Advertisement

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

  • Murat Arcak
  • John Maidens
Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, 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.

References

  1. 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)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Tabuada, P.: Verification and Control of Hybrid Systems: A Symbolic Approach. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-1-4419-0224-5CrossRefzbMATHGoogle Scholar
  3. 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)MathSciNetCrossRefGoogle Scholar
  4. 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. 5.
    Chutinan, A., Krogh, B.: Computational techniques for hybrid system verification. IEEE Trans. Autom. Control 48(1), 64–75 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Lin, Y., Stadtherr, M.A.: Validated solutions of initial value problems for parametric ODEs. Appl. Numer. Math. 57(10), 1145–1162 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Neher, M., Jackson, K.R., Nedialkov, N.S.: On Taylor model based integration of ODEs. SIAM J. Numer. Anal. 45, 236–262 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lakshmikantham, V., Leela, S.: Differential and Integral Inequalities, vol. 1. Academic Press, New York (1969)zbMATHGoogle Scholar
  9. 9.
    Scott, J.K., Barton, P.I.: Bounds on the reachable sets of nonlinear control systems. Automatica 49(1), 93–100 (2013)MathSciNetCrossRefGoogle Scholar
  10. 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_16CrossRefGoogle Scholar
  11. 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. 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_16CrossRefGoogle Scholar
  13. 13.
    Maidens, J., Arcak, M.: Reachability analysis of nonlinear systems using matrix measures. IEEE Trans. Autom. Control 60(1), 265–270 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lohmiller, W., Slotine, J.J.: On contraction analysis for nonlinear systems. Automatica 34, 683–696 (1998)MathSciNetCrossRefGoogle Scholar
  15. 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_20CrossRefGoogle Scholar
  16. 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. 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. 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)CrossRefGoogle Scholar
  19. 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)CrossRefGoogle Scholar
  20. 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)MathSciNetCrossRefGoogle Scholar
  21. 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. 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_3CrossRefGoogle Scholar
  23. 23.
    Nishimura, D.G.: Principles of Magnetic Resonance Imaging. Lulu, Morrisville (2010)Google Scholar
  24. 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)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of California, BerkeleyBerkeleyUSA

Personalised recommendations