Adaptive reachability algorithms for nonlinear systems using abstraction error analysis

Abstract

In many reachability algorithms for nonlinear ordinary differential equations (ODEs), the tightness of the computed reachable sets mainly depends on abstraction errors and the choice of the set representation. One has to mitigate the resulting wrapping effects by suitable tuning of internally-used algorithm parameters since there exists no wrapping-free algorithm to date. In this work, we investigate the fundamentals governing the abstraction error in reachability algorithms – which we also refer to as set-based solvers – and its dependence on the time step size, leading to the introduction of a gain order. This order is related to measures for local and global abstraction errors and thus relates the well-known concept of convergence order from classical ODE solvers to set-based solvers. Furthermore, the simplification of the set representation is tackled by limiting the Hausdorff distance between the original and reduced sets; we demonstrate this for zonotopes. Both these theoretical advancements are incorporated in a modular adaptive parameter tuning algorithm suited for multiple classes of nonlinear ODEs whose efficiency is demonstrated on a wide range of benchmarks.