Lagrangian Approximations for Stochastic Reachability of a Target Tube
For control engineers, this provides a more scalable approach to stochastic reachability analysis, though it is incremental as it trades accuracy for computational efficiency.
The paper develops Lagrangian methods to compute under- and over-approximations of stochastic reach-avoid level sets for nonlinear systems without convexity assumptions, achieving faster computation and scalability at the cost of conservative approximations. The methods are demonstrated on double-integrator, chain of integrators, and 4D spacecraft rendezvous docking problems.
In this paper we examine how Lagrangian techniques can be used to compute underapproximations and overapproximation of the finite-time horizon, stochastic reach-avoid level sets for discrete-time, nonlinear systems. This approach is applicable for a generic nonlinear system without any convexity assumptions on the safe and target sets. We examine and apply our methods on the reachability of a target tube problem, a more generalized version of the finite-time horizon reach-avoid problem. Because these methods utilize a Lagrangian (set theoretic) approach, we eliminate the necessity to grid the state, input, and disturbance spaces allowing for increased scalability and faster computation. The methods scalability are currently limited by the computational requirements for performing the necessary set operations by current computational geometry tools. The primary trade-off for this improved extensibility is conservative approximations of actual stochastic reach set. We demonstrate these methods on several examples including the standard double-integrator, a chain of integrators, and a 4-dimensional space vehicle rendezvous docking problem.