Hamilton–Jacobi Formulation for Reach–Avoid Differential Games

A new framework for formulating reachability problems with competing inputs, nonlinear dynamics, and state constraints as optimal control problems is developed. Such reach-avoid problems arise in, among others, the study of safety problems in hybrid systems. Earlier approaches to reach-avoid computations are either restricted to linear systems, or face numerical difficulties due to possible discontinuities in the Hamiltonian of the optimal control problem. The main advantage of the approach proposed in this paper is that it can be applied to a general class of target-hitting continuous dynamic games with nonlinear dynamics, and has very good properties in terms of its numerical solution, since the value function and the Hamiltonian of the system are both continuous. The performance of the proposed method is demonstrated by applying it to a case study, which involves the target-hitting problem of an underactuated underwater vehicle in the presence of obstacles.

[1]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid , 2012 .

[2]  S. Shankar Sastry,et al.  Conflict resolution for air traffic management: a study in multiagent hybrid systems , 1998, IEEE Trans. Autom. Control..

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

[4]  P. Cardaliaguet.  A differential game with two players and one target , 1994 .

[5]  J. Sethian.  Level set methods : evolving interfaces in geometry, fluid mechanics, computer vision, and materials science , 1996 .

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

[7]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

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

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

[10]  Alexandre M. Bayen,et al.  A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games , 2005, IEEE Transactions on Automatic Control.

[11]  Marc Quincampoix,et al.  Pursuit Differential Games with State Constraints , 2000, SIAM J. Control. Optim..

[12]  Nancy A. Lynch,et al.  Formal Verification of Safety-Critical Hybrid Systems , 1998, HSCC.

[13]  Tryphon T. Georgiou,et al.  Worst case analysis of nonlinear systems , 1999, IEEE Trans. Autom. Control..

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

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

[16]  John Lygeros,et al.  On reachability and minimum cost optimal control , 2004, Autom..

[17]  Ian M. Mitchell,et al.  Optimization Techniques for State-Constrained Control and Obstacle Problems , 2006 .

[18]  E. Barron,et al.  The Bellman equation for minimizing the maximum cost , 1989 .

[19]  Yan Gao,et al.  On the Reachability Problem for Uncertain Hybrid Systems , 2007, IEEE Transactions on Automatic Control.

[20]  Hasnaa Zidani,et al.  Reachability and Minimal Times for State Constrained Nonlinear Problems without Any Controllability Assumption , 2010, SIAM J. Control. Optim..

[21]  P. Varaiya.  On the Existence of Solutions to a Differential Game , 1967 .

[22]  Alexandre M. Bayen,et al.  Aircraft Autolander Safety Analysis Through Optimal Control-Based Reach Set Computation , 2007 .

[23]  John Lygeros,et al.  Verified hybrid controllers for automated vehicles , 1998, IEEE Trans. Autom. Control..

[24]  I. Kitsios,et al.  Final Glide-back Envelope Computation for Reusable Launch Vehicle Using Reachability , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[25]  John Lygeros,et al.  Air traffic management with target windows: An approach using reachability , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[26]  S. Shankar Sastry,et al.  Aerodynamic envelope protection using hybrid control , 1998, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207).

[27]  Jean-Pierre Aubin,et al.  Impulse differential inclusions: a viability approach to hybrid systems , 2002, IEEE Trans. Autom. Control..

[28]  J. Sethian,et al.  FRONTS PROPAGATING WITH CURVATURE DEPENDENT SPEED: ALGORITHMS BASED ON HAMILTON-JACOB1 FORMULATIONS , 2003 .

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

[30]  Pravin Varaiya,et al.  Ellipsoidal Techniques for Reachability Under State Constraints , 2006, SIAM J. Control. Optim..

[31]  John Lygeros,et al.  A Stochastic Hybrid Model for Air Traffic Management Processes , 2007 .

[32]  John Lygeros,et al.  A viability approach for the stabilization of an underactuated underwater vehicle in the presence of current disturbances , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[33]  Pravin Varaiya,et al.  Optimization Methods for Target Problems of Control , 2002 .

[34]  George J. Pappas,et al.  Hybrid control in air traffic management systems , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[35]  Jean-Pierre Aubin,et al.  Viability theory , 1991 .