A dimensionality reduction method for computing reachable tubes based on piecewise pseudo-time dependent Hamilton–Jacobi equation

Abstract

Reachability analysis is a powerful tool for studying the safety of nonlinear systems, in which one of the key points is the computation of reachable tubes. As a common method in engineering, the Hamiltonian Jacobi technique often faces the “curse of dimensionality”. Its computational complexity grows exponentially with the dimensionality of the system state space. This paper proposes a dimensionality reduction method for the computation of reachable tubes that can be used for problems with dynamical systems of a particular form and a columnar target set. In the proposed method, one state variable is considered a pseudo-time variable, and the remaining state variables are contained within a low-dimensional dynamical system. Multiple slices of the original reachable tube are obtained by solving the Hamilton–Jacobi equation constructed based on this low-dimensional dynamical system and then stacking these slices to reconstruct the original reachable tube. Since the solved Hamilton–Jacobi equation is one dimension lower than the Hamilton–Jacobi equation in the original problem, the complexity of the computation is significantly reduced. Furthermore, the proposed method can be combined with existing methods to further reduce the dimensionality of the reachability problem. The computational accuracy and efficiency of the proposed method are demonstrated by some examples.