Abstract
This paper describes three techniques for reachability analysis for systems modeled by ordinary differential equations (ODES). First, linear models with regions modeled by convex polyhedra are considered, and an exact algorithm is presented. Next, non-convex polyhedra are considered, and techniques are presented for representing a polyhedron by its projection onto two-dimensional subspaces. This approach yields a compact representation, and allows efficient algorithms from computational geometry to be employed. Within this context, an approximation technique for reducing non-linear ODE models to linear nonhomogeneous models is presented. This reduction provides a sound basis for applying methods for linear systems analysis to non-linear systems.
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References
D. Avis and K. Fukuda. A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra. Discrete Computational Geometry, 8:295–313, 1992.
Thomas M. Apostle. Calculus, volume 1. John Wiley and Sons, Inc., New York, second edition, 1967.
H.B. Bakoglu. Circuits, Interconnections, and Packaging for VLSI. Addison-Wesley, 1990.
R. W. Brockett. Smooth dynamical systems which realize arithmetical and logical operations. In Hendrik Nijmeijer and Johannes M. Schumacher, editors, Three Decades of Mathematical Systems Theory: A Collection of Surveys at the Occasion of the 50th Birthday of J. C. Willems, volume 135 of Lecture Notes in Control and Information Sciences pages 19–30. Springer, 1989.
Lance A. Glasser and Daniel W. Dobberpuhl. The Design and Analysis of VLSI Circuits. Addison-Wesley, 1985.
Mark R. Greenstreet and Ian Mitchell. Reachability with discrete and ODE models. In Michael Lemmon, editor, Fifth International Hybrid System Workshop, Notre Dame, September 1997.
Mark R. Greenstreet. Verifying safety properties of differential equations. In Proceedings of the 1996 Conference on Computer Aided Verification, pages 277–287, New Brunswick, NJ, July 1996.
T.A. Henzinger and P.-H. Ho. HYTECH: The Cornell Hybrid Technology Tool. In P. Antsaklis, A. Nerode, W. Kohn, and S. Sastry, editors, Hybrid Systems II, Lecture Notes in Computer Science 999, pages 265–293. Springer-Verlag, 1995.
Morris W. Hirsch and Stephen Smale. Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press, San Diego, CA, 1974.
L.W. Nagel. SPICE2: a computer program to simulate semiconductor circuits. Technical Report ERL-M520, Electronics Research Laboratory, University of California, Berkeley, CA, May 1975.
Christos H. Papadimitriou and Kenneth Steiglitz. Combinatorial Optimization: Algorithms and Complexity. Prentice Hall, Englewood Cliffs, NJ, 1982.
Franco P. Preparata and Michael I. Shamos. Computational Geometry: An Introduction. Texts and Monographs in Computer Science. Springer, 1985.
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© 1998 Springer-Verlag Berlin Heidelberg
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Greenstreet, M.R., Mitchell, I. (1998). Integrating projections. In: Henzinger, T.A., Sastry, S. (eds) Hybrid Systems: Computation and Control. HSCC 1998. Lecture Notes in Computer Science, vol 1386. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-64358-3_38
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DOI: https://doi.org/10.1007/3-540-64358-3_38
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