# Validating a Hamilton-Jacobi Approximation to Hybrid System Reachable Sets

## Abstract

We develop a general framework for solving the hybrid system reachability problem, and indicate how several published techniques fit into this framework. The key unresolved need of any hybrid system reachability algorithm is the computation of continuous reachable sets; consequently, we present new results on techniques for calculating numerical approximations of such sets evolving under general nonlinear dynamics with inputs. Our tool is based on a local level set procedure for boundary propagation in continuous state space, and has been implemented using numerical schemes of varying orders of accuracy. We demonstrate the numerical convergence of these schemes to the viscosity solution of the Hamilton-Jacobi equation, which was shown in earlier work to be the exact representation of the boundary of the reachable set. We then describe and solve a new benchmark example in nonlinear hybrid systems: an auto-lander for a commercial aircraft in which the switching logic and continuous control laws are designed to maximize the safe operating region across the hybrid state space.

## Keywords

Hybrid System Total Variation Diminish Reachability Problem Continuous State Space Linear Hybrid Automaton## Preview

Unable to display preview. Download preview PDF.

## References

- 1.C. Tomlin, J. Lygeros, and S. Sastry, “Controller design for hybrid systems,”
*Proceedings of the IEEE*, vol. 88, no. 7, July 2000.Google Scholar - 2.I. Mitchell and C. Tomlin, “Level set methods for computation in hybrid systems,” in
*Hybrid Systems: Computation and Control*(B. Krogh and N. Lynch, eds.), LNCS 1790, pp. 310–323, Springer Verlag, 2000.CrossRefGoogle Scholar - 3.C. J. Tomlin,
*Hybrid Control of Air Traffic Management Systems*. PhD thesis, Department of Electrical Engineering, University of California, Berkeley, 1998.Google Scholar - 4.O. Maler, A. Pnueli, and J. Sifakis, “On the synthesis of discrete controllers for timed systems,” in
*STACS 95: Theoretical Aspects of Computer Science*(E. W. Mayr and C. Puech, eds.), no. 900 in LNCS, pp. 229–242, Munich: Springer Verlag, 1995.Google Scholar - 5.E. Asarin, O. Maler, and A. Pnueli, “Symbolic controller synthesis for discrete and timed systems,” in
*Proceedings of Hybrid Systems II*, Volume 999 of LNCS (P. Antsaklis, W. Kohn, A. Nerode, and S. Sastry, eds.), Cambridge: Springer Verlag, 1995.Google Scholar - 6.H. Wong-Toi, “The synthesis of controllers for linear hybrid automata,” in
*Proceedings of the IEEE Conference on Decision and Control*, (San Diego, CA), 1997.Google Scholar - 7.T. Dang,
*Vérification et synthèse des systèmes hybrides*. PhD thesis, Institut National Polytechnique de Grenoble (Verimag), 2000.Google Scholar - 8.E. Asarin, O. Bournez, T. Dang, O. Maler, and A. Pnueli, “Effective synthesis of switching controllers for linear systems,”
*Proceedings of the IEEE*, vol. 88, no. 7, pp. 1011–1025, July 2000.CrossRefGoogle Scholar - 9.B. Silva and B. H. Krogh, “Formal verification of hybrid systems using
*CheckMate*: A case study,” in*Proceedings of the American Control Conference*, (Chicago, IL), pp. 1679–1683, 2000.Google Scholar - 10.O. Botchkarev and S. Tripakis, “Verification of hybrid systems with linear differential inclusions using ellipsoidal approximations,” in
*Hybrid Systems: Computation and Control*(B. Krogh and N. Lynch, eds.), LNCS 1790, pp. 73–88, Springer Verlag, 2000.CrossRefGoogle Scholar - 11.E. Asarin, O. Bournez, T. Dang, and O. Maler, “Approximate reachability analysis of piecewise-linear dynamical systems,” in
