Using Redundant Constraints for Refinement
This paper is concerned with a method for computing reachable sets of linear continuous systems with uncertain input. Such a method is required for verification of hybrid systems and more generally embedded systems with mixed continuous-discrete dynamics. In general, the reachable sets of such systems (except for some linear systems with special eigenstructures) are hard to compute exactly and are thus often over-approximated. The approximation accuracy is important especially when the computed over-approximations do not allow proving a property. In this paper we address the problem of refining the reachable set approximation by adding redundant constraints which allow bounding the reachable sets in some critical directions. We introduce the notion of directional distance which is appropriate for measuring approximation effectiveness with respect to verifying a safety property. We also describe an implementation of the reachability algorithm which favors the constraint-based representation over the vertex-based one and avoids expensive conversions between them. This implementation allowed us to treat systems of much higher dimensions. We finally report some experimental results showing the performance of the refinement algorithm.
KeywordsHybrid System Supporting Point Directional Distance Reachability Analysis Sharp Angle
Unable to display preview. Download preview PDF.
- 1.Althoff, M., Stursberg, O., Buss, M.: Reachability Analysis of Nonlinear Systems with Uncertain Parameters using Conservative Linearization. In: CDC 2008 (2008)Google Scholar
- 3.Asarin, E., Bournez, O., Dang, T., Maler, O.: Approximate reachability analysis of piecewise linear dynamical systems. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 21–31. Springer, Heidelberg (2000)Google Scholar
- 7.Cameron, S.A., Culley, R.K.: Determining the Minimum Translational Distance between Two Convex Polyhedra. Proceedings of International Conference on Robotics and Automation 48, 591–596 (1986)Google Scholar
- 8.Dang, T.: Verification and Synthesis of Hybrid Systems. PhD Thesis, INPG (2000)Google Scholar
- 9.Frehse, G.: PHAVER: Algorithmic Verification of Hybrid Systems past HYTECH. International Journal on Software Tools for Technology Transfer (STTT) 10(3) (June 2008)Google Scholar
- 11.Le Guernic, C., Girard, A.: Reachability Analysis of Hybrid Systems Using Support Functions. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 540–554. Springer, Heidelberg (2009)Google Scholar
- 20.Lin, M.C., Manocha, D.: Collision and proximity queries. In: Handbook of Discrete and Computational Geometry (2003)Google Scholar
- 21.Pappas, G., Lafferriere, G., Yovine, S.: A new class of decidable hybrid systems. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC 1999. LNCS, vol. 1569, pp. 29–31. Springer, Heidelberg (1999)Google Scholar
- 24.Varaiya, P.: Reach Set computation using Optimal Control. In: KIT Workshop, Verimag, Grenoble, pp. 377–383 (1998)Google Scholar