Tight reachability bounds for constrained nonlinear systems using mean value differential inequalities

This paper presents a new method for bounding the set of trajectories consistent with a system of nonlinear ordinary differential equations, a compact set of admissible time-invariant uncertainties, and a set of state constraints. Such reachability bounds are important in set-based state estimation, fault detection, robust control, etc. Interval reachability methods based on differential inequalities (DI) are desirable in these applications due to their high efficiency, but they often produce extremely conservative bounds. Here, we extend the DI approach to compute tighter enclosures using the mean value form, which is widely used to mitigate the conservatism of interval methods in non-dynamic settings. We present a general mean-value DI bounding theorem along with an efficient algorithmic implementation. We also prove that the overestimation error of mean-value DI converges to zero quadratically as the uncertainty set diminishes, rather than linearly as for interval DI methods, which is important for algorithms based on uncertainty set partitioning. Finally, we present numerical results for two challenging test problems. (C) 2021 Elsevier Ltd. All rights reserved.

Automated summary

This paper presents a new method for tighter enclosures of trajectories consistent with a system of nonlinear ordinary differential equations by extending the DI approach with mean value form. Mean-value DI method converges quadratically to zero overestimation error as uncertainty set diminishes, which is crucial for algorithms based on uncertainty set partitioning. Additionally, numerical results for challenging test problems are provided to demonstrate the effectiveness of the proposed method.