Polytopic Approximations of Reachable Sets Applied to Linear Dynamic Games and a Class of Nonlinear Systems

This chapter presents applications of polytopic approximation methods for reachable set computation using dynamic optimization. The problem of computing exact reachable sets can be formulated in terms of a Hamilton-Jacobi partial differential equation (PDE). Numerical solutions which provide convergent approximations of this PDE have computational complexity which is exponential in the continuous variable dimension. Using dynamic optimization and polytopic approximation, computationally efficient algorithms for overapproximative reachability analysis have been developed for linear dynamical systems tikya[1]. In this chapter, we extend these to feedback linearizable nonlinear systems, linear dynamic games, and norm-bounded nonlinear systems. Three illustrative examples are presented.

[1]  Claire J. Tomlin,et al.  Nonlinear systems: analysis, stability, and control, Shankar Sastry, Springer‐Verlag, New York, NY, 1999, 667 pages. ISBN 0‐387‐98513‐1. , 2001 .

[2]  Pravin Varaiya,et al.  Reach Set Computation Using Optimal Control , 2000 .

[3]  Oded Maler,et al.  Reachability Analysis via Face Lifting , 1998, HSCC.

[4]  Ian M. Mitchell,et al.  Overapproximating Reachable Sets by Hamilton-Jacobi Projections , 2003, J. Sci. Comput..

[5]  Bruce H. Krogh,et al.  Computational techniques for hybrid system verification , 2003, IEEE Trans. Autom. Control..

[6]  J. Meditch,et al.  Applied optimal control , 1972, IEEE Transactions on Automatic Control.

[7]  Pravin Varaiya,et al.  On Reachability Under Uncertainty , 2002, SIAM J. Control. Optim..

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

[9]  Ian M. Mitchell,et al.  Games of Two Identical Vehicles , 2007 .

[10]  Rodney Teo,et al.  Decentralized overlapping control of a formation of unmanned aerial vehicles , 2004, Autom..

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

[12]  A. Kurzhanski,et al.  Ellipsoidal Calculus for Estimation and Control , 1996 .

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

[14]  Inseok Hwang,et al.  Reachability Analysis of Delta-NotchLateral Inhibition Using Predicate Abstraction , 2002, HiPC.

[15]  P. Varaiya,et al.  Dynamic Optimization for Reachability Problems , 2001 .

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

[18]  Inseok Hwang,et al.  Computation of an over-approximation of the backward reachable set using subsystem level set functions , 2003, 2003 European Control Conference (ECC).

[19]  S. Osher,et al.  Algorithms Based on Hamilton-Jacobi Formulations , 1988 .

[20]  Alexander B. Kurzhanski,et al.  Control Synthesis for State Constrained Systems and Obstacle Problems , 2004 .

[21]  P. Varaiya,et al.  On Some Nonstandard Dynamic Programming Problems of Control Theory , 2005 .

[22]  W. Delashmit.  Dynamics of Continuous , Discrete & Impulsive Systems Series A : Mathematical Analysis , 2007 .

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

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

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

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

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

[28]  Eugene Asarin,et al.  Abstraction by Projection and Application to Multi-affine Systems , 2004, HSCC.

[29]  E. K. Kostousova.  State estimation for dynamic systems via parallelotopes optimization and parallel computations , 1998 .

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