Fast Reachable Set Approximations via State Decoupling Disturbances

With the recent surge of interest in using robotics and automation for civil purposes, providing safety and performance guarantees has become extremely important. In the past, differential games have been successfully used for the analysis of safety-critical systems. In particular, the Hamilton-Jacobi (HJ) formulation of differential games provides a flexible way to compute the reachable set, which can characterize the set of states which lead to either desirable or undesirable configurations, depending on the application. While HJ reachability is applicable to many small practical systems, the curse of dimensionality prevents the direct application of HJ reachability to many larger systems. To address computation complexity issues, various efficient computation methods in the literature have been developed for approximating or exactly computing the solution to HJ partial differential equations, but only when the system dynamics are of specific forms. In this paper, we propose a flexible method to trade off optimality with computation complexity in HJ reachability analysis. To achieve this, we propose to simplify system dynamics by treating state variables as disturbances. We prove that the resulting approximation is conservative in the desired direction, and demonstrate our method using a four-dimensional plane model.Comment: in Proceedings of the IEE Conference on Decision and Control, 201

Reach Set Approximation through Decomposition with Low-dimensional Sets and High-dimensional Matrices

Approximating the set of reachable states of a dynamical system is an algorithmic yet mathematically rigorous way to reason about its safety. Although progress has been made in the development of efficient algorithms for affine dynamical systems, available algorithms still lack scalability to ensure their wide adoption in the industrial setting. While modern linear algebra packages are efficient for matrices with tens of thousands of dimensions, set-based image computations are limited to a few hundred. We propose to decompose reach set computations such that set operations are performed in low dimensions, while matrix operations like exponentiation are carried out in the full dimension. Our method is applicable both in dense- and discrete-time settings. For a set of standard benchmarks, it shows a speed-up of up to two orders of magnitude compared to the respective state-of-the art tools, with only modest losses in accuracy. For the dense-time case, we show an experiment with more than 10.000 variables, roughly two orders of magnitude higher than possible with previous approaches

Koopman-Hopf Hamilton-Jacobi Reachability and Control

The Hopf formula for Hamilton-Jacobi Reachability analysis has been proposed for solving viscosity solutions of high-dimensional differential games as a space-parallelizeable method. In exchange, however, a complex, potentially non-convex optimization problem must be solved, limiting its application to linear time-varying systems. With the intent of solving Hamilton-Jacobi backwards reachable sets (BRS) and their corresponding online controllers, we pair the Hopf solution with Koopman theory, which can linearize high-dimensional nonlinear systems. We find that this is a viable method for approximating the BRS and performs better than local linearizations. Furthermore, we construct a Koopman-Hopf controller for robustly driving a 10-dimensional, nonlinear, stochastic, glycolysis model and find that it significantly out-competes both stochastic and game-theoretic Koopman-based model predictive controllers against stochastic disturbance