PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2013 | 23 | 4 | 731-747
Tytuł artykułu

A verified method for solving piecewise smooth initial value problems

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In many applications, there is a need to choose mathematical models that depend on non-smooth functions. The task of simulation becomes especially difficult if such functions appear on the right-hand side of an initial value problem. Moreover, solution processes from usual numerics are sensitive to roundoff errors so that verified analysis might be more useful if a guarantee of correctness is required or if the system model is influenced by uncertainty. In this paper, we provide a short overview of possibilities to formulate non-smooth problems and point out connections between the traditional non-smooth theory and interval analysis. Moreover, we summarize already existing verified methods for solving initial value problems with non-smooth (in fact, even not absolutely continuous) right-hand sides and propose a way of handling a certain practically relevant subclass of such systems. We implement the approach for the solver VAL E NC IA-IVP by introducing into it a specialized template for enclosing the first-order derivatives of non-smooth functions. We demonstrate the applicability of our technique using a mechanical system model with friction and hysteresis. We conclude the paper by giving a perspective on future research directions in this area.
Rocznik
Tom
23
Numer
4
Strony
731-747
Opis fizyczny
Daty
wydano
2013
otrzymano
2012-12-05
poprawiono
2013-04-30
poprawiono
2013-06-19
Twórcy
  • Department of Computer Science and Applied Cognitive Science, University of Duisburg-Essen, 47048 Duisburg, Germany
autor
  • Department of Computer Science and Applied Cognitive Science, University of Duisburg-Essen, 47048 Duisburg, Germany
autor
  • Chair of Mechatronics, University of Rostock, Justus-von-Liebig-Weg 6, 18059 Rostock, Germany
Bibliografia
  • Acary, V. and Brogliato, B. (2008). Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics, Lecture Notes in Applied and Computational Mechanics, Vol. 35, Springer-Verlag, Berlin/Heidelberg.
  • Alefeld, G. and Herzberger, J. (1983). Introduction to Interval Computations, Computer Science and Applied Mathematics, Academic Press, New York, NY.
  • Auer, E., Albassam, H., Kecskeméthy, A. and Luther, W. (2011). Verified analysis of a model for stance stabilization, in A. Rauh and E. Auer (Eds.), Modeling, Design and Simulation of Systems with Uncertainties, Mathematical Engineering, Springer-Verlag, Berlin/Heidelberg, pp. 294-308.
  • Barboteu, M., Bartosz, K. and Kalita, P. (2013). An analytical and numerical approach to a bilateral contact problem with nonmonotone friction, International Journal of Applied Mathematics and Computer Science 23(2): 263-276, DOI: 10.2478/amcs-2013-0020.
  • Bernardo, M., Budd, C., Champneys, A. and Kowalczyk, P. (2007). Piecewise-smooth Dynamical Systems: Theory and Applications, Applied Mathematical Sciences, Springer-Verlag, London.
  • de Figueiredo, L.H. and Stolfi, J. (2004). Affine arithmetic: Concepts and applications, Numerical Algorithms 37(1-4): 147-158.
  • Dötschel, T., Auer, E., Rauh, A. and Aschemann, H. (2013). Thermal behavior of high-temperature fuel cells: Reliable parameter identification and interval-based sliding mode control, Soft Computing (17): 1329-1343.
  • Eble, I. (2007). Über Taylor-Modelle, Ph.D. thesis, University of Karlsruhe, Karlsruhe.
  • Eggers, A., Fränzle, M. and Herde, C. (2009). Application of constraint solving and ODE-enclosure methods to the analysis of hybrid systems, in E. Simos, G. Psihoyios and Ch. Tsitouras (Eds.), Numerical Analysis and Applied Mathematics 2009, American Institute of Physics, Melville, NY, pp. 1326-1330.
  • Filippov, A. (1988). Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers, Dordrecht.
  • Galias, Z. (2012). Rigorous study of the Chua's circuit spiral attractor, IEEE Transactions on Circuits and Systems 59I(10): 2374-2382.
  • Goldsztejn, A., Mullier, O., Eveillard, D. and Hosobe, H. (2010). Including ordinary differential equations based constraints in the standard CP framework, in D. Cohen (Ed.), Principles and Practice of Constraint Programming CP 2010, Lecture Notes in Computer Science, Vol. 6308, Springer, Berlin, pp. 221-235.
  • Granas, A. and Dugundji, J. (2003). Fixed Point Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, NY.
  • Hansen, E. and Walster, G. (2004). Global Optimization Using Interval Analysis: Revised and Expanded, Pure and Applied Mathematics, Marcel Dekker, New York, NY.
  • Henzinger, T. A., Horowitz, B., Majumdar, R. and Wong-Toi, H. (2000). Beyond HYTECH: Hybrid systems analysis using interval numerical methods, in N.A. Lynch and B.H. Krogh (Eds.), Proceedings of the Third International Workshop on Hybrid Systems: Computation and Control, HSCC'00, Springer-Verlag, London, pp. 130-144.
  • Ishii, D. (2010). Simulation and Verification of Hybrid Systems Based on Interval Analysis and Constraint Programming, Ph.D. thesis, Waseda University, Tokyo.
  • Ishii, D., Ueda, K. and Hosobe, H. (2011). An interval-based SAT modulo ODE solver for model checking nonlinear hybrid systems, International Journal on Software Tools for Technology Transfer 13(5): 449-461.
  • Jaulin, L., Kieffer, M., Didrit, O. and Walter, E. (2001). Applied Interval Analysis, Springer-Verlag, London.
  • Kearfott, R.B. (1996). Rigorous Global Search: Continuous Problems, Kluwer, Boston, MA.
