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Adaptive step size numerical integration for stochastic differential equations with discontinuous drift and diffusion

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Abstract

Stochastic hybrid systems (SHSs) are a modelling framework for a cyber-physical system (CPS), used to simulate, validate, and verify safety critical controllers under uncertainty. Popular simulation tools can miss detecting discontinuities when simulating SHS, thereby producing incorrect outputs during simulation. We propose a novel adaptive step size simulation/integration technique for a subset of SHS—stochastic differential equations (SDEs) with discontinuous drift and diffusion coefficients. Each integration step, of the Euler-Maruyama numerical solution of the SDEs, is made dependent upon the values of the continuous variables inducing the discontinuity. This in turn guarantees convergence of the system trajectory towards the discontinuity without missing it. A thorough analysis and extensive benchmarking of the proposed integration technique shows the efficacy of the approach when simulating complex SHSs.

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Notes

  1. More than one Wiener process.

References

  1. Pola, G, Bujorianu, M.L., Lygeros, J., Di Benedetto, M.D.: Stochastic hybrid models: an overview. In: ADHS, pp 45–50 (2003)

  2. Cassandras, C.G., Lygeros, J.: Stochastic Hybrid Systems. CRC Press, Boca Raton (2018)

    Book  MATH  Google Scholar 

  3. Alur, R.: Principles of Cyber-Physical Systems. MIT Press, Cambridge (2015)

    Google Scholar 

  4. Glover, W., Lygeros, J.: A stochastic hybrid model for air traffic control simulation. In: International Workshop on Hybrid Systems: Computation and Control, pp 372–386. Springer (2004)

  5. Poznyak, A.: Stochastic sliding mode control: what is this?. In: 2016 14th International Workshop on Variable Structure Systems (VSS), pp 328–333. IEEE (2016)

  6. Hespanha, J.P.: Stochastic hybrid systems: application to communication networks. In: International Workshop on Hybrid Systems: Computation and Control, pp 387–401. Springer (2004)

  7. Hu, J., Lygeros, J., Sastry, S.: Towards a theory of stochastic hybrid systems. In: International Workshop on Hybrid Systems: Computation and Control, pp 160–173. Springer (2000)

  8. Fränzle, M., Gao, Y., Gerwinn, S.: Constraint-solving techniques for the analysis of stochastic hybrid systems. In: Provably Correct Systems, pp 9–38. Springer (2017)

  9. Hahn, E.M., Hartmanns, A., Hermanns, H., Katoen, J.-P.: A compositional modelling and analysis framework for stochastic hybrid systems. Formal Methods in System Design 43(2), 191–232 (2013)

    Article  MATH  Google Scholar 

  10. David, A., Du, D., Larsen, K.G., Legay, A., Mikučionis, M., Poulsen, D.B., Sedwards, S.: Statistical model checking for stochastic hybrid systems. arXiv:1208.3856 (2012)

  11. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations, vol. 23. Springer, Berlin (2013)

    Google Scholar 

  12. Itô, K.: 109. Stochastic integral. Proceedings of the Imperial Academy 20(8), 519–524 (1944)

    MathSciNet  MATH  Google Scholar 

  13. Gaines, J.G., Lyons, T.J.: Variable step size control in the numerical solution of stochastic differential equations. SIAM J. Appl. Math. 57(5), 1455–1484 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rackauckas, C., Nie, Q.: Adaptive methods for stochastic differential equations via natural embeddings and rejection sampling with memory. Discrete and Continuous Dynamical Systems, Series B 22(7), 2731 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. MathWorks Corp: Simulink: Simulation and Model-Based Design. https://www.mathworks.com/products/simulink.html/ (2019)

  16. Alur, R., Courcoubetis, C., Henzinger, T.A., Ho, P.-H.: Hybrid automata: an algorithmic approach to the specification and verification of hybrid systems. In: Hybrid Systems, pp 209–229. Springer, London (1993)

