ABSTRACT
We investigate quantifying the difference between two hybrid dynamical systems under noise and initial-state uncertainty. While the set of traces for these systems is infinite, it is possible to symbolically approximate trace sets using \emph{reachpipes} that compute upper and lower bounds on the evolution of the reachable sets with time. We estimate distances between corresponding sets of trajectories of two systems in terms of distances between the reachpipes.
In case of two individual traces, the Skorokhod distance has been proposed as a robust and efficient notion of distance which captures both value and timing distortions. In this paper, we extend the computation of the Skorokhod distance to reachpipes, and provide algorithms to compute upper and lower bounds on the distance between two sets of traces. Our algorithms use new geometric insights that are used to compute the worst-case and best-case distances between two polyhedral sets evolving with time.
- H. Abbas, B. Hoxha, G.E. Fainekos, J.V. Deshmukh, J. Kapinski, and K. Ueda. Conformance testing as falsification for cyber-physical systems. CoRR, abs/1401.5200, 2014.Google Scholar
- H. Abbas, H. D. Mittelmann, and G. E. Fainekos. Formal property verification in a conformance testing framework. In MEMOCODE 2014, pages 155--164. IEEE, 2014.Google ScholarDigital Library
- H. Alt and M. Godau. Computing the Fréchet distance between two polygonal curves. Int. J. Comput. Geometry Appl., 5:75--91, 1995.Google ScholarCross Ref
- D. Avis, D. Bremner, and R. Seidel. How good are convex hull algorithms? Comput. Geom., 7:265--301, 1997. Google ScholarDigital Library
- S. Basu, R. Pollack, and M.F. Roy. Algorithms in Real Algebraic Geometry. Springer-Verlag, 2006. Google ScholarDigital Library
- X. Chen, E. Abrahám, and S. Sankaranarayanan. Taylor model flowpipe construction for non-linear hybrid systems. In RTSS 2012, pages 183--192. IEEE Computer Society, 2012. Google ScholarDigital Library
- A. Chutinan and B. H. Krogh. Computational techniques for hybrid system verification. IEEE Trans. Automat. Contr., 48(1):64--75, 2003.Google ScholarCross Ref
- M. Colón and S. Sankaranarayanan. Generalizing the template polyhedral domain. In ESOP 2011, LNCS 6602, pages 176--195. Springer, 2011. Google ScholarDigital Library
- J. V. Deshmukh, R. Majumdar, and V. S. Prabhu. Quantifying conformance using the Skorokhod metric. In CAV 2015, LNCS 9207, pages 234--250 Part(II). Springer, 2015.Google ScholarCross Ref
- G. Frehse, C. Le Guernic, A. Donzé, S. Cotton, R. Ray, O. Lebeltel, R. Ripado, A. Girard, T. Dang, and O. Maler. Spaceex: Scalable verification of hybrid systems. In CAV 2011, LNCS 6806, pages 379--395. Springer, 2011. Google ScholarDigital Library
- A. Girard. Reachability of uncertain linear systems using zonotopes. In HSCC 2005, LNCS 3414, pages 291--305. Springer, 2005. Google ScholarDigital Library
- A. Girard and C. Le Guernic. Zonotope/hyperplane intersection for hybrid systems reachability analysis. In HSCC, LNCS 4981, pages 215--228. Springer, 2008. Google ScholarDigital Library
- A. Girard, C. Le Guernic, and O. Maler. Efficient computation of reachable sets of linear time-invariant systems with inputs. In HSCC 2006, LNCS 3927, pages 257--271. Springer, 2006. Google ScholarDigital Library
- G.M.Ziegler. Lectures on Polytopes. Springer, 1995.Google ScholarCross Ref
- C. Le Guernic and A. Girard. Reachability analysis of linear systems using support functions. Nonlinear Analysis: Hybrid Systems, 4(2):250--262, 2010.Google ScholarCross Ref
- Z. Han and B. H. Krogh. Reachability analysis of large-scale affine systems using low-dimensional polytopes. In HSCC 2006, LNCS 3927, pages 287--301. Springer, 2006. Google ScholarDigital Library
- A. B. Kurzhanski and P. Varaiya. Ellipsoidal techniques for reachability under state constraints. SIAM J. Contr. & Optim., 45(4):1369--1394, 2006. Google ScholarDigital Library
- R. Majumdar and V. S. Prabhu. Computing the Skorokhod distance between polygonal traces. In HSCC 2015, pages 199--208. ACM, 2015. Google ScholarDigital Library
- R. Majumdar and V.S. Prabhu. Computing distances between reach flowpipes. CoRR, 1602.03266, 2016. Google ScholarDigital Library
- P. Prabhakar and M. Viswanathan. A dynamic algorithm for approximate flow computations. In HSCC 2011, pages 133--142. ACM, 2011. Google ScholarDigital Library
- S. Sankaranarayanan, T. Dang, and F. Ivancic. A policy iteration technique for time elapse over template polyhedra. In HSCC 2008, LNCS 4981, pages 654--657. Springer, 200 Google ScholarDigital Library
Index Terms
- Computing Distances between Reach Flowpipes
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