Skip to main content
SpringerLink
Log in

Simulation-Based Reachability Analysis for Nonlinear Systems Using Componentwise Contraction Properties

  • Chapter
  • First Online:
Principles of Modeling

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 10760))

Abstract

A shortcoming of existing reachability approaches for nonlinear systems is the poor scalability with the number of continuous state variables. To mitigate this problem we present a simulation-based approach where we first sample a number of trajectories of the system and next establish bounds on the convergence or divergence between the samples and neighboring trajectories that are not explicitly simulated. We compute these bounds using contraction theory and reduce the conservatism by partitioning the state vector into several components and analyzing contraction properties separately in each direction. Among other benefits this allows us to analyze the effect of constant but uncertain parameters by treating them as state variables and partitioning them into a separate direction. We next present a numerical procedure to search for weighted norms that yield a prescribed contraction rate, which can be incorporated in the reachability algorithm to adjust the weights to minimize the growth of the reachable set. The proposed reachability method is illustrated with examples, including a magnetic resonance imaging application.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Kapinski, J., Deshmukh, J.V., Jin, X., Ito, H., Butts, K.: Simulation-based approaches for verification of embedded control systems: an overview of traditional and advanced modeling, testing, and verification techniques. IEEE Control Syst. 36(6), 45–64 (2016)

    Article  MathSciNet  Google Scholar 

  2. Tabuada, P.: Verification and Control of Hybrid Systems: A Symbolic Approach. Springer, Heidelberg (2009). https://doi.org/10.1007/978-1-4419-0224-5

    Book  MATH  Google Scholar 

  3. Mitchell, I., Bayen, A., Tomlin, C.: A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games. IEEE Trans. Autom. Control 50(7), 947–957 (2005)

    Article  MathSciNet  Google Scholar 

  4. Althoff, M., Stursberg, O., Buss, M.: Reachability analysis of nonlinear systems with uncertain parameters using conservative linearization. In: IEEE Conference Decision Control, pp. 4042–4048 (2008)

    Google Scholar 

  5. Chutinan, A., Krogh, B.: Computational techniques for hybrid system verification. IEEE Trans. Autom. Control 48(1), 64–75 (2003)

    Article  MathSciNet  Google Scholar 

  6. Lin, Y., Stadtherr, M.A.: Validated solutions of initial value problems for parametric ODEs. Appl. Numer. Math. 57(10), 1145–1162 (2007)

    Article  MathSciNet  Google Scholar 

  7. Neher, M., Jackson, K.R., Nedialkov, N.S.: On Taylor model based integration of ODEs. SIAM J. Numer. Anal. 45, 236–262 (2007)

    Article  MathSciNet  Google Scholar 

  8. Lakshmikantham, V., Leela, S.: Differential and Integral Inequalities, vol. 1. Academic Press, New York (1969)

    MATH  Google Scholar 

  9. Scott, J.K., Barton, P.I.: Bounds on the reachable sets of nonlinear control systems. Automatica 49(1), 93–100 (2013)

    Article  MathSciNet  Google Scholar 

  10. Donzé, A., Maler, O.: Systematic simulation using sensitivity analysis. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 174–189. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-71493-4_16

    Chapter  Google Scholar 

  11. Huang, Z., Mitra, S.: Computing bounded reach sets from sampled simulation traces. In: Hybrid Systems: Computation and Control, pp. 291–294 (2012)

    Google Scholar 

  12. Julius, A.A., Pappas, G.J.: Trajectory based verification using local finite-time invariance. In: Majumdar, R., Tabuada, P. (eds.) HSCC 2009. LNCS, vol. 5469, pp. 223–236. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-00602-9_16

    Chapter  Google Scholar 

  13. Maidens, J., Arcak, M.: Reachability analysis of nonlinear systems using matrix measures. IEEE Trans. Autom. Control 60(1), 265–270 (2015)

    Article  MathSciNet  Google Scholar 

  14. Lohmiller, W., Slotine, J.J.: On contraction analysis for nonlinear systems. Automatica 34, 683–696 (1998)

    Article  MathSciNet  Google Scholar 

  15. Sontag, E.D.: Contractive systems with inputs. In: Willems, J.C., Hara, S., Ohta, Y., Fujioka, H. (eds.) Perspectives in Mathematical System Theory, Control, and Signal Processing, pp. 217–228. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-540-93918-4_20

    Chapter  Google Scholar 

  16. Rungger, M., Zamani, M.: SCOTS: a tool for the synthesis of symbolic controllers. In: Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control HSCC 2016 (2016)

    Google Scholar 

  17. Desoer, C., Vidyasagar, M.: Feedback systems: input-output properties. In: Society for Industrial and Applied Mathematics, Philadelphia (2009). Academic Press, New York (1975)

    Google Scholar 

  18. Kapela, T., Zgliczyński, P.: A Lohner-type algorithm for control systems and ordinary differential equations. Discret. Continuous Dyn. Syst. Ser. B 11(2), 365–385 (2009)

    Article  Google Scholar 

  19. Russo, G., di Bernardo, M., Sontag, E.D.: A contraction approach to the hierarchical analysis and design of networked systems. IEEE Trans. Autom. Control 58(5), 1328–1331 (2013)

    Article  Google Scholar 

  20. Reissig, G., Weber, A., Rungger, M.: Feedback refinement relations for the synthesis of symbolic controllers. IEEE Trans. Autom. Control 62(4), 1781–1796 (2017)

    Article  MathSciNet  Google Scholar 

  21. Fan, C., Kapinski, J., Jin, X., Mitra, S.: Locally optimal reach set over-approximation for nonlinear systems. In: Proceedings of the 13th International Conference on Embedded Software. EMSOFT 2016, pp. 6:1–6:10 (2016)

    Google Scholar 

  22. Aminzare, Z., Shafi, Y., Arcak, M., Sontag, E.D.: Guaranteeing spatial uniformity in reaction-diffusion systems using weighted \(L^2\) norm contractions. In: Kulkarni, V.V., Stan, G.-B., Raman, K. (eds.) A Systems Theoretic Approach to Systems and Synthetic Biology I: Models and System Characterizations, pp. 73–101. Springer, Dordrecht (2014). https://doi.org/10.1007/978-94-017-9041-3_3

    Chapter  Google Scholar 

  23. Nishimura, D.G.: Principles of Magnetic Resonance Imaging. Lulu, Morrisville (2010)

    Google Scholar 

  24. Edelstein, W.A., Glover, G.H., Hardy, C.J., Redington, R.W.: The intrinsic signal-to-noise ratio in NMR imaging. Magn. Reson. Med. 3(4), 604–618 (1986)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of California, Berkeley, Berkeley, USA

    Murat Arcak & John Maidens

Authors
  1. Murat Arcak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John Maidens
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Murat Arcak .

Editor information

Editors and Affiliations

  1. University of California, Berkeley, Berkeley, California, USA

    Marten Lohstroh

  2. National Instruments, Berkeley, California, USA

    Patricia Derler

  3. Mälardalen University, Västerås, Sweden, Reykjavík University, Reykjavik, Iceland

    Marjan Sirjani

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Arcak, M., Maidens, J. (2018). Simulation-Based Reachability Analysis for Nonlinear Systems Using Componentwise Contraction Properties. In: Lohstroh, M., Derler, P., Sirjani, M. (eds) Principles of Modeling. Lecture Notes in Computer Science(), vol 10760. Springer, Cham. https://doi.org/10.1007/978-3-319-95246-8_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-95246-8_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-95245-1

  • Online ISBN: 978-3-319-95246-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation