Software & Systems Modeling

, Volume 14, Issue 1, pp 121–148 | Cite as

Improving the SAT modulo ODE approach to hybrid systems analysis by combining different enclosure methods

  • Andreas EggersEmail author
  • Nacim Ramdani
  • Nedialko S. Nedialkov
  • Martin Fränzle
Special Section Paper


Aiming at automatic verification and analysis techniques for hybrid discrete-continuous systems, we present a novel combination of enclosure methods for ordinary differential equations (ODEs) with the iSAT solver for large Boolean combinations of arithmetic constraints. Improving on our previous work, the contribution of this paper lies in combining iSAT with VNODE-LP, as a state-of-the-art interval solver for ODEs, and with bracketing systems, which exploit monotonicity properties allowing to find enclosures for problems that VNODE-LP alone cannot enclose tightly. We apply the combined iSAT-ODE solver to the analysis of a variety of non-linear hybrid systems by solving predicative encodings of reachability properties and of an inductive stability argument, and evaluate the impact of the different enclosure methods, decision heuristics and their combination. Our experiments include classic benchmarks from the literature, as well as a newly-designed conveyor belt system that combines hybrid behavior of parallel components, a slip-stick friction model with non-linear dynamics and flow invariants and several dimensions of parameterization. In the paper, we also present and evaluate an extension of VNODE-LP tailored to its use as a deduction mechanism within iSAT-ODE, to allow fast re-evaluations of enclosures over arbitrary subranges of the analyzed time span.


Analysis of hybrid discrete-continuous systems Satisfiability modulo theories Enclosure methods for ODEs Bracketing systems 



We would like to thank Stefan Ratschan, Christian Herde, Tino Teige, Jens Oehlerking, and Corina Mitrohin for discussions on the region-stability-related proof scheme utilized for the experiments in this paper and all colleagues from the transregional research center AVACS, project H1/2 “Constraint-based Verification for Hybrid Systems” for the joint development of the iSAT core. Additionally, we are grateful to the reviewers of [6] for their detailed comments. Especially by insisting on a more thorough experimental evaluation and by pointing out shortcomings in our presentation, the SoSyM reviewers have helped tremendously to improve the quality of this paper. Thank you!


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Andreas Eggers
    • 1
    Email author
  • Nacim Ramdani
    • 2
  • Nedialko S. Nedialkov
    • 3
  • Martin Fränzle
    • 1
  1. 1.Department of Computing ScienceCarl von Ossietzky UniversitätOldenburgGermany
  2. 2.Université d’OrléansBourgesFrance
  3. 3.McMaster UniversityHamiltonCanada

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