*Hybrid Systems: Computation and Control*(N. Lynch and B. Krogh, eds.), no. 1790 in LNCS, pp. 21–31, Springer Verlag, 2000.CrossRefGoogle Scholar - 12.O. Bournez, O. Maler, and A. Pnueli, “Orthogonal polyhedra: Representation and computation,” in
*Hybrid Systems: Computation and Control*(F. Vaandrager and J. van Schuppen, eds.), no. 1569 in LNCS, pp. 46–60, Springer Verlag, 1999.CrossRefGoogle Scholar - 13.A. Chutinan and B. H. Krogh, “Verification of polyhedral-invariant hybrid automata using polygonal flow pipe approximations,” in
*Hybrid Systems: Computation and Control*(F. Vaandrager and J. H. van Schuppen, eds.), no. 1569 in LNCS, pp. 76–90, New York: Springer Verlag, 1999.CrossRefGoogle Scholar - 14.A. Chutinan and B. H. Krogh, “Approximating quotient transition systems for hybrid systems,” in
*Proceedings of the American Control Conference*, (Chicago, IL), pp. 1689–1693, 2000.Google Scholar - 15.M. Greenstreet and I. Mitchell, “Reachability analysis using polygonal projections,” in
*Hybrid Systems: Computation and Control*(F. Vaandrager and J. van Schuppen, eds.), no. 1569 in LNCS, pp. 103–116, Springer Verlag, 1999.CrossRefGoogle Scholar - 16.A. B. Kurzhanski and P. Varaiya, “Ellipsoidal techniques for reachability analysis,” in
*Hybrid Systems: Computation and Control*(B. Krogh and N. Lynch, eds.), LNCS 1790, pp. 202–214, Springer Verlag, 2000.CrossRefGoogle Scholar - 17.M. G. Crandall, L. C. Evans, and P.-L. Lions, “Some properties of viscosity solutions of Hamilton-Jacobi equations,”
*Transactions of the American Mathematical Society*, vol. 282, no. 2, pp. 487–502, 1984.MATHCrossRefMathSciNetGoogle Scholar - 18.L. Evans,
*Partial Differential Equations*. Providence, Rhode Island: American Mathematical Society, 1998.MATHGoogle Scholar - 19.C.-W. Shu and S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes,”
*Journal of Computational Physics*, vol. 77, pp. 439–471, 1988.MATHCrossRefMathSciNetGoogle Scholar - 20.S. Osher and C.-W. Shu, “High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations,”
*SIAM Journal on Numerical Analysis*, vol. 28, no. 4, pp. 907–922, 1991.MATHCrossRefMathSciNetGoogle Scholar - 21.J. A. Sethian,
*Level Set Methods and Fast Marching Methods*. New York: Cambridge University Press, 1999.MATHGoogle Scholar - 22.R. Fedkiw, T. Aslam, B. Merriman, and S. Osher, “A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method),”
*Journal of Computational Physics*, vol. 152, pp. 457–492, 1999.MATHCrossRefMathSciNetGoogle Scholar - 23.M. Kang, R. Fedkiw, and X.-D. Liu, “A boundary condition capturing method for multiphase incompressible flow,”
*Journal of Computational Physics*, 2000. Submitted.Google Scholar - 24.M. G. Crandall and P.-L. Lions, “Two approximations of solutions of Hamilton-Jacobi equations,”
*Mathematics of Computation*, vol. 43, no. 167, pp. 1–19, 1984.MATHCrossRefMathSciNetGoogle Scholar - 25.D. Peng, B. Merriman, S. Osher, H. Zhao, and M. Kang, “A PDE based fast local level set method,”
*Journal of Computational Physics*, vol. 165, pp. 410–438, 1999.CrossRefMathSciNetGoogle Scholar - 26.I. M. Kroo,
*Aircraft Design: Synthesis and Analysis*. Stanford, California: Desktop Aeronautics Inc., 1999.Google Scholar - 27.J. Anderson,
*Fundamentals of Aerodynamics*. New York: McGraw Hill Inc., 1991.Google Scholar - 28.United States Federal Aviation Administration,
*Federal Aviation Regulations*, 1990. Section 25.125 (landing).Google Scholar