7, 481-491, (2000) · Zbl 0967.46031
[23] Nijmeijer, H.; van der Schaft, A.J., Nonlinear dynamical control systems, (1990), Springer Berlin · Zbl 0701.93001
[24] V. Polyakov, R. Ghanadan, G. L. Blankenship, Symbolic numerical computational tools for nonlinear and adaptive control, in: Proceedings of the IEEE/IFAC Joint Symposium on Computer-Aided Control System Design, Tucson, Arizona, 1994, pp. 117-122. · Zbl 0925.93057
[25] J. D. Pryce, J. K. Reid, AD01, a Fortran 90 code for automatic differentiation, Technical Report RAL-TR-1998-057, Rutherford Appleton Laboratory, Computing and Information Systems Department, 1998.
[26] Röbenack, K., Computation of Lie derivatives of tensor fields required for nonlinear controller and observer design employing automatic differentiation, Proc. appl. math. mech., 5, 1, 181-184, (2005) · Zbl 1391.93102
[27] Röbenack, K.; Reinschke, K.J., The computation of Lie derivatives and Lie brackets based on automatic differentiation, Z. angew. math. mech., 84, 2, 114-123, (2004) · Zbl 1037.93029
[28] R. Rothfuß, Anwendung der flachheitsbasierten Analyse und Regelung nichtlinearer Mehrgrößensysteme, VDI-Forschrittsberichte, vol. 664, Reihe 8: Meß-, Steuerungs- und Regelungstechnik, VDI-Verlag, Düsseldorf, 1997.
[29] Rothfuß, R.; Rudolph, R.; Zeitz, M., Flatness based control of a nonlinear chemical reactor model, Automatica, 32, 10, 1433-1439, (1996) · Zbl 0865.93046
[30] Rugh, W.J., Nonlinear system theory: the Volterra/Wiener approach, (1981), The Johns Hopkins University Press Baltimore, MD · Zbl 0666.93065
[31] O. Stauning, Flexible automatic differentiation using templates and operator overloading in \(\operatorname{C} ++\), Talk Presented at the Automatic Differentiation Workshop at Shrivenham Campus, Cranfield University, June 2003.
[32] Wambacq, P.; Sansen, W., Distortion analysis of analog integrated circuits, (1998), Kluwer Academic Publishers Dordrecht
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.