  • Kofman, E. (2004). Discrete event simulation of hybrid systems, SIAM Journal on Scientific Computing 25(5): 1771-1797.
  • Kunze, M. (2000). Non-Smooth Dynamical Systems, Springer, Berlin/Heidelberg.
  • Lohner, R. (1988). Einschließung der Lösung gewöhnlicher Anfangs- und Randwertaufgaben und Anwendungen, Ph.D. thesis, Universität Karlsruhe, Karlsruhe.
  • Lunze, J. and Lamnabhi-Lagarrigue, F. (2009). Handbook of Hybrid Systems Control-Theory, Tools, Applications, Cambridge University Press, Cambridge.
  • Magnus, K. and Popp, K. (2005). Schwingungen, Leitfäden der angewandten Mathematik und Mechanik, Teubner, Wiesbaden.
  • Mahmoud, S. and Chen, X. (2008). A verified inexact implicit Runge-Kutta method for nonsmooth ODEs, Numerical Algorithms 47(3): 275-290.
  • Makino, K. (1998). Rigorous Analysis of Nonlinear Motion in Particle Accelerators, Ph.D. thesis, Michigan State University, East Lansing, MI.
  • Mannshardt, R. (1978). One-step methods of any order for ordinary differential equations with discontinuous right-hand sides, Nimerische Mathematik 31(2): 131-152.
  • McLeod, R. M. (1964/1965). Mean value theorems for vector valued functions, Proceedings of the Edinburgh Mathematical Society 14: 197-209.
  • Moore, R. (1966). Interval Arithmetic, Prentice-Hall, Englewood Cliffs, NJ.
  • Munoz, H. and Kearfott, R.B. (2004). Slope intervals, generalized gradients, semigradients, slant derivatives, and csets, Reliable Computing 10(3): 163-193.
  • Myśliński, A. (2012). Topology optimization of quasistatic contact problems, International Journal of Applied Mathematics and Computer Science 22(2): 269-280, DOI: 10.2478/v10006-012-0020-y.
  • Nedialkov, N.S. (2002). The Design and Implementation of an Object-Oriented Validated ODE Solver, Kluwer Academic Publishers, Dordrecht.
  • Nedialkov, N. and von Mohrenschildt, M. (2002). Rigorous simulation of hybrid dynamic systems with symbolic and interval methods, Proceedings of the American Control Conference, Anchorage, AK, USA, Vol. 1, pp. 140-147.
  • Orlov, Y. (2004). Finite time stability and robust control synthesis of uncertain switched systems, SIAM Journal on Control and Optimization 43(4): 1253-1271.
  • Patton, R.J., Chen, L. and Klinkhieo, S. (2012). An LPV pole-placement approach to friction compensation as an FTC problem, International Journal of Applied Mathematics and Computer Science 22(1): 149-160, DOI: 10.2478/v10006-012-0011-z.
  • Ramdani, N. and Nedialkov, N.S. (2011). Computing reachable sets for uncertain nonlinear hybrid systems using interval constraint-propagation techniques, Nonlinear Analysis: Hybrid Systems 5(2): 149-162.
  • Ratschan, S. (2012). An algorithm for formal safety verification of complex heterogeneous systems, Proceedings of REC 2012, Brno, Czech Republic, pp. 457-468.
  • Rauh, A. and Auer, E. (2011). Verified simulation of ODEs and DAEs in VAL E NC IA-IVP, Reliable Computing 5(4): 370-381.
  • Rauh, A., Brill, M. and Günther, C. (2009). A novel interval arithmetic approach for solving differential-algebraic equations with VAL E NC IA-IVP, International Journal of Applied Mathematics and Computer Science 19(3): 381-397, DOI: 10.2478/v10006-009-0032-4.
  • Rauh, A., Kletting, M., Aschemann, H. and Hofer, E.P. (2006). Interval methods for simulation of dynamical systems with state-dependent switching characteristics, IEEE CCA 2006, Munich, Germany, pp. 355-360.
  • Rauh, A., Siebert, C. and Aschemann, H. (2011). Verified simulation and optimization of dynamic systems with friction and hysteresis, Proceedings of ENOC 2011, Rome, Italy.
  • Rihm, R. (1992). Enclosing solutions with switching points in ordinary differential equations, in L. Atanassova and J. Herzberger (Eds.), Computer Arithmetic and Enclosure Methods. Proceedings of SCAN 91, North-Holland, Amsterdam, pp. 419-425.
  • Rihm, R. (1993). Über Einschließungsverfahren für gewöhnliche Anfangswertprobleme und ihre Anwendung auf Differentialgleichungen mit unstetiger rechter Seite, Ph.D. thesis, Universität Karlsruhe, Karlsruhe.
  • Rihm, R. (1998). Implicit methods for enclosing solutions of ODEs, Journal of Universal Computer Science 4(2): 202-209.
  • Schnurr, M. (2007). Steigungen höherer Ordnung zur verifizierten globalen Optimierung, Ph.D. thesis, Universität Karlsruhe, Karlsruhe.
  • Smirnov, G. (2002). Introduction to the Theory of Differential Inclusions, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI.
  • Stauning, O. (1997). Automatic Validation of Numerical Solutions, Ph.D. thesis, Technical University of Denmark, Kgs. Lyngby.
  • Stewart, D. (1990). A high accuracy method for solving ODEs with discontinuous right-hand side, Numerische Mathematik 58(1): 299-328.
  • Walter, W. (1972). Gewöhnliche Differentialgleichungen, Springer, Berlin/Heidelberg/New York, NY.
  • Zgliczynski, P. and Kapela, T. (2009). A Lohner-type algorithm for control systems and ordinary differential inclusions, Discrete and Continuous Dynamical Systems B 11(2): 365-385.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-amcv23z4p731bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.