Abstract
Differentiator blocks are useful in many applications. However, their use leads to the simulation of a numerically sensitive system since they use difference formulas. We suggest a new method which provides exact derivative values. Our approach is applicable to simulators based on object oriented programming languages.
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
A. Isidori. Nonlinear Control Systems: An Introduction. Springer-Verlag, 3 edition, 1995.
H. Nijmeijer and A. J. van der Schaft. Nonlinear Dynamical Control systems. Springer Verlag, 1990.
M. Fliess, J. Lévine, P. Martin, and P. Rouchon. Flatness and defect of non-linear systems: Introductory theory and examples. Int. J. Control, 61: 1327–1361, 1995.
J. Rudolph, editor. Fortbildungskurs Flachheitsbasierte Regelung, Institut für Regelungs-und Steuerungstheorie, TU Dresden, 1997.
R. Rothfuß, J. Rudolph, and M. Zeitz. Flachheit: Ein neuer Zugang zur Steuerung und Regelung nichtlinearer Systeme. Automatisierungstechnik, 45: 517–525, 1997.
J. v. Löwis, J. Rudolph, and K. J. Reinschke. Neues Regelungskonzept für SensorAktor-Systeme: Lageregelung eines Langhubmagneten. Technical Report SFB 358-D5–1/98, TU Dresden, Sonderforschungsbereich 358, 1998.
U. Feldmann, M. Hasler, and W. Schwarz. Communication by chaotic signals: the inverse system approach. Int. J. of Circuit Theory and Applications, 24(5):551579, 1996.
U. Feldmann, M. Hasler, and W. Schwarz. On the design of a synchronizing inverse of a chaotic system. In Proc. Europ. Conf. Circ. Th. Design (ECCTD), Istanbul, Turkey, volume 1, pages 479–482, 1995.
O. E. Rössler. An equation for continuous chaos. Phys. Lett., 57A: 397, 1976.
O. E. Rössler. Continous chaos — four prototyp equations. In O. Gruel and O. E. Rössler, editors, Bifurcation Theory and Applications in Scientific Disciplines, volume 316 of Ann. N.Y. Acad. Sci., pages 376–392, 1979.
G. Jetschke. Mathematik der Selbstorganisation. VEB Deutscher Verlag der Wissenschaften, Berlin, 1989.
H. G. Schuster. Deterministic Chaos - An Introduction. VCH Verlagsgesellschaft rnbH, Weinheim, 1989.
B. Speelpenning. Compiling Fast Partial Derivatives of Functions Given by Algorithms. PhD thesis, Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana-Champaign, Ill., Jan. 1980.
G. F. Corliss. Automatic differentiation bibliography. In A. Griewank and G. F. Corliss, editors, Automatic Differentiation of Algorithms: Theory, Implementation, and Application, pages 331–353. SIAM, Philadelphia, Penn., 1991.
G. Corliss. Automatic differentiation bibliography. Technical Memorandum ANL/MCS-TM-167, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Ill., July 1992.
A. Griewank and G. F. Corliss, editors. Issus in Parallel Automatic Differentiation. 1991.
M. Iri. History of automatic differentiation and rounding error estimation. In A. Griewank and G. F. Corliss, editors, Automatic Differentiation of algorithms: Theory, implementation, and application, pages 3–16, Breckenridge, Colorado, 1991. Proc. of the 1st SIAM Workshop on Automatic Differentiation.
B. Christianson. Reverse accumulation and accurate rounding error estimates for Taylor series. Optimization Methods and Software, 1: 81–94, 1992.
A. Griewank. On automatic differentiation. In M. Iri and K. Tanabe, editors, Mathematical Programming: Recent Developments and Applications, pages 83108. Kluwer Academic Publishers, 1989.
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. http://www.math.tu-dresden.de/wir/project/adolc/index.html.
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.
M. Grundmann and M. Masmoudi. MIPAD: an AD package. AD’2000, Nice (France), 2000.
D. Werner. Funktionalanalysis. Springer-Verlag, 1997.
T. F. Coleman and A. Verma. Structured automatic differentiation. http://www.cs.cornell.edu/Info/People/verma/AD/research.html.