  17. Open Source Modelica Consortium (OSMC): Openmodelica. https://openmodelica.org/index.php/ (2019)

  18. Burrage, P.M., Burrage, K.: A variable stepsize implementation for stochastic differential equations. SIAM J. Sci. Comput. 24(3), 848–864 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ngo, H.-L., Taguchi, D.: On the Euler–Maruyama approximation for one-dimensional stochastic differential equations with irregular coefficients. IMA J. Numer. Anal. 37(4), 1864–1883 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Fridman, L., Levant, A., et al.: Higher order sliding modes. Sliding Mode Control in Engineering 11, 53–102 (2002)

    Google Scholar 

  21. Cruz, G.L., Alazki, H., Hernández, R.G.: Super twisting control for thermo’s catalyst-5 robotic arm. IFAC-PapersOnLine 51(13), 303–308 (2018)

    Article  Google Scholar 

  22. Kumari, K., Chalanga, A., Bandyopadhyay, B.: Implementation of super-twisting control on higher order perturbed integrator system using higher order sliding mode observer. IFAC-PapersOnLine 49(18), 873–878 (2016)

    Article  Google Scholar 

  23. Göttlich, S., Lux, K., Neuenkirch, A.: The Euler scheme for stochastic differential equations with discontinuous drift coefficient: a numerical study of the convergence rate. Advances in Difference Equations 2019(1), 429 (2019)

    Article  MathSciNet  Google Scholar 

  24. Malik, A: Benchmarks. https://github.com/amal029/eha. Last Accessed 05 April 2020 (2020)

  25. Lamba, H.: An adaptive timestepping algorithm for stochastic differential equations. J. Comput. Appl. Math. 161(2), 417–430 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ilie, S., Jackson, K.R., Enright, W.H.: Adaptive time-stepping for the strong numerical solution of stochastic differential equations. Numerical Algorithms 68(4), 791–812 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Neuenkirch, A., Szolgyenyi, M., Szpruch, L.: An adaptive Euler–Maruyama scheme for stochastic differential equations with discontinuous drift and its convergence analysis. SIAM J. Numer. Anal. 57(1), 378–403 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lamperti, J.: A simple construction of certain diffusion processes. J. Math. Kyoto Univ. 4(1), 161–170 (1964)

    MathSciNet  MATH  Google Scholar 

  29. Møller, JK, Madsen, H: From State Dependent Diffusion to Constant Diffusion in Stochastic Differential Equations by the Lamperti Transform, ser. IMM-Technical Report-2010-16. Technical University of Denmark, DTU Informatics, Building 321 (2010)

  30. Kofman, E., Junco, S.: Quantized-state systems: a DEVS approach for continuous system simulation. Transactions of The Society for Modeling and Simulation International 18(3), 123–132 (2001)

    Google Scholar 

  31. Malik, A., Roop, P.: A dynamic quantized state system execution framework for hybrid automata. Nonlinear Analysis: Hybrid Systems 36, 100870 (2020)

    MathSciNet  MATH  Google Scholar 

  32. Lygeros, J., Prandini, M.: Stochastic hybrid systems: a powerful framework for complex, large scale applications. Eur. J. Control. 16(6), 583–594 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Abate, A., Katoen, J.-P., Lygeros, J., Prandini, M.: Approximate model checking of stochastic hybrid systems. Eur. J. Control. 16(6), 624–641 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Fränzle, M., Hahn, E.M., Hermanns, H., Wolovick, N., Zhang, L.: Measurability and safety verification for stochastic hybrid systems. In: Proceedings of the 14th International Conference on Hybrid Systems: Computation and Control, pp 43–52 (2011)

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  1. Department of Electrical, Computer and Software Engineering, University of Auckland, Auckland, New Zealand

    Avinash Malik

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  1. Avinash Malik
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Correspondence to Avinash Malik.

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Malik, A. Adaptive step size numerical integration for stochastic differential equations with discontinuous drift and diffusion. Numer Algor 87, 849–872 (2021). https://doi.org/10.1007/s11075-020-00990-x

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