Overapproximating Reachable Sets by Hamilton-Jacobi Projections

In earlier work, we showed that the set of states which can reach a target set of a continuous dynamic game is the zero sublevel set of the viscosity solution of a time dependent Hamilton-Jacobi-Isaacs (HJI) partial differential equation (PDE). We have developed a numerical tool—based on the level set methods of Osher and Sethian—for computing these sets, and we can accurately calculate them for a range of continuous and hybrid systems in which control inputs are pitted against disturbance inputs. The cost of our algorithm, like that of all convergent numerical schemes, increases exponentially with the dimension of the state space. In this paper, we devise and implement a method that projects the true reachable set of a high dimensional system into a collection of lower dimensional subspaces where computation is less expensive. We formulate a method to evolve the lower dimensional reachable sets such that they are each an overapproximation of the full reachable set, and thus their intersection will also be an overapproximation of the reachable set. The method uses a lower dimensional HJI PDE for each projection with a set of disturbance inputs augmented with the unmodeled dimensions of that projection's subspace. We illustrate our method on two examples in three dimensions using two dimensional projections, and we discuss issues related to the selection of appropriate projection subspaces.

[1]  S. Osher,et al.  Motion of curves in three spatial dimensions using a level set approach , 2001 .

[2]  Stanley Osher,et al.  Geometric Optics in a Phase-Space-Based Level Set and Eulerian Framework , 2002 .

[3]  Olivier Bournez,et al.  Approximate Reachability Analysis of Piecewise-Linear Dynamical Systems , 2000, HSCC.

[4]  Mark R. Greenstreet,et al.  Hybrid Systems: Computation and Control , 2002, Lecture Notes in Computer Science.

[5]  Randal E. Bryant,et al.  Graph-Based Algorithms for Boolean Function Manipulation , 1986, IEEE Transactions on Computers.

[6]  S. Osher,et al.  High-order essentially nonsocillatory schemes for Hamilton-Jacobi equations , 1990 .

[7]  Ian M. Mitchell,et al.  Reachability Analysis Using Polygonal Projections , 1999, HSCC.

[8]  Alberto Bemporad,et al.  Optimization-Based Verification and Stability Characterization of Piecewise Affine and Hybrid Systems , 2000, HSCC.

[9]  P. Lax,et al.  Systems of conservation equations with a convex extension. , 1971, Proceedings of the National Academy of Sciences of the United States of America.

[10]  J. Lygeros,et al.  A game theoretic approach to controller design for hybrid systems , 2000, Proceedings of the IEEE.

[11]  Ian M. Mitchell,et al.  Level Set Methods for Computation in Hybrid Systems , 2000, HSCC.

[12]  David L. Dill,et al.  Improved approximate reachability using auxiliary state variables , 1999, DAC '99.

[13]  Marko Subasic,et al.  Level Set Methods and Fast Marching Methods , 2003 .

[14]  James A. Sethian,et al.  The Fast Construction of Extension Velocities in Level Set Methods , 1999 .

[15]  Alan J. Hu,et al.  Approximate reachability with BDDs using overlapping projections , 1998, Proceedings 1998 Design and Automation Conference. 35th DAC. (Cat. No.98CH36175).

[16]  Ian M. Mitchell,et al.  Safety verification of conflict resolution manoeuvres , 2001, IEEE Trans. Intell. Transp. Syst..

[17]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods , 1999 .

[18]  Ian M. Mitchell,et al.  Integrating Projections , 1998, HSCC.

[19]  Danping Peng,et al.  Weighted ENO Schemes for Hamilton-Jacobi Equations , 1999, SIAM J. Sci. Comput..

[20]  S. Shankar Sastry,et al.  Decidable Controller Synthesis for Classes of Linear Systems , 2000, HSCC.

[21]  John Lygeros,et al.  Controllers for reachability specifications for hybrid systems , 1999, Autom..

[22]  Björn Engquist,et al.  High-Frequency Wave Propagation by the Segment Projection Method , 2002 .

[23]  P. Lions,et al.  Two approximations of solutions of Hamilton-Jacobi equations , 1984 .

[24]  P. Lions,et al.  Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations. , 1984 .

[25]  Stephen P. Boyd,et al.  A class of Lyapunov functionals for analyzing hybrid dynamical systems , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

[26]  A. Merz.  The game of two identical cars , 1972 .

[27]  Alberto L. Sangiovanni-Vincentelli,et al.  Optimal Control Using Bisimulations: Implementation , 2001, HSCC.

[28]  Ronald Fedkiw,et al.  Level set methods and dynamic implicit surfaces , 2002, Applied mathematical sciences.

[29]  Stavros Tripakis,et al.  Verification of Hybrid Systems with Linear Differential Inclusions Using Ellipsoidal Approximations , 2000, HSCC.

[30]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[31]  Stefania Bortoletto.  The Bellman equation for constrained deterministic optimal control problems , 1993 .

[32]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[33]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[34]  Thomas A. Henzinger,et al.  Beyond HYTECH: Hybrid Systems Analysis Using Interval Numerical Methods , 2000, HSCC.

[35]  Pravin Varaiya,et al.  Ellipsoidal Techniques for Reachability Analysis , 2000, HSCC.

[36]  B. Krogh,et al.  Approximating quotient transition systems for hybrid systems , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[37]  David L. Dill,et al.  Counterexample-guided choice of projections in approximate symbolic model checking , 2000, IEEE/ACM International Conference on Computer Aided Design. ICCAD - 2000. IEEE/ACM Digest of Technical Papers (Cat. No.00CH37140).

[38]  John Lygeros,et al.  Controlled Invariance of Discrete Time Systems , 2000, HSCC.

[39]  Ashish Tiwari,et al.  Series of Abstractions for Hybrid Automata , 2002, HSCC.

[40]  Alexandre M. Bayen,et al.  Validating a Hamilton-Jacobi Approximation to Hybrid System Reachable Sets , 2001, HSCC.

[41]  P. Souganidis,et al.  Differential Games and Representation Formulas for Solutions of Hamilton-Jacobi-Isaacs Equations. , 1983 .

[42]  P. Saint-Pierre,et al.  Set-Valued Numerical Analysis for Optimal Control and Differential Games , 1999 .

[43]  C. Tomlin,et al.  Ellipsoidal approximations of reachable sets for linear games , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).