T. F. Coleman and A. Verma. ADMIT-1: Automatic differentiation and MATLAB interface toolbox, user guide, release alpha 1. Technical Report CTC97TR271, Cornell Theoy Center, 1997.
T. F. Coleman and A. Verma. ADMIT-1: Automatic differentiation and MAT-LAB interface toolbox. Available as Cornell Computer Science technical report CS TR 98–1663 and as Cornell CCOP Technical report TR 98–1, 1998.
A. Verma. ADMAT: Automatic differentiation for MATLAB using object oriented methods. Submitted to SIAM workshop on object oriented methods, 1999.
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.
G. F. Corliss and Y. F. Chang. Solving ordinary differential equations using Taylor series. ACM Trans. Math. Software, 8: 114–144, 1982.
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, pages 61–94. Springer Verlag, New York, 1974.
Y. F. Chang. The ATOMCC toolbox. BYTE, 11 (4): 215–224, 1986.
K. E. Brenan, S. L. Campbell, and L. R. Petzold. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. North-Holland, 1989.
E. Griepentrog. The index of differential-algebraic equations and its significance for the circuit simulation. In Mathematical Modelling and Simulation of Electrical Circuits and Semiconductor Devices, Conference held at the Mathematisches Forschungsinstitut, Oberwolfach, pages 11–25. Birkhäuser Verlag, 1990.
E. Griepentrog and R. März. Differential-Algebraic Equations and Their Numerical Treatment, volume 88 of Teubner-Texte zur Mathematik. Teubner Verlagsgesellschaft, Leipzig, 1986.
E. Hairer, C. Lubich, and M. Roche. The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods, volume 1409 of Lecture Notes in Mathematics. Springer-Verlag, 1989.
B. Straube, K. Reinschke, W. Vermeiren, K. Röbenack, B. Müller, and C. Clauß. DAE-index increase in analogue fault simulation. In Proc. Workshop on System Design Automation SDA’2000, Rathen (Germany), March 13–14, pages 99–106, 2000.
P. Schwarz et al. KOSIM ein Mixed-Mode, Multi-Level-Simulator. Informatik- Fachberichte, 225: 207–220, 1990.
C. W. Gear and L. R. Petzold. ODE methods for the solution of differential/algebraic systems. SIAM J. Numer. Anal., 21 (4): 717–728, August 1984.
C. W. Gear. Differential-algebraic equation index transformations. SIAM J. Sci. Stat. Comput., 9 (1): 39–47, 1988.
P. J. Rabier and W. C. Rheinboldt. A general existence and uniqueness theory for implicit differential-algebraic equations. Differential and Integral Equations, 4 (3): 563–582, May 1991.
S. Reich. On a geometrical interpretation of differential-algebraic equations. Circuits Systems Signal Processing, 9 (4): 367–382, 1990.
K. Röbenack. Nutzung des Automatischen Differenzierens in der nicht-linearen Regelungstheorie. In Gemeinsamer Workshop des GAMM-Fachausschusses “Dynamik und Regelungstheorie” und des VDI/VDE-GMAAusschusses 1.40 “Theoretische Verfahren der Regelungstechnik”, Kassel, 01.-02. März, 1999.
K. Röbenack and K. J. Reinschke. Trajektorienplanung und Systeminversion mit Hilfe des Automatischen Differenzierens. In Workshop des GMA-Ausschusses 1.4 “Neuere theoretische Verfahren der Regelungstechnik”, Thun, Sept. 26–29, pages 232–242, 1999.
K. Röbenack and K. J. Reinschke. Reglerentwurf mit Hilfe des Automatischen Differenzierens. Automatisierungstechnik, 48 (2): 60–66, 2000.
K. Röbenack and K. J. Reinschke. Nonlinear observer design using automatic differentiation. AD’2000, Nice, 2000.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Röbenack, K., Reinschke, K.J. (2001). Simulation of Numerically Sensitive Systems by Means of Automatic Differentiation. In: Merker, R., Schwarz, W. (eds) System Design Automation. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6666-0_18
Download citation
DOI: https://doi.org/10.1007/978-1-4757-6666-0_18
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4886-1
Online ISBN: 978-1-4757-6666-0
eBook Packages: Springer Book Archive