Fixed Point Iteration for Computing the Time Elapse Operator
We investigate techniques for automatically generating symbolic approximations to the time solution of a system of differential equations. This is an important primitive operation for the safety analysis of continuous and hybrid systems. In this paper we design a time elapse operator that computes a symbolic over-approximation of time solutions to a continuous system starting from a given initial region. Our approach is iterative over the cone of functions (drawn from a suitable universe) that are non negative over the initial region. At each stage, we iteratively remove functions from the cone whose Lie derivatives do not lie inside the current iterate. If the iteration converges, the set of states defined by the final iterate is shown to contain all the time successors of the initial region. The convergence of the iteration can be forced using abstract interpretation operations such as widening and narrowing.
We instantiate our technique to linear hybrid systems with piecewise-affine dynamics to compute polyhedral approximations to the time successors. Using our prototype implementation TimePass, we demonstrate the performance of our technique on benchmark examples.
KeywordsHybrid System Continuous System Invariant Region Initial Region Reachable State
Unable to display preview. Download preview PDF.
- 5.Cousot, P., Cousot, R.: Abstract Interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: ACM Principles of Programming Languages, pp. 238–252 (1977)Google Scholar
- 7.Davey, B.A., Priestly, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990)Google Scholar
- 12.Lafferriere, G., Pappas, G., Yovine, S.: Symbolic reachability computation for families of linear vector fields. J. Symbolic Computation 32, 231–253 (2001)Google Scholar
- 18.Silva, B., Richeson, K., Krogh, B.H., Chutinan, A.: Modeling and verification of hybrid dynamical system using checkmate. In: ADPM 2000 (2000), available online from: http://www.ece.cmu.edu/~webk/checkmate
- 20.Tiwari, A., Khanna, G.: Non-linear systems: Approximating reach sets. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 477–492. Springer, Heidelberg (2004)Google Scholar