Incremental Search Methods for Reachability Analysis of Continuous and Hybrid Systems

In this paper we present algorithms and tools for fast and efficient reachability analysis, applicable to continuous and hybrid systems. Most of the work on reachability analysis and safety verification concentrates on conservative representations of the set of reachable states, and consequently on the generation of safety certificates; however, inability to prove safety with these tools does not necessarily result in a proof of unsafety. In this paper, we propose an alternative approach, which aims at the fast falsification of safety properties; this approach provides the designer with a complementary set of tools to the ones based on conservative analysis, providing additional insight into the characteristics of the system under analysis. Our algorithms are based on algorithms originally proposed for robotic motion planning; the key idea is to incrementally grow a set of feasible trajectories by exploring the state space in an efficient way. The ability of the proposed algorithms to analyze the reachability and safety properties of general continuous and hybrid systems is demonstrated on examples from the literature.

[1]  Lydia E. Kavraki,et al.  On finding narrow passages with probabilistic roadmap planners , 1998 .

[2]  Stavros Tripakis,et al.  The Tool KRONOS , 1996, Hybrid Systems.

[3]  John H. Reif,et al.  Complexity of the mover's problem and generalizations , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

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

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

[6]  Vijay Kumar,et al.  Hierarchical modeling and analysis of embedded systems , 2003, Proc. IEEE.

[7]  Rajeev Motwani,et al.  Path Planning in Expansive Configuration Spaces , 1999, Int. J. Comput. Geom. Appl..

[8]  Ed Dawson,et al.  Cryptography: Policy and Algorithms , 1996, Lecture Notes in Computer Science.

[9]  Thomas A. Henzinger,et al.  HYTECH: The Cornell HYbrid TECHnology Tool , 1994, Hybrid Systems.

[10]  S. LaValle,et al.  Randomized Kinodynamic Planning , 2001 .

[11]  Gerardo Lafferriere,et al.  A New Class of Decidable Hybrid Systems , 1999, HSCC.

[12]  Panos J. Antsaklis,et al.  Hybrid Systems II , 1994, Lecture Notes in Computer Science.

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

[14]  Pravin Varaiya,et al.  What's decidable about hybrid automata? , 1995, STOC '95.

[15]  Rajeev Alur,et al.  Counterexample-guided predicate abstraction of hybrid systems , 2003, Theor. Comput. Sci..

[16]  Thomas A. Henzinger,et al.  The Algorithmic Analysis of Hybrid Systems , 1995, Theor. Comput. Sci..

[17]  T. Henzinger,et al.  Algorithmic Analysis of Nonlinear Hybrid Systems , 1998, CAV.

[18]  Hassen Saïdi,et al.  Construction of Abstract State Graphs with PVS , 1997, CAV.

[19]  S. LaValle.  Rapidly-exploring random trees : a new tool for path planning , 1998 .

[20]  David L. Dill,et al.  Experience with Predicate Abstraction , 1999, CAV.

[21]  Thomas A. Henzinger,et al.  Hybrid Systems: Computation and Control , 1998, Lecture Notes in Computer Science.

[22]  Jean-Claude Latombe,et al.  Randomized Kinodynamic Motion Planning with Moving Obstacles , 2002, Int. J. Robotics Res..

[23]  Lydia E. Kavraki,et al.  Path planning using lazy PRM , 2000, Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065).

[24]  Steven M. LaValle,et al.  On the Relationship between Classical Grid Search and Probabilistic Roadmaps , 2004, Int. J. Robotics Res..

[25]  Lydia E. Kavraki,et al.  Probabilistic roadmaps for path planning in high-dimensional configuration spaces , 1996, IEEE Trans. Robotics Autom..

[26]  Edmund M. Clarke,et al.  Verification Tools for Finite-State Concurrent Systems , 1993, REX School/Symposium.

[27]  Rajeev Alur,et al.  A Theory of Timed Automata , 1994, Theor. Comput. Sci..

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

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

[30]  Edmund M. Clarke,et al.  Model Checking , 1999, Handbook of Automated Reasoning.

[31]  Arjan van der Schaft,et al.  Non-linear dynamical control systems , 1990 .

[32]  Gerardo Lafferriere,et al.  Symbolic Reachability Computation for Families of Linear Vector Fields , 2001, J. Symb. Comput..

[33]  J. Tsitsiklis,et al.  The boundedness of all products of a pair of matrices is undecidable , 2000 .

[34]  Bruce H. Krogh,et al.  Verification of Polyhedral-Invariant Hybrid Automata Using Polygonal Flow Pipe Approximations , 1999, HSCC.

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

[36]  J. Aubin.  A survey of viability theory , 1990 .

[37]  S. Sastry.  Nonlinear Systems: Analysis, Stability, and Control , 1999 .

[38]  Edmund M. Clarke,et al.  Counterexample-Guided Abstraction Refinement , 2000, CAV.

[39]  Rajeev Alur,et al.  Reachability Analysis of Hybrid Systems via Predicate Abstraction , 2002, HSCC.

[40]  David P. Dobkin,et al.  The quickhull algorithm for convex hulls , 1996, TOMS.

[41]  Steven M. LaValle,et al.  Resolution complete rapidly-exploring random trees , 2002, Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No.02CH37292).

[42]  Rajeev Alur,et al.  Progress on Reachability Analysis of Hybrid Systems Using Predicate Abstraction , 2003, HSCC.

[43]  John N. Tsitsiklis,et al.  Complexity of stability and controllability of elementary hybrid systems , 1999, Autom..