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Verified Parameter Identification for Dynamic Systems with Non-Smooth Right-Hand Sides

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Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9553))

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Abstract

Modeling of systems in engineering involves two major stages. First, a system structure is derived that is based on the fundamental laws from physics that characterize the relevant processes. Second, specific parameter values are determined by minimizing the distance between the measured and simulated system outputs. In previous work, strategies for verified parameter identification using techniques from interval analysis were developed. These techniques are extended in this paper to a verified estimation for systems with non-smooth ordinary differential equations. Suitable experimental results for parameter estimation of a mechanical system with friction conclude this contribution to highlight the practical applicability of the developed identification procedure.

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Notes

  1. 1.

    Optional: Trisectioning of \(\left[ {x_2^{\langle {l^*} \rangle }}\right] \) around the value zero if static and sliding friction are possible simultaneously in the simulation of the uncertain system model.

  2. 2.

    The presented integration procedure is implemented by using the toolbox IntLab, where a parallelization of the evaluation can be achieved in a straightforward manner if the state equations are evaluated after a distribution onto multiple CPU cores. The Parallel Computing Toolbox can be utilized for this purpose in Matlab.

  3. 3.

    The increase of the list length from \(L^\prime \) to \(3L^\prime \) has the advantage that information about the parameter splitting before the reset is not lost. Usually, the static friction is similar after standstill, even if it does not remain identical. In this case, the splitting information speeds up the identification and elimination of inconsistent subdomains.

  4. 4.

    Note that the interval replacement (Step I6) and the reduction of the interval number (Step I7) can be employed interchangeably.

References

  1. Auer, E., Kiel, S., Rauh, A.: A verified method for solving piecewise smooth initial value problems. Int. J. Appl. Math. Comput. Sci. AMCS 23(4), 731–747 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Hofer, E.P., Rauh, A.: Applications of interval algorithms in engineering. In: CD-Proceedings of 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN 2006. IEEE Computer Society, Duisburg, Germany (2007)

    Google Scholar 

  3. Jaulin, L., Kieffer, M., Didrit, O., Walter, É.: Applied Interval Analysis. Springer, London (2001)

    Book  Google Scholar 

  4. Kalman, R.E.: A new approach to linear filtering and prediction problems. Trans. ASME - J. Basic Eng. 82(Series D), 35–45 (1960)

    Article  MathSciNet  Google Scholar 

  5. Lohner, R.: Enclosing the solutions of ordinary initial and boundary value problems. In: Kaucher, E.W., Kulisch, U.W., Ullrich, C. (eds.) Computer Arithmetic: Scientific Computation and Programming Languages, pp. 255–286. Wiley-Teubner Series in Computer Science, Stuttgart (1987)

    Google Scholar 

  6. Nedialkov, N.S.: Interval tools for ODEs and DAEs. In: CD-Proceedings of 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN 2006. IEEE Computer Society, Duisburg, Germany (2007)

    Google Scholar 

  7. Nedialkov, N.S., von Mohrenschildt, M.: Rigorous simulation of hybrid dynamic systems with symbolic and interval methods. In: Proceedings of the American Control Conference ACC, Anchorage, USA, pp. 140–147 (2002)

    Google Scholar 

  8. Ramdani, N., Nedialkov, N.S.: Computing reachable sets for uncertain nonlinear hybrid systems using interval constraint-propagation techniques. Nonlinear Anal. Hybrid Syst. 5(2), 149–162 (2011)

    Article  MathSciNet  Google Scholar 

  9. Rauh, A., Dötschel, T., Auer, E., Aschemann, H.: Interval methods for control-oriented modeling of the thermal behavior of high-temperature fuel cell stacks. In: Proceedings of 16th IFAC Symposium on System Identification SysID 2012. Brussels, Belgium (2012)

    Google Scholar 

  10. Rauh, A., Kletting, M., Aschemann, H., Hofer, E.P.: Interval methods for simulation of dynamical systems with state-dependent switching characteristics. In: Proceedings of the IEEE International Conference on Control Applications CCA 2006, pp. 2243–2248. Munich, Germany (2006)

    Google Scholar 

  11. Rauh, A., Kletting, M., Aschemann, H., Hofer, E.P.: Reduction of overestimation in interval arithmetic simulation of biological wastewater treatment processes. J. Comput. Appl. Math. 199(2), 207–212 (2007)

    Article  MathSciNet  Google Scholar 

  12. Rauh, A., Senkel, L., Aschemann, H.: Experimental comparison of interval-based parameter identification procedures for uncertain odes with non-smooth right-hand sides. In: CD-Proceedings of IEEE International Conference on Methods and Models in Automation and Robotics MMAR 2015. Miedzyzdroje, Poland (2015)

    Google Scholar 

  13. Rauh, A., Siebert, C., Aschemann, H.: Verified simulation and optimization of dynamic systems with friction and hysteresis. In: Proceedings of ENOC 2011. Rome, Italy (2011)

    Google Scholar 

  14. Rihm, R.: Enclosing solutions with switching points in ordinary differential equations. In: Computer Arithmetic and Enclosure Methods. Proceedings of SCAN 91. North-Holland, Amsterdam, pp. 419–425 (1992)

    Google Scholar 

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Authors and Affiliations

  1. Chair of Mechatronics, University of Rostock, Justus-von-Liebig-Weg 6, D-18059, Rostock, Germany

    Andreas Rauh, Luise Senkel & Harald Aschemann

Authors
  1. Andreas Rauh
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  2. Luise Senkel
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  3. Harald Aschemann
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Correspondence to Andreas Rauh .

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Editors and Affiliations

  1. Institute of Computer Science, University of Würzburg, Würzburg, Germany

    Marco Nehmeier

  2. Universität Würzburg, Würzburg, Germany

    Jürgen Wolff von Gudenberg

  3. Department of Mathematics, Uppsala University, Uppsala, Sweden

    Warwick Tucker

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Rauh, A., Senkel, L., Aschemann, H. (2016). Verified Parameter Identification for Dynamic Systems with Non-Smooth Right-Hand Sides. In: Nehmeier, M., Wolff von Gudenberg, J., Tucker, W. (eds) Scientific Computing, Computer Arithmetic, and Validated Numerics. SCAN 2015. Lecture Notes in Computer Science(), vol 9553. Springer, Cham. https://doi.org/10.1007/978-3-319-31769-4_19

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  • DOI: https://doi.org/10.1007/978-3-319-31769-4_19

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